Ioannis Tanis
email: tanis.ioannis@gmail.com

Kostas Karatasos

Thomas Salez

Molecular Dynamics Simulation of the Capillary Leveling of a Glass-Forming Liquid

Motivated by recent experimental studies probing i) the existence of a mobile layer at the free surface of glasses, and ii) the capillary leveling of polymer nanofilms, we study the evolution of square-wave patterns at the free surface of a generic glass-forming binary Lennard-Jones mixture over a wide temperature range, by means of molecular dynamics simulations. The pattern's amplitude is monitored and the associated decay rate is extracted. The evolution of the latter as a function of temperature exhibits a crossover between two distinct behaviours, over a temperature range typically bounded by the glass-transition temperature and the mode-coupling critical temperature. Layer-resolved analysis of the film particles' mean-squared displacements further shows that diffusion at the surface is considerably faster than in the bulk, below the glass-transition temperature. The diffusion coefficient of the surface particles is larger than its bulk counterpart by a factor that reaches 10 5 at the lowest temperature studied. This factor decreases upon heating, in agreement with recent experimental studies.

I. Introduction

Thin films and their free interfaces are used in a variety of applications such as catalysis [START_REF] Ulbricht | Advanced functional polymer membranes[END_REF] , fast crystal growth [START_REF] Hasebe | Fast surface crystal growth on molecular glasses and its termination by the onset of fluidity[END_REF][START_REF] Sun | Glasses crystallize rapidly at free surfaces by growing crystals upward[END_REF] and the formation of low-energy glasses [START_REF] Ediger | Vapor-deposited glasses provide clearer view of two-level systems[END_REF][START_REF] Mangalara | Three-layer model for the emergence of ultrastable glasses from the surfaces of supercooled liquids[END_REF] . Apart from their applications in industry and technology, molecular mobility and relaxation at surfaces and interfaces have been recently a subject of growing interest for fundamental research. To this end, several experimental approaches have been used to probe the surface evolution and mobility of thin films [START_REF] Yang | Glass transition dynamics and surface layer mobility in unentangled polystyrene films[END_REF][START_REF] Yang | Glass transition dynamics and surface mobility of entangled polystyrene films at equilibrium[END_REF][START_REF] Mapes | Self-diffusion of supercooled oterphenyl near the glass transition temperature[END_REF][START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF][START_REF] Teisseire | Confinement and flow dynamics in thin polymer films for nanoimprint lithography[END_REF] . More specifically, an efficient way to gain insight into the surface mass transport and the associated dynamics, is to deposit nanoparticles on the film surface and subsequently to monitor the host material's response [START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF][START_REF] Fakhraai | Measuring the surface dynamics of glassy polymers[END_REF] . Another powerful and versatile technique widely utilized for the determination of the surface-diffusion coefficient D s , characterizing the in-plane translation of molecules at the film surface, is the capillary-driven leveling of surface-gratings [START_REF] Yang | Glass transition dynamics and surface layer mobility in unentangled polystyrene films[END_REF][START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF][START_REF] Zhang | Surface diffusion of polymer glasses[END_REF][START_REF] Brian | Surface self-diffusion of organic glasses[END_REF][START_REF] Hamdorf | Surface-rheological measurements on glass forming polymers based on the surface tension driven decay of imprinted corrugation gratings[END_REF] . Several studies suggest that the mechanism governing the relaxation of these gratings is temperature and material dependent 12, 15. More specifically, recent works on glass-forming molecular liquids reported a transition from bulk viscous flow at temperatures higher than the glass-transition temperature T g , to surface diffusion below T g 13, 16 . Analogous studies in polymeric liquids revealed that bulk viscous flow was the only mechanism governing the surface decay, even at the highest viscosities studied [START_REF] Hamdorf | Surface-rheological measurements on glass forming polymers based on the surface tension driven decay of imprinted corrugation gratings[END_REF][START_REF] Kim | Surface dynamics of polymer films[END_REF][START_REF] Wang | Investigation of surface mass transport in Al-Si-Ca-oxide glasses via the thermal induced decay of submicron surface gratings[END_REF] . According to Mullin's pioneering work [START_REF] Mullins | Flattening of a nearly plane solid surface due to capillarity[END_REF] , the different mechanisms that could flatten the surface pattern (surface and bulk diffusions, evaporation-condensation, bulk viscous flow) can be distinguished by determining the decay rate dependence on spatial frequency. This method has been applied in crystalline metals [START_REF] Bonzel | Surface self-diffusion on Ni(110): Temperature dependence and directional anisotropy[END_REF] , amorphous silica [START_REF] Keeffe | Surface self-diffusion on Si from the evolution of periodic atomic step arrays[END_REF] and molecular glasses [START_REF] Zhu | Surface self-diffusion of an organic glass[END_REF] .

Measurements of diffusion coefficients at the surface report values considerably higher than the respective bulk coefficients at the glass-transition temperature [START_REF] Zhang | Surface diffusion of polymer glasses[END_REF][START_REF] Brian | Surface self-diffusion of organic glasses[END_REF] .

Furthermore, large variations in surface diffusion coefficients among different molecular glass formers bearing a similar bulk mobility were linked to the strength of the intermolecular forces [START_REF] Chen | Hydrogen bonding slows down surface diffusion of molecular glasses[END_REF] .

Further insight into glass-forming systems has been provided by numerical simulations. Several studies have addressed the factors that govern dynamical heterogeneity and spatial correlations in the bulk [START_REF] Donati | Spatial correlations of mobility and immobility in a glass-forming Lennard-Jones liquid[END_REF][START_REF] Kob | Dynamical heterogeneities in a supercooled Lennard-Jones liquid[END_REF][START_REF] Andersen | Molecular dynamics studies of heterogeneous dynamics and dynamic crossover in supercooled atomic liquids[END_REF][START_REF] Kob | Computer simulations of supercooled liquids and glasses[END_REF][START_REF] Murthy | Molecular dynamics in supercooled liquids: A study of the relaxation in binary solutions[END_REF][START_REF] Pakula | Collective dynamics in simple supercooled and polymer liquids[END_REF][START_REF] Vogel | Slow molecular dynamics in binary organic glass formers[END_REF] . Furthermore, other works addressing systems under confinement have managed to elucidate the effects of confinement by rough or smooth walls on the liquid dynamics and to extract associated length-scales [START_REF] Hocky | Crossovers in the dynamics of supercooled liquids probed by an amorphous wall[END_REF][START_REF] Scheidler | Cooperative motion and growing length scales in supercooled confined liquids[END_REF][START_REF] Varnik | Molecular dynamics simulation of confined glass forming liquids[END_REF][START_REF] Scheidler | The relaxation dynamics of a simple glass former confined in a pore[END_REF][START_REF] Scheidler | The relaxation dynamics of a supercooled liquid confined by rough walls[END_REF] . Extensive simulation studies have also been conducted on ultrastable vapor-deposited glasses [START_REF] Ediger | Vapor-deposited glasses provide clearer view of two-level systems[END_REF][START_REF] Singh | Ultrastable glasses from in silico vapour deposition[END_REF][START_REF] Lin | Molecular modeling of vapor-deposited polymer glasses[END_REF] . However, to our knowledge, a small number of simulation studies probing the surface mobility in glass formers have been reported so far.

Surface-diffusion-mediated decay of two-dimensional nanostructures has been recently studied by a combined analytical and kinetic Monte-Carlo approach [START_REF] Marcos | Continuous and discrete modeling of the decay of two-dimensional nanostructures[END_REF] Lennard-Jones (LJ) liquid [START_REF] Haji-Akbari | The effect of substrate on thermodynamic and kinetic anisotropies in atomic thin films[END_REF] , whereas Malshe and co-workers extracted surfacediffusion coefficients by the aid of the grating-decay approach [START_REF] Malshe | Evolution of glassy gratings with variable aspect ratios under surface diffusion[END_REF] in free standing films of the aforementioned liquid type. Results were consistent with data obtained from the mean-squared displacements of the liquid particles. In a very recent study, Kuan and co-workers examined single-particle dynamics in the surface and in the bulk of supported films prepared by vapour deposition. Both bulk and surface particle displacements showed evidence of heterogeneous dynamics [START_REF] Kuon | Comparison of single particle dynamics at the center and on the surface of equilibrium glassy films[END_REF] .

The aim of the present work is to provide additional insight into the surface mobility of a glass-forming liquid by means of molecular dynamics simulations. Similarly to a recent work where we probed the viscoelastic behaviour of nonentangled polymer films above the glass-transition temperature, through the capillary leveling of a square-wave surface pattern [START_REF] Tanis | Molecular dynamics simulation of the capillary leveling of viscoelastic polymer films[END_REF] , we examine here the evolution of a square-wave surface pattern atop a generic binary LJ mixture supported by an attractive substrate.

This new study is carried out over a wide temperature range, sampling both the glassy and liquid states of the film. The associated decay rates are extracted whereas surface and bulk mobilities are examined. Further insight into the surface and bulk mobilities is gained by examining the self part of the van Hove function, as well as via the layerresolved analysis of the liquid particles' mean-squared displacements.

II. Computational Details

We examine a binary mixture of 80% A and 20% B particles that interact via a LJ potential The Van der Waals parameters were chosen as

The potential was truncated at a cut-off radius The numbers of particles of type A and B are 2938 and 734, respectively. The binary mixture interacting with this parameter set corresponds to the Kob-Andersen liquid model which is not prone to crystallization or to phase separation; it is extensively utilized in studies of glass-forming liquids [START_REF] Kob | Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture[END_REF][START_REF] Kob | Aging effects in a Lennard-Jones glass[END_REF] . The liquid film is supported by a strongly attractive substrate whose atoms (henceforth designated by the type S) are located on the sites of an hexagonal lattice and are kept fixed to their lattice positions.

The substrate particles' radius is and they attract A and B atoms with the following energy magnitudes [START_REF] Li | Insights into the adsorption of oxygen and water on low-index Pt surfaces by molecular dynamics simulations[END_REF] : and . The length of the cubic box is 16.56 in the lateral directions x and y, and 441.17 in the z dimension. The value of the latter is chosen large enough to avoid interaction of the particles with periodic images of the substrate, and ensure a system pressure p = 0. All quantities are reported in LJ units, i.e. length is measured in units of , energy in units of , time in units of and temperature in units of where stands for the Boltzmann constant. For the case of argon, these units correspond to a length of 3.4 Å, an energy of 120 and a time of 3•10 -13 s. All simulations are performed in the canonical (NVT) ensemble using the DL_POLY code [START_REF] Smith | A short description of DL_POLY[END_REF] . Temperature is maintained constant through the use of the Nosé-Hoover thermostat. Simulations are conducted at the temperature range T = 0.20-0.525, and the length of each run is 3 .

