Laura Vinué

Thomas Jové

Carmen Torres

Marie-Cécile Ploy

Diversity of class 1 integron gene cassette Pc promoter variants in clinical Escherichia coli strains and description of a new P2 promoter variant

Keywords:

A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t ABSTRACT Gene cassettes of class 1 integrons may be differently expressed depending on the Pc promoter variant as well as occasionally from a second promoter located downstream of Pc, named P2. So far, the distribution of the variants has only been described in an in silico study. In this study, the prevalence of these variants in vivo was analysed in a population of 85 Escherichia coli strains from a variety of phylogenetic groups isolated from healthy subjects and clinical samples in Spain and France from 2004 to 2007. The weakest variants (PcW and PcH1) prevailed (variants associated with the integrase having the most efficient excision activity), whilst the two strongest variants, PcW and PcS, were less frequent. Furthermore, a new variant of P2 associated with PcW was characterised in one integron (harbouring the gene cassette bla OXA-1 -aadA1) from a French strain of a healthy subject. This variant was hereafter named P2m3 and shows a G→A substitution in its -10 element (TACAGT to TACAAT), a mutation that doubled the strength of P2 and approached the level of expression of the strong PcW TGN-10 variant. When the correlation between the Pc variants and the origin of the strains was analysed, no significant difference (P < 0.05) was observed in the Pc variant distribution according to the geographic origin or clinical setting.

A c c e p t e d M a n u s c r i p t

Introduction

Integrons are genetic elements strongly involved in the dissemination of antibiotic resistance among Gram-negative bacteria [START_REF] Cambray | Integrons[END_REF]. Integrons can acquire, exchange and express genes embedded within gene cassettes (GCs). The integron functional platform is composed of an intI gene encoding an integrase, a recombination site attI and a promoter Pc. The integrase is the key element, able to catalyse through site-specific recombination either the insertion of a GC within the integron or its excision. To date, more than 130 different GCs encoding resistance to almost all antibiotic families have been identified in integrons. Several classes of integrons have been characterised, with class 1 being the most prevalent. As GCs are usually promoterless, they are transcribed from the integron promoter Pc, and the level of transcription is dependent upon their position within the integron [START_REF] Collis | Expression of antibiotic resistance genes in the integrated cassettes of integrons[END_REF]. Pc variants with different strengths have been described in class 1 integrons based on the sequence of the -35 and -10 elements. Eight variants were initially described, with the four most prevalent being named PcS for 'strong', PcW for 'weak' (PcS being 30-fold stronger than PcW), PcH1 for hybrid 1 and PcH2 for hybrid 2, these two latter Pc variants containing the -35 and -10 hexamers of PcW and PcS in opposite combinations and having intermediate strengths [START_REF] Papagiannitsis | Relative strengths of the class 1 integron promoter hybrid 2 and the combinations of strong and hybrid 1 with an active p2 promoter[END_REF][START_REF] Brizio | High-level expression of IMP-5 carbapenemase owing to point mutation in the -35 promoter region of class 1 integron among Pseudomonas aeruginosa clinical isolates[END_REF][START_REF] Bunny | New mobile gene cassettes containing an aminoglycoside resistance gene, aacA7, and a chloramphenicol resistance gene, catB3, in an integron in pBWH301[END_REF]. In 10% of class 1 integrons a second promoter (P2) has been described 119 bp downstream of Pc [START_REF] Collis | Expression of antibiotic resistance genes in the integrated cassettes of integrons[END_REF][START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF].

We recently performed an in silico study and reported 13 Pc variants, including a new variant with a 'TGN' extended -10 motif, a motif known to increase the transcription efficiency of σ 70 promoters in Escherichia coli. These 13 variants expressed different strengths and the most frequent ones were the weak variants PcW and PcH1 and the Thus, class 1 integron-containing E. coli strains isolated in Spain and France from hospital patients and from healthy subjects were analysed to determine the Pc distribution in circulating strains.

Materials and methods

Bacterial isolates

In total, 85 class 1 integron-containing E. coli strains isolated from 2004 to 2007 were studied. The strains were geographically distinct (from Spain and France), from various origins (clinical strains from hospital patients and strains from faecal carriage of healthy subjects) and belonging to different phylogenetic groups (A, B1, B2 and D) (Table 1) [START_REF] Clermont | Rapid and simple determination of the Escherichia coli phylogenetic group[END_REF].

Study of the integron and characterisation of the gene cassette promoter variants

The GC arrays of the Spanish strains (n = 41) had previously been characterised [START_REF] Sáenz | Class 1 integrons lacking qacEΔ1 and sul1 genes in Escherichia coli isolates of food, animal and human origins[END_REF][START_REF] Vinué | Genetic environment of sul genes and characterisation of integrons in Escherichia coli[END_REF][10], whereas those of the French strains (n = 44) were characterised in this study by polymerase chain reaction (PCR) mapping as described elsewhere [10]. To characterise the Pc-P2 promoters, a 691-bp fragment obtained by PCR with primers intI1-LC9 (5'-CCA AGC TCT CGG GTA ACA TC-3') and orf11-R (5'-GCG GGA TCC AGG TAA CTT TGT TTT AGG GCG ACT GCC CT-3') located, respectively, in intI1 and attI sequences was sequenced.

A c c e p t e d M a n u s c r i p t

Statistical analysis was performed by means of Pearson's χ 2 test (with Yates' corrections) to check whether there were differences in the prevalence of promoter types among isolates of different origins or countries.

Estimation of the strength of promoters

The relative strength of a GC promoter was estimated in a transcriptional fusion in which both Pc and P2 were upstream of a promoterless lacZ gene in a reporter vector, as described previously [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. The 'pPcW-P2m3' fusion was constructed by PCR assembly (see protocol in [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]) of the 'pPcW+P2' fusion with P2mut3L (5'-CTG TTT TTT TGG GGT ACA ATC TAT GCC TCG GGC ATC CAA G-3') and P2mut3R (5'-CCC GAG GCA TAG ATT GTA CCC CAA AAA AAC AGT CAT AAC-3') as internal primers. The efficiency of this promoter to drive lacZ gene expression was then estimated by the βgalactosidase assay [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF].

Results and discussion

Integron detection and characterisation of gene cassette promoter variants

The 85 E. coli strains contained 88 class 1 integrons (3 strains had 2 integrons) with 22 different GC arrays (Table 2). Two integrons had no GC, and PCR failed to produce products for five strains. A new GC array (dfrB1-aadA11) was identified in a French clinical strain (GenBank accession no. GQ900749). All other GC arrays have been previously described in class 1 integrons and, as in other studies, the dfr and aadA GCs A c c e p t e d M a n u s c r i p t associated with trimethoprim and streptomycin resistance, respectively, were the most prevalent [START_REF] Ajiboye | Global spread of mobile antimicrobial drug resistance determinants in human and animal Escherichia coli and Salmonella strains causing community-acquired infections[END_REF][START_REF] Iversen | Characterization of antimicrobial resistance, plasmids, and gene cassettes in Shigella spp. from patients in Vietnam[END_REF].

