Nelly Couzon

Laurence Bois
email: laurence.bois@univ-lyon1.fr

Clémentine Fellah

Cristian Loestean

Fernand Chassagneux

Rodica Chiriac

Francois Toche

Lhoussain Khrouz

Arnaud Brioude

Ovidiu Ersen

François Toche

Lucian Roiban
email: lucian.roiban@insa-lyon.fr

Manganese oxidation states repartition in a channel-like mesoporous zirconium oxide

Keywords: manganese, zirconium, mesoporous, analytical electron tomography, self-assembly block copolymer

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

Introduction

The development of catalysis area during last decades has led to an increase interest in the use of transition metals as catalysts, proposing a great alternative to noble metals, being cheaper and as efficient. More specifically, the development of manganese containing mesoporous oxide is an important challenge for oxidation catalysis applications. Mesoporous materials possess interesting properties such as narrow pore size distribution, high specific surface area or high pore volume allowing an easy diffusion of guest species [START_REF] Wang | Recent advances in ordered meso/macroporous metal oxides for heterogeneous catalysis: a review[END_REF][START_REF] Zhang | Templated Growth of Crystalline Mesoporous Materials: From Soft/Hard Templates to Colloidal Templates[END_REF][START_REF] Taguchi | Ordered mesoporous materials in catalysis[END_REF]. These properties make them particularly adapted for catalysis applications. Zirconia is a really attractive transition metal oxide for catalysis thanks to its outstanding properties: high corrosion resistance, high mechanical and thermal resistance and excellent oxygen mobility [START_REF] Tsoncheva | Cobalt and iron oxide modified mesoporous zirconia: Preparation, characterization and catalytic behaviour in methanol conversion[END_REF]. Thanks to these properties, ZrO 2 is commonly used as a support and solid solutions based on ZrO 2 possess high catalytic activity [START_REF] Tanabe | Surface and catalytic properties of ZrO2[END_REF][START_REF] Zippel | Electrical transport and optical emission of MnxZr1-xO2(0≤x≤0.5)thin films[END_REF][START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF]. However, to improve its characteristics and catalytic efficiency, ZrO 2 doping is often developed, leading to interesting results [START_REF] Moharreri | Comprehensive Magnetic Study of Nanostructured Mesoporous Manganese Oxide Materials and Implications for Catalytic Behavior[END_REF].

Manganese as supported catalyst is particularly interesting because it possesses various valence states and five stable stoichiometric oxides (MnO, Mn 3 O 4 , Mn 2 O 3 , Mn 5 O 8 , and MnO 2 ) allowing a wide range of applications [START_REF] Moharreri | Comprehensive Magnetic Study of Nanostructured Mesoporous Manganese Oxide Materials and Implications for Catalytic Behavior[END_REF][START_REF] Wei | Manganese oxide-based materials as electrochemical supercapacitor electrodes[END_REF][START_REF] Miller | Nanoscale metal oxide-based heterojunctions for gas sensing: A review[END_REF]. Manganese containing catalysts have received much attention for oxidation reactions such as the oxidation of carbon monoxide or benzene [START_REF] Chen | High catalytic performance of mesoporous zirconia supported nickel catalysts for selective CO methanation[END_REF][START_REF] Biswas | Heterogeneous mesoporous manganese/cobalt oxide catalysts for selective oxidation of 5-hydroxymethylfurfural to 2,5diformylfuran[END_REF][START_REF] Afonasenko | Effect of the calcination temperature and composition of the MnO x -ZrO 2 system on its structure and catalytic properties in a reaction of carbon monoxide oxidation[END_REF][START_REF] Lee | Silver and manganese oxide catalysts supported on mesoporous ZrO2 nanofiber mats for catalytic removal of benzene and diesel soot[END_REF] or for the combustion of toluene [START_REF] Hou | The performance of manganese-based catalysts with Ce0.65Zr0.35O2 as support for catalytic oxidation of toluene[END_REF]. Also, lot of studies have shown that manganese-based catalysts exhibit good NH 3 -selective catalytic reduction (SCR) activity at low temperature [START_REF] Gao | 2D, 3D mesostructured silicas templated mesoporous manganese dioxide for selective catalytic reduction of NOx with NH3[END_REF], which is a promising method for the treatment of NOx [START_REF] Gao | 2D, 3D mesostructured silicas templated mesoporous manganese dioxide for selective catalytic reduction of NOx with NH3[END_REF][START_REF] Jia | Study of NO removal and resistance to SO 2 and H 2 O of MnOx/TiO 2 , MnO x /ZrO 2 and MnO x /ZrO 2 -TiO 2[END_REF].

Therefore, the development of new synthesis methods of manganese oxide supported on mesoporous zirconia is the main challenge that could lead towards new applications. Such challenge is most important because the way of introduction of manganese during the synthesis of the mesoporous oxide supports can strongly affect the global characteristics of the final catalyst.

Manganese is known for its ability to penetrate the lattice of ZrO 2 forming thus some solid solutions Zr 1-x Mn x O, which induces a higher lattice oxygen mobility and high reactivity [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF][START_REF] Afonasenko | Effect of the calcination temperature and composition of the MnO x -ZrO 2 system on its structure and catalytic properties in a reaction of carbon monoxide oxidation[END_REF][START_REF] Gutierrez-Ortiz | Structure of Mn-Zr mixed oxides catalysts and their catalytic performance in the gas-phase oxidation of chlorocarbons[END_REF][START_REF] Miao | Mesoporous Mn-Zr composite oxides with a crystalline wall: synthesis, characterization and application[END_REF][START_REF] Fanelli | MnO x /ZrO 2 gel-derived materials for hydrogen peroxide decomposition[END_REF][START_REF] Hernandez | Influence of the synthesis method on the nanostructure and reactivity of mesoporous Pt/Mn-WO x -ZrO 2 catalysts[END_REF][START_REF] Carrillo | Hydrogen production by methane decomposition over MnOx/YSZ catalysts[END_REF][START_REF] Zaki | CO and CH4 total oxidation over manganese oxide supported on ZrO 2 , TiO 2 , TiO 2 -Al 2 O 3 and SiO 2 -Al 2 O 3 catalysts[END_REF][START_REF] Döbber | MnO x /ZrO 2 catalysts for the total oxidation of methane and chloromethane[END_REF]. Manganese oxides on zirconia support have been often studied in heterogeneous oxidation catalysis such as gas-phase oxidation of hydrocarbons or chlorocarbons. Mesoporous zirconia synthesis is well described [START_REF] Yuan | Facile synthesis of Zr-based functional materilas with highly ordered mesoporous structures[END_REF][START_REF] Yuan | Electrospun mesoporous zirconia ceramic fibers for catalyst supporting applications[END_REF][START_REF] Gu | Surface-Casting Synthesis of Mesoporous Zirconia with a CMK-5-Like Structure and High Surface Area[END_REF], but few studies concern the synthesis of mesoporous mixed manganese zirconium oxide and the impact of manganese on the mesoporous zirconia characteristics [START_REF] Lee | Silver and manganese oxide catalysts supported on mesoporous ZrO2 nanofiber mats for catalytic removal of benzene and diesel soot[END_REF][START_REF] Miao | Mesoporous Mn-Zr composite oxides with a crystalline wall: synthesis, characterization and application[END_REF]. Impregnation of zirconia with manganese salt was often used [START_REF] Lee | Silver and manganese oxide catalysts supported on mesoporous ZrO2 nanofiber mats for catalytic removal of benzene and diesel soot[END_REF] while the one-pot evaporation induced self-assembly (EISA) method based on the evaporation of a sol containing the F127 block copolymer and a mixture of manganese chloride and zirconium oxychloride salt is rarely described [START_REF] Miao | Mesoporous Mn-Zr composite oxides with a crystalline wall: synthesis, characterization and application[END_REF].

Herein, a solid solution of manganese oxide in mesoporous zirconia has been synthesized by an one-pot EISA process using the evaporation of a sol containing the F127 block copolymer and a mixture of manganese acetylacetonate and zirconium oxychloride. The mesoporous texture of Mn incorporated oxide was analyzed by X-Ray Diffraction, Transmission Electron Microscopy and N 2 adsorption-desorption isotherm. Analytical electron tomography analysis was done to evaluate the manganese dispersion inside zirconia. Manganese oxidation states have been studied by thermal analysis, X-Ray photoelectron spectroscopy and Electronic paramagnetic resonance.

Experimental 2.1 Materials

A poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) block copolymer, Pluronic F-127 ((EO) 106 -(PO) 70 -(EO) 106 ), zirconium oxychloride octahydrate, ZrOCl 2 ,8H 2 O, manganese II acetylacetonate [CH 3 CO=CHCOCH 3 ] 2 Mn, were all purchased from Sigma-Aldrich and used as received.

Synthesis of the mesoporous Mn-ZrO 2 sample

Mixed MnZr oxide was prepared by modifying the method of the mesoporous zirconia synthesis previously described [START_REF] Zaki | CO and CH4 total oxidation over manganese oxide supported on ZrO 2 , TiO 2 , TiO 2 -Al 2 O 3 and SiO 2 -Al 2 O 3 catalysts[END_REF]. ZrOCl 2 ,8H 2 O (1.7 g, 5.10 -3 mole) was mixed in ethanol (10 mL) with 1 g of block copolymer F-127. Manganese salt, Mn(acetylacetonate) 2 (Mn(acac) 2 ), was added in Mn/Zr = 0.14 proportion to avoid any phase separation of manganese oxide. Slow evaporation step was performed until a gel was obtained. A thermal treatment was realized at 450°C during 5 h, with a heating rate of 1°C/min. Sample was named MnZr or Zr for the sample without manganese addition. Another thermal treatment was done at 600°C during 5 h to study the impact of manganese on the crystallization of zirconia. In Table 1 are summarized the synthesized samples.

Characterization

X-ray diffraction (XRD) patterns were recorded on a PANalytical Xpert Pro diffractometer (The Netherlands) equipped with a monochromator, using Cu K radiation (wavelength of 1.54 Å). Fullprof (FullProf.2k, Version 5.30 -Mar2012-ILL JRC) has been used for Rietveld refinement analysis on some of the XRD patterns.

