Edithe Selwa

Bogdan I Iorga

In silico self-assembly of nanoparticles with applications in drug delivery

published or not. The documents may come

Introduction

During the recent years, nanotechnology has generated a great deal of interest throughout the biological sciences, due to its potential to provide novel approaches to therapeutic and diagnostic drug delivery. Traditional drug formulations are often associated with poor protection of the active compound from the biological environments and a lack of controlled release. [START_REF] Pearson | Smart polymersomes: Formation, characterization and applications[END_REF] In this chapter we will present briefly the computational methods that are currently chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 2 used to investigate the self-assembly process, as well as the drug loading and release from nanoparticles. In the second part, representative in silico studies describing these processes will be presented, with a special emphasis on the techniques that are specific for each step and on the innovative approaches (multiscale simulations, doubly thermoresponsive nanoparticles, simultaneous delivery of multiple drugs, etc.). A more detailed presentation and discussion of some of these studies can be found in a recent review by Thota and Jiang. [START_REF] Thota | Computational amphiphilic materials for drug delivery[END_REF]

Methods

The systems that are to be studied can be represented at different levels of precision, depending on the size and the complexity of the system, the duration of the self-assembly process, etc.

Full-atom (FA) representations allow a description of the system with atomistic details. Among the all-atom force fields available, CHARMM [START_REF] Brooks | CHARMM: The biomolecular simulation program[END_REF] and the associated CHARMM General Force Field (CGenFF), [START_REF] Vanommeslaeghe | CHARMM general force field: A force field for drug-like molecules compatible with the CHARMM all-atom additive biological force fields[END_REF] as well as AMBER [START_REF] Case | The Amber biomolecular simulation programs[END_REF] and GAFF, [START_REF] Wang | Development and testing of a general Amber force field[END_REF] are the most often employed for such molecular dynamics (MD) simulations, together with GROMOS, [START_REF] Scott | The GROMOS biomolecular simulation program package[END_REF] which is a typical united-atom force field. However, the processes involving self-assembly of nanoparticles, drug loading and release occur over a long-time scale, on the order of microseconds or longer. Therefore, these processes are relatively difficult to reproduce by allatom simulations, since these time-consuming approaches are computationally very expensive. However, recent advances in computation algorithms, especially in GPU-based computation, expanded significantly the scope of these simulations.

Coarse-grain (CG) models overcome these limitations by replacing fullatomistic details by lower resolution particles, while retaining most of the chemical specificity. MARTINI [START_REF] Marrink | Coarse grained model for semiquantitative lipid simulations[END_REF][START_REF] Marrink | The MARTINI force field: Coarse grained model for biomolecular simulations[END_REF] is one of the most used CG force field, which can well describe peptides, lipids, membranes, a wide variety of copolymers and other organic molecules. The main limitations of this force field are the rapid freezing and nucleation of water that may be observed at about 290 K, [START_REF] Marrink | Coarse grained model for semiquantitative lipid simulations[END_REF] and the fact that transition of secondary structures cannot be accessed.

Dissipative Particle Dynamics (DPD) [START_REF] Hoogerbrugge | Simulating microscopic h y d r o d y n a m i c phenomena with dissipative particle dynamics[END_REF][START_REF] Koelman | Dynamic simulations of hard-sphere suspensions under steady shear[END_REF][START_REF] Espanol | Statistical mechanics of dissipative particle dynamics[END_REF] is a MD-based approach with higher level CG representation that is commonly applied to simulate drug loading and delivery in large nanoparticles. This technique s i m u l t a n e o u s l y resolves both the molecular and continuum scales and reproduces the hydrodynamic behavior, to examine phase segregation in multicomponent lipid vesicles. [START_REF] Aydin | Bioinspired vesicles encompassing two-tail phospholipids: Self-assembly and phase segregation via implicit solvent coarse-grained molecular dynamics[END_REF] In this way, the simulation length scales that are accessible are much larger than those by conventional MD. The size of dissipative particles can be freely tuned, but it cannot always be controlled with precision. The parameters are not transferable. Energy-preserving dissipative particle dynamics (eDPD) [START_REF] Li | Energyconserving dissipative particle dynamics with temperature-dependent properties[END_REF] i s a n extension of the classical DPD method to non-isothermal fluid systems, predicting the diffusivity and viscosity consistent with experimental data. In this model, each particle's internal energy is explicitly modeled as a degree of freedom.