Four different Pc variants were found in the 88 class 1 integrons: two weak variants (PcW and PcH1) predominated, whilst two strong variants (PcW TGN-10 and PcS) were less frequent (Table 3). These four variants were also the most prevalent in the in silico study, but strong variants have been more frequently found (16.8% for PcW TGN-10 and 23.7 % for PcS) [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. Only a few studies have examined the prevalence of GC promoter variants in clinical strains. The PcW variant has been found to be the most prevalent in studies of clinical Enterobacteriaceae isolates [START_REF] Cocchi | Distribution and characterization of integrons in Escherichia coli strains of animal and human origin[END_REF][START_REF] Lindstedt | Characterization of class I integrons in clinical strains of Salmonella enterica subsp. enterica serovars Typhimurium and Enteritidis from Norwegian hospitals[END_REF]. However, the frequency of the PcW variant could have been overestimated in these studies because differentiation between PcW and PcW TGN-10 was not performed. Indeed, it has been recently shown that >40% of the PcW-containing integrons harboured the PcW TGN-10 variant, in which a -10 extended region increases the efficiency of PcW by 15-fold [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. In this study, it was found that in 15 of 51 PcW-containing integrons, PcW was associated with the P2 form, a configuration known to be more efficient than PcW alone [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. In the in silico study, P2 was also mostly found associated with PcW [START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF].

Among the 22 GC arrays identified, 8 were found in only one integron (Table 2). The 14 remaining GC arrays were found in at least two integrons, in strains from Spain and/or France belonging to different phylogenetic groups. Among these 14 GC arrays, 9 were always associated with the same Pc-P2 combination, whilst the other 5 could be transcribed from two or three different Pc-P2 combinations. In the previous in silico A c c e p t e d M a n u s c r i p t study, we found that there was no link between a specific Pc variant and specific GC arrays (unpublished data). Here, the most frequent GC array (dfrA1-aadA1) was associated with three Pc-P2 combinations, suggesting also that Pc variants are not associated with specific GC arrays. All of the integrons containing dfrA17 as the first GC bore the PcH1 promoter, which is in agreement with findings for Salmonella enterica serovar Typhimurium [START_REF] Lindstedt | Characterization of class I integrons in clinical strains of Salmonella enterica subsp. enterica serovars Typhimurium and Enteritidis from Norwegian hospitals[END_REF]. After statistical analysis, no significant difference (P < 0.05) was observed in the Pc variant distribution according to the geographic origin or clinical setting (Table 3).

Estimation of the strength of promoters

One integron (harbouring the GC bla OXA-1 -aadA1) from a French strain of a healthy subject presented a new P2 variant associated with PcW (Tables 2 and3). This variant shows a G→A substitution in its -10 element (TACAGT to TACAAT), which is closer to the consensus sequence of the E. coli promoter, and it was named P2m3. When the strength of this promoter variant was determined by β-galactosidase assay, it was found that this mutation doubled the strength of P2 and approached the level of expression of the strong PcW TGN-10 variant (data not shown).

This work showed interesting data, with a predominance of the two weak variants PcW and PcH1 in the circulating E. coli strains studied and a low prevalence of the strong Pc variants (PcW TGN-10 and PcS), unlike in the in silico study. The population of strains studied was considered to be representative of the E. coli diversity as they came from different geographical and contextual origins. Moreover, they belonged to different A c c e p t e d M a n u s c r i p t phylogenetic groups, which are known to have a good correlation to multilocus sequence type (MLST)-based phylogroups [START_REF] Tenaillon | The population genetics of commensal Escherichia coli[END_REF]. However, further studies with more isolates are needed to determine whether this higher prevalence of weaker GC promoter variants is widespread.

dfrA12-orfF-aadA2 3 (1/2) 3 dfrA12-orfF-aadA2- cmlA1-aadA1-qacH- IS10-IS440-sul3 1 (1/0) 1 dfrA12-orfF-aadA2- cmlA1-aadA1-qacH- IS440-sul3 2 (2/0) 1 1 dfrA12-orfF-aadA2- cmlA1-aadA1-IS440- sul3 1 (1/0) 1 dfrA14 1 (0/1) 1 dfrA15-aadA1 1 (1/0) 1 dfrA16-aadA2-aadA2 3 (2/1) 1 2 2

A

  c c e p t e d M a n u s c r i p t strong variants PcW TGN-10 and PcS[START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. However, such data might contain a bias because of the subjective selection of the deposited integrons.

  Interestingly, as the Pc sequence lies within the intI1 gene, different IntI1 integrases correspond to different Pc variants, with PcW and PcH1 being associated with the integrase having the most efficient excision activity[START_REF] Jové | Inverse correlation between promoter strength and excision activity in class 1 integrons[END_REF]. Thus, a better ability to excise cassettes might compensate for weak expression of antibiotic resistance factors and could theoretically enhance the capacity of the integron to adapt to antibiotic pressure and thus represent a survival advantage. If so, this may be one reason for the high prevalence of the weakest Pc variants observed here. It remains to be shown whether environmental factors (antibiotic pressure, stress, etc.) influence the selection of weak variants during bacterial evolution.

Table 1

 1 Distribution of 85 Escherichia coli strains included in the study according to origin and

a

  Included in GenBank under accession no. GQ900749. b One of these integrons presented the new P2 variant named P2m3. c Class 1 integrons for which polymerase chain reaction (PCR) failed to amplify the gene cassette array.

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Table 2

 2 Gene cassette arrays in 88 class 1 integrons from 85 Escherichia coli strains and their

	relationship with the promoter variant
	Gene cassette array	No. of integrons	Pc variant
		(Spain/France)	PcW PcH1 PcW+P2 PcW TGN-	PcS
				10
	aadA1	8 (3/5)		8
	aadA2	1 (1/0)		1
	aadB-aadA1-cmlA1	2 (1/1)		2
	dfrA1	3 (2/1)	3
	dfrA1-aadA1	20 (4/16)	15	3

Table 3

 3 Pc and P2 promoter variants detected among 88 class 1 integrons detected in 85Escherichia coli strains from Spain and France

	Pc-P2	Total	Healthy	Clinical	Spain (n	France (n
	combinations a	no.	subjects (n =	samples (n =	= 43)	= 45)
			27)	61)
			n (%)	n (%)
	PcW	36	14 (51.9)	22 (36.1)	18	18
	PcH1	29	8 (29.6)	21 (34.4)	14	15
	PcW+P2	14	3 (11.1)	11 (18.0)	6	8
	PcW+P2m3	1	1 (3.7)	0 (0.0)	0	1
	PcW TGN-10	6	1 (3.7)	5 (8.2)	4	2
	PcS	2	0 (0.0)	2 (3.3)	1	1

a P2 is indicated when present.

Acknowledgments

The authors thank Sandra Da Re and Sophie Raherison for critical reading of the manuscript, and Fernando Antoñanzas and Reyes Lorente (University of La Rioja, Logroño, Spain) for statistical analysis.

Funding

This work was supported by Project SAF2006-14207-C02 (Spanish Ministry of Education and Science), FEDER, and grants from the French Ministère de la

Competing interests

None declared.

Ethical approval

Not required.