Transmission Electron Microscopy (TEM) images have been performed on a Topcon instrument at 200 kV. Samples for the TEM characterization were ground and deposed directly on a copper grid coated with holey carbon film. Analytical Electron Tomography (AET) [START_REF] Midgley | Electron tomography and holography in materials science[END_REF][START_REF] Friedrich | Electron tomography for heterogeneous catalysts and related nanostructured materials[END_REF] was also used, which is an imaging mode of electron tomography [30] allowing the 3D quantification of the chemical distribution of the components in a sample [START_REF] Messaoudi | Three-Dimensional Chemical Mapping by EFTEM-TomoJ Including Improvement of SNR by PCA and ART Reconstruction of Volume by Noise Suppression[END_REF] by recording filtered images tilted series [32]. AET was performed in Energy Filtered TEM (EFTEM) mode using a Jeol 2100F electron microscope equipped with a GATAN Tridiem energy filter. The samples were crushed in a mortar, then dispersed in ethanol, and ultrasonicated for 10 min. A droplet of the suspension was deposited on a microscopy holey carbon grid on which previously 5 nm Au fiducial markers was added, and the grid was dried 10 min under a heating lamp. The tilt series were recorded by using the GATAN EFTEM tomography plugin of Digital Micrograph that allows the recording of several tilt series containing images with 512 x 512 pixels each. AET requires the uses of intense electron beam that can induce irreversible damages in the sample; therefore, as ZrO 2 is the matrix, to limit these damages, only the spatial distribution of Mn was followed. At each tilt angle was recorded a filtered image of 10 eV centered on the Zero Loss peak (ZL) with an exposure time of 0.07 s. Then three filtered images wide of 36 eV were recorded on the Mn L (640 eV) peak, centered on 573 eV, 600 eV and 640 eV with an exposure time of 10 s. The last image recorded was an unfiltered image recorded with an exposure time of 0.07 s. The chemical distribution in the 2D maps of the Mn was computed using the three-windows method implemented in the EFTEM-TOMOJ [32] plugin of ImageJ. A fine alignment was done on the ZL tilt series then applied on the computed chemical map of the Mn. The volume was computed using 15 algebraic reconstruction techniques (ART) implemented in TomoJ [START_REF] Friedrich | Electron tomography for heterogeneous catalysts and related nanostructured materials[END_REF].

The protocol employed on the data analysis from AET is published in [START_REF] Roiban | Three-Dimensional Analytical Surface Quantification of Heterogeneous Silica-Alumina Catalyst Supports[END_REF] and the 3D quantification was done using ImageJ tools [START_REF] Doube | BoneJ: Free and extensible bone image analysis in ImageJ[END_REF][START_REF] Colombo | Characterisation of the morphology of porous ceramics by 3D image processing of X-ray tomography data[END_REF][36] and 3D Slicer software [START_REF] Fedorov | 3D Slicer as an image computing platform for the Quantitative Imaging Network[END_REF].

Characterization of porosity of the samples was performed using nitrogen adsorption/desorption isotherms on a BELSORP mini (Bel Japan). Prior to analysis, degassing samples under vacuum at 100°C was performed during 4 h. The pore size distribution (d max) and mesoporous volume were calculated from the adsorption branch of the isotherms using the Barrett-Joyner-Halenda (BJH) method [START_REF] Barrett | The Determination of Pore Volume and Area Distributions in Porous Substances. I. Computations from Nitrogen Isotherms[END_REF], and the surface area was determined by the Brunauer-Emmett-Teller (BET) approach [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF].

Thermogravimetric analyses (TGA) with simultaneous differential thermal analysis (SDTA) signal have been realized on a TGA/SDTA 851 e Mettler Toledo apparatus. Around 20 mg of precursors were placed inside an 150 µL alumina crucible with a pre-pierced cover.

Temperature was increased from ambient up to 1200°C at 20°C/min. Air of high purity was used as reactive gas at a rate of 50 mL. min -1 .

Custom-build SPECS spectrometer was used for X-Ray Photoelectron Spectroscopy (XPS).

The core level lines were recorded with Al anode (1486.6eV) radiation. The etching was performed with Ar ions accelerated at 1000V. For quantitative analysis of samples, the following XPS core-level lines were recorded: Zr 3d, Mn 2p, O 1s and C 1s. Charge was referenced to adventitious C 1s peak at 284.8eV. Shirley background was subtracted from the spectra and the deconvolutions were done by using Gaussian/Lorentzian line shapes.

Electron Paramagnetic Resonance (EPR) measurements were carried out using an X-band spectrometer Bruker EMX plus, double mode.

Results and discussion

X-ray diffraction

The X-ray diffraction patterns of the Zr-450 and MnZr-450 oxides are given in Fig. 1a, showing the formation of tetragonal zirconia (t-ZrO 2 ) (JCPDS 00-042-1164). Comparison between them shows that manganese addition slows down the crystallization of tetragonal ZrO 2 , since the peaks of the Mn-450 sample are broader [START_REF] Fanelli | MnO x /ZrO 2 gel-derived materials for hydrogen peroxide decomposition[END_REF][START_REF] Ozawa | Structural and ESR characterization of heat-stable manganese -alumina lean NO removal catalyst[END_REF][START_REF] Conlan | Photo-Catalytic oxidation of a di-nuclear manganese centre in an engineered bacterioferritin reaction centre[END_REF][START_REF] Gutierrez-Ortiz | Structure of Mn-Zr mixed oxides catalysts and their catalytic performance in the gas-phase oxidation of chlorocarbons[END_REF]. The absence of any manganese oxide phases in the pattern of the MnZr-450 sample indicates that manganese ions are substituted or inserted in the zirconium oxide matrix [START_REF] Valigi | Manganese oxide-zirconium oxide solid solutions. An X-ray diffraction, Raman spectroscopy, thermogravimetry and magnetic study[END_REF][START_REF] Tao | Mn(III) species formed by the multi-copper oxidase MnxG investigated by electron paramagnetic resonance spectroscopy[END_REF]. It has been described in the literature that with concentrations of Mn below 30 at%, it is possible to obtain samples with single-phase solid solutions (Mn x Zr 1-x O 2-δ ) based on a ZrO 2 structure [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF][START_REF] Pomykalska | Electrical conductivity of MnO x -Y 2 O 3 -ZrO 2 solid solutions[END_REF][START_REF] Nandy | Microstructure correlated electrical conductivity of Manganese alloyed nanocrystalline cubic zirconia synthesized by mechanical alloying[END_REF]. The lower oxide crystallinity induced by manganese addition can be described more precisely by comparing the crystallite size measured from the Scherrer law (Table 2). As shown in the pattern obtained from the pure zirconia sample, a higher crystallized state is reached with a crystallite size of 9.7 nm, instead of 4.1 nm when manganese salt is added. Another expected effect of the manganese addition is the modification of the cell parameter of zirconia. In our case, we observe a small contraction of the zirconia cell size. Various cell size evolutions are reported in the literature. The zirconium substitution by manganese is sometimes related to a cell contraction. The Mn introduction in zirconia resulted in a decrease in the lattice parameter of zirconia, related to the substitution with smaller cations Mn 3+ (ionic radius 0.66 Å) of bigger cations Zr 4+ (ionic radius 0.84 Å) [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF][START_REF] Valigi | Manganese oxide-zirconium oxide solid solutions. An X-ray diffraction, Raman spectroscopy, thermogravimetry and magnetic study[END_REF]. But, from the published works [START_REF] Tao | Mn(III) species formed by the multi-copper oxidase MnxG investigated by electron paramagnetic resonance spectroscopy[END_REF][START_REF] Pattanaik | Enhancement of room temperature ferromagnetism in nanocrystalline Zr 1-x Mn x O 2 by the suppression of monoclinic structure of zirconia[END_REF] this decrease in the lattice parameter is observed in spite of the substitution of Zr 4+ ions by larger Mn 2+ ions (Mn 2+ , radius ionic 0.93 Å), because of the increasing proportion of oxygen vacancies in the lattice. Thus, the cell expansion is explained either by the Mn 2+ substitution or by the insertion of small Mn 4+ (Mn 4+ , radius ionic 0.53 Å) in the zirconia. In the work of Fanelli et al. [START_REF] Fanelli | MnO x /ZrO 2 gel-derived materials for hydrogen peroxide decomposition[END_REF], at 5% Mn, a cell expansion was evidenced which is no more observed for 30% Mn. This down-shift of the tetragonal t-ZrO 2 lines agrees with the substitution of Zr 4+ by larger, Mn 2+ ions. The lack of this shift in the 30% Mn was related to the competitive effect of smaller ions, mainly Mn 3+ . In our case, the Rietveld refinement, done by supposing the zirconium substitution with manganese (Table 3), shows that the parameters a and b of the tetragonal cell are slightly smaller than those of zirconia but the c parameter is much smaller.

The cell volume (66.45 +/-0.05. 10 6 pm 3 ) is slightly smaller (69.83. 10 6 pm 3 ). This analysis also reveals the existence of oxygen vacancies (95.4% of anionic sites are occupied with oxygen anions) which means that an important part of manganese ions substituted to zirconium are in the Mn II state (75% if there is a mixture Mn II-Mn IV and 50% if there is a mixture Mn II and Mn III).

The X-ray diffraction patterns after a thermal treatment at 600°C are presented in Fig. 1b.

In the pure zirconia sample, Zr-600, the monoclinic zirconia crystallization was evidenced, while the mixed MnZr-600 oxide was still under the tetragonal zirconia form. The addition of Mn improves the stabilization of the zirconia in the tetragonal crystalline phase [START_REF] Hernandez | Influence of the synthesis method on the nanostructure and reactivity of mesoporous Pt/Mn-WO x -ZrO 2 catalysts[END_REF], without any clear evolution between 450°C and 600°C.