These methodologies can also be combined to represent the same system, either sequentially [START_REF] Rad-Malekshahi | The supramolecular organization of a peptide-based nanocarrier at high molecular detail[END_REF] (CG or DPD representations are used for the self-assembly step, then converted into a FA representation for more detailed analysis) or simultaneously [START_REF] Mackenzie | Multiscale modeling of drug-polymer nanoparticle assembly identifies parameters influencing drug encapsulation efficiency[END_REF] (multiscale models combining CG or DPD representations for solvent and a FA representation for solute).

Applications

Peptide nanoparticles

Some synthetic polymers possess cell toxicity and therefore are not useful for clinical applications. Ideal carriers for drug delivery must be non-toxic, nonimmunogenic, biocompatible, biodegradable and k i n e t i c a l l y s t a b l e . The amphiphilic peptides possess most of these properties and are therefore increasingly used in drug delivery. [START_REF] Thota | Computational amphiphilic materials for drug delivery[END_REF] By controlling the ratio of hydrophobic and hydrophilic r e s i d u e s i n a m p h i p h i l i c p e p t i d e s , the peptide sequence or the concentration in solution, v a r i o u s morphologies such as vesicles, micelles, fibers, and hydrogels can be obtained by self-assembly. Typically, their sizes are below 100 nm, small enough to avoid an immune reaction from the human body. [START_REF] Thota | Computational amphiphilic materials for drug delivery[END_REF][START_REF] Sun | Self-assembled peptide nanomaterials for biomedical applications: Promises and pitfalls[END_REF]

Short peptides

Extensive CGMD simulations of diphenylalanine (FF) peptides assembly showed the formation of nanovesicles and nanotubes as the main products, together with new ordered planar bilayers and a rich variety of other vesicle-like structures including toroid, ellipsoid, discoid, and pot-shaped vesicles (Figure 1). The assembly pathways are concentration-dependent, the fusion of small vesicles and bilayers being observed at low concentration, whereas at high concentration the formation of a bilayer seems to be favored. Subsequent allatom MD simulations on a fine-grained structure of a spherical vesicle from the CGMD trajectory shows that FF nanostructures are stabilized by a combination of T-shaped aromatic stacking, interpeptide head-to-tail hydrogen-bonding, and peptide water hydrogen-bonding interactions. 18 For clarity, a close-up of one dipeptide together with its atomistic structure is shown in the top-left corner. In the atomistic structure, carbon atoms are in cyan, oxygen atoms in red, hydrogen atom in white, and nitrogen atom in blue. Reproduced from reference [START_REF] Guo | Probing the self-assembly mechanism of diphenylalanine-based peptide nanovesicles and nanotubes[END_REF] .

In a complementary study, all-atom MD simulations in explicit water were performed on both zwitterionic and capped (uncharged) versions of the FF peptide to understand the early stages of self-assembly. The amphiphilic nature of FF seems to be the key for understanding its self-assembly, and the early precursors to nanotube structures are likely to involve substantial hydrophobic clustering, rather than hexamer ring motifs. [START_REF] Jeon | Molecular insights into diphenylalanine nanotube assembly: All-atom simulations of oligomerization[END_REF] Similarly, detailed atomistic MD simulations of FF and dialanine (AA) peptides were carried out in two solvents, water and methanol. An important self-assembling propensity of FF in water was observed, while in methanol this propensity was very weak. For AA the chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 5 tendencies are the same, but the differences between the two solvents are substantially smaller. [START_REF] Rissanou | Effect of solvent on the self-assembly of dialanine and diphenylalanine peptides[END_REF] Guo et al. d e s c r i b e d a s t u d y o f t r i p h e n y l a l a n i n e ( F F F ) p e p t i d e s s e l fassembly using CGMD simulations, showing that solid nanometer-sized nanospheres and nanorods with substantial β-sheet content are formed. In contrast with the formation mechanism of water-filled FF nanovesicles and nanotubes, intermediate bilayers were not observed during the self-assembly process of FFF nanospheres and nanorods. [START_REF] Guo | Triphenylalanine peptides selfassemble into nanospheres and nanorods that are different from the nanovesicles and nanotubes formed by diphenylalanine peptides[END_REF] The ability to predict the aggregation propensity (AP) of short peptides was evaluated by the screening of 400 dipeptide c o m b i n a t i o n s using CGMD simulations, in order to determine the best precursors for self-assembly. Systems with the best scores were selected for longer simulations, and overall the results are in good agreement with data from literature. [START_REF] Frederix | Virtual screening for dipeptide aggregation: Toward predictive tools for peptide self-assembly[END_REF] An extension of this work on a dataset containing 8,000 tripeptides indicated only a weak correlation between hydrophilicity and AP, confirming that self-assembly propensity is not simply a measure of hydrophobicity. The analysis of these results provided a set of rules for the design of tripeptides with improved self-assembly propensities: aromatic amino acids are most favorable in positions 2 and 3 in a tripeptide, and positive and hydrogen-bonding residues favor p o s i t i o n 1 ( N -terminus) and negative residues position 3 (C-terminus). Following these rules, a number of peptides were synthesized and characterized, allowing the identification of the first unprotected full-L tripeptides (KYF, KYY, KFF and KYW) able to form a hydrogel at neutral pH. [START_REF] Frederix | Exploring the sequence space for (tri-)peptide self-assembly to design and discover new hydrogels[END_REF]