The initial configuration of the glassy film was obtained by the aid of the Aten program [START_REF] Youngs | Aten-An application for the creation, editing, and visualization of coordinates for glasses, liquids, crystals, and molecules[END_REF] . To create the 3-dimensional pattern at the free surface of the film, a flat film was initially equilibrated in the canonical ensemble at a temperature of T = 0.3.

Afterwards, particles were removed from both edges of the free surface in order to obtain the 3D pattern, henceforth designated as the 'top layer' (see Fig. 1). The lateral dimensions of the latter were L x = L y = 4.41, whereas its thickness and initial height (counted from the substrate) were respectively d = 4.36 and h 0 = 15.67

To examine the behaviour in the bulk, a configuration comprising 5076 particles was constructed at an elevated temperature, T = 0.80, and at a density of . After removing close contacts, the system was cooled to T = 0.30 by conducting runs in the isothermal-isobaric ensemble (NPT) at pressure p = 0. The temperature step between the isothermal runs ranged from dT = 0.1 for the highest temperatures to dT = 0.025 and dT = 0.0125 at temperatures around the bulk glass-transition temperature T g . The duration of each isothermal run ranged from 1.6•10 4 (at T > 0.5) to 3•10 5 (at T  0.5).

Defining the cooling rate as the difference between the starting temperature and the final temperature divided by the time of the quench, the cooling rate was 0.75 •10 -7 .

This amounts to half the cooling rate used in the original model of Kob and Andersen and, although it remains quite fast, it is slower than the fastest cooling rate in experiments [START_REF] Kob | Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture[END_REF] . The glass-transition temperature of the binary mixture was determined by extrapolating and intersecting the affine dependencies of the low-and hightemperature specific-volume-vs-temperature curves. This yielded a value of T g = 0.392 which is below the reported mode-coupling critical temperature for this system, T c ≈ 0.435 [START_REF] Kob | Scaling behavior in the beta-relaxation regime of a supercooled Lennard-Jones mixture[END_REF] .

Figure 1 displays the initial configuration of the samples, as well as snapshots from a run at T = 0.3 at times t = 20000 and 233333. All snapshots were generated by the aid of the VMD software [START_REF] Humphrey | Visual molecular dynamics[END_REF] .The particles that evaporated at elevated temperatures were disregarded from post-analysis.

III. Results

To investigate the mobility of the liquid's molecules, we examine the self-part of the van Hove space-time correlation function defined as :

(1)

with representing the position of particle i along any of the x, y and z directions, N being the number of particles that constitute the top layer, being the ensemble average over realizations and time origins and is the Dirac's distribution. In order to account for the natural anisotropy of the system induced by the interfaces, the van Hove function was evaluated in each direction separately. In Fig. 2 A visual inspection of Fig. 2 reveals that the top layer particles are significantly more mobile than those corresponding to the bulk. Focusing on the top left panel, we observe that in the bulk, and at low temperature, the distribution shows a vanishing probability for x-displacement larger than the near-neighbour distance (x 1). As expected for a glassy behaviour, this implies that the particles have not managed to escape from the cages in which they were trapped initially. In contrast, the distribution for the top layer particles exhibits non-negligible values up to x 4, thus providing evidence for an enhanced mobility at the free interface of the film. As expected, the difference between surface and bulk mobilities grows with time. Bulk x mobility remains practically unchanged at the timescales examined, since the system at the temperatures shown in Fig. 2 resides below the estimated glass-transition temperature. Focusing on the top layer particles' distribution at early times, one can discern, apart from the peak at x 0.1, the existence of a "shoulder" at distance x . The distribution at later times further confirms the existence of two different populations, as reflected in the peaks at x 0.2 and at x 1.1. The appearance of these two peaks reflects a clear distinction in mobility for the surface and internal particles belonging to the top-layer and lies in accordance with the findings of Malshe et al. [START_REF] Malshe | Evolution of glassy gratings with variable aspect ratios under surface diffusion[END_REF] . On the other hand, the distribution also shows that the peak belonging to the more mobile particles gains significantly in intensity as time lapses. This is consistent with the fact that an increasing fraction of the top-layer particles gets closer to the free surface with time (as demonstrated by the snapshots in Fig. 1, blue particles).

The previous two-peak fine-structure separation of relaxation processes is no longer detected at late times for T > 0.30, thus indicating a more homogenous mobility at higher temperature. Nevertheless, the "shoulder" observed in panels b and c at at early times, is maintained and gains significantly in intensity at higher temperatures.

Assuming that the mechanism of diffusion does not change appreciably at the sub-T g temperatures examined, it is reasonable to surmise that this behavior arises from the same origin as in the lowest temperature case, i.e. the increase with time of the number of top-layer particles gaining access to the free surface. In sharp contrast, there is no sign of enhanced mobility in the bulk samples as temperature rises. Only at T = 0.375, one can discern a marginal increase in the mobility of the bulk particles at long times, as reflected by the tail of the van Hove distribution. Since the bimodal distribution at low temperatures which is indicative for the dynamic heterogeneity in the motion of the top layer particles, is no longer detected at elevated temperatures, it may be considered as a signature for the transition for these particles to the glassy state.

To examine the global consequences of the enhanced surface mobility on the film leveling, we have monitored the average height h av (t) of the top layer particles. The initial value of h av (counted from the substrate) was h av0 = 13.5. An example of the time evolution of the rescaled amplitude, , for a temperature T = 0.45, is displayed in Fig. 3a, where = 11.31 is the average height of the top layer particles after the film leveling. It should be noted here that is smaller than the value predicted from the volume conservation. This is due to the fact that, in this work, it is the average height of the square pattern that is monitored rather than its maximum height. On these grounds, does not correspond the value of h av (t) at infinite time and, consequently, it is not an equilibrium value. We stress that the total number of evaporated particles did not exceed 0.35% of the total number of particles constituting the film, for all simulated systems. As stated earlier, the evaporated particles were excluded from all analysis calculations. As it can be inferred from Fig. 3a, the temporal decay is exponential. Figure 3b displays the associated decay constant as a function of temperature. We note that the minimum temperature at which full leveling of the square pattern was observed within the total simulation time, was T = 0.30. Examining Fig. 3b, it appears that in between T g = 0.392 and T c =0.435, a crossover takes place. At temperatures above T c , a weak slope characteristic of a liquid-like behaviour is observed, while below T g a stronger dependence corresponding to a glassy-like behaviour is noticed. This observation is consistent with the fact that the onset of supercooling effects is generally observed near T c . Interestingly, recent experimental studies report different rates for the decay of patterned surfaces in supported films above and below the glass-transition temperature [START_REF] Yang | Glass transition dynamics and surface layer mobility in unentangled polystyrene films[END_REF][START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF][START_REF] Fakhraai | Measuring the surface dynamics of glassy polymers[END_REF][START_REF] Brian | Surface self-diffusion of organic glasses[END_REF][START_REF] Chen | Hydrogen bonding slows down surface diffusion of molecular glasses[END_REF] . The flattening of profile patterns at the free surface of films below T g has been ascribed to surface diffusion as suggested by Mullin's theory [START_REF] Mullins | Flattening of a nearly plane solid surface due to capillarity[END_REF] . It must be noted that, since the mobile layer can still be considered at equilibrium within a certain temperature range below T g , and thus the Stokes-Einstein relation is valid there, one can also describe the surface mobility as a surface flow which leads to an identical partial differential equation for the profile's evolution [START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF] . Surface diffusion levels a sinusoidal surface pattern at a decay rate proportional to q [START_REF] Ediger | Vapor-deposited glasses provide clearer view of two-level systems[END_REF] , where q is the angular-wavenumber. Other mechanisms, such as bulk viscous flow and evaporation-condensation exhibit weaker q dependences (except for the limiting lubrication case [START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF] , that accidently shows the same q 4 dependence for a bulk viscous flow above T g ). We have not investigated the angular-wavenumber dependence of the decay rate here but, instead, we have carried out a layer-resolved analysis of the particles' z-dependent mean-squared displacements (MSDs) as defined by: where is the position of particle i at time t and is an ensemble average. This definition takes into account only the n t particles which are at all times within the slab centred at z and bearing a width of . The results for the runs carried out at T = 0.35 are presented in Fig. 4. For comparison purposes, the results for runs carried out above T g , at T = 0.46, are also shown. We note here that the simulation runs at T = 0.46 were significantly shorter than the runs below T g due to the evaporation of the surface particles. larger than the one for the particles in the second outermost film layer. There is no sign of major propagation of the enhanced surface mobility into the bulk. This observation corroborates a scenario in which the relaxation of the square pattern is governed by surface diffusion below the glass transition. A similar behaviour was observed at all sampled temperatures below T g . Furthermore, a visual inspection of the MSDs of the innermost particles (black lines in Fig. 4), demonstrates enhanced dynamics near the solid substrate at short times. This behaviour lies in accordance with simulation studies of polymers near solid interfaces [START_REF] Daoulas | Molecular dynamics simulation of a polymer melt/solid interface: local dynamics and chain mobility in a thin film of polyethylene melt adsorbed on graphite[END_REF][START_REF] Kritikos | Molecular dynamics study of polyethylene under extreme confinement[END_REF][START_REF] Baschnagel | Mobility of polymers near surfaces[END_REF] .

Finally, the apparent total diffusion coefficients, of the top layer particles were extracted from the MSD curves at different temperatures, and compared with the corresponding bulk values, D b in Fig. 5. We should note that, although the diffusion of the top layer particles is anisotropic, we have calculated the total diffusion coefficient in order to be consistent with the calculations in the bulk case. As can be discerned, the diffusion coefficient of the top layer particles seems to exhibit an Arrhenius-type temperature dependence, which is reminiscent of previous observations for oligomer glasses [START_REF] Yang | Glass transition dynamics and surface layer mobility in unentangled polystyrene films[END_REF][START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF] . Moreover, diffusion at the surface is faster than in the bulk by a factor that ranges from 10 2 to 10 5 at the lowest temperature studied.