A c c e p t e d M a n u s c r i p t


Gérald Bardan
email: gbardan@imft.fr

Franck Plouraboué

Mokhtar Zagzoule
email: zagzoule@cict.fr

Olivier Balédent
email: olivier.baledent@chu-amiens.fr

Simple Patient-Based Transmantle Pressure and Shear Estimate From Cine Phase-Contrast MRI in Cerebral Aqueduct

Keywords: Aqueduct of sylvius, cerebrospinal fluid (CSF), electric impedance modeling, Fourier decomposition, intracranial pressure, pulsatile flows, transmantle pressure (TRP). I

From measurements of the oscillating flux of the cerebrospinal fluid (CSF) in the aqueduct of Sylvius, we elaborate a patient-based methodology for transmantle pressure (TRP) and shear evaluation. High-resolution anatomical magnetic resonance imaging first permits a precise 3-D anatomical digitalized reconstruction of the Sylvius's aqueduct shape. From this, a very fast approximate numerical flow computation, nevertheless consistent with analytical predictions, is developed. Our approach includes the main contributions of inertial effects coming from the pulsatile flow and curvature effects associated with the aqueduct bending .

Integrating the pressure along the aqueduct longitudinal center line enables the total dynamic hydraulic admittances of the aqueduct to be evaluated, which is the pre-eminent contribution to the CSF pressure difference between the lateral ventricles and the subarachnoidal spaces also called the TRP. The application of the method to 20 healthy human patients validates the hypothesis of the proposed approach and provides a first database for normal aqueduct CSF flow. Finally, the implications of our results for modeling and evaluating intracranial cerebral pressure are discussed.

pressure measurements can be found from control volume compartmental image analysis and could contribute to a better understanding of intracranial dynamics and progression to disorder [START_REF] Cohen | Development of a theoretical framework for analyzing cerebrospinal fluid dynamics[END_REF]. Furthermore, since some in vivo measurements are possible, it is tempting to use computational fluid dynamics (CFD) for a deeper understanding of the cerebrospinal fluid (CSF) flow [START_REF] Linninger | Cerebrospinal fluid flow in the normal and hydrocephalic human brain[END_REF]- [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], and for a better description of the CSF pressure variations inside the brain as well as a more complete understanding of the hydraulic coupling within the various intracerebral compartments. However, even though numerical computations have already been developed for CSF flow inside the third or fourth ventricles [START_REF] Kurtcuoglu | Computational investigation of subjectspecific cerebrospinal fluid flow in the third ventricle and aqueduct of Sylvius[END_REF], it should be borne in mind that complete computations of CSF flow fields inside brain cavities have to face difficult obstacles, if not challenges.

First, the numerical evaluation of temporally oscillating flows inside very complex cavity shapes is a 4-D problem that needs massive data storage and heavy and lengthy computations, the precision of which is difficult to control. It is important to realize that, since the size of the smallest (e.g., intrathecal space and Sylvius aqueduct) and largest cavities (ventricles) differ by a large factor and adaptive mesh is necessary. This involves various issues such as mesh generation, regularization, and refinement, the quality of which necessitates much effort and case-specific treatments. CSF is mainly produced by the choroid plexus of the lateral ventricle (LV) and it is mainly reabsorbed by the Paccioni granulation pathway in the venous blood through the sagittal Sinus [START_REF] Penn | Ventricule wall movements and cerebrospinal fluid flow in hydrocephalus[END_REF], [START_REF] Johanson | Multiplicity of cerebrospinal fluid functions: New challenges in health and disease[END_REF]. Under normal circumstances, CSF oscillates in and out of the ventricular system with a net outward flow Q s equal to the physiological production rate of CSF. The aqueduct, a slightly curved pipe with a variable elliptic section, is the smallest cavity between the LV and SAS. This is where most of the viscous dissipation, and thus most of the pressure drop, occurs as illustrated in [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], Fig. 5]. The pressure in large cavities is found to be spatially uniform while most variation occurs within the cerebral aqueduct. Pressure variations inside the third ventricle are typically smaller than 0.5 Pa [START_REF] Kurtcuoglu | Computational investigation of subjectspecific cerebrospinal fluid flow in the third ventricle and aqueduct of Sylvius[END_REF], while transmantle pressure (TRP) is close to 4 Pa [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF]. The aqueduct influence thus deserves special attention, especially in stenotic conditions [START_REF] Fin | Three dimensional modeling of the cerebrospinal fluid dynamics and brain interactions in the aqueduct of sylvius[END_REF], [START_REF] Jacobson | Computer modelling of the cerebrospinal fluid flow dynamics of aqueduct stenosis[END_REF]. In the field of fluid mechanics, oscillating pulsed flow in curved pipes has received considerable attention [START_REF] Lyne | Unsteady viscous flow in a curved pipe[END_REF]- [START_REF] Gammack | Flow in pipes with non-uniform curvature and torsion[END_REF]. It involves three independent parameters: 1) the ratio between the typical radius R of the pipe and its radius of curvature R c , λ = R/R c ;2 ) the product between viscous diffusion time and the main cardiac forcing pulsation ω described by the so-called Womersley number α 2 = ωρR 2 /µ, where ρ and µ are, respectively, the density and dynamic viscosity of the CSF(very close to water), 3) the Reynolds number Re = ρU a/µ, where U is the averaged longitudinal velocity inside the aqueduct and a is the mean cross-sectional ellipse's major axis averaged along the aqueduct centerline, which is approximated by the average over all major axis a i of ellipses Ω i in all sections i (see Fig. 2), i.e., a =(1/L) L 0 a(s)ds ≃ (1/n) n i=1 a i . The average longitudinal velocity U = Q/πab is defined as the measured flux (which is constant along the aqueduct) over the average section πab, where, again, b is the average ellipse Ω i minor axis in each section i. Usually, two supplementary parameters, which are a combination of the above, are also considered: the Dean number De = Re • √ λ and the reduced Reynolds number Re s =(Re/α) 2 λ. Depending on the value of these parameters, some complex flow patterns and case-specific approximations can be built [START_REF] Lyne | Unsteady viscous flow in a curved pipe[END_REF]- [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF]. Since the physiological Womersley number α typically lies between 5 and 30, the flow field variations inside the cavities are generically localized in the vicinity of the walls, within so-called Stokes layers. These layers can be either large or small, so a finer grid discretization is also needed inside them if velocity is to be evaluated properly. We show below that the Stokes layers develop around 10% of the radius (mean value of the Womersley number in Table II).

Finally, from the acquisition point of view, as pointed in [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], it is difficult to obtain anatomically reliable data for the smallest cavities (e.g. SAS and intrathecal spaces) and the mechanical parameters used to describe soft tissue deformations and poroelastic behaviors. For all these reasons, it is interesting to develop alternative approaches to full CFD, such as [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], for CSF modeling and case-specific analysis. Such alternative approaches are not always possible: they depend on the geometry involved and on the flow conditions.