Long peptides

The self-assembly of medium length amphiphilic peptides with the generic structures F m D n and F m K n was investigated by μs-scale CGMD simulations. The most interesting results were obtained with F 3 K 4 and F 6 K 8 , which formed quasispherical micelles with distinct core/shell structure. This result suggests that the optimal ratio of hydrophobic/hydrophilic residues is 3/4. Increasing this ratio led to elliptical aggregates, whereas decreasing this ratio resulted in smaller and less stable micelles. [START_REF] Thota | Molecular insights into the self-assembly of short amphiphilic peptides F m D n and F m K n[END_REF] Three ionic peptides EAK16, containing the same 16 residues but different amino acid sequences, were studied by CGMD simulations in bulk solution or in the presence of a hydrophobic surface at different pH values. In bulk solution, two peptides showed aggregation into fibrillar structures at all pH ranges. The third one formed globular assemblies around the neutral pH and fibrillar aggregates in low and high pH conditions. However, on the hydrophobic surface, this last peptide formed disc-shaped assemblies under neutral pH conditions. [START_REF] Emamyari | pH-dependent self-assembly of EAK16 peptides in the presence of a hydrophobic surface: Coarse-grained molecular dynamics simulation[END_REF] chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 6 CGMD simulations were used to explore the self-assembling behavior of a FA32 amphiphilic peptide, (AF) 6 H 5 K 15 . The formation of micelles is observed, with Ala and Phe residues in the hydrophobic core, Lys in the hydrophilic shell, and amphiphilic His at the interface. The self-assembly is a three-step process, involving the formation of small clusters, their aggregation and merging into large clusters, and finally the formation of micelles. Higher p e p t i d e concentrations induces the formation of bigger micelles and a change in their shape, from quasi-spherical into elongated forms. [START_REF] Thota | Self-assembly of amphiphilic peptide (AF) 6 H 5 K 15 : Coarse-grained molecular dynamics simulation[END_REF] A follow-up study from the same group provided more detailed information about the role of hydrophilic and hydrophobic residues in the self-assembly of FA32-like peptides. The selfassembly capability is reduced by increasing hydrophilicity, with the formation of smaller micelles or the presence of individual peptide chains, whereas increasing hydrophobicity leads to an increase in micelle size and to a morphology shift from spherical to fiber-like. [START_REF] Thota | Self-assembly of amphiphilic peptide (AF) 6 H 5 K 15 derivatives: Roles of hydrophilic and hydrophobic residues[END_REF] The loading and the release of ibuprofen (IBU) in FA32 and its derivatives F 12 H 5 K 15 and F 16 H 5 K 15 was studied using CGMD simulations. IBU loading in FA32 led to quasi-spherical core/shell structured micelles, where IBU was predominantly located at the core along with Phe and Ala, while Lys and His were in the shell and at the core/shell interface, respectively. After loading, due to increased hydrophobic interactions, the micelles become l a r g e r , proportionally with the increase in the ratio drug/peptide. The morphology of IBU-loaded nanoparticles depends on the type of peptide and has a significant effect on IBU release profile. In FA32 derivatives, IBU loading led to different morphologies, a w e l l -structured nanofiber being f o r m e d i n F 16 H 5 K 15 . IBU release from FA32 micelles upon pH change is slower than from F 16 H 5 K 15 nanofiber, suggesting that the former might be more appropriate for the controlled release. [START_REF] Thota | Ibuprofen loading and release in amphiphilic peptide FA32 and its derivatives: A coarse-grained molecular dynamics simulation study[END_REF] The amphiphilic self-assembling decapeptide 2 (SA2 peptide, sequence Ac-AAVVLLLWEE-COOH) was shown to spontaneously self-organize into hollow spheres. CGMD s i m u l a t i o n s w e r e u s e d t o g e t m o r e i n s i g h t i n t o t h e supramolecular structure of SA2 nanocarriers and the intermediates involved in the self-assembly process. The coarse-grained system was then converted into an atomistic representation, with a system containing 41 millions atoms. Subsequent FAMD simulations provided a more accurate knowledge on the atomic level geometry and dynamics of assembled peptides in relation to the supramolecular organization of the SA2 nanocarrier. It was shown that this organization involves interdigitated antiparallel β-sheets and not the liposomelike tail-to-tail orientation that is generally observed for t h e a m p h i p h i l i c molecules. [START_REF] Rad-Malekshahi | The supramolecular organization of a peptide-based nanocarrier at high molecular detail[END_REF] MD simulations were carried out to provide insight into the effect of secondary structure on the self-assembly of β-peptides u s i n g h y d r o p h o b i c blocks (composed of 10-mer or 20-mer of β-valine residues) and hydrophilic blocks (composed of 10-mer or 20-mer of positively charged 2-aminomethylchapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 7 3,3-dimethyl β-amino acid residues). In a good agreement with experimental data, all-atom free energy calculations showed that the association of a pair of homochiral hydrophobic blocks promotes self-assembly compared to the same hydrophobic blocks in a heterochiral form. CGMD simulations of mixtures containing hydrophobic and hydrophilic blocks showed that these molecules spontaneously form spherical micelles. [START_REF] Mondal | Effect of secondary structure on the self-assembly of amphiphilic molecules: A multiscale simulation study[END_REF] A mixture of 15 and 23-residue branched, amphiphilic peptides that mimic phosphoglycerides in molecular architecture was shown to produce supramolecular self-assembly and form solvent-filled vesicles with 50-200 nm diameters. In contrast, CGMD simulations of the same system produced a welldefined bilayer. [START_REF] Gudlur | Peptide nanovesicles formed by the selfassembly of branched amphiphilic peptides[END_REF] Further theoretical investigations would be needed, using bigger systems and possibly different peptide/water ratio, to determine the optimal conditions for obtaining an in silico model of the vesicles.

