M.-P Noutary

N Biboulet

A A Lubrecht
email: ton.lubrecht@insa-lyon.fr

Dimple influence on load carrying capacity of parallel surfaces

Keywords:

In the case of parallel surfaces, such as the oil control ring, no load carrying capacity is generated without texture. A flow conserving multigrid code is used to solve the hydrodynamic lubrication equation described by the Reynolds equation. This code is used to investigate the influence of texture geometry (dimples) and its position along the contact. In particular the behavior of the load carrying capacity according to the texture position on the central line of the contact is studied. To generate positive load carrying capacity, only partial texturing must be used. A significant difference can be observed between inlet texturing and outlet texturing. A multigrid code has been developped that allows a comparison between Dirichlet and periodic results. A comparison with equivalent step bearings is made to analyse the load carrying capacity.

Introduction

Texturing greatly affects the performance of lubricated contacts in terms of pressure, load carrying capacity and friction. For parallel surfaces, it can generate a load carrying capacity by creating the equivalent of a Raleigh step bearing. Full texturing induces a negative load carrying capacity, but inlet texture generates a positive one. Hence the texture parameters (size, density, texture position) have to be carefully designed in order to optimize the contact performance.

Early research was conducted by Etsion, Kligerman, Halperin, Brizmer [START_REF] Etsion | Experimental investigation of laser textured parallel thrust bearings[END_REF][START_REF] Kligerman | Improving tribological performance of piston rings by partial surface texturing[END_REF][START_REF] Brizmer | A laser surface textured parallel thrust bearing[END_REF][START_REF] Etsion | Analytical and experimental investigation of laser-textured mechanical seal faces[END_REF][START_REF] Ryk | Experimental investigation of laser surface texturing for reciprocating automotive components[END_REF][START_REF] Etsion | A laser surface textured hydrostatic mechanical seal[END_REF]. They performed experimental but also analytical investigations and showed the benefits of surface texturing for the enhancement of lubrication and in particular of partial texturing for parallel surfaces. In Ref. [START_REF] Brizmer | A laser surface textured parallel thrust bearing[END_REF] for a simplified model, the distribution of the local pressure for partial texturing in the inlet and for full texturing was shown. Tonder [START_REF] Tonder | Hydrodynamics effects of tailored inlet roughness: extended theory[END_REF] introduced the idea that performance improvement can be gained by introducing artificial roughness in the contact inlet by means of micro-structures instead of macro-structures, inducing low coefficients of friction coexisting with high hydrodynamic stiffness and damping parameters. In an experimental study, Vladescu et al. [START_REF] Vladescu | Lubricant film thickness and friction force measurements in a laser surface textured reciprocating line contact simulating the piston ring-liner pairing[END_REF] analysed the influence of texturing in reciprocating contacts for various lubrication regimes. Surface texture just after reversal helps to build up the film and reduces friction. Texture pockets should entirely lie inside the contact zone. No pocket should be at the reversal point. An experimental study of Lu et al. [START_REF] Lu | The friction reducing effect of square-shaped surface textures under lubricated line-contacts -an experimental study[END_REF] for reciprocating sliding line contacts analysed the effect of surface texturing by square dimples in various lubrication regimes. A lower friction coefficient was obtained in the boundary regime, due to the fact that the dimples act as lubricant reservoirs. They observed that texture at the reversal point can reduce friction, this observation contradicts [START_REF] Vladescu | Lubricant film thickness and friction force measurements in a laser surface textured reciprocating line contact simulating the piston ring-liner pairing[END_REF]. Many 1D analytical or numerical studies have been performed. Using a 1D model Tomanik [START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF] has investigated the influence of laser textured profiles for cylinder bore and rings and particularly for the oil control ring. Fowell et al. [START_REF] Fowell | Entrainment and inlet suction: two mechanisms of hydrodynamic lubrication in textured bearings[END_REF] explained the inlet suction phenomenon on a 1D analytical model. Fowell et al. [START_REF] Fowell | Parametric study of texturing in convergent bearings[END_REF] have used a 1D mass conserving numerical model to study the influence of the texture parameters according to the convergence ratio of the two main surfaces on load support and friction. Pascovci et al. [START_REF] Pascovici | Analytical investigation of a partially textured parallel slider[END_REF] have performed a 1D analysis without cavitation to optimize the texture parameters for parallel sliders and made a comparison with Raleigh step bearings. Rahmani et al. [START_REF] Rahmani | Optimization of partially textured parallel thrust bearings with square-shaped micro-dimples[END_REF] have used an analytical solution of the 1D Reynolds equation to optimize micro dimple texture for parallel thrust bearings.

The validity of the Reynolds model was discussed in several papers. Sahlin et al. [START_REF] Sahlin | Two-dimensional CFD-analysis of micro-patterned surfaces in hydrodynamic lubrication[END_REF] used a commercial CFD code to solve the Navier-Stokes system for a single micro-groove on one of 2 parallel surfaces. For them, fluid inertia is an important factor for the load carrying capacity and they have shown that for deep and wide grooves a vortex occurs that limits the increase in load carrying capacity. In their sequel, Menon et al. [START_REF] Menon Dileep | An analysis on the influence of oil pocket shape and distribution on the reduction of friction in hydrodynamic lubrication[END_REF] also think that fluid advection is a major factor for generating load carrying capacity. In both works, cavitation was not considered. They tend to show that the Reynolds model is not adequate to study these problems. However, in Ref. [START_REF] Dobrica | About the validity of Reynolds equation and inertia effects in textured sliders of infinite width[END_REF], Dobrica and Fillon have studied the range of validity of the Reynolds equation also neglecting cavitation. They conclude that the Reynolds number and the dimple aspect ratio (dimple depth/dimple width) must be sufficiently small but nevertheless in a range that covers most useful cases. https://doi.org/10.1016/j.triboint.2018. [START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF] Non mass-conserving models underestimate the cavitation area leading to poor estimates of the pressure as explained by Ausas et al., [START_REF] Ausas | The impact of the cavitation model in the analysis of microtextured lubricated journal bearings[END_REF]. Ausas et al. [START_REF] Ausas | Conservative one-dimensional finite volume discretization of a new cavitation model for piston-ring lubrication[END_REF] proposed a finite volume implementation of the Elrod-Adams model.

To study cavitation in dimples, Qiu and Khonsari [START_REF] Qiu | On the prediction of cavitation in Dimples using a massconservative algorithm[END_REF] developped a mass-conservative cavitation multigrid algorithm based on the work by Vijayaraghavan and Keith [START_REF] Vijayaraghavan | Development and evaluation of a cavitation algorithm[END_REF]. Bayada, Martin and Vazquez [START_REF] Bayada | An average flow model of the Reynolds roughness including a mass-flow preserving cavitation model[END_REF] developped a mass-conserving algorithm using homogenization techniques. Using the pressure and the void fraction, Giacoponi et al. [START_REF] Giacopini | A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation[END_REF] built a mass-conserving complementary formulation of the 1D cavitation problem. Bertocchi et al. [START_REF] Bertocchi | Fluid film lubrication in the presence of cavitation: a mass-conserving two-dimensional formulation for compressible, piezoviscous and non-Newtonian fluids[END_REF] have extended it to 2D domains and for compressible, piezoviscous and non-Newtonian fluids and used finite element methods to numerically solve the problem. An implementation of a finite volume method derived from thep θ Elrod- Adams cavitation model [START_REF] Elrod | A computer program for cavitation and starvation problems[END_REF][START_REF] Elrod | A cavitation algorithm[END_REF] was proposed by Profito et al. [START_REF] Profito | A general finite volume method for the solution of the Reynolds lubrication equation with a mass-conserving cavitation model[END_REF]. Woloszynski, Podsiadlo and Stachowiak [START_REF] Woloszynski | Efficient solution to the cavitation problem in hydrodynamic lubrication[END_REF] have reformulated the complementary constraint on = ≥ ≥ p θ p θ . 0, 0, 0 by a continuous differentiable system = F p θ ( , ) 0 to which Newton's method can be efficiently applied. Biboulet [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF] has successfully reduced the storage requirements and improved robustness through the use of a multigrid strategy.

Using a mass-conserving algorithm, Dobrica et al. [START_REF] Dobrica | Optimizing surface texture for hydrodynamic lubricated contacts using a mass-conserving numerical approach[END_REF] have studied the influence of full and partial texturing for parallel and plane inclined sliders, verifying in particular the benefit of inlet partial texturing and the importance of the cavitation phenomenon to generate load carrying capacity in parallel or nearly parallel structures. Cupillard et al. [START_REF] Cupillard | Computational fluid dynamic analysis of a journal bearing with surface texturing[END_REF][START_REF] Cupillard | Pressure buildup mechanism in a textured inlet of a hydrodynamic contact[END_REF] used the full Navier-Stokes equation with a multi-phase flow cavitation model to study the performance of a dimple textured journal bearing and analyse the pressure buildup mechanism. In Ref. [START_REF] Tala-Ighil | Effect of textured area on the performances of a hydrodynamic journal bearing[END_REF], Tala-Ighil, Fillon and Maspeyrot have analysed various texture configurations on the performance of a journal bearing. ) have numerically and experimentally studied parallel textured surfaces. For macroscopic dimples,the influence of the cavitation pressure in steady state lubrication can experimentally be significantly lower than the ambient pressure. Its value can influence simulation results. In the case of macroscopic dimples, the influence of the size and shape of the dimples on the load carrying capacity was investigated. They tried to optimize the texture shape for parallel surfaces. Gherca et al. [START_REF] Gherca | Influence of texture geometry on the hydrodynamic performances of parallel bearings[END_REF][START_REF] Fatu | Influence of surface geometry on the hydrodynamic performances of parallel bearings in transient flow condition[END_REF][START_REF] Gherca | Effects of surface texturing in steady-state and transient flow conditions: two-dimensional numerical simulation using a mass-conserving cavitation model[END_REF] have recently developed a mass-conserving algorithm to study the influence of surface texturing in steady-state but also transient hydrodynamic lubrication of parallel sliders.

