Keltoum Chahour

Rajae Aboulaich
email: aboulaich@gmail.com

Abderrahmane Habbal

Cherif Abdelkhirane

Nejib

Nejib Zemzemi

Numerical simulation of the fractional flow reserve (FFR)

Keywords: Mathematics Subject Classification. 35Q30, 92C50, 97M60, 74F10 Atherosclerosis, fractional flow reserve, generalized flow model, fluid-structure interaction

come

Introduction

Atherosclerosis is a chronic inflammatory disease that affects the entire arterial network and especially the coronary arteries. It is an accumulation of lipids over the arterial surface due to a dysfunction of this latter.

The objective of clinical intervention, in this case, is to establish a revascularization using different angioplasty techniques, among which the implantation of stents is the most widespread. This intervention involves intro-Q2 ducing a stent into the damaged portion in order to allow the blood to circulate in a normal way over all the vessels. Revascularization is based on the principle of remedying ischemia, which is a decrease or an interruption of the supply of oxygen to the various organs. This anomaly is attenuated by the presence of several lesions (multivessel disease patients), which can lead to several complications. The key of a good medical intervention is the fact of establishing a good diagnosis, in order to decide which lesion requires to be treated. In the diagnosis phase, the clinician uses several techniques, among which angiography is the most popular. Angiography is an X-ray technique to show the inside (the lumen) of blood vessels, in order to identify vessel narrowing: stenosis. Despite its widespread use, angiography is often imperfect in determining the physiological significance of coronary stenosis. If the problem remains simple for non-significant lesions (≤ 40%) or very severe (≥ 70%), a very important category of intermediate lesions must benefit from a functional evaluation which will determine the strategy of treatment [START_REF] Cuisset | Qu'est-ce que la FFR? Comment l'utiliser? Réalités Cardiologiques[END_REF].

The technique of the fractional flow reserve (FFR) has derived from the initial coronary physical approaches decades ago. Since then, many studies have demonstrated its effectiveness in improving the patients prognosis, by applying the appropriate approach. Its contribution in the reduction of mortality was statistically proved by the Fractional Flow Reserve Versus Angiography for Multivessel Evaluation (FAME) study [START_REF] Van Nunen | Fractional flow reserve versus angiography for guidance of PCI in patients with multivessel coronary artery disease (FAME): 5-year follow-up of a randomised controlled trial[END_REF]. It is established that the FFR can be easily measured during coronary angiography by calculating the ratio of distal coronary pressure P d to aortic pressure P a . These pressures are measured simultaneously with a special guide-wire. FFR in a normal coronary artery equals to 1.0. FFR value of 0.80 or less identifies ischemia-causing coronary lesions with an accuracy of more than 90% [START_REF] Van Nunen | Fractional flow reserve versus angiography for guidance of PCI in patients with multivessel coronary artery disease (FAME): 5-year follow-up of a randomised controlled trial[END_REF].

Obviously, from an interventional point of view, the FFR is binding since it is invasive. It should also be noted that this technique induces additional costs, which are not covered by insurances in several countries. For these reasons, it is used only in less than 10% of the cases.

In this perspective, a new virtual version of the FFR, entitled virtual fractional flow reserve (VFFR), has emerged as an attractive and non-invasive alternative to standard FFR, see [START_REF] Morris | Virtual (computed) fractional flow reserve: current challenges and limitations[END_REF][START_REF] Tu | Fractional flow reserve calculation from 3-dimensional quantitative coronary angiography and TIMI frame count[END_REF]. However, there are key scientific, logistic and commercial challenges that need to be overcome before VFFR can be translated into routine clinical practice.

Blood circulation is generated by the heart "pump" that produces consecutive contraction/relaxation movements. These movements are performed during what we call a cardiac cycle. It consists of two phases: the systole, that is the phase of contraction. It occupies about one third of the cardiac cycle. The diastole, during which the heart muscle relaxes and refills with blood. It lasts the two remaining thirds of the cardiac cycle.

Assuming a healthy heart and a typical rate of 70-75 beats per minute, each cardiac cycle takes about 0.8 sec.

A flow and pressure wave starts from the heart to cross all major arteries in which it is damped, dispersed and reflected due to changes in vessel sizes, as well as the properties of the tissues and branches. The two coronary arteries cover the surface of the heart and represent the first derivations of the general circulation, see Figure 1.

Fractional flow reserve

In order to provide a good estimation of the VFFR, a good understanding of the FFR technique as well as the various medical verifications preceding the test is required. In this section, we give details about the invasive FFR. The patient is initially placed in the supine position. To start the measure of the FFR the operator crosses the coronary lesion with an FFR-specific guide wire. This guide wire is designed to record the coronary arterial pressure distal to the lesion (Fig. 2 left). Once the transducer is distal to the lesion (approximately 20 mm), a hyperemic stimulus is administered by injection through the guide catheter, and here the FFR value is subject to a wide variation. The operator waits for few minutes so that the FFR value becomes constant, this value corresponds to the maximal vasodilatation.

The mean arterial pressures from the pressure wire transducer P aortic and from the guide catheter P distal are then used to calculate FFR ratio: FFR = P distal /P aortic (Fig. 2

right).

The pressure values given by the FFR instrument are calculated as temporal mean pressures over small time intervals, depending on the frequency of acquisition of the pressure sensor p s (t). Assuming that T c is the duration of a cardiac cycle, these pressures are given as follows: Left, represents schema of the invasive FFR technique [START_REF] Cuisset | Qu'est-ce que la FFR? Comment l'utiliser? Réalités Cardiologiques[END_REF]. Right, a typical example of FFR measurement. Automated calculation of FFR corresponds to the ratio of mean distal coronary pressure (green) to mean aortic pressure (red) during maximal hyperemia, see [START_REF] Toth | Standardization of fractional flow reserve measurements[END_REF].

P = 1 T c Tc 0 p s (t)dt. (1.1)
An FFR value lower than 0.75 indicates a hemodynamically significant lesion. An FFR value higher than 0.8 indicates a lesion that is not hemodynamically significant. Values between 0.75 and 0.80 are indeterminate and should be considered in the context of patient's clinical history to determine if revascularization is necessary.

Modeling and simulation of blood flow have attracted a lot of attention. A realistic modeling of blood velocity and pressure is mandatory in view of virtual design and analysis of the performance of some medical devices, such as heart valves, filters and stents, on one hand. On the other hand, it provides clinicians with a set of non-invasive and reliable tools to guide the diagnostic phase.

