Member IEEE Emilie Franceschini
email: franceschini@lma.cnrs-mrs.fr

Ratan K Saha

Guy Cloutier

Comparison of three scattering models for ultrasound blood characterization

Ultrasonic backscattered signals from blood contain frequency-dependent information that can be used to obtain quantitative parameters reflecting the aggregation level of red blood cells (RBCs). The approach consists in estimating structural aggregate parameters by fitting the spectrum of the backscattered radio-frequency echoes from blood to an estimated spectrum considering a theoretical scattering model. In this study, three scattering models were examined: a new implementation of the Gaussian Model (GM), the Structure Factor Size Estimator (SFSE) and the new Effective Medium Theory combined with the Structure Factor Model (EMTSFM). The accuracy of the three scattering models in determining mean aggregate size and compactness was compared by two-and three-dimensional (2D and 3D) computer simulations where RBC structural parameters are controlled. Two clustering conditions were studied: (1) when the aggregate size varied and the aggregate compactness was fixed in both 2D and 3D cases, and (2) when the aggregate size was fixed and the aggregate compactness varied in the 2D case. For both clustering conditions, the EMTSFM was found more suitable than the GM and SFSE for characterizing RBC aggregation.

Comparison of three scattering models for ultrasound blood characterization I. INTRODUCTION Quantitative ultrasound (US) techniques are mainly based on the frequency analysis of backscattered signals by biological tissues to determine physical properties of the average tissue microstructure. These techniques rely on theoretical scattering models to fit the spectrum of backscattered echoes to an estimated spectrum using an appropriate model. The theoretical scattering model most frequently used for this purpose is the Gaussian model (GM) [START_REF] Lizzi | Relationship of ultrasonic spectral parameters to features of tissue microstructure[END_REF] [2] that yields two tissue properties: the average scatterer size and the acoustic concentration (i.e., the product of the scatterer number density by the square of the relative impedance difference between scatterers and the surrounding medium). This approach was used to characterize dilute scattering media such as the eye [START_REF] Feleppa | Diagnostic spectrum analysis in ophthalmology: a physical perspective[END_REF], prostate [START_REF] Feleppa | Ultrasonic spectral-parameter imaging of the prostate[END_REF] and breast [START_REF] Oelze | Differentiation and characterization of rat mammary fibroadenomas and 4T1 mouse carcinomas using quantitative ultrasound imaging[END_REF]. Blood has also been studied with this technique [START_REF] Kitamura | Roles of hematocrit and fibrinogen in red cell aggregation determined by ultrasonic scattering properties[END_REF], although estimations could be biased.

An important contribution of ultrasonic blood characterization is to assess the level of red blood cell (RBC) aggregation, which is a surrogate marker of inflammation [START_REF] Ami | Parameters of red blood cell aggregation as correlates of the inflammatory state[END_REF]. It is well known that when RBCs are under low shear rates (<10 s -1 ), they interact strongly with each other and form complex rouleaux or three-dimensional (3D) structures. When the shear rate increases, these rouleaux or compact structures disaggregate. The aggregation phenomenon in human blood is normal, however hyperaggregation is a pathological state associated with several circulatory diseases, such as deep venous thrombosis, atherosclerosis and diabetes mellitus. Blood characterization using US backscatter techniques provides the unique opportunity of monitoring RBC aggregation non-invasively and in vivo within blood vessels. This quantification may help to elucidate the role of RBC aggregation in the etiology of such diseases. US backscatter by blood is mainly due to RBCs that constitute the vast majority (97%) of the blood cellular content. Blood can thus be described as a biphasic fluid composed of RBCs immersed in plasma. Since RBCs are acoustically considered as weak scatterers (impedance contrast between RBCs and plasma being around 13%), multiple scattering can be neglected. However, for such tissue, it is not straightforward to develop a theoretical scattering model because of the high density of RBCs (their volume fraction or hematocrit varies between 30 and 50%) and their ability to form aggregates. The Structure Factor Model (SFM) [START_REF] Savery | A point process approach to assess the frequency dependence of ultrasound backscattering by aggregating red blood cells[END_REF] [START_REF] Fontaine | Simulation of ultrasound backscattering by red blood cell aggregates: effect of shear rate and anisotropy[END_REF] is an US scattering model proposed to simulate the backscatter coefficient (BSC) of aggregated RBCs. The SFM sums the acoustic contributions from individual RBCs and models their interaction by a statistical mechanics structure factor, which is by definition the Fourier transform of the spatial distribution of RBCs [START_REF] Savery | A point process approach to assess the frequency dependence of ultrasound backscattering by aggregating red blood cells[END_REF] [START_REF] Fontaine | Simulation of ultrasound backscattering by red blood cell aggregates: effect of shear rate and anisotropy[END_REF]. However, the SFM can hardly be implemented to estimate the structural aggregate parameters in the framework of an inverse problem formulation because of the intensive computational time to assess the structure factor by realizing distributions of aggregating RBCs. That is why Yu and Cloutier [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] developed the SFSE scattering theory that approximates the SFM by August 20, 2013 DRAFT using a second-order Taylor expansion of the structure factor. The SFSE is thus not as accurate as the SFM. The SFSE estimates two physical parameters describing the microstructure of RBC aggregates: the packing factor that increases with erythrocyte clustering and the average aggregate isotropic radius. However, experiments with pig blood in controlled flow devices [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] and 3D numerical simulations of isotropic monodisperse aggregates [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] recently showed that the two indices are correlated and follow a quadratic relationship, thus reducing the BSC parameterization to one structural index.

Another scattering model called the Effective Medium Theory combined with the SFM (EMTSFM) was recently proposed to better approximate the SFM [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. It assumes that aggregates of RBCs can be treated as individual homogeneous scatterers, which have effective properties determined by the acoustical characteristics and concentration of RBCs within aggregates. The EMTSFM allows characterizing the radius, and for the first time in the quantitative US field, the compactness of RBC aggregates [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. In the field of clinical hemorheology [START_REF]Handbook of hemorheology and hemodynamics[END_REF], assessing the compactness of RBC aggregates is of high clinical importance since it is related to the binding energy between cells. Normal RBC aggregates form rouleaux type structures, whereas pathologies associated with stronger binding energies result in clumps of RBCs (close to a spherical isotropic packing) [14] [15].

In our previous study [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF], the EMTSFM and the SFM were compared in the framework of a forward problem study to determine the BSC from a known distribution of RBCs with known acoustical parameters. The goodness of the approximation of the EMTSFM in comparison with the SFM was examined as a function of frequency and structural aggregate parameters (aggregate size and compactness). Based on a two-dimensional (2D) simulation study, the EMTSFM was found to approximate the SFM with relative errors less than 30% for a product of the wavenumber times the mean aggregate radius kr ag ≤1.32 [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. The aims of the present paper are :

1) to evaluate the EMTSFM in an inverse problem framework, i.e. to determine RBC structural features from the measured BSC, and

2) to compare the EMTSFM with two other scattering models: the SFSE and a new implementation of the GM slightly modified to treat aggregating scatterers.

To our knowledge, there is no means to experimentally assess aggregate sizes at a normal physiological hematocrit of 40% because RBCs at that hematocrit are opaque to light. It would thus not be feasible to quantitatively evaluate the performance of the different models with real experimental data. In the field of blood imaging and characterization, the assessment of accuracy of a scattering model was only performed at a low hematocrit of 6% by comparing optical and acoustic measurements of RBC aggregate sizes [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF]. In the current paper, we thus aim to determine the performance of three theoretical scattering models (the new implementation of the GM, the SFSE and the EMTSFM) to extract the aggregation parameters from computer simulations where RBC structural parameters (such as the hematocrit, the aggregate size and compactness) are known.

The important contribution of the EMTSFM is the parameterization of the BSC with the aggregate compactness [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF], which is a structural parameter not available in any other modeling strategies proposed in quantitative US.

The potential of the EMTSFM and of the two other scattering models in estimating the aggregate compactness was examined by 2D computer simulations based on the SFM in controlled clustering configurations (when the August 20, 2013 DRAFT aggregate compactness varies and the aggregate radius is fixed). This clustering condition was only conducted in 2D because of the computational load required to generate three-dimensional RBC distributions with various aggregate compactnesses with the SFM [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. Some 3D computer simulations were also used in the controlled clustering configurations as those performed in 2D (when the aggregate size vary and the aggregate compactness is fixed) in order to compare the BSC behavior between 2D and 3D simulations, and estimated structural aggregate parameters with the three scattering models.

II. COMPUTER SIMULATIONS BASED ON THE STRUCTURE FACTOR MODEL (SFM)

This section presents computer simulations performed to predict the frequency dependence of the BSC from aggregated RBCs based on the SFM. In the following, it is assumed that the incident wavelength λ is large compared to the RBC size. Consequently, the RBC shape could be approximated by a simple geometry having an equivalent surface in 2D or having an equivalent volume of a RBC in 3D [START_REF] Savery | High-frequency ultrasound backscattering by blood: Analytical and semi-analytical models of the erythrocyte cross section[END_REF]. RBCs were modeled as parallel infinite cylinders in the 2D case and as spheres in the 3D case of radius a, that have small contrast in acoustical properties relatively to the plasma (see Table I). This RBC shape approximation has some limitations for larger frequencies (>20 MHz) and will be discussed later in Section V-C.

