Loic Guillot

PhD Pierre-Yves Boëlle

PhD Dominique Debray

PhD Harriet Corvol
email: harriet.corvol@aphp.fr

SERPINA1 Z allele is associated with cystic fibrosis liver disease

Keywords: SERPINA1 Z, cystic fibrosis, liver disease, cirrhosis, portal hypertension

Purpose: The SERPINA1 Z allele has been associated with cystic fibrosis (CF)-related liver disease (CFLD), a common manifestation in patients with CF. We estimated the incidence of CFLD according to the SERPINA1 genotype among 3,328 CF patients with CFLD phenotype information.

Methods:

The associations of SERPINA1 Z (rs28929474) and S (rs17580) alleles with age at CFLD onset and the development of CFLD-related complications (severe liver disease with cirrhosis, portal hypertension, esophageal varices) were analyzed.

Results: Overall, 3% of patients carried the SERPINA1 Z allele and 13% the S allele. The cumulative incidence of CFLD increased more rapidly in patients carrying the Z allele (HR = 1.6; 95% CI = 1.1-2.4, P = 0.019), reaching 47% at age 25 compared to 30% for non-carriers. Although the increase in risk was similar for patients with severe CFLD (HR = 1.5, 95% CI = 0.7-3.2, P = 0.31) it did not reach significance due to limited number of severe cases among the Z allele carriers. No significant effect was found for the S allele.

Conclusion:

CF patients carrying the SERPINA1 Z allele experience an increased risk of developing CFLD and related complications compared to non-carriers patients. Routine SERPINA1 Z genotyping upon CF diagnosis is warranted for selecting patients worthy of closer liver disease monitoring.

INTRODUCTION

Cystic fibrosis (CF) is the most common, severe, autosomal recessive genetic disease in Caucasians caused by pathogenic variants in the gene encoding the cystic fibrosis transmembrane conductance regulator (CFTR) protein, a chloride channel expressed in epithelial cells. [START_REF] Riordan | Identification of the cystic fibrosis gene: cloning and characterization of complementary DNA[END_REF] The disease affects several organs, including the lungs, pancreas, intestine, and liver. Cystic fibrosis-related liver disease (CFLD) is a manifestation of CF in the liver involving a wide range of hepatobiliary abnormalities. Focal biliary cirrhosis is the most clinically relevant form of CFLD, since extension of the initially focal fibrogenic process may lead to multilobular biliary cirrhosis with subsequent portal hypertension and related complications. [START_REF] Debray | Best practice guidance for the diagnosis and management of cystic fibrosis-associated liver disease[END_REF][START_REF] Stonebraker | Features of Severe Liver Disease With Portal Hypertension in Patients With Cystic Fibrosis[END_REF] Currently, multilobular cirrhosis ranks as the third cause of death after respiratory failure and transplantation-related complications in patients with CF. [START_REF] Rowland | Outcome in patients with cystic fibrosis liver disease[END_REF] Although CF is recognized as a monogenic disease, considerable phenotypic diversity exists among patients with the same CFTR pathogenic variants. [START_REF] Cutting | Cystic fibrosis genetics: from molecular understanding to clinical application[END_REF][START_REF] Guillot | Lung disease modifier genes in cystic fibrosis[END_REF][START_REF] Mckone | Effect of genotype on phenotype and mortality in cystic fibrosis: a retrospective cohort study[END_REF] Besides environmental factors, additional genetic modifiers have been shown to contribute to this variability. [START_REF] Cutting | Cystic fibrosis genetics: from molecular understanding to clinical application[END_REF][START_REF] Guillot | Lung disease modifier genes in cystic fibrosis[END_REF] We have assembled a large cohort of patients as part of the French CF Modifier Gene Study, and recently reported the incidence of CFLD and severe CFLD in these patients. [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF] We further found that the risk of CFLD increased with age, with a frequency up to 32% by age 25, and was associated with several risk factors such as male sex, CFTR F508del homozygosity, and history of meconium ileus at birth. SERPINA1 gene has been implicated in the development and progression of CFLD. 9 SERPINA1 encodes the alpha-1 antitrypsin (AAT) protein, which is synthesized in large quantities by the liver and then secreted into the serum. Several pathogenic variants within the SERPINA1 gene have been identified to cause AAT deficiency that predisposes the individual to liver disease and early-onset emphysema. The most common variants are the Z and S alleles, each caused by a single nucleotide polymorphism. [START_REF] Bartels | Real time PCR detection of the PI*Z and PI*S mutations associated with alpha-1 antitrypsin deficiency[END_REF] The Z variant is the allele overwhelmingly associated with liver disease. [START_REF] Zorzetto | SERPINA1 gene variants in individuals from the general population with reduced alpha1-antitrypsin concentrations[END_REF] Indeed, the Z-type AAT protein folds abnormally during biogenesis in the endoplasmic reticulum of hepatocytes, and is retained intracellularly rather than efficiently secreted, resulting in low serum AAT levels. [START_REF] Stoller | A review of alpha1-antitrypsin deficiency[END_REF] The intracellular accumulation of AAT mutant Z proteins within hepatocytes can cause liver injury, cirrhosis, and hepatocellular carcinoma by triggering a cascade of chronic hepatocellular apoptosis, regeneration, and end-stage organ injury. [START_REF] Teckman | Liver disease in alpha-1 antitrypsin deficiency: current understanding and future therapy[END_REF] Individuals who are heterozygous for AAT, carrying one protease inhibitor normal M allele and one mutant Z allele (PiMZ or MZ) are generally considered asymptomatic and healthy with regard to liver disease. It is commonly accepted that those who are compound heterozygotes for the S and the Z alleles of AAT (PiSZ) may develop liver disease with identical manifestations to that of PiZZ patients, whereas liver disease is absent in PiSS individuals. [START_REF] Teckman | Liver disease in alpha-1 antitrypsin deficiency: current understanding and future therapy[END_REF] In CF, a case-control study revealed that carriers of the SERPINA1 Z allele were more common among patients with severe CFLD, with a large odds ratio of 5. [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] However, the study focused on patients with severe liver disease defined as cirrhosis showing signs of portal hypertension, preventing determination of the actual risk of developing CFLD associated with the variant.

Therefore, in the present study, we estimated the cumulated incidence of CFLD according to SERPINA1 genotype in a cohort of French CF patients of unprecedented size (n = 3,328).