Transmission electron microscopy and Analytical electron tomography of MnZr sample

The MnZr samples general morphology was firstly observed by classical TEM (Fig. 2). The oxide mesostructure of MnZr-450 is clearly observed, with parallel channels of diameter 5 nm (Fig. 2a). HR TEM images show that MnZr-450 sample is constituted by crystallites of 4-6 nm size of tetragonal zirconia with d 101 equal to 0.30 nm (Fig. 2b). Scarce crystallites of monoclinic zirconia can be also observed with d 001 = 0.50 nm.

The TEM images of the MnZr sample treated at 600°C (Fig. 2c) show no major evolution: the oxide mesostructure and the channels of 5 nm are still clearly visible and HR TEM image shows that the MnZr-600 sample is constituted by crystallites of 4-6 nm size having mainly tetragonal zirconia. The presence of few crystallites of monoclinic zirconia can be also detected (Fig. 2d).

Analytical Electron Tomography (AET) analysis has been done only on the MnZr-450 sample. Within the estimated spatial resolutions in the mean density reconstruction (calculated from the ZL tilt series) and in the chemical selective one (deduced from the Mn 2D maps) which are 3 and 7 nm respectively, the localization of manganese inside zirconia grain seems to be quite homogenous and any segregation effect cannot be observed at the grain level (Fig. 3). In Fig. 3a is showed the EELS spectrum recorded on the analyzed sample.

The position of the energy windows are showed by the black rectangles. In insert are showed the Pre-edge1, Pre-edge2 and Post-edge energy windows recorded at 0° tilt used to compute the Mn chemical map. The initial ZL projection at 0° tilt is shown (Fig 3b), followed by a typical slice extracted from the mean density volume (Fig. 3c), the corresponding slices through the Mn chemical volume (Fig. 3d) and through the "combined" volume obtained by superposing the morphological (in red) and chemical volumic information (in green) (Fig. 3e).

By analyzing this last image, we can unambiguously conclude on the homogeneity of the Mn distribution within the zirconia grain at the nanometer scale. Finally, Fig. 3f illustrates the 3D model of the analyzed grain with in insert the pores size distribution. The analysis of the ZL volume confirms the channel like structure of the pores for which the tortuosity, representing the ratio between the estimated length of the pores and the Euclidian length between the edges of the pores, is estimated at an average of 1.32±0.53. However, the analysis of the 3D data shows also that the mesoporous structure is not very well ordered at long range and can be characterized by a lot of defects regarding the spatial arrangement and orientation of the pores, in agreement with the structural disorder induced by the Mn insertion in the zirconia structure. From a quantitative point of view, this analysis allowed us also to calculate that the pores volume which represents 21% of the total volume of the grain.

Nitrogen adsorption/desorption isotherms

Textural characterizations were done using nitrogen adsorption/desorption isotherms (Table 4 and Fig. 4) on Zr-450 and MnZr-450. Specific surface area of MnZr-450 sample is 107 m².g -1 and 67 m².g -1 for Zr-450. The specific surface area of the mixed oxide is higher than the simple oxide [START_REF] Valigi | Manganese oxide-zirconium oxide solid solutions. An X-ray diffraction, Raman spectroscopy, thermogravimetry and magnetic study[END_REF]. The hysteresis loop indicated the presence of mesopores for both samples (Fig. 4). The pore size distribution calculated by Barret-Joyner-Halenda (BJH) from the adsorption branch showed that the mean pore diameter was around 5 nm for both samples and the mesoporous volume was equal to 0.10 cm 3 /g for MnZr-450 and 0.08 cm 3 /g for Zr-450. Mesoporosity is therefore improved by the presence of Mn species, even if the pores size remains unchanged. Porosities of the MnZr-450 sample, deduced from nitrogen adsorption experiments, is around 34%, higher than the porosity deduced from analytical electronic tomography. This difference is explained by the fact that nitrogen adsorption allows to measure both inter and intra-particle porosity whereas analytic electron tomography gives intra particle porosity and has, in addition, a quite limited spatial resolution in the present case which doesn't allow to consider the smaller pores in the quantification procedure.

Thermal analysis of MnZr sample

The thermal treatment temperatures (450 °C and 600 °C) have been chosen according to the TGA/SDTA analyses, which were done under air (Fig. 5). To understand the signification of the obtained TGA and SDTA curves, we have considered a reference MnO 2 sample. The MnO 2 TGA (Fig. 5a) curve revealed a weight loss at around 400°C, attributed to the reduction of MnO 2 into Mn 2 O 3 and another weight loss, at around 900°C, related to the subsequent endothermic reduction of Mn 2 O 3 to -Mn 3 O 4 . Two endothermic peaks can be observed at 430°C and 890°C (Fig. 5b), which correspond to the reduction of Mn IV into Mn III species and Mn III into Mn II species, as described previously.

The TGA curve of the Zr-450 sample (Fig. 5c), shows a 2% mass loss between 600°C and 1000°C. This small weight loss occurred during the crystallization of the monoclinic zirconia and corresponds to the small exothermic peak observed on SDTA signal at 600°C (Fig. 5d).

The TGA of the MnZr-450 oxide (Fig. 5e) shows a small and sharp weight loss (0.3 %) at 500°C followed by a second weight loss (1.4 %) between 600°C and 1000°C. The first weight loss is associated to the endothermic peak at 500°C, followed by a strong exothermic peak at 560°C on SDTA curve (Fig. 5f). Then, we can make the hypothesis that Mn IV inside zirconia is reduced into Mn III with some delay (compared to simple MnO 2 ), since the endothermic peak happens at 500°C instead of 430°C in a simple manganese oxide. The endothermic peak is followed immediately by an exothermic peak, which could be explained by some residual carbon oxidation. Around 800°C, the noted 1.4 % weight loss is associated to a small exothermic peak, which can be related to the crystallization of monoclinic zirconia. At the end of the experiment, ZrO 2 monoclinic and Mn 3 O 4 are clearly observed by X-ray diffraction on the TGA residue (not shown here).

X-ray Photoelectron Spectroscopy characterization

XPS spectrum of the MnZr-600 sample is shown in Fig. 6. The peaks at 641, 532, 436, 346 and 182 eV correspond to the binding energies of Mn 2p, O 1s, Zr 3s, Zr 3p and Zr 3d, respectively. The Mn2p 3/2 spectra of the MnZr-600 sample at different sputtering times are shown in Fig. 7. MnO, Mn 2 O 3 and MnO 2 possess respectively a Mn 2p 3/2 binding energy in the range of 640.6-641.7eV, 641.7-641.9eV and 641.9-642.6 eV [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF][START_REF] Wang | Recent advances in ordered meso/macroporous metal oxides for heterogeneous catalysis: a review[END_REF]. All oxidation states are observed as highlighted by the vertical lines at 641.9 eV for Mn 4+ /Mn 3+ and 640.6eV for Mn 2+ . At the surface of the sample mainly Mn 4+ /Mn 3+ oxidation states are present. Mn 2+ increases by depth, however the ratio is low considering that the specific "shake-up" satellite (~647 eV) is weak [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF]48].

The Zr 3d spectra for sample MnZr-600 at different sputtering times are shown in Fig. 8. The spectra are deconvoluted with only one specific doublet. The observed Zr 3d 5/2 binding energies are presented in Table 5, and from 181.8 to 182.1 eV correspond to Zr 4+ oxidation state [START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF]48].

The quantitative analysis of the XPS spectra at different sputtering time (or different etching depth) allows the evaluation of the Mn/Zr ratio. For MnZr-600 sample (Table 5) the zirconia is homogeneous.

Electron Paramagnetic Resonance

Acquisitions in perpendicular mode of the MnZr-450 sample (Fig. 9a) showed a main signal at g=2 characteristic of the Mn II presence in an isolated or aggregated form [START_REF] Ozawa | Structural and ESR characterization of heat-stable manganese -alumina lean NO removal catalyst[END_REF]. The small signal at g=4 may be due to the presence of Mn II species with different distortion or minor Mn IV species. The parallel mode (Fig. 9b) gave other information. A small signal at g=8 was due to the presence of some Mn III species [START_REF] Conlan | Photo-Catalytic oxidation of a di-nuclear manganese centre in an engineered bacterioferritin reaction centre[END_REF][START_REF] Pomykalska | Electrical conductivity of MnO x -Y 2 O 3 -ZrO 2 solid solutions[END_REF]. This analysis indicates the presence of Mn III. A strong signal at g=4 may be attributed to different Mn III species. From the EPR analysis, dissolved manganese species inside zirconia were in the possible forms Mn II, Mn III species and probably also in Mn IV.

Discussion

By using the EISA process, a solid solution of zirconium and manganese oxide was obtained characterized by a well-defined mesoporous texture. Within the spatial resolution estimated to few nanometers, the analytical electron tomography shows a homogeneous repartition of manganese species in the zirconium oxide grains and allows to estimate the intra-particle porosity to 21%. A higher porosity (34%) is measured by nitrogen adsorption which is due to the slight difference in the size of the pores the two analyses may take into account and also to the fact that AET provides only intra-particle porosity. The tortuosity measured by AET is quite low and this is coherent with the channel-like morphology observed by TEM, but the texture is characterized by the presence of lots of defects in the arrangement and orientations of the pores (Fig. 2a &2c). EPR shows that Mn III and Mn II were present and probably Mn IV. XPS, in agreement with EPR, shows that manganese is present under II, III and IV states. The states III and IV, observed before etching, are present at / or near the crystallites surface. XPS reveals that manganese is mostly under the Mn II inside the crystallites, since this state is very clearly observed state after the etching process.