The SFM describing US backscatter by biological tissues consists of summing contributions from cells and modeling the cellular interaction by a statistical mechanics structure factor [8] [9]. By considering a collection of N identical and weak scattering RBCs, the BSC expression can be written as:

BSC SF M (-2k) = mσ b (-2k)S(-2k), (1)
where k is the wavenumber and m the number density of RBCs that is related to the systemic hematocrit φ as m = φ/A p (where A p is the RBC area) for 2D modeling, or as m = φ/V p (where V p is the RBC volume) for 3D

modeling. The backscattering cross section σ b of a single weak RBC was calculated using the fluid infinite cylinder expression in the 2D case [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF] or using the fluid-filled sphere expression in the 3D case [17] [18] given by:

σ b (-2k) = k 3 A 2 p γ 2 z 2π J 1 (2ka) ka 2 in the 2D case = k 4 V 2 p γ 2 z 4π 2 3 sin(2ka) -2ka cos(2ka) (2ka) 3 2 in the 3D case (2)
where J 1 is the first order Bessel function of the first kind and γ z is the relative impedance difference between the RBC and its suspending medium (i.e., the plasma). The function S is the structure factor representing the spatial positioning of RBCs and is defined by:

S(-2k) = E   1 N N i=1 e -i2kri 2   (3)
where E represents the expected value of a random variable and r i the position vectors defining the center of the ith RBC in space. In general, the structure factor of a medium containing RBCs distributed in the 2D space (or in the 3D space) can be determined from the 2D Fourier transform (or 3D Fourier transform) of the spatial distribution August 20, 2013 DRAFT of particles (see the Appendix of Ref. [START_REF] Fontaine | A system-based approach to modeling the ultrasound signal backscattered by red blood cells[END_REF]).

The computation of the BSC SF M using the SFM requires an intensive computation because of the calculation of the structure factor S as described in Eq. ( 1). Since the structure factor is by definition a statistical quantity, an average of structure factors from several RBC spatial distributions can give an estimated value of S. Because of the computational load to generate aggregating RBC distributions, a simple and fast method was used to randomly generate non-overlapping RBC aggregates which were isotropic and similar in size. For the 2D and the 3D computer simulations, the simulated BSC SF M were obtained from the method described in section III of Refs. [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF] and [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF], respectively. Note that 2D simulations are computationally less expensive but significant insights can be gained by studying 2D systems. On the other hand, 3D simulations are intuitively appealing because they better mimic experimental situations but are computationally expensive. These methods are briefly summarized in the following.

Random distributions for aggregating RBCs were computed within the simulated surface area of 600 × 600 µm 2 in the 2D case and within the simulated volume of 1000 × 125 × 125 µm 3 in the 3D case. The RBC radius a was set to 2.75 µm for all simulations. We first specified the systemic hematocrit φ, the aggregate radius r ag and the aggregate compactness φ i (i.e. the RBC concentration within aggregate). Aggregates of identical radii r ag and of identical compactness φ i were then randomly distributed with non-overlapping positions to give the desired concentration of aggregates φ ag = φ/φ i . Note that in the case of the 3D study, a small number of non-aggregated RBCs was added to reach the desired systemic hematocrit. It means that all the RBCs were aggregated in blood in the 2D case, whereas a fraction of RBCs were aggregated while the rest remained disaggregated in the 3D case.

Finally, RBC distributions within aggregates were then generated as follows:

• in the 2D case, the locations of the RBCs were generated using external and repulsive forces to obtain random RBC positions inside each aggregate, such that the distribution of RBCs within each aggregate was different [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. This technique allowed to study several aggregation configurations (1) when the aggregate size varied and the aggregate compactness was fixed to 0.6, and (2) when the aggregate compactness varied from 0.3 to 0.6 and the aggregate size was fixed.

• in the 3D case, the RBCs were stacked by following a hexagonal close packing (HCP) structure for each aggregate, such that the distribution of RBCs within each aggregate was identical. This HCP structure provides the highest compactness that is about 0.74 for spheres [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]. Therefore, this technique allowed to study several aggregation configurations when the aggregate size varied and the aggregate compactness was fixed to 0.74.

For each distribution of RBCs, the 2D or 3D Fourier transformation of the spatial organization of RBCs was then computed to obtain the corresponding structure factor. A mean structure factor was determined from 400 different tissue realizations in the 2D case (see section III.B in Ref. [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]) and from 250 different tissue realizations in the 3D case (see section III.B in Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]).

August 20, 2013 DRAFT

III. ULTRASOUND BACKSCATTERING MODELING FOR THE ESTIMATION OF STRUCTURAL AGGREGATE

PARAMETERS

As seen previously in section II, the SFM allows to simulate the BSC from RBCs whatever the RBC spatial distribution (i.e. disaggregated or aggregated RBCs and/or with various aggregate sizes and compactnesses). However, the SFM can hardly be implemented to estimate structural parameters in the framework of an inverse problem formulation because of the intensive computational time to assess the structure factor by realizing distributions of RBCs with simulations. That is why two scattering theories, named the SFSE and the EMTSFM, have been recently developed to approximate the SFM for practical assessments of RBC structural features (i.e., in an inverse problem formulation). This section presents these two scattering theories (the SFSE and the EMTSFM) as well as the GM also used for tissue characterization. In this work, we present a new implementation of the GM model inspired by our development on EMTSFM. All three theories fit a curve to the simulated BSC SF M from blood to estimate aggregation parameters using the minimization routine "fminsearch" in MATLAB (The MathWorks, Inc., Natick, MA), i.e. a Nelder-Mead simplex method. Note that this fit was realized in the frequency bandwidth from 4 MHz to the frequency corresponding to the first minimum of the BSC SF M (i.e., after the frequency-dependent increase in BSC followed by a peak and a reduction to its first minimum).

A. The Structure Factor Size Estimator (SFSE)

The SFSE developed by Yu and Cloutier [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] approximates the SFM with a second-order Taylor expansion of the structure factor in k as follows:

S(-2k) ≈ W -4(kR g a) 2 in the 2D case ≈ W -4(kR g a) 2 ≈ W - 12 5 (kR sp a) 2 in the 3D case ( 4
)
where W is the low-frequency limit of the structure factor (S(k)| k→0 ) called the packing factor [START_REF] Twersky | Low-frequency scattering by correlated distributions of randomly oriented particles[END_REF] [21] and R g is the radius of gyration of RBC aggregates assumed to be isotropic and expressed in number of RBCs [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF].

Note that in the 3D case, R g is related to the isotropic radius R sp of an aggregate (expressed in number of RBCs)

by

R g = 3 5
R sp [11] [22]. By assuming identical RBCs, and recombining Eqs. ( 1) and ( 4), the SFSE model approximates the BSC as follows:

BSC SF SE (-2k) = 1 2π mk 3 A 2 p γ 2 z J 1 (2ka) ka 2 W -4(kR g a) 2 in the 2D case = 1 4π 2 mk 4 V 2 p γ 2 z 3 sin(2ka) -2ka cos(2ka) (2ka) 3 2 W - 12 5 (kR sp a) 2 in the 3D case (5)
The SFSE assumes that the hematocrit φ, the RBC radius a and the acoustical properties of plasma and RBCs are known a priori. Therefore, Eq. ( 5) presents only two unknowns that characterize the aggregate structure: W and R g (or equivalently, W and R sp in the 3D case). Estimated values of W * and R * g (or equivalently, W * and R * sp in the 3D case) were determined by fitting the simulated BSC SF M given by Eq. ( 1) with BSC SF SE given by Eq.

(5). August 20, 2013 DRAFT

B. The Effective Medium Theory Combined with the Structure Factor Model (EMTSFM)

The EMTSFM assumes that all the scatterers are aggregated, that the aggregates are identical and isotropic and that the scatterers within aggregates are evenly distributed [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]. In the case of blood backscatter, the model consists of treating the RBC aggregates as individual homogeneous particles of radius r ag . These homogeneous particles are characterized by a density ρ ag and a compressibility κ ag derived from the acoustical properties of the two fluids constituting them (i.e., ρ 1 , ρ 2 , κ 1 and κ 2 , where 1 indicates properties of RBCs and 2 those of plasma), and from the internal concentration of RBCs within the aggregates, defined as the aggregate compactness φ i , as follows:

ρ ag = φ i ρ 1 + (1 -φ i )ρ 2 1 κ ag = φ i κ 1 + 1 -φ i κ 2 (6)
The BSC from blood is then obtained by summing contributions from individual effective particles of radius r ag and modeling the effective particle interaction by a statistical mechanics structure factor S ag . The equivalent BSC expression is thus given by [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]:

BSC EM T SF M (-2k) = m ag σ ag (-2k)S ag (-2k), (7)
where S ag is the structure factor of a collection of N ag identical particles of radius r ag randomly distributed and m ag is the number density of aggregates that is related to the effective aggregate concentration φ ag . The effective aggregate concentration is equal to the RBC concentration in blood φ divided by the aggregate compactness φ i :

φ ag = φ/φ i .
The backscatter cross-section of an effective single particle σ ag was calculated using the fluid infinite cylinder expression in the 2D case [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF] or using the fluid-filled sphere expression in the 3D case [17] [18] given by:

σ ag (-2k) = k 3 πr 4 ag γ 2 zag 2 J 1 (2kr ag ) kr ag 2 in the 2D case = 4k 4 r 6 ag γ 2 zag 9 3 sin(2kr ag ) -2kr ag cos(2kr ag ) (2kr ag ) 3 2 in the 3D case (8)
where z ag is the impedance of the equivalent particle and γ zag the relative impedance difference between the equivalent particle and the plasma (γ zag = (z ag -z 2 )/z 2 ). For a random distribution of hard-cylinders in 2D, the structure factor was numerically computed as described in the appendix A. For a random distribution of hard-spheres in 3D, the structure factor can be analytically calculated as established by Wertheim [START_REF] Wertheim | Exact solution of the Percus-Yevick integral equation for hard spheres[END_REF]. The analytical expression for the structure factor in the 3D case is described in the appendix B.

By assuming that the hematocrit φ, the RBC radius a and the acoustical properties of plasma and RBCs are known a priori, the unknown parameters are the radius of aggregates r ag and their compactness φ i . The unknown parameters were estimated by matching the simulated BSC SF M given by Eq. ( 1) with the theoretical BSC EM T SF M

given by Eq. [START_REF] Ami | Parameters of red blood cell aggregation as correlates of the inflammatory state[END_REF].