Atomistic MD simulations of an amphiphilic peptide with the sequence SLSLAAAEIKVAV [START_REF] Lee | Atomistic molecular dynamics simulations of peptide amphiphile self-assembly into cylindrical nanofibers[END_REF] a n d t w o s i m p l i f i e d v a r i a n t s ( V 2 A 4 E 3 a n d V 4 A 2 E 3 ) [START_REF] Lee | Molecular dynamics simulation of βsheet formation in self-assembled peptide amphiphile fibers[END_REF] showed a self-assembly process resulting in the formation of cylindrical micelles and nanofibers. In a subsequent study, the self-assembly of the same peptide was investigated using CGMD simulations. A three-step mechanism was evidenced in the formation of nanofibers: i) formation of spherical micelles, ii) their aggregation into a three-dimensional network, and finally iii) the formation of a cylindrical nanofiber (Figure 2). [START_REF] Lee | Modeling the self-assembly of peptide amphiphiles into fibers using coarse-grained molecular dynamics[END_REF] A c a r e f u l a n a l y s i s o f t h e f r e e -energy landscape for this process showed the formation of a pillar-like intermediate state before the final cylindrical semifiber structure. Additionally, the head group assembly was found to occur on a much shorter time scale than tail group assembly. Although CGMD simulations might not be precise enough to draw such conclusions, it was proposed that the driving force leading to the formation of the micelle structure is enthalpic, originating from several factors, including the conformational energy of amphiphilic peptides and the electrostatic and van der Waals interaction energy between solvent molecules and between solvent and peptides. [START_REF] Yu | Free-energy landscape for peptide amphiphile selfassembly: Stepwise versus continuous assembly mechanisms[END_REF][START_REF] Yu | Free energy profile and mechanism of self-assembly of peptide amphiphiles based on a collective assembly coordinate[END_REF] Figure 2. Coarse-grained molecular dynamics simulations showing selfassembly of peptide amphiphiles into spherical micelles, a three-dimensional network of micelles, and finally a cylindrical micelle fiber. Reproduced from reference [START_REF] Lee | Modeling the self-assembly of peptide amphiphiles into fibers using coarse-grained molecular dynamics[END_REF] .

chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 8

Amphiphilic copolymer nanoparticles

Amphiphilic polymeric nanostructures have long been well-recognized as an excellent candidate for drug delivery applications. With the recent advances in the "top-down" and "bottom-up" approaches, development of well-defined polymeric nanostructures of different shapes has been possible. [START_REF] Venkataraman | The effects of polymeric nanostructure shape on drug delivery[END_REF] Under appropriate conditions, the amphiphilic copolymers can assemble into a variety of morphologies (spheres, cylinders, bicontinuous structures, lamellae, vesicles, micelles, fibers, and many other complex or hierarchical assemblies). These morphologies may be affected by polymer concentration, hydrophobic/hydrophilic ratio, end-group, and solvent content.

