Emmanuel Moyse
email: moyse.e@chu-toulouse.fr

Maxime Ros

Fouad Marhar

Pascal Swider

Eric Albert Schmidt

Characterisation of Supra-and Infratentorial ICP Profiles

Keywords: Intracranial pressure, Posterior fossa, Transtentorial gradient, Monitoring, Signal processing

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

Introduction

The skull is rostrocaudally segmented by the tentorium into two major compartments: the supratentorial space occupied by the cerebrum and the infratentorial space occupied by the cerebellum. We know that the pressure within the skull is unevenly distributed, with demonstrated intracranial pressure (ICP) gradients [START_REF] Sahuquillo | Interhemispheric supratentorial intracranial pressure gradients in head-injured patients: are they clinically important?[END_REF]. In pathophysiology and clinical practice, little is known about the ICP profi les in the supratentorial and infratentorial compartments. An increase in ICP in the posterior fossa can lead to herniation of the cerebellar tonsils and compression of the brain stem can provoke the patient's death. A supratentorial hydrocephalus can also result from a collapsed fourth ventricle or aqueduct. We tried to characterise the supra-and infratentorial ICP pattern to understand what drives the transtentorial ICP gradient.

Materials and Methods

A 70-year-old man was admitted to our neurosurgery unit with acute cerebellar infarction and infratentorial mass effect. Poor clinical tolerance quickly led to surgical treatment. Before surgery, an ICP probe was inserted into the right frontal region, which is a normal procedure. No external ventricular drainage was placed before resection of the left cerebellar lobe infarction. We used a small craniectomy and, at the end of the surgery, a second ICP probe was left in the posterior fossa. The dura was not sutured and the bone fl ap was not left in place. After the surgery, the patient was sedated and ventilated in our critical care unit. Both ICP signals were recorded concurrently for 48 h and analysed offline with ICM+ software. We calculated mean ( m ), amplitude (amp), respiratory (resp), slow waves and correlation coefficient ( R ) between the amplitude of the fundamental component ( A ) and mean pressure ( P ), RAP, values of the supra-and infratentorial ICP signals [START_REF] Czosnyka | Monitoring and interpretation of intracranial pressure[END_REF]. We then performed correlation analysis of the various indices. We also analysed the transtentorial ICP gradient by computing absolute values of supra-infra differences ( d ) among the various ICP-derived indices.

Results

The supra-and infratentorial results are presented in Table 1 . ICP was always lower in the supratentorial space. The mean (standard deviation) absolute differences between the supraand infratentorial space were as follows: dICPm = 8.5 (1.5) mmHg; dICPamp = 0.01 (0.03) mmHg; dICPresp = 0.01 (0.01) mmHg; dICPslow = 0.11 (0.19) mmHg; and dRAP = 0.22 (0.18).

We further analysed the signals to detail the supra-/ infratentorial profi les of every parameter. Figure 1 displays a 1-h recording of all ICP-derived indices. We also performed correlation analysis of the supra-and infratentoria and calculated the linear Pearson's coeffi cient. For the ICPm signal (Fig. 2 ) very good agreement was found with an ≈10 mmHg offset. For the other signals (Fig. 3 ), the agreement was also very strong.

Discussion

Supratentorial ICPm was 8.5 mmHg lower than infratentorial ICP, which is expected because of the physiology; the changes in ICPm across the tentorium also showed close correlation. The supra-infratentorial gradient was reduced for higher ICPm values. Supra-and infratentorial pulse amplitudes showed strong correlation. Slow waves and respiratory waves also showed a high degree of correlation across the tentorium (Fig. 3 ). The compensatory reserve measured with RAP did not show a perfect match during the whole recording, but correlated well within the physiological range of ICP (Fig. 3 ).

There is sparse literature on infratentorial ICP. Rosenwasser et al. [START_REF] Rosenwasser | Intracranial pressure monitoring in the posterior fossa: a preliminary report[END_REF] reports a clinical series suggesting that during the fi rst 12 h the posterior fossa pressure might be 50 % greater than that of the supratentorial space in all patients. Over the next 12 h the supratentorial pressure was 10 and 15 % higher than the posterior fossa pressures in all patients, and after 48 h of monitoring, the pressures had equilibrated. Slavin and Misra also report dynamic changes over time in a small series of patients [START_REF] Slavin | Infratentorial intracranial pressure monitoring in neurosurgical intensive care unit[END_REF]. Rieger et al. published an experimental study of animal models of supratentorial ICP elevation [START_REF] Rieger | Is it useful to measure supratentorial ICP in the presence of a posterior fossa lesion? Absence of transtentorial pressure gradients in an animal model[END_REF] and demonstrated that infratentorial ICP elevation led to a uniform ICP elevation in the intracranial space without development of a considerable pressure gradient below and above the tentorium. In the low pressure part of the ICP curve, cerebrospinal fl uid connected the compartments and contributed to the pressure equilibrium.

Our results are in accordance with those of the published data. During the signal analysis of this case report, we have been very surprised by the strength of correlation between supraand infratentorial profi les. In this particular case, the posterior fossa was left open, and we thought that the mechanical properties should be altered. Although we found a signifi cant transtentorial difference in terms of mean pressure, supra-and infratentorial ICP profi les were very much in accordance. The mean value of ICP is higher in the posterior fossa, but the difference lessens for higher values of all other ICP-derived parameters, which display the same symmetrical profi le. For higher values, the signal seems lower in the posterior fossa.

However, this study deals with only one patient who shows a pathological condition of the posterior fossa. Thus, while surgery restores the physiological state as far as possible, the skull approach and the craniectomy, even if very small, could have led to disturbance of the data. The same applies to the resection of the cerebellar infarction by modifying the volume of the infratentorial space. This highlighted the diffi culties of routinely monitoring ICP in the cerebellum in a clinical situation. In fact, the placement of any probe in the infratentorial compartment is not as easy and as safe as in supratentorial the compartment and the method of using non-invasive techniques could be of interest in many cases. Zhang evaluated the correlation between increased ICP in supratentorial, infratentorial and transcranial Doppler (TCD) parameters of basal arteries [START_REF] Zhang | Study of acute supratentorial and infratentorial localized lesions by transcranial Doppler[END_REF]. He concluded that there were intracranial pressure gradients in mass lesions; thus, the probe should be positioned where the mass lesion is situated. To perform non-invasive ICP monitoring using TCD, the basal artery should be chosen instead of the middle cerebral artery, which is usually used. However, we have been unable to detail the supra-infratentorial profi les for every ICP parameter.

Overall, in one recording, ICPm was 8.5 mmHg lower in the cerebrum than in the cerebellum and all the ICP profi les show a high degree of correlation.

Confl ict of Interest Statement

There is no confl ict of interest to declare.

Table 1 Fig. 1 Fig. 2 Fig. 3

 1123 Fig. 1 Example of a 1-h recording of supra-and infratentorial intracranial pressure (ICP) signals. Note the very good agreement along the timeline of the various indices


Frédéric Moisy
email: moisy@fast.u-psud.fr

Supercritical bifurcation of a hula hoop

The motion of a hoop hung on a spinning wire provides an illustrative and pedagogical example of a supercritical bifurcation. Above a certain angular velocity threshold Ωc, the hoop rises, making an angle θ ≃ (Ω-Ωc) 1/2 with the vertical. The equation of motion is derived in the limit of a long massless wire, and the calculated steady states are compared to experimental measurements. This simple experiment is suitable for classroom demonstration, and provides an interesting alternative to the classical experiment of the bead sliding on a rotation hoop.

I. INTRODUCTION

The rotation of rigid bodies often displays interesting instability problems. Bodies with three different inertia moments are well known to have unstable rotation about the intermediate axis 1 , as commonly observed from acrobatic jumps or dives. Throwing a tennis racket provides an easy illustration of this instability of the free rotation. Such purely inertial instability has no threshold, i.e. it can be observed for very low rotation rates. In contrast, for axisymmetric bodies such as hoops, disks, rockets, eggs etc., the free rotations remain stable, but different mechanisms may also lead to instability when external forces are present. Under some circumstances, spectacular and unexpected instabilities may originate from frictional forces. This is the case for the tippe-top, a popular toy that flips over and rotates on its stem, or for the hardboiled egg problem, which has recently received a nice analysis [START_REF] Moffatt | Spinning eggs -a paradox resolved[END_REF] . More classically, the competition between gravity and centrifugal forces may lead to an instability with a finite rotation rate threshold [START_REF] Philip | Nonlinear systems[END_REF] , as illustrated by the simple experiment described in this paper.

A hoop is hung on a long wire, whose upper end is spun. At low rotation rate, the hoop is vertical and simply spins about its diameter. Increasing the rotation rate, the hoop progressively rises and becomes horizontal, spinning about its symmetry axis. This situation may appear paradoxical, since the horizontal position maximizes both kinetic and potential energy. It is similar to the conical pendulum problem 1 , commonly illustrated in the classical demonstration experiment of the bead sliding along a vertically rotating hoop [START_REF] Sivardière | A simple mechanical model exhibiting a spontaneous symmetry breaking[END_REF][START_REF] Fletcher | A mechanical analog of first-and second-order phase transitions[END_REF][START_REF] Mancuso | A working mechanical model for first-and second-order phase transitions and the cusp catastrophe[END_REF] . The popular 'hula hoop' game, where the wire rotation is replaced by the hips oscillations of the player, is a common illustration of this phenomenon. Once the hoop spins horizontally, its rotation is maintained by a parametric oscillation mechanism [START_REF] Thomas | Hula-Hoop: An Example of Heteroparametric Excitation[END_REF] .