C. The Gaussian Model (GM)

Using the GM, the BSC is modeled with a spatial autocorrelation function describing the shape and distribution of scatterers in the medium. The scattering sites are assumed to be randomly distributed and of simple geometric August 20, 2013 DRAFT shapes, represented as gaussian scatterers mimicking continuous changes in impedance. In this framework, the BSC can be written as the product of the theoretical BSC under Rayleigh scattering and the backscatter form factor (see Eqs. ( 74)-(76) of Ref. [START_REF] Insana | Acoustic scattering theory applied to soft biological tissues[END_REF] for the GM formulation in 3D). The form factor describes the frequency dependence of BSC attributed to the size and shape of the prototype scatterer. The Gaussian form factor has been used for many applications [START_REF] Feleppa | Diagnostic spectrum analysis in ophthalmology: a physical perspective[END_REF]- [START_REF] Kitamura | Roles of hematocrit and fibrinogen in red cell aggregation determined by ultrasonic scattering properties[END_REF]. It represents tissue structures as continuously varying distributions of acoustic impedance fluctuations about the mean value, and the effective radius is related to the impedance distribution of the scatterers.

The BSC for the GM formulation is written as the product of the BSC in the Rayleigh limit and the backscatter form factor as follows [START_REF] Insana | Acoustic scattering theory applied to soft biological tissues[END_REF]:

BSC GM (-2k) = k 3 S 2 s n z 2π e -2k 2 d 2 = πk 3 a 4 ef f n z 2 e -k 2 a 2 ef f in the 2D case = k 4 V 2 s n z 4π 2 e -2k 2 d 2 = 4k 4 a 6 ef f n z 9 e -0.827k 2 a 2 ef f
in the 3D case [START_REF] Fontaine | Simulation of ultrasound backscattering by red blood cell aggregates: effect of shear rate and anisotropy[END_REF] where n z is the acoustic concentration (i.e., the product of the number density of scatterers times the square of the relative impedance difference γ z between scatterers and the surrounding tissue). In the 2D case (or respectively in the 3D case), the characteristic dimension d is related to the area of the effective scatterer S s by: S s = 2πd 2

(or related to the volume of the effective scatterer V s by: V s = (2πd 2 ) 3/2 ). Continuous isotropic media can be characterized by the correlation distance d, in the same way as discrete isotropic media are characterized by a scatterer radius [START_REF] Insana | Acoustic scattering theory applied to soft biological tissues[END_REF]. The effective radius of the scatterer a ef f is related to the correlation distance d by setting values of S s (or V s , respectively) for a continuum model equal to the area of an effective cylinder (or equal to the volume of an effective scatterer) of radius a ef f : S s = 2πd 2 = πa 2 ef f or V s = (2πd 2 ) 3/2 = (4/3)πa 3 ef f . Estimates of the effective radius a * ef f and acoustic concentration n * z were determined by fitting the simulated BSC SF M given by Eq. ( 1) with BSC GM given by Eq. ( 9). Effective radii a ef f estimated with the GM have been hypothesized to be related to the aggregate radii, and the acoustic concentration n z is postulated to be the product of the number density of aggregates times the square of the relative impedance difference between aggregates and the plasma as follows:

n z = γ 2 zag φ ag πa 2 ef f = z ag -z 2 z 2 2 φ φ i πa 2 ef f , in the 2D case = γ 2 zag 3φ ag 4πa 3 ef f = z ag -z 2 z 2 2 3φ 4φ i πa 3 ef f ,
in the 3D case [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] where z ag is the effective impedance of the aggregates approximated by the mixing law:

z ag = φ i z 1 + (1 -φ i )z 2 .
Since the hematocrit φ and the acoustical properties of plasma and RBCs are assumed to be known a priori, the aggregate compactness can be deduced from the estimated parameters a * ef f and n * z by using Eq. ( 10) as follows:

φ * i = πa * ef f 2 n * z z 2 2 φ(z 2 -z 1 ) 2 in the 2D case = 4πa * ef f 3 n * z z 2 2 3φ(z 2 -z 1 ) 2 in the 3D case (11)
It means that the new proposition of the GM was employed in our study as an effective medium model, but contrary to the EMTSFM, the GM is not combined with the SFM (such that the GM is assumed to be accurate only at low August 20, 2013 DRAFT systemic hematocrits). In the following, we thus give the estimated parameters a * ef f and φ * i with the GM, instead of the classical estimated parameters a * ef f and n * z .

IV. RESULTS

This section gives the results of the inverse problem obtained for 2D and 3D computer simulations with the three backscattering models aforementioned: SFSE, EMTSFM and GM.

A. Results obtained from the 2D computer simulations

For the 2D computer simulations, we first studied clustering configurations where the aggregate compactness was fixed to φ i =60% and the aggregate radius r ag /a varied, and then clustering configurations where the aggregate radius was fixed to r ag /a=6.32 and the aggregate compactness φ i varied.

1) Results for the SFSE:

The SFSE was first examined for systemic hematocrits of 10, 20 and 30% when the aggregate size varied and the aggregate compactness was fixed to a high value: φ i =60%. Figure 1 shows II for systemic hematocrits of 10, 20 and 30%. In this table, the relative error for parameter R * g corresponds to: ǫ R * g = R * g -(r ag /a) /(r ag /a). Figure 2a shows the estimated values of R * g as a function of the actual aggregate radii r ag /a for all hematocrits. Also represented are the corresponding linear regression lines showing good correlation r 2 ≥0.95 at all hematocrits. For radii r ag /a between 4.47 and 7.95, relative errors ǫ R * g were less than 30% for hematocrits of 10 and 20%. It is interesting to notice that estimated parameters W * and R * g follow a linear relation for all hematocrits (see Fig. 2b). The SFSE was also evaluated at systemic hematocrits of 10 and 20% when the aggregate size r ag /a was fixed to 6.32 and the aggregate compactness φ i varied from 30 to 60%. It is important to emphasize that 2D random particle distributions could be easily generated using a random number generator up to an area fraction of approximatively 0.5. For the 20% systemic hematocrit, aggregate compactnesses smaller than 40% could not be computed because the corresponding area fractions of aggregates were too high: φ ag > 0.5. Similarly, the variation of the aggregate compactness could not be performed at a systemic hematocrit of 30% because the area fractions of aggregates is already equal to 0.5 for an aggregate compactness φ i =60%. Figure 3 displays BSC SF SE in dashed lines for the following clustering conditions: r ag /a=6.32 and φ i varying from 30 to 60%. One can notice large differences August 20, 2013 DRAFT between simulated and fitted SFSE curves, especially at low frequencies where the fitted curves over-estimate the BSC SF M amplitude. These differences are larger at φ=20%. The estimated values of R * g for different aggregate compactnesses are plotted in Fig. 4a. Although the true radius is fixed, estimated R * g increases with the aggregate compactness at both hematocrits. We found no correlation between the actual fixed radius and the estimated radii (r 2 <0.06). Notice the linear relation between W * and R * g when the aggregate compactness varies (see Fig. 4b), as observed previously in Fig. 2b when the aggregate radius r ag /a was changed.

BSC SF M
2) Results for the EMTSFM: The BSC curves fitted with the EMTSFM are shown in solid lines in Fig. 1 for the case of varying values of r ag /a, and in Fig. 3 for varying φ i . In both cases, the EMTSFM provided good fittings to the simulated BSC SF M curves for all systemic hematocrits. For the clustering conditions where the aggregate radius varied and the aggregate compactness was constant, the estimated values r * ag /a and φ * i , and corresponding relative errors are given in Fig. 5 for systemic hematocrits of 10, 20 and 30%. For the clustering conditions where the aggregate compactness varied and the aggregate radius was constant, the results are shown in Fig. 6 for systemic hematocrits of 10 and 20%. For the EMTSFM, the relative errors for each parameter correspond to:

ǫ r * ag = (r * ag /a) -(r ag /a) (r ag /a) and ǫ φ * i = φ * i -φ i φ i . (12)
In both sub-studies where the aggregate radius and compactness varied, a very good correspondence can be observed between true and estimated aggregate sizes and compactnesses. The relative errors for the estimated aggregate radii and compactnesses were less than 13% and 14%, respectively, for all hematocrits and for all studied aggregating configurations.

3) Results for the GM: Figure 7 presents the BSC SF M curves fitted with the GM for several aggregate sizes at the same clustering conditions as in Fig. 1. The GM provided over-estimates in the low frequency range for all systemic hematocrits. Excellent correlations (r 2 ≥0.92) were found between the estimated and true aggregate radii for all hematocrits (data not shown). The estimated values a * ef f /a and φ * i from the GM, and corresponding relative errors are given in Fig. 8. For systemic hematocrits of 10 and 20%, the estimated radii and compactnesses are quantitatively satisfactory with relative errors less than 15%. However, for φ=30%, the relative errors increase up to 40%.

For the clustering conditions where the aggregate compactness varied and the aggregate radius was constant, the results are shown in Fig. 9 for systemic hematocrits of 10 and 20%. As previously observed with the SFSE, the estimated effective radius increases as the aggregate compactness increases. The estimated radii and compactnesses matched the true parameters at φ=10% with relative errors less than 17%. However, for φ=20%, large relative errors (up to 74%) were obtained.

4) Comparison of the errors between the simulated BSC and the fitted curves with the three scattering models:

The errors (i.e. differences) between the simulated BSC and the fitted curves with the three scattering models (SFSE, EMTSFM and GM) are presented in Fig. 10. The logarithm of the error is shown to enhance reading. The error reveals how the models fit the data. It is clear from the figure that errors were smaller with the EMTSFM and larger with the GM, for each hematocrit. For the aggregating conditions where the aggregate radius varied, the August 20, 2013 DRAFT error decreases as the radius increases. When the aggregate radius increases, the bandwidth frequency used for the fit becomes smaller and therefore, the number of frequencies used for the error computation decreases.

B. Results obtained from the 3D computer simulations

For the 3D computer simulations, the GM, SFSE and EMSTFM were examined when the aggregate size varied and the aggregate compactness was fixed to a high value: φ i =74%.