MATERIAL AND METHODS

We assembled the French CF Modifier Gene Study cohort, including CF patients treated in French CF centers since 2004 as previously described (see also Supplementary Information). [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF] In brief, 4,798 patients with CF were recruited for the study, corresponding to approximately 80% of all French patients with CF. [START_REF] Bellis | Registre Français de la mucoviscidose -Bilan des données[END_REF] Among this group, 3,328 CF patients with pancreatic insufficiency born after 1985 were available to evaluate the CFLD incidence and risk factors, including patients with severe CFLD. [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF] CFLD was defined according to the European Best Practice Guidance by Debray et al. [START_REF] Debray | Best practice guidance for the diagnosis and management of cystic fibrosis-associated liver disease[END_REF] , and patients with cirrhosis, portal hypertension, and/or esophageal varices were classified as having "severe CFLD" (see also Supplementary Information). [START_REF] Debray | Cystic Fibrosis-related Liver Disease: Research Challenges and Future Perspectives[END_REF] The study was approved by the French ethical committee (CPP no. 2004/15), and the information collection was approved by the Commission Nationale de L'informatique et des Libertés (no. 04.404). Written informed consent was obtained from each patient and/or guardian.

Genotyping of the SERPINA1 Z (rs28929474) and S (rs17580) alleles was carried out using Kompetitive Allele Specific PCR (KASP) genotyping chemistry (LGC, Teddington, UK). In dbSNP (https://www.ncbi.nlm.nih.gov/projects/SNP/), rs28929474 is identified as an A/G variant with G being the ancestral allele, and rs17580 is identified as an A/T variant, with A being the ancestral allele.

Descriptive statistics were compiled as the mean ± standard deviation (SD) or percentages as appropriate. All patients were considered to be at risk of CFLD since birth and were censored at the time of the last visit without a CFLD diagnosis before January 2017. For other patients, the age at CFLD diagnosis was determined using the date of the first report in the medical record, allowing for "interval censoring" between birth and age of the first report when the date of onset was not precisely known (this was the case for 142 out of 605 patients with CFLD [24%]). Likewise, severe CFLD onset was defined as the first date that cirrhosis, portal hypertension, and/or esophageal varices were reported, and uncertainty regarding this date was considered as described above (the date of severe CFLD was interval censored in 19 out of 175 patients [11%]). We used the log-rank test adapted for interval-censored data for comparisons between the cumulative incidences curves, [START_REF] Fay | Rank invariant tests for interval censored data under the grouped continuous model[END_REF] and Cox regression adapted to interval-censored data to determine the association of factors linked to age at CFLD onset. [START_REF] Pan | Extending the Iterative Convex Minorant Algorithm to the Cox Model for Interval-Censored Data[END_REF] Confidence intervals were computed by the bootstrap method.

RESULTS

Clinical characteristics of the 3,328 CF patients included in this study along with the distribution of the SERPINA1 Z and S alleles are shown in Table 1. The minor allele frequencies were similar to those reported for Europeans (SERPINA1 Z (A): 1.4% vs. 2%, SERPINA1 S (T): 6.4% versus 6%). There were no patients homozygous for SERPINA1 Z. Details on the number of CF patients at risk at developing CFLD and severe CFLD, as well the cumulated number of CFLD events and of severe CFLD events for the whole cohort and according to SERPINA Z and S genotypes are provided in the supplementary information (Tables S1, S2 and S3).

Overall, 3% of patients carried the SERPINA1 Z allele and 13% the S allele. The cumulative incidence of CFLD increased more rapidly in patients carrying the SERPINA1 Z allele (HR = 1.6; 95% CI = 1.1-2.4, P = 0.019), reaching 47% at age 25 in the Z allele carriers versus 30% at age 25 in the others (Table 1 and Figure 1). The increase in risk was similar for patients with severe CFLD (HR = 1.5, 95% CI = 0.7-3.2, P = 0.31), although it did not reach significance. In this last case, there were only seven severe cases among the Z allele carriers, making the cumulated incidence curve difficult to estimate accurately. Adjusting on CFTR variants and meconium ileus did not change the strength of the association.

The effect of carrying one SERPINA1 S allele on CFLD risk was not statistically significant (Table 1 and Figure 1).

DISCUSSION

The French CF Modifier Gene Study provided an unprecedentedly large cohort of 3,328 pancreaticinsufficient patients with CF born after 1985. This gave an opportunity to obtain a more accurate estimate of the incidence of CFLD and severe CFLD with sufficient power to detect associations of clinical relevance. [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF] We found that the SERPINA1 Z allele was associated with an increased risk of developing CFLD, although the association was weaker than that previously reported. [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] Nevertheless, the incidence of CFLD increased more rapidly in patients carrying the SERPINA1 Z allele, with up to 47% of the Z allele carriers developing liver disease before the age of 25 compared to only 30% for the non-carriers patients.

In CF, the role of SERPINA1 in liver disease was first identified in a two-stage case-control study including CF patients from several countries worldwide. [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] Both the initial and the replication studies showed that severe CFLD was associated with the SERPINA1 Z allele (odds ratio of 4.72 and 3.42, respectively). [START_REF] Bartlett | Genetic modifiers of liver disease in cystic fibrosis[END_REF] Our cohort of patients with CF confirms this association, but demonstrates a smaller difference in risk than reported previously. For example, using the cumulative incidence at age 25, the odds ratio for the Z allele would only be 2 in our cohort. Yet, the difference in cumulative risk with age is still clinically relevant, as SERPINA1 Z carriers have a 50% greater risk of developing CFLD compared to non-carriers. Not surprisingly, we did not observe any association with the SERPINA1 S allele, which is recognized to be associated with reduced levels of AAT protein but no liver manifestations. [START_REF] Torres-Duran | Alpha-1 antitrypsin deficiency: outstanding questions and future directions[END_REF] Our study has limitations related to its design and the use of medical data records as a primary source of information. However, we previously described that selection bias due to differential mortality was likely small. [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF] A second potential issue is the rarity of the SERPINA1 Z allele in our cohort, which reduced the precision of estimation despite the large cohort size. Indeed, there were only 84 carriers of the SERPINA1 Z allele (PiMZ), and no homozygous PiZZ patients in the cohort. Furthermore, only seven of these patients experienced a severe disease form, making the cumulative incidence difficult to estimate accurately.

Obtaining a more accurate prediction of the risk of CFLD in CF patients remains an important issue, especially since the treatment commonly prescribed for its prevention, ursodeoxycholic acid, seems to have little to no effect. [START_REF] Boelle | Cystic Fibrosis Liver Disease: Outcomes and Risk Factors in a Large Cohort of French Patients[END_REF][START_REF] Cheng | Ursodeoxycholic acid for cystic fibrosis-related liver disease[END_REF] Given this situation, identifying biomarkers to predict the occurrence of CFLD is fundamental to improve monitoring disease progression and assessing the effects of novel therapies such as bile salt analogues, anti-fibrotics, and CFTR correctors and potentiators. SERPINA1 Z genotyping at the time of CF diagnosis could thus help single out a population of patients who should be more closely screened for liver disease. Gaining a better understanding of the genetic profiles of patients with CFLD will undoubtedly open new therapeutic avenues and help to develop prospective therapies focusing on high-risk groups.