In this paper, the equation of motion is derived from the Lagrange's equation. An alternate derivation, from the angular momentum equation, is also presented as a good illustration of dynamics of a rigid body. The hoop is shown to rise following a supercritical bifurcation for the wire rotation rate Ω above a critical value Ω c = (2g/R) 1/2 , where g is the gravitational acceleration and R the hoop radius. The stable solution θ ∼ (Ω -Ω c ) 1/2 coincides with that of the conical pendulum or the bead-on-a-hoop. The nonlinear oscillations are briefly described by means of phase portraits, and compared to that of the bead-on-a-hoop. Simple considerations from wire torsion In this experiment, the angular rotation threshold being of order of a few rad/s, the wire can be simply spun with the fingers. Since it only requires a wire and a hoop, this experiment can be conveniently used as a classroom illustration of spontaneous symmetry breaking and bifurcation. Although the detail of the calculation is somewhat more subtle than that of the classical bead-on-a-hoop problem, since it deals with rigid body dynamics, the physics is basically the same and does not require the much heavier apparatus of the bead-on-ahoop demonstration.

II. THEORY

A. Equation of motion

We consider a thin uniform hoop, of radius R and mass m, fixed at a point P of its periphery by a massless wire of length L ≫ R (see figure 1). The wire is spun from its other end O at a constant angular velocity Ω. The center of mass G is assumed to remain at the vertical of O. Let us first consider a torsionless wire, so that the hoop angular velocity φ about the vertical axis e z instantaneously follows the imposed angular velocity Ω.

Let us consider the rotating frame of reference of the hoop (e 1 , e 2 , e 3 ). We note θ the pitch angle between the hoop plane and the vertical axis e z . We consider for simplicity a zero roll angle about e 1 (i.e. we assume that the point P remains the highest point of the hoop), so that θ is the only degree of freedom for this problem. The absolute angular velocity ω of the hoop has two contributions. The first one, of magnitude Ω, is imposed from O through the torsionless wire, and is about e z = cos θ e 1 + sin θ e 3 (the points O and G are at rest, so that OG is the instantaneous axis of rotation when θ is kept constant). The second one comes from the pitch variation θ, and is about the axis e 2 . We therefore obtain

ω =   Ω cos θ θ Ω sin θ   . (1)
The potential energy is V = mgz G , where z G ≃ R(1cos θ) is the center of mass elevation to first order in R/L. The kinetic energy T has two contributions: one from the rotation 1 2 ω • Ĩ • ω, where Ĩ is the inertia matrix of the hoop relative to G, and one from the vertical translation of the center of mass, 1 2 m ż2 G ≃ 1 2 mR 2 θ2 sin 2 θ. In the reference frame of the hoop, the inertia matrix Ĩ has diagonal components

I 1 = I 2 = 1 2 mR 2 and I 3 = mR 2 . The Lagrangian function L = T -V then finally writes L = 1 4 mR 2 Ω 2 (1 + sin 2 θ) + θ2 (1 + 2 sin 2 θ) +mgR cos θ.
(2) Writing the Lagrange's equation for the coordinate θ, we end up with the differential equation of motion

θ(1 + 2 sin 2 θ) -(Ω 2 -2 θ2 ) sin θ cos θ + Ω 2 c sin θ = 0. ( 3
)
where

Ω c = 2g/R.
This equation of motion can also be obtained from the angular momentum equation and Newton's law. In the rotating frame of reference, the angular momentum equation reads

dL dt + ω×L = Γ, (4)
where L = Ĩω is the angular momentum and Γ = GP×N is the torque of the wire tension relative to the center of mass (the gravitational torque vanishes). The wire tension is obtained from Newton's law,

mz G e z = N + mg. (5)
To first order in R/L, N is vertical, so its torque is simply

Γ ≃ -RN sin θ e 2 . Using z G ≃ R(1 -cos θ), we obtain Γ ≃ -R mg + mR( θ2 cos θ + θ sin θ) sin θ e 2 . (6)
Replacing into (4) and projecting on e 2 , we finally recover the equation of motion (3).

B. Steady states and small oscillations

We are interested in the steady states and the natural frequency for small oscillations about them. To first order in θ, equation ( 3) reduces to

θ -Ω 2 sin θ cos θ + Ω 2 c sin θ = 0. (7)
It is worth noting that this linearized equation is exactly the same as the one from the bead-in-a-hoop problem [START_REF] Sivardière | A simple mechanical model exhibiting a spontaneous symmetry breaking[END_REF] . The difference between equations ( 3) and ( 7) originates from the kinetic energy of the center of mass translation, which is not present in the bead-on-a-hoop.

In addition to the trivial solution θ eq = 0, equation ( 7) has a non trivial solution for Ω ≥ Ω c ,

θ eq = ± cos -1 Ω c Ω 2 . (8)
Following the usual terminology, the pitch angle θ eq is the order parameter, and we introduce the reduced angular velocity ǫ = Ω/Ω c -1 as the control parameter. The linearization of (8) finally leads to the classical form of a supercritical pitchfork bifurcation for ǫ ≪ 1:

θ eq ≃ 2 √ ǫ. (9)
Stability and natural frequency for small oscillations are obtained by introducing in equation ( 7) a small perturbation in the form

θ(t) = θ eq + δθ e (σ+iω)t , (10)
where θ eq stands for the trivial or non-trivial solution. The trivial solution is stable for Ω < Ω c (σ = 0), with a natural frequency ω = (Ω 2 c -Ω 2 ) 1/2 , and unstable for Ω > Ω c , with a growth rate given by σ = (Ω 2 -Ω 2 c ) 1/2 (and ω = 0). The non-trivial solution (8) is found to be always stable, with the same natural frequency. In terms of ǫ, one can see that the period of the oscillations,

T ∼ |ǫ| -1/2 , ( 11
)
diverges as one approaches the transition from both side. This critical slowing down is a usual signature of supercritical bifurcation [START_REF] Philip | Nonlinear systems[END_REF] .

C. Nonlinear oscillations

When the oscillation amplitudes about the steady states are not small, the nonlinear terms in (3) become important and may affect the dynamics of the system. The orbits in the phase space (θ, θ) then provide a useful tool to characterize the nonlinear dynamics of the hula hoop, and to compare it with the one of the bead-on-a-hoop (7). The equation of motion can be integrated using the Painlevé invariant,

C = θ ∂L ∂ θ -L, (12)
which takes a constant value along an orbit (this invariant corresponds to the total energy E = T + V in the case of a conservative system). Around the stable fixed points, both the two systems show nearly elliptic orbits, as expected from small harmonic oscillations. Larger oscillations of the hula hoop show orbits with sharper corners around θ ≃ 0, associated to larger angular velocities | θ|. This discrepancy originates from the vertical translation of the center of mass, which is responsible for an additional pitch angle acceleration when the hoop is nearly vertical.

D. Timescales

Two timescales are relevant for a practical experiment: the startup timescale of rotation τ s and the damping timescale of oscillations τ d . The startup timescale may be obtained considering the wire torsion. We start from an initially vertical hoop (θ = 0), and let now its rotation angle φ about e z be free. The wire communicates a torque κ(φ -Ωt) = κ δφ to the hoop, where κ is the torsion constant and δφ the wire torsion. The angular momentum equation about e z applied to the hoop then reads

1 2 mR 2 δ φ + κ δφ = 0, (13)
leading to a startup timescale τ s ≃ R(m/2κ) 1/2 (which corresponds also to the period of the wire torsion oscillations if no damping were present). The damping timescale τ d is relevant both for the φ oscillations due to the wire torsion, and for the θ oscillations around the equilibrium states. It can be obtained considering the dissipation with the surrounding air. With typical velocity of order m/s and hoop thickness w of a few millimeters, we can make use of a turbulent estimate for the drag force. Neglecting θ compared to φ, a unit length dl of the hoop experiences a drag force df ≃ ρR 2 φ2 w dl, where ρ is the air density. Integrating over the hoop perimeter leads to the frictional torque

Γ r ≃ -ρR 4 φ2 w Ω |Ω| . (14)
The angular momentum equation about e z reads

φ + ρ 0 ρ w R φ2 = 0. ( 15
)
where ρ 0 is the hoop density. Taking φ ≃ Ω and φ ≃ Ω/τ d , the damping timescale finally writes

τ d ≃ ρ 0 ρ w R Ω -1 . ( 16
)
For values of practical interest, τ d is of order of a few tens of the rotation period Ω -1 . Note here that in the case of the real hula hoop game, the rolling friction on the player hips is fortunately dominant, leading to a damping timescale of order Ω -1 . An experiment has been carried out using a wood hoop (ρ 0 ≃ 0.67 g.cm -3 ) of mass m = 15.5 g, radius R = 86 mm and section 4 × 10 mm 2 . The expected frequency threshold is then Ω c /2π ≃ 2.40 Hz (classical hula hoops have Ω c /2π ≃ 1 Hz). A simple cotton thread, of length L = 0.68 m and mass less than 3% of the hoop mass, was fixed to a constant current motor. The startup timescale for this wire can be estimated from the free oscillation period, τ s ≃ 5 s (corresponding to a torsion constant κ ≃ 2 × 10 -6 N.m). This wire is far from being torsionless, and during the early stage of the rotation, torsional energy is stored into the wire. The wire progressively communicates rotation to the hoop, and the fluctations (due to a varying imposed rotation rate at its upper end, or to pitch angle variation at the other end) are smoothed down on a timescale τ d .