It is important to note that the 3D simulated aggregates were highly packed leaving small numbers of particles as non-aggregated RBCs. For each tissue realization, the actual mean aggregate radius r ag was computed using Eq. ( 6) in Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF], and then the concentration of aggregated RBCs φ ′ was computed as:

φ ′ = φ i N ag (4/3)πr 3 ag 1000 × 125 × 125 × (10 -6 ) 3 . ( 13
)
Figure 11a shows the values of φ ′ as a function of the mean aggregate radius r ag /a for the three systemic hematocrits of 20, 30 and 40%. The percentage of disaggregated RBCs was between 20 and 30% for the systemic hematocrit of 20% and between 27 and 37% for the systemic hematocrit of 40%. Note that the three models presented in section III assumed that all RBCs were aggregated in blood and that aggregates had identical shape and size. Consequently, during the inversion procedure of the 3D BSC data, we neglected the contribution of the disaggregated RBCs on the simulated BSC SF M and we replaced the hematocrit φ by the value of the concentration of aggregated RBCs φ ′ .

Figure 11b andc shows BSC SF M as a function of frequency for several aggregate sizes and systemic hematocrits of 30 and 40%. Also represented in Fig. 11b andc are corresponding fitted curves obtained with the SFSE, EMTSFM and GM. The fitted GM and SFSE curves did not produce good fits to the 3D data and overestimated the BSC SF M amplitude (especially in the low frequency range), as observed in the 2D case (see Figures 1 and7). At the opposite, the EMTSFM provided good fittings to the simulated BSC SF M curves.

The results obtained with the SFSE were already presented in a previous article [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]. Excellent correlations (r 2 ≥0.94) were found between the estimated and true aggregate radii for all hematocrits (see Fig. 5a in Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]).

It can also be seen in Fig. 5a of Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] that for each hematocrit there is an aggregate size range where the SFSE method works at its best. For example, relative errors for estimated radii were less than 20% for true radius values between 14 and 17 µm at the hematocrit of 40%. The parameters W * and R * sp followed a quadratic relationship (as in Fig. 5b of Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]).

Figure 12 gives the values of r * ag and φ * i estimated with the EMTSFM and corresponding relative errors that were less than 15% and 23%, respectively, for all hematocrits. Figure 13 gives the values of a * ef f and φ * i estimated with the GM and corresponding relative errors. The estimated radii with the new formulation of the GM are quantitatively satisfactory with relative errors less than 9% for all hematocrits. The relative errors for the estimated compactnesses with the GM are larger with relative errors up to 32% for the hematocrits of 20 and 30%, and up to 76% for the hematocrit of 40%. Three scattering models for the characterization of RBC aggregation were examined. From these models, the gold standard simulated BSC SF M was fitted and aggregation parameters were extracted. The SFSE has been developed

for blood characterization and the GM is a model that has been used in various tissue studies. Herein, the radius estimates R g from the SFSE and a ef f from the GM were hypothesized to represent the aggregate size.

A. Clustering conditions where the aggregate radius varied and the aggregate compactness was constant (2D and 3D computer simulations)

The 2D and 3D computer simulations were performed on the same clustering configuration where the aggregate radius varied and the aggregate compactness was constant. It is interesting to observe the same BSC SF M behavior for both 2D and 3D studies. Indeed, the simulated BSC SF M amplitude increases with the size of aggregates and the BSC SF M first peaks occur at lower frequencies as the aggregate radius increases (see Fig. 1 in the 2D case, and Fig. 11b andc of the present paper and Fig. 4 of Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] in the 3D case). Moreover, as it can be observed in Figures 1 and7 in the 2D case and in Figure 11 in the 3D case, the data fitting quality obtained with the three models were quite similar. In both 2D and 3D cases, it is clear that the GM and the SFSE are insufficient to model the complex behavior of BSC and that the EMTSFM was the model that better fitted the BSC data for all studied hematocrits.

Although the SFSE model did not produce good spectral fits to the BSC data for 2D and 3D computer simulations, significant correlations were found between the estimated and true radii with r 2 superior to 0.95 at all hematocrits (see Fig. 2a in the present paper and see Fig. 5a of Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF]). However, the estimated aggregation parameters W * and R * g followed a linear relationship in our 2D simulation study. This relation was also found to be quadratic in 3D numerical simulations [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] and under experimental conditions [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF]. It means that the BSC parameterization can be reduced to one parameter and that no new information can be obtained with the parameter W * .

The EMTSFM and the GM used as effective medium models gave quantitatively satisfactory radius estimates with relative errors less than 15% for the 10 and 20% hematocrits in the 2D case, and for all hematocrits in the 3D case. For the highest systemic hematocrit, the aggregate compactnesses were better estimated with the EMTSFM with relative errors less than 14% in the 2D case (and less than 23% in the 3D case), whereas the relative errors were between 19 and 36% in the 2D case (and between 59 and 76% in the 3D case) for the GM. These results with the EMTSFM and the GM were somewhat anticipated since the assumption of a random distribution of scatterers used by the GM fails due to the spatial correlation between scatterers in a dense medium [START_REF] Franceschini | Experimental assessment of four ultrasound scattering models for characterizing concentrated tissuemimicking phantoms[END_REF]. To conclude, the EMTSFM was more suitable than the GM and SFSE for characterizing the aggregate microstructure in both 2D and 3D studies. August 20, 2013 DRAFT

B. Clustering conditions where the aggregate compactness varied and the aggregate radius was constant (2D computer simulations)

For the highest simulated hematocrit of 20%, the aggregate radii normalized by the RBC radius were estimated between 4.34 and 8.55 using the SFSE model and between 3.83 and 5.33 using the GM (see Fig. 4a and Fig. 9), whereas the actual aggregate radius was r ag /a=6.32. Therefore, we found no correlation between the actual fixed aggregate radius and the estimated radii. The GM and SFSE cannot take into account a variation in the aggregate compactness at a large hematocrit, since it is interpreted as a change in the aggregate size.

In the case of the SFSE, one could have expected a fixed value of the estimated radius R * g and a variation of the estimated packing factor W * , when the aggregate radius was fixed and the aggregate compactness varied. However, both R * g and W * increased as the true aggregate compactness was raised. The estimated parameters R * g and W * followed linear relations for all hematocrits (see Fig. 4b), as observed previously in Fig. 2b when the aggregate radius r ag /a was changed. It means that W * and R * g carry the same information and that the BSC parametrization is reduced to one parameter.

The estimated parameters using the EMTSFM presented in Fig. 6 show that the model gave quantitatively satisfactory estimates for all aggregate compactnesses and for all studied hematocrits. Contrary to the GM and SFSE, the EMTSFM provided a quasi-constant aggregate radii between 5.7 and 5.9 for both studied hematocrits. Moreover, the aggregate compactnesses were estimated with relative errors less than 12% at both studied hematocrits for that model. The errors between simulated BSC SF M and the fitted curves were also smaller with the EMTSFM, as it can be observed in Fig. 10b. To conclude, the EMTSFM was the model that explained better the simulated

BSC SF M .

C. Computation of RBC distributions and of BSC SF M

The two methods we used here to obtain the RBC spatial distributions did not take into consideration realistic interactions between RBCs. These methods were already presented in Ref. [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF] and [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] for the 2D and 3D computer simulations, respectively. They were simple and fast methods to generate samples containing non-overlapping, identical and isotropic aggregates. The 3D computer simulations allowed to better mimic real data but they are time consuming (see section III.A and section V of Ref. [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectrum backscattering coefficient[END_REF] to obtain a quick review of different approaches to simulate compact RBC aggregates). In order to simulate the BSC data with the SFM reference model, the method we chose to distribute RBCs in the 3D case allowed (1) to study various aggregate sizes with the same aggregate compactness and (2) to reach the physiological hematocrit of 40% by mixing identical RBC aggregates and disaggregated RBCs. Since studied scattering models assumed that all RBCs were aggregated in blood and since the averaged percentage of disaggregated RBCs was small (around 25%), the influence of the disaggregated RBCs on the simulated BSC SF M was neglected during the inversion procedure of the 3D BSC data. Contrary to the 3D modeling, the method we chose to distribute RBCs in the 2D case allowed (1) to study the clustering condition where the aggregate compactness vary and the aggregate size is fixed and (2) to have only aggregated RBCs in blood. However, the 2D computer simulations were limited to a maximum hematocrit of 30% because of August 20, 2013 DRAFT the difficulty to simulate with the SFM values greater than 30%. To clarify, the main difficulty in the 2D case was to distribute compact aggregates and to have only aggregated RBCs in blood. The maximum value of the aggregate area fraction φ agmax was fixed to 0.5, corresponding to the maximum particle area fraction that can be easily generated using a random number generator. The procedure we chose to distribute the RBCs within aggregates allows reaching a maximum value of aggregate compactness φ imax equal to 0.6 (see section III.B. in Ref. [START_REF] Franceschini | Forward problem study of an effective medium model for ultrasound blood characterization[END_REF]).

As a consequence, the maximum value of the systemic hematocrit was limited to:

φ max = φ agmax φ imax =0.3.
We also modeled individual biconcave RBCs as spheres of equivalent volume in the 3D study and studied BSC between 4 and 45 MHz. The impact of modeling a RBC by a sphere on the frequency dependence of the backscatter cross-section has been studied and errors are introduced above 20 MHz [16] [25]. The impact of this simplification on the simulated BSC SF M and structural aggregate estimates with the three models (SFSE, GM and EMTSFM)

is unknown and still needs to be explored.

D. On the use of the EMTSFM in vivo

The EMTSFM assumes that all RBCs are aggregated in blood and that aggregates are identical and isotropic.