In this paper, we aim at presenting a preliminary two-dimensional (2D)-based study to understand the flow distribution in a stenosed coronary artery and to enumerate the factors that affect the value of the VFFR. We give a special concern to the influence of the lesion's parameters.

Since we are interested in studying the flow in the coronary arteries, we decided to use a non-Newtonian flow model like in [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF] and [START_REF] Boujena | Generalized Navier-Stokes equations with Non-standard conditions for blood flow in atherosclerotic artery[END_REF]. In fact, the coronary arteries have a small caliber (0.5 cm) compared to the aorta for example, where the use of non-Newtonian flow model is not really crucial. In a first place, we assumed that the arterial wall is rigid. This is justified by the fact that the FFR value taken into account by the clinician during the test is established into a domain corresponding to the maximal vasodilation.

In a second place, we considered a simplified weakly coupled fluid-structure interaction model to include the arterial wall elastic behavior, as presented in [START_REF] Fernandez | Fully decoupled time-marching schemes for incompressible fluid/thin-walled structure interaction[END_REF]. The coronary arteries are subject to two different displacements:

-Large displacements: Since they are partially attached to the myocardium, they are directly influenced by the myocardium contraction/relaxation, and by the movements induced by the respiratory system.

-Small displacements: Due to the propagation of the blood wave generated by the heart pulse.

In this work, we chose to restrain our study to the small displacements. Moreover, only the upper face of the arterial portion is involved since the lower one is fixed (glued to the myocardium), see [START_REF] Lassila | A reduced computational and geometrical framework for inverse problems in haemodynamics[END_REF]. We also assume that the displacements of the shell are infinitesimal. As for the boundary conditions, even if their choice is crucial for this kind of studies, we decided to make few simplifications in order to be able to address the problem. At the inlet, we impose a sinusol wave function to illustrate the pulsatile property of the flow, as in many works [START_REF] Gostaf | Pressure boundary conditions for blood flows[END_REF], [START_REF] Fernandez | Fully decoupled time-marching schemes for incompressible fluid/thin-walled structure interaction[END_REF] and [START_REF] Lassila | A reduced computational and geometrical framework for inverse problems in haemodynamics[END_REF]. At the outlet, we assume that the vessel following the portion of interest is long enough before getting to the small tissues, or having a change in the vessel caliber, so there is no resistance effect. This justifies the choice of natural outlet boundary condition.

We implement from scratch, within the FreeFem++ environment, a finite element solver for both the generalized flow model and the coupled arterial wall/blood flow model. To describe the 2D mobile domain, we used an Arbitrary Lagrangian-Eulerian (ALE) coupling scheme. Like in [START_REF] Lassila | A reduced computational and geometrical framework for inverse problems in haemodynamics[END_REF], we proceed by decoupling the problem into two parts: solid and fluid, while introducing coupling boundary conditions at the interface. We then introduce and implement an algorithm for computing the VFFR following the industrial manufacturer protocol for the analogic FFR estimation. Using the solvers, we lead a study of the VFFR with respect to the stenosis dimensioning parameters (degree of stenosis and lesion radius).

In this study, our goal is to provide a first estimation of the coronary fractional flow reserve VFFR in a simplified 2D geometry. We conduct different simulations in order to identify the impact of the lesion's parameters on the value of VFFR. In Section 1, we present the non-Newtonian flow model used to carry all the simulations, as well as the boundary conditions. In Section 2, we give some numerical results considering that the arterial portion is rigid. The flow and pressure distributions are given in two different geometries: a single stenosis case (presence of only one lesion), and a multi-stenosis case. In these two configurations of the domain, we plot VFFR variations according to some parameters of influence: degree of stenosis, lesion's radius and the spacing between the two lesions in the multi-stenosis case. In Section 3, we present the fluid-structure interaction model, and the different VFFR variations corresponding to it.

Generalized non-Newtonian flow model

In large arteries, blood flow can be modeled by the Navier-Stokes equation. In our case, the blood cannot be assimilated to a Newtonian fluid, since the coronary vessels caliber is very small (0.5 cm). We choose a non-Newtonian flow model, as in [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF]. The mathematical model was studied in [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF] and authors proved the existence of a solution to this type of problems. In this paper, we are more interested in giving a bi-dimensional based estimation of the VFFR on the one hand. On the other hand, we lead different simulations, in order to explore the impact of the plaque's characteristics on the velocity and pressure fields.

We consider the Carreau law and we suppose that the viscosity varies as a function of the second invariant of the deformation tensor s(u):

(s(u)) 2 = 2Du : Du = 2 i,j (Du) ij (Du) ji , (2.1)
with:

Du = 1 2 (∇u + ∇ T u). (2.2)
Following the Carreau law, µ is given by:

µ = µ ∞ + (µ 0 -µ ∞ )(1 + (λs(u)) 2 ) (n-1)/2 , (2.3)
where µ 0 = 0.0456 Pa.s and µ ∞ = 0.0032 Pa.s, are the values of the viscosity for the lowest and highest shear rates. λ = 10.03 s and n = 0.344.

The problem considered involves the blood velocity u = (u 1 , u 2 ) and pressure p defined in Ω f × (0, T c ) as follows (the considered domain Ω f is shown in Fig. 3):

ρ f ∂u ∂t + ρ f (u.∇)u -∇.(2µ(s(u))Du) + ∇p = 0, sur Ω f × (0, T c ) ∇.u = 0, sur Ω f × (0, T c ) (2.4)
where ρ f is the blood density, we impose ρ f = 1060 kg.m -3 like in [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF][START_REF] Fernandez | Fully decoupled time-marching schemes for incompressible fluid/thin-walled structure interaction[END_REF].

These equations are completed with the following boundary conditions on Ω f (n is the normal):

2µ(s(u))Du.n -pn = h, sur Γ in × (0, T c ), (2.5) 2µ(s(u))Du.n -pn = 0, sur Γ out × (0, T c ), (2.6)
u = 0, sur Γ ω1 ∪ Γ ω2 × (0, T c ). (2.7)
The blood flow is initially at rest and enters the vessel by the left side Γ in where a sinusol pressure-wave with a maximum P max = 10 4 Pa is prescribed during T * = 5.10 -3 sec. The wave's profile is set equal to a stress vector of magnitude h, oriented in the negative normal direction given by the equation:

h = (P max × (1 -cos(2πt/T * )), 0) t , x ∈ Γ in , 0 ≤ t ≤ T * (0, 0) t , x ∈ Γ in , T * ≤ t ≤ T c . (2.8)
We consider that the fixed geometry at t = 0 corresponds to a maximal vasodilation.