Seen from the side, the hoop rotation appears as Lissajous ellipses, whose principal axis makes an angle θ with the vertical axis. Pictures acquired from a simple CCD camera (see figure 3) allows us to easily measure the pitch angle θ within 1 o from these ellipses. The time aperture of the camera may blur the picture, but facilitate in some case the angle measurement (see for instance picture c).

The measured pitch angles θ, shown in figure 4, are in excellent agreement with the exact solution (8), except for low rotation rate, where non zero angles are measured below the expected transition. The experimental frequency threshold, obtained by extrapolating the curve down to θ ≃ 0, is Ω c /2π ≃ (2.30 ± 0.03) Hz, which agrees within 5% with the theoretical value. The discrepancy at low rotation rate is typical of an imperfect bifurcation, where small asymmetry in the apparatus (such as the position of the knot) slightly anticipates the destabilization of the basic state. An interesting observation is that, for Ω ≥ 2.6Ω c (see the dashed line in figure 4), the bifurcated state is no stable any more: a secondary instability appears, in the form of a slow precession of the center of mass, with a period of about ten rotation periods. This new behavior is probably an effect of the non zero mass of the wire, and can clearly not be described in our calculation, where the hoop center of mass is constrained to remain aligned with the vertical axis.

Transient phenomena may also be investigated, by means of usual video processing. An illustration is given in figure 5, showing a spatio-temporal diagram obtained by collecting the light intensity recorded on a horizontal line passing through the wire. In this example, the imposed frequency Ω/2π has been suddenly increased from 0 to 5.5 Hz ≃ 2.4 Ω c /2π (first arrow). One can see that, after a transient time of around 7 s (second arrow) during which the wire rotation propagates down to the hoop, the amplitude increases and saturates to a finite value.

From this diagram, the instantaneous pitch angle as well as the instantaneous oscillation frequency φ/2π may be extracted, as shown in figure 6. The frequency is obtained from averaging over 6 successive oscillations. Due to the progressive torsion of the wire, this frequency slowly approaches its imposed value 5.5 Hz. As a consequence, the increase of the pitch angle towards its stationary value θ eq ≃ 1.42 rad is rather slow.

It is interesting to note that, when plotting the instantaneous pitch angle as a function of the instantaneous frequency (see figure 7), the bifurcation diagram of figure 4 is recovered to a high degree of accuracy. This suggests that the pitch angle follows 'adiabatically' the instantaneous frequency, so that the apparent growth rate is essentially controlled by the wire torsion more than the intrinsic dynamics of the instability (at least far from the transition).

IV. DISCUSSION

This simple experiment provides an interesting alternative to the bead-on-a-hoop [START_REF] Sivardière | A simple mechanical model exhibiting a spontaneous symmetry breaking[END_REF][START_REF] Fletcher | A mechanical analog of first-and second-order phase transitions[END_REF][START_REF] Mancuso | A working mechanical model for first-and second-order phase transitions and the cusp catastrophe[END_REF] experiment as a mechanical analog to bifurcation and second-order phase transition in physics. The poor attention paid to the choice of the material and the experimental conditions illustrates the robustness of the phenomenon, and makes this experiment easy to work out by undergraduate students. Further experiments can be performed, e.g. studying the transient phenomena resulting from a small perturbation. As a suggestion, restrain the point P on the vertical axis when Ω > Ω c by means of a small hook around the wire. Releasing the hook allows us to measure the growth rate and to characterize its divergence as ǫ → 0. This method can be hardly achieved with a bead-on-a-hoop apparatus due to its inherent difficult access.

Another motion may compete with the bifurcation described in this paper: the hoop can rotate as a whole about the vertical axis, its center of mass remaining aligned with the wire. This is the usual motion for the conical pendulum, which also leads to a supercritical bifurcation with a threshold simply given by the natural frequency ω 0 ≃ g/(L + R). The conical pendulum motion overcomes the hula hoop motion for ω 0 of order of Ω c , i.e. for a wire length of order of the hoop radius. Moving away the center of mass from the vertical axis may allow us to observe the competition between the two regimes, even for a longer wire.

Similar bifurcations as the result of a competition between centrifugal force and gravity are present in a number of situations. Spinning plates provides an interesting illustration: as for the real hula hoop, the motion here is forced by the precession of the rigid rod rather than its rotation. The imposed precession frequency has to overcome the natural frequency for the plate to stand up, and then the horizontal state is maintained by a parametric oscillation mechanism [START_REF] Thomas | Hula-Hoop: An Example of Heteroparametric Excitation[END_REF] . Lasso roping is another example, where the rigid hoop is replaced by a deformable loop. Here again the motion is maintained by the precession of the spoke rather than rotation. As a consequence, the knot on the loop makes the spoke to rotate as well, so that the cord has to be continuously untwisted at its other end.

It is worth pointing out that this experiment can be carried out with any rigid body, not necessarily axisymmetric.

In this case, the angle of the equilibrium state (8) remains unchanged, and the angular velocity threshold just becomes Ω c = mga/(I 3 -I 1 ), where a = |GP| is the distance between the center of mass and the knot. Note however that, for bodies with three different inertia moments, in addition to the bifurcation described here, inertial instabilities may also occur. Such instability involves the roll angle about e 1 as an additional degree of freedom, and is not described by our cal-culation.

  FIG.1:A hoop of center of mass G, attached to a wire OP, spins about the vertical axis OG, making a pitch angle θ with the vertical.

FIG. 2 :

 2 FIG.2: Orbits in the phase space (θ, θ/Ωc) for the hula hoop (solid lines) compared to the bead-on-a-hoop system (dashed lines). (a): Ω = 0.8Ωc. (b): Ω = 1.2Ωc.

Figure 2

 2 shows isolines of C for two forcing frequencies, below (Ω/Ω c = 0.8, fig. a) and above (Ω/Ω c = 1.2, fig. b) the transition, together with the orbits of the bead-on-a-hoop problem (dashed lines). One can clearly see from these phase portraits the trivial solution θ eq = 0 for Ω/Ω c = 0.8 and the two non trivial solutions (8) separated by a saddle point at θ = 0 for Ω/Ω c = 1.2. Note that we are only interested in the closed orbits in the domain |θ| ≤ π/2: orbits that cross θ = ±π/2 are not consistent with the assumption that the point P remains the highest point of the hoop.

FIG. 3 :

 3 FIG. 3: Pictures of the hoop. (a) The hoop at rest. (b) and (c) θ ≃ 37 o . (d) θ ≃ 78 o .

FIG. 4 :

 4 FIG. 4: Bifurcation diagram of the pitch angle θ as a function of the normalized angular velocity Ω/Ωc. The solid line is the calculated solution (8), with Ωc adjusted to fit the experimental data.

FIG. 5 :FIG. 6 :

 56 FIG. 5: Spatio-temporal diagram of the light intensity recorded on an horizontal line, showing the wire oscillation, when the frequency Ω/2π is increased from 0 to 5.5 Hz.

FIG. 7 :

 7 FIG.7: Instantaneous pitch angle θ as a function of the instantaneous rotation frequency φ/Ωc (same data as in figure6). The solid line is the calculated solution (8), as in figure4. The cross indicates the stationary solution θeq ≃ 1.42 rad for Ω = 2.4 Ωc.

Acknowledgments

This work has benefited from fruitful discussions with M. Rabaud and B. Perrin.


Conor-James Macdonald
email: conor.macdonald@gustaveroussy.fr

Anne-Laure Madika
email: annelaure.madika@chru-lille.fr

Fabrice Bonnet
email: fabrice.bonnet@chu-rennes.fr

Guy Fagherazzi
email: guy.fagherazzi@gmail.com

Martin Lajous
email: mlajous@insp.mx

Marie-Christine Boutron-Ruault

Cholesterol and Egg Intakes, and Risk of Hypertension in a Large Prospective Cohort of French Women

Keywords: eggs, hypertension, cholesterol, epidemiology, prospective studies

Purpose:

The relationship between egg and cholesterol intakes, and cardiovascular disease is controversial. Meta-analyses indicate that egg consumption is associated with increased cardiovascular disease and mortality, but reduced incidence of hypertension, a major risk factor for cardiovascular disease. This study aims to investigate the associations between consumption of egg and cholesterol, and hypertension risk in a cohort of French women. Methods: We used data from the E3N cohort study, a French prospective population-based study initiated in 1990. From the women in the study, we included those who completed a detailed diet history questionnaire, and who did not have prevalent hypertension or cardiovascular disease at baseline, resulting in 46,424 women. Hypertension cases were self-reported. Egg and cholesterol intake was estimated from dietary history questionnaires. Cox proportional hazard models with time-updated exposures were used to calculate hazard ratios. Spline regression was used to determine any dose-respondent relationship. Results: During 885,321 person years, 13,161 cases of incident hypertension were identified. Higher cholesterol consumption was associated with an increased risk of hypertension: HR Q1-Q5 = 1.22 [1.14:1.30], with associations similar regarding egg consumption up to seven eggs per week: HR 4-7 eggs = 1. 14 [1.06:1.18]. Evidence for a non-linear relationship between hypertension and cholesterol intake was observed. Conclusions: Egg and cholesterol intakes were associated with a higher risk of hypertension in French women. These results merit further investigation in other populations.