This ratio drastically rises with cooling as it has been observed in both experimental and theoretical studies [START_REF] Yang | Glass transition dynamics and surface layer mobility in unentangled polystyrene films[END_REF][START_REF] Chai | A direct quantitative measure of surface mobility in a glassy polymer[END_REF][START_REF] Brian | Surface self-diffusion of organic glasses[END_REF][START_REF] Malshe | Evolution of glassy gratings with variable aspect ratios under surface diffusion[END_REF] .

IV. Conclusion

Molecular dynamics simulations have been conducted in order to examine the decay process of a periodic square-wave pattern at the free surface of a binary Lennard-Jones film. Contrary to previous simulation studies of the capillary leveling of thin films, the evolution of the height was examined over a wide temperature range sampling both the liquid and glassy states of the film. Different mechanisms appeared to control the pattern leveling in those two states. By the aid of layer-resolved analysis of the particles' mean-squared displacements, we showed that diffusion is much more efficient in the layer close to the free surface than elsewhere in the bulk. Specifically, at the lowest temperature studied, the diffusion coefficient of the surface particles is found to be 10 5 times larger than the bulk counterpart and this difference decreases upon heating.

  , whereas Kayhani et al. performed molecular dynamics (MD) simulations to examine the coalescence of platinum nanoclusters 39 . Concerning simulation work on the surface mobility of molecular glass formers, Hoang et al. performed MD studies on freestanding monatomic glass-forming liquids 40 . The study of Akbari et al. addressed the effects of the substrate in kinetic and thermodynamic properties of a binary

Figure 1 :

 1 Figure 1: Top: Initial configuration of the sample bearing a total number of 3672 particles. Periodic boundary conditions are applied in all directions. L x stands for the x-dimension of the simulation box and h 0 is the maximum vertical height (colour code : blue = top layer, yellow = bottom layer, purple = substrate). Middle: Snapshot of the film at t = 20000 where h av is the average height value. Bottom: Snapshot of the film at t = 233333; h |v| the average height of the fully leveled film.

  , the x dependencies of for the top layer particles and two times t, are displayed. A practically indistinguishable behaviour (not shown for clarity reasons) was observed in the y direction, as well. Also shown in the figure, are the corresponding curves for the bulk sample. The van Hove functions were evaluated at the temperatures T = 0.30 and T = 0.35, which lie below the bulk T g as well as at a temperature T = 0.375 near the bulk T g .

Figure 2 :

 2 Figure 2: Self part of the van Hove function (eq.1) for the top layer particles, as a function of the molecular displacement in the x direction, evaluated at a) T = 0.30, b) T = 0.35 and c) T = 0.375. The different line colours correspond to different times and the references for a bulk sample are provided for comparison.

Figure 3 :

 3 Figure 3: a) Time evolution of the rescaled pattern amplitude at T = 0.45. The red line is an affine fit. b) Decay constant K extracted from the affine fits as a function of temperature. Error bars in K are on the order of 10 -6 .

Figure 4 :

 4 Figure 4: Layer-resolved mean-squared displacements (see eq.2) as a function of time, for various distances z from the substrate as indicated, and at temperatures a) T = 0.35 and b) T = 0.46. The dashed line corresponds to the scaling law Δr 2 ~t.

Figure 5 :

 5 Figure 5: Apparent diffusion coefficients of the top layer (D s ) and bulk (D b ) particles versus inverse temperature. At temperatures data points correspond to estimations in the non-Fickian regime. Temperature is scaled by the bulk T g . The solid line is an affine fit in log-linear representation.

Acknowledgements

The authors gratefully acknowledge financial support from ANR WAFPI and ANR FSCF, as well as the Global Station for Soft Matter, a project of Global Institution for Collaborative Research and Education at Hokkaido University. Hendrik Meyer is kindly thanked for providing the layer-resolved analysis code. The authors also thank Anthony Maggs, Elie Raphaël, Kari Dalnoki-Veress, James Forrest, Joshua McGraw, Oliver Bäumchen and Jörg Baschnagel for stimulating discussions. K.K. would like to thank FO.R.TH.-IESL for the warm hospitality during his stay there.


Lucas Soustelle

Julien Lamy

François Rousseau

Jean-Paul Armspach

Paulo Loureiro De Sousa

A diffusion-based method for long-T 2 suppression in steady state sequences: Validation and application for 3D-UTE imaging

Keywords: ultrashort echo time (UTE), steady state, diffusion, long-T2 suppression, short-T2 contrast, preclinical

Purpose: To introduce a novel method for long-T 2 signal physical suppression in steadystate based on configuration states combination and modulation using diffusion weighting.

Its efficiency in yielding a high contrast in short-T 2 structures using an ultrashort echo time acquisition module (Diff-UTE) is compared to the adiabatically prepared Inversion-

Recovery-UTE sequence (IR-UTE).

Theory and methods: Using a rectangular-pulse prepared 3D-UTE sequence, the possibility of long-T 2 component signal cancellation through diffusion effects is addressed, and the condition met for sets of sequence parameters. Simultaneously, the short-T 2 component signal is maximized using a Bloch equation-based optimization process. The method is evaluated from simulations, and experiments are conducted on a phantom composed of short and long-T 2 components, as well as on an ex vivo mouse head.

Results: Within equal scan times, the proposed method allowed for an efficient long-T 2 signal suppression, and expectedly yielded a higher signal to noise ratio in short-T 2 structures compared to the IR-UTE technique, although an intrinsic short-T 2 signal loss is expected through the preparation module.

Conclusion:

The Diff-UTE method represents an interesting alternative to the IR-UTE technique. Diffusion weighting allowing for a long-T 2 suppression results in a less penalizing method to generate a high and selective contrast in short-T 2 components.

INTRODUCTION

Ultrashort echo time (UTE) MRI methods are used to acquire and quantify NMR signals from tissues with sub-millisecond transverse relaxation time (T 2 ). Among them, one can cite cortical bone [START_REF] Du | Qualitative and quantitative ultrashort echo time (UTE) imaging of cortical bone[END_REF][START_REF] Chen | Measurement of bound and pore water T1 relaxation times in cortical bone using three-dimensional ultrashort echo time cones sequences[END_REF][START_REF] Chen | Fast volumetric imaging of bound and pore water in cortical bone using three-dimensional ultrashort-TE (UTE) and inversion recovery UTE sequences[END_REF][START_REF] Horch | Characterization of 1H NMR signal in human cortical bone for magnetic resonance imaging[END_REF] with strong perspective for CT-like use [START_REF] Johnson | Improved cortical bone specificity in UTE MR Imaging[END_REF][START_REF] Wiesinger | Zero TE MR bone imaging in the head[END_REF][START_REF] Weiger | ZTE imaging with long-T 2 suppression[END_REF], meniscus [START_REF] Nebelung | Ex vivo quantitative multiparametric MRI mapping of human meniscus degeneration[END_REF][START_REF] Williams | UTE-T2* mapping detects sub-clinical meniscus injury after anterior cruciate ligament tear[END_REF], tendon [START_REF] Chang | Singleand Bi-component T2* analysis of tendon before and during tensile loading, using UTE sequences[END_REF][START_REF] Ma | Ultrashort echo time magnetization transfer (UTE-MT) imaging and modeling: magic angle independent biomarkers of tissue properties[END_REF][START_REF] Du | Magic angle effect in magnetic resonance imaging of the Achilles tendon and enthesis[END_REF][START_REF] Wright | Comparison of two ultrashort echo time sequences for the quantification of T1 within phantom and human Achilles tendon at 3 T[END_REF], lung parenchyma [START_REF] Johnson | Optimized 3D ultrashort echo time pulmonary MRI[END_REF][START_REF] Tibiletti | Pulmonary perfusion quantification with flow-sensitive inversion recovery (FAIR) UTE MRI in small animal imaging[END_REF][START_REF] Gai | Three-dimensional T1 and T2* mapping of human lung parenchyma using interleaved saturation recovery with dual echo ultrashort echo time imaging (ITSR-DUTE)[END_REF], and myelin [START_REF] Du | Ultrashort echo time (UTE) magnetic resonance imaging of the short T2 components in white matter of the brain using a clinical 3T scanner[END_REF][START_REF] Du | Measurement of T1 of the Ultrashort T2* Components in White Matter of the Brain at 3T[END_REF][START_REF] Sheth | Magnetic resonance imaging of myelin using ultrashort Echo time (UTE) pulse sequences: Phantom, specimen, volunteer and multiple sclerosis patient studies[END_REF]. Since UTE images are mainly proton-density weighted, an efficient method for suppressing water and fat signals (referred hereafter to as long-T 2 components (T 2 > 10 ms)) is commonly used with an UTE acquisition scheme to properly exhibit a positive contrast over the solid-like tissues of interest. Several schemes for long-T 2 signal suppression or attenuation have been proposed, such as saturation modules [START_REF] Du | Short T2 contrast with three-dimensional ultrashort echo time imaging[END_REF][START_REF] Larson | Designing long-T2 suppression pulses for ultrashort echo time imaging[END_REF], dual echo subtraction [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF][START_REF] Lee | Weighted subtraction in 3D ultrashort echo time (UTE) imaging for visualization of short T2 tissues of the knee[END_REF], or even specific excitation patterns [START_REF] Deligianni | Water-selective excitation of short T2 species with binomial pulses[END_REF] to either null the undesired signal or to selectively excite the component of interest. One of the most frequently used methods to achieve an efficient suppression consists in using an inversion-recovery module in pair with an appropriate inversion time (T I) to physically null the long-T 2 component, followed by an UTE acquisition [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF][START_REF] Tyler | Magnetic resonance imaging with ultrashort TE (UTE) PULSE sequences: Technical considerations[END_REF]. The IR-UTE sequence has often been adapted into enhanced versions for varied purposes (suppressing fat and water signal [START_REF] Du | Short T2 contrast with three-dimensional ultrashort echo time imaging[END_REF][START_REF] Du | Dual inversion recovery, ultrashort echo time (DIR UTE) imaging: Creating high contrast for short-T2 species[END_REF][START_REF] Larson | Using adiabatic inversion pulses for long-T2 suppression in ultrashort echo time (UTE) imaging[END_REF], filtering short-T 2 signals [START_REF] Horch | Clinically compatible MRI strategies for discriminating bound and pore water in cortical bone[END_REF], and acceleration [START_REF] Carl | UTE imaging with simultaneous water and fat signal suppression using a time-efficient multispoke inversion recovery pulse sequence[END_REF]), and has already demonstrated promising results in various applications [START_REF] Chen | Fast volumetric imaging of bound and pore water in cortical bone using three-dimensional ultrashort-TE (UTE) and inversion recovery UTE sequences[END_REF][START_REF] Sheth | Magnetic resonance imaging of myelin using ultrashort Echo time (UTE) pulse sequences: Phantom, specimen, volunteer and multiple sclerosis patient studies[END_REF][START_REF] Larson | Using adiabatic inversion pulses for long-T2 suppression in ultrashort echo time (UTE) imaging[END_REF][START_REF] He | Direct magnitude and phase imaging of myelin using ultrashort echo time (UTE) pulse sequences: A feasibility study[END_REF][START_REF] Gai | Long T2 suppression in native lung 3-D imaging using k-space reordered inversion recovery dual-echo ultrashort echo time MRI[END_REF].