The outflow is the right boundary Γ out where a zero pressure is imposed. A no-slip condition is enforced on the lower and upper boundaries Γ ω1 and Γ ω2 , which assume that the fluid is not moving with respect to these boundaries. given by the following equation:

u 0 (y) = u 0m × y/H × (1 -y/H), (2.9)
where u 0m = 0.4 m/s.

Arterial coronary plaques present a large variability in their configuration. We chose a simplified axisymmetric 2D configuration in order to address our problem, following [START_REF] Lassila | A reduced computational and geometrical framework for inverse problems in haemodynamics[END_REF]. The shape of the plaque in this case is modeled as a sinusol function: The variational formulation of the problem can be written as follows:

ω s (x) = D × cos(π(x -x s )/2 * δ) if x s -δ < x < x s + δ 0 otherwise. ( 2
   ρ f Ω f ∂u ∂t vdx + (Au, v) + ρ f b(u, u, v) = Γin hvdσ + Γω 1 gvdσ, ∀v ∈ V u(0) = u 0 , sur Ω f (2.11) with: (Au, v) = Ω f 2µ(s(u))Du : Dvdx, (2.12) b(u, v, w) = 2 i,j=1 Ω f u i ∂v j ∂x i w j dx. (2.13)
Simulations are performed using the finite element solver Freefem++, based on a semi-implicit time discretization scheme. Fluid velocity and pressure are calculated at each time step. A comparison with the Newtonian In order to visualize the impact of the plaque's characteristics on the flow over the diseased portion of the artery, the degree of stenosis R varies from 40% to 70% (focusing only on the intermediate lesions). The plaque's radius also varies from 2.5 mm to 5 mm. Since the presence of many lesions is clinically frequent, we have also considered a geometrical model with two plaques to get an estimation of the velocity and pressure field variations in this case.

Numerical results

The case of single stenosis

The simulation of blood flow in the presence of stenosis in a 2D geometry has been the subject of several works [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF][START_REF] Lassila | A reduced computational and geometrical framework for inverse problems in haemodynamics[END_REF][START_REF] Li | How critical is fibrous cap thickness to carotid plaque stability?: a flow-plaque interaction model[END_REF]. These works were based on the Navier-Stokes model and the arterial wall was considered to be rigid. In our work, we consider a non-Newtonian flow model, as in [START_REF] Boujena | A 2D mathematical model of blood flow and its interactions in an atherosclerotic artery[END_REF]. In this first simulation, the arterial wall is considered to be rigid. In Figure 4, we give velocity and pressure distribution using Navier-Stokes model, in a first place, and using the generalized flow model in the second. The length of the plaque is 10 mm, and the degree of stenosis is 40%. Velocity arrows show the flow profile across the portion of the vessel. We can see reverse flow on the distal side of the plaque. Severe stenosis leads to high flow velocity, high pressure at the throat of the lesion, and a large re-circulation region distal to it.

For the calculation of the VFFR ratio, the aortic pressure P a is calculated at each time step by the spatial mean pressure of the nodes at 1 cm from the inlet of the vessel: x a = 1 cm. Whereas the distal pressure P d is obtained at 1 cm after the lesion: x d = x s + 1 cm. Then a temporal mean is performed during the cardiac cycle: mean P a and mean P d are then used to calculate the VFFR ratio. The following setting (Fig. 5) describes the approach.

In order to take into account the time variations in the value of the VFFR, this value is calculated during 5 consecutive cardiac cycles. We notice that starting from the third cardiac cycle, this value becomes constant.

The VFFR takes values in the neighborhood of 0.67 for a lesion with a degree of stenosis equal to 75%.

The following graphic (Fig. 6) gives the variation of the VFFR during the five cardiac cycles.

For all the graphics in the next sections, the VFFR value considered is calculated during the third cycle of the simulation. The two preceding cycles are run in order to reach stable pressure distribution. The following figures give the aortic pressure P a , the distal pressure P d and the VFFR respectively according to the degree of stenosis (Fig. 7 left) and the plaque's radius (Fig. 7 right).

The most common parameter considered to evaluate the significance of a lesion is the degree of stenosis.

However, the lesion length (or radius) is also significant for this evaluation, especially when the degree of stenosis is in the intermediate value range [START_REF] Lopez-Palop | Correlation between intracoronary ultrasound and fractional flow reserve in long coronary lesions. A three-dimensional intracoronary ultrasound study[END_REF]. The linear regression models presented in [START_REF] Lopez-Palop | Correlation between intracoronary ultrasound and fractional flow reserve in long coronary lesions. A three-dimensional intracoronary ultrasound study[END_REF] give the correlation between the FFR value (obtained after the invasive test) and different plaque's parameters. Particularly, the lesion radius and the degree of stenosis were considered. The graphics given in paper [START_REF] Lopez-Palop | Correlation between intracoronary ultrasound and fractional flow reserve in long coronary lesions. A three-dimensional intracoronary ultrasound study[END_REF] were obtained from a statistical study of medical data. The results in Figure 7 cannot be quantitatively compared to those presented in this paper; however, we can see qualitatively that the graphics have approximately the same trend.

Figure 8 shows the simulated VFFR corresponding to different values of the lesion radius according to the degree of stenosis. We can note from this figure that the curve describing the VFFR according to the degree of stenosis changes with the value of the lesion's radius. Thus, there is an important change in classification, especially for the lesions with a degree of stenosis lower than 45%. For example, for a degree of stenosis of 40%, VFFR value is equal to 0.82 in the case of a lesion's radius of 0.25 cm, and to 0.75 in the case of a lesion's radius of 0.5 cm. As a consequence, there is a change in the lesion's classification from not hemodynamically significant to hemodynamically significant. Top, velocity and pressure field corresponding to identic lesions of 40% stenosis, with a spacing 'a' of 0.5 cm. Bottom, velocity and pressure field with a spacing 'a' of 1.5 cm.

Mutli-stenosis case

In the case of a multi-stenosis diseased patient, many lesions might be considered in the arterial wall. The following figures describe the blood velocity and pressure in this case.

The distance between the two lesions influences the flow, and particularly the micro-circulation downstream of the stenosis. Thus, the values of the VFFR obtained in the two cases given in Figure 9 are different, even if the lesion is somehow similar. The VFFR obtained for a spacing of 0.5 cm is equal to 0.73, while the VFFR with a spacing of 1.5 cm is equal to 0.81.