In this paper, a multigrid flow-conserving algorithm based on Alcouffe's paper [START_REF] Alcouffe | The multi-grid method for the diffusion equation with strongly discontinuous coefficients[END_REF] and on the Elrod and Adams [START_REF] Elrod | A computer program for cavitation and starvation problems[END_REF][START_REF] Elrod | A cavitation algorithm[END_REF] model is built. It does not show full multigrid efficiency. This is probably due to the discontinuity of the Poiseuille and Couette flow which were not well accounted for when the pressure builds up again. However, the code converges sufficiently fast and is flow conserving. It can therefore be used to study and discuss dimple textured parallel surfaces, even with deep dimples.

Problem formulation and algorithm

Notations

The following parameters are used.

Mass complementary formulation

The hydrodynamic lubrication of the piston ring-cylinder liner contact can be described by the Reynolds equation. According to Elrod, because of cavitation and in order to have a mass conserving algorithm, the following complementary problem is solved:

⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ + ∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ - ∂ ∂ - ∂ ∂ = x ρh η p x y ρh η p y u ρθh x ρθh t 12 12
( ) ( ) 0 m 3 3 (1)
with

> = = < < p p θ p p θ ( and 1
) or ( and 0 1)

cav cav ( 2
)
Where p is the hydrodynamic pressure, h, the geometry, u m the mean velocity. The lubricant density, ρ and the viscosity, η can be considered constant in the studied regime. Considering that p in equation ( 1) is the difference between the pressure and the cavitation pressure, p cav , p cav is set to 0 in equation ( 2). This problem will be solved on a rectangle Ω.

The difference between the ambient pressure and the cavitation pressure is denoted is denoted p amb . Here the generated pressures are not very high because the considered surfaces are mainly parallel. Therefore the difference between the ambient pressure and the cavitation pressure is significant. No pressure would be generated if the cavitation pressure and the ambient pressure were identical. The pressure would not be able to decrease when entering the dimple and no extra oil would flow inside the dimple. In the cavitated area the fluid becomes a mixture of liquid, vapor and gas and θρ can be viewed as the lubricant density.Here we prefer to consider the variable = h θh oil , that represents the fluid level or an equivalent of the fluid level in the whole domain = -× -

x x y y Ω [ , ] [ , ]
0 0 0 0 . Hence = h h oil in the pressurized zones and lies between 0 and h elsewhere. In this paper only the stationary case is considered and therefore, we solve the following problem:

⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ + ∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ - ∂ ∂ = x h η p x y h η p y u h x 12 12 0 m oil 3 3 (3)
with

> = = < < p h h p h h ( 0 and
) or ( 0 and 0 )

oil oil (4a) = ∂ p p on Ω amb (4b)
Two kinds of boundary conditions are considered:

• Dirichlet conditions: the pressure is set to p amb , the ambient pres- sure, on the boundaries of the domain.

• Periodic conditions: the pressure and the geometry are periodic in the y-direction that is perpendicular to the sliding direction, while the pressure is p amb on the x-boundaries of the domain.

In what follows the fluid will be considered to flow between two parallel surfaces of constant gap h 0 . The stationary surface will be in- dented by circular dimples of radius r and depth a. For a single dimple, the cross section of the two surfaces by a plane passing through the middle of the dimple is sketched in Fig. 1. The geometry h is defined as follows:

= + h x y h ϕ x y ( , ) ( , ) 0 with
= ⎛ ⎝ + ⎛ ⎝ - + - ⎞ ⎠ ⎞ ⎠ ϕ x y a π r x x y y ( , ) 2 1 cos ( ) ( ) i i 2 2
if there is a dimple of depth a centered at x y ( , )

i i and if - + - ≤ x x y y r ( ) ( ) i i 2 2
2 and = ϕ x y ( , ) 0 otherwise.

Dimensionless equations

Upper letters are used for the associated dimensionless parameters: The lengths in directions orthogonal to the surfaces are made dimensionless using h 0 . In the x and y directions, a reference length λ is used. It could be = λ x 0 or = λ r.

∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ + ∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ - ∂ ∂ = X H P X Y H P Y H X ( ) 0 oil 3 3 (5)
with the following dimensionless complementary and boundary conditions:

> = = < < P H H P H H ( 0 and
) or ( 0 and 0 )

oil oil (6a) = ∂ =- ×- P P X X Y Y on Ω with Ω [ , ] [ , ] amb 0 0 0 0 (6b) and = + H X Y X Y ( , ) 1 Φ( , ) with = ⎛ ⎝ + ⎛ ⎝ - + - ⎞ ⎠ ⎞ ⎠ X Y A π R X X Y Y Φ( , ) 2 1 cos ( ) ( ) i i 2 2 (7)
if there is a dimple of depth A centered at X Y ( , )

i i and if - + - ≤ X X Y Y R ( ) ( ) i i 2 2
2 and = X Y Φ( , ) 0 otherwise.

Algorithm

A multigrid code using Alcouffe et al. ideas [START_REF] Alcouffe | The multi-grid method for the diffusion equation with strongly discontinuous coefficients[END_REF] was adapted to have flow conservation. A volume discretization is still used to discretize the problem and the residual represents the flow balance through a cell centered at the node i j ( , ) where the pressure is eval- uated. To eliminate the residual it is possible to act on the pressure or on the equivalent oil level. This is done in agreement with the physics of the problem for more details see Ref. [START_REF] Noutary | A robust Reynolds solver for textured surfaces in the piston-ring cylinder liner contact INSA Lyon[END_REF]. This enables one to build a satisfying mono-grid discretization scheme that correctly describes the flow behavior through an elementary cell. Afterwards, to properly represent the flow behavior through a coarse grid cell made of four fine grid cells, many difficulties were encountered. When using multigrid techniques, the problem to be solved is written as

= Lu f .
Here, we have two options.

• Option 1: set = f 0 and the unknown vector u contains P and Hoil.

• Option 2: set = ∂ ∂ f
Hoil X / which means that the Couette part of the flow is put in the right hand side of the equation.

In the first case we need to define coarsening and refining procedures for Hoil as well as for P while in the second case this is not needed but it will be necessary to update the right hand side f on the finest grid of each V-cycle each time the value of Hoil is changed. The coarsening routines for f should induce the correct right-hand side for the coarse grid equations.

In both cases an interpolation of Hoil will be needed if FMG (full multigrid see Refs. [START_REF] Brandt | multigrid guide, lightly revised 2011[END_REF][START_REF] Venner | Multilevel methods in lubrication[END_REF]) is used. Option 2 is the simplest one and was the one we finally chose. In that case it was also decided not to change the free boundaries on the coarser grids. Henceforth, the coarse grid relaxation will consist in only changing the pressure in the pressurized zone.

Computing time

The convergence is not the expected convergence of a multigrid process. Convergence difficulties are located at the film reformation boundary. While the frontiers of the cavitation zone stabilized rather rapidly, the flow equilibrium is slowly reached at the film reformation boundary. The multigrid routines are efficient in solving the pressure in the non cavitated zone but the equation changes a little especially at the film reformation boundary on the fine grid and this slows down the process. However the obtained convergence is sufficiently robust and fast and allows us to study the influence of dimples in the hydrodynamical lubrication of parallel surfaces. Table 1 shows the obtained residual and computing time for a single dimple of radius 1 and depth 1 at the center of a square domain = -× -Ω [ 4,4] [ 4,4] for various values of the ambient pressure P amb with Dirichlet conditions for a 1024 2 grid. Table 2 shows the obtained residual and computing time for a single dimple of radius 1 at the center of a square domain = -× -Ω [ 4,4] [ 4,4] for various depths with = P 0.01 amb and Dirichlet conditions for a 1024 2 grid. It takes about 40 min. Afterwards most computations will be done using a 640 2 grid that will ensure in each considered case that there are enough points to properly describe the dimple geometry. The computing time will then be around 10 min to get a very small residual. In the periodic case, for some texture patterns, it is also possible to work on a domain that is much smaller in the Y-direction (see 7.2) and this will greatly diminish the calculation time. It seems that the cavitation boundary is rapidly found but then, especially at film reformation, the oil film level and henceforth the pressure distribution are slow to converge. In Fig. 2, the dimple and the cavitation zone obtained for various values of P amb are shown. The convergence seems to depend on the height of the jump for the oil film when it reforms. It also corresponds to the proximity of the frontier at film reformation with the Fig. 1. Geometry: cross section through the middle of a dimple. . In these Tables, LCC stands for the load carrying capacity and corresponds to the integral of -P P amb over the domain. It can be noted that the load carrying ca- pacity is always negative for a centered dimple. An explanation is provided in section 5. Table 3 shows the grid convergence of the load carrying capacity: the LCC has a relative accuracy of 0.1%.