ACKNOWLEDGEMENTS

Figure legend Figure 1 .

 legend1 Figure legend

The authors would like to thank the clinical research assistants involved in the data collection, namely, Pauline Touche, Tania Kombi, and Julie Mésinèle. We would like to especially thank the patients and families who participated in this study and the contributing French CF Centers (see the list of investigators in the Supplementary Information).

Financial Support

This work was supported by the Institut National de la Santé et de la Recherche Médicale (INSERM), (APHP), Agence Nationale de la Recherche (ANR), DGS, Association Vaincre La Mucoviscidose, Cystic Fibrosis Foundation (CORVOL16G0), Association Agir Informer Contre la Mucoviscidose, GIS-Institut des Maladies Rares, and Société Française de la Mucoviscidose.

Conflict of interest notification

The authors certify that they have no potential conflict of interest to declare in the subject matter or materials discussed in this manuscript.


Romain De Monchy

Julien Rouyer

François Destrempes

Boris Chayer

Guy Cloutier

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

Estimation of polydispersity in aggregating red blood cells by quantitative ultrasound backscatter analysis

Quantitative ultrasound techniques based on the backscatter coefficient (BSC) have been commonly used to characterize red blood cell (RBC) aggregation. Specifically, a scattering model is fitted to measured BSC and estimated parameters can provide a meaningful description of the RBC aggregates' structure (i.e., aggregate size and compactness). In most cases, scattering models assumed monodisperse RBC aggregates. This study proposes the Effective Medium Theory combined with the polydisperse Structure Factor Model (EMTSFM) to incorporate the polydispersity of aggregate size. From the measured BSC, this model allows estimating three structural parameters: the mean radius of the aggregate size distribution, the width of the distribution, and the compactness of the aggregates. Two successive experiments were conducted: a first experiment on blood sheared in a Couette flow device coupled with an ultrasonic probe, and a second experiment, on the same blood sample, sheared in a plane-plane rheometer coupled to a light microscope. Results demonstrated that the polydisperse EMTSFM provided the best fit to the BSC data when compared to the classical monodisperse models for the higher levels of aggregation at hematocrits between 10-40%.

Fitting the polydisperse model yielded aggregate size distributions that were consistent with direct light microscope observations at low hematocrits.

I. Introduction

Quantitative ultrasound (QUS) techniques providing insight into tissue microstructure may be based on spectral analysis of signals backscattered from biological tissues. QUS methods used to determine red blood cell (RBC) aggregate structures may rely on theoretical scattering models in order to fit a theoretical BackScatter Coefficient (BSC) to experimentally measured BSC. The BSC is defined as the power backscattered by a unit volume of scatterers per unit incident intensity per unit solid angle. Two scattering theories, called the Structure Factor Size Estimator (SFSE) and the Effective Medium Theory combined with the Structure Factor Model (EMTSFM), have been developed for practical assessment of RBC structural features. 1;2;3 The SFSE scattering theory approximates the structure factor with its second-order Taylor expansion. [START_REF] Yu | Experimental ultrasound characterization of red blood cell aggregation using the structure factor size estimator[END_REF] The SFSE estimates two physical parameters describing the microstructure of RBC aggregates: the packing factor, which increases with erythrocyte clustering, and the average aggregate isotropic diameter. However, experiments with porcine blood 1 and three-dimensional (3D) numerical simulations of RBC aggregates 4;5 recently revealed that these two structural parameters are correlated, thus reducing the BSC parameterization to one structural index.

The SFSE is therefore empirical and only allows assessing one feature of RBC aggregation, i.e. its mean fractal size. This is the reason why the EMTSFM was recently proposed. 3;5 The EMTSFM assumes that aggregates of RBCs can be treated as individual homogeneous de Monchy et al., JASA, p. 2 scatterers, which have effective properties determined by the acoustical characteristics and concentration of RBCs within aggregates. The approximation of RBC aggregates as homogeneous effective particles is combined with the structure factor to model acoustic wave interferences in concentrated blood medium. The EMTSFM allows characterizing the aggregate radius, and for the first time in QUS, the compactness of RBC aggregates. Based on 3D computer simulations of monodisperse aggregates, the EMTSFM was demonstrated to be more suitable than the former scattering theories (i.e., the SFSE and the classical Spherical Gaussian Model) for characterizing RBC aggregation. [START_REF] Franceschini | Comparison of three scattering models for ultrasound blood characterization[END_REF] Both scattering models SFSE and EMTSFM assume that all RBCs are aggregated in blood, and that aggregates are identical with sphere-shaped structures. Therefore, the BSC behavior obtained from the monodisperse EMTSFM 3 or numerical simulations 4;5 show pronounced frequency peaks. However, in experimental conditions, 1;2 the BSC behavior turned out to be smoother and peaks were less pronounced. The reason behind this discrepancy might be that blood presents a variable size of aggregates, and since the location of BSC peaks is expected to be different for different aggregate sizes, a relatively smoother BSC curve as a function of frequency might occur. Moreover, optical observations of flowing RBC aggregates at a low hematocrit showed that the aggregates are polydisperse in size. 6;7 In the field of spectral-based QUS techniques, the local monodisperse de Monchy et al., JASA, p. 3 approximation 8;9 or the polydisperse structure function 10;11 were experimentally studied to consider polydisperse scatterers in concentrated medium. Preliminary experimental results

on aggregating blood at a low hematocrit of 20% suggest that taking into account polydispersity can help better understand BSC from aggregating RBCs. [START_REF] De Monchy | Effective Medium Theory combined with a polydisperse Structure Factor Model for characterizing red blood cell aggregation[END_REF] Nonetheless, this work was limited by using the local monodisperse approximation, which is valid only in the case of moderate polydispersity. [START_REF] De Monchy | Développement et évaluation d'une théorie de milieu effectif combinée à un facteur de structure polydisperse pour la caractérisation ultrasonore de l'agrégation érythrocytaire[END_REF] Moreover, the benefits of polydisperse modeling was not evaluated against classical monodisperse modeling. Therefore, the present work combines the Effective Medium Theory (EMT) with the polydisperse structure function, and compares the capacities of the polydisperse EMTSFM and the monodisperse models (i.e., the SFSE and the monodisperse EMTSFM) to characterize the aggregate structure in the context of in vitro experiments. Two successive experiments were conducted: a first experiment on blood sheared in a Couette flow device coupled with an ultrasonic probe at hematocrits of 10, 20 and 40%, and a second experiment, on the same blood sample, sheared in a plane-plane rheometer coupled to a light microscope. The optical measurements were limited to low hematocrits of 10 and 20% because RBCs are opaque to light, which prevents optical measurements at physiological hematocrit of 40%. The aggregate size distributions estimated by ultrasound and by optical microscopic image segmentation were compared on the same blood samples. This allowed evaluating the performance of the ultrasonic technique using optical detection of aggregate boundaries as de Monchy et al., JASA, p. 4 the gold standard measure of aggregate structure.