Coupling scheme: fluid-structure interaction

To achieve more realistic simulations, we consider the fluid-structure interaction between the arterial wall and the blood. We assume that the displacements of the shell are infinitesimal, and that only the upper face of the arterial portion is able to move. In a first place, a generalized linear Koiter model is adopted for the structure, as in [START_REF] Gostaf | Pressure boundary conditions for blood flows[END_REF]. In this case, the arterial wall is a 1D layer with a thickness .

The problem is to find the solid vertical displacement η and the solid vertical velocity η = ∂ t η of the solid such that:

       ρ s ∂ t η -c 1 ∂ 2 x η + c 0 η = -σ(u, p)n.n overΓ ω1 × (0, T c ), η = ∂ t η over Γ ω1 × (0, T c ), u.n = η, u.τ = 0 over Γ ω1 × (0, T c ), η = 0 over ∂Γ ω1 × (0, T c ).
where:

σ(u, p) = -pI + 2µ(s(u))Du.
u and p are respectively the fluid velocity and pressure, solutions of problem (2.4). ρ s = 1.1 is the solid density.

Conclusions

In this paper, we led different simulations to study the flow through a sclerotic artery. Firstly, we considered a generalized flow model in a fixed 2D domain. We assumed that the initial configuration of the domain corresponds to the maximal vasodilation of the portion of interest. Our purpose was to give a first estimation of the VFFR. We studied the variation of the VFFR with respect to some lesion's parameters: the degree of stenosis and the lesion's radius in the case of a single stenosis. In the case of multi-stenosis (the presence of two parallel lesions), we also studied the VFFR variations according to the distance between the two lesions, since this value also modifies the blood circulation through the diseased portion. Secondly, we introduced a generalized fluid-structure interaction model, in order to take into account the infinitesimal displacements of the upper arterial wall. Large displacements due to the myocardium movements were not considered. Each one of these models: rigid and elastic has a particular importance in the quantification of the VFFR. In medical practice, it is not the degree of stenosis that modifies the FFR value, even if this parameter is dominant to judge the severity of a lesion. But it is the coronary micro-circulation downstream of the stenosis and its importance that must impact the FFR. This micro-circulation depends not only on the shape of the plaque (flattened, large, sharp...), but also on the flow and the wall properties as well. The VFFR model presented in this paper is able to take into account modifications of the shape of the lesions, through certain parameters on the one hand. On the other hand, it is able to take into account/or not the elastic behavior of the upper wall. We highlight that the rigid model, fitted with patient-specific flow parameters, can give useful VFFR estimation in the case of non-viable vessels, while the elastic model, in general, is supposed to be more adapted to estimate the VFFR.

The values of VFFR obtained by this model are certainly different from those obtained by the rigid one. This could be explained by the fact that the elastic property of the upper wall compensates the over-pressure before the lesion and thus modifies the pressure distribution in the whole domain.

The numerical results for the VFFR presented in this paper are not yet clinically usable. However, the aim is to enumerate some of the factors influencing the FFR value and to insist on the necessity of going beyond the degree of stenosis in the evaluation of a lesion. In this view, from the comparison between the Figures 8 and 14 we can conclude that the VFFR decreases with the degree of stenosis for both rigid and elastic models. As for the lesion's radius, it influences the VFFR in the rigid model only for lesions with a degree of stenosis lower than 60%. Beyond this value, the VFFR is not subject to a big change using this model. On the contrary, for the elastic model, the lesion's radius influences the VFFR for lesions with degree of stenosis higher than 50%.

The geometry of the lesion, dimensioned by the two parameters: R the degree of stenosis and r the lesion's radius, affects the flow through the lesion, and depending on whether the domain is rigid or elastic, the pressure distribution is modified, and the VFFR variation as well.

Our aim for future works is to compare the results from modeling with in vivo measurements recorded during an interventional procedure based on 2D or 3D geometries issued from image reconstruction techniques. There are different approaches to reconstruct 3D vessels from 2D images. The most widespread are those that are based on images corresponding to different slices of the 3D object: mainly CT scans or MRI. Another 3D reconstruction technique is based on different projections of the 3D vessel, issued from 2D angiography, like presented in [START_REF] Morris | Virtual (computed) fractional flow reserve: current challenges and limitations[END_REF] and [START_REF] Yang | Novel approach for 3-D reconstruction of coronary arteries from two uncalibrated angiographic images[END_REF]. The choice of suitable conditions is crucial in order to obtain a good estimation of the VFFR. In this work, we chose simplified boundary conditions. Various studies have tackled the problem of boundary conditions in the case of blood flow like [START_REF] Pironneau | Simplified fluide-structure interactions for hemodynamics[END_REF], [START_REF] Gostaf | Pressure boundary conditions for blood flows[END_REF] and [START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF]. In the case of complex geometries, or 3D domains reconstructed from medical imaging, there are several evacuations. This produces a resistant effect at the exit due to the change in the vessels caliber, bifurcations and the variation in the properties of the wall. Therefore, the boundary conditions based on impedance and resistance models, commonly known in the literature as the Winkessel effect [START_REF] Catanho | Model of aortic blood flow using the Windkessel effect[END_REF], are the most adapted to incorporate this reflected wave effect into the model [START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF].

The goal of this prospective study is to obtain a mathematical model integrating measurable clinical parameters with simulation performances which gives a very close estimation to the realistic FFR measure. The validation method of this calculation usually consists on matching the values resulting from simulation with those of the clinical invasive FFR test. That gives rise to a statistical study, as in [START_REF] Morris | Virtual (computed) fractional flow reserve: current challenges and limitations[END_REF] and [START_REF] Lopez-Palop | Correlation between intracoronary ultrasound and fractional flow reserve in long coronary lesions. A three-dimensional intracoronary ultrasound study[END_REF]. Finally, FFR simulation in the cath lab could be a helpful tool for cardiologists, since it provides essential information during the coronary intervention. For this reason, FFR simulation should help in improving decision making as well as the prognosis of patients.

Figure 1 .

 1 Figure 1. The coronary tree [19].

Figure 2 .

 2 Figure 2. Left, represents schema of the invasive FFR technique[START_REF] Cuisset | Qu'est-ce que la FFR? Comment l'utiliser? Réalités Cardiologiques[END_REF]. Right, a typical example of FFR measurement. Automated calculation of FFR corresponds to the ratio of mean distal coronary pressure (green) to mean aortic pressure (red) during maximal hyperemia, see[START_REF] Toth | Standardization of fractional flow reserve measurements[END_REF].

Figure 3 .