Generated pressure and oil film

In this section some figures of the pressure and oil film obtained for several dimple configurations are shown. In the chosen examples = P 0.01. amb Fig. 3 shows the results for a single dimple of depth = A 1

and radius = R 1 in the domain = - × - Ω [ 4,4] [ 4,4] whose center is located at = - X Y ( , ) ( 2,0) i i
. The geometry is given in Fig. 3(a): the dimple is located at the center of the first half of the domain. The resulting pressure is given in Fig. 3(c) and (d) respectively in the Dirichlet and periodic cases. The pressure rapidly decreases towards the cavitation pressure inside the dimple. At the dimple exit which corresponds to the convergent part of the surfaces, it grows very rapidly to reach its maximum value. In Fig. 3(b), the equivalent oil level is given for the Dirichlet case. In the cavitation zone the representation is not adequate since, in reality, there is a mixture of gas, oil and vapor. We have decided to represent an equivalent oil level that is to say = Hoil θH where θ is the filling ratio. The oil flow inside the dimple before cavitation can be viewed. Inside the cavitation zone it remains constant in the X-direction. At the exit of the dimple, where the film reforms, a discontinuity can be observed. The full film zone inside the dimple is narrow around the dimple edges. Fig. 5 shows the geometry, the oil level and the pressure for a set of nine dimples of depth

= A 1 and radius = R 1 in the domain = - × - Ω [ 10,10] [ 10,10] whose center is located at = - X Y ( , ) ( 2,0) i i
. The dimple pattern is also shown in that figure . In each dimple the pressure shows the same behavior as in a single dimple but at the exit of the first 2 rows of dimples the pressure does not reach very high values since rapidly it will decrease towards cavitation because of the next row of dimples. The maximum pressure is reached at the exit of the last row of dimples because afterwards no cavitation occurs. Here the limitation is mainly given by the exit boundary condition and therefore depends on the distance to the contact exit. Related results can be seen on the oil film. The last row of dimples is the fullest of the three. The main difference between the Dirichlet and the periodic case lies in the fact that the pressure does not reach the ambient pressure on the Y-boundaries enlarging the zone where the pressure is above the ambient pressure in the case of a dent or a pattern of dents in the contact inlet or enlarging the zone of pressure smaller than the ambient pressure in the case of a texture in the contact outlet. It may be noticed that the cavitated areas are not significantly different in those 2 cases as shown in Fig. 4. Henceforth it seems that the distance of the dimple to the boundaries and the boundary conditions must have a large influence on the load carrying capacity.

Load carrying capacity

Influence of the textured area on the load carrying capacity

The load carrying capacity is defined as:

∫ = - LCC P P dX dY ( ) .
amb Ω [START_REF] Vladescu | Lubricant film thickness and friction force measurements in a laser surface textured reciprocating line contact simulating the piston ring-liner pairing[END_REF] When the dimples are near the inlet, the load carrying capacity is positive and then it becomes negative as noticed by many authors. Considering a single dimple, it may be noted that the pressure will first decrease. This decrease will be limited by the cavitation pressure. To conserve the flow, oil must enter the dimple leading to the "inlet suction" phenomenon. Afterwards, in the convergent part of the dimple, the pressure will rise again starting from a point determined by the extra oil inside the dimple and then it will decrease to reach the ambient pressure at the contact exit. If the dimple is situated near the contact entrance, the zone where the pressure is less than the ambient pressure is limited and if it is located near the exit, the zone where it is above the ambient pressure is more limited. Besides, in case of cavitation, the pressure is not anti-symmetric with respect to P amb and if P max denotes the maximum pressure, -P P max amb is usually higher than -P P amb cav . This is illustrated in Fig. 6 where, on each row, the middle figure shows the part of the pressure above P amb while the last one shows the part of the pressure below the ambient pressure to highlight the zone where the pressure is less that the ambient pressure. In the last figure we have plotted -P and -P amb . Again it may be noted that in the case of a dent in the inlet, the zone of pressure above P amb significantly increases. If the texture symmetrically covers the domain the generated load carrying capacity becomes zero in case of no cavitation and negative otherwise. Hence partial texturing is needed to induce a positive load. To illustrate this, rows of three dimples of depth 1 spaced by 4 centered at

= - = X Y 18, 0 i i , then = - = X Y 16, 0 i i , then = - = X Y 14, 0 i i , etc … up to = = X Y 18, 0 i i
where they cover the whole domain = -× -Ω [ 20,20] [ 20,20] in the X-direction were progressively added and the generated load was computed. The pressure distribution can be viewed in Fig. 7. It can be noticed that at the exit of each row the pressure progressively increases except in the last figure where the whole range of values in X is covered. The largest pressures are obtained on the last row of dimples when the dimples remain in the first part of the domain. The load carrying capacity in terms of the number, N, of dimple rows is shown in Fig. 8. The maximum load carrying capacity is obtained with 3 rows of dimples in the inlet of the contact. When adding more rows it rapidly decreases to reach a negative value when the entire domain is covered. When rows are added at the end of the contact, there is always a loss in LCC. The variations in LCC are more limited than when rows are added in the inlet. The behavior of the maximum pressure is also quite different. Considering the different behaviors of the load carrying capacity and of the maximum pressure according to whether the inlet or the outlet is textured (see Figs. 7 and8, it can be noticed that the loss of load carrying capacity when the outlet is textured is lower than the gain when the inlet is textured. The maximum pressure is much higher if the inlet is textured rather than the outlet and its variations are also less important. For a one-dimensional simple model, Fowell et al. [START_REF] Fowell | Entrainment and inlet suction: two mechanisms of hydrodynamic lubrication in textured bearings[END_REF] show that

≈ P P L/Δ max amb
where L is the contact length and Δ, the distance of the pocket to the inlet. Here it seems that a similar trend can be observed. It can be noticed that in the periodic case these effects are increased due to the larger zone where the pressure is greater that the ambient pressure. In the periodic case the maximum LCC is roughly twice the maximum LCC in the Dirichlet case. The size of the domain in the direction that is perpendicular to the sliding direction probably plays an important part in this phenomenon. Besides, in the periodic case the flow remains constant.

Partial texturing at both ends

Because of the previous remarks, it may be worthwhile for reciprocating contacts to have partial texturing at both ends of the contact. This type of partial texturing for piston rings was experimentally tested by Ryk and Etsion [START_REF] Ryk | Testing piston rings with partial laser surface texturing for friction reduction[END_REF] and Etsion and Sher [START_REF] Etsion | Improving fuel efficiency with laser surface textured piston rings[END_REF]. Tomanik [START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF] also investigated that kind of partial texturing but using a 1D code without mass-conservation. In 2016, Shen and Khonsari [START_REF] Shen | The effect of laser machined pockets on the lubrication of piston ring prototypes[END_REF] have also conducted experimental as well as numerical studies with a mass-conserving algorithm and for the transient case on the effects of different positions of pockets (rather patches of micro-dimples) in the inlet and in the outlet for a flat piston ring. Tala-Ighil, Fillon and Maspeyrot [START_REF] Tala-Ighil | Effect of textured area on the performances of a hydrodynamic journal bearing[END_REF] have also tested various configurations of patches of dimples but for a hydrodynamic journal bearing. As can be seen in Fig. 9, a positive load carrying capacity is always generated. Its value is indicated for each case on the corresponding image of the pressure distribution. For our example the maximum load carrying capacity is obtained for two rows of indents at both ends. The maximum pressure is obtained at the exit of the last row of indents in the inlet. It will reach a sufficiently high value if the distance between the 2 sets of dimples is large enough and if the first set of dimples lies sufficiently near the inlet boundary inducing a total positive load carrying capacity. Fig. 10 shows similar results with similar conclusions for the periodic case.

Load carrying capacity in terms of the texture location

The behavior of the load carrying capacity in terms of the position of the set of dimples along the line = Y 0 is now studied. In Fig. 11, the results are shown for = -× -Ω [ 10,10] [ 10,10] and for = -× -Ω [ 20,20] [ 20,20] for a pattern of nine dimples of radius 1, depth 1 and equally spaced by 4. The load carrying capacity becomes negative when the texture approaches the middle of the contact. A linear behavior is observed in the center of the domain although this is less clear on the smaller domain. In that case, the domain is not large enough to ensure that no dimple is close to the inlet or the outlet of the contact. Close to the boundaries and especially near the inlet, the dimple influence becomes non linear. The maximum pressure and the size of the zone behind the texture induce the positive part in the load carrying capacity while the cavitation pressure and the size of the zone ahead the texture determine the negative part.

Homogenization

In this section it is shown that the behavior of a pattern of dimples can be investigated using a single well chosen dimple. Choosing = -× -Ω [ 10,10] [ 10,10], we tried to replace the previous pattern of nine dimples by a square dimple. The volume of the chosen dimples (see equation ( 7)) of radius R and depth A is given by =

- V AR π π ( /2 2/ ) 2 .
We considered a single rectangular dimple of the same volume and the same center as the nine dimples. We had two obvious choices for that 12. The square of size 10 is the best choice in the inlet while the square of size 8 is better in the outlet. Similar trends as for step bearings are observed (see Fig. 13).

Another comparison with the results shown in Figs. 789was done by considering rectangular dimples of the same volume as the dimple patterns tangent to the external dimples (see [START_REF] Pascovici | Analytical investigation of a partially textured parallel slider[END_REF].

Results can be viewed in Figs. 141516. Similar behavior can be observed but with a roughly doubled load carrying capacity. Again in the periodic case, larger load carrying capacity than in the Dirichlet case is generated. Here too the maximum LCC is roughly doubled (from Dirichlet to the periodic case) as with the pattern of dimples. The flow and therefore the load carrying capacity is influenced by the geometry. The flow behavior depends on the "conductance" values generated by the geometry. These conductances defined by H 3 have been evaluated for the two geometries, rows of three dimples and equivalent rectangular dimples. Using conductance rules, the ratio of the additional conductance CR generated by the equivalent rectangular dimple and the dimples was computed.

With our discretization, the conductance is computed by

∑ ∑ j 1 i H i j 1 , 3 .
In the smooth case, it simplifies to H 0 3 . The values are given in Table 4.

It shows a ratio around 1.8. The other columns of the table show the values that are obtained by multiplying the LCC for a row of three dimples and the LCC for the associated rectangular dimple of the same volume for the periodic case (columns 3 and 4) and for the Dirichlet case (columns 5 and 6). For a number N of rows between 2 and 5, there is less than 10% difference.