II. Ultrasound backscattering theory

The incident wavelength λ is assumed to be larger than the RBC size. Consequently, the biconcave RBC shape can be approximated by a sphere of radius a having an equivalent volume V c = [START_REF] Saha | Assessment of accuracy of the structure-factor-size-estimator method in determining red blood cell aggregate size from ultrasound spectral backscatter coefficient[END_REF] 3 πa 3 (typically, V c ≈87µm 3 and a ≈2.75µm for human RBCs, whereas V c ≈61µm 3 and a ≈2.45µm for porcine RBCs). RBCs are described in terms of their mass density ρ c and compressibility κ c , and their surrounding medium (i.e., the plasma) is characterized by its mass density ρ 0 and compressibility κ 0 . We introduce the parameters γ κ = (κ c -κ 0 )/κ 0 and γ ρ = (ρ c -ρ 0 )/ρ c corresponding to the fractional variations in compressibility and mass density, respectively. It is also assumed that the mean distance between the red blood cells is not larger than the wavelength and that the aggregates of RBCs are spherical as this can occur in pathological cases. Indeed, the structures of RBC aggregates have a tendency to form clumps in pathological human blood, such as in diabetes mellitus,

σ ag (k) = σ ag,coh (k) + σ ag,inc (k), (1)
where the coherent component σ ag,coh corresponds to the average wave emerging from the effective scatterer, and the incoherent component σ ag,inc describes the fluctuation of the scattering wave around its average within the effective scatterer.

Using Born and far-field approximations, the backscattering amplitude from a single RBC aggregate Φ ag can be expressed as (see Eq. ( 5) in Ref. (13))

Φ ag (k) = k 2 (γ κ -γ ρ ) 4π V c F 0 (k; a) N j=1 e 2ik n 0 •r j , ( 2
)
where k is the wavenumber, n 0 is the incident wave direction, r j is the position in three dimensional space of the j th RBC (among the N RBCs inside an aggregate) and

F 0 (k; a) 2 = F (k; a)
is the spherical form factor of a RBC expressed as 17

F 0 (k; a) = 3 sin(2ka) -2ka cos(2ka) (2ka) 3 . ( 3
)
By assuming that the RBCs are grouped into spherical aggregates, of radius r ag and of

compactness φ i = N V c /V
σ ag,coh (k) = | Φ ag (k) | 2 ≈ k 4 (γ κ,ag -γ ρ,ag ) 2 16π 2 V 2 ag F (k; r ag ), (4)
where γ κ,ag = (κ ag -κ 0 )/κ 0 and γ ρ,ag = (ρ ag -ρ 0 )/ρ ag are the fractional variations in compressibility and in mass density of the effective scatterer, with ρ ag and κ ag defined as: 18

ρ ag = φ i ρ c + (1 -φ i )ρ 0 κ ag = [φ i /κ c + (1 -φ i )/κ 0 ] -1 . (5)
Note that the scatterer radius used in the spherical form factor of Eq. ( 4) was directly related to r ag because the RBCs' positions inside an aggregate are unknown in experimental conditions.

Inside an aggregate, each sphere of radius a produces a scattered wave, with density ρ c and compressibility κ c , in a surrounding medium of density ρ ag and compressibility κ ag throughout the aggregate. Therefore, the wavenumber inside the aggregate is modified as [START_REF] Morse | Theoretical Acoustics[END_REF] k ag = k √ ρ ag κ ag / √ ρ 0 κ 0 . The incoherent component of the differential backscattering cross-section of the effective scatterer is then modeled as (see Eq. ( 15) in Ref. (13)):

σ ag,inc (k) = |Φ ag (k ag ) -Φ ag (k ag ) | 2 ≈ k 4 ag (γ κ -γ ρ ) 2 16π 2 V ag V c φ i F (k ag ; a)S(k ag ; a, φ i ), ( 6
)
where S is the structure factor for a collection of randomly distributed identical RBCs of de Monchy et al., JASA, p. 7 radius a and of concentration φ i within an aggregate (analytically calculated as established by Wertheim [START_REF] Wertheim | Exact solution of the Percus-Yevick integral equation for hard spheres[END_REF] ).

Previous computer simulations demonstrated that the theoretical differential backscattering cross-section σ ag of an aggregate (given by Eqs. ( 1), ( 4) and ( 6)) allows to approximate satisfactorily the simulated σ ag within a large frequency bandwidth ranging from 1 to 100 MHz for spherical aggregate size up to r ag /a=9 (see Fig. 5 in Ref. (13)).

The relative errors between simulated and theoretical σ ag were shown to be less than 15%

for a product of the wavenumber times the mean aggregate radius kr ag ≤2.

Polydisperse and monodisperse structure functions

The EMT combined with the polydisperse structure function (named in the sequel the polydisperse EMTSFM) assumes that RBC aggregates are polydisperse in size and that the distribution p of the aggregates' radius follows a gamma probability density function (PDF): 10;20 p (rag,z) (r ag ) = r z ag Γ(z + 1)

z + 1 r ag z+1 e -(z+1)rag/rag , (7)
where r ag is the mean aggregate radius and z is the gamma width factor which is inversely related to the width of the distribution (a large value of z corresponds to a narrow size distribution). In the sequel of the paper, p will be assumed to be gamma distributed and the subscript (r ag , z) will be omitted.

Assuming that the aggregate compactness φ i is identical for all aggregates (except in de Monchy et al., JASA, p. 8 the case of a single RBC having an aggregate compactness of 1), the BSC can be written as the sum of an incoherent scattering contribution that sums up the intensities of the backscattered waves of each effective scatterer independently and of a coherent scattering contribution originating from the spatial correlation in effective scatterer positions:

BSC EMTpoly (k) = BSC inc (k) + BSC coh (k), (8)
with

BSC inc (k) = m ag ∞ 0 |Φ ag (k, r ag )| 2 p(r ag )dr ag , BSC coh (k) = m ag ∞ 0 Φ ag (k, r ag1 ) Φ ag (k, r ag2 ) H 12 (k)p(r ag1 )p(r ag2 )dr ag1 dr ag2 , (9)
where H 12 is the partial structure function between the effective scatterers of radii r ag1 and r ag2 , and m ag is the number density of effective spheres defined as:

m ag = φ ag (4/3)π ∞ a p(r ag )r 3 ag dr ag . (10)
The volume fraction of an effective scatterer φ ag is computed as follows:

φ ag = ψφ + (1 -ψ)φ/φ i , (11)
where φ denotes the hematocrit (i.e., the volume fraction of RBCs) and ψ is the ratio between the number of disaggregated RBCs (for which the aggregate compactness is

φ i = 100%
) and the total number of RBCs.

de Monchy et al., JASA, p. 9

When all RBCs are disaggregated, one has ψ = 1 and φ ag = φ. When all RBCs are aggregated with a compactness φ i , one has ψ = 0 and φ ag = φ/φ i . Note that, when r ag /a = 1 (disaggregated case), the differential backscattering cross-section is that of a single RBC:

σ b (k) = k 4 (γ κ -γ ρ ) 2 16π 2 V 2 c F (k; a). (12)
In the proposed study, the term BSC coh was computed using the structure function (denoted SF ) of the effective particle defined as the ratio between the total BSC and the incoherent BSC inc : 10;20

BSC coh (k) = BSC inc (k) SF (k) -1 (13)
where the expression of the structure function SF is given in Appendix B of Ref. (10).