An interesting multiscale (dual resolution) approach that combines atomistic (GROMOS96) and coarse-grain (MARTINI) force fields was used to simulate the drug-polymer nanoparticle assembly by nano-precipitation from mixed solvents. The interaction between three poly(glycerol adipate) (PGA) polymer variants and the anticancer drug dexamethasone phosphate (DXMP) was studied (Fi g u re 3), and the relative e n c a p su l a t i o n e ffi c i e n c y a n d dru g loading retrieved for t h e PGA polymers were in agreement with the experimental data. The water and acetone co-solvents were modeled as purely coarse-grained entities, whereas PGA and DXMP were modeled at both atomistic and coarse-grained levels, using virtual sites. In this way, all solutesolute interactions were treated fully atomistically, and the solute-solvent and solvent-solvent interactions were t r e a t e d a t t h e c o a r s e -grain level. The simulations results showed that a balance between hydrophobic and hydrophilic moieties in the polymer is important for optimal drug incorporation. The C18 chains are required to interact favorably with DXMP, but having too many C18 chains decreases solubility, preventing the polymer from interacting with surrounding DXMP molecules. [START_REF] Mackenzie | Multiscale modeling of drug-polymer nanoparticle assembly identifies parameters influencing drug encapsulation efficiency[END_REF]

Figure 3. Snapshots taken during the 80 ns multiscale simulation of PGA20-co-C18PGA80 with DXMP: (A) 5 ns (with acetone shown), (B) 5 ns, (C) 30 ns, (D) 50 ns, (E) 80 ns (with acetone shown), and (F) 80 ns. DXMP (multi), PGA (white with C18 chains yellow), acetone (blue). For clarity, water and coarse-

grain virtual sites are not shown. Reproduced from reference [START_REF] Mackenzie | Multiscale modeling of drug-polymer nanoparticle assembly identifies parameters influencing drug encapsulation efficiency[END_REF] .

CGMD simulations were used to explore the self-assembly process of polyamidoamine (PAMAM) amphiphilic dendrimers (ADs) with different alkyl chains and generations, and the interaction of the resulting aggregates with siRNA. These simulations evidenced t h a t h y d r o p h o b i c t a i l l e n g t h a n d hydrophilic head size, or dendron generation, have a critical impact on the size, shape, and self-organization of the aggregates, as well as on interactions with nucleic acids. [START_REF] Márquez-Miranda | Self-assembly of amphiphilic dendrimers: The role of generation and alkyl chain length in siRNA interaction[END_REF] A method, based on all-atom MD simulations using the AMBER/GAFF force field, [START_REF] Case | The Amber biomolecular simulation programs[END_REF][START_REF] Wang | Development and testing of a general Amber force field[END_REF] was developed for the calculation of Flory-Huggins interaction parameters as a measure for the loading capability (solubility) of lipophilic drugs into polymer micelles. Very good agreement with experimental data was observed for four different drugs with significantly different lipophilicities. [START_REF] Kasimova | Validation of a novel molecular dynamics simulation approach for lipophilic drug incorporation into polymer micelles[END_REF] In a similar work, FAMD simulations allowed the prediction of solubilities for t

w o d r u g s , f e n o f i b r a t e a n d n i m o d i p i n e , in a series of micelle-forming poly(ethylene oxide)-b-poly(ε-caprolactone) (PEO-b-PCL) b l o c k c o p o l y m e r s with combinations of blocks having different molecular weights. The results