In this paper, we show that a slightly curved aqueduct of Sylvius with low reduced Reynolds number is a fair description that leads to a fast estimation of the pressure drop, shear, and hydraulic admittance. This analysis provides an equivalent frequency-dependent electric impedance description of the TRP/flux relationship, and is, to our knowledge, the first explicit evaluation available in a healthy cohort of patients. Section II describes the four-step method [aqueduct reconstruction (see Section II-A), fluid flow (see Section II-B), Fourier decomposition and hydraulic admittances (see Section II-C), and physiological validation of the model hypothesis (see Section II-D)] that we put forward here associated with asymptotic evaluation of the aqueduct of Sylvius hydraulic admittances. Section III shows the evaluation of these hydraulic admittances for three out of 20 real patients, resulting from the first Fourier modes decomposition of their cardiac cycle variations. We, finally, discuss the relevance of the results obtained for the derivating realistic electrical models of complete CSF hydraulic models in Section III.

II. METHODS

A. Aqueduct Reconstruction

MRI acquisitions were obtained using the method described in [START_REF] Balédent | Relationship between cerebrospinal fluid and blood dynamics in healthy volunteers and patients with communicating hydrocephalus[END_REF] and [START_REF] Balédent | Cerebrospinal fluid dynamics and relation with blood flow : a magnetic resonance study with semiautomated cerebrospinal fluid segmentation[END_REF] with isotropic voxels of ∆x =∆y =∆z = 0.55 mm. The anatomical orientation of the aqueduct is sagittal and mostly vertical. Thus, it was possible to obtain a reliable 3-D digitalized reconstruction of it by analyzing each successive horizontal section (cf., Fig. 2). In each horizontal section, we extracted the elliptical contour that best fitted the gray-level anatomical image of the aqueduct (cf., Fig. 3). First, we drew (dotted line) a contour plot of the gray-scale image with equally spaced contour levels in the plot. Then, in the least squares sense, we plotted (continuous line) the best fit ellipse. For each ellipse, the center, major radius a(s), and minor radius b(s) were extracted. From the evaluation of the ellipse's center in each successive horizontal section, the aqueduct centerline was reconstructed. Over the human patients examined, we found that, to a good approximation, this centerline lies within the sagittal plane. Hence, it was easy to evaluate the main radius of curvature of the resulting planar centerline curve. Using a quadratic minimization among the discrete set of n points of the centerline, it was possible to extract the main radius of curvature R c of the aqueduct centerline, which quantified its bending in the sagittal plane as well as the curvilinear length L of the centerline. For some patients, a double bending of the centerline was observed, with two radii of curvature of opposite sign (cf., Fig. 4). In this case, each radius of curvature R c+ (respectively, R c-) was fitted using a properly chosen number of points n + (respectively, n -) among the total number of points n = n + + n -,g i v i n gl e s s than 5% relative difference with the centerline in every case. The average curvature was thus computed from

R c = n + R c+ + n -R c- n , λ = a R c (1)
where the influence of curvature on the CSF flow can be quantified by the dimensionless parameter λ. Here, again, a is the longitudinally averaged major axis of the elliptical cross section of the aqueduct over cross-sectional images, i.e., a =(1/L) L 0 a(s)ds ≃ (1/n) n i=1 a i . Other useful parameters, of interest, for quantitative comparison with previous analyses [START_REF] Topakoglu | On the steady laminar flow of an incompressible viscous fluid in a curved pipe of elliptical cross-section[END_REF], [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF], were also derived from the major and minor radii a and b, such as the mean eccentricity e and the mean ellipticity m, defined as

e = 1 - b 2 a 2 , m = a -b a + b . (2)
The values observed for parameter m (m ≃ 0.6, as seen in Fig. 5 and Table II) showed that slightly noncircular models of the cerebral aqueduct were not relevant, and called for fully elliptic cross sections geometries. Also, a Fourier mode decomposition of the flow rate during the cardiac cycle, similar to [START_REF] Robertson | Flow in elliptical vessels calculated for a physiological waveform[END_REF], is proposed in the next sections.

B. Fluid Flow, Pressure, and Shear Evaluation Within the Aqueducts

In this section, we use an asymptotic approach for flow evaluation. This model is derived from an asymptotic approximation of the Navier-Stokes equations without considering elastic deformation, consistently with [START_REF] Gammack | Flow in pipes with non-uniform curvature and torsion[END_REF]- [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF]. It is necessary to consider a dimensionless formulation of the problem, which can then provide a patient-independent formulation whatever the particular aqueduct shape or cardiac temporal forcing. Taking * for dimensional quantities, let us first discuss a formulation for curvilinear coordinates, s, associated with the centerline arc length following [START_REF] Gammack | Flow in pipes with non-uniform curvature and torsion[END_REF]. Since the aqueduct is slightly bent, s is nondimensionalized by the radius of curvature R c , s * = R c s, while the radial coordinate r * = ar is nondimensionalized by the typical transverse length a. The forcing pulsation is used to define dimensionless time t * = t/ω, where the typical longitudinal velocity provides the scaling for the longitudinal velocity component u * = Uu. A viscous pressure is used for the pressure nondimensionalization since we consider moderate reduced Reynolds numbers, as will be seen in Section II-D, so that p * =(µU L/a 2 )p. The dimensionless flux is defined by Q * = U a 2 Q, where Q * is given by MRI acquisitions. Following previous authors and seeking the leading order, slightly curved, low Reynolds number flow description, the longitudinal velocity component forced by a single sinusoidal mode forcing with associated local longitudinal pressure gradient ∂p ∂s (s) satisfies

∇ 2 u -α 2 ∂u ∂t = - ∂p ∂s (3)
where ∇ 2 is the Laplacian operator in the plane locally orthogonal to the centerline at curvilinear distance s,tobesolvedwith boundary condition u =0 on the frontier ∂Ω(s) of the local transverse elliptic section Ω(s) having major and minor radii a(s) and b(s). The local pressure gradient is decomposed into a stationary K S (s) and unstationary K U (s). In subsequent notations, index S (respectively, U) is associated with stationary (respectively, unstationary) cases

∂p ∂s = K S (s)+ℜ K U (s)e it . ( 4
)
1) Stationary forcing: Solving (3) for α =0 and ∂P ∂s = K S (s) leads to

u S (x, y, s)=- 1 2 a 2 b 2 a 2 + b 2 1 - x 2 a 2 - y 2 b 2 K S (s) (5)
in the coordinates x, y locally orthogonal to the centerline arc length s, oriented toward the principal axis of the ellipse Ω(s) (see Fig. 2). The associated longitudinal flux Q S = Ω u S dS with infinitesimal elementary surface dS = dxdy in Ω i reads

Q S = -y S (s)K S (s) (6)
where we have introduced the local complex hydraulic admittance of the stationary contribution y S (s)=C S (s)+iS cS (s) which is, in this case, a pure real conductance

y S (s)=C S (s)= π 4 a 3 b 3 a 2 + b 2 (7)
since, for nonoscillating flow, the imaginary hydraulic susceptance contribution is zero, S cS (s)=0. The local pressure gradient K S (s) is adapted so that constant flux is preserved in each successive section Q S . By considering the leading order contribution to the viscous stress tensor of this longitudinally dominated flow, we also evaluate the shear stress τ S . It can be expressed (see the Appendix) at leading order O(λ) in the coordinates locally orthogonal to the centerline arc length s,inthe principal axis coordinates of the ellipse Ω(s), along its frontier ∂Ω(s)