In the case of a monodisperse size distribution (i.e., an ensemble of identical aggregates of radius r ag ), the monodisperse EMTSFM allows expressing the BSC as follows:

5 BSC EMTmono (k) = m ag σ ag (k)S ag (k), (14)
where σ ag is the differential backscattering cross-section of an aggregate computed using Eqs. ( 1), ( 4) and ( 6), and S ag corresponds to the structure factor of a collection of N ag identical effective particles of radius r ag randomly distributed (analytically calculated as established by Wertheim 19 ). approximates the SFM with a second-order Taylor expansion of the structure factor in k.

By assuming identical RBCs and identical isotropic aggregates, the SFSE model approximates the BSC as follows:

BSC SFSE (k) = mσ b (k) W - 12 5 (kaD) 2 ( 15
)
where m is the number of RBCs per unit volume (m = φ/V s ), σ b is the differential backscattering cross-section of a single RBC computed using Eq. ( 12), W is the low-frequency limit of the structure factor (S(k)| k→0 ) called the packing factor and D is the mean aggregate diameter expressed in number of RBCs. 1

III. Methods

A. Blood samples preparation Fresh porcine whole blood was obtained from a local slaughter house and anticoagulated with 3 g/L of ethylene diamine tetra acetic acid (EDTA). The whole blood was centrifuged and the plasma and buffy coat were removed.

The RBCs were then washed three times in a saline solution.

Three aggregating blood samples, denoted S10, S20 and S40, were prepared with an hematocrit of 10, 20 and 40%, respectively. The sample preparation was performed as C. Measured backscatter coefficients and attenuation First, attenuation of blood samples was estimated using a standard substitution method. [START_REF] Kuc | Estimating the acoustic attenuation coefficient slope for de Monchy et al., JASA, p. 29 liver from reflected ultrasound signals[END_REF] For the highest level of aggregation (i.e., for the shear rate of 3 s -1 ), averaged attenuation coefficients were 0.09, 0.38 and 0.81 dB/cm/MHz for the samples S10, S20 and S40, respectively. For the lowest level of aggregation (i.e., for the shear rate of 90 s -1 ), averaged attenuation coefficients were 0.03, 0.11 and 0.20 dB/cm/MHz for the samples S10, S20 and S40, respectively. For the disaggregated sample H4, averaged attenuation coefficient was 0.02 dB/cm/MHz.

The measured BSC values reported in this study were computed using the reference phantom technique. 24;25 The reference scattering medium makes it possible to compensate de Monchy et al., JASA, p. 14 the measured backscattered power spectrum P meas for the electromechanical system response and the depth-dependent diffraction and focusing effects caused by the ultrasound beam. The reference scattering medium was a disaggregated blood sample [START_REF] Wang | An approach for measuring ultrasonic backscattering from biological tissues with focused transducers[END_REF] at a low hematocrit of 4% (i.e., Rayleigh scatterers) sheared at 22 s -1 in the Couette flow device.

Echoes from the reference scattering medium were acquired and windowed as with the aggregating blood sample, and their power spectra were averaged to obtain P ref .

The measured BSC was thus computed as follows

BSC meas (k) = BSC ref (k) P meas (k) P ref (k) e 4d(αs-α ref ) k 8.68 c 2π (16)
where d is the distance from the interface agar/blood to the center of the windowed

BSC ref (k) = mσ b (k)W PY = mσ b (k) (1 -φ) 4 (1 + 2φ) 2 . ( 17
)
The Percus-Yevick packing factor theory was previously experimentally assessed on disaggregated blood sample using a substitution method with a planar reflector (see Fig. 1 in Ref. (1)). This procedure yielded a BSC ref for each probe. The two BSC meas were then

F = i BSC meas (k i ) -BSC theo (k i ) BSC meas (k i ) 2 , (18)
where BSC theo is given by Eq. ( 15) for the SFSE, by Eq. ( 14) for the monodisperse EMTSFM and by Eq. ( 8) for the polydisperse EMTSFM. In the case of the SFSE, a quadratric function of the two variables W and D 2 is obtained, and by calculating its critical points, a system of two linear equations with two unknowns can be solved. In the case of the monodisperse EMTSFM, 5 the routine fminsearch of MATLAB (The Mathworks Inc., Natick, MA), i.e., the Nelder-Mead simplex method, was employed to minimize the cost function F; whereas in the case of the polydisperse EMTSFM, we used the routine fmincon of MATLAB, with the constraints 0 ≤ φ i ≤ 1 and p (rag,z) (r ag < a) = 0 (i.e., the aggregate radius should be larger than the RBC radius a). For both monodisperse and de Monchy et al., JASA, p. 16 polydisperse EMTSFM, ten random initializations were used in our implementation. In the sequel of the paper, the standard deviation of the aggregate radius distribution σ * ag = r * ag / √ z * + 1 will be reported instead of the gamma width factor z * .

For each studied shear rate and sample, the goodness of fit of the three models to the experimentally measured BSC meas was assessed by the goodness-of-fit statistic, R 2 , given by: 29

R 2 = 1 - j ||log(BSC meas (k j )) -log(BSC theo (k j ))|| 2 j ||log(BSC meas (k j )) -log(BSC meas )|| 2 (19)
where BSC theo represents the theoretical BSC computed with the estimated values of the QUS parameters obtained by the minimization procedure, and log(BSC meas ) represents the mean value of log(BSC meas ) over the studied frequency range. The BSC in log scale was used (instead of the BSC in linear scale) in order to give approximately the same weight to all frequencies. The closer the R 2 value is to 1, the better is the fit of the model to the data. A negative value for R 2 means that a mere horizontal line passing through the average value of the log(BSC meas ) would fit data better than does the model.