From XRD data, the tetragonal form of zirconia was stabilized by manganese addition. Indeed, the consequence of the manganese presence is a delay in the growth of the crystallites with increasing temperature. Then, at 600°C, the crystallite size is maintained at a value of 5 nm, corresponding to the thermodynamic field of existence of the quadratic phase [START_REF] Sharma | Size driven thermodynamic crossovers in phase stability in zirconia and hafnia[END_REF]. The Rietveld analysis also confirms that the Mn II state is preponderant. Then, in our case, XPS and Rietveld analyses support Nandy et al. [START_REF] Nandy | Microstructure correlated electrical conductivity of Manganese alloyed nanocrystalline cubic zirconia synthesized by mechanical alloying[END_REF] and Valigi et al. [START_REF] Valigi | Manganese oxide-zirconium oxide solid solutions. An X-ray diffraction, Raman spectroscopy, thermogravimetry and magnetic study[END_REF] hypothesis that Zr IV is substituted by Mn in II oxidation state which induces the presence of oxygen vacancies to equilibrate the charge. The small change of the cell size in the solid solution MnZr-450 oxide is due to competitive effects of the Mn II substitution on one hand, and the presence of oxygen vacancies on the other hand [START_REF] Valigi | Manganese oxide-zirconium oxide solid solutions. An X-ray diffraction, Raman spectroscopy, thermogravimetry and magnetic study[END_REF]. From the thermal analysis, the experimental weight loss (0.3 %) is coherent with an equal repartition between Mn II, Mn III and Mn IV. Indeed, the reduction of Mn IV (30% of the complete Mn) into Mn III induces a calculated weight loss limited to 0.28 wt%.

The crystallization of monoclinic zirconia is delayed from 600°C to 800°C by the presence of manganese. At 1000°C, XRD analysis reveals the presence of haussmanite Mn 3 O 4 and monoclinic zirconia. Since the Mn II/Mn III ratio is higher to the one necessary for the Mn 3 O 4 formation, a significant part of Mn II may still be inside the monoclinic zirconia. Such substitution inside monoclinic zirconia has already been observed [START_REF] Pattanaik | Enhancement of room temperature ferromagnetism in nanocrystalline Zr 1-x Mn x O 2 by the suppression of monoclinic structure of zirconia[END_REF]. All these results show that manganese ions are uniformly incorporated in the frameworks of the mesoporous zirconia.

Conclusion

A reaction involving manganese and zirconium salts using a block copolymer in an EISA process has been used to synthesize a mixed mesoporous manganese zirconium oxide. This mixed mesoporous oxide was studied by XRD, TEM, electron tomography, nitrogen adsorption, electron paramagnetic resonance, XPS and thermal analysis. A channel-like mesoporous oxide has been obtained with an intraparticular porosity of 21%, pores size of 5 nm and a tortuosity of 1.32. Manganese oxide was not observed as separate phase and was forming a solid solution with zirconia, which decreased zirconia crystallinity. Analytical electron tomography has shown that a homogeneous repartition at nanometric scale of the manganese was found in zirconia. EPR, TGA and XPS show that Mn II, III and IV are present in such solid solution. More specifically, XPS reveals that Mn II is mostly present in the core of crystallites while Mn III and IV are on the surface. All these results show that the partial substitution of zirconium with manganese II inside zirconia allows to maintain the tetragonal form and from this fact the textural characteristics up to 600°C. This stability opens the way towards catalytic applications of such a solid solution.

Tables Table 1 Summary

Figures Caption

Fig.

1 .

 1 Fig. 1. X-ray diffraction patterns of a: Zr-450 (black) and MnZr-450 (blue), b: Zr-600 (black) (blue

Fig. 2 .

 2 Fig. 2. TEM (a) and HRTEM (b) images of MnZr-450 sample. TEM (c) and HRTEM (d) images of MnZr -

Fig. 3 .

 3 Fig. 3. Analytical electron tomography: a) Recorded EELS spectrum on the analysed sample showing

Fig. 4 .

 4 Fig. 4. N 2 adsorption-desorption isotherms and pore size distribution (BJH) curves (inserts) for a:

Fig. 5 .

 5 Fig. 5. TGA & SDTA curves of the MnO 2 (a,b), (c,d) and MnZr-450 (e,f) oxides.

Fig. 6 .

 6 Fig. 6. XPS spectrum of the MnZr-600 sample.

Fig. 7 .

 7 Fig.7. XPS spectra of Mn 2p at different sputtering times (up to 1.78 nm depth), corresponding to

Fig. 8 .

 8 Fig. 8. XPS spectra of Zr 3d at different sputtering times (up to 1.78 nm depth), corresponding to

Fig. 9 .

 9 Fig. 9. EPR spectra (15 K) of the MnZr-450 sample, a: perpendicular mode and b: parallel

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1.

Fig. 3 . 25 Fig. 4 .

 3254 Fig. 3. Analytical electron tomography: a) Recorded EELS spectrum on the analysed sample showing the position of the energy windows. In insert are showed the energy filtered projections recorded at 0° from which the Mn chemical map was computed using the three windows method. b) TEM projection of the MnZr-450 sample. c) typical cross sections through the reconstructed volume obtained from the zero loss tilt series, d) corresponding chemical Mn slice, at the same depth and orientation, e) superposition of the chemical selective slice (in green) and the mass selective one (in red) and f) 3D mode of the analysed grain with, in insert, the pore size distribution calculated from the ZL volume.

Fig. 5 .Fig. 6 .

 56 Fig. 5.

Fig. 7 & 8 Fig. 7 .

 787 Fig. 7 & 8

Fig. 8 .

 8 Fig. 8. XPS spectra of Zr 3d at different sputtering times (up to 1.78 nm depth), corresponding to MnZr-600 sample.

29 Fig. 9

 299 Fig.9

Table 2

 2 of MnZr oxide materials prepared Summary of the Zr and MnZr oxide crystallite size (calculated from the (101) position and width).

	Sample	Molar ratio of Mn: Zr
	Zr-450	0: 1
	Zr-600	0: 1
	MnZr-450	0.14: 1
	MnZr-600	0.14: 1

Table 3

 3 Summary of the Rietveld refinement of the MnZr-450 sample

*OccupZr and OccupMn: cationic site occupancy rate for Zr and Mn cations. *OccupO: anionic site occupancy rate for O anions.

Table 4

 4 Summary of the textural properties of the MnZr oxide from nitrogen adsorption/desorption measurements.

	Sample	Specific surface area (m 2 /g)	Porous volume (cm 3 /g)	Mesoporous volume (cm 3 /g)	pore size (nm)
	MnZr-450	107	0.13	0.10	5.0
	Zr-450	67	0.10	0.08	4.8

Table 5 :

 5 Atomic ratios of elements and binding energies (eV) observed during depth profile analysis at different sputtering times (Sp. Time) with their corresponding oxidation state according to[START_REF] Bulavchenko | Reduction of mixed Mn-Zr oxides: in situ XPS and XRD studies[END_REF], for sample MnZr-600.

	Sample	2Theta	width (°)	Crystallite size (nm)
	Zr-450	30.34		0.90		9.7
	MnZr-450	30.42		2.11		4.1
	Zr-600	30.23		0.50		17.5
	MnZr-600	30.41		1.70		5.0
		a	b		c		Occup	Occup	Occup	O/(Zr+Mn) Bragg	RF-
	Sample	(nm)	(nm)		(nm)		Zr*	Mn*	O*		R-	Factor
							(%)	(%)	(%)		Factor
	MnZr-450	+/0.0001 0.3632	+/0.0001 0.3632	+/-0.0002 0.5041	87.7	12.3	95.4	1.908	2.09	1.26
		Sp Time	Depth	Mn/Zr Zr3d5/2 (eV)		Mn 2p3/2 (eV)
		(min)	(nm)		ZrO2		MnO	Mn2O3/MnO2
		0		0.00	0.16	182.0		640.6	642
		10		0.11	0.16	182.1		640.6	642
		40		0.45	0.16	182.1		640.6	642
		100	1.11	0.15	181.8		640.6	641.8
		160	1.78	0.15	181.8		640.6	641.9

Acknowledgments

This work has been supported at the Laboratory of Multimaterials and Interfaces -UMR 5615. This work was financially supported by the Région Rhône-Alpes. The authors gratefully acknowledge Florian Molton and Carole Duboc, GHMFL-LCMI, Université Joseph Fourier Grenoble 1, for their help during the EPR experiments and the national EPR network TGE RENARD, FR 3443 -CNRS. The authors gratefully acknowledge the national EPR federation RENARD (IT CNRS 3443) and the Département de Chimie Moléculaire, Université Grenoble Alpes. LB and LR thank to the METSA (FR CNRS 3507) network for the financial support of the AET experiments.

Highlights

Block copolymer self-assembly process is used to form a mixed manganese and zirconium oxide A robust mesoporous channel-like texture of MnZr oxide is obtained Analytical Electron tomography proves that manganese is homogeneously distributed Mn II substitutes Zr IV Graphical abstract


domain: spi phys

text:
Baptiste Pialot

Jérémie Gachelin

Mickael Tanter

Jean Provost

Olivier Couture

Flow rate and low hematocrit measurements for in-vitro blood processing with Doppler ultrasound

In-vitro techniques for processing of flowing blood and its components have recently emerged from microfluidics. The blood flow rate and hematocrit are two keys parameters to monitor for guaranteeing the reliability of these techniques. But, there is a lack of monitoring methods adapted to low flow rates and small tubing. In this study, we exploit minimization approaches of continuous Doppler measurements to survey blood flow rate. Combined with a packing factor model, we also estimate hematocrit from the Doppler spectrum. The presented method is implemented with a CW Doppler probe mounted on a 3D-printed support. Accuracy of flow rate was measured in a range from 0.5mL/min to 1.5mL/min. For each of 4 different blood bags, hematocrit in a range under 8% was estimated from Doppler spectrum using a packing factor model derived from the other three bags. Flow rate estimation shows a mean measurement error under 3% for a measurement time of 2s. The mean error is still under 5% for a measurement time of 0.5s. Hematocrit estimation for the 4 blood bags shows errors of 1.4, 1.1, 0.67 and 0.70 %Hct for a measurement time of 5s. The versatility and simplicity of the method make it highly valuable for in-vitro blood processing, in particular for low hematocrit blood fractionation techniques derived from microfluidics, as it can be performed through sterile tubing.