τ S = λ a 2 b 2 a 2 + b 2 x 2 a 4 + y 2 b 4 K S (s) (8)
where index s refers to stationary flow contribution and t is the local tangent vector to the centerline at location s. By integrating around the elliptic contour ∂Ω(s), we can see that the simple result for the leading order total shear at the wall

τ TS = ∂ Ω τ S dl is τ TS (s)=-λ πab K S (s). (9)
2) Unstationary forcing: Solving (3) for nonzero α with complex velocity u U and forcing ∂P ∂s = K U (s)e it is performed numerically using a finite-element discretization of Ω(s) with a classical variational formulation using P 0 elements [START_REF] Mejdi Azaïez | Eléments Finis Pour Les Fluides Incompressibles[END_REF]. From the numerical evaluation of the complex velocity component u U , the flux Q U can be computed numerically from a simple spatial integration Q U = Ω u U dS. On the other hand, Haslam and Zamir [START_REF] Haslam | Pulsatile flow in tubes of elliptic cross sections[END_REF] provide an analytical expression for the flux

Q U = -ℜ y U (s)K U (s)e it (10)
which can be expressed using the local complex admittance y U (s):

y U (s)= 2πi η a 3 b 3 a 2 + b 2 1 - 2J 1 ( √ -iη) √ -iη J 0 ( √ -iη) (11)
with η =2 a 2 b 2 a 2 + b 2 α 2 and J 0 , J 1 the zeroth-and first -order Bessel functions. Consistently with previous results [START_REF] Kurtcuoglu | Computational investigation of subjectspecific cerebrospinal fluid flow in the third ventricle and aqueduct of Sylvius[END_REF] tends to [START_REF] Linninger | A mathematical model of blood, cerebrospinal fluid and brain dynamics[END_REF] as α → 0. The analytical result (11) (derived from [START_REF] Haslam | Pulsatile flow in tubes of elliptic cross sections[END_REF]) is fully consistent with finite-element numerical evaluation, as represented in Fig. 10. Similar to the stationary case, the oscillating part of the shear leading order wall shear stress is approximately given by (see the Appendix)

τ U = -λℜ Ω K U (s) -iα 2 u U dS e it (12)
or, using the flux expressions ( 10) and ( 11)

τ U = -λπab ℜ 2J 1 ( √ -iη) √ -iη J 0 ( √ -iη) e it K U (s) (13)
which obviously tends to (9) as α → 0 and η → 0.

C. Fourier Decomposition and Hydraulic Admittances 1) Fourier Decomposition of the Flux:

The experimental flux is decomposed into discrete Fourier modes from a signal having 32 points per cardiac cycle. The zeroth mode corresponds to the stationary flux Q S . The constant flux amplitude is generally smaller than the first oscillatory mode by almost three orders of magnitude. It is thus difficult to obtain a reliable Fourier decomposition, and this is why a dedicated acquisition sequence has previously been proposed for its evaluation [START_REF] Linninger | Cerebrospinal fluid flow in the normal and hydrocephalic human brain[END_REF], [START_REF] Huang | Supratentorial cerebrospinal fluid production rate in healthy adults: Quantification with twodimensional cine phase-contrast MR imaging with high temporal and spatial resolution[END_REF]. From the f k = 2π ω k frequency, we compute the sequence of Womersley numbers α k for which the admittance y k U (s) is evaluated for a flux amplitude Q k U (t). A truncated reconstruction of the variation of the flux Q n (t)=Σ n 0 Q k U (t) over the complete cardiac cycle [0,T c ] taking only the first n modes has been tested. The L 2 norm error E n recovered from the truncated evaluation is

E n = T c 0 (Q exp -Q n ) 2 dt T c 0 (Q exp ) 2 dt .
It is represented in Table I and Fig. 6. This computation shows that keeping only the first three modes provides a accurate description to within 10% of the flux temporal variation. The saturation of the error for n>3 arises since there is a modest number of points per cycle (32 here).

2) Hydraulic Admittances and TRP Reconstruction: Equations ( 6) and (10) describe a local linear relation between the flux and the pressure gradient associated with a local admittance. The local admittance is also the inverse of the local hydraulic impedance z(s)=y -1 . As for the electrical description, the integral of relations ( 6) and [START_REF] Linninger | Cerebrospinal fluid flow in the normal and hydrocephalic human brain[END_REF] leads to QU

y U (s) ds = -K U (s)ds =∆P U ( 14
)
QU is constant along the aqueduct (incompressible Newtonian fluid) such that

QU = Y ∆P U = 1 L 0 y U (s) -1 ds ∆P U (15)
where ∆P U is the pressure difference obtained between the two extremities of the aqueduct of length L for a flux amplitude QU .

A discrete version of (15) over each successive ellipses Ω i has been implemented with finite-difference integration. This was done for each mode k to evaluate the admittance Y k of each flux Q k U . The real and imaginary components of Y k = C k + iS k c are called conductance and susceptance, respectively. The resulting complex admittance results are discussed in the next section. We also evaluate the complex impedance Z k = 1

Y k and find that, for all patients, ℑ(Z k ) > 0. Thus, the electric model of the aqueduct is a self-inductance with

Z k = R k + L k ω k i with a phase lag φ k = tan -1 ( L k ω k R k ).
Using the truncation to the first three modes, described in the previous section, leads to the reconstruction of the total pressure drop for the aqueduct

∆P = R 0 Q S + 3 k =1 ℜ(Z k Qk U e iω k t ) , (16)
which is the main contribution to the TRP drop, i.e., ∆P ≈ TRP, as will be discussed in Section III. The values of Q S , Qk U , R k , and L k are reported in Table II.

D. Discussion of the Model's Validity

In this section, we investigate the validity of the creeping flow regime that has been developed for CSF in the previous sections. As mentioned in Section I, many detailed studies [START_REF] Lyne | Unsteady viscous flow in a curved pipe[END_REF]- [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF], [START_REF] Dean | Note on the motion of a fluid in a curved pipe[END_REF] Where units are not stated, the quantity is dimensionless. The resistances R k are nondimensionalized by µ L/a 4 with k = 0,1,2,3 andtheinductances

L k by µ L/a 4 ω k with k =1, 2, 3,w i t hω k =2πf k .
conditions considered, so that it is reasonable to neglect inertial corrections to the leading order viscous contributions. Most studies have focused on the occurrence and structure of secondary transverse counterrotating flows arising from the conjugated effects of inertia and curvature. Systematic Dean number and reduced Reynolds number expansions, respectively, are proposed for those corrections [START_REF] Lyne | Unsteady viscous flow in a curved pipe[END_REF]- [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF], [START_REF] Dean | Note on the motion of a fluid in a curved pipe[END_REF]. Here, we are mainly interested in the leading order viscous contribution to the flux for a given longitudinal pressure gradient. Following previous authors, we looked for the leading order flow description in the range of the fluid flow parameters evaluated by MRI (see, for example, Table II). According to [START_REF] Zalosh | Pulsating flow in a curved tube[END_REF] and [START_REF] Gammack | Flow in pipes with non-uniform curvature and torsion[END_REF], small reduced Reynolds number Re s ≈ 1 (see Table III for the mean value of Re s ) allows us to assume that the CSF flow is well approximated by [START_REF] Tain | Compliance effect on amplitude and phase of cranio-spinal CSF Flow Measured by MRI[END_REF]. Moreover, the pressure/flux relationship in a curved pipe with an elliptical cross section has been thoughtfully analyzed in [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF], where inertial deviations from the stationary creeping flow regime are systematically provided for each family of parameter m = (ab)/(a + b), Reynolds number Re, and dimensionless curvature λ

E = λ 2 A(m)+B(m)Re 2 + C(m)Re 4 (17)
where coefficients A(m), B(m), and C(m) can be extracted from figures given in [START_REF] Topakoglu | Viscous laminar flow in a curved pipe of elliptical cross-section[END_REF]. E is the dimensionless relative difference (the relative error) between the creeping flow flux and the real flux, which includes inertial effects up to third-order corrections that are valid for Reynolds numbers as large as 1000.

