Son Ho-Van
email: son.ho-van@emse.fr

Dr Baptiste Bouillot
email: bouillot@emse.fr

Daniel Garcia

Jérome Douzet

Ana Cameirao

Saheb Maghsoodloo-Babakhani

Jean-Michel Herri

Crystallization mechanisms and rates of Cyclopentane Hydrates formation in Brine

Keywords: Cyclopentane, Clathrate Hydrates, Crystallization, Brine, Mass transfer

Clathrate hydrates most often grow at the interface between liquid water and another fluid phase (hydrocarbon) acting as a provider for the hydrate guest molecules, and some transfer through this shell is required for the hydrate growth to proceed, thus self-limiting the reaction rate. An optical microscope and a horizontal reaction cell are utilized to capture the shell growth phenomenology and to estimate the hydrate layer growth rates from sequential pictures. Cyclopentane (CP) is chosen as the hydrate-forming molecule to obtain hydrates at low pressure.

Experimental hydrate layer growth rates are provided for the CP + brine system, using various combinations of salts and various degrees of subcooling.

Introduction

Clathrate hydrates are ice-like and non-stoichiometric compounds consisting of water molecules forming polyhedral hydrogen-bonded cavities that enclose guest molecules such as CO2, CH4, C2H6, C3H8, or even Cyclopentane (CP) [START_REF] Sloan | Clathrate Hydrate of Natural Gases[END_REF]. The clathrate hydrates usually form at high pressures and at low temperatures depending on the guest molecules. These compounds have been widely studied over decades to understand and prevent hydrate plugging in pipelines [START_REF] Sloan | Natural Gas Hydrates in Flow Assurance[END_REF].

More recently, clathrate hydrates have gained increasing attention due to their many potential applications for example: gas transportation [START_REF] Mimachi | [END_REF][4][5], gas storage [6][7][8][9][10][11][12], gas separation [13][14][15], refrigeration [16,17], or desalination [18][19][20][21][22][START_REF] Ho-Van | Proc. of the Int. Conf. on Integrated Petroleum Engineering[END_REF].

In order to develop innovative clathrate hydrate-based techniques, and to better manage hydrate formation in multiphase flow, not only the formation conditions, phase equilibrium, are necessary, but also the crystallization mechanisms: nucleation, growth, agglomeration.

In former studies, our group focused on cyclopentane hydrates for water treatment/desalination application [18,19]. The formation temperature in brine have therefore been experimentally obtained and modelled. The present contribution focuses on cyclopentane hydrate formation mechanisms and rates using a small unstirred cell under microscope.

Cyclopentane hydrates (CPH) form structure II clathrates at 7.1°C under atmospheric pressure.

They are composed of 17 water molecules per cyclopentane molecule. Since cyclopentane is not miscible with water (solubility of 156 mg/l at 25°C [START_REF] Mcauliffe | [END_REF]), the hydrate formers can be easily separated in a hydrate-based industrial application. Moreover, CPH formation from water + liquid cyclopentane (CP) may be viewed as a model analogue of gas hydrate crystallization in a water/oil emulsion under mild conditions. Therefore, it is an interesting system to study hydrate crystallization mechanisms, especially the mass transfer of the former species in the different phases, without the need of a pressurized experimental setup.

In the literature, many studies addressed mass transfer phenomena in clathrate hydrate formation [25][START_REF] Mori | Proc. of 2nd Int. Conf. on Nat. Gas Hydrates (Eds: J.P.Monfort)[END_REF][START_REF] Lv | [END_REF][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. In the most usual case, crystallization occurs at an interface between water, and another liquid phase, acting as a reservoir of the guest species. Therefore, a specific attention has been paid to mass transfer phenomena through this interface and the crystal layer that is formed. In quiescent systems, the crystalline layer prevents direct contact between water and guest molecules, and inhibits the hydrate formation (hydrate crust, or hydrate shell [34,[START_REF] Hobeika | Gas Hydrates 1[END_REF][START_REF] Daniel-David | [END_REF]). Hence, the mass transfer of water and/or guest molecules across the hydrate layer is a potential ratelimiting factor that controls the hydrate growth. Understanding the mass transfer through the hydrate layer is thus needed.

It is worth mentioning here the early experimental studies that addressed the crystal growth rates and mas transfer rates through the hydrate layer [31,[44][45][46][47][48][49][50], provide models [25,[START_REF] Mori | Proc. of 2nd Int. Conf. on Nat. Gas Hydrates (Eds: J.P.Monfort)[END_REF]28,34,35,41,51], or perform molecular dynamics (MD) simulations [33,38]. The pioneering work of Mori and co-workers [25,[START_REF] Mori | Proc. of 2nd Int. Conf. on Nat. Gas Hydrates (Eds: J.P.Monfort)[END_REF] explored theoretically the evolution of a hydrate layer thickness at the interface between water and another non-miscible phase of guest molecules. Their showed that a porous solid plate model is more realistic than the diffuse layer model due to the permeability of the hydrate film [25,[START_REF] Mori | Proc. of 2nd Int. Conf. on Nat. Gas Hydrates (Eds: J.P.Monfort)[END_REF].

Later, in 2010, Sun et al. [34] measured experimentally the hydrate film growth, and considered Mori's model as well as later developments (Peng et al. [52]). They concluded that the numerous factors that influence the hydrate layer growth rate in the model make the simulation very challenging. Abe et al. [45] also measured the hydrate film thickness growth for CO2 hydrates and suggested a model.

In the 2010s, Davies and co-worker have conducted valuable experimental studies. They used differential scanning calorimetry method to study the mass transfer phenomenon through methane hydrate layer at the interface in a non-mixing reactor [31]. They found that, at the beginning, hydrate layers at gas-water interface are porous, which facilitates the mass transfers.

In addition, the mass transfer rate decreases with time, as the hydrate cavities are filled with guest molecules. Then, they utilized Raman spectroscopy to investigate the mass transfer mechanism across a methane hydrate film [30]. Their results indicate that water molecules are more moveable than gas molecules in the hydrate layer, and that the growth of gas hydrate is controlled by the water movement in the interior of the hydrate layer.

Li and co-workers [47,48] reported the crystal growth of gas hydrates from methane and ethane by suspending a single gas bubble in water under different subcoolings. Their showed that, under low subcoolings, the growth of hydrate film is dominated by the film thickening, but not by lateral film growth. Importantly, simples kinetic models, as function of the subcoooling, could be used to correlate film growths In another of their efforts, Li et al [32] also concluded that hydrate layer thickening at the interface is controlled by the outward transport of water across hydrate film, while the lateral growth is controlled by the mass transfer of the guest molecules (CH4). Based this understanding, Kishimoto et al. [35] proposed a correlation of the lateral growth rate to mass transfer resistance.