Comparison with the behavior of one dimensional step bearing

Analytical results for a one dimensional step bearing

The 1D geometry and the notations used are described in Fig. 17. In this case the dimensionless Reynolds equation reads:

∂ ∂ ⎛ ⎝ ∂ ∂ ⎞ ⎠ - ∂ ∂ = X H P X H X ( ) 0 oil 3 (10)
In Fig. 18, the step bearing is represented in black. The red line represents the pressure distribution in the case of cavitation. P 0 corre- sponds to P amb .

In the 1D case analytical formulas for the pressure and the oil film can easily be found using flow conservation. In case of cavitation and fully flooded conditions, the following results are obtained: It may be noted that the maximum pressure does not depend on A, the step depth. The influence of A results from the position X r of the film reformation boundary. Similarly, in the non cavitated case, the following results are obtained:

= - - = ∈ = = = + ∈ = - - = + ∈ = - - - + = ∈ P X P X X l H H X X X P X P H H H P l X X X P X P X X X X H H A X X X P X P P X X X X P H H X X X ( ) (1 ( )/ ) [ , ] ( ) 0 / [ , ] ( ) ( )/( ) [ , ] ( ) ( )( ) /( ) [ , ]
= - + - = ∈ = - + - = + ∈ = - + - = ∈ P X P X X l P X X l H H X X X P X P X X ls P X X ls H H A X X X P X P X X L P X X L H H X X X ( ) ( )/ ( )/ [ , ] ( ) ( )/ ( )/ [ , ] ( ) ( )/ ( )/ [ , ]
0 1 0 1 2 1 0 1 2 0 2 0 2 (14)
with, for In this case, P min and P max depend on A. There is no cavitation if P min is positive, that is to say if is sufficiently large or A sufficiently small or sufficiently large there is no cavitation. This is also true in the 2D case. These analytical results can be used to validate our code in Fig. 18, the numerical pressure (in blue) and the analytical pressure in (red) are represented together. The two curves superimpose.

= H 1 0 , = - + + + P P ll A l l L A /( ( ) (1 )
+ + + < ll A l l L A P /( ( ) (1 )
An analytical expression of the load carrying capacity can easily be deduced from the above expression of the pressure. In the cavitated case when the cavitation is important enough, it can be approximated by the following expression: Where X mil is the position of the middle of the step bearing. It may be noted that this approximated value does not depend on the step bearing depth. The curve of the approximated LCC as a function of the position of the middle of the step and computed values are shown in Fig. 19.

⎜ ⎟ ⎜ ⎟ ≈ ⎛ ⎝ --+ ⎞ ⎠ = ⎛ ⎝ - + - - - - ⎞ ⎠ LCC P l l L l P X X X X l X X l 2 2 0.5 ( ) 0.5
There is a good agreement between them. One also notes a good agreement between the calculated results and predicted values for the maximum pressure (see Fig. It can be noted that the 1D model provides a good approximation of the LCC behavior after the first quarter of the domain. There is an asymptotic line corresponding to the position of X mil for which the dimple starts to completely lie in the contact. Before this position, the analytical approximated value is not valid. The approximation is much better with rectangular dimples and of course also better when they get larger in the Y-direction becoming closer to the 1D case. The geometry of the dimple seems to affect the LCC mostly in the contact inlet. . For a periodic pattern in the ydirection of period 0.3 mm with N rows of three dents at the inlet and at the outlet of the contact (see Fig. 10), the obtained dimensionless and total dimensional load carrying capacity are given in Table 5. For a periodic pattern in the y-direction of period 0.3 mm with N rows of 10 dents at the inlet and at the outlet of the contact, the obtained dimensionless and total dimensional load carrying capacity are given in Table 6.

Dimensional results

Conclusion

For parallel surfaces in hydrodynamic lubrication, no pressure can be generated without a surface texture. A multigrid flow preserving algorithm is used to analyse the influence of the texture on the load carrying capacity. Partial texturing with micro dimples can induce positive pressure when the texture is located in the contact inlet. However, the loss of pressure generation is less important for the outlet texture than the gain for the inlet texture. By texturing both contact ends, a positive load carrying capacity is generated which may be useful for reciprocating contacts as for the flat ring in the piston ring cylinder liner contact. Partial texturing generates a pressure distribution similar to that of a step bearing. This behavior was analysed by comparison with a rectangular dimple of the same volume as the indent texture pattern and similar location. Higher LCC is generated by the equivalent rectangular dimple. This was explained by the influence of the geometry on the conductances. Besides in the periodic case and in the case of cavitation, the analytical LCC expression for the 1D model for a step bearing as a function of the position of the texture provides a good estimate of the load carrying capacity for dimples in the 2 dimensional case. This can provide a way of designing micro-textures for the flat ring without further calculations.

Appendix. comparison with literature results

A validation of our code was previously mentioned by comparison with the 1D analytical results in section 7.2. Our results were also compared with the results obtained by Biboulet and Lubrecht in Ref. [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF] with a code based on the Woloszynski, Podsiadlo and Stachowiak [START_REF] Woloszynski | Efficient solution to the cavitation problem in hydrodynamic lubrication[END_REF] approach. The dimensionless domain Ω is a square -× - . The dimensionless parameters are the same but [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF] computes the mean value of -P P amb and henceforth our results have to be divided by the domain area. Results are given in Table 7. Our results and those of [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF] are converged up to machine accuracy. Hence the difference stems from the discretization schemes used. Both are second order accurate but our scheme has an error that is 3/8 of the error in Ref. [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF]. This result is coherent with a grid convergence analysis performed.

Finally, our results were compared with the results obtained by Profito et al. in [START_REF] Profito | A general finite volume method for the solution of the Reynolds lubrication equation with a mass-conserving cavitation model[END_REF] for a textured sliding bearing. They consider parallel bearings with the characteristics described in Table 8. The dimples have a parabolic geometry. For the load carrying capacity they compute ∫ = W P dX dY.

Ω

The stationary bearing is textured with N parallel rows of 10 dimples in the inlet spaced by r 4 . This texture will be denoted by × N 10 . The results are shown in Table 9 Our results are calculated up to machine precision on a × 1025 1025 grid. It leads to about 2500 nodes per local dimple cell and 10 4 nodes per full dimple cell instead of respectively 900 and 3600 in [START_REF] Profito | A general finite volume method for the solution of the Reynolds lubrication equation with a mass-conserving cavitation model[END_REF]. We keep a uniform mesh all over the domain even in the non textured area which is not the case in [START_REF] Profito | A general finite volume method for the solution of the Reynolds lubrication equation with a mass-conserving cavitation model[END_REF]. This can explain the observed difference which increases with increasing non textured area. However, the difference is smaller than 1% for the worst case.

Fig. 2 .

 2 Fig. 2. : Dimple section (black) and cavitation zones for a dimple of radius 1, depth 1 at the center of = -× -Ω [ 4,4] [ 4,4] for = P 0.001, 0.005, 0.01, 0.05 amb in red, green, blue and pink. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3 .

 3 Fig. 3. Pressure and oil film generated on -× -[ 4,4] [ 4,4] by a single dimple of depth 1 centered at -( 2,0). (a) Geometry, (b) Film height, (c) Pressure (Dirichlet), (d) Pressure (periodic).

Fig. 4 .

 4 Fig. 4. Cavitation zone: zone inside the dimple (blue disk), cavitation zone: Dirichlet (red + green), Periodic green. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 5 .

 5 Fig. 5. Pressure and oil film generated on -× -[ 10,10] [ 10,10] by a pattern of 9 dimples of depth 1 and spaced by = D 4 centered at -( 2,0).(a) Geometry (b) Film height (Dirichlet) (c) Pressure (Dirichlet), (d) Pressure (periodic) (e) Nine dimple pattern.

Fig. 6 .

 6 Fig. 6. : (a, d) Pressure generated on -× -[ 4,4] [ 4,4] by a single dimple of depth 1 centered at -( 2,0). (b, e) Part above the ambient pressure (c, f) Part below the ambient pressure (Here -P and -P amb are plotted). First row: Dirichlet case, second row: periodic case.

Fig. 7 .

 7 Fig. 7. Pressure generated on -× -[ 20,20] [ 20,20] as a function of the number of rows of 3 dimples of depth 1 in the inlet or in the outlet. Left: 3 rows in the inlet - Middle: 10 rows -Right: 3 rows in the outlet. First row: Dirichlet conditions -Second row: periodic conditions.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Load carrying capacity as a function of the number N of rows of three dimples of depth 1 spaced by = D 4 on = -× -Ω [ 20,20] [ 20,20]. In blue the first row is located at -( 18,0), in red the last row is located at (18,0). (a) Dirichlet -(b) Periodic. (Note the different vertical axes.). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 10 .

 10 Fig. 10. Periodic case: Pressure generated on -× -[ 20,20] [ 20,20] with N rows of 3 dimples of depth 1 spaced by 4 at both ends, = P 0.01 amb .(a) = N 1 (b) = N 2 (c) = N 3 (d) = N 4.

Fig. 11 .

 11 Fig. 11. Load carrying capacity generated by a pattern of nine dimples of depth 1 spaced by 4 as a function of the position of the center of the pattern on the line = Y 0.(a) and (c) = - × -Ω [ 10,10] [ 10,10] -(b) and (d) = -× -Ω [ 20,20] [ 20,20]. First Row: Dirichlet case -Second row: periodic case.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Dirichlet case: Load carrying capacity of a square dimple of identical volume as the nine indent pattern (in blue) compared with the load carrying capacity (in red) of the pattern of nine dimples of depth 1 spaced by 4 on = -× -Ω [ 10,10] [ 10,10], (a) a square dimple of size 10, (b) a square dimple of size 8. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 14 .

 14 Fig. 14. Pressure generated on -× -[ 20,20] [ 20,20] with an equivalent rectangular dimple of position similar to the indent position of Fig. 7. First row: Dirichlet conditions -Second row: periodic conditions.

Fig. 15 .