Introduction

The relationship between cholesterol or egg consumption and the onset of cardiovascular diseases (CVD) is unclear, with observational studies and trials finding inconsistent results [START_REF] Berger | Dietary cholesterol and cardiovascular disease: A systematic review and meta-analysis[END_REF][START_REF] Zhong | Associations of Dietary Cholesterol or Egg Consumption With Incident Cardiovascular Disease and Mortality[END_REF][START_REF] Soliman | Dietary Cholesterol and the Lack of Evidence in Cardiovascular Disease[END_REF][START_REF] Larsson | Dietary fats and dietary cholesterol and risk of stroke in women[END_REF]. Eggs are a main source of dietary cholesterol, and their consumption has been associated with an increased risk of cardiovascular disease [START_REF] Li | Egg consumption and risk of cardiovascular diseases and diabetes: A meta-analysis[END_REF], all-cause mortality [START_REF] Xu | Egg consumption and the risk of cardiovascular disease and all-cause mortality: Guangzhou Biobank Cohort Study and meta-analyses[END_REF], haemorrhagic stroke [START_REF] Tong | The associations of major foods and fibre with risks of ischaemic and haemorrhagic stroke: A prospective study of 418 329 participants in the EPIC cohort across nine European countries[END_REF], and diabetes [START_REF] Lajous | Egg and cholesterol intake and incident type 2 diabetes among French women[END_REF][START_REF] Tamez | Egg consumption and risk of incident type 2 diabetes: A dose-response meta-analysis of prospective cohort studies[END_REF]; paradoxically, in meta-analysis eggs are associated with a decreased risk of hypertension [START_REF] Zhang | Red meat, poultry, and egg consumption with the risk of hypertension: A meta-analysis of prospective cohort studies[END_REF], which is a major risk-factor for cardiovascular disease [START_REF] Kjeldsen | Hypertension and cardiovascular risk: General aspects[END_REF]. Further, results are inconsistent between regions, and results are conflicting when studying specific CVD such as stroke or ischemic heart disease [START_REF] Hu | A Prospective Study of Egg Consumption and Risk of Cardiovascular Disease in Men and Women[END_REF][START_REF] Shin | Egg consumption in relation to risk of cardiovascular disease and diabetes: A systematic review and meta-analysis[END_REF][START_REF] Alexander | Meta-analysis of Egg Consumption and Risk of Coronary Heart Disease and Stroke[END_REF][START_REF] Qin | Associations of egg consumption with cardiovascular disease in a cohort study of 0.5 million Chinese adults[END_REF][START_REF] Djoussé | Egg consumption and risk of heart failure in the Physicians' Health Study[END_REF][START_REF] Abdollahi | Egg consumption, cholesterol intake, and risk of incident stroke in men: The Kuopio Ischaemic Heart Disease Risk Factor Study[END_REF][START_REF] Virtanen | Associations of egg and cholesterol intakes with carotid intima-media thickness and risk of incident coronary artery disease according to apolipoprotein E phenotype in men: The Kuopio Ischaemic Heart Disease Risk Factor Study[END_REF][START_REF] Dehghan | Association of egg intake with blood lipids, cardiovascular disease, and mortality in 177,000 people in 50 countries[END_REF][START_REF] Zhuang | Egg and egg-sourced cholesterol consumption in relation to mortality: Findings from population-based nationwide cohort[END_REF]. Aside from cholesterol, eggs are a source of essential nutrients, vitamins, and high-quality proteins [START_REF] Song | Nutritional contribution of eggs to American diets[END_REF], which may be a reason for the inconsistencies, as overall diet quality may be important in the relationship between eggs and CVD.

It has been demonstrated that higher dietary cholesterol can raise serum cholesterol [START_REF] Vincent | Meta-regression analysis of the effects of dietary cholesterol intake on LDL and HDL cholesterol[END_REF], which is in turn associated with blood pressure [START_REF] Ferrara | Serum cholesterol affects blood pressure regulation[END_REF]. High levels of dietary cholesterol have been associated with increased systolic blood pressure in the INTERMAP study [START_REF] Sakurai | Relationship of dietary cholesterol to blood pressure: The INTERMAP study[END_REF] conducted in China, Japan, UK and USA, but not in the Finnish Kuopio Ischaemic Heart Disease Risk Factor Study (KIHD) [START_REF] Abdollahi | Egg consumption, cholesterol intake, and risk of incident stroke in men: The Kuopio Ischaemic Heart Disease Risk Factor Study[END_REF]. Recently, Mazidi et al. observed increased systolic and diastolic blood pressure in higher consumers of eggs, but found no relationship with cardiovascular mortality [START_REF] Mazidi | Egg Consumption and Risk of Total and Cause-Specific Mortality: An Individual-Based Cohort Study and Pooling Prospective Studies on Behalf of the Lipid and Blood Pressure Meta-analysis Collaboration (LBPMC) Group[END_REF]. The complexity of the association between egg/cholesterol intake and CVD is reflected in confusing guidelines for egg and cholesterol consumption. As of 2015, guidelines from the American Heart Association [START_REF]2015-2020 Dietary Guidelines for Americans[END_REF] do not include recommendations for cholesterol intake, but note that individuals should consume as little cholesterol as possible. French guidelines recommend limiting red meat and eggs to 500 g per week, but do not give guidelines for eggs or cholesterol specifically [START_REF] Van Dooren | Food Parterns and Dietary Recommendatio in Spain, France and Sweden[END_REF][START_REF]L'agence Nationale de Sécurité Sanitaire de L'alimentation, de L'environnement et du Travail. Actualisation des Repères du PNNS: Révision des Repèresde Consommations Alimentaires[END_REF].

In view of inconsistencies in previous studies, we sought to investigate the associations between egg consumption, dietary cholesterol, and incident hypertension in a large prospective study of French women, and assessed these relations over a number of potential lifestyle and dietary factors.

Materials and Methods

Study Population

The Etude Epidémiologique de femmes de la Mutuelle Générale de l Education (E3N) [START_REF] Clavel-Chapelon | E3N, a French cohort study on cancer risk factors. E3N Group. Etude Epidémiologique auprès de femmes de l'Education Nationale[END_REF] is a French prospective cohort started in 1990 comprising 98,995 women aged 40-65 years at baseline and insured by the MGEN (Mutuelle Générale de l´Education Nationale), a health insurance plan for workers in the National Education System and their families. The objective of E3N was to study the main risk factors of cancer and chronic diseases. The E3N is the French component of the European Prospective Investigation into Cancer and Nutrition. The cohort received ethical approval from the French National Commission for Computerized Data and Individual Freedom (Commission Nationale Informatique et Libertés), and all participants in the study signed an informed consent.

Participants returned mailed questionnaires on lifestyle information and disease occurrence every 2 to 3 years. The average response rate at each questionnaire cycle was 83%, and the total loss to follow-up was 3%.

We included women who responded to a dietary questionnaire in 1993 (n = 74,522), and excluded women with prevalent hypertension, coronary disease or stroke (n = 26,974) before or at the 1993 questionnaire and those with unrealistic energy consumption (the 1st and 99th percentiles of the distribution in the population, n = 938). The final study population included 46,501 women. A flow diagram of exclusions (Supplementary Figure S1), as well as a comparison between the excluded and included women is included in the supplementary material (Supplementary Table S1).

Assessment of Egg and Cholesterol Intake

In 1993, dietary data was collected using a two-part questionnaire detailing consumption of 208 food items which has been shown to be valid and reproducible [START_REF] Van Liere | Relative validity and reproducibility of a French dietary history questionnaire[END_REF]. Women were asked to answer questions about quantities and frequencies of consumption of food groups. Eleven possible responses were available, never or less than once a month; 1 to 3 times a month, and 1 to 7 times a week. A photo booklet was added to help estimate portion sizes [START_REF] Lucas | Estimation of food portion size using photographs: Validity, strengths, weaknesses and recommendations[END_REF]. From this questionnaire and using a detailed food composition table, mean daily intakes of energy (excluding energy from alcohol), alcohol, and nutrients including cholesterol were estimated.

Total egg consumption was based on the number and frequency of consumption of cooked eggs (i.e., poached or boiled eggs) or eggs from mixed dishes (i.e., omelettes). Egg consumption per week was estimated as the weekly intake in grams estimated from the questionnaire, divided by 60 g (the estimated weight of one egg in grams). The validity and reproducibility of the dietary questionnaire was previously assessed in a sample of 119 women who had completed two diet history questionnaires and twelve monthly 24-h dietary recalls over a one-year period [START_REF] Van Liere | Relative validity and reproducibility of a French dietary history questionnaire[END_REF]. The correlation coefficient between the dietary questionnaire and the 24-h recalls was 0.49 for eggs and 0.40 for dietary cholesterol. Cholesterol content for eggs was estimated as 380 mg/100 g of egg [START_REF] Genet | ANSES-CIQUAL French Food Composition Table Version[END_REF].