Very recently, Sun et al. [28,41,51] published a remarkable series of articles for modelling the mass transfer in the hydrate shell. While their system is not a liquid-liquid interface, they looked into details at the interface between gas bubbles and water, and of course the inside of the hydrate shell. All mass and heat transfer equations are written to represent the problem of the hydrate crystallization and growth at the interface between a gas bubble and water.

Finally, MD simulations have been performed to investigate molecular mobility of interstitial H2O molecules across hydrate cavities [33,38]. Liang et al [38] found that interstitial molecules diffusion could be a key mechanism for the mass transfer of H2O across the hydrate layer. Lo et al. [33] explored the water vacancy driven diffusion of guest molecule through hydrate layer, and estimated the guest (CO2) diffusivity. This short review highlights two important points: there are not so many experimental studies that provide a local insight of the hydrate growth at the interface and furnish interface growth rates, or mass diffusion rates. Secondly, the modelling of such system is challenging.

In the present contribution, a new experimental insight is explored. Unlike other studies, the experimental set-up is a non-agitated small system. Moreover, this system geometry presents a vertical interface between water and cyclopentane, allowing a top view of the lateral hydrate growth under a microscope. There are two objectives: to measure the CPH formation rate from a two-phase system (brine-cyclopentane), and to discuss the crystallization mechanism. Moreover, both the influence on the aqueous electrolyte (four saline solutions of {NaCl, NaCl-MgCl2, NaCl-KCl-MgCl2, Na2SO4}) and the influence of subcooling (2.5°C, 3.5°C, and 4.3°C) on the hydrate layer growth rate are explored.

Experimental section

Materials

In this study, materials utilized are cyclopentane (purity 98%), NaCl (purity 99.5%), KCl (purity 99.0%), Na2SO4 (purity 99.5%), and MgCl2 (purity 99.5%). They were all furnished by Sigma-Aldrich. Distillated water (conductivity <0.055 μS.cm 1 ) was obtained from a decontamination system, Mili-Q®.

Experimental apparatus

Simplified diagram of experimental apparatus is presented in Figure 1.

The main device is a horizontal jacketed reactor cell (1) with a volume of approximately 1.23 ml. This reactor has a height of 4mm, and a diameter of 19.8mm. The metal on the bottom and sides is inox. On the top is a sapphire window (8) in order to observe the system with both optical microscope and Raman microscope. A chiller (6), Ministat 240-Huber, with a temperature uncertainty of 0.02°C, is used to control the temperature of the inside solution, thanks to a cooling jacket (6). The coolant is a mixture of water and absolute ethanol (50% v/v). A temperature probe (4) is utilized to monitor the solution temperature inside the reactor. Input [START_REF] Sloan | Natural Gas Hydrates in Flow Assurance[END_REF] and output (3) lines are used to introduce, and also reject, the solutions. A microscope (10), connected to a camera (11), is employed to observe the crystallization of CPH through the transparent sapphire window (8) from the top. Because this reactor system usually works at low temperatures (0-5°C), grams of silica gel ( 7) is used to absorb condensed water on the sapphire window surface in order to achieve quality images of crystallization. A transparent tube ( 9) is also used to prevent air from coming in, and to keep a low humidity level above the reactor.

Digital data, including microscope images and temperature, are recorded by a computer (13) with a rate of 1 image every 5 minutes.

Experimental protocol

Before each experiment, chiller is set at a temperature according to the desired subcooling. Note that subcooling, ΔTsub, is defined as the difference between the equilibrium temperature and the reactor temperature. The equilibrium temperatures of CPH in the presence of salts have been previously determined [18,19]. They are presented in Table 1.

At the beginning, in order to reduce the induction time of the crystallization, the saline solution is first used to form CPH in a bigger reactor (1L). Once CPH have been crystallized that way, hydrates are melted slowly with the heating rate of 0.1°C/hour, up to a temperature of 0.1°C above equilibrium. At this temperature, there should not be any crystals left the brine. Therefore, the initial salt concentration is regenerated in a solution close to thermodynamic equilibrium with liquid CP. Then, approximately 0.8 ml of the solution is introduced into the reactor cell, followed by about 0.4 ml of CP. It was observed that this "water preconditioning " reduces significantly the induction time, compared to the use of fresh water [START_REF] Martinez De Baños | Mechanisms of formation and dissociation of cyclopentane hydrates[END_REF].

After stabilization, the microscope is set-up to observe the water/CP interface. Photos are taken every 5min. Hopefully, the first moments of the nucleation and growth are detected.

After 3-4 days, when CPH has formed completely, videos are made from the previous pictures.

Experimental Observations

Experimental layout

7 mixtures at 3 different initial subcoolings have been considered. However, since the occurrence of crystallization is not easy to detect, only three selected runs will be discussed in details: in pure water at 2.5°C subcooling, in 5% equi-mass mixture of NaCl-KCl-MgCl2 at 3.5°C and 4.3°C initial subcoolings. Note that, in the case of salty mixtures, only the initial subcooling is controlled. Indeed, with the CPH crystallization, the average salinity of the aqueous phase changes, hence the subcooling. Since the objective is to investigate a batch process at constant temperature, this is normal. Finally, note that recorded videos are available online in "Supplementary information" (S1, S2, and S3). The speed of these videos is 50 min/s, corresponding to 10 frames per second, one picture every 5 min.

Interface shape

Note that whatever the experience, the interface shape is similar (see Figure 2). Unexpectedly, the interface is not horizontal, even though CP is less dense than water. In the next pictures, CP phase is always on the left side, while the water phase is on the right side. The liquid-liquid interface clearly separates the two phases into two adjacent regions. Apparently, there is neither a droplet of CP into water, nor a droplet of water into CP. This is based on two observations: from microscopy, since the interface can be optically tracked by varying the focal length, but also from RAMAN spectroscopy. Indeed, the use of a RAMAN spectroscope (Horiba Xplora, 532nm laser) showed that, close to the CP/water interface, no water was detected between the substrates and the CP phase (points A and B on Figure 2), and likewise no CP was found between the substrates and brine (point C). Therefore, this infers that there is not a water film, respectively CP film, onto the substrate surfaces in the CP phase, respectively brine phase.

The physico-chemistry of components and interfaces (water/CP, walls/components) is the probable reason for such an atypical geometry. Of course, the configuration of the reactor cell (non-mixing, flat cylinder-shaped, very small height of 4mm), is the plausible reason why phase inversion cannot occur. Moreover, note that the process of injection is another determining factor. Indeed, the sequence of injection can change the final geometry. Hence, in order to obtain a clear vertical interface, water is always injected first.