E. Optical images of flowing RBC aggregate in a plane-plane rheometer After

the ultrasound measurements, a plane-plane rheometer (Anton Paar MCR 301, Saint Laurent, Canada) coupled with a light microscope (objective X20) was used to observe the flowing RBC aggregates (see Fig. 1). The blood sample was first sheared at 300 s -1 to disrupt RBC aggregates and then sheared at residual values of 3, 6, 11, 22, 45 and 90 s -1 , de Monchy et al., JASA, p. 17 as performed previously in the US measurements. Because the RBCs are opaque to light, the optical measurements were only performed for the lower hematocrits of 10% and 20% and the rheometer gap was adjusted depending of the studied shear rates in order to observe the aggregate boundaries. The rheometer gap was equal to 100, 100, 80, 60, 40 and 20 µm, respectively for the shear rates 3, 6, 11, 22, 45 and 90 s -1 . Images were acquired using the optical microscope and a camera CCD and were segmented manually to obtain the aggregate contour. Segmentation consisted in identifying an aggregate and putting five points on its contour. The ellipse passing through these five points was traced and the disc radius having the equivalent ellipse area was calculated. When the level of aggregation is high, aggregates are superimposed on the optical images, and it becomes difficult to identify aggregate contours on a single image. In that case, aggregates were tracked on several consecutive optical images to facilitate contour delimitation (see Fig. 2). Around two hundred radii were registered for each studied shear rate and each hematocrit. Then, the histogram h of the aggregate radius was fitted with a gamma distribution to estimate the mean aggregate size r * opt and the distribution width z * opt by minimizing the cost function F defined as:

F (r * opt , z * opt ) = i h(r i ) -p (r * opt ,z * opt ) (r i ) 2 . ( 20
)
We employed the routine fmincon of MATLAB, with the constraint p (ropt,zopt) (r opt < a) = 0 (i.e., the aggregate radius should be larger than the RBC radius a), in order to be de Monchy et al., JASA, p. 18 consistent with the constraint that we adopted in the case of ultrasound signals. given in Table 1 for all the studied hematocrits. The goodness-of-fit statistic reveals that the monodisperse EMTSFM had the worst fit to the data, especially for shear rates smaller than 11 s -1 at hematocrits of 20 and 40%. For all values of hematocrit, the polydisperse EMTSFM provided the best fit to the data, especially for the lower shear rates 3 s -1 and 6 s -1 . The polydisperse EMTSFM gave overall satisfactory fitting curves with goodness-of-fit statistic comprised between 0.89 and 0.99 for the shear rates greater than 6 s -1 , whatever the considered hematocrit (against goodness-of-fit statistics comprised between 0.49 and 0.99 for the SFSE model).

IV. Results

A. Ultrasonic characterization of RBC aggregation

The QUS parameters estimated from the monodisperse and polydisperse EMTSFM are reported in Fig. 4 for hematocrit values of φ =10, 20 and 40%. For the highest shear goodness-of-fit values occurred at the lowest shear rate of 3 s -1 . The polydisperse EMTSFM seems thus to reach its limit of applicability at the lowest shear rate of 3 s -1 , corresponding to a large width distribution and a large aggregate size. Several assumptions made in the polydisperse EMTSFM might explain this limitation. The assumption of spherical aggregates used in the EMT theory might be less reliable for modeling a single RBC aggregate of variable shape at high frequencies. In the proposed in vitro experiments, the blood sample was prepared by using polymer dextran 500 kDa to mimic pathological blood and to form clumps. The resulting aggregates have an ellipsoidal shape (see Fig. One can also notice that negative values of the goodness-of-fit statistic were obtained with the SFSE modeling for the lowest shear rate of 3 s -1 . Moreover, the estimated aggregate diameter D * estimated by the SFSE did not agree with the expected tendency:

the aggregate size is expected to increase while the shear rate decreases, whereas the estimated aggregate diameter D * decreased for the lowest shear rate of 3 s -1 . This limitation of the SFSE for hyperaggregating blood was previously observed in experimental studies for D * >7.2 corresponding to kR * >2 (R * being the aggregate radius estimated by the SFSE). [START_REF] Franceschini | Ultrasound characterization of red blood cell aggregation with intervening attenuating tissue-mimicking phantoms[END_REF] One can also notice the linear relationship between W * et D * in Fig. 5(c), as observed previously under experimental conditions 1 and in 3-D numerical simulations. 4;5 This means that the BSC parameterization can be reduced to one parameter when using the SFSE.

Although the monodisperse EMTSFM did not produce a satisfactory goodness-of-fit statistic, correlation was found with the optical estimated radii, with r 2 superior to 0.85. It is interesting to observe that both monodisperse and polydisperse EMTSFM gave similar values of the aggregate radius, and that the mean radii estimated with the monodisperse EMTSFM were slightly larger than those estimated with the polydisperse EMTSFM. For the highest shear rate of 90 s -1 , the aggregate radius PDF estimated with the polydisperse de Monchy et al., JASA, p. 23 EMTSFM was quasi-monodisperse, and the monodisperse and polydisperse EMTSFM gave similar estimated radii, as expected. For shear rates less than 45 s -1 (i.e., for the higher levels of aggregation), the monodisperse EMTSFM provided radius estimates that fall within the radius range estimated with the polydisperse EMTSFM, as observed previously by Lavarello and Oelze 32 when the effective scatterer size is estimated from a population of scatterers with a continuous size distribution. As an example, for the shear rate of 3 s -1 , the polydisperse EMTSFM provided aggregate radii r ag /a varying between 1 and 8, with r ag /a=4, for the S10 sample (and varying between 1 and 10, with r ag /a=5.2, for the S40 sample), whereas the monodisperse EMTSFM estimated a normalized aggregate radius approximatively equal to 5 for the S10 sample (and approximatively equal to 8 for the S40 sample) (see Fig. 4).

When considering the polydisperse EMTSFM, the estimates r * ag /a and σ * ag /a are similar for the two series of experiments and show low variance, whatever the studied hematocrit (see Fig. 4). Whereas, the estimates φ * i differ from the two series of experiments and show high variance (Figs. 4(a3) and 4(c3)). Netherveless, the aggregate compactness φ * i estimated with the polydisperse EMTSFM were found comprised between 0.8 and 1, and seem to be in a reasonable range of values for deformable packed red blood cells.

To our knowledge, there is no known way to experimentally characterize the real size of RBC aggregates at a physiological hematocrit. That is why the optical measurements de Monchy et al., JASA, p. 24 were limited to hematocrits of 10% and 20%. We observed a good agreement between optical and ultrasound measurements for the mean aggregate radius and the standard deviation of the aggregate radius distribution. However, the rheometer gap used in optics varies between 20 and 100 µm (depending on the studied shear rate) in order to observe the aggregate boundaries. So, the set-up may suffer from wall effects, since the rheometer gap is about eight or nine times the mean aggregate radius. The gap should be around twenty times the mean aggregate radius to avoid wall effects. In comparison, the Couette gap of 2 mm used for ultrasonic experiments did not suffer from wall effects. This is why the comparison between ultrasonic and optical estimates is only qualitative. Netherveless, a good correlation was found between optics and ultrasound for the mean aggregate radius and distribution width, as reported in Table II and in Figure 6.