I -INTRODUCTION

. A significant part of these technologies operate with diluted blood, below physiological hematocrit values. For being fully operational, employed devices have to be maintained at a constant flow rate with a high accuracy. Also, measuring the hematocrit in such devices without extracting additional blood samples is of a great interest for monitoring their effectiveness and stability. Currently, most flow meters designed for in-vitro blood circuits techniques such as dialysis or heart-lung machines are based on ultrasonic [4], [5] and electromagnetic [6], [7] methods. Optical methods have also been developed [8], [9]. However, the slow flow rates and narrow tubing used in microfluidic-derived blood processing devices are outside the scope of most available commercial flow-meters. Among others, we can cite high-throughput acoustophoresis blood fractionation chips, with flow rates that from

a few hundred 𝜇L/min to a few mL/min, are now developed aiming at a clinical use [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF]- [START_REF] Dutra | A Novel Macroscale Acoustic Device for Blood Filtration[END_REF]. Moreover, in acoustophoresis and in blood fractionation in general, the hematocrit is a direct indicator of the blood fractionation efficiency.

In comparison to optic and electromagnetic methods, ultrasonic methods have the combined advantages of being biocompatible, non-invasive and cheap while dealing with a relatively simple physical framework. More importantly, ultrasound waves preserve their coherence while propagating through tissue and can thus measure flow in optically opaque fluids such as blood. For in-vitro blood flow, they consist mainly in two distinct approaches: the transit-time method and the Doppler method. Whereas the transit-time method is more frequently used in-vitro, Doppler ultrasound is widely used in vivo [START_REF] Satomura | Ultrasonic Doppler Method for the Inspection of Cardiac Functions[END_REF], [START_REF] Evans | Doppler Ultrasound: Physics, Instrumentation and Signal Processing[END_REF], which has led to a wide gamut of Doppler systems potentially usable in-vitro [START_REF] Hartley | An ultrasonic pulsed Doppler system for measuring blood flow in small vessels[END_REF]- [START_REF] Huang | Design and implementation of a smartphonebased portable ultrasound pulsed-wave doppler device for blood flow measurement[END_REF]. However, most of these systems have not been optimized for an in-vitro environment or have been limited to relatively large flow rates [START_REF] Dantas | Ultrasonic pulsed Doppler blood flowmeter for use in extracorporeal circulation[END_REF], [START_REF] Christmann | Realtime DSP system for ultrasonic blood flow measurement[END_REF].

From a physical point of view, the ultrasonic Doppler power spectral density (PSD) of blood flow is a continuous distribution of frequencies arising from the Doppler effect due to a continuous distribution of velocities [START_REF] Brody | Theoretical analysis of the CW Doppler ultrasonic flowmeter[END_REF], [START_REF] Angelsen | A Theoretical Study of the Scattering of Ultrasound from Blood[END_REF]. For the flow rate of a stationary, fully developed and laminar blood flow in a cylindrical tube, the corresponding mean velocity is half of the peak velocity. With the constant cross-section area of the tube, estimating the blood flow rate is thus equivalent to finding the maximum frequency of the Doppler PSD. However, this measurement is not straightforward due to intrinsic spectral broadening [START_REF] Jones | Fundamental sources of error and spectral broadening in Doppler ultrasound signals[END_REF]. As this phenomenon is the combination of several effects coming from different sources (finite dimension of the transducer, focusing, transit-time of the scatterers), its understanding for proper peak velocity estimation remains complex [START_REF] Guidi | Intrinsic spectral broadening (ISB) in ultrasound Doppler as a combination of transit time and local geometrical broadening[END_REF]. Thereby, multiple estimators that are robust to spectral broadening for the Doppler maximum frequency have been proposed [START_REF] Tortoli | Improved blood velocity estimation using the maximum Doppler frequency[END_REF]- [START_REF] Kathpalia | Adaptive Spectral Envelope Estimation for Doppler Ultrasound[END_REF]. Recently, Vilkomerson et al. proposed a new theoretical model for the Doppler PSD that provides a better understanding of the position of the maximum Doppler frequency in the Doppler spectrum [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF]. This model has been shown to accurately predict the maximum Doppler frequency of a fully insonified laminar blood flow. Moreover, ultrasound backscattered intensity is correlated to blood hematocrit [START_REF] Shung | Scattering of Ultrasound by Blood[END_REF]. Hence, besides the flow-rate measurement, Doppler ultrasound is a potential simple and lowcost alternative to existing methods such as impedance spectroscopy [START_REF] Trebbels | Hematocrit Measurement -A high precision on-line measurement system based on impedance spectroscopy for use in hemodialysis machines[END_REF] or optical methods [START_REF] Ekuni | Non-invasive measurement method of blood hematocrit for management of extracorporeal circulation[END_REF], [START_REF] Zeidan | In vitro hematocrit measurement using spectrally encoded flow cytometry[END_REF] for directly measuring hematocrit without opening the blood circulation. To our knowledge, there are relatively few studies focused on ultrasound-based hematocrit measurement. Secomski et al. [START_REF] Secomski | Non-invasive measurement of blood hematocrit in artery[END_REF] showed that ultrasonic attenuation in an artery is a function of hematocrit. Franceschini and Guillermin [START_REF] Franceschini | Experimental assessment of four ultrasound scattering models for characterizing concentrated tissue-mimicking phantoms[END_REF] used an inverse formulation of a backscattering coefficient (BSC) model for determining the concentration of a tissue phantom. In the Rayleigh scattering regime, the BSC of blood follows a non-linear relationship with hematocrit consisting in a quasi-linear slope that peaks before slowly decreasing [33], [34]. Notably, it was shown that the BSC of non-aggregated porcine and bovine blood peak around 13% hematocrit at a frequency of 7.5 MHz [35]. For modeling this non-linearity, a parameter called the packing factor has been extensively used [35]- [37]. In these studies, the packing factor was used to fit theoretical curves to experimentally measured BSC. Also, the packing factor has been used for modeling the Doppler signal coming from blood [38], [39]. A simplified expression of the Doppler PSD using a packing factor has been thus developed by Bascom and Cobbold [40]. The main limitation of such a model is to not take into account the shear-rate dependent aggregation tendency of human red blood cells. More precisely, it is known that red blood cells aggregation decreases with increasing shear rate [41], [42]. Thus, more elaborate models have been developed since for characterizing the scattering of ultrasound by human blood [43].

In this study, we present a novel Doppler ultrasound method for non-invasively measuring in-vitro blood flow rate and hematocrit. The method has been designed for the low flow rate and hematocrit ranges encountered in blood processing devices derived from microfluidics. For reaching a high accuracy for flow rate measurement, we use the theoretical model from Vilkomerson et al. [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF] in order to reduce the Doppler measure to a straightforward minimization problem. The hematocrit measurement is also performed from Doppler signals using a simplified packing factor model derived from the study of Bascom and Cobbold [40] that we show to be reproducible for different blood samples. The method is applied on a simple and low-cost dual-probe continuous Doppler apparatus with blood flowing in a sub-millimeter silicon tube. We first detail the theory involved in our measurement method in section II. After describing the experimental system and acquisition (section III), we demonstrate in section IV its accuracy for the measurement of flow rates rate from 0.5 mL/min to 1.5mL/min on one blood bag. This flow rate range is based on the one used for the acoustophoretic blood fractionation device presented by Bohec et al. [START_REF] Bohec | Acoustophoretic purification of platelets: Feasibility and impact on platelet activation and function[END_REF]. Furthermore, we show on four different blood bags the system capacity to measure hematocrit. For each blood bag, hematocrit measurement is performed in a range under 8%, using a parameterization of the packing factor model estimated from the three other bags. This hematocrit range also corresponds to the one used by Bohec et al. [START_REF] Bohec | Acoustophoretic purification of platelets: Feasibility and impact on platelet activation and function[END_REF].

II -Theory

We consider a laminar and steady blood flow in a medical tube with a constant radius R. The blood flow rate is imposed at the entrance of the tube. The laminar blood flow exhibits an ideal parabolic velocity profile characterized by the following equation:

𝑣(𝑟) = 𝑣 𝑚𝑎𝑥 (1 - 𝑟 2 𝑅 2 ) (1)
Where 𝑣(𝑟) is the velocity corresponding to the radial position 𝑟 in the tube and 𝑣 𝑚𝑎𝑥 is the peak velocity of the parabola.

The blood flow is assumed to be fully and uniformly insonified with a Doppler dualprobe consisting of an emitting and a receiving transducer. It is assumed that the flow is fullydeveloped within the ultrasound beam. The peak velocity 𝑣 𝑚𝑎𝑥 is related by the Doppler equation to the peak Doppler frequency 𝑓 𝑚𝑎𝑥 of the Doppler PSD. If the internal angle between the transducers is neglected, the Doppler equation for 𝑣 𝑚𝑎𝑥 and 𝑓 𝑚𝑎𝑥 is:

𝑣 𝑚𝑎𝑥 = 1 2 𝑐 𝑓 𝑚𝑎𝑥 𝑓 𝑒 𝑐𝑜𝑠𝜃 (2)
Where 𝑐 is the speed of sound in blood, 𝑓 𝑒 is the frequency of the emitting transducer and 𝜃 is the Doppler angle between the probe and the tube.