Experimental phenomenology of CPH growth

Figure 3 illustrates the first interesting example (see also video S1). At the beginning appears a CPH nucleus in the CP phase (left). Apparently, some droplets of water are present in the CP phase, but without hydrates around them. There seem to be no CP droplets into the water phase (although this is possible out of the focus range). From stage 1 to stage 3, a triangle-shaped "single crystal" grows, and successive layers can be observed onto the surfaces of the crystal with time. At some moment, there is a sudden crystallization at the water-CP interface.

Remember that the solution is supersaturated in CPH, and it is not surprising to see such a rapid crystallization in the first steps. Then, step 4 corresponds to a growth of this interface, mostly toward CP phase. Note that the growth is not parallel to the interface, showing some preferential direction for the crystallization in the beginning. Somewhat later on, the interface shifts abruptly to the right. Later, step 5, there is a CPH growth on both sides, but still quicker toward the CP phase. At this moment, the growth seems to be parallel to the interface, at the same speed in all directions, although there are some exceptions. Moreover, there is a dramatic decrease of the crystal growth on the left side at the very end of the video, when the hydrate thickness is the most significant. In the end, CPH covers the whole observation plan.

A second key experiment (video S2) is illustrated in Figure 4. This is CPH formation from a 5% equi-mass NaCl-KCl-MgCl2 aqueous phase under a 3.5°C initial subcooling. In this scenario, a CPH nucleus appears first in the water phase. Its shape is hexagonal; it grows rapidly in the very beginning, hence its size stabilizes (step 2-3). During steps 3, there is still a quick crystallization at the water/CP interface and the interface becomes rough. The CPH is then observed to grow volumetrically, but mostly at the expense of the CP liquid phase, between a CP-CPH "front" moving to the left and a CPH-bine interface that almost preserves the position of the initial CPbrine contact. Suddenly (stage 4), the CP-CPH and the CPH-brine interfaces are shifted to the right, and their abrupt displacement is recorded thanks to the fixed reference given by the early formed hexagonal crystal. Later on, the crystallization proceeds mostly towards the CP phase (stage 5) and less so towards the water phase, layer by layer, at a varying speed, with occasional oscillation. Finally, note that the crystallization does not occur only in the microscope focus plan, showing blurry crystals above, or beneath, the crystallization front.

A last experiment, illustrated in figure 5, explores the same system as before, but at 4.3°C initial subcooling. Most of the previous observation can be made, and steps are quite similar. However, there is an exception to that. Between the quick interface crystallization and the interface shift + growth, there is a significant pause. This suggests that the interface shift is required by the volumetric changes linked to the ongoing reaction, but delayed by some mechanical resistance to displacement: a shear stress must be superrated for the displacement to occur; hence, when this stress is superated, the volume variation is acomodated and the interface shifts to the right.

After this breakage, a growth toward the CP phase is observed. Again, the reaction front remains roughly parralel to the interface, although maybe not everywhere. Also, at this moment, the interface shift is not really noticeable. Nevertheless, at the final stage (6) there is a blurry phase growing and moving to the right, probably CPH formation above or below.

CPH -forming front displacement velocity

In order to investigate the kinetics of CPH formation, and the effect of salt on it, the hydrate layer growth rate of crystallization with and without salts, under different concentrations and initial subcoolings, has been measured. The CHP layer growth rate, r, is calculated as follows:

r = l/Δt ( 1
)
where l is the distance of the hydrate layer propagation, and Δt the duration for this propagation.

Note that, this rate r represents either a time-averaged growth rate, by considering a significant Δt, or an instantaneous growth rate if 𝑟(𝑡) = 𝑑𝑙/𝑑𝑡. If the first possibility is easy to obtain, the calculation of the instantaneous hydrate layer growth rate 𝑟(𝑡) is more complicated since all the pictures need to be workable, which is rarely the case. In these calculations, the starting time is taken to be the beginning of the crystallization at the CP/water interface. Moreover, an average of ten different value of the CPH layer thickness l at ten different positions of the CP-CPH interfaces in plain view have been used. Figure 6 illustrates such calculations in pure water at subcooling of 4.3°C, and Table 2 provides numerical results of all calculations for different salt concentration and initial subcoolings, for three crystallization durations (60 min, 120min, and 180 min).

As expected, the CPH layer growth rate decreases over time. Note that our results agree well with the report on the thickness of CPH versus time according to Taylor et al. [START_REF] Taylor | [END_REF]. Their results pointed out that the CPH layer thickening rate is much higher at the first 0-40 min compared to that at the later periods. Moreover, CPH layer growth rate increases with increasing of the initial subcooling for all aqueous systems (pure water and electrolyte solutions), at the same crystallization time.

Discussion on growth rate of the CPH layer

Subcooling is related to the driving force of crystallization. Therefore, when it increases, the speed of hydrate formation increases. This phenomenon has been described in details elsewhere [47,48,[52][START_REF] Martinez De Baños | Mechanisms of formation and dissociation of cyclopentane hydrates[END_REF][START_REF] Taylor | [END_REF][55][56][57].

The decrease of the CPH layer growth rate when increasing the crystallization time can be explained as follows: In salty systems, the driving force for the crystallization changes with the increase in salinity due to the water consumption. Hence, the growth rate is expected to decrease with time. Also, a decrease of the growth speed is expected because of the increasing resistance to the mass transfers, especially water.

Furthermore, the growth rates in pure water are almost unsurprisingly higher than those in the presence of salts (except some particular cases: NaCl-MgCl2 3.5% mass and Na2SO4 5% mass) at the same subcoolings and the same crystallization durations. Indeed, salts affect strongly both thermodynamic [19,44,[57][58][59] and kinetic [57] of CPH formation. The exception of the NaCl-MgCl2 3.5% mass and Na2SO4 5% mass found no explanation.

In addition, the hydrate layer growth rates are different when considering different brine at the same subcoolings and same crystallization durations. This indicates that salts affect differently the kinetic of CPH formation. The reason of this variance is probably attributed to the different role of each salt in inhibiting the hydrate formation. Indeed, each salt has a different influence in weakening water hydrogen bonding and reducing CP solubility due to the difference in their solubility, salt ion charge density, and importantly water activity [60,61]. Thus, the formation of CPH is inhibited differently. Certainly, more experiments are required to discover clearly each salt effect on the kinetic of CPH formation.

In comparison with literature, the lateral CPH crystal growth rate in pure water reported by Sakemoto et al [44] is approximately 0.0070-0.0090 mm/min and 0.0420-0.0440 mm/min under a subcooling of 2.4°C and 4.2°C, respectively. While the first number is quite similar to our results (0.0035 mm/min at 2.5°C subcooling at 60 min), the second is significantly lower (0.0055 mm/min under subcooling 4.3°C at 60 min, see Table 2). Nevertheless, their system is different and presents a horizontal interface in a bigger volume.