VI. Conclusion

This study proposed a novel scattering ultrasound model for considering polydispersity in terms of aggregate size. Fitting the polydisperse model yielded aggregate size distributions that were consistent with direct light microscope observations at low hematocrits. It is interesting to notice that the width of the aggregate radius PDF estimated by the polydisperse EMTSFM is larger as the level of aggregation increases, whatever the studied hematocrit. This trend was observed not only on the QUS estimates but also on the optical observations. To our knowledge, it is the first experimental study showing that an increase in the level of aggregation corresponds to an increase in the size distribution width. Further studies should be conducted on blood with different aggregation tendencies (promoted by different macromolecules, for instance) to confirm this observation.

Table I. Goodness-of-fit statistic (averaged over the twelve BSC measurements) of the polydisperse EMTSFM (EMT poly ), the monodisperse EMTSFM (EMT mono ) and the SFSE modeling for the blood samples sheared at several shear rates and for hematocrits of 10, 20 and 40% (denoted S10, S20 and S40).

Table II. Correlation coefficient r 2 between mean aggregate radius estimated by optics and by ultrasound (using the three scattering models).

de

de

  Monchy et al., JASA, p. 10 B. Structure Factor Size Estimator (SFSE) The structure factor model (SFM) sums up the acoustic contributions from individual RBCs and models their interaction by a statistical mechanics structure factor S. The SFSE developed by Yu and Cloutier 1

  Experiments were conducted with a focused transducer with a center frequency of 20 MHz (with -10 dB bandwidth of 11-35 MHz in water), a focus of 25 mm, and an f-number of 2.1. The rotating inner cylinder was used as a reflector. The transducer was used in emission/reception and the transducer focus was positioned at the interface between the blood sample under investigation and the rotating inner cylinder. The substitution method consists in comparing the spectra of the echoes reflected by the inner cylinder both with blood and with water inserted in the ultrasound path and sheared in the Couette flow device. The attenuation was found to depend linearly on frequency, within the 11-35 MHz frequency bandwidth.

  analysis region, α s and α ref are the predetermined attenuation coefficients (in dB/mm/MHz) of the aggregating blood sample and of the reference disaggregated blood sample. The coefficient 8.68 expresses unit conversion from decibels to neper. The BSC ref corresponds to the BSC of the disaggregated blood sample estimated by using the expression of the Perkus-Yevick packing factor for spheres W PY as follows:27;28

Figure 3

 3 represents a typical measured BSC meas for different shear rates at hematocrit φ=20%, as well as the corresponding fitted curves obtained with the monodisperse models (EMTSFM and SFSE), and the polydisperse EMTSFM. The QUS parameters were estimated from twelve measured BSC meas (corresponding to the twelve acquired B-mode images). The values of the goodness-of-fit statistic of each model to the experimental data were computed and are

  Fig. 7 in Ref. (13)). Moreover, the polydisperse EMTSFM assumed polydispersity in

de

  Monchy et al., JASA, p. 25

Figure 1 .

 1 Figure 1. (Color online) Schematic illustration of the plane-plane rheometer

Figure 2 .

 2 Figure 2. (Color online) (On the left) Consecutive optical images in the plane-plane

Figure 3 .

 3 Figure 3. (Color online) Examples of measured BSC meas at different shear rates with

Figure 4 .

 4 Figure 4. (Color online) (a) Values of r * ag /a, σ * ag /a and φ * i estimated by the

Figure 5 .

 5 Figure 5. (Color online) (a) Packing factor W * and (b) aggregate diameter D *

Figure 6 .Figure 1 :

 61 Figure 6. (Color online) (a) Comparison of aggregate radius PDF estimated by US

  Aggregate radius rag/a 100 μm

Figure 2 :Figure 3 :

 23 Figure 2: (Color online) (On the left) Consecutive optical images in the plane-plane rheome-

Figure 4 : 1 )Figure 5 :

 415 Figure 4: (Color online) (a) Values of r * ag /a, σ * ag /a and φ * i estimated by the polydisperse

Figure 6 :

 6 Figure 6: (Color online) (a) Comparison of aggregate radius PDF estimated by US (solid

  The Effective Medium Theory (EMT) detailed in Ref.(13) for modeling one aggregate of RBCs is briefly recalled as follows. The EMT model assumes that an aggregate of cells can be treated as an effective homogeneous spherical scatterer of radius r ag . As previously studied by Morse and Ingard,[START_REF] Morse | Theoretical Acoustics[END_REF] the differential backscattering cross-section σ ag of an aggregate can be decomposed into two components:

14;15

whereas aggregates of RBCs form "rouleaux" in normal blood (i.e., anisotropic structures).

A. Effective Medium Theory combined with polydisperse Structure Factor

Model 1. Effective Medium Theory for one single RBC aggregate de Monchy et al., JASA, p. 5

  de Monchy et al., JASA, p. 11 follows. The RBCs were resuspended in a mixture containing 75% of saline solution and 25% of Optiprep TM (iodixanol solution, Progen Biotechnik), and 1.5 g/L of dextran 500 kDa (Sigma Aldrich, Saint Quentin Fallavier, France) was then dissolved in the reconstituted blood. The Optiprep TM allows adjusting the density of the suspending medium at around 1080 kg/m 3 and to prevent sedimentation of RBCs. The dose and molecular weight of dextran allowed producing highly aggregated blood 21 with a tendency to form quasi-spherical aggregates and to mimic pathological blood.

	probes, the oscillating single-element focused circular transducers had center frequencies of
	16 and 30 MHz, focuses of 17.5 and 10 mm and f-numbers of 2.1 and 2.5, respectively. Raw
	RF data were digitized at a sampling frequency of 250 MHz (12-bit precision) using a
	high-speed acquisition card (Gagescope, model CS12501, Montreal, Canada). The selected
	probe was mounted in the side wall of the fixed outer cylinder and was positioned to have

A disaggregated blood sample, denoted H4, was prepared with an hematocrit of 4%.

The H4 sample consisted of RBCs resuspended in a mixture containing 75% of saline solution and 25% of Optiprep TM to prevent the aggregation process in the absence of macromolecules (dextran or plasma proteins).

B. Ultrasound measurements in the Couette flow device US measurements were performed in a Couette device to produce a linear blood velocity profile at a given rotational speed to obtain a constant shear rate. The system consists of a rotating inner cylinder with a diameter of 160 mm surrounded by a fixed concentric cylinder of diameter 164 mm (see Fig.

1

in Ref.