 3 Figure 3. Considered geometry for the problem.

. 10 )

 10 We consider a portion of a length L = 60 mm from the diseased artery. The lumen diameter is considered equal to H = 5 mm (coronary artery). The plaque is assumed to be 100% eccentric and it is characterized by three parameters: D the height of the plaque, x s the position of the center of the plaque and 2 × δ its length. R = D/H indicates the degree of stenosis, it varies by changing the value of D, see 3.

Figure 4 .

 4 Figure 4. Left, velocity and pressure field with Navier-Stokes equation at time t = 0.3 s. Right, velocity and pressure field with the generalized flow model at time t = 0.3 s.

Figure 5 .

 5 Figure 5. FFR calculation. In this case, the degree of stenosis is equal to 40% and the VFFR is equal to 0.81.

Figure 6 .

 6 Figure 6. VFFR variation during 5 cardiac cycles for a lesion with 75% stenosis.

Figure 7 .

 7 Figure 7. Left, P a , P d and VFFR variation according to the degree of stenosis R. Right, P a , P d and VFFR variation according to the lesion radius δ.

Figure 8 .

 8 Figure 8. VFFR variation for lesions with different radius according to the degree of stenosis.

Figure 9 .

 9 Figure 9. Top, velocity and pressure field corresponding to identic lesions of 40% stenosis, with a spacing 'a' of 0.5 cm. Bottom, velocity and pressure field with a spacing 'a' of 1.5 cm.

Figure 10 .Q3Figure 11 .

 1011 Figure 10. Left, P a , P d and VFFR variation according to the degree of stenosis R in the case of multiple stenoses. Right, P a , P d and VFFR variation according to the lesion radius δ in the case of multiple stenosis.

Figure 12 .

 12 Figure 12. Pressure profile with the fluid-structure interaction model.

Figure 13 .

 13 Figure 13. Left, P a , P d and VFFR variation according to the degree of stenosis R using the fluid-structure interaction model. Right, P a , P d and VFFR variation according to the lesion radius δ using the fluid-structure interaction model.

Figure 14 .

 14 Figure 14. VFFR variation using the fluid-structure interaction model according to the degree of stenosis.

c 1 R 2 ( 1 -ν 2 ) 6 and

 12126 and c 0 are defined by: c 1 = E 2(1+ν) and c 0 = E , solid thickness = 0.1, Young modulus E = 0.75 × 10 Poisson coefficient ν = 0.5.The fluid-structure interaction model gives different results compared to the one with rigid boundaries (presented in Sect. 2). Therefore, we obtain different values for the VFFR. It should be expected that this model gives better values since it is more adapted to the physiology of the arterial wall. However, to validate the values obtained using this model, we should consider a realistic geometry, reconstructed from clinical images.

K. CHAHOUR ET AL.

Acknowledgements. This work has been supported by EPICARD cooperative research program, funded by the INRIA international laboratory LIRIMA.

Author Query

Q1 Please check the change made of full form of 'VFFR' for correctness. Q2 Please provide the city and country name of affiliation 3. Q3 Please provide the citation of Figs. 10-13 in text suitably. Q4 Please verify the journal title, volume, year and page range of ref. [START_REF] Boujena | Generalized Navier-Stokes equations with Non-standard conditions for blood flow in atherosclerotic artery[END_REF]. Q5 Please verify the journal title, volume and page range of ref. [START_REF] Lopez-Palop | Correlation between intracoronary ultrasound and fractional flow reserve in long coronary lesions. A three-dimensional intracoronary ultrasound study[END_REF]. Q6 Please check the author names of ref. [START_REF] Van Nunen | Fractional flow reserve versus angiography for guidance of PCI in patients with multivessel coronary artery disease (FAME): 5-year follow-up of a randomised controlled trial[END_REF]. Q7 Please check, refs. [START_REF] Pironneau | Simplified fluide-structure interactions for hemodynamics[END_REF] and [START_REF] Yang | Novel approach for 3-D reconstruction of coronary arteries from two uncalibrated angiographic images[END_REF] are identical, so ref. [20] has been deleted from the list and text as well. Q8 Please provide the access date of ref. [19].


J.-M Hartmann
email: jean-michel.hartmann@lmd.polytechnique.fr

C Boulet

H Zhang

F Billard

O Faucher

B Lavorel

Collisional dissipation of the laser-induced alignment of ethane gas. A requantized classical model

We present the first theoretical study of the collisional dissipation of the alignment of a symmetric-top molecule (ethane gas) impulsively induced by a linearly polarized nonresonant laser field. For this, Classical Molecular Dynamics Simulations (CMDS) are carried out for an ensemble of C 2 H 6 molecules based on knowledge of the laser-pulse characteristics and on an input intermolecular potential. These provide, for a given gas pressure and initial temperature, the orientations of all molecules at all times from which the alignment factor is directly obtained. Comparisons with measurements show that these CMDS well predict the permanent alignment induced by the laser pulse and its decay with time but, as expected, fail in generating alignment revivals. However, it is shown that introducing a simple requantization procedure in the CMDS "creates" these revivals and that their predicted dissipation decay agrees very well with measured values. The calculations also confirm that, as for linear molecules, the permanent alignment of ethane decays more slowly than the transient revivals. The influence of the intermolecular potential is studied as well as that of the degree of freedom associated with the molecular rotation around the symmetry axis. This reveals that ethane practically behaves as a linear molecule because the intermolecular potential is only weakly sensitive to rotation around the C-C axis.

I. INTRODUCTION

Aligning molecules with intense laser pulses is a well-established technique 1,2 . The alignment-dependent potential felt by anisotropic polarizable molecules immersed in a strong laser field, confines the most polarizable axis along the field polarization direction. When the pulse duration is much shorter than the rotational period of the molecule, the laser-molecule interaction is non-adiabatic and molecules exhibit periodic revivals of alignment after the pulse 3,4 . This so-called field-free molecular alignment is highly desirable in a large variety of fields such as molecular tunnel-ionization 5,6 and orbital imaging 7 , high-order harmonic generation 8 , attosecond electronics 9 , and molecule solvation dynamics 10 . Various approaches have been used to measure the field-free molecular alignment, mainly categorized as either ionization-fragmentation imaging techniques 11,12 or nonlinear pure optical techniques 13,14 .