However, in presence of NaCl 3.5% mass, Sakemoto et al [44] indicates a lateral hydrate crystal growth rate of 0.0020-0.0040 mm/min at an initial subcooling of 2.4°C, while our result calculation is 0.0024 mm/min at 2.5°C initial subcooling at 60 min. If the number are quite close (same order of magnitudes), this also reveals that the hydrate crystal growth rates reported by Sakemoto et al [44] are usually higher than our observations.

In addition, Taylor el al [START_REF] Taylor | [END_REF] reported the final CPH film thickness is approximately 15.2 μm after 200 min at a subcooling of 3.8°C in pure water, indicating that the average thickening rate is about 0.076×10 -3 mm/min. Our observations (see table 2) present a rate of 0.0039mm/min at 60 min or 0.0036mm/min at 120 min at a subcooling of 3.5°C, which is much higher than the value from Taylor et al [START_REF] Taylor | [END_REF].

These differences in the growth rate can be attributed to the different experimental systems and calculation. Indeed, Sakemoto et al. [44] defined this growth rate as an one-dimension growth rate of an individual hydrate crystal. Taylor el al [START_REF] Taylor | [END_REF] measured the final thickness of the interface in 200 min, the average thickening rate is then calculated based on this measurement. In this work, the 2D polycrystalline hydrate layer growth rate in the nearly perpendicular direction to the water surface and CP surface are considered and not individual crystals (Figure 6). It was found that the hydrate growth in the perpendicular direction to the liquid-liquid interface is usually much more difficult than the lateral growth rate due to the slow mass transfer rate of water and CP across the hydrate layer [START_REF] Martinez De Baños | Mechanisms of formation and dissociation of cyclopentane hydrates[END_REF]. Consequently, the growth rates from this present study and from Taylor el al [START_REF] Taylor | [END_REF] are significant lower than the individual crystal growth rate reported by Sakemoto et al [44].

Discussion on CPH formation mechanism

First steps

To begin with, Figure 7 is an attempt to capture what has been observed previously, except the apparent "hydrate breakage".

At the beginning, some nucleation sites near the liquid-liquid interface, either in CP phase or in water phase are witnessed. Sometimes, it can even be in both phases (1 st stage). Indeed, the nucleation site is stochastic [62][63][64][65], and any impurity near the liquid-liquid interface can be an active nucleation center. Moreover, remember that the water used for the experiment is taken from a previous crystallization process. Therefore, it is a possibility that hydrate clusters remain in the water phase. Also, traces of CP (CP droplets) on the reactor surfaces, in the water phase, could be a source of formation of an individual hydrate crystal (hexagonal shaped on figures 4 and 7). Note that water droplets can also be present in the CP phase, leading to a nucleus (see figure 5). In order to simplify the mechanism, figure 7 only illustrates a nucleus in the water phase.

Later, the individual crystal grows by consuming the CP traces and water (2 nd stage). Quick nucleation and crystallization along the interface are then observed (3 rd stage). At this stage, the individual crystal also reaches its final size since all CP traces could have been consumed

completely.

The hydrate layers grow mainly toward the CP phase by consuming initially water in the brine phase and also water dissolved in the CP phase near the liquid-liquid interface. Obviously, over time, the dissolved water becomes depleted, and water is hence required from the aqueous phase (discussed further). Accordingly, at the very beginning of the interface growth, there is only a growth toward CP phase, without interface shift. This comes right after.

Interface shift and volume balance

In the next step, a shift of the water-CPH interface into the water phase is observed (4 th stage). This pressure difference has not been investigated in details in this effort. An evaluation would be needed in the case of a modelling. Nevertheless, this is consistent with the study of Li et al. [32] reporting the sinking of the hydrate shell into a water droplet, in which the difference between inner and outside pressure is large enough due to the water permeation out the shell for crystallization. Finally, it is not surprising that an interface shift toward the water phase is observed since the ration 17:1 for the CPH crystallization corresponds to nearly 76% of the volume change occurring in the water phase.

Hydrate growth and mass transfers

Later (step 5 and after), CPH hydrates grow into both CP and water phases. However, the growth toward CP phase is obviously faster than toward the water phase. As discussed before, more water is needed to form CPH. However, this growth on both sides demonstrates that water and CP molecules are migrating, probably inside de hydrate phase.

Several possibilities can be considered. First, water phase and CP phase could always be in contact. This particular situation corresponds to a droplet of water in CP, or the opposite, based on the kind of substrates. This is the situation of the lateral growth on a substrate. Martínez de Baños et al. [START_REF] Martinez De Baños | Mechanisms of formation and dissociation of cyclopentane hydrates[END_REF]68] especially observed and discussed this phenomena for water/CP system.

They witnessed an initial fast crystallization at the water/CP interface, and then, the spread of an "halo" on a non-hydrophobic substrate, due to a water film between the substrate and CP. In this case, the limiting factor is the water mass transfer in this film. Since such water film was not observed in our system (see the discussion in "interface shape section"), this is not the mechanism expected in our apparatus, unless the water amount is so low that the Raman signal was not noticeable.

In the other case CPH form a wall between the two former species. The growth is therefore not substrate related. In order to observe the CPH growth, on both sides, former molecules need to cross this wall. Accordingly, this means that the hydrate phase is a porous medium, which is not surprising and documented in literature [30].

Let us have a closer look at CPH phase. CPH is believed to be more water-wettable than oilwettable [59,[69][70][71]. This means that water phase can move more easily through the hydrate pore network than CP, proving much more H2O for crystallization on the CP side than CP molecules for crystallization on the brine side, hence a faster growth toward CP phase. This observation agrees well with other studies. Davies et al [30] stated that the hydrate layer is initially porous and could be favorable to water transportation. In addition, their results illustrate that the growth of hydrate is controlled by the water migration within the hydrate layer. Also, Li et al [32] concluded from their experiments that hydrate layer thickening is controlled by the outward mass transport of water molecules across hydrate layer.

Another transport mechanism can be considered. Based on MD simulations, Liang et al [38] reported that the interstitial water molecules are mobile entities in the interior a hydrate.

However, the rate difference of these two phenomena is not determined to our knowledge.

An illustration of the mass transfers across hydrate layer is illustrated in Figure 8. With the hydrate formation at the interface appears a wall. The interface is all covered by the polycrystalline hydrate and then it plays as a barrier to prevent water from contacting with guest molecules (CP) and vice versa, as presented in literature [START_REF] Sloan | Natural Gas Hydrates in Flow Assurance[END_REF]15,34,51,[START_REF] Taylor | [END_REF][72][73][74][75][76][77]. Moreover, the salinity then increases in the bulk of the brine phase because salts are excluded from hydrate crystallization [29,78,79]. Consequently, this shifts the thermodynamic equilibrium temperatures of CPH to lower values [18,19,[START_REF] Ho-Van | Proc. of the Int. Conf. on Integrated Petroleum Engineering[END_REF]. Thus, the crystallization rate is expected to decrease over time [44,57].