Therefore, the BSC behavior obtained in our simulations have pronounced frequency peaks. In experimental conditions [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF], the BSC behavior was smoother and the peaks were less pronounced. The reason behind this might be that real blood contains several sizes of aggregates, and since the location of BSC peaks are different for different aggregate populations, a relatively smoother BSC curve can be obtained. Another important aspect to consider is the assumption of isotropic aggregates. In human blood, low shear rates can promote the formation of RBC aggregates having anisotropic (i.e. rouleaux) or isotropic (i.e. clump) structures. The rouleaux like pattern is typically associated to normal blood. However, as the binding energy between RBCs increases with inflammation [START_REF] Weng | Influence of acute-phase proteins on erythrocyte aggregation[END_REF], aggregates form clump structures such as in diabetes mellitus [14] [15]. The assumption of isotropic aggregates in the EMTSFM is thus valid as far as we are concerned with the study of pathological states. In the case of normal human rouleaux of RBCs, if the EMTSFM is applied to estimate structural parameters such as the aggregate size and compactness, this assumption would obviously create a bias against the parameter estimation. Therefore, future improvements should consider incorporating the aggregate anisotropy and the polydispersity in terms of aggregate size and compactness to provide an optimal model for the inversion of experimental data. Future validations will also evaluate the EMTSFM in a controlled Couette flow experiment with ghost RBCs (i.e., optically visible RBCs with no hemoglobin and viable membrane properties) coated with dextran polymers to change attractive energies between erythrocytes and thus modulate the aggregate compactness and size.

Another difficulty to apply the EMTSFM in vivo is that the spectral content of backscattered echoes is also affected by attenuation caused by intervening tissue layers (such as the skin) between the probe and the blood flow. To evaluate correctly microstructural parameters, it is thus of major interest to take into account tissue attenuation effects.

Note that the SFSE was slightly modified to introduce the attenuation term in the BSC expression and was named the Structure Factor Size and Attenuation Estimator (SFSAE) [START_REF] Franceschini | Ultrasound characterization of red blood cell aggregation with intervening attenuating tissue-mimicking phantoms[END_REF]. The SFSAE allows to determine simultaneously Since there is no analytical expression of the structure factor S ag for hard-cylinders in 2D [START_REF] Tejero | Hard-sphere and hard-disk freezing from the differential formulation of the generalized effective liquid approximation[END_REF] [31], S ag was numerically computed for several values φ ag varying from 0.01 to 0.5 with a step of 0.01. It means that, in the 2D case, the cylinder concentration φ ag was rounded to the second decimal for the computation of S ag in Eq. [START_REF] Ami | Parameters of red blood cell aggregation as correlates of the inflammatory state[END_REF].

Note that the computation of S ag depends not only on the area fraction φ ag but also on the effective particle radius r ag . That is why S ag that depends on r ag was computed in a dimensionless way as described next.

For each specified value of φ ag , aggregates of an arbitrarily normalized (dimensionless) radius of 1/60 were randomly distributed within a dimensionless surface area L ′2 = 1 × 1 using a random number generator with nonoverlapping positions. The corresponding density matrix D ′ was computed by dividing the square simulation plane

L ′2 in N 2
p pixels (herein, N p =512) and by counting the number of particles falling into each pixel. The Fourier transformation of the density matrix D ′ was then computed to generate a structure factor S ag . A mean S ag was determined by repeating 400 times this procedure. When the value of the effective particle radius r ag was specified, the centered grid of wavevectors for the structure factor S ag was computed between ± πN p 2 × 60r ag with a step of ∆k = π 60r ag (i.e., by putting the simulated surface length L ′ = 60r ag ).

APPENDIX B ANALYTICAL EXPRESSION OF THE STRUCTURE FACTOR S ag FOR HARD-SPHERES IN 3D

The structure factor S for hard-spheres is given by [START_REF] Wertheim | Exact solution of the Percus-Yevick integral equation for hard spheres[END_REF] [32]

S(k) = 1 1 -4πmd 3 1 0 z 2 sin(2kz) 2kz c(z)dz (14)
where m is the number density of hard-spheres, d is the hard-sphere diameter and c(z) is the direct correlation function given by [START_REF] Wertheim | Exact solution of the Percus-Yevick integral equation for hard spheres[END_REF] [32]:

-c(z) = c 0 + c 1 z d + c 3 z d 3 for z ≤ d = 0 for z > d (15)
The coefficients c 0 , c 1 and c 3 are given by [START_REF] Wertheim | Exact solution of the Percus-Yevick integral equation for hard spheres[END_REF] [32]

c 0 = (1 + 2φ) 2 (1 -φ) 4 c 1 = - 6φ(1 + φ/2) 2 (1 -φ) 4 c 3 = φ 2 c 0 = φ(1 + 2φ) 2 2(1 -φ) 4 (16)
August 20, 2013 DRAFT

  figure that the SFSE provided better fits for the lower hematocrit of 10%. As the hematocrit increases, the SFSE model is insufficient to predict the behavior of BSC SF M , especially in the low frequency range. The estimated values of W * and R * g are given in TableIIfor systemic hematocrits of 10, 20 and 30%. In this table, the relative error for parameter R * g corresponds to: ǫ R * g = R * g -(r ag /a) /(r ag /a). Figure2ashows the estimated values of R * g as a function of the actual aggregate radii r ag /a for all hematocrits. Also represented are the corresponding linear regression lines showing good correlation r 2 ≥0.95 at all hematocrits. For radii r ag /a between 4.47 and 7.95,

Figure 1 .

 1 Figure 1. Frequency-dependent BSCs for different aggregate sizes and a constant aggregate compactness φ i =60% at systemic hematocrits of 10, 20 and 30%. The symbols represent the BSC SF M computation.The dashed lines represents the corresponding fitting with the SFSE, whereas the solid lines expresses the fitting with the EMTSFM.

Figure 2 .

 2 Figure 2. a) Comparison of R *g estimated with SFSE and the actual aggregate size r ag /a for the three systemic hematocrits 10, 20 and 30%. b) Linear relationships between W * and R * g . Results presented here correspond to the configuration where r ag /a varies and φ i is fixed.

Figure 3 .

 3 Figure 3. Frequency-dependent BSCs computed with the SFM for different aggregate compactnesses and a constant aggregate size r ag /a=6.32 at systemic hematocrits of 10 and 20%, and corresponding fitting with the SFSE (in dashed lines) and with the EMTSFM (in solid lines).

Figure 4 .

 4 Figure 4. a) Aggregate size R * g estimated with the SFSE as a function of different aggregate compactnesses for systemic hematocrits of 10 and 20%. The solid line represents the actual aggregate size r ag /a=6.32. b)Linear relationships between W * and R * g . Results presented here correspond to the configuration where φ i varies and r ag /a is fixed.

Figure 5 .

 5 Figure 5. a) Values of r * ag /a and φ * i estimated by the EMTSFM as a function of the actual aggregate radius for the three systemic hematocrits of 10, 20 and 30%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of r * ag /a and φ * i . Figure 6. a) Values of r * ag /a and φ * i estimated by the EMTSFM as a function of the actual aggregate compactness for the systemic hematocrits of 10 and 20%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of r *ag /a and φ * i . Figure7. Frequency-dependent BSCs computed with the SFM for different aggregate sizes and a constant aggregate compactness φ i =60% at systemic hematocrits of 10, 20 and 30%, and corresponding fitting with the GM.

Figure 8 .

 8 Figure 8. a) Values of a * ef f /a and φ * i estimated by the GM as a function of the actual aggregate radius for the three systemic hematocrits of 10, 20 and 30%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of a * ef f /a and φ * i . Figure 9. a) Values of a * ef f /a and φ * i estimated by the GM as a function of the actual aggregate compactness for the systemic hematocrits of 10 and 20%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of a * ef f /a and φ * i . Figure 10. Logarithm of the error between the simulated BSC SF M and the fitted curves with the three scattering models GM, SFSE and EMTSFM. a) As a function of the actual aggregate size for the clustering configuration where r ag /a varies and φ i is fixed. b) As a function of the actual aggregate compactness for the clustering configuration where φ i varies and r ag /a is fixed.

Figure 12 .Fitted line, r 2

 122 Figure 12. a) Values of r * ag /a and φ * i estimated by the EMTSFM as a function of the actual aggregate radius for the three systemic hematocrits of 20, 30 and 40%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of r * ag /a and φ * i . Figure 13. a) Values of a * ef f /a and φ * i estimated by the GM as a function of the actual aggregate radius for the three systemic hematocrits of 20, 30 and 40%. Also represented are actual values of r ag /a and φ i . (b) Corresponding relative errors of a * ef f /a and φ * i .

  Fig. 12.

  August 20, 2013 DRAFT blood structural parameters (i.e., W * and R * g ) and the total attenuation[28] [29]. Future improvements of the EMTSFM should incorporate the tissue attenuation as for the SFSAE. It means that the EMTSFM should be slightly modified by introducing the attenuation term to estimate simultaneously the RBC aggregate size, compactness and the total attenuation.

	APPENDIX A
	NUMERICAL COMPUTATION OF THE STRUCTURE FACTOR S ag FOR HARD-CYLINDERS IN 2D

TABLE CAPTION Table I .

 CAPTIONI Acoustical properties of blood found in[START_REF] Savery | High-frequency ultrasound backscattering by blood: Analytical and semi-analytical models of the erythrocyte cross section[END_REF] and[START_REF] Shung | On the ultrasound scattering from blood as a function of hematocrit[END_REF].

Table II .

 II Values of the aggregate radius and compactness used for computation of the simulated BSC SF M , and values of parameters found with the SFSE. Aggregating conditions: r ag /a varies, φ varies, φ i =60% (except in the case of diaggregated RBCs where φ i =100%). The parameter ǫ indicates the relative error.

	August 20, 2013	DRAFT

ACKNOWLEDGMENTS

This work was supported by the Fonds Incitatifs Recherche of the Université de Provence and by an operating grant from the Canadian Institutes of Health Research (♯MOP-84358).