were in a good agreement with experimental data. [START_REF] Patel | Application of molecular dynamics simulation to predict the compatability between water-insoluble drugs and self-associating poly(ethylene oxide)-b-poly(ε-caprolactone) block copolymers[END_REF] The same protocol applied for two water insoluble anticancer drugs, cucurbitacin B (CuB) and cucurbitacin I (CuI), showed that in both cases the drug affinity increases with increasing PCL/PEO ratio. [START_REF] Patel | Roles of nonpolar and polar intermolecular interactions in the improvement of the drug loading capacity of PEO-b-PCL with increasing PCL content for two hydrophobic cucurbitacin drugs[END_REF] When the branched PEO-b-3PCL block copolymer was used, the solubility of cucurbitacin drugs CuB and CuI increased considerably chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 10 compared to PEO-b-PCL, whereas the opposite behavior was observed in the case of the fenofibrate and nimodipine molecules. [START_REF] Patel | Molecular dynamics study of the encapsulation capability of a PCL-PEO based block copolymer for hydrophobic drugs with different spatial distributions of hydrogen bond donors and acceptors[END_REF] Diffusion of the hydrophobic drug CuB in linear PEO-b-PCL and branched PEO-b-3PCL was studied using FAMD simulations in conjunction with different amounts of water. It was found that the drug diffusivity is insensitive to the water amount and double in linear PEO-b-PCL compared with those in the branched PEO-b-3PCL, due to a cumulative effect of stronger hydrogen bonding and lower swelling. [START_REF] Razavilar | Molecular dynamics study of the diffusivity of a hydrophobic drug cucurbitacin B in pseudo-poly(ethylene oxide-bcaprolactone) micelle environments[END_REF] Subtle structural characteristics (size, flexibility, rigidity) play a significant role in the efficiency of drug loading. A computational model to predict the affinity between the drug and its nanoparticulate carrier, e.g. D T O -SA/5K nanospheres, was developed by combining FAMD simulations with docking calculations. The model was able to successfully rank order t h r e e d r u g s (nutraceutical curcumin, anti-cancer drug paclitaxel and pre-hormone vitamin D3) in terms of experimentally observed drug loading at maximum into the nanospheres, by comparing it with computationally calculated binding energies. [START_REF] Costache | Polymerdrug interactions in tyrosine-derived triblock copolymer nanospheres: A computational modeling approach[END_REF] DPD simulations were used t o get more insight into t h e structural organization and properties of pure paclitaxel and paclitaxel-loaded PEO-b-PLA systems. Alone, paclitaxel molecules formed ribbon or fiber-like structures in water. With the addition of the copolymer block, paclitaxel acted as a template and polymer molecules assembled around the paclitaxel structure to form core/shell structured fibers having a paclitaxel/PLA core and a PEO shell. [START_REF] Guo | Computational studies on self-assembled paclitaxel structures: Templates for hierarchical block copolymer assemblies and sustained drug release[END_REF] DPD s i m u l a t i o n s w e re also carried out by the same group to study drug diffusion abilities into the core of micelles after the core-shell structure of the micelle is formed, which is the main issue affecting the drug loading efficiency. Important parameters such as topological structure of the drug, the hydrophobic block length of the polymer, and the compatibility between the drug and the hydrophobic block were shown to have significant effects on drug loading efficiencies and drug distributions inside micelles. [START_REF] Guo | Can drug molecules diffuse into the core of micelles?[END_REF] 47-48 micelles. More advanced in silico studies tried to rationalize the relationship between the structure of pH-sensitive polymers and drug release behaviors from the micelles, s h o w ing that the "fireworks-like" three layer structure is very important for the drug release, and different drug distributions (e.g. i) in the pH-sensitive layer, ii) both in the pHsensitive layer and in the core, or iii) in the core of micelles) generally induce different drug release processes. [START_REF] Nie | Drug release from pH-sensitive polymeric micelles with different drug distributions: Insight from coarse-grained simulations[END_REF] In some cases, a loading-dependent morphology transition can be observed, suggesting the maximum payload that the considered copolymer nanocarrier can accommodate under the selected conditions. In a representative study, the selfassembly of (D-L)-PLA/PEO di/triblock copolymers in aqueous environment and in the presence of a model drug (nifedipine) was investigated, based on a hybrid molecular simulation recipe. Atomistic MD simulations were used to obtain DPD input parameters, and this mesoscale technique was employed to derive the entire phase diagrams for these systems. It was observed that spherical aggregates are formed at low drug concentrations, which rearrange their assembly into cylindrical or eventually membrane-like aggregate as the payload amount increases. [START_REF] Posocco | Morphology prediction