Figure 8 illustrates that there are three likely mass movements through the hydrate layer: water advection toward CP phase, CP advection toward water phase, and salt diffusion toward water phase. Water and/or CP phase migration can only be driven by pressure gradients within the phase of interest. By definition, a capillary pressure is not a pressure gradient, but a pressure difference between two phases. Water or CP molecules (not phases) can migrate by transport and/or diffusion within any phase. The drive for advective transport is always a pressure gradient, and a chemical potential gradient for diffusive transport. Salt diffusion is driven by a gradient in chemical potential of salt in the bulk of the brine phase and in the filled hydrate pores. Certainly, because salts are excluded from crystallization, the salinity in the hydrate layer is higher than in bulk of the brine phase. Thus, salts tend to diffuse from hydrate pores to saline solution.

As aforementioned, there is a quick growth toward CP phase and a slow growth toward water phase. This difference in growth rate is related to the migration mechanism of CP and water, and also salts across hydrate layer. Indeed, during hydrate formation process, individual hydrate grains can join and generate some spaces, pores or channel (unknown geometry), in which water, CP, and salts can move. According to the numerical study of Mori et al [25], water permeates through hydrate layer filling the capillaries. The driving force for this water permeation is the capillary pressure due to the hydrophilic characteristic of the hydrate surface. This agrees well with the observations using Raman spectrometer reported by Davies et al [30]. They stated that, in the polycrystalline hydrate layer, numerous visible pores or capillaries could be the mean for relatively high mobility water and also hydrate former migration across the hydrate film.

Our observations also indicate that the growth of the crystal interface remained somehow parallel to the CPH interface, without any obvious preferential direction, especially in the later steps.

Therefore, it is suspected that the limiting step is the mass transfer. Pressure gradients are not negligible. When it is not the case, probably in the first steps, the crystal growth to be the limiting step can be expected.

From a modelling viewpoint, mass transfer across a hydrate layer was numerically explored by Mori et al. [25,[START_REF] Mori | Proc. of 2nd Int. Conf. on Nat. Gas Hydrates (Eds: J.P.Monfort)[END_REF]. However, these authors assumed a steady state where the hydrate layer forms at the hydrate-guest interface while it melts at the water-hydrate interface, thus maintaining a constant thickness. This is not strictly speaking a growth model, but the main concept to be retained from Mori et al 1997 [25], is that because gas hydrates are (presumably) hydrophilic, the pore space in the hydrate layer is (presumably) water-wet, hence a sustained water flux may be driven through the hydrate layer by the combined effects of capillary forces and the consumption of water at the hydrate-guest interface, where the hydrate actually grows.

In the presence of salts, there is also a mass transport of salt within the hydrate layer and the bulk of saline water. Obviously, hydrate formation excludes salt into saline solution and enhances the local salinity. Recently, the salt diffusion during hydrate crystallization was investigated by Meyer et al [29]. Salt diffusion is necessary to observe the hydrate growth in CP/CPH interface, and therefore needs to be taken into account.

Conclusion

The formation of CP hydrate between brine and liquid CP was explored using a non-agitated small cell and a microscope allowing a top view of the processes and an evaluation of the lateral growth rate of the hydrate layer, under a variety of driving conditions and brine compositions.

The phenomenology of the hydrate formation involves the early formation of single crystals near the interface, as a likely consequence of the preparation of the system, where water is produced by melting previous CP hydrate. The volumetric Hydrate layer formation starts by a flash nucleation at the water-CP interface, then hydrate grows mostly, but not exclusively, at the expense of the CP phase, at a rate well recorded by the position of the CP-CPH interface in time.

Accordingly, the hydrate formation mechanism involves the advective transfer of the water phase through the porous hydrate layer which is, presumably, water-wet.

Not surprisingly, the measured growth rates increase linearly with increasing driving force (i.e.

undercooling), and decreases with salt concentrations. This rate also decreases with time, in relation to the increased thickness of the hydrate layer and its increased resistance to the water transfer.

Table of Contents: Graphical Abstract

Cyclopentane hydrates are considered to be a candidate for hydrate-based desalination. Thus, mechanism of cyclopentane hydrate crystallization in pure water and in brine is crucial. In this contribution, varied salts at different concentrations and under three driving forces (subcooling)

were considered to discover the phenomenology of hydrate formation, utilizing an optical microscope. An effort of measuring the hydrate growth speeds was also conducted.

Cyclopentane hydrates crystallization near the liquid-liquid interface between water and cyclopentane

Figure 1 .

 1 Figure 1. Schematic illustration of the experimental set-up Reactor cell; (2) input line; (3) output line; (4) Temperature probe; (5) Cooling jacket; (6) Chiller; (7) Silica gel; (8) Sapphire window; (9) Transparent tube; (10) Objective lens; (11) Camera; (12) Temperature transmitter; (13) Computer

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Schematic illustration of the hydrate layer in the reactor-cell

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. CPH crystallization mechanism in the presence of NaCl-KCl-MgCl2 5% at an initial subcooling of 4.3°C

Figure 8 .

 8 Figure 8. Mass transfer at the CPH interface during crystallization (illustration based on the photo taken from the experiment with NaCl-KCl-MgCl2 5% mass at subcooling 2.5°C)

Figure 1 ( 1 )

 11 Figure 1 Schematic illustration of the experimental set-up (1) Reactor cell; (2) input line; (3) output line; (4) Temperature probe; (5) Cooling jacket; (6) Chiller; (7) Silica gel; (8) Sapphire window; (9) Transparent tube; (10) Objective lens; (11) Camera; (12) Temperature transmitter; (13) Computer.

Figure 2 Figure 7 Figure 8

 278 Figure 2 Schematic illustration of the hydrate layer in the reactor-cell Figure 3 CPH crystallization mechanism in the presence of pure water at an initial subcooling of 2.5°C. Arrows indicate moving crystal boundaries (single crystal growth) or interface boundary Figure 4 CPH crystallization mechanism in the presence of NaCl-KCl-MgCl2 5% at an initial subcooling of 3.5°C Figure 5 CPH crystallization mechanism in the presence of NaCl-KCl-MgCl2 5% at an initial subcooling of 4.3°C Figure 6 CPH layer growth rate in pure water at 4.3°C subcooling after 60min, 120min, and 180min Figure 7 Simplified schematic of the CPH crystallization mechanism of near the interface in the reactor cell Figure 8 Mass transfer at the CPH interface during crystallization (illustration based on the photo taken from the experiment with NaCl-KCl-MgCl2 5% mass at subcooling 2.5°C)