Sandra S Bigot

Guy Louarn

Nasreddine Kébir

Fabrice Burel

Click grafting of seaweed

Keywords: PVC, ionic liquid, polysaccharides, click chemistry

come

Introduction

In the context of the world ecological and economical crisis, the development of new straightforward and green chemistry process is of great interest. During the last decade, room temperature ionic liquids (ILs) had known a great development as media and catalysts for organic reactions [1][2][3][4][5][6] . Their recyclable characteristics and their catalytic activities make them a green alternative to organic solvents. Moreover, ILs exhibit many others attributes like chemical-thermal stability and ability to solvate a broad range of inorganic and organic compounds. Polysaccharides are one of the most important reactive and active biopolymers. They have many applications in the medical field. In particular, heparin and chitosan, were grafted onto biomaterial surfaces in order to improve biocompatibility and/or to prevent bacteria, blood platelets or protein attachment. Often, the grafting strategies were multi-step ones, including uncontrolled surface physical treatments or multi-step chemical modifications of the surface and/or of the polymer 7 . Poly(vinylchloride) (PVC) represents 25% of plastics used in medical applications 8 . It is widely used as biomaterial. However, as far as we know, only few examples of grafting of polysaccharides onto PVC surfaces were described in the literature [9][10] . Actually, C. Mao et al 9 grafted O-butyrylchitosan (OBCS) onto PVC surface. OBCS was previously bound to 4-azidobenzoic acid and then it was covalently immobilized onto PVC surfaces by ultraviolet irradiations. Besides, Asadinezhad et al 10 functionalized PVC surface by plasma treatment in air radical and graft copolymerization of acrylic acid through grafting from pathway. Then, chitosan monolayer was bound onto surfaces. Z. Zhou et al 11 grafted heparin onto aminated PVC after modification of the polysaccharide chains with N-hydroxysuccinimide, using 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide as coupling agent.

Recent studies have reported the dissolution and chemical modification of polysaccharides, especially cellulose, in ILs such as N,N'-diimidazolium chloride salts 12 , N,N'-diimidazolium formate, phosphate or phosphonate [13][14] and 1-butyl-3-methylimidazolium chloride 15 . Indeed, because of high degree of intra-molecular hydrogen bonds, cellulose exhibit a high crystallinity, this makes difficult its dissolution in common organic solvents. ILs cleave these hydrogen bonds by their ionic interaction and consequently, they dissolve the polymer 16 . Moreover, it has been reported that some seaweed polysaccharides such as alginates, carrageenan, aragose and fucans [17][18][19][20][21] exhibited biological properties in good correlation with their chemical structures. Especially, sulfated polysaccharides from algae possess important pharmacological activities such as anticoagulant, antioxidant, antiproliferative, antitumoral, anticomplementary, anti-inflammatory, antiviral, antipeptic and antiadhesive activities [21][22][23][24] . In this work, we describe a green and click chemical process of grafting bioactive seaweed polysaccharides as well as methylcellulose, a strong bacteria and protein repellent polymer 25 , onto PVC surfaces. A first step consisted on displacement of chloride groups within the PVC surface by isothiocyanate groups, in water/DMSO mixture, to afford PVC-NCS surfaces. Then, unmodified polysaccharides were directly grafted onto these surfaces using [C2mim][(MeO)(H)PO 2 ] ionic liquid as green solvent and catalyst (Scheme 1).

Experimental

Commercial bulk polymerized PVC was obtained from Sigma-Aldrich, Lyon, France. The average molecular weights determined by GPC were w M =80 000 g/mol and n M =47 000 g/mol. Seaweed polysaccharides were provided from the CEVA (www.ceva.fr), Paimpol, France. Their characteristics are depicted in Table 1. Methylcellulose ( w M = 88000 g/mol, degree of methylation (DM) = 1.5-1.9), tetrabutylammonium hydroxide (TBAH) (40 wt% in water) and N-ethylimidazole (≥ 95%) were purchased from Sigma-Aldrich. Potassium thiocyanate (KSCN) was purchased from Merck, and dimethylphosphite (98%) from Alfa Aesar.

Characterizations

NMR spectra were recorded on a Bruker Avance (300MHz) Spectrometer. IR spectra were recorded on a FTIR (Perkin-Elmer) equipped with a diamond ATR device (Attenuated Total Reflection). XPS measurements were carried out at room temperature. An Axis Nova instrument from Kratos Analytical spectrometer with Al Kα line (1486.6 eV) as excitation source has been used. The core level spectra were acquired with an energy step of 0.1 eV and using a constant pass energy mode of 20 eV, to obtain data in a reasonable experimental time (energy resolution of 0.4 eV). The pressure in the analysis chamber was maintained lower than 10-7 Pa. The background spectra are considered as Shirley type and curve fitting is carried on with a mixture of Gaussian-Lorentzian functions [A.K. Berreshein, M. Mattern-Klosson, W. Fresenius, J. Anal. Chem. 341 (1991) 121]. The error in defining the position of peaks is estimated at about 0.1 eV. No surface cleaning, using Ar sputtering for example was made. We know from experience that carbonaceous atmospheric contamination on material usually occurs, but in our case, ion sputtering can be suspected of changing chemical composition and inducing structural damage. Surface energy of the materials were evaluated using Digidrop Goniometer (GBX, France), by static contact angle (θ i ) measurements on the sample surfaces with 3 liquids: water, diiodomethane and glycerol. The static contact angles were measured at the equilibrium time. At least five measurements on different surfaces were performed to calculate the mean contact angle and its standard deviation. According to the Owens-Wendt relationship ( 1 (

) d s d i p i p s d i i i 2 ) cos 1 ( γ + γ γ × γ = γ γ θ + 1
Where γ i , γ i d , γ i p are the solvent superficial tension parameters. Surface polarities (P) of the materials are calculated using equation ( 2):

(2) 100 P s p s × γ γ = Dynamic contact angle measurements with water were also performed to get the Advancing Angle (AA) and the Receding Angle (RA) as well as the surface hysteresis (Hys=AA-RA).

An AFM microscope (JPK instruments, NanoWizard, Berlin, Germany) was used for imaging and roughness measurements. AFM images were obtained by using the intermittent contact mode AFM in air. Classical silicon cantilevers were used (NanoWorld Pointprobe® NCHR). The average force constant and resonance were approximately 42 N.m -1 and 320 kHz, respectively.Topographic images were taken at different locations on the surface, in order to account for any heterogeneity in the coverage of the polysaccharides. AFM data were analyzed by the software SPM Image Processing v.3 from JPK Instruments, to extract the surface roughness (calculated on a 10 µm 2 area).

Preparation of PVC films

About 1g of PVC powder was dissolved in 40 mL of dichloromethane and poured into a Teflon mould. PVC films were then obtained by slow solvent evaporation under nitrogen atmosphere.

Preparation of PVC-isothiocyanate surfaces (PVC-NCS)

In a round bottom flask, KSCN (9.0×10 -2 mol) and tetrabutylammonium hydroxide (TBAH) (4.5×10 -2 mol) were dissolved in a 60 mL of DMSO/water (4/1) mixture. PVC sheets (1x1 cm 2 ) were immersed and the medium was heated at 60°C for 17h under slow-stirring. Then, the PVC sheets were thoroughly washed with water, rinsed with diethyl ether and dried at 60°C. The same reaction mixture could be recycled 4 to 5 times. FTIR: ν max /cm -1 2150 (S-CN), 2051 (N=C=S), 1600 (C=C).

Synthesis of 1-ethylimidazolium phosphate ([C2mim][(MeO)(H)PO 2 ])

The ionic liquid, i.e. 1-ethylimidazolium phosphate, was prepared as described in the literature [5]. Briefly, in a round bottom flask containing 225 mL of THF, N-ethylimidazole (622 mmol) and dimethyl phosphite (622 mmol) were added dropwise under nitrogen atmosphere at room temperature. Then, the mixture was stirred at 90°C for 2 days. Then, a biphasic system was obtained. The lower phase containing the IL was separated from the upper phase (THF) and washed with diethyl ether. The obtained ionic liquid (IL) was dissolved in dichloromethane, purified on neutral activated alumina and then dried under vacuum. 1 H NMR (300 MHz, CDCl 3 , δ ppm) 1.58 (t, 3H, J = 7.4 Hz, NCH 2 CH 3 ), 3.44 (d, 3H, J = 11.8 Hz, POCH 3 ), 4.01 (s, 3H, NCH 3 ), 4.32 (q, 2H, J = 7.4 Hz, NCH 2 CH 3 ), 6.96 (d, 1H, J = 5.41 Hz, PH), 7.32 (d, 2H, J = 11.4 Hz, NCHCHN), 10.40 (s, 1H, NCHN). 13 C NMR (75 MHz, CDCl 3, δ ppm) 15.8, 36.7, 45.5, 50.7, 121.7, 123.6, 139.3.

Grafting of polysaccharides onto PVC-NCS surfaces

Each polysaccharide was dissolved in the ionic liquid (C = 1 mg/mL) under appropriate conditions (Table 1). Then, the PVC-NCS sheets were added to these solutions at 60°C. After stirring for 24h, the PVC sheets were thoroughly washed with water, rinsed with diethyl ether and dried at 60°C. The Ionic liquid was recycled by dialysis in water. The polysaccharides were retained in the dialysis tube and the IL migrated outside the tube. Then, the IL was dried under vacuum.

Table 1

Results and discussion

Scheme 1

The chemical grafting of polysaccharide onto PVC surfaces was performed in two steps (Scheme 1). In a first step, chloride groups of PVC surface were substituted by isothiocyanate groups (-NCS) via a reaction with KSCN, in presence of TBAH as phase transfer catalyst 26 and DMSO/water (4/1) as solvent. The isothiocyanate groups resulted from isomerisation of thiocyanate ions. Nevertheless, Kameda et al. reported that phase transfer agents, such as TBAH, favored the substitution (SN 2 ) over the elimination of HCl 27 .

FTIR spectra of the treated surfaces (PVC-NCS) showed a characteristic band of isothiocyanate groups at ~2050 cm -1 . However, two bands at ~2150 cm -1 and at ~1600 cm -1 were also observed and were ascribed to thiocyanate groups and to double bonds resulting from the elimination, respectively.