A second similar dietary questionnaire sent in 2005 allowed follow-up of dietary data. This questionnaire was completed by 82% of the study population. If a participant did not complete the second dietary questionnaire, the dietary data were imputed as the value from the first dietary questionnaire.

The outcome for this study was self-reported hypertension, validated through identification of an anti-hypertensive medication. Participants were asked to report whether they had hypertension at baseline (1993) and in each follow-up questionnaire (1994, 1997, 2000, 2002, 2005, 2008, 2011, and 2014), the date of diagnosis, and the use of antihypertensive treatments. The month and year of diagnosis were provided for most cases (69%). For individuals who were missing the month of diagnosis (14% of cases), it was imputed to June of the year of diagnosis. The median time between the date of diagnosis and the date of response to the first questionnaire after diagnosis was 12 months. Thus, for the cases with no year of diagnosis (n = 17%), we assigned it to be 12 months before they reported hypertension in a questionnaire. In 2004, a drug reimbursement database became available for 97.6% of participants. We used the self-reported date of diagnosis or the first date of drug reimbursement for antihypertensive medications (Anatomical Therapeutic Chemical Classification System codes C02, C03, C07, C08, and C09) whatever happened first, as the date of diagnosis for cases identified after 2004. Using the information of the MGEN health insurance plan drug claim database, we assessed the validity of self-reported hypertension within the E3N cohort, up to 2008. We compared hypertension self-report to antihypertensive drug reimbursement (any of the above specified codes). A positive predictive value of 82% was observed [START_REF] Lajous | Flavonoid intake and incident hypertension in women[END_REF].

Assessment of Covariates

Family history of hypertension, education (elementary school education, high school education, and university education), and smoking (ever smoker, current smoker, or never smoker) were based on self-reports, and for treated diabetes and treated dyslipidaemia (i.e., use of lipid lowering medications) we used cases which had been validated through the use of the drug reimbursement database [START_REF] Lajous | Processed and unprocessed red meat consumption and incident type 2 diabetes among French women[END_REF]. Age at menopause was determined on a combination of variables as previously reported [START_REF] Fournier | Use of different postmenopausal hormone therapies and risk of histology-and hormone receptor-defined invasive breast cancer[END_REF]. The use of menopausal hormone therapy (MHT) was assessed at baseline, using a booklet containing photos of all types of oestrogens and progestogens. A Prudent/Mediterranean diet score was determined from dietary data using principal component analysis as previously described [START_REF] Varraso | Dietary patterns and asthma in the E3N study[END_REF].

We assessed usual physical activity with a questionnaire in 1993 that included questions on weekly hours spent walking, cycling and performing light and heavy household chores, or recreational activities and sports considering the winter and summer seasons. It included questions on the time spent walking (to work, shopping, and leisure time), cycling (to work, shopping, and leisure time), housework, and sports activities (such as racket sports, swimming). Metabolic equivalents (METs) per week were estimated by multiplying the hourly average METs for each item based on values from the Compendium of Physical Activities [START_REF] Ainsworth | Compendium of physical activities: An update of activity codes and MET intensities[END_REF] by the reported activity duration.

Self-reported height and weight were used to calculate body mass index (BMI), defined as weight (kg) divided by squared height (m 2 ). In the cohort, self-reported anthropometry is considered reliable from a validation study [START_REF] Tehard | Anthropometric measurements and body silhouette of women: Validity and perception[END_REF].

Variables considered in the analyses, including egg consumption, diabetes, dyslipidaemia, physical activity, smoking, menopausal status, use of MHT, and BMI were considered at baseline and then updated using data from the 2005 questionnaire.

Statistical Analysis

Participants were split into groups depending on egg consumption as <1, 1-1.9, 2-2.9, 3-3.9, 4-6.9 and ≥7 eggs/week. Similarly, participants were grouped into quintiles of cholesterol intake defined by the population distributions. Time at entry was the age at the beginning of follow-up (1993), exit time was the age when participants were diagnosed with hypertension, died (dates of death were obtained from the participants' medical insurance records), were lost to follow-up, or were censored at the end of the follow-up period (15 June 2014), whichever occurred first. Hazard ratios and 95% confidence intervals were estimated from Cox regression models with age as the time scale. P-values for trends were calculated using the median category value as a semi-continuous variable in the models.

A model was first assessed with time-updated egg consumption, and cholesterol consumption, and age as timeline (Model 0). Models were first adjusted for BMI (Model 1), then various known risk-factors for hypertension: total physical activity (MET-hours/week), total calories (KCAL), diabetes and dyslipidaemia status, menopausal status, use of MHT, family history of cardiovascular disease (yes/no), smoking (never, former, and current at baseline), education (no high school diploma, high school diploma) (Model 2), and dietary variables (salt, potassium, saturated fats, fruit and vegetables, alcohol, and cholesterol containing foods: unprocessed meat, processed meat, fish, shellfish) (Model 3); and in the case of cholesterol as exposure, Model 2 + intake of vegetables, fruits, salt, potassium and saturated fats (Model 3*). Finally, models were adjusted as model 2 plus a Mediterranean dietary pattern in place of other nutritional variables (Model 4). We considered Model 4 as the main model. As dyslipidaemia and diabetes may affect both exposure and outcome, models with and without these variables were assessed, but no large difference in estimates was obtained, thus the variables were retained. All covariates were updated at the 2005 time-point.

Subgroup analyses were performed for the following stratifications in order to assess the relations in groups associated with lifestyle factors and cholesterol intake: non-obese/excess weight, dyslipidaemia/diabetes at baseline, low reported processed meat intake/high reported processed meat intake, low/high Mediterranean diet score, current and ex-smokers/never smokers, and education level. A number of sensitivity analyses were performed. In order to account for reverse causation, we excluded cases diagnosed within five and then within 10 years.

Correlations between dietary variables were assessed using Spearmans correlation coefficient (r). Missing values occurred in less than 5%, and were imputed using the mean for continuous, or median for categorical variables. All statistical analyses used R version 3.5.1 (www.r-project.org) and the survival package, with an alpha of statistical significance equal to 0.05. Results from cox-models were interpreted as hazard ratio (HR) (95% confidence interval (CI)). The proportional hazards assumption was assessed using the cox.zph function in R.

Results

A total of 46,424 women were included in this study. After an average of 19.1 years of follow-up and 885,321 person years, 13,161 cases of incident hypertension were identified, corresponding to an incidence rate of 14.9 per 1000 person-years. At baseline, the mean (standard deviation) age was 50.1 (6.2) years, the mean BMI was 22.2 (2.8), and the mean physical activity was 54.4 (29.8) MET-hours/week.

At baseline, the mean egg consumption in the cohort was 2.7 (2.0) eggs/week, including 2.0 (1.5) eggs/week from omelettes and mixed dishes, and 0.8 (1.0) eggs/week from boiled eggs. The mean daily cholesterol intake was 366 (126) mg/day. Cholesterol from eggs constituted on average 22.4 (12.8)% of total dietary cholesterol at baseline. Compared to low consumption, high consumers of egg and cholesterol were younger and were less likely to be menopausal, more likely to smoke, and less likely to report treated dyslipidaemia (Table 1). Cholesterol consumption correlated strongly with intakes of eggs (r = 0.67) saturated fat (r = 0.72), processed meat (0.41), shellfish (r = 0.37), and liver (r = 0.34). Egg consumption correlated with intakes of shellfish (r = 0.42), processed meat (r = 0.23), Mediterranean diet score (r = 0.20), and saturated fat (r = 0.22).

In cox models using updated exposures (Table 2), the highest consumers of cholesterol were associated with a higher risk of hypertension than the lowest consumers (M4, HR Q1-Q5 = 1.22 [1.14: 1.30], p < 0.05). Associations were weakened, but remained significant when controlling on saturated fat intake (M3, HR Q1-Q5 = 1.10 [1.03: 1.18], p < 0.05). Similarly, egg consumption up to 7 eggs/week was associated with a higher risk of hypertension (M4, HR 4-7 eggs = 1.14 [1.06: 1.18]), whereas over 7 eggs/week was not (M4, HR >7 eggs = 1.04 [0.96: 1.14]). Considering a mutually adjusted model for egg and cholesterol intake, associations were consistent for cholesterol (HR Q1-Q5 = 1.21 [1.12: 1.31], p < 0.05), but associations with eggs were null (HR 4-7 eggs = 1.04 [0.97: 1.12], p = 0.11) suggesting that cholesterol was driving the associations between egg consumption and hypertension.

Spline analysis (Figure 1) revealed a steep sharp dose-response relationship in low cholesterol consumers, which decreased in steepness at around 400 mg/day. For eggs, spline analysis showed a linear response from zero to around 7 eggs/week, after which a return to no extra risk was observed at around 10 eggs/week.

Subgroup Analyses

In analysis of groups we hypothesised as less likely to be confounded, a number of stratified models were assessed (Tables 3 and4). Results were similar amongst most subgroups of women. Considering egg intake, only amongst the high consumers of processed meat was there no clear evidence of an association between egg consumption and hypertension risk (p-interaction <0.005) (Table 3). When considering cholesterol, associations were not evident for high consumers of processed meat, and those with a low Mediterranean diet score (both p-interaction <0.005) (Table 4). Amongst smokers, associations were null for both egg and cholesterol intake, but the statistical test for interaction for this variable was not significant. All models adjusted for BMI, total physical activity, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, diabetes, intake of alcohol, total calorific intake, Mediterranean diet score and age as the timescale.