The flow rate Q is related to 𝑣 𝑚𝑎𝑥 by 𝑄 = 1 2 𝜋𝑅 2 𝑣 𝑚𝑎𝑥 . Substituting in equation ( 2):

𝑄 = 𝜋𝑅 2 𝑐 4 𝑓 𝑚𝑎𝑥 𝑓 𝑒 𝑐𝑜𝑠𝜃 (3)
We will now use the theoretical model for the Doppler PSD developed by Vilkomerson et al. [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF]. This model consists of dividing the velocity profile into M concentric shells having M velocities. The shells are then separated into two groups based on their velocity values. The "slow" group contains the shells with a velocity such that corresponding red blood cells cannot cross the whole ultrasound beam during the acquisition of the Doppler signal. The "fast" group contains all the others shells. From this dichotomy the model has the following general expression:

𝐷𝑃𝑆𝐷 = ∑ 𝑆𝑆 𝑚 + ∑ 𝑆𝐹 𝑚 (4) 𝑀-1 𝑚=𝑚 𝑡 𝑚 𝑡 -1 𝑚=0
Where 𝐷𝑃𝑆𝐷 is the theoretical Doppler PSD, 𝑆𝑆 𝑚 is the spectral contribution of the slow shells, 𝑆𝐹 𝑚 is the spectral contribution of the fast shells and 𝑚 𝑡 is the index of the first fast shell. The detailed expressions for 𝑆𝑆 𝑚 and 𝑆𝐹 𝑚 are:

𝑆𝑆 𝑚 =
Here 𝜌 𝑚 is the red blood cells density in the m-th shell, 𝑣 𝑚 is their velocity and 𝑓 𝑚 is the associated Doppler frequency calculated with the Doppler equation. A is the width of the emitter aperture, 𝑇 0 is the acquisition time of the Doppler signal and N is the number of samples on which the Doppler PSD is estimated. The spectral contribution of the 𝑆𝐹 𝑚 shells corresponds to red blood cells staying in the ultrasound beam during the time 𝑡𝑡 𝑚 = 𝐴 𝑣 𝑚 .

These red blood cells are thus producing a temporal pulse of 𝑤 𝑚 = 𝑡𝑡 𝑚 . 𝑓 𝑠 samples with 𝑓 𝑠 being the sampling frequency. Finally, P is a function that expresses the spectral contribution for one frequency bin and for which the detailed expression can be found in the original paper.

We suppose now that all measurements parameters are known and fixed except for the flow rate. In this case, the model shows for an ideal laminar blood flow that a single parabolic velocity profile will correspond to a single theoretical Doppler PSD and conversely, that a given peak velocity will correspond to a single Doppler PSD. (Note that this statement is also true for the corresponding maximum Doppler frequency calculated with (2)). Moreover, per equation ( 3), a single flow rate will correspond to a single theoretical Doppler PSD and conversely. It is thus possible to estimate blood flow rate from a measured Doppler PSD by finding the flow rate value Q that minimizes the Euclidean distance (L² norm) with the theoretical model following:

𝑎𝑟𝑔𝑚𝑖𝑛 ∥ 𝐷𝑃𝑆𝐷 𝑚 -𝐷𝑃𝑆𝐷 𝑡ℎ (𝑄) ∥ (7)
Where 𝐷𝑃𝑆𝐷 𝑚 is the measured Doppler PSD and 𝐷𝑃𝑆𝐷 𝑡ℎ (𝑄) is the Vilkomerson et al.

theoretical model [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF] for which the maximum Doppler frequency corresponds to the flow rate Q.

In the next part, we are going to further model the amplitude of the Doppler PSD. The blood flow is still assumed to be steady, fully and uniformly insonified, laminar and fully developed.

With the spectral broadening being neglected and the packing factor considered to be spatially invariant, the model given by Bascom and Cobbold [40] for the Doppler PSD is:

𝐷𝑃𝑆𝐷(𝑓(𝑟)) = 2𝜋Ω𝑙𝜎 𝑟𝑏𝑐 𝐻 𝑉 𝑟𝑏𝑐 𝑊(𝐻)𝑟 𝑑𝑟 𝑑𝑣(𝑟) (8)
Here, 𝑓(𝑟) is the Doppler frequency corresponding to the radial velocity 𝑣(𝑟) through the Doppler equation. Ω is the solid angle between the probe and the blood sample, 𝑙 is the width of the ultrasound beam, 𝜎 𝑟𝑏𝑐 is the red blood cells backscattering cross-section and 𝑉 𝑟𝑏𝑐 is the red blood cells mean volume. 𝐻 is the hematocrit and 𝑊(𝐻) is the packing factor for which the expression is:

𝑊(𝐻) = (1 -𝐻) 𝑚+1 [1 + 𝐻(𝑚 -1)] 𝑚-1 (9)
With m being a parameter whose values are 1,2 or 3 if red blood cells are modeled respectively by a slab, a circle or a sphere. Notably, Yuan and Shung [35] demonstrated that a packing factor calculated with m = 3 corresponds to the BSC of disaggregated red blood cells in a laminar flow. When turbulences are present in the flow, the value of m decreases, corresponding to an increase of the BSC [33], [35]. This relation appears in equation [START_REF] Nilsson | Evaluation of a Laser Doppler Flowmeter for Measurement of Tissue Blood Flow[END_REF] through the quantity 𝜎 𝑟𝑏𝑐 𝐻 𝑉 𝑟𝑏𝑐 𝑊(𝐻), which is the expression of the BSC given by Mo and Cobbold [38]. Also, Bascom and Cobbold [36] proposed that m could take non-integer value, possibly superior to 3 [36]. For a quantitative approach it is more convenient to take the frequency integral of the Doppler PSD rather than its amplitude. Because the blood flow is steady, the integral of the quantity 𝑟 𝑑𝑟 𝑑𝑣(𝑟) is constant. We can then write from (8) the following approximated expression for the frequency integral 𝐼 𝐷𝑃𝑆𝐷 : 𝐼 𝐷𝑃𝑆𝐷 = 𝑎𝐻𝑊(𝐻) + 𝑏 [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF] The constant 𝑎 refers to parameters in [START_REF] Nilsson | Evaluation of a Laser Doppler Flowmeter for Measurement of Tissue Blood Flow[END_REF] whereas the constant 𝑏 allows to take into account the noise floor of the measurement set-up.

It is clear from equation [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF] that the measurement of the frequency integral of the Doppler PSD could give access to the hematocrit of the blood flow. Furthermore, equation [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF] indicates that the Doppler PSD frequency integral 𝐼 𝐷𝑃𝑆𝐷 is following the same non-monotonic behavior than the blood backscattering as a function of the hematocrit 𝐻. Thus, as long as the hematocrit range is restrained to any monotonic region of the curve, measuring the hematocrit from the Doppler PSD frequency integral can be done by finding the hematocrit 𝐻 that verifies:

argmin ||𝐼 𝐷𝑃𝑆𝐷 -𝑎𝐻 (1 -𝐻) 𝑚+1 [1 + 𝐻(𝑚 -1)] 𝑚-1 -𝑏 || (11)
Where parameters a, 𝑏 and m are estimated through measurements.

III -Materials and methods

a) Ethics statement

All blood donors were healthy volunteers who gave written consent for this research study in accordance with the Declaration of Helsinki. Legal and ethical authorization for research use of collected blood was obtained through a national convention between the company AEnitis Technologies and the French Blood Institute (EFS) (convention number 15/EFS/028).

b) Experimental set-up

As described in Fig. 1, an 8-MHz continuous-wave (CW) Doppler pencil dual-element probe (diameter 1cm, quality factor: 1.10 6 ) was used for ultrasound emission and reception. The blood flow was imposed by a precisely calibrated syringe pump (KD Scientific Inc., Holliston, USA) into a 20-cm-long silicon medical tube (Freudenberg, Kaiserslautern, Germany, internal diameter: 760 micrometers). The silicon tube and the CW Doppler probe were mounted into a 3D-printed support in a way that the Doppler angle between the probe and the tube was 40°. The probe insonified all the cross-section of the tube. The support, the probe and the tube were then placed in a small water tank for ultrasound coupling. The probe was supplied in voltage by a function generator (Rigol, Beijing, China). RF signals were amplified with a 40dB analog amplifier and then acquired using a USB acquisition card (HS6, Tiepie engineering, Sneek, Netherlands) interfaced with Matlab (Mathworks, Natick, MA, USA). The postprocessing of RF signals was also done using Matlab.

c) Flow rate measurement and signal processing

For the flow measurement validation, ten acquisitions of 10s have been performed for a flow rate ranging from 0.5 mL/min to 1.5 mL/min with a step of 0.1 mL/min using blood samples coming from a single donor blood bag and diluted in its own plasma to a hematocrit of 8%. The syringe was reloaded with blood and carefully agitated manually each 50s.

The function generator was set to deliver a 10-V peak-to-peak voltage resulting in a 10-kPa peak-to-peak acoustic pressure. The sampling frequency was fixed at 20 MHz and acquisitions have been performed on a 2-V peak-to-peak, 8-bit voltage range, leading to a measurement voltage resolution of 2 2 8 𝑉 = 0.0078 Volts. It was checked that the received RF signals presented no saturation effect for this voltage range.

The following post-processing was performed for several signal acquisition durations ranging between 0.5s and 10s. The acquisition time was set by truncating the 10-s-long signals to the desired value.

First, signals were filtered with a Butterworth filter between 8MHz-5000Hz and 8MHz+5000Hz. Signals were then demodulated using quadrature demodulation and down sampled with a rate of 2000. A clutter filter was then applied to each signal to discard unwanted signals for frequencies below 80Hz.

Each signal was next segmented and the Doppler PSD was estimated for each segment. The number of segments was chosen such that each segment had a temporal duration corresponding to a frequency resolution of 1 Hz, except in the case of 0.5-s-long signals that had a 2-Hz resolution. For each signal the Doppler PSD was computed as the mean of the Doppler PSD of all segments. A control acquisition of PSD was performed with distilled water using the previous procedure. It allowed to verify that no unwanted frequency components were present beyond the 80 Hz threshold.

The fitting between each Doppler PSD and its corresponding theoretical model was performed using the Nelder-Mead optimization algorithm with equation ( 7) as a cost function. Initial estimates were chosen at random between 0.4 mL/min and 1.6 mL/min and the number of iterations was limited to 15. The cost function was computed with the theoretical model being expressed as a function of the flow rate using equation [START_REF] Antfolk | Continuous flow microfluidic separation and processing of rare cells and bioparticles found in blood -A review[END_REF] with 𝑓 𝑚𝑎𝑥 corresponding to the highest value of the 𝑓 𝑚 set in the Vilkomerson et al. model. At the end of the 15 iterations, the algorithm was launched again with the estimated flow rate as a new starting point and so on until the flow rate estimation became stable. Measurement errors relative to the flow rate value given by the syringe pump were finally calculated for each PSD.