Nevertheless, the IR-UTE sequence shows limitations in terms of short-T 2 signal to noise ratio (SNR) generation. Through the use of adiabatic inversion pulses that often saturate the short-T 2 components [START_REF] Larson | Designing long-T2 suppression pulses for ultrashort echo time imaging[END_REF][START_REF] Li | Comparison of optimized soft-tissue suppression schemes for ultrashort echo time MRI[END_REF], the available magnetization in the component of interest is governed by its longitudinal recovery during the inversion time delay, itself depending on the T 1 value of the component to be suppressed. Another issue that has to be considered for clinical applications is the elevated Specific Absorption Rate (SAR) induced by the use of adiabatic inversion pulses for long-T 2 signal suppression and short-T 2 signal filtering [START_REF] Horch | Clinically compatible MRI strategies for discriminating bound and pore water in cortical bone[END_REF], especially for currently developing ultra-high field scanners.

This study presents a novel method for long-T 2 suppression in a steady-state 3D-UTE sequence through the use of diffusion weighting induced by spoiling gradients, while acquiring maximized short-T 2 signals [START_REF] Soustelle | Long-T2 suppression based on saturation and diffusion in a steady-state 3D-UTE sequence[END_REF]. The behavior of the method was evaluated from simulations and phantom experiments as a proof of concept on a preclinical scanner. Comparison of the proposed sequence to the IR-UTE sequence was conducted on the same phantom and on an ex vivo mouse head.

THEORY

In the following, the superscripts L and S will indicate the long and short-T 2 component, respectively. Simulated signal relative amplitude will be introduced as S = |M xy |/M 0 , with M xy transverse magnetization, and M 0 its corresponding magnetization amplitude at thermal equilibrium. To provide a distinction between the two components having respective unknown proton density, it is further quantified in %M 0 .

Diff-UTE sequence

The employed pulse sequence (Fig. 1.a) consists in a long (> T S

2 ) rectangular preparation pulse followed by a short one, whose flip angle is computed to maximize the short-T 2 signal.

In steady-state, the approach is somehow similar to the Actual Flip Angle method [START_REF] Yarnykh | Actual flip-angle imaging in the pulsed steady state: A method for rapid three-dimensional mapping of the transmitted radiofrequency field[END_REF], except that the two consecutive pulses have different flip angles (α 1 = α 2 ) and durations, and the readout is performed exclusively after the second pulse. Hence, gradient spoiling, RF spoiling, and delays are optimized to ensure a steady-state of the long-T 2 component to be suppressed, as well as minimizing the impact of potential static gradients (e.g. B 0 inhomogeneities) [START_REF] Nehrke | On the steady-state properties of actual flip angle imaging (AFI)[END_REF]. The sequence aims at removing the long-T 2 signal, while simultaneously maximizing the short-T 2 one, as described in the following sections.

Short-T 2 signal description

Considering the signal of interest arising from the short-T 2 component being fully spoiled before each excitation, its quantity can be tracked using the Bloch equations in steady-state immediately after the readout pulse (T E = 0 ms):

S S = (1 -E s ) + E s (1 -E w )f z 1 1 -f z 1 f z 2 E s E w f xy , [1]
with E s = e -T R 1 /T S 1 , E w = e -T R 2 /T S 1 (T R 1 and T R 2 referring to the internal delays after excitations),

f z i = e -w i cos γ i + w i sinc γ i , f xy = e -w 2 α 2 sinc γ 2 (with γ i = α 2 i -w 2 i and w i = τ i /2T S 2
), α i and τ i being the respective flip angles and pulse durations. A detailed derivation of the Bloch equations solutions is given in the Appendix. The functions f z i and f xy account for the signal loss during a hard RF pulse caused by T S 2 relaxation (mapping functions), as described in [START_REF] Sussman | Design of practical T2-selective RF excitation (TELEX) pulses[END_REF][START_REF] Torrey | Transient Nutations in Nuclear Magnetic Resonance[END_REF][START_REF] Hu | Pulsed saturation transfer contrast[END_REF]. Including the mapping functions is advantageous and helps quantifying the impact of the durations of the consecutive pulses [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF][START_REF] Carl | Optimization of RF excitation to maximize signal and T2 contrast of tissues with rapid transverse relaxation[END_REF]. It therefore allows to adequately model the available S S signal. Thus, once delays and pulses characteristics are set, the actual short-T 2 signal can be maximized with respect to α 2 as α2 = argmax α 2 (S S ).

Long-T 2 signal description

Once sequence parameters are set, we make use of the steady-state condition in a periodic MR sequence to suppress the water signal. The long-T 2 signal can be modeled by any simulation framework taking diffusion into account [START_REF] Gudbjartsson | Simultaneous calculation of flow and diffusion sensitivity in steady-state free precession imaging[END_REF][START_REF] Weigel | Extended phase graphs with anisotropic diffusion[END_REF]. Thus, using the expression for configuration states in [START_REF] Weigel | Extended phase graphs: Dephasing, RF pulses, and echoes -pure and simple[END_REF], the signal to be suppressed can be written:

F + 0 = cos 2 (α 2 /2)F - 0 + e 2iΦ sin 2 (α 2 /2) 2 F - * 0 -ie iΦ sin(α 2 )Z - 0 , [2] with F - 0 and Z - 0 being functions of α 1 , α 2 , RF-phase Φ, T R 1 , T R 2 , T L 1 , T L 2 , gradient
spoiling areas Gδ and Gδ in respective internal segments T R 1 and T R 2 , and tissue diffusion characteristics. In this case, having |F + 0 | = 0 would imply a complete long-T 2 suppression.

Since no trivial analytical expression exists for the F - 0 and Z - 0 states in steady-state, we numerically explored the tissues and sequence parameter space in order to assess whether the diffusion weighting induced by the spoiling gradients would combine the F - 0 and Z - 0 states in order to satisfy |F + 0 | = 0. This condition can be met for sets of parameters and

RF-phase increment Φ 0 = k × 360/(n + 1) (k ∈ Z, n = T R 2 /T R 1 ≥ 2)
, offering a signal falling to 0 (referred hereafter to as "signal minimum").

In the following, the apparent diffusion coefficient D will be assimilated to that of the single spoiling direction used in experiments (i.e. z-direction). Hence, and for better readability, diffusion weighting is quantified in terms of gradient spoiling moment Gδ, equal to the gradient area in the first internal delay T R 1 . To maintain a steady-state condition, the spoiling gradient areas ratio Gδ /Gδ is set to be equal to n [START_REF] Nehrke | On the steady-state properties of actual flip angle imaging (AFI)[END_REF].

Finally, given a specific isotropic geometry and receiving bandwidth, the minimum spoiling moment to provide a 2π spin dephasing across a voxel in T R 1 amounts to:

(Gδ) min = G read τ read . [3]
The resulting minimum spoiling moment in T R 2 is equal to n×(Gδ) min , yielding a zero-order b-value difference between the readout (considering an isotropic diffusion for simplification purpose since the readout gradients vary within each repetition) and the spoiling gradients in T R 2 of about n 3 [START_REF] Mattiello | Matrix in Diffusion Tensor Echo-Planar Imaging[END_REF]. Hence, the diffusion weighting contributions from the readout gradients will further be neglected.

METHODS

Long and short-T 2 magnetizations behavior hypotheses

Throughout the paper, the following hypotheses (summarized in Table 1) regarding the long and short-T 2 components are made:

-the short-T 2 component being considered to have a fully decayed transverse magnetization prior to the excitations, any contribution of diffusion weighting by spoiling gradients is assumed negligible as it does not affect the magnetization. Hence, its relative amplitude can be simulated using Eq. [START_REF] Du | Qualitative and quantitative ultrashort echo time (UTE) imaging of cortical bone[END_REF], and exclusively depends on τ 1 , τ 2 , flip angles α 1 and α 2 , T S 1 , T S 2 and sequence delays. Alternatively, the adiabatic inversion pulse used in IR-UTE is supposed to saturate the short-T 2 component given its assumed sub-millisecond T 2 value in simulations and experiments [START_REF] Carl | UTE imaging with simultaneous water and fat signal suppression using a time-efficient multispoke inversion recovery pulse sequence[END_REF][START_REF] Li | Comparison of optimized soft-tissue suppression schemes for ultrashort echo time MRI[END_REF].

-the long-T 2 component is affected by diffusion weighting through the spoiling gradients

Gδ and Gδ , flip angles α 1 and α 2 , apparent diffusion coefficient D, T L 1 , T L 2 and sequence delays. Similarly, relaxation during excitations is assumed negligible, and adiabatic inversion is supposed to fully invert the component's magnetization.

Optimization of the Diff-UTE sequence parameters

Once long and short-T 2 components parameters and sequence delays have been set, generating the highest contrast in the short-T 2 component consists in stretching τ 1 to take advantage of the relaxation occurring during excitation, contracting τ 2 to ensure a maximum excitation, and adapting α 1 , α 2 and Gδ to maximize the short-T 2 components signal while nulling the long-T 2 one through the states combination process.

The following is an example of algorithm for the sequence parameters choice: (i) provide expected tissues/materials properties T L 1 /T L 2 , D and T S 1 /T S 2 ; (ii) set system parameters RFphase Φ, T R 2 /T R

Particular considerations

The sequence repetition time in the experiments are equal to (Fig. 1):

T R = τ 1 + T R 1 + τ 2 + T R 2 in Diff-UTE [4]
T R = τ inv + T I + τ IR + RD in IR-UTE [START_REF] Johnson | Improved cortical bone specificity in UTE MR Imaging[END_REF] with τ inv and τ IR preparation and readout pulse durations in IR-UTE, respectively. Internal delays after preparation and readout pulses are respectively designated as T I and RD in the same sequence. In addition, when both methods are compared to each other, the repetition time is kept constant in both sequences so that the acquisition times are identical, and hence global signals comparable to each other.