Subgroup Analyses

In analysis of groups we hypothesised as less likely to be confounded, a number of stratified models were assessed (Tables 3 and4). Results were similar amongst most subgroups of women. Considering egg intake, only amongst the high consumers of processed meat was there no clear evidence of an association between egg consumption and hypertension risk (p-interaction <0.005) (Table 3). When considering cholesterol, associations were not evident for high consumers of processed meat, and those with a low Mediterranean diet score (both p-interaction <0.005) (Table 4). Amongst smokers, associations were null for both egg and cholesterol intake, but the statistical test for interaction for this variable was not significant. In sensitivity analyses excluding cases diagnosed within five and 10 years in order to account for reverse causation bias, associations were unchanged (data not tabulated).

Discussion

The results from this large prospective study suggest that high cholesterol and egg consumption is associated with a higher risk of hypertension. These results were independent of saturated fat, salt, processed meat/fish intakes, protective dietary factors such as vegetable and dietary fibre intake, and Mediterranean diet adherence. Associations between egg consumption and hypertension were null when adjusting for total cholesterol, indicating that cholesterol was driving the associations with eggs in this cohort. Using splines, we observed a non-linear response between cholesterol consumption and hypertension risk, with the steepest increase in low cholesterol consumers. For eggs, the risk increased linearly until around seven eggs per week, then decreased.

Few epidemiological studies have assessed the relationship between dietary cholesterol and blood pressure; however, they do find positive associations, which are in agreement with our results. Two previous studies involving American males have identified positive associations between cholesterol consumption and blood pressure increases [START_REF] Stamler | Relationship to blood pressure of combinations of dietary macronutrients. Findings of the Multiple Risk Factor Intervention Trial (MRFIT)[END_REF][START_REF] Stamler | Eight-year blood pressure change in middle-aged men: Relationship to multiple nutrients[END_REF]. The Multiple Risk Factor Intervention Trial [START_REF] Stamler | Relationship to blood pressure of combinations of dietary macronutrients. Findings of the Multiple Risk Factor Intervention Trial (MRFIT)[END_REF] trial found that cholesterol intake was positively associated with blood pressure, even after adjustment for multiple dietary variables including saturated fat. The Chicago Western Electric study [START_REF] Stamler | Eight-year blood pressure change in middle-aged men: Relationship to multiple nutrients[END_REF] confirmed these observations. Similarly, the multi-centre INTERMAP study reported positive associations between cholesterol intake and blood pressure in a multi-national cohort [START_REF] Sakurai | Relationship of dietary cholesterol to blood pressure: The INTERMAP study[END_REF].

The observations regarding egg consumption agree with the observations from the Lipid and Blood Pressure Meta-analysis Collaboration (LBPMC) Group [START_REF] Mazidi | Egg Consumption and Risk of Total and Cause-Specific Mortality: An Individual-Based Cohort Study and Pooling Prospective Studies on Behalf of the Lipid and Blood Pressure Meta-analysis Collaboration (LBPMC) Group[END_REF] (n = 23,524), who observed higher blood pressure in higher consumers of eggs, and a recent European study that identified associations between higher blood pressure and high consumers of eggs [START_REF] Tong | The associations of major foods and fibre with risks of ischaemic and haemorrhagic stroke: A prospective study of 418 329 participants in the EPIC cohort across nine European countries[END_REF]. However, our observations disagree with four previous prospective investigations into egg consumption and hypertension [START_REF] Golzarand | Protein Foods Group and 3-Year Incidence of Hypertension: A Prospective Study from Tehran Lipid and Glucose Study[END_REF][START_REF] Steffen | Associations of plant food, dairy product, and meat intakes with 15-y incidence of elevated blood pressure in young black and white adults: The Coronary Artery Risk Development in Young Adults (CARDIA) Study[END_REF][START_REF] Circs Investigators; Umesawa | Association between dietary behavior and risk of hypertension among Japanese male workers[END_REF][START_REF] Lee | Diet-Related Risk Factors for Incident Hypertension during an 11-Year Follow-Up: The Korean Genome Epidemiology Study[END_REF] and one meta-analysis which reported lower associations of hypertension amongst egg consumers [START_REF] Zhang | Red meat, poultry, and egg consumption with the risk of hypertension: A meta-analysis of prospective cohort studies[END_REF]. In the prospective CARDIA study [START_REF] Steffen | Associations of plant food, dairy product, and meat intakes with 15-y incidence of elevated blood pressure in young black and white adults: The Coronary Artery Risk Development in Young Adults (CARDIA) Study[END_REF] (n = 4304, cases = 997), higher egg intake was associated with lower risk of hypertension. An Iranian study [START_REF] Golzarand | Protein Foods Group and 3-Year Incidence of Hypertension: A Prospective Study from Tehran Lipid and Glucose Study[END_REF] (n = 1708, cases = 144) observed a decreased incidence of hypertension in participants consuming upwards of around half an egg per day. Trends were similar in a Korean (n = 6792, cases = 1212) [START_REF] Lee | Diet-Related Risk Factors for Incident Hypertension during an 11-Year Follow-Up: The Korean Genome Epidemiology Study[END_REF] and a Japanese study [START_REF] Circs Investigators; Umesawa | Association between dietary behavior and risk of hypertension among Japanese male workers[END_REF] (n = 3486, cases = 846). The Finnish KIHD study identified null associations between egg/cholesterol consumption [START_REF] Abdollahi | Egg consumption, cholesterol intake, and risk of incident stroke in men: The Kuopio Ischaemic Heart Disease Risk Factor Study[END_REF], blood pressure, and stroke, but included only men, with a higher baseline cholesterol intake than our population. Eggs may vary in size and cholesterol content depending on geographic location, which could be a reason for differing results. It is likely that the overall diet, population characteristics, and regional differences in chronic disease epidemiology also affect the results.

Associations between dietary cholesterol and hypertension are possibly caused by higher levels of serum cholesterol in those with a high cholesterol intake. A recent meta-regression [START_REF] Vincent | Meta-regression analysis of the effects of dietary cholesterol intake on LDL and HDL cholesterol[END_REF] using data solely from randomised trials indicates that a 100-mg/d increase in dietary cholesterol would result in a 4.52-mg/dL increase in circulating LDL-C, with the effect more pronounced in women. Higher serum cholesterol could be associated with increased blood pressure due to effects on nitric oxide (NO). Studies have shown that high cholesterol is strongly correlated with higher serum dimethylarginine (ADMA), and upregulates its synthesis in human endothelial cells [START_REF] Böger | LDL cholesterol upregulates synthesis of asymmetrical dimethylarginine in human endothelial cells: Involvement of S-adenosylmethionine-dependent methyltransferases[END_REF]. ADMA inhibits NO production, a naturally derived substance that produces vasodilation and modulates blood pressure [START_REF] Gamboa | Contribution of Endothelial Nitric Oxide to Blood Pressure in Humans[END_REF]. Inhibition of NO results in reduced dilatory capability of the blood vessels, endothelial dysfunction, and reduced salt sensitivity [START_REF] Ivanovic | Hypercholesterolemia and Hypertension: Two Sides of the Same Coin[END_REF], leading to increased blood pressure [START_REF] Haynes | Inhibition of nitric oxide synthesis increases blood pressure in healthy humans[END_REF] and arterial stiffening. Egg consumption has been associated with increases in both LDL-C and HDL-C [START_REF] Dimarco | Intake of up to 3 Eggs per Day Is Associated with Changes in HDL Function and Increased Plasma Antioxidants in Healthy, Young Adults[END_REF][START_REF] Mutungi | Dietary Cholesterol from Eggs Increases Plasma HDL Cholesterol in Overweight Men Consuming a Carbohydrate-Restricted Diet[END_REF][START_REF] Rouhani | Effects of Egg Consumption on Blood Lipids: A Systematic Review and Meta-Analysis of Randomized Clinical Trials[END_REF], without significantly altering the ratio of HDL:LDL-C [START_REF] Dimarco | Intake of up to 3 Eggs per Day Is Associated with Changes in HDL Function and Increased Plasma Antioxidants in Healthy, Young Adults[END_REF][START_REF] Mutungi | Dietary Cholesterol from Eggs Increases Plasma HDL Cholesterol in Overweight Men Consuming a Carbohydrate-Restricted Diet[END_REF][START_REF] Rouhani | Effects of Egg Consumption on Blood Lipids: A Systematic Review and Meta-Analysis of Randomized Clinical Trials[END_REF]. Increased HDL-C is implicated in the excretion of cholesterol via the liver in a process known as reverse cholesterol transport [START_REF] Ouimet | HDL and Reverse Cholesterol Transport[END_REF], and has various anti-oxidising effects, whereas high LDL-C is strongly associated with increased rates of CVD [START_REF] Silverman | Association Between Lowering LDL-C and Cardiovascular Risk Reduction Among Different Therapeutic Interventions: A Systematic Review and Meta-analysis[END_REF]. Lowering circulating cholesterol has been demonstrated to improve endothelial function [START_REF] Vogel | Cholesterol lowering and endothelial function[END_REF], thus potentially explaining why increased cholesterol intake from sources such as eggs may have negative implications for vascular health, and incident hypertension [START_REF] Dharmashankar | Vascular endothelial function and hypertension: Insights and directions[END_REF].