For an optimal accuracy, the pipe radius R and the beam aperture width A have also been added as cost function parameters in a first calibration step in addition to the flow rate. This calibration was performed with Doppler spectra acquired for the hematocrit estimation and described in the next paragraph. For each Doppler spectrum, the radius and the beam aperture width were thus jointly estimated by the optimization algorithm. The mean values of the radius and the beam aperture width for all the acquisitions were then retained as reference values.

d) Hematocrit measurement and signal processing

For the hematocrit measurement validation, measurements were performed on 4 donor blood bags provided by the EFS at a 1 mL/min flow rate. Each bag was diluted in its own plasma to obtain 10 values of hematocrit (1% to 18%) for each bag. Hematocrit was measured for each of the 40 blood samples using an ABX Pentra 80 (HORIBA, Kyoto, Japan). Acquisition time was reduced to 5s and 30 acquisitions were performed for each blood sample. The syringe was reloaded with blood and carefully agitated manually each 50s. The set-up was entirely dismounted and reassembled between each blood bag. Peak-to-peak acoustic pressure and acquisition parameters were the same than for flow measurement. Signal filtering and demodulation were also strictly equivalent and were performed for acquisition durations ranging between 1s and 5s.

Each signal was segmented into five segments, and the frequency integral was computed for each of them. Their median was used as the frequency integral of the Doppler PSD in the optimization process to avoid over-estimations of the integrals arising from the strong backscattered echo of rare, yet unwanted air bubbles. In addition, the flow rate for each Doppler PSD was estimated using the procedure detailed in b) and measurements errors relative to the syringe pump were calculated.

In order to assess the general response to our system to hematocrit changes, the parameters of the packing factor model were estimated for the 4 blood bags. For that, equation [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF] was fitted to the Doppler PSD frequency integrals of the four blood bags using an iterative reweighted least square algorithm with a and m as fitting parameters. For estimating the value of b, ten 5-s acquisitions of the Doppler frequency integral were made in distilled water. The mean value of all acquisitions was then retained as the value for b. Values for a and m for the four blood bags were then extracted from the fit. Measurement errors on the values of a and m were computed as the standard deviation of their estimation. The convergence of the estimation of a and m was evaluated by performing the fit 10 times with starting points ranging randomly between 2 and 4 for m and between 1.0.10 -4 and 2.2.10 -3 for a.

For performing the hematocrit measurement, the following procedure has been performed for each blood bag. The procedure was limited to hematocrits <8%. First, a fit between the Doppler PSD frequency integrals of the other 3 blood bags and equation [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF] was performed with m and a as fitting parameters such as in the previous paragraph. The hematocrit was then estimated for the Doppler PSD frequency integrals of the measuring blood bag through equation [START_REF] Bohec | Acoustophoretic purification of platelets: Feasibility and impact on platelet activation and function[END_REF] with the values for m and a extracted from the fit and using the same setting of the Nelder-Mead algorithm than for the flow rate. The value for b was kept from distilled water measurements. Starting points for the algorithm randomly ranged from a hematocrit of 0.5% to 8.5%. Measurement errors relative to hematocrits value given by the ABX Pentra 80 were finally calculated for each Doppler PSD frequency integrals.

IV -Results

An example of a measured Doppler PSD fitted with its Vilkomerson et al. theoretical model [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF] is shown in Fig. 2. The optimal values of the estimated beam aperture width and pipe radius averaged over all Doppler PSD were respectively 4±0.3 mm and 396±3 um (manufacturer-supplied value for the inner radius of the pipe was 380 um). The measured flow rate for Fig. 2 was estimated to be 0.81±0.01 mL/min, while the flow rate imposed by the syringe-pump was 0.8±0.04 mL/min. Fig. 3 reports the evolution of the flow rate measurement error averaged for all acquisitions and all flow rates as a function of the acquisition time. The error is quickly converging to a value under 3% for an acquisition time of 2 s and slowly decays to an error of approximately 2.6% for an acquisition time of 10s. Fig. 4 illustrates measured Doppler PSD frequency integrals for diluted blood samples with the fit of equation [START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF]. The value for b estimated from acquisitions in distilled water is 4.1.10 -6 V². Standard deviation for b is ±4.10 -7 . Values for a and m are respectively 1.10±0.01.10 -3 V² and 3.3±0.2. The fitted curve reaches its maximum value for a hematocrit of approximately 12%. The root mean square error of the fit is 6.0.10 -6 V². Standard deviations between each 10 iterations of the fitting algorithm with different starting points are respectively 2.10 -7 and 3.10 -4 for a and m.

Table . I resume the parameters used for flow rate measurement as well as the values of a, b and m used for the fit in Fig. 4.

The mean error of the hematocrit measurement from Doppler PSD frequency integrals as a function of acquisition time is displayed in Fig. 5 for each blood bags. The mean error decreases with acquisition time and is under 1.2% hematocrit for an acquisition time of 5s for the 4 blood bags. Fig. 6 shows the mean measured flow rate compared to the flow rate imposed by the syringe pump for an acquisition time of 2s. A linear correlation in the range 0.5 to 1.5 mL/min was obtained with a R² coefficient of 0.998 and a root mean square error of 0.013 mL/min. The flow rate measurement error is displayed in Fig. 7 for the same acquisition time of 2s. The mean value for the measurement error is 2.8%. The measure presents no bias and is less dispersed for flow rates under 0.8 mL/min. Fig. 8 gives the flow rate measurement error averaged for all 5-s-long acquisitions performed for the hematocrit estimation. The error is above 5% for hematocrits between 1% and 2% and stabilizes around 2% for hematocrits above 4%. Fig. 9 reports the hematocrit measurement from the Doppler PSD frequency integrals for an acquisition time of 5s. There is a linear correlation between the mean measured hematocrit and the hematocrit estimated by the ABX Pentra 80 for each blood bag. Bag 2, 3 and 4 exhibit R² coefficients close to 1 and root mean square error under 1% hematocrit. Bag 1 presents an outlier point for hematocrits at 8% giving a R² coefficient of 0.62 with a root mean square error of 1.1% hematocrit. The hematocrit mean measurement error for each blood bag at an acquisition time of 5s is given in Fig. 10. The errors averaged on all hematocrit values for blood bag 1, 2, 3 and 4 are respectively 1.4, 1.1, 0.67 and 0.70 % hematocrit. Except for bag 1 at 8% hematocrit, the error have a homogeneous distribution in the tested hematocrit range.

V -Discussion

The recent growth in new blood processing techniques derived from microfluidics increase the need of specific, low-cost, stable and rapid monitoring methods. In particular, new methods for sensing low blood flow rate and hematocrit in narrow medical tubing with a high accuracy are required. We wished to demonstrate in this paper a new Doppler ultrasound method for measuring in vitro blood flow rate and hematocrit. The principle of flow rate measurement is to extract the flow rate as an optimal parameter from the fit between the measured Doppler PSD and the theoretical model developed by Vilkomerson et al. [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF]. The method was tested in sub-milliliter medical tubing and for the flow rate range of the acoustophoresis blood fractionation chip designed by Bohec et al. [START_REF] Bohec | Acoustophoretic purification of platelets: Feasibility and impact on platelet activation and function[END_REF] .We also evaluated how the method can measure hematocrit with the same Doppler signals than for flow rate, using a simplified packing factor model.

The measured Doppler PSD presented the usual CW Doppler spectrum shape with an expected speckle noise due to the sum of a large number of random-phased signals. Despite the small tubing employed, the signal-to-noise ratio was sufficient for performing close fits with the theoretical model from Vilkomerson et al [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF], leading to a highly accurate estimation of the blood flow rate for the whole tested range. In addition, Doppler spectra converged to the theoretical model with a nearly stable error within a few seconds. This confirm the validity of our approach and demonstrate how the Doppler method can be reduced to a basic minimization problem in such an environment. Besides the flow rate, the slight discordance (4%) between the estimated pipe radius and the constructor value might be explained by a pipe dilation due to the pressure gradient generated by the syringe pump. Another explanation could be an incorrect value attributed to one of the measurement parameter and then reported by the optimization algorithm on the pipe value.

To estimate the pipe radius and the beam width through the optimization algorithm might be questionable since both can be respectively known from the constructor or estimated experimentally. However, it is difficult to have a full control on all parameters involved in a quantitative ultrasound experiment. For this reason, the optimization algorithm allows to verify if the fixed parameters used in the model correspond to their expected value when the estimated flow rate is close to the syringe pump flow rate. Here, the minimization process highlighted that our system was able to properly estimate the imposed flow rate provided that the value of the radius was slightly overestimated. The cause of the bias has not been identified but in any case it does not bother the accuracy of the method. For the beam width, it is shown in the study by Vilkomerson et al. [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF] that its value does not influence the maximum Doppler frequency in the model.

The high signal-to-noise ratio of Doppler spectra can be mainly attributed to the continuous digitalization of the Doppler signal over a large acquisition time. This combined with the steady and uniform nature of extracorporeal blood flows are the key to perform the fitting with the model. As a consequence, the method for flow rate estimation is restrained to well-controlled environments, where the model can be applied. Nonetheless, a limitation to the demonstrated method is the sampling frequency of 20MHz, which is close to the Nyquist frequency for the 8 MHz signal. In the same way, the 8bit rate would be increased with more resolved electronics. If these drawbacks are partially compensated by the use of a large acquisition time, the signal-to-noise ratio could be increased with more optimal acquisition parameters. As well, the Vilkomerson et al. [START_REF] Vilkomerson | Finding the peak velocity in a flow from its Doppler spectrum[END_REF] model is valid for a uniformly insonated flow. This might not be the case because of the curvature of the pipe walls which produces a lens effect [44]. For future studies, a correction based on the aberration induced by the propagation medium inhomogeneity could be implemented to improve accuracy.