Using the notation in Eq. [START_REF] Johnson | Improved cortical bone specificity in UTE MR Imaging[END_REF] and Bloch equations, it can be shown that signal suppression in IR-UTE is performed by setting the T I delay to:

T I = -T L 1 ln (1 + e -(T R-τ inv -τ IR )/T L 1 )/2 , [6]
with T L 1 longitudinal relaxation time of the fully inverted long-T 2 component. Additionally, due to the saturated condition of the short-T 2 magnetization after inversion (inversion efficiency Q = 0), the relative amplitude can be expressed as [START_REF] Du | Measurement of T1 of the Ultrashort T2* Components in White Matter of the Brain at 3T[END_REF] (T E = 0 ms):

S S IR-UTE = (1 -e -T I/T S 1 )f IR xy , [7]
with f IR xy the corresponding transverse mapping function of the readout pulse in IR-UTE.

Alternatively, an unprepared UTE sequence (α 1 = 0 • in Diff-UTE) with an echo-subtraction technique to suppress the long-T 2 signal is considered [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF] and used to quantify the expected short-T 2 signal loss through the preparation module in Diff-UTE. Using the notations of the Diff-UTE sequence, the resulting resting delay between two consecutive excitations is equal

to RD = τ 1 + T R 1 + T R 2 .
The short-T 2 signal in this sequence is therefore described by (T E = 0 ms):

S S UTE = 1 -E r 1 -E r f z f xy , [8]
with E r = e -RD/T S 1 , and f z and f xy the corresponding mapping functions of the unique excitation pulse. From Equation [START_REF] Nebelung | Ex vivo quantitative multiparametric MRI mapping of human meniscus degeneration[END_REF], an optimized excitation flip angle can be evaluated similarly to the Diff-UTE short-T 2 signal optimization process, as described in [START_REF] Carl | Optimization of RF excitation to maximize signal and T2 contrast of tissues with rapid transverse relaxation[END_REF]. Finally, the ratio of both optimized signal, is expressed as:

R = S S Diff-UTE S S UTE √ 2, [ 9
]
where the √ 2 factor takes account of the SNR loss through the echo subtraction process in the unprepared UTE sequence to yield a theoretical long-T 2 signal suppression [START_REF] Larson | Designing long-T2 suppression pulses for ultrashort echo time imaging[END_REF][START_REF] Li | Comparison of optimized soft-tissue suppression schemes for ultrashort echo time MRI[END_REF].

Numerical simulations

If not otherwise stated, all numerical simulations were done using the following nominal The short-T 2 parameters are that of the Lego brick sample used in experiments (T S 1 /T S 2 = 300/0.315 ms, assuming its T 2 value to be close to T * 2 [START_REF] Du | Qualitative and quantitative ultrashort echo time (UTE) imaging of cortical bone[END_REF][START_REF] Du | Measurement of T1 of the Ultrashort T2* Components in White Matter of the Brain at 3T[END_REF][START_REF] Sussman | Design of practical T2-selective RF excitation (TELEX) pulses[END_REF]). Simulations and analyses of the long-T 2 signal were performed using the Extended Phase Graph formalism [START_REF] Weigel | Extended phase graphs: Dephasing, RF pulses, and echoes -pure and simple[END_REF][START_REF] Weigel | Extended phase graphs with anisotropic diffusion[END_REF][START_REF] Hennig | Echoes-how to generate, recognize, use or avoid them in MR-imaging sequences. Part I: Fundamental and not so fundamental properties of spin echoes[END_REF], with adapted code available at http://epg.matthias-weigel.net/ge_ssfp_epg.html.

Long-T 2 signal simulations using Diff-UTE were performed. Its signal variability along gradient spoiling moments ranging from 0.0 to 500.0 mT/m.ms was evaluated for different preparation flip angles α 1 ranging from 60 • to 90 • , τ 1 = 1 ms, and an optimized α 2 for each value. Similarly, the short-T 2 signal is evaluated for optimized parameters given the same α 1 range, and τ 1 values from 0.5 to 3 ms.

In order to provide insights about the robustness of the suppression in case of inaccuracies Finally, in order to assess the maximal signal loss induced by the preparation in Diff-UTE, the R ratio is computed within respective optimized conditions of S S Diff-UTE and S S UTE , and with identical excitation pulse duration and repetition time. In Diff-UTE, τ 1 is ranging from 0.5 to 5 ms and α 1 from 60 to 90 • .

Experimental details

Experiments were conducted on a 7T BioSpec 70/30 USR small animal MRI system (Bruker BioSpin MRI GmbH, Ettlingen, Germany), operated with ParaVision 6.0.1. Gradient system provided a maximum amplitude of 442 mT/m, and a maximum slew rate of 3660 T/m/s.

Scans were performed using a 86-mm diameter transmitter, and a mouse surface coil for reception. The native UTE-3D Bruker sequence was modified and declined into the proposed sequence (Fig. 1.a), as well as in IR-UTE (Fig. 1.b). A high-order B 0 shimming was systematically estimated prior to acquisition series. To compensate for eddy currents and gradients non-idealities [START_REF] Brodsky | Rapid measurement and correction of phase errors from B0 eddy currents: Impact on image quality for non-cartesian imaging[END_REF], spokes trajectories were measured using the constructor's routine [START_REF] Duyn | Simple correction method for k-space trajectory deviations in MRI[END_REF].

Images were then reconstructed using the built-in software reconstruction pipeline.

Since the imaged phantom is considered as an homogeneous item, voxels size was chosen to be large compared to the ex vivo sample experiments in order to avoid a prohibitive total acquisition time. and T E was set to 1 and 0.02 ms, respectively, amounting an acquisition time per scan to 14 min and 52 s. To explore the diffusion effects, 24 gradient spoiling moment values linearly spaced from 120.2 to 488.2 mT/m.ms were used, the lowest value corresponding to a gradient spoiling moment sufficiently strong to yield an artifact-free volume.

A comparison of the short-T 2 SNR estimated in Diff-UTE and IR-UTE was conducted within long-T 2 suppression conditions. Short-T 2 signal was acquired using the same readout pulse duration τ 2 = τ IR = 0.07 ms and same T E = 0.02 ms. For an optimal Gδ value in Diff-UTE, a scan was acquired using the previously proposed protocol. An 8-ms effective 1.75 kHz single-band inversion hyperbolic secant adiabatic pulse [START_REF] Silver | Highly selective and π pulse generation[END_REF] was employed in IR-UTE, whose duration and bandwidth are set to be consistent to that employed in the literature which used 8.6 ms and 1.5 kHz pulses [START_REF] Chen | Fast volumetric imaging of bound and pore water in cortical bone using three-dimensional ultrashort-TE (UTE) and inversion recovery UTE sequences[END_REF][START_REF] Carl | UTE imaging with simultaneous water and fat signal suppression using a time-efficient multispoke inversion recovery pulse sequence[END_REF]. TR was kept identical in both sequences in order to equal acquisition times for each increment, while respective sequences delays obey

Eq. [START_REF] Horch | Characterization of 1H NMR signal in human cortical bone for magnetic resonance imaging[END_REF] and [START_REF] Johnson | Improved cortical bone specificity in UTE MR Imaging[END_REF]. In addition, readout flip angle α IR in IR-UTE was set to 90 • to ensure a maximum SNR of the short-T 2 component, and a sufficiently strong and long gradient spoiling was used to provide a dephasing moment superior to 2π after inversion and readout.

For

Ex vivo sample

A C57BL/6 adult mouse was perfused via intra-cardiac delivery of PBS followed by tissue fixation using 4% paraformaldehyde [START_REF] Cahill | Preparation of fixed mouse brains for MRI[END_REF]. All the procedures were done under deep anesthesia and in accordance to the local ethical committee rules. The head was collected and soaked into a proton-free solution (PFPE, Galden, Solvay) and scanned using both UTE sequences

RESULTS

Simulations

Signal minima occurrences are depicted in blue bands on figures relating simulation results.

It corresponds to an arbitrary threshold considered when the evaluated long-T 2 transverse magnetization has reached 0.05% of its value at thermal equilibrium.

As shown in Figure 2.a, signal minima are reachable given an initial tip angle α 1 threshold of 76 • given the proposed optimization pipeline and used set of parameters. For intermediary α 1 values between 79 • and 87 • , two Gδ-ranges for suppression can be theoretically yielded, and a single range is finally obtained for α 1 between 88 • and 90 • . This indicates a much higher signal generation in Diff-UTE for low flip angles values, but careful attention shall be given in that a long-T 2 suppression condition is not systematically reachable in Diff-UTE for such regimes.

Figure 6 shows the ratio R as a function of τ 1 and α 1 . Given a constant α 1 , R increases as τ 1 does due to the increasing short-T 2 relaxation occurring during the excitation. Even though this effect allows for more short-T 2 signal generation in Diff-UTE as τ 1 increases, the signal generated by the unprepared UTE sequence is intrinsically higher since an additional T 1 relaxation effect occurs. The highest ratio is generated for α 1 = 60 • and τ 1 = 5 ms (R = 0.996). Within the conditions in experiments (i.e. α 1 = 90 • , τ 1 = 1 ms), the estimated loss amounts to 58%.

Experiments

The acquired signal in the single pulse sequence showed a spectral line's frequency width half magnitude resulting in a T S * an appropriate description of the long-T 2 signal. While the long-T 2 signal is made varying with Gδ, no noticeable impact over the short-T 2 component signal at a given {α 1 ; α 2 } pair is observed, suggesting that the proposed suppression scheme does not affect the component of interest. However, as expected, the higher the preparation flip angle α 1 , the lower the short-T 2 signal amplitude.

In a long-T 2 suppression condition, the effective SNR in the short-T 2 component is systematically higher in Diff-UTE than in IR-UTE, with respective ratios of 1.49, 1.70, 1.95, 2.03 and 2.31 for T R of 31.07, 31.57, 32.07, 32.57 and 33.07 ms. The ratio evolution along the increasing T R shows that the suppression scheme in Diff-UTE is less penalizing than in IR-UTE regarding the short-T 2 component signal. Figure 8 illustrates the results with views of the phantom acquired at T R of 31.07 and 33.07 ms in both sequences.