  ) indicates that one CP molecule combines with 17 H2O molecules to form one CPH cell. Therefore, a large quantity of water is needed for the crystallization, leading a huge decrease in the volume of water phase.Let us have a look at the volume balance of the process. Density of pure water at 4°C is close to 1000 kg/m 3 (presence of salt decreases this density). Standard density of cyclopentane is 745 kg/m 3 . At last, CPH density, considering full occupancy of large SII cavities, can be calculated to be 950 -970 kg/m 3[START_REF] Hobeika | Gas Hydrates 1[END_REF][START_REF] Martinez De Baños | Mechanisms of formation and dissociation of cyclopentane hydrates[END_REF]59,66,67], i.e. less than the liquid water phase, and more than liquid CP. Based on these estimations, the volume balance is:{ 𝑉 𝑚 (𝐶𝑃𝐻) -[𝑉 𝑚 (𝐶𝑃) + 17. 𝑉 𝑚 (𝐻 2 𝑂)] = ∆𝑉 𝑚 3.92.10 -1 -(9.41.10 -2 + 17.1.8.10 -2 ) = 8.33.10 -3 𝐿/𝑚𝑜𝑙(3)Therefore, the volume diminution is about 8.3 mL/mol of CP. Since about 0.8mL of water is here

	Equation (2
	This is probably due to water consumption of crystallization. Remember that CP and water form
	hydrate structure II according to the following reaction:
	C5H10 (CP) + 17.H2O = C5H10.17H2O (CPH)	(2)

introduced, a quick estimation indicates a volume change up to 22µL, for full water consumption. As consequence, this volume reduction can induce a pressure difference between the two sides of the interface.

Table 1 .

 1 CPH equilibrium temperature in different solutions[18,19]

	Solutions	Salinity a,b,c , % mass CPH equilibrium temperature, ±0.1°C
	Pure water	-	7.1
	NaCl	3.5	5.5
	NaCl-MgCl2	3.5	5.1
	NaCl-KCl-MgCl2	3.5	5.2
	NaCl-KCl-MgCl2	5.0	4.6
	Na2SO4	3.5	6.0
	Na2SO4	5.0	5.6

a Uncertainty due to weighing: ±0.002% mass. b Uncertainty due to drying oven: ±0.2% mass. c Relative uncertainty due to ion chromatography: 1.5% (see the Supporting Information from Ho-Van et al.

[19]

for all uncertainty estimations).

Table 2 .

 2 Measured CPH layer growth rate in different subcoolings and brine solutions with time.

					Propagation rate, mm.min -1
	Aqueous solution		ΔTsub = 2.5°C			ΔTsub = 3.5°C			ΔTsub = 4.3°C
		60min	120min	180min	60min	120min 180min	60min	120min 180min
	Pure water	0.0035	0.0033	0.0025	0.0039	0.0036	NA	0.0055	0.0049	0.0037
		±0.0004	±0.0002	±0.0002	±0.0002	±0.0004		±0.0005	±0.0006	±0.0003
	NaCl 3.5 % mass	0.0024	0.0018	0.0016	0.0040	0.0027	0.0020	0.0046	NA	NA
		±0.0004	±0.0003	±0.0003	±0.0005	±0.0002	±0.0002	±0.0005
	NaCl-MgCl2 3.5%	0.0022	0.0017	0.0014	0.0045	0.0033	NA	0.0063	0.0041	0.0041
	mass	±0.0003	±0.0005	±0.0002	±0.0005	±0.0002		±0.0006	±0.0006	±0.0008
	NaCl-KCl-MgCl2	0.0013	0.0009	NA	0.0039	0.0031	0.0023	0.0054	NA	NA
	3.5% mass	±0.0002	±0.0002		±0.0008	±0.0004	±0.0004	±0.0005
	NaCl-KCl-MgCl2	0.0011	0.0008	0.0006	0.0016	0.0011	0.0008	0.0020	0.0014	0.0011
	5% mass	±0.0002	±0.0001	±0.0001	±0.0003	±0.0004	±0.0001	±0.0001	±0.0002	±0.0001
	Na2SO4 3.5%	0.0009	0.0006	0.0006	0.0018	0.0012	0.0010	0.0020	0.0013	0.0010
	mass	±0.0001	±0.0001	±0.0001	±0.0003	±0.0002	±0.0002	±0.0002	±0.0003	±0.0001
	Na2SO4 5% mass	0.0009	0.0007	0.0005	0.0040	0.0038	NA	0.0089	NA	NA
		±0.0001	±0.0001	±0.0001	±0.0001	±0.0001		±0.0002
	NA: Not Acknowledge

Supporting information.

Video S1. CPH formation in pure water at a subcooling of 2.5°C Video S2. CPH formation in the presence of NaCl-KCl-MgCl2 5% mass at a subcooling of 3.5°C Video S3. CPH formation in the presence of NaCl-KCl-MgCl2 5% mass at a subcooling of 4.3°C

List of symbols used


Biswajit Das
email: biswajit.das@inria.fr

Khalid Daoudi
email: khalid.daoudi@inria.fr

Jiri Klempir
email: jiri.klempir@lf1.cuni.cz

Jan Rusz

Towards disease-specific speech markers for differential diagnosis in Parkinsonism

Keywords: Pathological speech processing, Speech disorder, Parkinsonism, Parkinson's disease, Machine learning

INTRODUCTION

Speech requires the integrity and integration of numerous neurocognitive, neuromotor, neuromuscular and muscu-loskeletal activities [START_REF] Duffy | Motor Speech Disorders Substrates, Differential Diagnosis, and Management[END_REF]. Deficit in any of this activity may be the cause of speech disorder. It is an early and prominent indication for different neurological disorder. Therefore, in such disorders, recognizing and understanding predictable patterns of speech disturbances and their underlying neurophysiological bases can help to understand nervous system organization for speech motor control.

Parkinson's disease (PD) is a neurological disorder caused by the degeneration of dopaminergic neurons, leading to clinical features characterized by bradykinesia, rigidity, resting tremor and postural instability. Atypical parkinsonian syndromes (APS) such as progressive supranuclear palsy (PSP) and multiple system atrophy (MSA) differ from PD by more widespread neuronal involvement, resulting in additional clinical signs, more rapid disease progression and poor response to dopamine replacement therapy [START_REF] Schrag | Prevalence of progressive supranuclear palsy and multiple system atrophy: a cross-sectional study[END_REF]. The majority of PSP and MSA patients develop clinical features that overlap those of PD and thus the correct diagnosis can be very challenging in early stages of the disease. However, an accurate early diagnosis is essential not only in assessing prognosis and making decisions regarding treatment, but also for understanding the underlying pathophysiology and for the development of new therapies [START_REF] Wenning | Milestones in atypical and secondary parkinsonisms[END_REF].