 15 Fig. 15. Pressure generated on -× -[ 20,20] [ 20,20] with an equivalent rectangular dimple of position similar to the indent position of Fig. 9. (a) Dirichlet (b) Periodic.

Fig. 16 .

 16 Fig. 16. Load carrying capacity for equivalent volume rectangular dimple in terms of the number N of associated rows of three dimples of depth 1 spaced by = D 4 on = -× -Ω [ 20,20] [ 20,20]. In blue the first row is located at -( 18,0), in red the last row is located at (18,0). (a) Dirichlet, (b) Periodic. (Note the different axes.). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 17 .

 17 Fig. 17. One dimensional step bearing: in black, geometry and in red pressure (cavitated case). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

20

 20

). 7 . 2 .

 72 Comparison with two dimensional numerical results for a single dimple in the periodic case Numerical 2D calculations were performed for the periodic case to determine the difference in LCC capacity generated by a row of dents in the Y-direction and an equivalent rectangular dimple of the same volume. Because of the periodicity it is sufficient to consider a single dimple. Four types of dimples where considered: our basic circular dimple of depth = A 1 of radius = R 1 and rectangular dimples of the same volume of section = l The domain Ω on which calculations were performed is = -× -Ω [ 20,20] [ 2,2]. The final LCC is divided by the domain width for comparison with the 1D results. Results are shown in Fig. 21.

Fig. 18 .Fig. 20 .Fig. 21 . 3 y

 1820213 Fig. 18. Computed 1D pressure (in red) generated by a step bearing of length 4 of depth 0.5 located at the center of the contact together with the analytical pressure (in blue) in = - Ω [ 20,20]. (a) Cavitated case = P 0.01 amb

For a viscosity = η 0 . 5 m 9 0

 059 02 Pa.s, a mean velocity = u MPa. In this case, using the dimple radius and the clearance as dimensionless factors, we have,

  [START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF][START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF] [START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF][START_REF] Tomanik | Modeling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF]. They consider a pattern of 25 dents of depth = A 1 spaced by = D 4 and radius = R 1 of center located at X c on the central line. The comparison was made for = P grid. It was also tested for a non cavitating case = P

  .033 Received 27 June 2018; Received in revised form 12 October 2018; Accepted 23 October 2018

Table 1

 1 Computing time, average residual (res) and load carrying capacity (LCC) for a 1024 2 grid on - × -[ 4,4] [ 4,4], single dimple of radius 1 and depth 1 at the center.

	P amb	0.1	0.05	0.01	0.005	0.001
	Time	s 34	s 2180	s 2229	s 2214	s 2297
	Res	-1.69 10 17	-2.94 10 12	-2.99 10 12	-4.95 10 13	-6.32 10 14
	LCC	-

-2.403 10 13 --1.108 10 2 --1.466 10 2 --9.211 10 3 --2.397 10 3

middle of the dimple. Without cavitation (high ambient pressure or very small or very large depth or small dimple radius), the algorithm recovers the multigrid efficiency and solution only takes a few seconds.

Table 2 :

 2 Computing time, average residual and LCC for a 1024 2 grid on -

	× -[ 4,4] [ 4,4],

Table 3 :

 3 Computing time, average residual and LCC for various level with a 8 2 coarse

	grid on -[ 4,4] [ 4,4], single dimple of radius 1 and depth 1 at the center, × -
	P amb	=	0.01	.
	Level		8		7		6	5
	Time		ss 2229	s 468	s 99	s 22
	Residual	-2.99 10 12	-9.98 10 13	-9.28 10 15	-2.23 10 18
	LCC		-	-1.466 10 2	-	-1.468 10 2	-	-1.468 10 2	-

-1.470 10 2

Table 4

 4 Conductance ratio between equivalent volume rectangular dimple and N rows of three dimples.

	N	Conductance	CR N ( )	×	LCC	LCC	CR N ( )	×	LCC	LCC
		ratio	N Rows		equivalent	N Rows		equivalent
		CR N ( )	of 3 dimples	rectangular	of 3 dimples	rectangular
						dimples				dimples
			Periodic		Periodic	Dirichlet		Dirichlet
	1	1.68	15.9			19.3	9.7			11.7
	2	1.79	30.3			30.1	16.8			16.5
	3	1.81	37.0			35.8	19.2			18.2
	4	1.82	37.9			36.0	19.2			17.8
	5	1.81	33.7			31.3	17.5			15.8
	6	1.80	26.2			22.8	14.7			12.5
	7	1.79	17.1			12.4	10.7			7.9
	8	1.78	8.3			2.3	6.0			2.2
	9	1.77	1.1			-5.9	1.2			-3.8
	10 1.76	-2.8			-10.5	-1.9			-8.1

Table 5

 5 Load Carrying capacity in the case of partial texturing with patterns of N rows of three dimples in the inlet and in the outlet of the contact.

	N	Dimensionless LCC	Dimensional total LCC
	1	6.12	2.07 N
	2	10.38	3.50 N
	3	9.35	3.16 N
	4	4.22	1.42 N

Table 6 :

 6 Load Carrying capacity in the case of partial texturing with patterns of N rows of ten dimples in the inlet and in the outlet of the contact.

	N	Dimensionless LCC	Dimensional total LCC
	1	33.26	11.23 N
	2	60.80	20.52 N
	3	41.02	13.84 N
	4	10.50	3.54 N

Table 7

 7 Comparison with the Biboulet and Lubrecht results in Ref.[START_REF] Biboulet | Efficient solver implementation for Reynolds equation with mass-conserving cavitation[END_REF].

	P amb	X c	[29] results	Present code results
	0.04	0	-	-1.0458 10 3	-	-1.04658 10 3
	0.04	-1	-	-16.7097 10 3	-	-16.0447 10 3
	0.04	1	-	-10.1009 10 3	-	-10.1116 10 3
	0.4	-1	-

-15.2877 10 3 --15.3012 10 3

Table 8

 8 Parallel bearings characteristics in Ref.[START_REF] Profito | A general finite volume method for the solution of the Reynolds lubrication equation with a mass-conserving cavitation model[END_REF].

	Bearing length:	20 mm
	Bearing width:	20 mm
	Clearance:	h	0	=	μ 4	m
	Velocity:	10 m/s
	Viscosity:	= η 10mPa.s
	Ambient pressure:	100 kPa
	Cavitation pressure:	90 kPa
	Dimple radius:	r	=	μ 500 m
	Dimple depth:	a	=	μ 10 m

Table 9

 9 Comparison with the Profito et al. results in Ref. [27].

	Texture	[27] results	Present code results
	× 10 10	38.91 N	38.95 N
	× 10 8	41.57 N	41.50 N
	× 10 6	44.84 N	44.61 N
	× 10 4	47.06 N	46.71 N
	× 10 2	48.24 N	47.81 N


Marcio M Marim

Elsa D Angelini

Jean-Christophe Olivo-Marin

J.-C Olivo-Marin

A Compressed Sensing Approach for Biological Microscopy Image Denoising

Compressed Sensing (CS) provides a new framework for signal sampling, exploiting redundancy and sparsity in incoherent bases. For images with homogeneous objects and background, CS provides an optimal reconstruction framework from a set of random projections in the Fourier domain, while constraining bounded variations in the spatial domain. In this paper, we propose a CS-based method to simultaneously acquire and denoise data based on statistical properties of the CS optimality, signal modeling and characteristics of noise reconstruction. Our approach has several advantages over traditional denoising methods, since it can under-sample, recover and denoise images simultaneously. We demonstrate with simulated and practical experiments on fluorescence images that we obtain images with similar or increased SN R even with reduced exposure times. Such results open the gate to new mathematical imaging protocols, offering the opportunity to reduce exposure time along with photo-toxicity and photo-bleaching and assist biological applications relying on fluorescence microscopy.

I. INTRODUCTION

In microscopy, observation of fluorescent molecules is challenged by the photo-bleaching and photo-toxicity, as these molecules are slowly destroyed by the light exposure necessary to stimulate them into fluorescence. Unfortunately, reducing exposure time drastically deteriorates the signal to noise ratio (SN R) and hence, the image quality. To improve the SN R, many denoising methods are available such as Non-Local Means (NL-means) [START_REF] Buades | A review of image denoising algorithms, with a new one[END_REF], Total Variation (TV) [START_REF] Rudin | Nonlinear total variation based noise removal algorithms*[END_REF], non-linear isotropic and anisotropic diffusion [START_REF] Weickert | Efficient and reliable schemes for nonlinear diffusion filtering[END_REF]. Other methods exploit the decomposition of the data into the wavelets, ridglets, curvelets bases and perform denoising by shrinking the transform coefficients [START_REF] Devore | Fast wavelet techniques for near-optimal processing[END_REF]. Recently, efficient denoising methods were also developed based on sparsity and redundant representations over learned dictionaries, where the image is denoised and a dictionary is trained simultaneously (e.g. the K-SVD [START_REF] Aharon | The K-SVD algorithm[END_REF]).

In this paper we propose an application of Compressed Sensing on fluorescence microscopy images, enabling the acquisition of high SN R images under reduced exposure times. Instead of postacquisition denoising methods, here we are interested in denoising as much as CS advantages. Our framework is based on the property of CS to efficiently reconstruct sparse signals with under-sampled acquisition rates, significantly below the Shannon/Nyquist theoretical bound. Similarly to recent experiments for MRI CS-based reconstruction [START_REF] Lustig | Sparse MRI: The application of compressed sensing for rapid MR imaging[END_REF], the acquisition protocol consists in measuring the image signal onto a random set of Fourier vectors [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] and constraining Total Variation (TV), which is incoherent to the Fourier domain. Indeed, the CS framework introduced by Candès [START_REF] Candès | Practical signal recovery from random projections[END_REF] provides theoretical results and shows that if a signal is sparse (i.e. has a small number of nonzero coefficients) in some basis, then with high probability, uniform random projections of this signal onto an unstructured domain, where the signal is not sparse, contains enough information to optimally reconstruct this signal. The incoherence property between the sparsity 1 M. Marim: marim@pasteur.fr 2 E. D. Angelini: elsa.angelini@telecom-paristech.fr 3 J.-C. Olivo-Marin: jcolivo@pasteur.fr, www.bioimageanalysis.org basis Ψ and the sampling basis Φ ensures that signals having sparse representations in Ψ must have a large support in the measurement domain described by Φ. Random selections of basis functions in Φ are typically suitable since random vectors are, with very high probability, incoherent with any sparsity-encoding basis functions from Ψ, defining orthogonal domains [START_REF] Donoho | Compressed sensing[END_REF].