In Asian cuisine, eggs are commonly incorporated into dishes, whereas in Europe, eggs are typically consumed with red or processed meat. Replacement effects could be in place for foods associated with hypertension in different regions, i.e., egg consumption was inversely related with salted seafood in the Korean study. Egg consumption is also difficult to compare between studies, for example, in the Japanese study, participants reported egg consumption as yes/no, whereas in the CARDIA study eggs were assessed as 'times eaten per day'. In the Korean study, the median egg consumption was lower than in our cohort, 12.5 g per day, which we would have estimated as 1.2 eggs per week. Additionally, previous studies into egg consumption and hypertension have failed to estimate the dietary cholesterol intake of their participants, thus it is possible that associations could differ in populations consuming different amounts of dietary cholesterol or other nutrients.

Studies into cardiovascular mortality have reported increased risk associated with egg consumption [START_REF] Zhong | Associations of Dietary Cholesterol or Egg Consumption With Incident Cardiovascular Disease and Mortality[END_REF][START_REF] Li | Egg consumption and risk of cardiovascular diseases and diabetes: A meta-analysis[END_REF][START_REF] Djoussé | Egg consumption and risk of heart failure in the Physicians' Health Study[END_REF]. A recent prospective analysis by Zhong et al. in the USA [START_REF] Zhong | Associations of Dietary Cholesterol or Egg Consumption With Incident Cardiovascular Disease and Mortality[END_REF] identified positive associations between egg consumption, cholesterol intake (with a level of cholesterol intake in their multiple cohorts similar to ours), and incident CVD/all-cause mortality. Conversely, a recent Chinese study observed an inverse relationship between egg consumption and CVD [START_REF] Qin | Associations of egg consumption with cardiovascular disease in a cohort study of 0.5 million Chinese adults[END_REF]. In meta-analysis of egg consumption and CVD, inverse associations have been reported in Asian populations, but not in American or European populations [START_REF] Drouin-Chartier | Egg consumption and risk of cardiovascular disease: Three large prospective US cohort studies, systematic review, and updated meta-analysis[END_REF].

In our study, the associations between cholesterol or egg intake and hypertension in subgroups were consistent. However, when considering those women who consumed over seven eggs per week, associations were null, perhaps due to the small sample size (~5% of the study population), or specific confounding with this group as they were significantly more likely to have attended university, and had a much higher Mediterranean diet score than the other groups of egg consumers. It is possible that egg intake this high represented a specific health-conscious behaviour in this population. Cholesterol and egg intake was associated with hypertension risk in those with a high Mediterranean diet score, suggesting that high cholesterol intake can have a negative impact on a high quality diet. In line with this, we observed weaker association between egg or cholesterol consumption and hypertension risk in high consumers of processed meat, a group who consumed more cholesterol, salt and saturated fats, and thus likely encountered no increased risk from egg-based or other sources of cholesterol.

Strengths and Limitations

The main strengths of this study are the time-updated cholesterol and egg data. The high number of egg consumers in this cohort, its prospective manner, large cohort and number of cases, limited loss to follow-up and detailed and validated dietary questionnaire, permitting us to assess updated coefficients in modelling, are also important factors. As with all observational studies, there are limiting factors. We relied on self-reported diagnosis, and prescription of anti-hypertensive drugs for the identification of hypertension cases. We observed a strong PPV of 82% for self-reported cases with drug reimbursements, but we could not identify undiagnosed cases. These cases should be randomly distributed, and would tend to attenuate associations.

The main weaknesses of the study are the self-reported diet, and the impossibility of claiming causality. The questionnaire, whilst validated using 24-h recalls, is subject to some variation. In this cohort, both eggs and cholesterol showed only modest recall correlation in 119 participants. It is difficult to rule out residual confounding due to dietary history, genetics, or lifestyle factors such as health awareness. Additionally, the E3N cohort is not representative of the French population, and is characterised by a high level of education, and a relatively low average BMI. However, we were able to adjust for the known risk-factors for hypertension onset, as well as various dietary factors associated with the disease such as fibre, salt, saturated fat and processed meat intake. We were also able to control for a Mediterranean dietary score, which has shown protective associations with various diseases in this cohort, the lack of which is a noted limitation of other studies in a recent meta-analysis.

Associations were weakened, but still consistent when controlling for saturated fat. We performed various subgroup analyses to investigate the likelihood of residual confounding from smoking status, weight, social-economic status, diet, and treatments for dyslipidaemia and diabetes, although we may have lacked statistical power amongst those with diabetes.

It is important to consider how eggs are consumed; we observed the highest correlation between eggs and processed meat, and no correlation with vegetables or dietary fibre. People who consume eggs as an omelette with vegetables would likely be associated with a different overall diet pattern than eggs consumed with ham and other processed meats. We did not consider eggs from other sources, such as baked goods, due to the low amount of egg per serving size that would have been estimated, and we were also unable to determine if participants were consuming egg whites or yolks differently.

Egg consumption has been associated with lifestyle characteristics but this varies according to regions and demographics. In the E3N cohort who consumed on average 366 mg of cholesterol daily, a higher consumption of eggs was associated with a lower prevalence of smoking, slightly higher activity levels, and slightly lower education attainment. Regarding other dietary factors, high egg consumption was associated with increased processed meat consumption, but not with fruit and vegetable consumption; it also tended to be associated with a higher Mediterranean diet score. As the cohort consists of women, results may not be generalizable to men due to differences in lipid metabolism and protective effects from different hormones [START_REF] Palmisano | Sex differences in lipid and lipoprotein metabolism[END_REF]; however, results in previous studies have been consistent across sex [START_REF] Steffen | Associations of plant food, dairy product, and meat intakes with 15-y incidence of elevated blood pressure in young black and white adults: The Coronary Artery Risk Development in Young Adults (CARDIA) Study[END_REF].

Conclusions

In conclusion, these results indicate that high cholesterol and egg consumption is associated with a higher risk of hypertension among French women. Evidence was found for non-linear relationships between the number of eggs consumed per week, cholesterol consumption, and the risk of hypertension.

Figure 1 .

 1 Figure 1. Spline regression curves detailing hypertension risk for associated with: (A) cholesterol consumption; (B) egg consumption. Dashed lines indicate 95% confidence interval.All models adjusted for BMI, total physical activity, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, diabetes, intake of alcohol, total calorific intake, Mediterranean diet score and age as the timescale.

Table 1 .

 1 Participant demographics and dietary data, depending on egg and cholesterol consumption at baseline.

  /33.3/13.4 53.1/33.7/13.2 52.0/34.3/13.7 51.8/33.8/14.4 50.0/34.2/15.8 49.7/32.2/18.1 53.9/32.3/13.8 51.9/34.0/14.1 51.7/34.6/13.7 51.6/34.2/14.2 50.8/33.8/15.3