The latter are particularly well suited when the alignment decreases quickly due to intermolecular collisions, as in high-pressure gases [15][16][17] , or to molecule-solution interactions, as in He nano-droplets 10,18 . The excitation of molecules with a non-resonant short pump pulse can be described through a cascaded rotational stimulated Raman effect that creates a rotational wavepacket and leads to energy deposition in the rotational motion. This energy then relaxes to the translational degrees of freedom through collisions, a process that was tracked in the time domain 19,20 . The concomitant elevation of temperature and pressure inside the sample produces a thermal expansion and an acoustic-wave which have been detected photo-acoustically 21 . Note that the exchange of energy between the pump electric field and the molecules can also be measured through dichroism 22 .

Field-free alignment, as a time-resolved technique, can provide a means to investigate collisional effects in molecular samples. An example is the determination of collisional linewidths from the recording of the time-dependent birefringence at moderate pressures 23 . It is thus an alternative to frequency-domain Raman spectroscopy. Moreover, as pointed out in the pioneering studies of Refs. 24,25, it could be possible to disentangle the effects of rotational population relaxation and pure phase decoherence by using the dissipation of field free alignment. This ability relies on the fact that the field-free molecular alignment exhibits two different contributions known as the transient and permanent alignments, which have different sensitivities to the various collisional processes. As discussed in Ref. 16, the permanent alignment corresponds to a population effect whereas the transient alignment reflects the coherence within the rotational excited wavepacket. While all collisions inducing decoherence (J-changing inelastic collisions, J-preserving elastic reorienting collisions, J-and M-preserving pure dephasing elastic collisions) reduce the transient alignment, the decrease of the permanent component only results from those which change the orientation of the rotational angular momentum. It is thus possible, in principle, to discriminate different contributions to the dissipation of alignment by simultaneously measuring the decays of permanent and transient alignments under the same gas conditions. The first work in this direction, made in pure CO 2 gas and CO 2 in mixtures with rare gases at low pressure 15 , was then followed by a study for nitrogen molecules 26 . The analysis of the data demonstrated that inelastic collisions dominate the decoherence in the gas phase. More interesting, the decay of both components of field-free molecular alignment was then measured in OCS excited by a strong terahertz single cycle pulse 27 , which showed that the permanent alignment decays 3 times slower that the transient one. The explanation for this difference was given in Ref. 28, for pure CO 2 gas, by using two independent models for the collisional relaxation of molecular alignment: A quantum one, based on the M-dependent Energy Corrected Sudden (ECS) model, and a nearly classical one, using requantized Classical Molecular Dynamics Simulations (rCMDS). The main feature exhibited by both approaches, which lead to results in excellent agreement, is a propensity of collisions to conserve the orientation of the molecular angular momentum. Consequently, the permanent alignment decays slower than the transient one. This was later on confirmed by the very good agreement between the computed and measured decays of the permanent and transient alignments in high-pressure pure CO 2 gas as well as CO 2 -He and CO 2 -Ar mixtures 16,17,29 . However, all the above mentioned experimental and theoretical studies of the collisional dissipation of the laserinduced alignment were made for linear molecules only, despite the obvious interest of studying the influence of the molecular symmetry. It is only very recently that dissipation for a nonlinear species (the symmetric-top molecule ethane C 2 H 6 ) has been investigated experimentally 30 . Again, the permanent alignment shows a slower decay compared to that of the transient alignment, indicating the potentially general applicability of the propensity rule to nonlinear polyatomic molecules. However, while quantum calculations of the alignment of symmetric tops under collision-free conditions have been made (eg Ref. 30 and those there in), no theoretical approach and thus no simulations have been proposed so far for (gas pressure) conditions making dissipation processes significant.. This lack of models and computations of the collisional dissipation of the alignment of symmetric-top molecules is the justification of this paper. The latter presents an approach based on rCMDS for symmetric-top species (Sec. II) which is applied to ethane gas with (Sec. III) comparisons with experimental results 30 , and a study of the influence of the symmetric-top geometry.

III. COMPUTATIONAL PROCEDURE AND DATA USED

A. Classical Molecular Dynamics Simulations (CMDS)

CMDS were performed for an ensemble of N ethane molecules at pressure P and initial temperature T as follows. First the molecules' orientations and center-of-mass positions [inside a cubic box of side length

m m t t E t µ = α
, where the polarizability tensor ( ) m t α in the laboratory frame is computed from knowledge of its value in the molecular frame and of the orientation of the molecule (i.e. of the quaternions). For the force and torque resulting from intermolecular collisions, they are easily obtained from the positions and orientations of the interacting molecules and a parameterization of the site-site forces 32.33 (see below). Using the input data described there after, computations were made for ethane gas initially at 295 K and for pressures of 2 and 6 bar. The alignment factor, i.e. the average

2 cos( ) ( ) 1/ 3 t < θ > - where
θ is the angle between the molecular axis of symmetry (i.e. the C-C axis) and the polarization of the laser, was computed since it is the quantity measured in Ref. 30. This was done with a time step of 1.5 fs for delays t ∆ with respect to the pulse ranging from -2 ps to 90 ps. Note that the molecules were treated as rigid rotors in the ground vibrational state, thus disregarding the influence of centrifugal distortion and internal torsion and the contribution of the first excited state (ν 4 ). These approximations make things much simpler and they likely have negligible consequences on the dissipation of the alignment. Indeed, the latter results from rotational-state changes governed by state-to-state rates that are almost insensitive to slight changes of the energies and wave functions of the levels.

B. Input data used

We used the equilibrium values of the C 2 H 6 geometry parameters, i.e. C-C=1.535 A, C-H=1.094 A and HCC=111.2°, and the anisotropy of the polarizability was set to γ=0.7 A 3 as in Ref. 30. For the description of intermolecular forces, two potential energy surfaces were used, denoted as PES1 and PES2 and taken from Refs. 32 and 33, respectively. They were both computed ab initio and represented by sums of site-site interactions described by analytical expressions. For the laser alignment pulse, we used the parameters reported in the experimental study 30 : a duration (FWHM) of 100 fs and a peak intensity of 80 TW/cm 2 . For comparisons with experiments, the computed alignment factor versus time was convoluted with a Gaussian function which takes into account the finite duration (150 fs) of the probe pulse and the crossing angle between the two beams.

C. Requantization procedure

While purely classical calculations can predict the response of the system during the laser pulse and the permanent alignment that it induces, they do not generate any alignment revivals since the latter are due to the quantum nature of the rotational motion (see Fig. 1a). A way to correct for this is to "requantize" the rotation in the course of the CMDS, following what was proposed 28 for CO 2 and successfully tested against quantum calculations 28 as well as experiments 29 . Before explaining how this was made in the case of the symmetric-top molecule ethane, recall that, under collision free conditions (zero pressure), the alignment factor computed quantum mechanically for a rigid rotor has a periodicity (see Fig. 1a of Ref.