Considering that a noisy signal x has a sparse representation in some basis Ψ, we want to recover the signal x ∈ R N from noisy measurements y = Φ(x + n) | y ∈ R M , the sampling matrix being defined by M vectors in Φ, with M ≪ N . The presence of noise in the acquired signal might alter its sparsity in the domain Ψ. By optimally reconstructing a signal with explicit sparsity constraints, CS offers a theoretical framework to remove non-sparse random noise components from a corrupted signal. Indeed, removing noise from x+ n will rely on the efficacy of Ψ on representing the signal x sparsely and the inefficacy on representing the noise n [START_REF] Donoho | Stable recovery of sparse overcomplete representations in the presence of noise[END_REF]. We propose to use the CS sampling and reconstruction framework to denoise and improve the SN R of microscopy fluorescence images acquired using shorter exposure times.

II. BACKGROUND

A. Sparsity and incoherence

A discrete signal is considered sparse when it has a large number of zero coefficients on some basis functions. We define a S-sparse signal if S elements are non-zero. Natural signals measuring discrete events or natural images with smooth and homogeneous objects can be considered approximately sparse in some basis and be accurately approximated with a small set of coefficients [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF], [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. In the general case, a signal has a S-sparse representation in the set of basis functions Ψ = {Ψi} N i=1 if:

x = X i ciΨi with c ℓ 1 ≈ S (1)
Here Ψi is a basis function and the ℓ1 norm well approximates the number of nonzero coefficients of c.

In our framework, for biological images, we choose the global Total Variation as sparsifying transform [START_REF] Candès | Sparsity and incoherence in Compressive Sampling[END_REF], [START_REF] Cohen | Nonlinear approximation and the space BV(R 2 ) 1[END_REF]. This measure is well-known in image processing and very popular in variational segmentation problems for its ability to limit high frequencies and to provide regularized segmented regions. This measure is also well suited for denoising where the goal is to restore an image with smooth objects and background. Minimization of the T V norm, introduced in [14], [START_REF] Rudin | Nonlinear total variation based noise removal algorithms*[END_REF] for signal denoising, corresponds to a constraint on the number of discontinuities in an image, and the homogeneity of the objects. T V sparsity constraints in CS were introduced by Candès in [START_REF] Candès | Practical signal recovery from random projections[END_REF], expressed as: It is interesting to note here, as pointed out in [START_REF] Candès | Practical signal recovery from random projections[END_REF], that the transform which produces the T V coefficients corresponds to the ℓ2 norm of a particular operation on the image, based on the computation of horizontal ∇x and vertical ∇y gradients and combination of these operators into a complex-valued image: ∇x + i∇y. The T V coefficients are then the ℓ2 norm of this transformed image (computed locally for each pixel) and the CS framework minimizes the global ℓ1 norm of these T V coefficients over the whole image. Such transform corresponds to the decomposition of the image onto Heavisides basis functions, which support is adaptive (similarly to the Basis Pursuit framework for decompositions on dictionaries [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]) and defined by the localization of high-gradient values in the image (i.e. edge maps). T V constraint is well suited for biological images, where structures and background provide small gradient values while a finite set of edges provides high gradient values as illustrated in Figure 1 for drosophila ovocytes.

T V (x) = ∇(x) ℓ 1 (2) T V (x) = X i,j q {x (i+1,j) -x (i,j) } 2 + {x (i,j+1) -x (i,j) } 2 (3)
The design of the sampling matrix Φ must enforce incoherence between the acquisition and the sparsity domain. This constraint has been formulated though several criteria, based on mutual coherence measures [START_REF] Candès | Sparsity and incoherence in Compressive Sampling[END_REF], [START_REF] Donoho | Stable recovery of sparse overcomplete representations in the presence of noise[END_REF], or matrix properties such as the Restricted Isometry Property (RIP) conditions described in [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF]. Random acquisition in the Fourier domain, encoded by a matrix with 1s and 0s at random frequency locations verifies this incoherence property when combined with T V spatial constraints.

III. METHOD

A. Reconstruction from noisy measurements

In the context of noisy measurements y = Φ(x + n), which is the case for microscopy images corrupted with acquisition noise, we wish to recover only the signal component x ∈ R N . If we make the assumptions that the noise energy is bounded by a known constant n ℓ 2 ≤ ǫ, the transformed signal Ψx is sparse, and Φ ∈ R M N is a random matrix sampling x in the Fourier domain, the true signal component x can be recovered nearly exactly using the following convex optimization:

x = arg min x∈R N Ψx ℓ 1 s.t. y -Φx ℓ 2 ≤ δ (4)
for some small δ ≥ ǫ. In [START_REF] Candès | Stable signal recovery from incomplete and inaccurate measurements[END_REF] it was shown that the solution x is guaranteed to be within Cδ of the original signal x.

x -x ℓ 2 ≤ Cδ with C > 0

We note here that this CS-based estimation framework, with noisy observations and T V spatial constraints, guarantees that no false component of x + n with significant energy is created as it minimizes its ℓ1 norm, which is particularly high for additive random noise components. More specifically, the T V -based spatial sparsity constraint, will lead to smooth edges and removal of noise components, resulting in an error:

x -x ℓ 2 ≤ α + β ( 6
)
where α reflects the desired error (responsible for noise removal) from the relaxation of the constrain δ in (4) and β reflects the undesired error from smooth edges of signal. If T V represents x efficiently and n inefficiently, the term β vanishes and α → Cδ.

B. The recovery algorithm

As an alternative to acquisition problems we focus on utilizing dual sparse and redundant representations in the CS framework for fluorescence microscopy image denoising via two separate schemes.

A first strategy consists in acquiring K images xi + ni (with Φi matrices) exposed T /K ms, restoring each one independently (equation 7) and combining the K restoration results into a single denoised image. This scheme exploits the fact that fluorescence signal Φixi should be strongly correlated in all acquisitions, while noisy components Φini should not be.

xi = arg min

x∈R N Ψxi ℓ 1 s.t. yi -Φixi ℓ 2 ≤ δ (7)
where yi = Φi(xi + ni) for i = 1...K.

A second CS denoising scheme is proposed, which consists in determining the exposure time X necessary to obtain, with a set of combined CS restorations, a target SN R level, corresponding to the SN R measured on the image exposed T ms. This scheme provides the potential advantage of requiring a single shorter acquisition time, limiting degradation of the biological material through photo-toxicity and photo-bleaching.

Combining CS reconstructions implies a sequence of CS reconstructions of a single noisy image acquisition x + n, using different sampling matrices Φi, as described below:

xi = arg min x∈R N Ψx ℓ 1 s.t. yi -Φix ℓ 2 ≤ δ (8)
for i = 1...K. The last step of both algorithms involves the combination of xi by averaging to generate a final denoised image x.

C. CS and scalability

The degrees of freedom in this series of CS experiments are in the design of the sampling matrix Φ: the number of random measurements M and their location in the Fourier domain. The CS theoretical framework states that the more measurements are used in the Φ domain, the closer is the reconstructed signal to the original measured signal. In the context of denoising (rather than estimation) we have a dual constraint on the estimation of true signal component and the risk to reconstruct noisy components. Indeed, for a single CS experiment, the fluorescence signal will generate, from a set of random measures of structured Fourier values, a restored image with high values depicting a good estimation of the true signal. At the same time, purely random noisy component will be interpreted, from a set of undifferentiated Fourier values, as a structured combination of oscillating components, extrapolated over the spatial domain into patches, under the regularizing T V effect. Noise patches and fluorescence spatial localization will be directly related to M , the number of CS measurements acquired by Φ. We illustrate in Figure 2 how this number of measurements can be naturally viewed as a scale parameter where small scales enable more noise reconstruction.

In the experiment on Figure 2, we observe that noise component is more uncorrelated than signal across scales while the signal component spatial resolution decreases.

We can make a connection here to the family of multi-scale transforms [START_REF] Lindeberg | Scale-Space Theory in Computer Vision[END_REF]. These transforms were theoretically defined as linear transforms with a scale parameter controlling the ability of the transform to simplify the signal. We know from the sparsity T V constraint that strong signal recovered by the CS framework will correspond to strong underlying true signal components. Therefore, CS does not introduce false signal components and fits well in the framework of multi-scale transforms, as illustrated in Figure 2. Indeed it appears that such multi-scale CS approach verify scale space properties such as simplifications, homogeneity, isotropy as well as rotational and shift invariance.

If results from CS recovery of a purely noise signal are combined, the mean intensity results in a nearly homogeneous signal, as illustrated in Figure 3. This observation clearly justifies the averaging operator introduced in Section III-B to remove noise from images.

IV. EXPERIMENTAL RESULTS

A. Noise removal and recovered images

Results from the first scheme of denoising are ilustrated in Figure 4 are detailed in table I and results from the second scheme of denoising are ilustrated in Figure 5 and are detailed in table II. In all experiments, important improvements of the SN R or the exposure time were achieved.