This relation provides an explicit quantitative test for the validity of our approximation, which we have illustrated in Fig. 7. Furthermore, this formula also gives an index for testing how good the proposed approximation is in certain diseases for which the Reynolds number might be large, or the aspect ratio λ might not be small. This can be done using the parameters A(m), B(m), and C(m) provided in the legend of Fig. 7.

Using E =0.1 when looking for an approximation to the flux with an error of < 10%, relation [START_REF] Zalosh | Pulsating flow in a curved tube[END_REF] provides a curve in the (λ,Re) plane for comparison. Most of patients (17 out of 20) had an ellipticity m between 0.57 and 0.70, so we chose the appropriate value of m for parameters A(m), B(m), and C(m) to compute the validity curve represented by the black line in Fig. 7. All the experimental data points lie below the 10% error curve, which indicates that our approximation for the linear pressure/flux relationship is accurate to within 10%. Since the mode truncation described in the previous section leaves out 15% of the total flow, the linear viscous approximation is more than a consistent approximation in this context. Thus, the validity of the approximation used in this paper compares well with the real physiological condition of the pool of patient considered.

III. RESULTS AND DISCUSSION

The very slight pressure variations observed in ventricles (lateral, third, and fourth) together with the pressure variations observed inside the cerebral aqueduct in the full CVD simulations illustrated in [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], Fig. 5] demonstrate the validity of the key hypothesis of our study: most of the pressure variations between LVs and spinal subarachnoid spaces arise within the thin passage of the cerebral aqueduct. Hence, the pressure difference amplitude obtained, of a few pascals, which is compatible with the TRP obtained in [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], is another confirmation that this hypothesis is relevant (see Fig. 8). Entrance effects discussed in [START_REF] Talbot | Pulsatile entrance flow in a curved pipe[END_REF] and numerically observed in [START_REF] Fin | Three dimensional modeling of the cerebrospinal fluid dynamics and brain interactions in the aqueduct of sylvius[END_REF] are expected to be small when considering realistic anatomical reconstruction of the cerebral aqueduct having a gently varying diameter.

As mentioned in [START_REF] Linninger | Cerebrospinal fluid flow in the normal and hydrocephalic human brain[END_REF], the hydrodynamic pressure, which can be reconstructed by solving CSF equations, is defined up to an unprescribed constant. In order to discuss our result quantitatively, it is interesting to consider how the cerebral aqueduct fits into a more global intracranial pressure distribution model. The electrical model depicted in Fig. 9 is very close to those proposed in [START_REF] Linninger | A mathematical model of blood, cerebrospinal fluid and brain dynamics[END_REF] and [START_REF] Czosnyka | Hydrocephalus shunts and waves of intracranial pressure[END_REF]. This linear model is easy to solve for the intracranial pressure P ICP

P ICP = RP V 3 + RZ 1 Cω 1 iP 0 + Z 1 P SAS R + RZ 1 Cω 1 i + Z 1 (18)
and thus evaluation of aqueduct impedance parameters Z 1 is useful but not sufficient to determine intracranial pressure. This conclusion is similar to the one drawn by previous authors [START_REF] Alperin | MR-intracranial pressure (ICP): A method to measure intracranial elastance and pressure noninvasively by means of MR imaging: Baboon and human study1[END_REF], who pointed out that cine phase-contrast MRI combined with fluid-pressure flow evaluation cannot provide intrinsic pressure values but necessitates supplementary information such as estimation of compliance and/or monitoring [START_REF] Czosnyka | Concept of true ICP in monitoring and prognostication in head trauma[END_REF]. Nevertheless, a positive correlation between TRP and ICP increase has been recognized [START_REF] Conner | Experimental normal-pressure hydrocephalus is accompanied by increased transmantle pressure[END_REF], [START_REF] Hoff | Transcerebral mantle pressure in normal pressure hydrocephalus[END_REF], so the evaluation of TRP is also interesting. As mentioned in Section I, the cerebral aqueduct is the main contributor to the total resistance between LVs and SAS. As R 1 ≫ R, using the electrical model of Fig. 9, a good approximation of P TRP is Hence, from an evaluation of the pressure drop between the third and fourth ventricles P V3 -P ICP , this simple electrical model leads to the conclusion that it is possible to obtain a good approximation for the TRP drop P TRP = P V3 -P SAS , and noninvasive estimation of this pressure from cine phasecontrast MRI is thus possible. We found a TRP magnitude of 4.3 ± 3.6 Pa and a mean TRP of 0.039 ± 0.029 Pa (respectively, ∆P max and ∆P S in Table III) which can be compared with typical literature values. In vivo studies on dogs with kaolin-induced hydrocephalus [START_REF] Conner | Experimental normal-pressure hydrocephalus is accompanied by increased transmantle pressure[END_REF] report a mean TRP of 34 Pa. In the cat kaolin model, Shapiro et al. [START_REF] Shapiro | Progressive ventricular enlargement in cats in the absence of transmantle pressure gradients[END_REF] found zero mean TRP. More recently, Penn et al. [START_REF] Penn | Pressure gradients in the brain in an experimental model of hydrocephalus[END_REF] have found that the mean TRP in dogs was not greater than the resolution of their sensors (66 Pa), and consequently, they concluded that the magnitude of such gradients could not be accurately determined in their study. A clinical study has been performed by Stephensen et al. [START_REF] Stephensen | There is no transmantle pressure gradient in communicating or noncommunicating hydrocephalus[END_REF] on ten human patients having communicating or noncommunicating hydrocephalus. The first microsensor was placed in the ventricle system, inside a ventricular catheter, and the second one in the subarachnoid space. When using a temporal average, they found a mean TRP of 1.3 ± 32 Pa, while the deviation from this temporal TRP pressure data (which should be related to the amplitude of TRP pressure oscillation) could be evaluated at 65 ± 22 Pa from their Fig. 3. The mean TRP value reported in [START_REF] Stephensen | There is no transmantle pressure gradient in communicating or noncommunicating hydrocephalus[END_REF] was compared to our stationary TRP ∆P S using a Mann-Whitney test. We found no significant difference with a p-value of 0.9 (nonparametric tests are insensitive to inhomogeneity of variance that may result from the use of different measurement methods). Nevertheless, when comparing the ten temporal standard deviations reported in [START_REF] Stephensen | There is no transmantle pressure gradient in communicating or noncommunicating hydrocephalus[END_REF] with ∆P max of our 20 healthy patients, we found them significantly different, with a p-value smaller than 10 -3 , which is consistent with the large departure from the statistical mean (65 Pa, as against, our 4.3 Pa). This difference between normal and pathological situations, and average value obtained for the oscillating part of the TRP, are also consistent with other estimates obtained from cine MRI measurements, 3-D reconstruction of brain geometry, and direct simulation by Penn [START_REF] Penn | Neurosurgical forum: Letters to the editor[END_REF]. They report a TRP magnitude below 10 Pa for healthy volunteers and close to 50 Pa in patients with hydrocephalus. The ICP in normal subjects is of the order of 500 Pa and the TRP smaller than 10 Pa. In abnormal conditions, the ICP can become as high as 3000 Pa, whereas the TRP does not exceed 30 Pa [START_REF] Linninger | Cerebrospinal fluid flow in the normal and hydrocephalic human brain[END_REF]. Our findings are thus consistent with all these considerations related to previous measurements. Furthermore, it is important to stress that previous direct experimental measurements of the TRP were obtained by evaluating pressure differences that were 10-100 times smaller than the actual value measured. As the resolution of the sensors was hardly a hundred times smaller than the actual measurement, the reliability of the direct measurements can be called into question. One major contribution of our TRP evaluation, apart from its painless and noninvasive nature, lies in the fact that it is much more precise since the limitation is related to the evaluated flux and, to a lesser extent, the spatial resolution of the reconstructed aqueduct.