30) equal to the rotational period

1 (2 ) 25.1 ps rev T Bc - = =
, where B=0.663 cm -1 . Note that the weak revivals which appear 30

I I I ⊥ = = leads to 2 2 2 2 2 ( ) ( 1)
x y

I J J K ⊥   ω + ω = + -   ℏ .
In order to requantize x ω and y ω we thus search for the integer J such that ( 1)

J J + is the closest to 2 2 2 2 2 ( ) / x y K I ⊥ + ω + ω ℏ . We then replace x ω by rq x ω with rq 2 2 2 ( / ) ( 1) ( / ) x x x y I J J K ⊥ ω = + - ω ω + ω ℏ
, an equivalent change being made for y ω .

This requantization scheme, denoted as RQ1, leads to a practically perfect agreement with the results of quantum calculations (carried as explained in Ref. 30) for the first revival (and all other ones of odd number), as shown by Fig. 1b, but Fig. 1c shows that it does not reproduce the shape of the second revival (and of all those of even number). We then introduced a second requantization scheme, denoted as RQ2, in which we search for the integer J such that ( / ) ( / )

x x x y I J K ⊥ ω = - ω ω + ω ℏ
, an equivalent change being made for y ω . Figures 1b and1c show that, while RQ1 leads to very good (resp. inaccurate) results for revivals of odd (resp. even) numbers, the situation is reversed when RQ2 is used.

1: Alignment factors for ethane gas under collision free conditions versus the time delay with respect to the laser pulse. The top panel (a) shows the results of a quantum calculation for rigid rotors in the ground vibrational state (blue line) and of fully classical MDS (red line). The lower panels display the same quantum results (circles) and those of requantized CMDS using requantization RQ1 (red line) and RQ2 (blue line) for the first (b) and second (c) revivals.

In order to understand the results in Figs. 1b and1c, first note that one can show that, with RQ1, the ethane C-C axis in the laboratory frame rotates with and angular speed

RQ1 ( ) ( 1) / ( 1/ 2) / J J J I J I ⊥ ⊥ ω = + ≈ + ℏ ℏ , while RQ2 leads to RQ2 ( ) / J J I ⊥ ω = ℏ .
Obviously, if we consider the evolution of the square of a cosine between the molecular axis and the laser polarization direction, associated angular speeds are

RQ1 2 ( ) ( 2
J J J Bc J J J J I J J J J +∆ ⊥ ω = π ∆ + ∆ ∆ + = ∆ + ∆ ∆ + ℏ |, this leads, for the 1 J ∆ = ± and 2 J ∆ = ± to [ ] , 1 (1 1) / 2 / J J J I ± ⊥ ω = + ± ℏ and , 2 (2 1 2) / J J J I ± ⊥ ω = + ± ℏ , respectively. Hence, , 2 RQ1 2 ( 1)
J J J + ω = ω + and , 2 RQ1 2 ( 1) J J J - ω = ω -.
This explains the good agreement between the quantum results and those obtained from rCMDS+RQ1 for the revivals of odd number (Fig. 1b) since only the with n any integer value] after the pulse implies that the molecule rotates with an angular speed that is an even multiple of that corresponding to T rev (i.e. to J=1) thus requiring that its quantum number J is even. For the revivals of even number, all the 1 J ∆ = ± and 2 J ∆ = ± transitions contribute and the situation is much more complicated. While numerical computations show that rCMDS+RQ2 lead to good agreement with quantum calculations for the revivals of even number (see Fig. 1c), we have not been able to understand this result from analytical developments. It is important to emphasize that the quality of the predictions obtained with the two requantization procedures is largely due to the fact that many rotational levels (up to 50 J ≈ ) are populated at room temperature. This "washes out", through averaging, the imperfections of RQ1 and RQ2. For the same reason, the total (rotational+translational) energy of the system remains remarkably constant after the laser pulse, despite the fact that the requantization changes the rotational speed of the molecules. Such a favorable situation would not occur if only few levels were populated, as it would be the case at (very) low temperature or for a molecule with a much larger rotational constant B. Indeed, comparisons similar to those made in Figs. 1b and1c show a good agreement between the results of rCMDs and of quantum calculations down to about 10 K with differences that increase quickly for temperatures below T<5 K as T/B approaches unity.

It is not possible to find a single way to change the angular speeds of classically rotating molecules (i.e. to requantize) that can lead to accurate predictions of the revivals of both odd and even number. Hence, following the previously presented elements, we carried two requantized CMDS of ω ω ω ) is only made, for a given molecule, at times for which the torque that it is experiencing (due to the laser and/or interaction with another molecule) is practically zero.

Indeed, requantizing at each time step would automatically forbid any change of the values of J and K and thus of the rotation of the molecule.

III. RESULTS AND COMPRISONS WITH MEASUREMENTS

A comparison between the measured alignment factor for pure ethane gas at 2 bar and initially at 295 K and that computed with the rCMDS approach and the intermolecular potential PES2 33 is shown in Fig. 2. As can be seen, the agreement between experiment and theory is good for the permanent alignment and the first two revivals but degrades for the higher order revivals. While it seems like the computations overestimate the decrease of the revivals amplitude with time, it is not truly the case. In fact, this does not come from an inaccurate modeling of the effect of collisions, as shown by the results for the decay time constants discussed below, but from the neglect of centrifugal distortion and of the contribution of the excited torsional state. Indeed, the comparison between quantum calculations displayed in Fig. 1 of Ref. 30 show that these contributions neglected in the rCMDS significantly reduce the amplitude of the 3 rd and 4 th revivals while broadening their extent but not those of the 1 rst and 2 nd one. The comparison of Figs. 1a and 1c of Ref. 30 also shows that these effects increase significantly as the revival number increases. In order to go further, we have determined the decay time constants of the permanent ( P τ ) and transient ( T τ ) alignments exactly as was done in Ref. 30. Namely, the permanent alignment was fit by an exponentially decaying function, yielding the value of P τ . This function was then subtracted from the rCMDS-computed alignment factor and the absolute value of the result was calculated. We then determined the areas below each revival and fit them by another exponentially decaying function yielding the value of T τ . The results of this exercise applied to rCMDS results at 2 bar are plotted in Fig. 3 which shows the quality of the exponential fits. A similar treatment was made for rCMDS results at 6 bar and, as expected from the binary nature of intermolecular collisions at such pressures, the time constants obtained are three times smaller than those given in the caption of Fig. 3. The values of P τ and T τ retrieved from the results of rCMDS carried using PES1 [START_REF] Rowley | [END_REF] and PES2 33 have thus been density-normalized. They are given in Table 1 together with those obtained 30 from experiments. As can be seen the agreement between theory and measurements is very good when the intermolecular potential PES2 33 is used, The predicted values of T τ and P τ then differ from experimentally determined ones by only -5.2% and +7.7% and are within the uncertainties of the measurements. Poorer results are obtained with PES1 [START_REF] Rowley | [END_REF] . This could be expected since, with respect to PES1, the very recent PES2 is more accurate for several reasons, including the use of a much larger basis set, the fact that complete basis set extrapolations were made, that zero-point vibrational effects were not neglected, etc. As noted in Ref. 30, and confirmed by the present theoretical results, ethane follows the rule, previously evidenced for linear molecules 16,17,27,29 , that the permanent alignment decays more slowly than do the transient revivals. This was explained, for CO 2 , by a "gyroscopic effect" 28 resulting from the fact that collisions, while they change the rotational speed (i.e. the value of J), tend to conserve the direction of the rotational angular momentum (i.e. the value of M/J). A preliminary analysis, whose results will be presented in a forthcoming paper, shows that, for ethane, the situation is the same since the collisional cross sections for the (J,K,M)→ (J',K',M') change have a maximum for M'/J'≈M/J.