of block copolymers for drug delivery by mesoscale simulations[END_REF] In another multiscale in silico study, FAMD simulations were used to estimate the Flory-Huggins interaction parameters and miscibility of binary components from a system representing the anti-cancer drug camptothecin (CPT) in a pH-sensitive amphiphilic copolymer, composed of hydrophobic poly(β-amino ester) (PAE) and hydrophilic methyl ether-capped poly(ethylene glycol) (PEG). DPD simulations were then employed to examine the micellization of PAE-PEG, CPT loading in PAE-PEG, and CPT release upon protonation from micelles/vesicles by a swelling-demicellization-releasing mechanism. [START_REF] Luo | pH-sensitive drug loading/releasing in amphiphilic copolymer PAE-PEG: integrating molecular dynamics and dissipative particle dynamics simulations[END_REF] An interesting multiscale approach was used to investigate the selfassembly of telodendrimer-based nanocarriers for anticancer drug delivery. CGMD simulations were carried out to represent the self-assembly process, followed by reverse mapping of the coarse-grained system to atomistic representation for structural analysis. This procedure provided important parameters like average size, drug-telodendrimer micellar stoichiometry, optimal drug loading capacity, and atomistic details such hydrogen-bonding and solvent accessible area of the nanocarrier. The computed results were in good agreement with the experimental data. [START_REF] Jiang | Multiscale approach to investigate selfassembly of telodendrimer based nanocarriers for anticancer drug delivery[END_REF] Recent findings showed that the nanoparticle shape can be an important factor for modulating the efficacy of drug delivery process. Flexible, worm-like nanocarriers (Worms) increase the amount of drug delivered to tumors and shrink the tumors more effectively than spherical micelles (Spheres). A rational coarse grain (rCG) model based on all-atom MD simulations was built and used to investigate the shape-dependent effects in the delivery of the anticancer drug Taxol by block copolymer micelles. In this way, it was evidenced that twice as much drug loads into Worms than Spheres, fully consistent with experiments. [START_REF] Loverde | Nanoparticle shape improves delivery: rational coarse grain molecular dynamics (rCG-MD) of taxol in worm-like PEG-PCL micelles[END_REF] The mechanism of transportation and delivery of hydrophilic drug components such as anticancer doxorubicin and mitoxantrone are of highest importance and poorly understood. CGMD simulations were employed to study how "patchy" polymeric micelle can assist the transportation of a hydrophilic drug across a lipid membrane and deliver it inside a biological lipid vesicle. Extensive simulations demonstrated successful encapsulation, transportation of the hydrophilic drug b y "patchy" p o l y m e r i c m i c e l l e s a n d t h e r e l e a s e o f a significant portion of drug inside the lipid vesicle (Figure 4). [START_REF] Srinivas | Polymer micelle assisted transport and delivery of model hydrophilic components inside a biological lipid vesicle: A coarse-grain simulation study[END_REF] chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 12 The interaction of Pluronic L64 block copolymer micelles with DPPC lipid bilayers was investigated using CGMD simulations showing, in agreement with experimental data, the release of triblock chains from the micelle inside the bilayer. The presence of a drug molecule inside the hydrophobic core of the micelle has a strong influence on this process, the micelle stability being the result of a complex interplay of drug-core and block-co-polymer-bilayer interactions modulating the structures of both the micelle and the bilayer. [START_REF] De Nicola | Micellar drug nanocarriers and biomembranes: How do they interact?[END_REF] A recent study investigated t h e b e h a v i o r o f d o u b l y t h e r m o r e s p o n s i v e micelles and vesicles through computer simulations, u s i n g the recentlydeveloped eDPD methodology. [START_REF] Li | Energyconserving dissipative particle dynamics with temperature-dependent properties[END_REF] Mesoscopic simulations were used to simulate chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 13 the thermally induced self-assembly of LCST-UCST diblock copolymers, process triggered by the evolution of temperature over time and space. Both external and intrinsic factors c a n i n f l u e n c e t h e a s s e m b l y p a t h w a y o f thermoresponsive micelles and thus determining the final morphology of the aggregate. It was found that thermoresponsive unilamellar vesicles can sustain repeated heating-cooling cycles in a thermal loading test, and the collapse probability and half-life of the vesicles under frequencies that cause vesicle destruction were quantified. The thermoresponsive bilayer membranes exhibit a very high ratio of molecular migration between the two leaflets during thermally induced inversion, process dominated by two molecular movement modes, flip and slip (Figure 5). [START_REF] Tang | Non-equilibrium dynamics of vesicles and micelles by self-assembly of block copolymers with double thermoresponsivity[END_REF]