P TRP = P V 3 -P SAS ≈ P V 3 -P ICP . ( 19
) Q1 P V 3 R 1 L 1 P ICP R P SAS C P 0
Q max mm 3 /s 357 ± 290 Q S mm 3 /s 9.1 ± 13 Q1 U mm 3 /s 270 ± 247 Q2 U mm 3 /s 67 ± 46 Q3 U mm 3 /s 25 ± 20 u max mm/s 32 ± 25 τ usmax 1.7 ± 2.2 τ uumax 5 ± 6.4 √ ab min mm 1.59 ± 0.3 α 1 (f 1 Hz) 9.5 ± 2.2 (1.2 ± 0.2) α 2 (f 2
Another consequence of our findings is to show that the stationary Hagen-Poiseuille approximation for the aqueduct flow used in [START_REF] Masoumi | Mathematical modeling of CSF pulsatile hydrodynamics based on fluid-solid interaction[END_REF] and [START_REF] Smillie | A hydroelastic model of hydrocephalus[END_REF] has to be corrected. Our results show that the leading order contribution comes from the first oscillatory mode k =1, for which the Womersley number is nonzero, and thus, the resulting admittance is complex (see Fig. 10). Analytical result (11) (derived from [START_REF] Haslam | Pulsatile flow in tubes of elliptic cross sections[END_REF]) is fully consistent with finite-element numerical evaluation, as represented in Fig. 10, where the two curves cannot be distinguished. Furthermore, our evaluation shows that the resistance part R k does not predominate over the inductance part L k . Our results (see Tables II andIII) show an average phase lag φ 1 of 63 ± 7 • between pressure and flow, which is consistent with the 58 • found in [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF] using 3-D numerical simulations. We also note the influence of α on the electric impedance. When α 1 is three times larger, the electric resistance and inductance are five and ten times larger, respectively. The mean values of R 1 and L 1 over the 20 patients are given in Table III. Finally, it is also interesting to note that our findings might contribute to the understanding of pathological evolutions associated with aqueduct stenosis, which is a common cause of hydrocephalus. Such stenosis increases wall shear stresses and can be easily detected by our method. Increased wall shear stresses occur with increased pressure drop and may result in ongoing damage to the aqueduct where the area of the sections is minimal, as mentioned in [START_REF] Jacobson | Computer modelling of the cerebrospinal fluid flow dynamics of aqueduct stenosis[END_REF]. This effect is also observed in healthy patients in Fig. 11(a)-(c) where the maximum for the total wall shear stress of nonstationary flow components τ U is shown to be maximal when the local section πab is minimal.

(c) Y Y Y α α α C S c α 1 α 2 α 3 C S c α 1 α 2 α 3 C S c α 1 α 2 α 3

IV. CONCLUSION

Our contribution has successfully tested the validity of a low curvature/low reduced Reynolds number approximation for CSF flow in real patients. Using this approximation, we propose a simple and very fast method for evaluating the aqueduct admittance from MRI anatomical images and cine acquisitions. This method can also evaluate the shear stress within the cerebral aqueduct, which is greatest in its smallest portion. The proposed approach provides the total pressure drop in the aqueduct, which permits a simple, noninvasive patient-based, approximate evaluation of the TRP.

Combined with other invasive (e.g., intrathecal pressure monitoring) or noninvasive (compliance estimation [START_REF] Cohen | Development of a theoretical framework for analyzing cerebrospinal fluid dynamics[END_REF], [START_REF] Alperin | MR-intracranial pressure (ICP): A method to measure intracranial elastance and pressure noninvasively by means of MR imaging: Baboon and human study1[END_REF], [START_REF] Tain | Noninvasive intracranial compliance from MRI-based measurements of transcranial blood and CSF flows: Indirect versus direct approach[END_REF]) measurements, our method could be useful for estimating intracranial pressure or evaluating a patient's disorder.

APPENDIX

This section provides a detailed derivation of the wall shear stress relations (8), ( 9), [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF], and [START_REF] Penn | Ventricule wall movements and cerebrospinal fluid flow in hydrocephalus[END_REF]. Let us call the total velocity field U. Due to the low-curvature geometry of the aqueduct and the incompressibility condition, U can be decomposed into a leading order term in the local longitudinal direction, ut,plus a small correction λv for the in-plane contribution in the elliptic section Ω(s), where v ∼ O(1), with λ ≪ 1 and v.t =0, so that U = ut + λv.

From the chosen nondimensionalization, the in-plane coordinates (x, y) in section Ω(s) are scaled by a and the longitudinal coordinate z by R c , so that the local dimensionless gradient operator reads ∇ =(∇ 2 , λ∂ z ), where ∇ 2 =(∂ x ,∂ y ). The dimensionless stress tensor σ then reads

σ = -p + λ 2 ∇⊗U + ∇ T ⊗ U
where we have consistently used the viscous pressure (µU L/a 2 ) to define the dimensionless stress σ, such that σ * =(µU L/a 2 )σ. By considering only the deviatoric part of the stress σ d , (since the viscous stress is the only contribution to the shear here) In the following, we neglect O(λ 2 ) terms and no longer mention them. In the stationary case, as given in (5), u can be computed analytically, and we have where we have used the same notation ∇ 2 = ∂ 2 x + ∂ 2 y as in (3). If we use (3) in the stationary case, this leads to relation [START_REF] Cohen | Development of a theoretical framework for analyzing cerebrospinal fluid dynamics[END_REF] τ TS = λ -K S Ω dS = -λπabK S .