  Fig. 1: Alignment factors for ethane gas under collision free conditions versus the time delay with respect to the laser pulse. The top panel (a) shows the results of a quantum calculation for rigid rotors in the ground vibrational state (blue line) and of fully classical MDS (red line). The lower panels display the same quantum results (circles) and those of requantized CMDS using requantization RQ1 (red line) and RQ2 (blue line) for the first (b) and second (c) revivals.

2 J

 2 ∆ = ± Raman transitions contribute to the latter. Indeed, a re-alignment of a molecule at

Fig. 2 :

 2 Fig. 2: Comparison between the measured (full blue line) alignment factor at 2 bar and that predicted by the requantized classical model (dashed red line) with the intermolecular potential PES2 33 . The top panel (a) shows the results over the full time delay investigated, while the lower ones show detailed view around the first (b), second (c), third (d), and fourth (e) revivals.

Finally, it is 2 2

 2 of interest to study how the symmetric-top nature of the ethane molecule affects the alignment and its collisional dissipation with respect to the behavior of linear species. In other words, what is the influence of the extra degree of freedom of rotation (that around the molecular axis of symmetry) brought by the fact that the molecule has a non zero moment of inertia I zz . In order to investigate this issue, rCMDS have been carried using the true value of I zz for ethane as well as with five times less and five times more. The results show that the collisional dissipation and the associated time constants P τ and P τ are practically insensitive to I zz . This can be explained by first noting that it is the component of the torque applied to a molecule in a collision with a partner that is perpendicular to the molecule symmetry (C-C) axis which changes the rotation of this axis and thus the alignment factor. Since I xx = I yy for symmetric-tops, I zz does not directly participate to changes of the associated speed of rotation [i.e. of shown by Eqs. (3.29a,b) of Ref. 31. The effect of I zz is thus only indirect, through its influence on the rotation around the C-C axis, which determines the relative positions of the H atoms of a colliding pair and thus the intermolecular forces. Now, it turns out that the potential energy surface of Ref. 33 is only weakly sensitive to the rotation around the C-C axes of the two molecules. Its anisotropy is essentially governed by the respective orientation of the two C-C axes so that this PES is more like that for two linear molecules. This explains the small sensitivity of the relaxation processes to the asymmetric-top nature of the considered molecule. This finding likely cannot be generalized to all symmetric-tops. IV. CONCLUSION This paper presents the first model for calculations of the collisional dissipation of the laser-induced alignment of a gas of symmetric-top molecules. It is based on Classical Molecular Dynamics Simulations which are requantized through proper changes of the molecules' rotational speeds. It was applied to ethane gas for which measurements of the alignment have been made using a recent and accurate intermolecular potential energy surface. The comparison between predicted and measured alignment signals demonstrates the quality of the model. Indeed, the predicted decay rates of the permanent and transient alignments differ from experimentally determined ones by only -5.2% and +7.7% and are within error bars. The study of the influence of the symmetric-top geometry reveals that pure ethane gas behaves, from the point of view of the alignment dissipation, like a linear molecule. This results from the small sensitivity of the C 2 H 6 -C 2 H 6 potential to rotations around the C-C axes, and is thus not a general rule for all symmetric-tops. It would thus be of interest to study, both experimentally and theoretically, the alignment of a gas species (pure or diluted gas) for which the potential sensitivity to rotations of the colliding molecules around their symmetry axes is large. However, finding a "good candidate" is not easy, very time consuming and my be unsuccessful because one needs to: (i) Find a symmetric-top + collision partner system for which a reliable intermolecular potential is available. (ii) This potential should be significantly anisotropic with respect to rotation around the molecule symmetry axis. (iii) The anisotropy of the polarizability of the symmetric-top molecule should be sufficient to enable a significant degree of alignment using current and reasonable laser pulse intensities. Although of interest, this search is beyond the scope of the present paper.

  alignment, without introducing any parameter adjustment,. Finally, the requantization (i.e., finding the values of K and J and then changing the rotational angular speeds

	x	, , y	z
			<	2 θ > -cos( ) ( ) 1/ 3 t	, using RQ1 and RQ2 respectively, for each given
	initial gas conditions and laser pulse characteristics. The final alignment factor was then
	obtained from a mix of the results obtained with the two requantizations. More precisely, we
	retained the alignment factor computed with RQ1 until	3 t T rev ∆ =	/ 4	and then in all the
	intervals [ ( 1/ 4) rev n T +	, ( 3/ 4) rev n T +	] where n=1,2,3,... while that obtained with RQ2 was
	used in the intervals [ ( 1/ 4) rev n T -	, ( 1/ 4) rev n T +	]. Alternately using RQ1 and RQ2 is

obviously not fully satisfactory, but this "trick" has the merit, as shown in Sec. III, to enable quantitative predictions of the collisional decays of the permanent and transient components of the

Acknowledgements: This work was supported by the Conseil Régional de Bourgogne under the PhotCom PARI program, the European Union (FEDER program), the Labex ACTION program (No. ANR-11-LABX-01-01), and the CoConiCS program (No. ANR-13-BS08-0013).