(23)

). A 60 mL blood sample was sheared in the annular space between both coaxial cylinders of length e=2mm. Ultrasonic data were acquired using a high frequency system (Vevo 770, Visualsonics Inc, Toronto, Canada). Two probes, RMV 716 and RMV 703, were used in B-mode imaging. For the RMV 716 and the RMV 703 de Monchy et al., JASA, p. 12 its focal zone at the center of the gap between both cylinders. To ensure ultrasonic coupling, the hole within the outer stationary cylinder (containing the probe) was filled with a liquid agar-agar gel based mixture in order to match the curvature of the cylinder and to avoid any flow disturbance.

All US measurements were made at room temperature. Prior to each measurement, the blood sample (S10, S20 or S40) was sheared at 300 s -1 during 30 s to disrupt RBC aggregates. The shear rate was then fixed to values of 3, 6, 11, 22, 45 and 90 s -1 for 60 s until a steady equilibrium in the state of aggregation was reached. The smallest shear rate produces the highest level of aggregation, and vice versa. For each shear rate, twelve B-mode images were successively constructed from acquired RF echoes for a total period of analysis of 90 s. For 100 scan lines in the center of B-mode images, echoes were selected within a rectangular window of axial length corresponding to twelve wavelengths. The power spectra of the 100 backscattered echoes were then averaged to provide P meas . The blood sample preparation and the ultrasonic measurement protocol was repeated twice de Monchy et al., JASA, p. 13 with different porcine bloods for hematocrits of 10, 20 and 40%.

  de Monchy et al., JASA, p. 15 combined to yield a single BSC meas over the combined bandwidth of the two transducers (i.e., 11-41 MHz).D. QUS parameters estimation The three tested models, namely, the monodisperse SFSE and EMTSFM, and the polydisperse EMTSFM, assume that the hematocrit φ, the RBC radius a and the acoustical properties of plasma and RBCs are prior knowledge. A theoretical model was fitted to the experimental BSC to estimate QUS parameters (i.e. W

*

and D * using the SFSE, r * ag and φ * i using the monodisperse EMTSFM, and r * ag , z * and φ * i using the polydisperse EMTSFM) by minimizing the cost function F defined as:

Table 1 :

 1 Monchy et al., JASA, p. 32 Goodness-of-fit statistic (averaged over the twelve BSC measurements) of the polydisperse EMTSFM (EMT poly ), the monodisperse EMTSFM (EMT mono ) and the SFSE mod-

	eling for the blood samples sheared at several shear rates (and for hematocrits 10, 20 and
	40% (denoted S10, S20 and S40) .
	Shear rate (s -1 )	3	6	11	22	45	90
		EMT poly	0.77	0.90 0.97 0.97 0.98 0.98
	S10	EMT mono	-0.44 0.68 0.96 0.96 0.97 0.98
		SFSE	-1.11 0.49 0.97 0.97 0.98 0.97
		EMT poly	0.69	0.89 0.97 0.98 0.98 0.97
	S20	EMT mono	-13.33 0.52 0.93 0.96 0.98 0.97
		SFSE	-1.18 0.62 0.97 0.96 0.96 0.97
		EMT poly	0.44	0.96 0.99 0.98 0.99 0.99
	S40	EMT mono	-25.23 -2.00 -0.02 0.95 0.99 0.99

SFSE

Table 2 :

 2 Correlation coefficient r 2 between mean aggregate radius estimated by optics and by ultrasound (using the three scattering models).

		SFSE	EMT mono EMT poly
	S10 -Series 1	0.75	0.85	0.94
	S10 -Series 2	0.84	0.92	0.98
	S20 -Series 1	0.82	0.95	0.92
	S20 -Series 2	0.76	0.85	0.90

Aggregate radius r ag /a Standard deviation σ ag /a Aggregate compactness φ i

	EMTpoly -Series 1	EMTpoly -Series 1
	EMTpoly -Series 2	EMTpoly -Series 2
	EMTmono -Series 1
	EMTmono -Series 2
	(a1)	(a2)	(a3)
	EMTpoly -Series 1	EMTpoly -Series 1
	EMTpoly -Series 2	EMTpoly -Series 2
	EMTmono -Series 1
	EMTmono -Series 2
			EMTpoly -Series 1
			EMTpoly -Series 2
			EMTmono -Series 1
			EMTmono -Series 2
	EMTpoly -Series 1	EMTpoly -Series 1	1.2
	EMTpoly -Series 2	EMTpoly -Series 2
	EMTmono -Series 1
	EMTmono -Series 2
			EMTpoly -Series 1
			EMTpoly -Series 2
			EMTmono -Series 1
			EMTmono -Series 2

Acknowledgment This research was supported by the French National Research Agency (Grant No. ANR-15-CE19-0017), and the Labex MEC (Grant No. ANR-10-LABX-0092) and the A*MIDEX project (Grant No. ANR-11-IDEX-0001-02) funded by the Investissements Avenir French Government program. Support was also obtained from the Canadian Institutes of Health Research (Grant #MOP-84358).

Parameters W * and D * estimated with the SFSE model are reported in Fig. 5.

Increasing the shear rates resulted overall in decreases of W * and D * (by excluding the shear rate 3 s -1 ). The SFSE could not follow the frequency dependence of the measured BSC meas at shear rates of 3 and 6 s -1 (i.e., for the higher level of aggregation), as reflected by drastically lower values of the goodness-of-fit statistic computed by Eq. ( 19) (see Table 1).

B. Comparison of ultrasonic and optical aggregate measurements Aggregate

size was also quantified using a plane-plane rheometer combined with an optical microscope (as described in section III.D) in order to evaluate the accuracy of the de Monchy et al., JASA, p. 20 aggregate size estimated by ultrasound. Figure 6 compares the aggregate radius PDF, the mean aggregate radius r opt /a and the standard deviation σ opt /a estimated by optics and the corresponding values estimated by QUS (using the polydisperse EMTSFM) for hematocrit φ=10%. Table 2 summarizes the correlation coefficients r 2 between mean aggregate radius estimated by optics and by ultrasound using the three scattering models, namely, the monodisperse SFSE and EMTSFM and the polydisperse EMTSFM. The mean aggregate radii estimated by the polydisperse EMTSFM agree well with those obtained from optical measurements, with correlation coefficients 0.90≤r 2 ≤0.98 (see Table 2) and with a slope of linear regression around 1, for both studied hematocrit values of 10% and 20%. Good correlation (r 2 ≥ 0.80) was also found between the standard deviation estimated by optics and ultrasound. In the case of the optical data, where estimates of radii of aggregates were available, we have also estimated the corresponding gamma distributions without imposing the constraint that aggregates radii be no smaller than an RBC radius (see Eq. ( 20)). Whether this constraint was imposed or not for the optical data, similar correlations were obtained between ultrasonic and optical data.

V. Discussion

The goodness-of-fit values given in Table I and the fitting curves shown in Fig. 3 reveal that the polydisperse EMTSFM provided the best fit to data, whatever the shear rates and the hematocrit studied. However, for all three models, the smallest