σ d = λ∇ 2 ⊗ ut + O(λ 2 )=λ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
τ S = λ 4 a 2 b 2 a 2 + b 2 K S (s)) t • ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
Alternatively, using (3) in the unstationary case leads to the complex version of [START_REF] Sweetman | Threedimensional computational prediction of cerebrospinal fluid flow in the human brain[END_REF] τ TU = λ -iα 2 QU -πabK U e it .

Fig. 1 .

 1 Fig. 1. Example of cerebral aqueduct (CA) joining the third (V3) and fourth (V4) ventricle visualized from a volume rendering in the sagittal plane of the anatomical MRI image of patient A. LVs can be seen on top.

Fig. 2 .

 2 Fig. 2. Sketch of the cerebral aqueduct centerline description with associated transverse elliptic section Ω(s) having minor axis b(s), major axis a(s),a n d ellipse frontier ∂Ω(s) at longitudinal curvilinear distance s.

Fig. 3 .

 3 Fig. 3. Gray-level anatomical slice of the aqueduct. Dotted line: gray-level contour profile of the aqueduct. Continuous line: ellipse extracted from best fit of the contour line (Patient A).

Fig. 4 .

 4 Fig. 4. Result of centerline segmentation in the sagittal plane (unit in voxel = 0.55 mm). Radius of curvature R c fitting in two cases for n + = n and n -=0 (left, Patient B)andforn + = 7 n 10 and n -= 3 n 10 (right, Patient C).

Fig. 5 .

 5 Fig. 5. Reconstructed aqueduct for three healthy patients anonymized as Patient D, Patient E, and Patient F.

Fig. 6 .

 6 Fig. 6. Solid line: MRI nondimensional flux versus times (over a cardiac cycle). dashed, dotted, and dot-dashed lines: reconstructed flux with n = 1, 2, and 3, respectively (Patient D).

Fig. 7 .

 7 Fig. 7. Plot of λ versus the Reynolds number Re for 0.57 <m<0.70. Validity of the low Dean number approximation. Lines correspond to E = 10% error on the TRP and are plotted for A(m)=-0.22, B(m)= -6.10 -5 , C (m)=2.10 -8 (m =0.54)a n df o rA(m)=3.12, B(m)= -2.10 -4 , C (m)=10 -9 (m =0.57). Each cross (+) corresponds to a patient measurement.

Fig. 8 .

 8 Fig. 8. (Top) experimental flux Q in mm 3 .s -1 . (Bottom) Reconstructed oscillating pressure ∆P in pascals for a cardiac cycle in seconds.

Fig. 9 .

 9 Fig. 9. Electrical model representing CSF flow.

  Hz) 13.7 ± 3.1 (2.5 ± 0.45) α 3 (f 3 Hz) 17 ± 3.8 (4 ± 0.8) R 0 32 ± 20 (2.2 ± 1.55 10 6 Pa.s.m 3 ) R 1 41.3 ± 27.1 (2.7 ± 1.710 6 P a.s.m 3 ) L 1 14.9 ± 10.5 (7.64 ± 410 5 P a.s 2 .m 3 ) R 2 51.9 ± 34.3 (3.5 ± 2.110 6 P a.s.m 3 ) L 2 14.4 ± 10.1 (3.5 ± 1.710 5 P a.s 2 .m 3 ) R 3 60.3 ± 40 (4 ± 2.410 6 P a.s.m 3 ) L 3 14.1 ± 10 (2.3 ± 1.310 5 P a.s 2 .m 3 ) φ 1 63 o ± 7 o ∆P max Pa 4.316 ± 3.6 ∆P S Pa 0.039 ± 0.029

Fig. 10 .

 10 Fig. 10. Hydraulic admittance Y = C + iS c versus Womersley number α. The hydraulic conductance C (α) is shown as a solid line, and the susceptance S c (α) as a dashed line. The vertical lines indicate the Womersley numbers α k for k =1, k =2,andk =3.

Fig. 11 .

 11 Fig. 11. (Top) Oscillating (solid line) τ TU and stationary (dashed line) τ TS wall shear stress in pascals. (Bottom) Surface area πa(s)b(s) in mm 2 of the aqueduct section versus curvilinear coordinate s.

+ O(λ 2 )

 2 we then compute the local shear stress τ (a scalar), at each point of the wall ∂Ω where the normal in-plane vector is denoted nτ = t • σ d • n = λ∇ 2 u • n + O(λ 2 ).

  obtain result[START_REF] Masoumi | Mathematical modeling of CSF pulsatile hydrodynamics based on fluid-solid interaction[END_REF] τ S = λ a 2 b 2 a 2 + b 2 x 2 a 4 + y 2 b 4 K S (s).Furthermore, we can estimate the total wall shear stress by integrating along the contour ∂Ωτ T = λ ∂ Ω ∇ 2 u • n dlwhich, from the Green formula also readsτ T = λ Ω ∇ 2 ud S(20)

  have already investigated moderate Reynolds number flow in curved pipes. In this context, inertial and curvature effects are properly described by a single dimensionless Dean number De = Re • √ λ for stationary flows or by a reduced Reynolds number Re s =(Re/α) 2 λ for oscillating flows. The latter parameter Re s is generally small (see Table III) in the normal

		Patient D	Patient E	Patient F
	λ	0.18	0.25	0.12
	e	0.91	0.92	0.86
	m	0.68	0.68	0.59
	Re	5.78	21.01	16.60
	De	2.49	10.52	5.87
	Re s	0.043	0.753	0.35
	ām m	4.21	4.45	3.54
	Lm m	14.22	9.85	9.3
	Q max mm 3 /s	99	371	254
	Q S mm 3 /s	6	5	3
	Q1 U mm 3 /s Q2 U mm 3 /s Q3 U mm 3 /s	84 26 9	322 49 56	203 58 8
	u max mm/s	9.08	21.28	34.24
	τ usmax	1.22	0.30	0.27
	τ uumax √ ab min mm	3.83 1.71	8.02 2.19	1.32 1.37
	α 1 (f 1 Hz)	12.00 (1.42)	12.14 (1.30)	9.87 (1.36)
	α 2 (f 2 Hz)	16.97 (2.84)	17.16 (2.60) 13.96 (2.71)
	α 3 (f 3 Hz)	20.78 (4.26)	21.02 (3.90) 17.10 (4.07)
	R 0	61.1	34.2	37.0
	(R 1 ; L 1 )	(77.7; 23.8)	(48.1; 19.7)	(45.7; 13.6)
	(R 2 ; L 2 )	(97.2; 22.5)	(63.2; 20.1)	(56.7; 13.2)
	(R 3 ; L 3 )	(114; 22.8)	(72.8; 18.9)	(64.6; 12.4)
	φ 1	69 o	73 o	68 o
	∆P max Pa	1.76	2.43	2.64

TABLE III TABLE OF NUMERICAL

 IIIOF MEAN VALUES MEASURED FOR 20 DIFFERENT PATIENTS,WITH STANDARD DEVIATION

	λ e m Re De Re s ām m Lm m	0.19 ± 0.06 0.84 ± 0.14 0.59 ± 0.14 29 ± 25 14 ± 13 0.49 ± 0.62 3.6 ± 0.9 10.2 ± 2.7