Figure 9 shows the fixed mouse head coronal and sagittal central views in FLASH, Diff-UTE and IR-UTE. Soft tissues are very well perceptible in the FLASH sequence, and no signal is visible in the head skull. On the contrary, a fairly good contrast on the cortical bone of the head skull is generated in both UTE sequences. SNR were calculated using the drawn ROI in the skull, and amounted respectively to 14.0 and 5.4 in Diff-UTE and IR-UTE, supporting simulations and previous experiments results about the short-T 2 signal generation efficiency using the proposed method.

DISCUSSION

A novel method for suppressing a long-T 2 component has been proposed. Similarly to the inversion-recovery approach, the method has proven to be efficient in suppressing contaminating signal, therefore yielding a high contrast in short-T 2 structures in an UTE acquisition sequence. Additionally, a compelling SNR in plastic polymer and cortical bone was generated through the proposed optimization process, making the approach all the more promising for further applications in imaging short-T 2 structures.

The higher SNR generated in pair with a 3D modality in the proposed method may be beneficial for various purposes. For example, sequence acceleration can be made possible through several already-established processes. Due to the expected higher short-T 2 transverse magnetization in Diff-UTE compared to the IR-UTE sequence, one can take advantage of the available short-T 2 magnetization to proceed to undersampling, or even implement a multispoke acquisition pattern within a single TR [START_REF] Carl | UTE imaging with simultaneous water and fat signal suppression using a time-efficient multispoke inversion recovery pulse sequence[END_REF][START_REF] Li | Selective in vivo bone imaging with long-T2 suppressed PETRA MRI[END_REF], however at the cost of a less efficient suppression, and although limited by the required short TR to ensure a steady-state. In a clinical application, these features may consequently reduce scan time for a whole 3Dcoverage. On another hand, small voxel sizes are also responsible for a loss of SNR, as is the case in high-resolution imaging or in preclinical scanner applications, even though ultra high fields used in the latter aim to compensate for it.

In this study, scan times may seem quite long for the ex vivo sample (≈ 31 min). This was due in this case i) to the relatively small voxels size yielding a rather low signal intensity and inducing SNR loss in case of a further increasing undersampling factor, and ii) to generate a distinguishable contrast in IR-UTE to be compared with the proposed method.

More sensitive devices, as well as more flexible acquisition schemes (2, 55) may help taking advantage of the required short TR to reduce the acquisition scan times.

As proposed in the literature [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF][START_REF] Rees | MRI of the brain with ultra-short echo-time pulse sequences[END_REF], a second echo may be acquired after reading and refocusing the first FID. This may come in handy if ones want to suppress fat signal through a magnitude subtraction process, for example, by setting an appropriate echo time in order to provide an in-phase signal refocusing. Additional fat suppression may also be performed by appropriately including an optimized off-resonance pulse centered at 3.5 ppm in the pulse sequence.

In ex vivo experiments, a strong global suppression and attenuation of the contaminating long-T 2 signals (white and grey matter) has been noticed. This is inherent to the short TR used in the Diff-UTE protocol, as is also the case in IR-UTE. For instance, the needed parameter -here T I -does not vary much for suppressing a wide range of T 1 , somehow yielding a large suppression of T 1 species, or at least a significant signal attenuation. When coupled with an efficient scheme for short-T 2 signal maximization, a substantially higher contrast to noise ratio in regions of interest is generated compared to the IR-UTE. These specific aspects may be of a great advantage in in vivo experiments as residual water signals are suppressed through an echo-subtraction technique [START_REF] Johnson | Improved cortical bone specificity in UTE MR Imaging[END_REF][START_REF] Du | Measurement of T1 of the Ultrashort T2* Components in White Matter of the Brain at 3T[END_REF][START_REF] Li | Comparison of optimized soft-tissue suppression schemes for ultrashort echo time MRI[END_REF], therefore diminishing the relative SNR in studied structures.

Another advantage of the radial 3D acquisition modality is the non-selective excitation: commonly used reshaped pulses (57) in 2D-UTE methods are sensitive to intrinsic sequence imperfections such as eddy currents inducing slice-select and readout gradients distortions [START_REF] Harkins | Simple and robust saturation-based slice selection for ultrashort echo time MRI[END_REF][START_REF] Josan | Double half RF pulses for reduced sensitivity to eddy currents in UTE imaging[END_REF], timing errors between slice-selective gradients and RF amplifier [START_REF] Soustelle | Minimum-Time VERSE Pulse Correction for Slice Selectivity Improvement in 2D-UTE Imaging[END_REF][START_REF] Harkins | Automatic Gradient Predistortion Applied to Clinical 2D-UTE[END_REF], and off-resonance effects that may cause slice distortions [START_REF] Pauly | Selective Excitation for Ultrashort Echo Time Imaging[END_REF]. These are also penalizing due to T 2 decay during the relatively long selective excitation [START_REF] Robson | Magnetic resonance: an introduction to ultrashort TE (UTE) imaging[END_REF].

No adiabatic pulse is used in Diff-UTE, but a global suppression of long relaxing signals has nonetheless been reached. This represents a great advantage in terms of RF power deposition, as well as compensating for clinical scanner limitations as the available B 1 peak is often low on such systems. For example, the pulses used in the ex vivo experiment yielded a relative SAR for the first order HS about 12.7 times higher in IR-UTE than the preparation rectangular pulse in Diff-UTE. This score was generated by using a B 1,max of 17 µT for the HS pulse, corresponding to the minimum RF amplitude to yield an efficient inversion (Q < -0.99 (63)). The latter was estimated by numerically solving the Bloch equations, assuming that no relaxation occurs during the inversion. The relative SAR expressions are given in Appendix. As a result, the method is well suited for ultra-high field scanners' applications where protocols can be greatly limited by the SAR.

So far, α 2 was set using the proposed optimization process where its estimation relies on the assumption of priors about T 1 and T 2 values of a single short relaxing component.

However, a wide short-T 2 distribution is often found in solid-like structures, essentially due to their complexity. For example, the transverse magnetization decay of the Lego brick indicates the presence of at least two components, as shown in Figure S7. This multicomponent behavior can also be regarded as a more realistic pattern of what can be found in biological tissues [START_REF] Horch | Characterization of 1H NMR signal in human cortical bone for magnetic resonance imaging[END_REF][START_REF] Horch | Origins of the ultrashort-T2 1H NMR signals in myelinated nerve: A direct measure of myelin content? Magnetic Resonance in[END_REF][START_REF] Wilhelm | Direct magnetic resonance detection of myelin and prospects for quantitative imaging of myelin density[END_REF][START_REF] Rad | Quantifying cortical bone water in vivo by three-dimensional ultra-short echo-time MRI[END_REF]. Hence, assuming a single-relaxing component in such systems may result in a short-T 2 selectivity pattern since its signal will be maximized with no particular regards to other short relaxing components. As such, SNR quantification in solid-like structures may vary with the respective proton density and relaxation time constants of the various excited pools, given sets of parameters. A more complex optimization scheme for α 2 optimization can be established under the condition of acknowledging the respective relaxing parameters, as well as their relative proton density fractions. An expected propagation of uncertainties regarding the experimental signal intensities may arise from such increasingly complexity. Nonetheless, a substantially higher SNR was found in our experiments when comparing both UTE sequences, supporting our simulations' trends.

Filtering short-T 2 and slightly short-T 2 components has also been reported in [START_REF] Horch | Clinically compatible MRI strategies for discriminating bound and pore water in cortical bone[END_REF] through adiabatic pulses combination by taking advantage of a partial inversion effect instead of a saturation of species having T 2 above 2 ms. While this filter is constituted by adiabatic inversion pulses in IR-UTE-like sequences, this can also be performed through steady-state and flip angles combinations in Diff-UTE. In the same manner as in IR-UTE and its adapted versions, a filtering effect of the short-T 2 components can be made through the long-T 2 suppression scheme, as well as the chosen short-relaxing compartment parameters in the maximization process, hence maximizing one component over the other. It has been shown

in Figure 9 that a higher SNR can be generated with the proposed sequence compared to the IR-UTE, but no attention was given about the multicomponent behavior of the cortical bone. Further investigations may allow to discriminate appropriately the various short-T 2 pools present in such structure.

The long-T 2 suppression method was proposed in the present paper for generating contrast in sub-millisecond relaxing transverse structures. Given the advantages discussed above, an adaptation of such technique offering new kinds of selectivities (relaxation and diffusion) may represent an interesting and rapid approach for novel contrasts generation in soft tissues. Further investigations will be led accordingly.

As a prepared sequence, a loss of magnetization of the short-T 2 component is to be expected compared to an unprepared sequence with equal T R. An echo subtraction technique may be included in the latter for long-T 2 suppression purpose, at the cost of an inherent reduced SNR by a √ 2 factor [START_REF] Larson | Designing long-T2 suppression pulses for ultrashort echo time imaging[END_REF][START_REF] Li | Comparison of optimized soft-tissue suppression schemes for ultrashort echo time MRI[END_REF]. Additionally, Equation [START_REF] Williams | UTE-T2* mapping detects sub-clinical meniscus injury after anterior cruciate ligament tear[END_REF] remains exclusively valid under the ideal condition of a fully relaxed short-T 2 component by the second echo. If not, as it might be the case for the longest T 2 component in cortical bone (T * 2 ≈ 1.1 ms at 3 T (1)) for T E about 2 ms for example, this will result in an increased short-T 2 signal penalization in the unprepared UTE sequence. In the same manner, the method proposed by a magnitude subtraction of an image acquired with a short pulse and a short T E, and an image acquired with a long pulse and a long T E, using the same flip angles. In spite of a doubled acquisition time, such process may yield a slightly higher short-T 2 SNR in the resulting unprepared and subtracted image since the residual short-T 2 components in the late T E are less excited within the long pulse. Other issues may occur, such as additional tissue-dependent long-T 2 relaxation between echoes, diffusion weighting through readout and refocusing gradients prior to the acquisition of the second echo [START_REF] Chaudhari | Imaging and T2 relaxometry of short-T2 connective tissues in the knee using ultrashort echo-time double-echo steady-state (UTEDESS)[END_REF][START_REF] Bieri | On the fluid-tissue contrast behavior of high-resolution steady-state sequences[END_REF], or even susceptibility effects [START_REF] Du | Short T2 contrast with three-dimensional ultrashort echo time imaging[END_REF]. These irreversible effects are therefore responsible for a physical impossibility of long-T 2 suppression, all the more important when dealing with small voxels sizes and increasing operating static field.