It turns out that speech impairment is frequently an early and prominent clinical feature of PD as well as APS. During the last decades, there has been a increasing interest in PD speech and voice analysis [START_REF] Brabenec | Speech disorders in parkinsons disease: early diagnostics and effects of medication and brain stimulation[END_REF]. However, very few attempts have been done in differential diagnosis between PD and APS or within APS [START_REF] Kluin | Perceptual analysis of speech disorders in progressive supranuclear palsy[END_REF][START_REF] Kluin | Characteristics of the dysarthria of multiple system atrophy[END_REF][START_REF] Sachin | Clinical speech impairment in parkinson's disease, progressive supranuclear palsy, and multiple system atrophy[END_REF][START_REF] Skodda | Acoustical analysis of speech in progressive supranuclear palsy[END_REF][START_REF] Huh | Differences in early speech patterns between parkinson variant of multiple system atrophy and parkinsons disease[END_REF]. A pioneer work [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF] provides a quantitative and objective analysis of speech characteristics for the discrimination between PD and APS and between MSA and PSP. The basic conclusion is that PD speakers manifest pure hypokinetic dysarthria, ataxic components are more affected in MSA whilst PSP subjects demonstrate severe deficits in hypokinetic and spastic elements of dysarthria. Using an SVM with a Gaussian radial basis kernel and an exhaustive search, [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF] reported a 95% accuracy in objective discrimination between APS and PD and 75% in discrimination between PSP and MSA. We emphasize that classification performance was not the main purpose of [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF], but rather a way to seek disease-specific dysarthric signs. In our recent study [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF], standard linear and generalized linear models were explored to address the curse of dimensionality problem in this setting. This study led to an 80% accuracy in classification between MSA and PSP.

This paper is a continuation of our previous work on discrimination between PSP and MSA [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF]. Here, we focus on defining new speech features which can objectively measure particular dysarthria attributes and which are disease-specific, in the sense that such features would have a (statistical) behavior for PSP which is significantly different than for MSA. Moreover, we want such features to be clinically interpretable in order to improve the understanding of speech impairments in PSP and MSA. Obviously, the first benefit of such investigation would be accurate and objective discrimination between PSP and MSA, given that subjective evaluation is quite challenging due to similar perceptual behavior [START_REF] Miller | Utility and accuracy of perceptual voice and speech distinctions in the diagnosis of parkinsons disease, psp and msap[END_REF]. The second and more important benefit is to potentially allow drawing hypothesis regarding the early stage of the diseases.

The paper is organized as follows. Section 2 describes the speech database. Acoustic features used in this study are described in Section 3. Methodology and experiments are presented in Section 4.

DATASET

From 2011 to 2014, 12 consecutive patients with the clinical diagnosis of probable PSP (10 men, 2 women) and 13 patients with the diagnosis of probable MSA (6 men, 7 women) were recruited. In this series, 9 PSP patients were diagnosed with the Richardson's syndrome (PSP-RS), 2 with PSPparkinsonism (PSP-P) and 1 with PSP-pure akinesia with gait freezing (PAGF), whereas 10 MSA patients were diagnosed as the parkinsonian type (MSA-P) and 3 as cerebellar type (MSA-C). The diagnosis of PSP was established by the NINDS-PSP clinical diagnosis criteria [START_REF] Litvan | Clinical research criteria for the diagnosis of progressive supranuclear palsy (steelerichardson-olszewski syndrome) report of the nindsspsp international workshop[END_REF], MSA according to consensus diagnostic criteria for MSA [START_REF] Gilman | Second consensus statement on the diagnosis of multiple system atrophy[END_REF]. Speech severity did not perceptually differ between PSP and MSA based on UPDRS speech item 18. For a detailed description of the patients we refer to [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF].

Speech recordings were performed in a quiet room with a low ambient noise level using a head-mounted condenser microphone (Bayerdynamic Opus 55, Heilbronn, Germany) situated approximately 5 cm from the mouth of each subject. Speech signals were recored with 48 kHz sampling frequency and 16-bit resolution. Each participant was instructed to perform sustained phonation of the vowel /a/ per one breath as long and steadily as possible, fast /pa/-/ta/-/ka/ syllable repetition at least seven times per one breath, a reading passage and a monologue on a given topic for approximately 90 s. All Table 1. List and description of the 13 features, grouped by dysarthria type participants performed the sustained phonation and syllable repetition tasks twice.

ACOUSTIC FEATURES

In earlier studies [START_REF] Rusz | Quantitative acoustic measurements for characterization of speech and voice disorders in early untreated parkinsons disease[END_REF][START_REF] Novotny | Automatic evaluation of articulatory disorders in parkinson's disease[END_REF] several acoustic parameters are explored, we refer to [START_REF] Rusz | supplementary material: Acoustic measurements for objective evaluation of motor speech disorders[END_REF] for the detailed description. In the same manner as in our recent study [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF], in order to allow easy future comparisons or reproduction, we consider in the paper the same set of 13 features that can be computed with existing and established scripts. In [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF], we adopted a "phonetic" point of view to group the feature because our main concern was the investigation the usability of linear models to address the curse of dimensionality. In this work, our concern is to find/define features which can be specific to each disease and which can lead to a clinical interpretation. We thus adopt a "symptomatic" point of view (as in [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF]) to categorize the features into the well-known 3 dysarthria groups: Hypokinetic, Spastic and Ataxic. The set of the 13 features we consider is presented in Table 1. All there features are computed using Python and Praat scripts [START_REF] Boersma | Praat: doing phonetics by computer [computer program[END_REF]. For phonation and syllable repetition, the final value of each feature is calculated as the mean of the values obtained from two vocal tasks.

In hypokinetic dysarthria, abnormality is most notably observed in respiratory or aerodynamic function, laryngeal function, and motor control. It leads to reduced vocal loudness, monopitch, disruptions of voice quality and abnormal fast rate of speech [START_REF] Duffy | Motor Speech Disorders Substrates, Differential Diagnosis, and Management[END_REF]. Conversely, in ataxic dysarthria, loudness may vary excessively, and increased effort is evident which can be described as explosive speech, pitch variation, irregular AMR and vocal tremor are notable components of this group. On the other hand, spastic dysarthria can be demonstrated as voice with strained or strangled, pitch breaks. Therefore, degree of voicelessness and slow AMR are grouped in spastic dysarthria.

METHODOLOGY AND EXPERIMENTS

In all experiments, the MSA and PSP data is used to normalize the acoustic features to zero mean and unit variance.

Univariate statistical analysis

Given that we are faced with the curse of dimensionality problem, we need to perform feature selection and/or reduction. The simplest way to do feature selection is to carry out a univariate statistical analysis. We use the latter as a first pass to discard the highly correlated (thus less discriminative) features. To do so, for each feature, we first compute the intra MSA (resp. PSP) class mean µ msa (resp. µ psp ) and variance σ msa (resp. σ psp ). We then use Mahalanobis distance, defined in Equation 1, as a measure of the the inter-class separation.