The surface elemental compositions of PVC and PVC-NCS surfaces were determined by XPS analysis. The XPS results are shown in Table 2. The contribution of poly(dimethylsiloxane) (PDMS), used as PVC plasticizer, was removed from the elemental analysis. As expected, XPS survey scans obtained from the PVC surface gave two signals corresponding to C and Cl with contributions of 66.6% and 33.4%, respectively. PVC-NCS surface revealed the presence of N (1.5%) and S (1.3%) suggesting that the thiocyanation reaction was successful. However, the percentage of Cl decreased dramatically confirming a PVC dechlorination (the elimination) under the operating conditions. The XPS survey scans for the main components were analyzed to assign the peaks C 1s , N 1s and S 2p to chemical functions (Fig. 1 and Table 2). The C 1s peak was decomposed into four peaks. The peaks ranging from 284 to 285 eV are ascribed to C of saturated and unsaturated hydrocarbon chains. The peak at 286.4 eV is attributed to C making single bonds with N, O, S or Cl. The contribution of this peak is too weak (6.2%) confirming the elimination of Cl during chemical treatment. The high resolution XPS spectra of the N 1s region revealed three peaks. The peak at 394.8 eV is owing to residual thiocyante ions adsorbed onto the surface. The peak at 399.9 eV is ascribed to grafted isothiocyanate and thiocyante groups. The peak at 401.5 is due to ammonium groups of residual TBAH, which exhibited also a very weak signal at 532.2 eV, undetectable in the survey spectra (<1%). The high resolution XPS spectra of the S 2p region revealed two peaks; each peak corresponds to two spin-orbit transitions. As usual, the main spin-orbit transition 3/2 was used for quantification. The peak at 136.8 eV is ascribed to the S in thiocyante groups and residual thiocyante ions. The peak at 168.1 eV is ascribed to the S in isothiocyante groups.

Static and dynamic contact angle measurements performed on modified and unmodified PVC surfaces are reported in Table 3. In comparison with PVC, the PVC-NCS surface showed a significant decrease of static and dynamic contact angles with water. The AA is characteristic of the wettability of the low-energy or hydrophobic part of the surface whereas the RA is more characteristic of the high-energy or hydrophilic part. The balance between the two parts was reduced as suggested by the decrease of hysteresis value (from 48 to 33%). This indicated that thiocyanation increased considerably the hydrophilicity of the PVC surface. In addition, the surface polarity increased from 2.3 to 32.7%. AFM data of the PVC and PVC-NCS surfaces are depicted in Figure 1 and Table 3. One can observe that the surface treatment with NCS led to changes in surface morphology without significant variation in the surface roughness.

In a second step, seaweed polysaccharides as well as MethylCellulose dissolved in an ionic liquid (IL), i.e. [C2mim][(MeO)(H)PO 2 ], were directly grafted onto PVC-NCS surface (Scheme 1).

The IL has been chosen for three reasons:

-It solubilized polysaccharides and not PVC.

-It was a good catalyst for the grafting reaction.

-The reaction could not be performed in the classical polysaccharide solvents such as water or alcohols, because these solvents reacted with NCS. In addition, in these conditions an external catalyst was need.

The seaweed polysaccharides used were laminarin, ulvans, fucans and zosterin. Ulvans and fucans are sulfonated polymers. These polysaccharides contain also a small amount of nitrogen (Table 1) owing to presence of residual proteins coming from their natural media.

The conditions of polysaccharide dissolution in IL were optimized and reported in Table 1.

After complete dissolution, the PVC-NCS samples were immersed in the polysaccharide solution in IL (1 mg/mL) during 24h at 60°C, to allow the chemical grafting. Besides, the addition of an external catalyst, namely the dibutyltin dilaurate (DBTL), to the reaction medium did not change the final surface properties (results not shown), which suggests a catalytic effect of the IL. Indeed, it is well known that ammonium salts have catalytic activities in nucleophilic addition reactions, such as the reaction of hydroxyle groups with isocyanate groups 28,29 . The obtained PVC grafted polysaccharide surfaces were thoroughly cleaned before characterizations.

In comparison with the PVC-NCS surfaces, the elemental analysis by XPS of the grafted surfaces showed the presence of O and the increase of N percentage, suggesting the presence of polysaccharides and their residual proteins onto the surface. However, the percentages of S being affected by the presence of proteins, the sulfonation degree of some polysaccharides, they varied in an unexpected manner. Furtheremore, the grafting of the Ulvan 816 seemed to be less efficient since it exhibited the lowest percentages of O and S. This result may be explained by the high molecular weight of this polymer limiting its mobility and reactivity onto the surface. Table 2 As a typical example, Figure 1 shows the high resolution XPS spectra of the C 1s , N 1s and S 2p regions for the PVC-NCS surface grafted with Lam812. The C 1s peak was fit with five components. The peaks between 284 and 285 eV arose from hydrocarbon groups (C-H, C-C) and C-Si bond which resulted from the PVC plasticizer (PDMS). The peak at 286.4 eV was assigned to carbons of C-Cl, C-O, C-N, C-S bonds as well as C-C-C bonds from polysaccharides rings. This peak contribution varied from 18 to 28% where about 6% was due to the surface control (PVC-NCS). The peak at 287.7 eV was ascribed to C in acetal or hemiacetal functions in polysaccharides. The peak at 288.9 eV was attributed to C in the thiocarbamate links (NH-(C=S)-O), in the carboxylic acid function of some polysaccharides (Ulvans) and proteins, and in the amide function of the residual proteins. The N1s peak was decomposed into three subpeaks. The peak at 401.5 eV resulted from C-NH 3 + and C-N + related to residual proteins and TBAH, respectively. The main peak at 399.9 eV was assigned to the thiocarbamate link and to the thiocyanate and isothiocyanate groups. The contribution of the peak at 384.8 eV, corresponding to free thiocyanate ions adsorbed onto the surface, was observed only in some grafted surfaces and with a weak contributions. The S 2p high resolution XPS spectra of the grafted surfaces exhibited the same peaks than PVC-NCS surface but with different contributions. The peak at 136.8 eV arose from thiocyanate groups and ions. The peak at 168.1 eV was more intense than the peak at 136.8 eV, in the case of surfaces grafted with the seaweed polysaccharides and was ascribed to unreacted isothiocyanate groups, to thiocarbamate functions and to sulfonate groups in some polysaccharides.

PVC-NCS surface grafted with polysaccharides exhibited increased contact angles with water, in comparison with the Ungrafted surfaces. The static contact angles varied in the range of 56° to 61°, suggesting that these surfaces kept their hydrophilic character. AA with water was very close to static angle, whereas RA was in the range of 15 to 20°. Thus, polysaccharide surfaces exhibited hysteresis in the range of 35 to 47° which was higher than the one of PVC-NCS. This result is mainly explained by the increase of the hydrophilic-hydrophobic balance after polysaccharide grafting. Besides, hysteresis can also be explained by surface heterogeneity, texture and roughness.

Table 3 AFM data of the PVC-NCS surface grafted with polysaccharides are also depicted in Table 3.

A typical example of AFM height images is given in Figure 1 (PVC-Lam822). These images revealed a modification of surface morphology after polymer grafting. The PVC grafted polysaccharide surfaces exhibited roughness values which were slightly higher than those of the PVC-NCS surface.

Conclusions

This work showed the possibility of grafting a range of bioactive polysaccharides onto PVC surfaces using a green and straightforward process. This process involved a reaction between alcohol groups of polysaccharides and isothiocyanate groups covalently bounded to PVC surfaces. 1-ethylimidazolium phosphate, an ionic liquid, was used as solvent for polysaccharides and as catalyst for the grafting reaction. This process is expected to be used to graft any specific polymer bearing alcohol, amine or thiol groups onto a surface bearing isothiocyanate groups.

M

  values were determined by High Performance Size Exclusion Chromatography (HP-SEC) using Pullulan calibration curve ranged from 5800 to 1600000 g/mol. ** Data supplied by the CEVA: N (%) was determined by the Kjeldhal technique, S (%) was determined by an internal method (turbidimetry after mineralization).

Figure 1 .

 1 Figure 1. C1s, N1s and S2p high resolution spectra of (a) PVC-NCS and (b) PVC-Lam822.

Figure 2 .

 2 Figure 2. IC-AFM height images (2 µm x 2 µm) of (a) PVC, (b) PVC-NCS and (c) PVC-Lam822, acquired in intermittent contact mode. The height scale was 10 nm for the 3 images.

Scheme 1 .

 1 The chemical approach for grafting polysaccharides onto PVC surfaces.

							S			S
							C			C
	Cl	Cl	Cl	Cl	KSCN/TBAH		N		Cl	N	Cl
		PVC			DMSO/H 2 O 60°C -17h				PVC
					OH	OH		IL =	O
										N	N	O	P	OCH 3
										H
					OH	OH			60°C -24h
									OH	OH
						S	C	O	S	C	O
							NH	Cl	NH	Cl
									PVC

Table 1 .

 1 Some characteristics of the used seaweed polysaccharides.

	Polysaccharide	Seaweed	Mw (g/mol) *	S (%) **	N (%) **	Conditions of dissolution in IL (C=1 mg/mL)
	Methylcellulose	/	80000	ND	ND	60°C (15 min)
	Laminarin 822	L. saccharina parmeat	6990	ND	ND	60°C (15 min)
	Ulvan 901	U. clathrata	360000	2.2	1.6	80°C (15 min)
	Ulvan 815	U. rotundata	620300	3.14	1.2	80°C (15 min)
	Ulvan 816	U. compressa	876250	2.6	1.9	80°C (15 min)
	Fucan 812	Fucus/Ascophyllum	438600	ND	ND	80°C (15 min)
	Zosterin 900	Zosteraceae	340200	ND	ND	95°C (30 min under sonication)
	* Data supplied by the CEVA: w

Acknowledgements

This work was a part of the project "Green Intelligent Material (GIMs)" which belongs to the "Interreg IV France (Channel) -England" European program. Authors thanks the Europe and the "haute Normandie" Region for their financial and logistic supports.