From our measurement sample, results tend to show that the packing factor model is sufficient for measuring hematocrit in a range <8%, where its relation with the BSC is close to be linear. For the 4 blood bags, the measure presented an error under 1.5% hematocrit for an acquisition time of 5s. The peak value around 12% in the packing factor model fitted to the Doppler frequency integrals of the 4 blood bags is in good agreements with observed values around 13% for non-aggregated porcine and bovine blood by Yuan and Shung [35]. The negligible standard deviations in the estimation of a and m using random starting points in the fitting algorithm furthermore demonstrated that the fit was robust. The error of ±0.2 for the value of m between the 4 blood bags shows that, in some cases, the estimation of m from our system will approach the value of 3 corresponding to spheres. This is in agreement with the previous study by Yuan and Shung [35]. In the case of values superior to 3, we might refer to the study performed by Bascom and Cobbold [36]. In this study, the authors suggested that m could take non-integer values, possibly superior to 3. More precisely, they showed that a value of 3.26 for m could correspond to red blood cells modeled by spherocylinders.

It can be clearly seen in Fig. 4 that a hematocrit measurement in a range approximately superior to 12% would have a poor precision because of the large variance of the Doppler frequency integrals. This large variance plus the strong non-linearity in the relationship between hematocrits around 10%-12% and Doppler PSD frequency integrals confirm the limitation of the hematocrit measure below values of 8%. In addition, because of the BSC dependence on flow conditions, it should be noted that the hematocrit estimation using the parameters extracted from the packing factor model is valid only for one flow rate a priori. The large variance of the measured Doppler PSD frequency integrals might be caused by red-blood cell aggregation. It is known that the aggregation become minor as the shear rate approach 100 s -1 [45]. Using the expression given in [41], the mean shear rate across the tube during hematocrit measurement at 1 mL/min was calculated to a value of 263 s -1 . On the other hand, De Monchy et al. [46] recently highlighted that the BSC of a mixture of disaggregated and aggregated red blood cells should be higher than for a mixture of disaggregated red blood cells. Following this development, an additional experiment has been conducted on washed red blood cells suspended in saline, where aggregation should be hampered. Ten Doppler frequency integrals have been acquired for hematocrit values of 5%, 10% and 15% (±0.2%) for red blood cells in both plasma and saline. The frequency integrals where then normalized by the following expression [39]:

[ 𝜅 𝑟𝑏𝑐 -𝜅 0 𝜅 0 + 𝜌 0 -𝜌 𝑟𝑏𝑐 𝜌 𝑟𝑏𝑐 ] 2 ( 12
)
Where 𝜅 0 and 𝜌 0 are the compressibility and the density of the suspending medium i.e plasma or saline. In the same way 𝜅 𝑟𝑏𝑐 and 𝜌 𝑟𝑏𝑐 are red blood cells density and compressibility. Values for the compressibility and density were 44.3.10 12 cm².dyne -1 and 1.005 g.cm -3 for saline, 40.9.10 12 cm².dyne -1 and 1.021 g.cm -3 for plasma and 34.1.10 12 cm².dyne -1 and 1.092 g.cm -3 for red blood cells. These values were selected from the study by Shung et al. [START_REF] Shung | Scattering of Ultrasound by Blood[END_REF]. This expression is found in the red blood cell backscattering cross-section 𝜎 𝑟𝑏𝑐 and quantifies the acoustic impedance rupture between red blood-cells and their suspending medium [39]. As this impedance rupture is different if red blood cells are suspended in saline or in plasma, the normalization allow to compare the Doppler PSD frequency integrals acquired for both. The result of the comparison is shown in supplementary materials (supplementary figure 1). It can be seen that Doppler PSD frequency integrals acquired for red blood cells in plasma are in average 1.4x higher than in saline. This result suggests that, in spite of the high mean shear rate across tube, it is probable that red blood cell aggregation cannot be neglected in our measurement system. Thus, further investigations will have to be conducted for a better understanding of the influence of red blood cell aggregation on the proposed method.

In the same way, an important aspect to question in order to extend the hematocrit measure to a higher range is the validity of the hypothesis of a spatially invariant packing factor. This hypothesis was initially stated by Bascom and Cobbold [40] for deriving equation [START_REF] Nilsson | Evaluation of a Laser Doppler Flowmeter for Measurement of Tissue Blood Flow[END_REF]. But, such an assumption considers non-interacting red-blood cells, which is known to be invalid for physiological hematocrits [47] . For a hematocrit measurement in a physiological range through ultrasonic backscattering, more elaborated models should be used [43], [46]. These models would notably allow to take into account red blood cells aggregation, which is not included in the packing factor model. Also, Yuan and Shung [48] demonstrated that the ultrasonic backscattered signal is dependent on the plasma composition and more particularly on the fibrinogen concentration. Hence, it would be interesting to assess how this dependence could be included in the model. Finally, another limitation of the hematocrit measurement is the inherent non-monotonic behavior of the relationship with backscattered ultrasonic signals. Indeed, it is not possible with this method to discriminate two different hematocrits with the same Doppler PSD frequency integral value. Efforts will have to be done for adding a supplementary indicator to the Doppler integral, while preserving its simplicity.

In the particular case of microfluidic blood processing using diluted blood, the reduction of the method to hematocrits below the peak value of Doppler PSD frequency integrals finds an application. For instance, many systems presented in [START_REF] Hou | Microfluidic Devices for Blood Fractionation[END_REF]- [START_REF] Antfolk | Continuous flow microfluidic separation and processing of rare cells and bioparticles found in blood -A review[END_REF] operate under 10% hematocrit. In this situation, our method presents the advantage to have a good accuracy for very low hematocrits , for which cited methods have not been demonstrated [START_REF] Trebbels | Hematocrit Measurement -A high precision on-line measurement system based on impedance spectroscopy for use in hemodialysis machines[END_REF]- [START_REF] Zeidan | In vitro hematocrit measurement using spectrally encoded flow cytometry[END_REF]. Equally, the capability of the method to sense very low hematocrits could be used as a monitoring tool for blood fractionation processes. As an illustration, we can highlight the acoustophoretic device presented by Bohec et al. [START_REF] Bohec | Acoustophoretic purification of platelets: Feasibility and impact on platelet activation and function[END_REF] which was used for platelet purification. Indeed, the system presented here could be employed for the detection of residual red blood cells in the platelet concentrate at the output of this acoustophoretic device.

For its part, the flow rate measurement method presents few needed improvements for being fully operational in a practical context. Some investigations should be performed to find an optimal coupling medium between the probe and the tube as well as for miniaturizing the system. However, the principle of the method is valid as it is presented. Theoretically, the only limit to the measure is that the signal-to-noise ratio has to be high enough to use the theoretical model. Thus, the method is not limited to flow rates above the tested upper value of 1.5mL/min. For lower flow rates, performances of the method can be expected to drop in the vicinity of 0.1mL/min because corresponding Doppler maximum frequencies in this range will be too close from the 80Hz clutter frequency (which corresponds to a flow rate of approximately 0.14mL/min). However, it is still possible to reduce the tube diameter for increasing blood speed and thus extending the measurement range to flow rate beyond this limit. The signal loss arising from a smaller tube diameter can then be compensated with a higher input voltage to the probe.

Finally, because of its aggregation tendency and the dense spatial arrangement of red blood cells, human blood is a complex medium to handle for ultrasound. The presented method can be thus expected to be usable for estimating cells concentration in a broad range of biological suspensions with less limitations than in blood. Combined with flow rate measurement, this concentration measurement makes our method an interesting tool for microfluidic bioprocessing in general, where slow flow rates are often encountered.

VI -Conclusion

We proposed in this paper a Doppler ultrasound method for measuring flow rate and hematocrit of an in vitro blood flow. More precisely, our method is conceived for the recent blood processing devices using slow flow rates in narrow medical tubing such as acoustophoresis blood fractionation. The novelty in our approach for flow rate measurement is to use recent theoretical developments for proposing an optimal, versatile and highly accurate adaption of the Doppler method. Furthermore, we showed that the system can be used as a hematocrit sensor for diluted blood based on the well know packing factor model used in the theory of quantitative ultrasound. This hematocrit sensing finds interesting applications in blood sorting technologies but also demonstrate the feasibility of a simple and non-invasive ultrasonic system for measuring hematocrit in in-vitro blood circuits. More generally, our study demonstrates how fundamental aspects of Doppler ultrasound can be easily adapted to the in vitro environment and how this adaptation can contribute to the rise of new blood processing technologies.

Fig. 1 .

 1 Fig. 1. Schema of the measurement set-up and picture of the probe and the tube mounted in the 3D-printed support.

Fig. 2 .

 2 Fig. 2. A measured Doppler power spectral density fitted to its corresponding Vilkomerson et al. theoretical model. The associated optimized values for the pipe radius and the beam width are respectively 392um and 4mm.

Fig. 3 .

 3 Fig. 3. Mean flow rate measurement error as a function of acquisition time.

Fig. 4 .

 4 Fig.4. Measured Doppler frequency integrals at diluted hematocrits for the 4 blood bags fitted with the packing factor model of[START_REF] Chen | High-throughput acoustic separation of platelets from whole blood[END_REF].

Fig. 5 .

 5 Fig. 5. Relative error of hematocrit measurement from Doppler frequency integrals as a function of acquisition time and averaged on all hematocrit values.

Fig. 6 .

 6 Fig. 6. Measured flow rate compared to the syringe pump flow rate and regression line for an acquisition time of 2s. The slope is 0.987 (1.6% error) and the offset is 0.493 (1.3% error).

Fig. 7 .

 7 Fig. 7. Flow rate relative measurement error for an acquisition time of 2s.

Fig. 8 .

 8 Fig. 8. Flow rate relative measurement error for the 4 blood bags at diluted hematocrits and with a flow rate at 1mL/min

Fig. 10 .

 10 Fig. 10. Mean relative error of hematocrit estimation for an acquisition time of 5s.

ACKNOWLEDGMENT

The work performed in this publication was completed with the support of the Inserm Technology Research Accelerator (Inserm ART) in Biomedical Ultrasound, Paris, France.

The authors would like to thanks Martin Flesch, Baptiste Heiles, Jonathan Porée and Kailiang Xu from Institute Physics for Medicine Paris for their valuable advices. This project has received funding from the European Union's Horizon 2020 research and innovation program, under approval number n°783667.

Supplementary Material 1.