CONCLUSION

A novel long-T 2 suppression method which employs diffusion induced weighting in a steadystate 3D-UTE sequence has been presented. Simultaneously, the sequence provides more degrees of freedom to maximize the short-T 2 signal quantity compared to the IR-UTE sequence. A good contrast was obtained with each sequence, and a compelling higher SNR was estimated in a mouse head skull in Diff-UTE compared to IR-UTE.

SAR expressions

With no particular regards about the duty cycle, the relative SAR Γ of the pulses used in experiments can be expressed as [START_REF] Idiyatullin | Gapped pulses for frequency-swept MRI[END_REF][START_REF] Bernstein | Handbook of MRI Pulse Sequences[END_REF] :

Γ HS ∝ B 2
1,max τ IR β for a first order HS pulse, [START_REF] Du | Magic angle effect in magnetic resonance imaging of the Achilles tendon and enthesis[END_REF] Γ BP ∝ B 2 1,max τ 1 for a rectangular pulse, [START_REF] Wright | Comparison of two ultrashort echo time sequences for the quantification of T1 within phantom and human Achilles tendon at 3 T[END_REF] with B 1,max the respective pulse maximum amplitude, and β ≈ 5.3 truncation factor of the HS pulse. Component

Short-T 2 Long-T 2 Parameters T S 1 ( ), T S 2 ( ) T L 1 , T L 2 , D RF Pulses α 1 ( ), α 2 α 1 , α 2 τ 1 ( ), τ 2 ( ) - Gradients influence - Gδ, Gδ
Adiabatic inversion effect saturation inversion

  relaxing and diffusion parameters: long-T 2 component: D = 1.00 µm 2 /ms, T L 1 /T L 2 = 1000/100 ms; short-T 2 component: T S 1 /T S 2 = 300/0.315 ms. In Diff-UTE, invariable parameters are intermediary delays T R 1 /T R 2 = 5/25 ms, and readout pulse duration τ 2 = 0.07 ms.

  of the estimated relaxing and diffusion parameters, signal minima occurrences are explored through the long-T 2 relaxing and diffusion parameters. One out of three parameters is made varying about ±50% of its nominal values along a constant spoiling gradient moment range from 0.0 to 300.0 mT/m.ms for α 1 = 80.0 • and an optimized α 2 = 57.0 • maximizing a T S 1 /T S 2 = 300/0.315 ms species. Specific tissues long-T 2 signal behavior were assessed in Diff-UTE. Relaxation and diffusion parameters were chosen from the literature at 3 T (45, 35, 46, 47): T 1 /T 2 = 1412/50 ms and D = 0.7 µm 2 /ms for skeletal muscle, T 1 /T 2 = 4000/2000 ms and D = 2.2 µm 2 /ms for cerebrospinal fluid, T 1 /T 2 = 1820/99 ms and D = 0.8 µm 2 /ms for grey matter, and T 1 /T 2 = 1084/69 ms and D = 0.8 µm 2 /ms for white matter. Preparation and excitation flip angles are α 1 /α 2 = 80/57 • . The short-T 2 signal ratio between Diff-UTE and IR-UTE is generated for the most penalizing preparation flip angle α 1 = 90 • and optimized α 2 , and for T S 1 and T S 2 respectively ranging from 100 to 1000 ms and 0.1 to 1 ms. Alternatively, the same quantity was generated for arbitrary α 1 and α 2 respectively ranging from 60 to 90 • and 1 to 90 • , and for T S 1 /T S 2 = 300/0.315 ms. For both simulations, the IR-UTE sequence parameters are optimized to yield a long-T 2 signal cancellation having a T L 1 of 1000 ms (i.e. T I = 11.5 ms), and with τ 1 /τ inv /τ IR = 1/8/0.07 ms.

PhantomA

  phantom composed of an ABS plastic Lego brick immersed in a doped distilled water with Ni 2+ , 2Cl -(C Ni 2+ = 1 mM; prior scans using gold-standard sequences allowed to estimate the following parameters at bore temperature (≈ 20 • C): D = 1.81 µm 2 /ms using a pulsed field gradient spin-echo sequence (PGSE), T L 1 = 590 ms using an inversion-recovery spinecho sequence (IR-SE), and T L 2 = 295 ms using a multi spin-echo sequence (MSE)). Relevant sequence parameters in phantom experiments were: RF phase increment Φ 0 = 0 • , receiver bandwidth = 150 kHz, matrix size = 96 × 96 × 96, voxel size = 0.26 mm isotropic, number of radial lines = 28733, dummy scans = 200. Again, in Diff-UTE, invariable parameters are intermediary delays T R 1 /T R 2 = 5/25 ms, and readout pulse duration τ 2 = 0.07 ms. A first estimation of the T S * 2 value of the Lego brick was made using a single-pulse sequence: TR = 4000 ms, 50 µs/90 • rectangular pulse, receiver bandwidth = 50 kHz, 64 averages. The possibility of long-T 2 signal cancellation in Diff-UTE is assessed for different optimized sets of flip angles {α 1 /α 2 } = {90.0 • /62.1 • , 80.0 • /57.0 • , 70.0 • /51.8 • , 60.0 • /46.7 • }. τ 1

1 and T 2

 12 along with a 2D Fast Low Angle SHot (FLASH; T R/T E = 50/5 ms, slice thickness = 0.5 mm, in-plane voxels size = 78 × 78 µm 2 , 32 averages, acquisition time/scan = 6 min 49 s).Prior scans (IR-SE, MSE and PGSE) allowed to estimate relaxing constants and diffusion characteristic in the gray matter, yielding a T 1 of 923 ± 8.1 ms, a T 2 of 70 ± 2.5 ms, and a diffusion coefficient along the spoiling direction of 0.328 ± 0.011 µm 2 /ms. Approximated T values of the cortical bone at room temperature found in[START_REF] Han | Quantifying temperature-dependent T1 changes in cortical bone using ultrashort echo-time MRI[END_REF] were used for Diff-UTE parameters optimization (T 1 = 120 ms, T 2 = 0.4 ms). Common parameters were T R/T E = 33.07/0.01 ms, RF phase increment Φ 0 = 0 • , voxel size = 0.104 mm isotropic, matrix size = 192 × 192 × 192, number of radial lines = 57527 (undersampling factor = 2) and receiver bandwidth = 200 kHz, acquisition time/scan = 31 min 42 s. Specific parameters were T R 1 /T R 2 = 5/25 ms, τ 1 /τ 2 = 3/0.07 ms, α 1 /α 2 = 90.0/40.0 • and Gδ = 88.4 mT/m.ms in Diff-UTE; 8-ms/1.75 kHz hyperbolic secant adiabatic inversion pulse, τ IR = 0.07 ms, α IR = 90.0 • and T I = 12.4 ms in IR-UTE.

Figure 2

 2 .b illustrates the relative short-T 2 signal amplitude S S as a function of α 1 and τ 1 . As τ 1 increases, the relaxation-excitation effect becomes all the more important, leading to a higher short-T 2 theoretical signal. Additionally, the black dashed line indicates the domain where long-T 2 signal suppression can be achieved, and illustrates the dependency of α 2 to τ 1 through the optimization process.

Figure 3

 3 Figure 3 illustrates the signal minima occurrences sensitivity about T L 1 , T L 2 and D. The

Figure 4

 4 Figure4shows the long-T 2 signal behavior of various biological tissues. Theoretical long-T 2 suppressions can rigorously be generated for every tissue, except for the skeletal muscle, most likely due to the relatively low T 2 and apparent diffusion coefficient D. Nonetheless, it can be considered suppressed under the threshold condition S L < 0.05%M 0 in a Gδ range between 97 and 172 mT/m.ms.

Figure 5 . 1 <

 51 Figure 5.a shows the ratio of short-T 2 signals generated in Diff-UTE and IR-UTE. Given the simulation parameters, the ratio reaches values below 1.0 in extreme parameters T S 1 < 262 ms and T S 2 > 0.91 ms, the lowest ratio being equal to 0.96 for T S 1 /T S 2 = 100/1 ms. Figure 5.b shows the same ratio as a function of arbitrary α 1 and α 2 for T S 1 /T S 2 = 300/0.315 ms.

2

  of about 315 µs. In IR-UTE, the appropriate T I (from Equation[START_REF] Wiesinger | Zero TE MR bone imaging in the head[END_REF]) nulling a T L 1 of 590 ms (T I = 11.48 ms) is close to that of a T L 1 of 1000 ms component (T I = 11.53 ms). Hence, the conditions of simulations from Figure 5.a being similar, the ratio of the short-T 2 signals generated in Diff-UTE and IR-UTE is systematically superior to 2.5 with the previously estimated T S 2 for T S 1 ranging from 100 to 1000 ms. As a result and given the operating field in experiments (7 T), T S 1 was further extrapolated from reported value at 105.6 MHz (53) (T 1 ≈ 150 ms) to 300 ms for simulations and parameters optimization purposes.

Figure 7

 7 Figure 7 shows doped-water and Lego brick signal intensities against spoiling gradient moment for four preparation flip angles α 1 , as well as relevant coronal views from the phantom. Maximal contrast between long-and short-T 2 components was obtained for intermediary Gδ amplitudes for suppression-yielding α 1 values of 80 • and 90 • . Signal suppression, if any, is obviously capped to the noise level, as depicted on the graphics. Theoretical and experimental long-relaxing signals tendencies are in good agreement with each other, suggesting

  1000 ms, D = 0.8 µm 2 /ms and T R 1 /T R 2 = 5/25 ms. Signal minima occurences (S L < 0.05%M 0 ) are depicted in the blue bands.

Fig. S3 .

 S3 Fig. S3. S L as a function of α 1 (in degrees) and Gδ (in mT/m.ms), for T L 2 = 25/50/100/1000 ms (columns), and α 2 = 15/30/45/60/75/90 • (rows). Other simulation parameters: T L 1 = 1000 ms, D = 2.0 µm 2 /ms and T R 1 /T R 2 = 5/25 ms. Signal minima occurences (S L < 0.05%M 0 ) are depicted in the blue bands.

Fig. S4 .

 S4 Fig. S4. S L as a function of α 1 (in degrees) and Gδ (in mT/m.ms), for T L 2 = 25/50/100/1000 ms (columns), and α 2 = 15/30/45/60/75/90 • (rows). Other simulation parameters: T L 1 = 1400 ms, D = 0.3 µm 2 /ms and T R 1 /T R 2 = 5/25 ms. Signal minima occurences (S L < 0.05%M 0 ) are depicted in the blue bands.

Fig. S5 .