D = (µ psp -µ msa ) 2 .( σ psp + σ msa 2 ) -1 (1)
The result of this univariate analysis is given in Figure 1. This analysis suggests that we can fairly discard monopitch from the hypokinetic set of features. Likewise, this analysis suggests that DUV and slow AMR can be disregarded. We will keep them however in our second stage of analysis in order to check whether this "first indication" stands in our second stage of analysis. As for the ataxic group, it seems like all features are contributing towards discrimination. We thus end up considering the following sets of features:

• H = {jitter, shimmer, HNR, intra-word pause, rapid AMR, no. of pauses, PPT}

• A = {F0 SD, irregular AMR, vocal tremor}

• S = {DUV, slow AMR}

Learning a new speech feature

In this section, we perform our second stage of feature selection and dimension reduction. Nowadays, most of "hot" machine learning problems and methods deal with large data Table 2. Classification accuracy for individual dysarthric groups sets (big data problems). In many applications however, such as biomedical engineering, data are rare and/or are collected with a (very) low time resolution. The problem we address in this paper falls in this setting; we are in a small dataset machine learning scenario. Indeed, only 12 PSP and 13 MSA patients are available. Thus, typically only a 1-dimensional feature space may provide acceptable statistics. Univariate statistical analysis, as performed in the previous section, is one solution. In fact, a more involved univariate analysis has been carried out [START_REF] Rusz | Speech disorders reflect differing pathophysiology in parkinsons disease, progressive supranuclear palsy and multiple system atrophy[END_REF], using the same dataset. This led to the argument that ataxic components are more affected in MSA whilst PSP subjects demonstrate severe deficits in hypokinetic and spastic elements of dysarthria. However, univariate analysis does not allow us to achieve the goal we target in this paper, that is finding a scalar speech feature which is disease specific and which is clinically interpretable.

In [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF], we presented a methodology to deal with the machine learning part of our small dataset problem. We showed that classical linear and generalized linear models, such as Factorial Discriminant Analysis (FDA) (also known as descriptive LDA) [START_REF] James | An introduction to statistical learning[END_REF], can provide a simple solution to this problem. We thus proceed now to investigate whether such models can lead us to our goal.

For classification, as the amount of data is small, a Leave-One-Speaker-Out (LOSO) training approach is adopted in all the experiments. In order to have a clear understanding of H+S+A H+S S+A H+A Accuracy (%) 72 60 24 84 The results we present in the following are the same using these two classifiers. We start (naturally) by evaluating FDA on each dysarthric group (H, A, S) individually. The idea here is to check whether a linear combination of a group feature can discriminate between MSA and PSP. Table 2 shows the FDA classification scores between PSP and MSA patients by individual dysarthric group. This result shows that FDA on A and S are extremely low and even worse than chance. This suggest that (linear combination of) the ataxic and spastic features we consider cannot be individually disease-specific. The same argument holds for hypokinetic features, though the score is significantly higher.

We now proceed to evaluate the combination of dysarthric groups. Table 3 presents the FDA classification scores for the 4 possible combinations. This result shows that the same argument we made for individual group holds also for H+S+A, H+S and S+A. Interestingly the score obtained by H+A is significantly high, even higher than the score in our recent work. This is a very good indication that measuring the "mutual amount" of hypokinetic and ataxic dysarthria can allow discrimination between MSA and PSP. Moreover, our result show that a relatively simple (weighted) averaging can measure this amount.

We now go deeper in the analysis of this averaging. Table 4 shows the weights obtained by FDA when training using all data (no LOSO). By looking at the weights, one observes that intra-word pause and no. of pauses have lower weights compared to the other feature. This suggests that they can be discarded in the linear FDA projection. We did so and rerun a LOSO training by discarding intra-word pause and no. of pause. This yields a classification score of 88%. This means that the arguments we stretched about hypokinetic and ataxic features (H+A) hold even better when discarding these two hypokinetic features. Figure 2 shows the values of the resulting variable for PSP (blue) and MSA (green) patients. The latter shows that a very good separation is indeed obtained. We mention however that these classification scores should be considered with precaution because of the LOSO bias (different weights at each iteration).

CONCLUSION

In this paper, we addressed the difficult problem of defining disease-specific speech features which is crucial in the perspective of early differential diagnosis in Parkinsonism. We Fig. 2. Values of the new speech feature for each patient focused on MSA and PSP and investigated this problem under the constraint of small dataset machine learning. Using FDA, we ended up defining a new scalar variable which can be considered as a disease-specific feature. This variable is learned from data as a linear combination of some hypokinetic and ataxic features. Using Gaussian or 1d SVM classifiers, we obtain a threshold which measures a certain degree of hypokinetic and ataxic "impairment". Above (resp. below) this threshold, the patient is classified as MSA (resp. PSP). Using LOSO training, we achieved 88% classification accuracy, which is an improvement w.r.t the 80% obtained in our recent work [START_REF] Li | Linear classification in speech-based objective differential diagnosis of parkinsonism[END_REF]. This achievement can be considered very successful as PSP and MSA are hardly perceptually distinguishable [START_REF] Miller | Utility and accuracy of perceptual voice and speech distinctions in the diagnosis of parkinsons disease, psp and msap[END_REF]. This result also suggests that hypokinetic and ataxic dysarthria convey considerable discriminative information when mutually considered. Moreover, it suggests that the hypokinetic and ataxic features we retained can be used as a vehicle to capture this information. We showed indeed that an appropriate weighted averaging of these features (the new variable) led to a high classification accuracy. We emphasize however that these arguments are not a "hard" conclusion. They need indeed to be confirmed by additional data and studies. This is the purpose of our ongoing research. We are actually in the process of collecting a (relatively) large speech corpus of French PD and APS patients (in the framework of a large pilot study involving French university hospitals).

Fig. 1 .

 1 Fig. 1. Feature-wise distance between PSP and MSA

Table 3 .

 3 Classification accuracy for combined dysarthric groupsHypokinetic Ataxic Jitter Shimmer HNR Intra-word pause No. of pauses PPT Rapid AMR F0 SD Irregular AMR Vocal tremor

	Weight	1.03	3.02	4.77	-0.10	-0.02	-1.11	0.80	2.14	-0.49	0.37

Table 4 .

 4 Feature weights obtained by FDA the data behavior, we use a simple Gaussian classifier and a 1d linear Support Vector Machine (SVM) with C = 1 as classifiers.

* This work is partly funded by the French research agency (ANR), project Voice4PD-MSA n o ANR-16-CE19-0010-01. † This work is partly funded by the Research Center for Informatics ( grant nr. CZ.02.1.01/0.0/0.0/16 019/0000765), and by the Czech Ministry of Education (PROGRES-Q27/LF1).