For experiments on drosophila cells image (Figure 4 and 5), the SN R was highly improved and details were very well preserved. The algorithm shows its efficacy and importance on microscopy applications, where photons detected are limited and image quality is normally degraded. The improvement can be provided in two different ways, fixing the desired SN R and reducing the exposure time, or fixing the exposure time and improving the SN R, as the denoised images in Figure 5 (middle) illustrate.

We have performed a comparison of our method with exclusive Total Variation denoising methods. The algorithm used for this comparison is the one presented by Gilboa et al. in [17]. This algorithm minimizes T V in two schemes constraints (scalar fidelity term) and 2. in order to preserve texture and small scale details, using an adaptive variational scheme that controls the level of denoising by local variance constraints (adaptive fidelity term). Residuals and SN R results in Figure 6 shows that CS is able to discriminate noise from signal more than exclusive T V minimization methods. This is justified by the reason that CS has both T V minimization constraint, and uncorrelated noise recovery. Differences on each set of under-sampled random projections establish strongest constraint for signal-measures fidelity than for noisemeasures fidelity.

V. CONCLUSION

We introduce a Compressed Sensing-based image acquisition and denoising method exploiting multiple reconstructions with random Fourier projections. Our approach presents several advantages over traditional denoising methods, joining image acquisition, CS advantages and denoising in one framework. In the context of biological imaging, experiments demonstrated improvements of SN R values up to a factor of 2 on microscopy images acquired with limited exposure times. These results also open the gate to new microscopy acquisition schemes, enabling for instance better control on cells photo-toxicity and photo-bleaching of fluorescent targets.

Fig. 1 .

 1 Fig. 1. Image sparsity illustrated in a two-dimensional spatial domain. Top left: original image. Top right: nearly sparse representation domain (using the gradient of the signal). Bottom: line profiles from corresponding blue lines in the spatial and gradient domains.

Fig. 3 .

 3 Fig. 3. Left: Pure noise image extracted from a microscopy image background. Right: Result obtained averaging 20 images recovered with different sets of measurements Φ i for (i = 1...20).

ACKNOWLEDGMENT

  We thank Anne-Laure Bouge of the "Génétique et Epigénétique de la Drosophile" Research Unit at Institut Pasteur for drosophila cells and Nathalie Sol-Foulon of the Virus and Immunity Research Unit for acquiring lymphocytes images. This work was funded by Institut Pasteur.

Fig. 5 .

 5 Fig. 5. Left: Noisy image x of drosophila cells imaged by fluorescent microscopy, SN R = 6.42 and exposure time t = 100ms. Middle: Denoised image x, composed from 10 images recovered with different measurements, as the second scheme proposed on Section III-B, SN R = 11.02. Right: Residual xx l2 .

Fig. 2 .

 2 Fig. 2. Average image recovered with six different numbers of measurements (i.e. 6 scales). Scales varies from a compression ratio exponentially increasing from M = 30% to M = 0.3%.

Fig. 4 .

 4 Fig. 4. Left: One noisy image x i of drosophila cells imaged by fluorescent microscopy, SN R = 3.61 and exposure time t = 10ms. Middle left: Image acquired with exposure time equal to t = 100ms and SN R = 6.42. Middle right: Denoised image x, composed from 10 images recovered from 10 images with exposure time t = 10ms, as the first scheme proposed on Section III-B, SN R = 13.31. Right: Residual x -I 100ms l2 .

Fig. 6 .

 6 Fig. 6. Top: Noisy original image x of lymphocytes imaged by fluorescent microscopy, SN R = 8.14. Middle left: Denoised image by scalar T V , SN R = 11.2. Middle: Denoised image by adaptive T V , SN R = 12.5. Middle right: Denoised image x with CS, composed from 10 images recovered with different sets of measurements, SN R = 14.28. Bottom: Images from residuals of all respective methods xx ℓ 2 .

  1. with global variance

	Exposure Time	CN R	SN R
	100 ms	30.00	6.42
	10 ms	5.17	3.61
	10 • 10 ms + 10 • CS	96.60	13.31

TABLE I RESULTS

 I FROM CS-BASED DENOISING OF THE IMAGE ON FIGURE 4 BY THE METHOD DESCRIBED ON SECTION III-B.

	Exposure Time	CN R	SN R
	100 ms	30.00	6.42
	100 ms + 1 • CS	36.50	7.24
	100 ms + 10 • CS	90.23	11.02

TABLE II RESULTS

 II FROM CS-BASED DENOISING OF THE IMAGE ON FIGURE 5 BY THE METHOD DESCRIBED ON SECTION III-B.


Dr Sophie Blaise
email: sblaise@chu-grenoble.fr

Joël Constans

Laure Pellegrini

Patricia Senet

Isabelle Lazareth

Jean-Luc Cracowski

Patrick Carpentier

Optimizing Finger Systolic Blood Pressure measurements with laser doppler: validation of the second phalanx site

Keywords: Finger Systolic Blood Pressure measurement systemic sclerosis, finger systolic blood pressure, laser-doppler. repeatability. arterial disease FSBP: Finger systolic blood pressure measurement SSc: systemic sclerosis patients DU: digital ulcer LDF: laser Doppler flowmetry

Objective: Finger systolic blood pressure measurement (FSBP) has been shown helpful in the detection of distal arterial insufficiency in upper limbs. This work assesses the possibility to measure FSBP on the 2 nd phalanx instead of the first one in order to improve its sensitivity and to verify this would not alter the repeatability of the measurement.

Methods:

In this multicenter study. FSBP was measured twice in all fingers but the thumbs in consecutive systemic sclerosis patients on the first phalanx and the second phalanx in alternate order using laser-doppler flowmetry.

Results. Thirty-seven patients were enrolled. The repeatability of FSBP was excellent and similar on the first and 2 nd phalanxes with coefficients of variation respectively of 7.1% and 7.6%. While the correlation between the FSBP at the two sites was fair (Pearson coefficient 0.69; p< 0.001). The agreement was poor, with a mean difference of 14 mmHg between the two sites. Significantly higher differences were found in fingers with digital ulcers. The ROC curves showed a better prediction of the 2 nd phalanx measurements.

Conclusion:

FSBP has an excellent repeatability whatever the site of phalanx. However, measurements performed on the 2 nd phalanx have a better sensitivity for the prediction of digital ulcers.

Introduction

Toe systolic blood pressure measurement is the reference technique for the hemodynamic evaluation of severe peripheral arterial disease of the lower limbs, and its methodology is well described [START_REF] Graaff | The usefulness of a laser Doppler in the measurement of toe blood pressures[END_REF][START_REF] Beinder | Laser Doppler technique for the measurement of digital and segmental systolic blood pressure[END_REF]. Finger systolic pressure measurement showed promising applications in patients with arterial deficiency of the upper limbs related to arterio-venous fistulae [START_REF] Odland | Management of dialysis-associated steal syndrome complicating upper extremity arteriovenous fistulas[END_REF][START_REF] Lazarides | Indications for Surgical Treatment of Angioaccess-Induced Arterial "Steal[END_REF][START_REF] Schanzer | Use of digital pressure measurements for the diagnosis of AV access-induced hand ischemia[END_REF][START_REF] Goff | Steal Syndrome Complicating Hemodialysis Access Procedures: Can It Be Predicted?[END_REF], and in systemic sclerosis [START_REF] Hirai | Arterial insufficiency of the hand evaluated by digital blood pressure and arteriographic findings[END_REF][START_REF] Herrick | Quantifying digital vascular disease in patients with primary Raynaud's phenomenon and systemic sclerosis[END_REF] where the distal arterial insufficiency is of crucial clinical relevance [START_REF] Watelet | Prevalence of macrovascular arterial involvement of the 4 limbs in systemic sclerosis: About a case series of 14 patients[END_REF][START_REF] Jeune | Arteriographic study of the hand in 52 cases of Raynaud's phenomenon of varying aetiology[END_REF][START_REF] Hirai | Arterial insufficiency of the hand evaluated by digital blood pressure and arteriographic findings[END_REF][START_REF] Hasegawa | Arteriographic evaluation of vascular changes of the extremities in patients with systemic sclerosis[END_REF][START_REF] Emad | Arterial vasculopathy in systemic sclerosis: computerized tomography (CT) angiographic features of macrovascular and microvascular upper limbs arteries[END_REF][START_REF] Allanore | Hand Vascular Involvement Assessed by Magnetic Resonance Angiography in Systemic Sclerosis[END_REF][START_REF] Lescoat | Vascular Evaluation of the Hand by Power Doppler Ultrasonography and New Predictive Markers of Ischemic Digital Ulcers in Systemic Sclerosis: Results of a Prospective Pilot Study[END_REF][START_REF] Lüders | Detection of severe digital vasculopathy in systemic sclerosis by colour Doppler sonography is associated with digital ulcers[END_REF][START_REF] Frerix | Ulnar artery occlusion is predictive of digital ulcers in SSc: a duplex sonography study[END_REF]. However, the methods used to measure finger systolic blood pressure are not yet standardized. Among the parameters varying according to the different protocols three seem of particular importance:

-the method used for flow detection, with plethysmographic techniques mostly used [START_REF] Nielsen Sl Lassen | Measurement of digital blood pressure after local cooling[END_REF][START_REF] Maricq | An objective method to estimate the severity of Raynaud phenomenon: digital blood pressure response to cooling[END_REF][START_REF] Bovenzi | Finger systolic blood pressure indices for the diagnosis of vibration-induced white finger[END_REF], but a recently increasing use of laserdoppler on the lower limbs. due to its better sensitivity for the measurement of low pressures [START_REF] Ubbink | Toe blood pressure measurements in patients suspected of leg ischaemia: a new laser Doppler device compared with photoplethysmography[END_REF]. which is of special interest in patients with severe arterial insufficiency [START_REF] Widmer | Reliability and repeatability of toe pressures measured with laser Doppler and portable and stationary photoplethysmography devices[END_REF][START_REF] Høyer | Reliability of laser Doppler flowmetry curve reading for measurement of toe and ankle pressures: intra-and interobserver variation[END_REF].