		5 (0.3)	1.5 (0.3)	2.5 (0.3)	3.3 (0.3)	5.0 (0.8)	8.5 (2.7)	1.0 (0.9)	1.7 (1.2)	2.4 (1.4)	3.3 (1.7)	4.9 (3.1)
	Median cholesterol (mg/day)	267 (93)	329 (95)	379 (99)	420 (102)	474 (116)	621 (154)	226 (42)	306 (19)	369 (19)	441 (25)	568 (105)
	Age (years)	51.8 (6.7)	50.5 (6.3)	49.9 (6.1)	49.5 (6.0)	49.6 (5.8)	49.9 (6.1)	51.8 (6.7)	50.5 (6.3)	49.8 (6.1)	49.5 (6.0)	49.1 (5.8)
	BMI (kg/m 2 )	21.8 (2.6)	22.1 (2.7)	22.2 (2.8)	22.4 (2.8)	22.7 (2.9)	22.9 (3.1)	21.8 (2.6)	22.1 (2.7)	22.2 (2.7)	22.4 (2.8)	22.8 (3.1)
	Total physical activity (METS/week)	53.3 (29.7)	53.5 (29.2)	54.6 (29.8)	55.5 (30.2)	55.6 (30.8)	56.2 (31.1)	53.4 (29.9)	53.3 (29.3)	54.1 (29.2)	54.5 (29.6)	56.3 (31.0)
	Diabetes (%)	0.4	0.4	0.6	0.6	0.6	0.7	0.5	0.5	0.5	0.5	0.6
	Dyslipidaemia (%)	9.0	6.0	4.5	4.1	3.4	2.4	8.9	5.3	5.0	3.9	3.3
	Family history CVD (%)	34.0	34.5	34.2	33.5	32.9	31.1	33.6	33.9	34.3	33.4	33.4
	Prior cancer (%)	6.4	6.8	6.8	6.4	6.8	6.5	6.6	6.8	6.8	6.4	6.4
	Menopausal at baseline (%)	50.2	43.2	40.3	38.4	40.1	42.1	52.5	44.4	40.5	38.3	36.5
	Ever use of MHT amongst menopausal women (%)	44.3	46.7	46.9	47.2	46.3	45.5	42.6	46.9	47.7	47.9	46.9
	High school education (%)	59.8	59.8	61.2	62.2	64.2	57.1	61.1	59.8	60.0	61.3	63.7
	University education (%)	40.2	40.2	38.8	37.8	35.8	42.9	38.9	40.2	40.0	38.7	36.3
	Smoking (Never/X/current) (%)	53.2Dietary variables (median, SD)
	Total energy (Kcal)	1882 (489)	1995 (484)	2104 (502)	2237 (560)	2267 (548)	2375 (603)	1626 (357)	1885 (369)	2079 (385)	2284 (416)	2642 (508)
	Salt (mg/day)	2478 (861)	2609 (836)	2760 (858)	2947 (924)	2987 (921)	3099 (1010)	2147 (706)	2463 (734)	2722 (755)	2983 (783)	3453 (918)
	Potassium (mg/day)	3474 (993)	3601 (953)	3720 (991)	3963 (1061)	4012 (1086)	4195 (1212)	3225 (934)	3475 (914)	36845 (905)	3908 (961)	4344 (1058)
	Saturated fats (g/day)	30 (12)	33 (12)	36 (12)	36 (12)	38 (13)	41 (15)	24 (7)	31 (7)	35 (8)	40 (10)	49 (13)
	Dietary fibre (g/day)	23 (8)	23 (7)	24 (8)	25 (8)	25 (9)	26 (9)	22 (8)	23 (7)	24 (7)	25 (8)	27 (8)
	Alcohol (g/day)	5 (13)	6 (12)	8 (13)	8 (14)	8 (15)	9 (16)	4 (11)	6 (13)	7 (13)	9 (14)	10 (16)
	Meat (g/day)	76 (45)	81 (42)	81 (43)	80 (42)	74 (43)	69 (46)	58 (35)	73 (38)	82 (40)	86 (44)	91 (49)
	Processed meat (sausage, ham, pate) (g/day)	10 (16)	13 (15)	16 (16)	20 (19)	21 (20)	23 (25)	9 (12)	12 (13)	16 (16)	19 (17)	26 (22)
	Fish (g/day)	21 (23)	25 (20)	25 (20)	25 (21)	25 (23)	25 (26)	18 (20)	24 (20)	25 (21)	25 (22)	25 (25)
	Shellfish/canned fish (g/day)	3 (9)	5 (10)	7 (10)	8 (11)	9 (12)	10 (15)	2 (7)	4 (8)	6 (9)	9 (11)	12 (15)
	Vegetables (g/day)	143 (102)	143 (98)	143 (98)	157 (103)	151 (104)	171 (121)	143 (104)	143 (99)	143 (100)	148 (100)	157 (107)
	Fruit (g/day)	231 (175)	228 (158)	227 (164)	219 (162)	229 (175)	233 (190)	237 (175)	231 (164)	226 (162)	219 (163)	219 (177)
	Mediterranean diet score [-1, 1]	-0.34 (0.92)	-0.28 (0.88)	-0.19 (0.90)	-0.06 (0.93)	0.14 (1.00)	0.50 (1.15)	-0.37 (0.87)	-0.27 (0.87)	-0.17 (0.90)	-0.05 (0.95)	0.26 (1.09)

Table 2 .

 2 BMI adjusted and multivariate adjusted cox proportional hazard models for incident hypertension risk based on time varying distributions of egg consumption and cholesterol consumption, with follow-up from 1993-2014.

	Weekly Egg Intake

M0 non adjusted, with age as timeline; M1 adjusted for BMI; M2 adjusted for total physical activity, total calories, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, and diabetes; M3 adjusted for M2 + intake of alcohol, unprocessed meat, processed meat, vegetables, fruits, shellfish, fish, salt, potassium and saturated fats; M3* adjusted for M2 + intake of vegetables, fruits, salt, potassium and saturated fats; M4 adjusted for M2 + Mediterranean diet score.

Figure 1. Spline regression curves detailing hypertension risk for associated with: (A) cholesterol consumption; (B) egg consumption.

Dashed lines indicate 95% confidence interval. All models adjusted for BMI, total physical activity, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, diabetes, intake of alcohol, total calorific intake, Mediterranean diet score and age as the timescale.

Table 3 .

 3 Subgroup analysis: stratified multivariate cox proportional hazard models for incident hypertension risk related to weekly egg consumption.

	Weekly Egg Consumption

Models adjusted for BMI, total physical activity, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, diabetes, intake of alcohol, total calorific intake, and Mediterranean diet score, with age as the timeline, minus the stratification variable.

Table 4 .

 4 Subgroup analysis: stratified multivariate cox proportional hazard models for incident hypertension risk related to daily cholesterol consumption.

			Quintiles of Cholesterol Intake (mg/Day)
		P for Interaction	≤270	271-336	337-401	402-487	≥488	P for Trend
	Normal weight (mean BMI = 21.4, n = 39,966)	0.85	ref	1.16 [1.09: 1.24]	1.22 [1.15: 1.30]	1.25 [1.17: 1.34]	1.24 [1.15: 1.34]	<0.0005
	Overweight/obese (mean BMI = 27.1, n = 6635)	-	ref	1.07 [0.93: 1.23]	1.05 [0.91: 1.21]	1.11 [0.97: 1.28]	1.21 [1.05: 1.40]	0.006
	No dyslipidaemia (n = 36,461)	0.25	ref	1.15 [1.08: 1.23]	1.19 [1.12: 1.27]	1.23 [1.15: 1.32]	1.25 [1.16 1.34]	<0.0005
	Dyslipidaemia (n = 9963)	-	ref	1.07 [0.94: 1.21]	1.10 [0.96: 1.26]	1.08 [0.93: 1.25]	1.09 [0.94: 1.28]	0.25
	No diabetes (n = 45,670)	0.79	ref	1.13 [1.07: 1.20]	1.17 [1.11: 1.24]	1.21 [1.13: 1.28]	1.22 [1.14: 1.30]	< 0.05
	Diabetes (n = 754)	-	ref	1.08 [0.66: 1.76]	0.94 [0.58: 1.54]	1.17 [1.73: 1.89]	1.01 [0.61: 1.66]	0.93
	High school education (n = 25,555)	0.28	ref	1.14 [1.06: 1.23]	1.20 [1.11: 1.29]	1.22 [1.13: 1.32]	1.27 [1.17: 1.38]	<0.0005
	University education (n = 18,065)	-	ref	1.12 [1.02: 1.23]	1.14 [1.04: 1.26]	1.15 [1.04: 1.27]	1.16 [1.04: 1.20]	0.02
	Smokers (n = 6622)	0.35	ref	1.09 [0.93: 1.27]	1.02 [0.87: 1.19]	1.03 [0.88: 1.21]	1.04 [0.88: 1.23]	0.90
	Ex-smokers (n = 15,677)	-	ref	1.16 [1.05: 1.28]	1.25 [1.13: 1.38]	1.26 [1.14: 1.40]	1.26 [1.12: 1.41]	<0.0005
	Never smokers (n = 24,215)	-	ref	1.13 [1.04: 1.22]	1.17 [1.08: 1.26]	1.21 [1.12: 1.32]	1.24 [1.14: 1.36]	0.0005
	Low processed meat (median = 7.9 g/day, n = 23,309)	<0.005	ref	1.17 [1.09: 1.26]	1.21 [1.12: 1.31]	1.21 [1.11: 1.32]	1.20 [1.08: 1.33]	<0.0005
	High processed meat (median = 25.9 g/day, n = 23,292)	-	ref	1.03 [0.94: 1.13]	1.05 [0.96: 1.15]	1.08 [0.99: 1.18]	1.06 [0.97: 1.16]	0.26
	High Mediterranean diet score (median = 0.5, n = 23,300)	<0.005	ref	1.21 [1.10: 1.32]	1.32 [1.21: 1.44]	1.35 [1.23: 1.48]	1.35 [1.24: 1.48]	<0.0005
	Low Mediterranean diet score (median = -0.7, n = 23,301)	-	ref	1.07 [1.00: 1.16]	1.05 [0.97: 1.14]	1.09 [1.00: 1.18]	1.11 [1.01: 1.22]	0.03

Models adjusted for BMI, total physical activity, smoking, family history of CVD, education level, menopausal status, use of MHT, dyslipidaemia, diabetes, intake of alcohol, total calorific intake, and Mediterranean diet score, with age as the timeline, minus the stratification variable.

Acknowledgments:

We gratefully acknowledge the contribution of all the participants in the E3N study for their diligence and their answers. The authors have no conflicts of interest to declare.

Supplementary Materials:

The following are available online at http://www.mdpi.com/2072-6643/12/5/1350/s1, Figure S1: Flow chart of participant exclusion and inclusion, Table S1: Comparison between characteristics of included and excluded population. Funding: The E3N cohort is being carried out with the financial support of the "Mutuelle Générale de l'Education Nationale" (MGEN); European Community; French League against Cancer (LNCC); Gustave Roussy Institute (IGR); French Institute of Health and Medical Research (INSERM). This present study was also supported by the French Research Agency (Agence Nationale de la Recherche) via an "Investissement d'Avenir" grant (investment for the future grant, ANR-10-COHO-0006). CJM is supported by funding from the Federation Française de Cardiologie.

Conflicts of Interest:

The authors declare no conflict of interest.