* not detected from the XPS survey spectra.


Christophe Noger
email: christophe.noger@iat.cnam.fr

Patrick Gilliéron
email: patrick.gillieron@renault.com

Banc expérimental d'analyse des phénomènes aérodynamiques générés par le dépassement de deux véhicules automobiles

Keywords: Aérodynamique, Automobile, Transitoire, Vent latéral. 1

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

Introduction

Lors du processus de dépassement, les champs aérodynamiques qui s'exercent sur les véhicules dépassé et dépassant interfèrent de manière transitoire et les véhicules sont soumis à de rapides sollicitations autour des axes de lacet, de roulis et de tangage. Ces sollicitations peuvent avoir des incidences significatives sur la stabilité dynamique, le confort, la sécurité et aucun moyen d'essais n'est aujourd'hui disponible pour analyser ces phénomènes. Les ingénieurs restent donc démunis pour évaluer les comportements dynamiques des véhicules lors des processus transitoires de dépassement avec ou sans vent latéral. Des approches analytiques et des simulations numériques basées sur des géométries simplifiées et des configurations quasistatiques ont été développées à la Direction de la Recherche de RENAULT, (Gilliéron, 2003). L'expertise développée reste à valider par l'expérience et les influences des effets transitoires doivent être caractérisées et quantifiées.

Une collaboration de recherche a donc été développée entre l'Institut Aérotechnique de Saint-Cyr l'École et la Direction de la Recherche de RENAULT pour mettre au point un banc expérimental de mesures et d'analyses. Les phénomènes physiques sont simulés en soufflerie automobile à l'aide de deux maquettes à échelle 1/5 dont une reste fixe par rapport au référentiel de la veine d'essais. Le dépassement s'obtient en déplaçant une seconde maquette à vitesse 2 constante dans une direction parallèle à l'axe longitudinal (X) de la maquette fixe. Une balance aérodynamique, reliée à la maquette fixe, permet de mesurer les torseurs aérodynamiques des maquettes en position de véhicules dépassant et dépassé. Le torseur aérodynamique instationnaire qui s'exerce sur le véhicule dépassé (resp. dépassant) s'obtient en déplaçant la maquette mobile dans le sens opposé au (resp. même sens que le) sens du vent incident. Les expérimentations peuvent s'effectuer pour différentes vitesses de vent o V de la soufflerie et différentes vitesses de déplacement r V de la maquette mobile. Le banc permet également d'analyser les influences du vent latéral (angle de dérapage ) et de l'espacement transversal (Y) entre les deux maquettes. Les différents systèmes de mesure et d'acquisition développés pour ce banc ainsi que les premiers résultats sont successivement présentés.

2

Éléments théoriques L'évolution des efforts et des moments aérodynamiques appliqués sur les véhicules dépassé et dépassant est pseudo harmonique de fréquence f donnée par (Gilliéron et al (2000) et Gilliéron ( 2003)) :

h r L V f  (1) où h
L est la distance qui sépare la face avant du véhicule dépassant à deux instants séparés d'une période T (T= 1/f), et r V la vitesse relative du véhicule dépassant par rapport au véhicule dépassé. En pratique et pour les résultats présentés dans ce papier, la distance h L est comprise entre 3 et 4 fois la longueur des véhicules dépassant ou dépassé (Gilliéron, 2003).

Le rapport des vitesses relative r V et de déplacement o V du véhicule à dépasser sur route et en soufflerie devant être respecté pour satisfaire les conditions de similitudes en fréquence réduite, la fréquence du phénomène pseudo harmonique pour les essais en soufflerie est donnée par :

route o h o V V L V f r soufflerie          (2)
Pour les souffleries utilisées, la vitesse o V de l'écoulement autour des maquettes est égale à la vitesse o V de déplacement autorisée sur route et autoroute. Les fréquences f des phénomènes pseudo harmoniques en soufflerie et sur route sont alors égales pour une même longueur L de véhicule. En pratique les valeurs du rapport r V / o V se situent entre 0.1 et 0.5.

Souffleries et maquettes

La simulation du processus de dépassement peut indifféremment s'effectuer dans l'une des deux souffleries automobiles S4 ou S10 de l'Institut Aérotechnique de Saint Cyr l'École, voir figure 1. Les veines d'essais de ces souffleries sont à fentes longitudinales et les hauteurs, les largeurs et les longueurs sont respectivement égales à 3, 5 et 10m. Les vitesses de vent peuvent atteindre 40m.s -1 pour la soufflerie S4 et 55m.s -1 pour la soufflerie S10.

La maquette mobile doit être légère et résister aux accélération et décélération (40 m.s -2 ) associées aux phases d'augmentation et de réduction de vitesse. Cette maquette est alors réalisée en carbone. La maquette fixe est réalisée en bois ou en PVC et est reliée à une balance aérodynamique qui permet de mesurer les six composantes du torseur instationnaire qui s'exerce sur la maquette. Elle peut être équipée sans difficulté de systèmes de mesure capables de relever 3 les évolutions instationnaires de grandeurs physiques à la surface ou dans le sillage, (pressions et vitesses). Enfin, les dimensions du banc et les caractéristiques des matériels utilisés permettent d'analyser les interactions aérodynamiques entre une berline et une berline, une berline et un poids lourd ou un poids lourd et un poids lourd.

5

Premiers résultats d'essais

Les processus de dépassement, décrits à partir de positions statiques ( r V nulle) et dynamiques ( r V  10 m.s -1 ), sont analysés pour différents espacements latéraux et différentes vitesses o V de l'écoulement incident. C de la maquette en position de véhicule dépassé pour un dépassement décrit à partir de positions statiques ( r V = 0 m.s -1 ).

Fig. 5 -Évolution du coefficient de traînée aérodynamique x C de la maquette en position de véhicule dépassé pour différentes vitesses de dépassement r V . V égale à 10 m.s -1 . Ces premiers résultats confirment les résultats numériques obtenus par Gilliéron (2003).

X/L

Cy

-5 -4 -3 -2 -1 0 1 2 3 4 5 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Y= 0.25l -= 0°Y = 0.5l -= 0°Y = 0.7l -= 0° X/L Cy aller -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.3 -0.2 -0

Conclusion et perspectives

Un banc d'essais permettant de simuler et d'analyser les phénomènes transitoires associés au processus de dépassement de deux véhicules a été réalisé et est présenté. Ce banc d'essais, conçu et réalisé par l'Institut Aérotechnique de Saint Cyr l'École avec le soutien de la Direction de la Recherche de RENAULT, permet de simuler le processus transitoire de dépassement à partir de deux maquettes de véhicule à échelle 1/5 pour des vitesses relatives entre les deux maquettes qui varient entre 0 et 10 m.s -1 . Sa conception permet d'étudier l'influence de l'espacement transversal et du vent latéral sur le torseur aérodynamique instationnaire qui s'exerce sur chacune des deux maquettes. Les différents systèmes de mesure et d'acquisition développés pour ce banc ainsi que les premiers résultats obtenus sont successivement présentés. Ces résultats confirment le caractère pseudo harmonique des évolutions des coefficients aérodynamiques relevés pour les véhicules dépassé et dépassant et permettent de déterminer des temps caractéristiques pour les évolutions des efforts et des moments autour des axes de roulis, de tangage et de lacet.

Ce banc d'essais fournit dès à présent aux ingénieurs et aux chercheurs un nouveau moyen d'investigation qui permet de mieux comprendre les phénomènes physiques associés aux processus transitoires. Ce moyen d'essais permettra de développer de nouvelles stratégies et de nouvelles techniques pour améliorer le confort et la sécurité des véhicules automobiles. Des recherches sont en cours pour prendre en compte et simuler les réponses mécaniques de la caisse et des trains des véhicules lors du processus de dépassement.

Références

Fig. 1 -4

 1 Fig. 1 -Soufflerie automobile S4 de l'Institut Aérotechnique (IAT) de Saint-Cyr l'École. 4 Banc d'essais Le banc d'essais et d'analyse du processus de dépassement est réalisé à partir d'un faux plancher en bois qui est fixé sur le sol de la soufflerie et qui occupe toute la largeur et la longueur de la veine d'essais, voir figure 2. Ce faux plancher, de hauteur 215 mm, permet de loger l'ensemble des matériels de mesure ainsi que les différents systèmes de fixation et de déplacement des maquettes fixe et mobile.

Fig. 2 -Fig. 3 -

 23 Fig. 2 -Banc d'essais avec un dérapage nul (vue de dessus).

Fig. 4 -

 4 Fig. 4 -Évolution du coefficient de traînée aérodynamique xC de la maquette en position de véhicule dépassé pour un dépassement décrit à partir de positions statiques ( r V = 0 m.s -1 ).

Fig. 6 -Fig. 7 -

 67 Fig. 6 -Évolution du coefficient aérodynamique d'effort latéral y C de la maquette en position de véhicule dépassé pour un dépassement décrit à partir de positions statiques ( r V = 0 m.s -1 )

  Gilliéron P., 2003, Detailed Analysis of the Overtaking Process, PREDIT2 Project, "Unsteady Aerodynamics of Ground Vehicles and their Environment", Journal Mechanical Engineering, JME 1-2003, ISSN 0039-2472. Noger C., 2002, "Étude expérimentale du processus instationnaire de dépassement de deux véhicules", Contrat IAT-CNAM/ RENAULT, Rapport d'essais n°2446/A. Gilliéron P. et Chometon F., 2000, "Simulation du coup de vent latéral, Attentes du secteur automobile et État de l'art", Société des Hydrauliciens de France, Interactions Mécaniques entre Fluides et Structures, Chatou, France, 23 et 24 novembre, La Houille Banche, Revue Internationale de l'Eau, n°3/4. Ahmed S.R., Ramm R. and Faltin G., 1984, Some salient features of the time-averaged ground vehicle wake SAE Technical Paper Series 840300.