-the warming of the measurement site in order to get freed from vasomotor artefacts in the pressure measurements. This is an issue for patients with Raynaud's phenomenon that may present the so-called closing phenomenon related to cold exposure [START_REF] Nielsen Sl Lassen | Measurement of digital blood pressure after local cooling[END_REF].

Several methods are currently used to overcome this difficulty, such as the use of a built-in warming probes, finger warming through the pressure cuff [START_REF] Maricq | An objective method to estimate the severity of Raynaud phenomenon: digital blood pressure response to cooling[END_REF] deviceindependent warming through an hair-dryer, or global body warming obtained in a heated room or through the application of an electric blanket.

-the exact location where the pressure is measured is also an important parameter. While the cuff was usually placed on the first phalanx in order to avoid noise related to physical contact between the probe and the cuff during inflation and deflation; the miniaturization of laser probes now allows positioning the cuff on the second phalanx without physical interaction with the sensor. This should theoretically increase the sensitivity of the measurement to digital arterial disease since arterial lesions can occur distally to the first phalanx. as already illustrated in the early works of Hirai [START_REF] Hirai | Arterial insufficiency of the hand evaluated by digital blood pressure and arteriographic findings[END_REF].

We took the opportunity of a longitudinal prospective study of capillaroscopy and digital pressure measurements in patients with systemic sclerosis, the SCEROCAP study, to assess the interest of a measurement on the second vs the first phalanx in a subset of SCLEROCAP patients from three different vascular centers. All using the same laser-doppler device, but with three different warming techniques, checking its repeatability and predictive value for digital ulcers in different settings.

Materials and Methods

Design of the study

This study is an offshoot of the SCLEROCAP study, which is a French prospective observational multicenter study aiming at validating prospectively the prognostic value of Maricq and Cutolo's capillaroscopic classifications and finger systolic blood pressure with laser Doppler in systemic sclerosis patients (SSc) during a three years longitudinal follow-up. This study was in compliance with Good Clinical Practices and the Declaration of Helsinki. Formal Ethic Committee approval was obtained on March 7 th 2014. The methods were already described in detail in previous publications [START_REF] Boulon | Reproducibility of capillaroscopic classifications of systemic sclerosis. Results from SCLEROCAP Study[END_REF][START_REF] Boulon | Correlation between capillaroscopic classifications and disease severity in systemic sclerosis: results from SCLEROCAP study at inclusion[END_REF]Blaise et al., submitted 2020).

Setting and participants

Thirty-seven scleroderma patients undergoing FSBP measurements with laser Doppler flowmetry were included in three vascular centers experienced in those measurements from April 2018 to October 2018. Main SCLEROCAP inclusion criteria were age above 18 years and diagnosis of SSc according to LeRoy and Medsger's definition [START_REF] Leroy | Criteria for the classification of early systemic sclerosis[END_REF]. Main non-inclusion criteria were association with other connective tissue disease and severe disease with low life expectancy. Presence of digital ulcers, pitting scars or fingertip amputation was carefully recorded.

Study Protocol

Standardized methodology for the FSBP measurements was used but for the warming technique. The patient was resting for 15 minutes in a room with a comfortable temperature between 20 and 25°C according to the seasonal requirements for a comfortable environment before the pressure measurements. Pressure measurements were carried out on eight fingers, excluding the thumb, with laser Doppler flowmetry (LDF) Periflux System 5000 (Perimed, Järfälla, Sweden) using a laser probe (Probe 457, Perimed, Järfälla, Sweden) attached to the digital pulp. A cuff of suitable size (UDC 1.6 cm or 2.5 cm, Hokanson, Bellevue WA, USA) was fitted either on the first or on the second phalanx. Before inflating the cuff, the pulp of the finger was emptied by gentle manual compression. The application of a supra systolic pressure (200 mmHg) was followed by an automatically regulated slow deflation of the cuff, reappearance of the laser Doppler flow signal was detected the FSBP (Figure 1). The measurement was repeated three times for both sites on each finger in order to obtain a more robust estimate of the FSBP with the median value, and this allowed to assess the repeatability.

For each patient, the measure was randomly taken with the cuff placed on the first or second phalanx. and after a 2 min interval, on the other measurement site. Each center used its customary warming technique: an electric blanket heated at 38°C in one center applied 15 minutes before the recording (one center; N=12). local warming with a laser doppler probe with integrated local heaters (Probe 457, Perimed, Järfälla, Sweden) (42°C during 10 minutes) (one center; N=12) and the last center with a hair dryer during during few minutes until the cutaneous temperature on the site reached to 28°C minimum (measured by BAT-10 Thermometer, Physitemp, USA) (N=13).

Statistical analysis

Variables are described as mean (Standard deviation). Repeatability was assessed using within subject coefficient of variations. calculated using the square root method according to Bland [START_REF] Bland | Statistical methods for assessing agreement between two methods of clinical measurement[END_REF]. ROC curves were used to assess the prediction of the two sites of measurement for digital ulcers. Statistical analysis was performed with IBM SPSS Statistics version 21.

Results

Among 37 patients with SSc, two presented active digital ulcers (DU) one of whom had two ulcers on different fingers. Four patients had pitting scars, one of whom had three scars on different fingers. No patient was amputated. The total number of fingers evaluated was 296, nine of which showed active DU or scar. Table 1 provides the within subject coefficient of variation (CV), calculated using the square root method according to Bland, on each finger except the thumb, on the second and first phalanges [Table 1]. Measurement of FSBP shows excellent repeatability with a CV of 7.6% for the second phalanx and 7.1% for the first one [START_REF] Fleiss | The Equivalence of Weighted Kappa and the Intraclass Correlation Coefficient as Measures of Reliability[END_REF][START_REF] Landis | The measurement of observer agreement for categorical data[END_REF], similar whatever the finger and the position of the cuff with no statistically significant difference. No significant influence of the method of warming was found on the variability itself [Table 2]. However, an order effect was observed as a small but consistent increase in FSBP between the first and the third measurement was found when using hair-dryer or non significantly using a local probe warming, showing a better stabilization of the measurement with the warming blanket procedure [Table 3].

The mean FSBP was 99.5mmHg +/-22 (N = 288) on the second phalanx versus 113.8 mmHg +/-23 (N=288) (p< 0.0001) on the first one. While their correlation was fair (coefficient of Pearson 0.69; p< 0.001) (Figure 2), their agreement was poor on the Bland and Altman plot, with an average difference of 14.22 mm Hg (confidence interval (IC) 95% [12.18-16.25]) between the two sites (Figure 3). This difference was two-times higher on fingers with ulcers/pitting scars (28.6 +/-8.7 vs 13.8 +/-1.0 mm Hg; P=0.013).

In spite of the low number of digital ulcers, the ROC curve of FSBP clearly shows a better ulcer prediction by the second phalanx measurements (Figure 4) Discussion:

To our knowledge, this is the first study to explore specifically the methodology of Finger Systolic Blood Pressure Measurements using laser doppler as flow signal. It was performed in the context of a multicenter prospective study performed by different experienced centers using the same laser doppler device and used to harmonize their practices through long-lasting collaborations on scleroderma patients. This study clearly shows that the use of Laser Doppler for the measurements of FSBP allows for reliable and repeatable measurements when the cuff is placed on the second phalanx, thereby increasing its diagnostic sensitivity without hampering its repeatability. Interestingly, we also showed that the observed pressure gradient between proximal and distal sites is higher in fingers with digital ulcers/pitting scars, which is consistent with the classical knowledge of arterial obstructions anywhere along the digital arteries in systemic sclerosis, hence the interest of making measurements as distal as possible. Our preliminary results using ROC curves suggest a better prediction of digital ulcers/spitting scars with the second phalanx cuff placement, even if the numbers are small and deserve confirmation on a larger scale.

As a limitation, for this study, we whose for each team to use their own warming procedure in order to avoid learning curve effects. Although we did not find an influence of the techniques on the repeatability itself. The existence of a small order effect is showing the delay of 15 minutes used for stabilization before the test in this study is probably too short when using local warming with the laser Doppler probe or the heat-dryer procedure. The electric blanket, however, seems to achieve a quicker stabilization, but our study was not designed to compare the heating procedures.

Conclusion:

This study confirms the ability of FSBP measured with laser doppler to detect clinically significant arterial disease in patients with systemic sclerosis, and the excellent repeatability of the measurements whatever the warming method used and the site of measurement. It shows that measurements performed with the cuff place on the second phalanx exhibit an excellent repeatability, lower pressure values when compared to proximal cuff placement, and have a better sensitivity for the prediction of digital ulcers. Additional studies will have to be undertaken in order to assess the best diagnostic threshold level and diagnostic performances.

Acknowledgments

We thank all the investigators of the Sclerocap study.

Sources of Funding

This study was supported by the Société Française de Médecine Vasculaire, and the Société Française de Microcirculation.

This study was supported by the Société Française de Médecine Vasculaire and the Société Française de Microcirculation.

Disclosures: None regarding this work

Perspectives

Finger systolic blood pressure measurement measured with laser doppler detect clinically significant arterial disease in upper limb in patients with systemic sclerosis.

There is an excellent repeatability of the measurements whatever the warming method used and the site of measurement.

Measurements performed on the 2 nd phalanx have a better sensitivity for the prediction of digital ulcers.

Additional studies will have to be undertaken in order to assess the best diagnostic threshold level and diagnostic performances. Table 1: Digital arterial pressure was recorded with laser Doppler flowmetry (LDF) on each finger except the thumb. The patient was comfortably relaxing in supine position for 15 minutes in a room with a stable temperature between 20 and 25°C before the start of pressure measurement.

Tables

A cuff of suitable size was fitted on 2 nd phalanx.