 S5 Fig. S5. S L as a function of α 1 (in degrees) and Gδ (in mT/m.ms), for T L 2 = 25/50/100/1000 ms (columns), and α 2 = 15/30/45/60/75/90 • (rows). Other simulation parameters: T L 1 = 1400 ms, D = 0.8 µm 2 /ms and T R 1 /T R 2 = 5/25 ms. Signal minima occurences (S L < 0.05%M 0 ) are depicted in the blue bands.

Fig. S6 .

 S6 Fig. S6. S L as a function of α 1 (in degrees) and Gδ (in mT/m.ms), for T L 2 = 25/50/100/1000 ms (columns), and α 2 = 15/30/45/60/75/90 • (rows). Other simulation parameters: T L 1 = 1400 ms, D = 2.0 µm 2 /ms and T R 1 /T R 2 = 5/25 ms. Signal minima occurences (S L < 0.05%M 0 ) are depicted in the blue bands.

Fig. S7 . 2 = 0 . 18 ± 2 = 1 .

 S7201821 Fig. S7. Fittings of the Lego brick experimental signal (circles). Bi-component fitting is depicted in dashed line, mono-component in dotted line, and noise level in crosses. Diff-UTE parameters were: TR = 31.07 ms (T R 1 /T R 2 = 5/25 ms, τ 1 /τ 2 = 1/0.07 ms), α 1 /α 2 = 80.0/57.0 • , Gδ = 159.12 mT/m.ms, and T E = 0.02, 0.03, 0.05, 0.10, 0.15, 0.25, 0.40, 1.10, 1.80 ms. Bi-component estimation yielded T * S,1 2

7

  Effect of the Diff-UTE suppression scheme for four preparation flip angle α 1 values:α 1 = 60 • (a), α 1 = 70 • (b), α 1 = 80 • (c) and α 1 = 90

Figure 1 :

 1 Figure 1: Diff-UTE (a) and IR-UTE (b) pulse sequences. The gradients inducing diffusion weighting in Diff-UTE correspond respectively to the short one after the preparation pulse (relative area Gδ), and to the long one after the readout gradients refocusing (relative area Gδ ).

Figure 2 :

 2 Figure 2: Simulation of long-T 2 signals along preparation flip angle α 1 and spoiling gradient moment Gδ for τ 1 = 1 ms (a), and short-T 2 signals along α 1 and τ 1 (b). In both cases, α 2 was set to its optimized value given α 1 and τ 1 . The dashed line in (b) indicates the lowest α 1 limit value as a function of τ 1 for the proposed optimization scheme to yield a long-T 2 suppression (S L < 0.05%M 0 ), and the white crosses in (a) and (b) relates this aspect for τ 1 = 1 ms. Simulation parameters are T L 1 /T L 2 = 1000/100 ms, T S 1 /T S 2 = 300/0.315 ms, D = 1.00 µm 2 /ms, τ 2 = 0.07 ms, T R 1 /T R 2 = 5/25 ms. Signal minima occurrences are depicted in the blue bands.

Figure 3 :

 3 Figure 3: Simulated long-T 2 signals as functions of spoiling gradient moment and T L 1 ((a) -nominal T L 1 = 1000 ms), T L 2 ((b) -nominal T L 2 = 100 ms) and D ((c) -nominal D = 1.00 µm 2 /ms) deviations. Simulation parameters: T S 1 /T S 2 = 300/0.315 ms, τ 1 /τ 2 = 1/0.07 ms, α 1 /α 2 = 80/57.0 • and T R 1 /T R 2 = 5/25 ms. Signal minima occurrences (S L < 0.05%M 0 ) are depicted in the blue bands.

Figure 4 :

 4 Figure 4: Simulated long-T 2 signals as functions of spoiling gradient moment for biological tissues: skeletal muscle (MSK, dashed line, T 1 /T 2 = 1412/50 ms, D = 0.7 µm 2 /ms), cerebrospinal fluid (CSF, solid line, T 1 /T 2 = 4000/2000 ms, D = 2.2 µm 2 /ms), grey matter (GM, dotted line, T 1 /T 2 = 1820/99 ms, D = 0.8 µm 2 /ms) and white matter (WM, dashed-dotted line, T 1 /T 2 = 1084/69 ms, D = 0.8 µm 2 /ms). Other simulation parameters: T S 1 /T S 2 = 300/0.315 ms, τ 1 /τ 2 = 1/0.07 ms, α 1 /α 2 = 80/57.0 • and T R 1 /T R 2 = 5/25 ms. The horizontal blue line indicates the suppression threshold (S L < 0.05%M 0 ).

Figure 5 :

 5 Figure 5: Simulated short-T 2 signal ratio as a function of T S 1 and T S 2 for the most penalizing case of a preparation flip angle α 1 = 90 • and optimized α 2 (a), and as a function of α 1 and α 2 for T S 1 /T S 2 = 300/0.315 ms (b). The black dashed line in (a) corresponds to the estimated T S 2 = 315 µs value in the Lego brick. The black solid line in (b) corresponds to the optimized α 2 maximizing Eq. [1]. Other simulation parameters are τ 1 /τ 2 = 1/0.07 ms, and T R 1 /T R 2 = 5/25 ms in Diff-UTE, τ inv /τ IR = 8/0.07 ms, and T I/RD = 11.5/11.5 ms in IR-UTE (optimized delays to suppress a component having a T L 1 of 1000 ms).

Figure 6 :

 6 Figure 6: Simulated short-T 2 signals ratio R = (S S Diff-UTE /S S UTE ) × √ 2 as a function of τ 1 and α 1 for respective optimized α 2 flip angle. Simulation parameters are T S 1 /T S 2 = 300/0.315 ms, τ 2 = 0.07 ms and T R 1 /T R 2 = 5/25 ms.

Figure 7 :

 7 Figure 7: Effect of the Diff-UTE suppression scheme for four preparation flip angle α 1 values: α 1 = 60 • (a), α 1 = 70 • (b), α 1 = 80 • (c) and α 1 = 90 • (d) against spoiling gradient moment Gδ. Doped-water (circle) and short-T 2 component (cross) correspond to the mean signal intensities estimated respectively in ROIs A and B drawn in the view (e). The solid line describes the simulated water signal amplitude given the experimental parameters. The noise ROI was drawn away in an artifact-free area (not shown), and its evolution is depicted in bullets. All experimental signals on plots of the different scenarios are normalized to the water signal's intensity corresponding to α 1 = 60• and Gδ = 488 mT/m.ms, the latter being considered to be fully spoiled (the signal evolution being very slow afterwards). Coronal views (e-h) were chosen given relevant spoiling gradient moment values (dashed vertical lines on corresponding plots {a,e}, {b,f}, {c,g}, {d,h}) where the contrast of the short-T 2 component is either negative (e), null (f) or positive (g,h), consecutively to the long relaxing component suppression.

Figure 8 :

 8 Figure 8: Coronal slices acquired in Diff-UTE for τ 1 of 1.0 (a) and 3.0 ms (b) in Diff-UTE, and for T I of 11.5 (c) and 12.3 ms (d) in IR-UTE. The same contrast window levels is used in each view. T R is kept constant in both sequences in each column {a,c} and {b,d}. Experimental SNR ratios of the short-T 2 component generated between Diff-UTE and IR-UTE amount to 1.49 for T R = 31.07 ms, and 2.31 for T R = 33.07 ms.

Figure 9 :

 9 Figure 9: Coronal (top row) and sagittal (bottom row) views of the fixed mouse head in FLASH (a), Diff-UTE (b) and IR-UTE (c). The same dynamic range was used in all four UTE images. The bone ROI used for SNR analysis between Diff-UTE and IR-UTE is in red on the sagittal view of the Diff-UTE column (b). The noise ROI (not shown) was drawn far from the head in an artifact-free area.

  • (d) against spoiling gradient moment Gδ. Doped-water (circle) and short-T 2 component (cross) correspond to the mean signal intensities estimated respectively in ROIs A and B drawn in the view (e). The solid line describes the simulated water signal amplitude given the experimental parameters. The noise ROI was drawn away in an artifact-free area (not shown), and its evolution is depicted in bullets. All experimental signals on plots of the different scenarios are normalized to the water signal's intensity corresponding to α 1 = 60 • and Gδ = 488 mT/m.ms, the latter being considered to be fully spoiled (the signal evolution being very slow afterwards). Coronal views (e-h) were chosen given relevant spoiling gradient moment values (dashed vertical lines on corresponding plots {a,e}, {b,f}, {c,g}, {d,h}) where the contrast of the short-T 2 component is either negative (e), null (f) or positive (g,h), consecutively to the long relaxing component suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8 Coronal slices acquired in Diff-UTE for τ 1 of 1.0 (a) and 3.0 ms (b) in Diff-UTE, and for T I of 11.5 (c) and 12.3 ms (d) in IR-UTE. The same contrast window levels is used in each view. T R is kept constant in both sequences in each column {a,c} and {b,d}. Experimental SNR ratios of the short-T 2 component generated between Diff-UTE and IR-UTE amount to 1.49 for T R = 31.07 ms, and 2.31 for T R = 33.07 ms.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9 Coronal (top row) and sagittal (bottom row) views of the fixed mouse head in FLASH (a), Diff-UTE (b) and IR-UTE (c). The same dynamic range was used in all four UTE images. The bone ROI used for SNR analysis between Diff-UTE and IR-UTE is drawn in red on the sagittal view of the Diff-UTE column (b). The noise ROI (not shown) was drawn far from the head in an artifact-free area. . . . . . . . . . . . . . . 42

Table 1 :

 1 Summary of the components' relevant parameters influencing their magnetization. Symbols in parenthesis, if any, indicate the tendencies of the corresponding parameter that would maximize the short-T 2 component signal in Diff-UTE.

ACKNOWLEDGEMENTS

The authors thank Dr. Arnaud Duchon for providing the fixed mouse head used in experiments, as well as Drs. Laura Harsan and Ericky C. A. Araujo for the careful manuscript reviews.

The first author wishes the manuscript to be considered for the Young Investigator Award program.

Supporting Information

Additional Supporting Information may be found in the online version of this article.

noting M S z (t i ) the longitudinal magnetization after the delay t i , and with

(T R 1 and T R 2 referring to the internal delays after excitations),

), α i and τ i being the respective flip angles and pulse durations. It therefore yields :

providing the normalized transverse magnetization after the excitation pulse in equation [START_REF] Du | Qualitative and quantitative ultrashort echo time (UTE) imaging of cortical bone[END_REF].

List of Figures