Figure 5. Thermoresponsive vesicles invert by diffusion and respond differently

to various thermal loading frequencies. A vesicle formed by L2N5U2 thermoresponsive block copolymer can invert repeatedly when subjected to thermal loading cycles and may collapse irreversibly. Adapted from reference [START_REF] Tang | Non-equilibrium dynamics of vesicles and micelles by self-assembly of block copolymers with double thermoresponsivity[END_REF] .

Lipid nanoparticles

Implicit solvent CGMD s i m u l a t i o n s w e r e u s e d t o i n v e s t i g a t e t h e s e l fassembly and phase segregation of stable single and binary vesicles composed of two-tail phospholipids. It was shown that the degree of dissimilarity between the tail groups of the lipid species composing the vesicle can be used to tune the phase segregation in the vesicle bilayer. Accordingly, the interfacial tension increases with the difference between the tail groups, driving the phase segregation process to minimize the interfacial tension. [START_REF] Aydin | Bioinspired vesicles encompassing two-tail phospholipids: Self-assembly and phase segregation via implicit solvent coarse-grained molecular dynamics[END_REF] A coarse-grained DPPC liposome model was loaded with varying amounts of hypericin, a natural compound proposed for use in photodynamic therapy. In chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 14 agreement with experimental data, CGMD simulations performed on t h e microsecond time scale of liposomes with varying hypericin concentration showed that the liposome expands as the amount of hypericin is increased. These simulations also provided valuable information about the distribution of hypericins, their orientations within the lipid bilayer, and the potential of mean force for transferring a hypericin molecule from the interior aqueous "droplet" through the liposome bilayer (Figure 6).

Hybrid nanoparticles

Using a novel coarse-grained peptide/polymer model, the spontaneous selfassembly process of a hybrid, lipid-like peptide amphiphile (palmitoyl-V 3 A 3 E 3 ) was studied using large-scale CGMD simulations. The formation of nanostructures of various sizes and shapes is observed as a function of the electrostatics and temperature. At optimal conditions, the self-assembly process starts by a quick micellization whose driving force is the hydrophobic interactions between alkyl tails, followed by a slow ordering process of neighboring peptide residues within a micelle and the formation of β-sheets exposing the hydrophobic core. Finally, spherical micelles merge together chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 15 through an end-to-end mechanism to form cylindrical nanofibers. [START_REF] Fu | The role of electrostatics and temperature on morphological transitions of hydrogel nanostructures self-assembled by peptide amphiphiles via molecular dynamics simulations[END_REF] T h e assembly process was found to occur through multiple steps with transient intermediates that ultimately leads to the stabilized nanostructures including open networks of β-sheets, cylindrical nanofibers, and elongated micelles. [START_REF] Fu | Solvent effects on kinetic mechanisms of self-assembly by peptide amphiphiles via molecular dynamics simulations[END_REF] Further work showed the important role of an increased hydrophobicity, which favors the progression from open networks of secondary structures toward closed cylindrical nanostructures containing either β-sheets or random coils. [START_REF] Fu | Role of hydrophobicity on self-assembly by peptide amphiphiles via molecular dynamics simulations[END_REF] DPD simulations are generally the most appropriate tools for studying the loading and release of drugs from nanoparticles. In a representative study, micelles self-assembled from cholesterol-conjugated His 10 Arg 10 (HR20-Chol) were used for the encapsulation of doxorubicin (DOX) at different pH conditions. Stronger DOX loading ability was observed at pH>6.0, due to the hydrophobicity of histidine residues. At pH<6.0, the dense structures of micelles adopted swollen conformations, thus facilitating the release of DOX from the core of micelles. [START_REF] Guo | Dissipative particle dynamics studies on microstructure of pH-sensitive micelles for sustained drug delivery[END_REF] Further atomistic simulations were performed to study the compatibility between each pair of components and to calculate the Flory-Huggins interaction parameters between water, DOX, and other components in this system. [START_REF] Guo | Systematic multiscale method for studying the structure-performance relationship of drug-delivery systems[END_REF] A similar study was carried out later using doxorubicin-loaded micelles prepared from block polymers His(x)Lys10 (x = 0, 5, 10) conjugated with docosahexaenoic acid. [START_REF] Wang | Mesoscale simulations and experimental studies of pH-sensitive micelles for controlled drug delivery[END_REF] The nanoparticles can accommodate simultaneously several drugs and release them specifically in different experimental conditions, as shown in a recent study in which a dual-drug-loaded micelle was designed and constructed by self-assembly from a mixture of poly(propylene oxide)-b-poly(γ-benzyl-Lglutamate)-b-poly(ethylene glycol) (PPO-b-PBLG-b-PEG) triblock terpolymers and two model drugs, doxorubicin (DOX) and naproxen (Nap). In the micelles, the DOX is covalently linked to PBLG through an acid-cleavable hydrazone bond, whereas the Nap is physically encapsulated in the cores. DPD simulations were used for studying the drug loading and releasing behaviors of these micelles (with one or two drugs loaded simultaneously), which revealed that the micelles possess a core-shell-corona structure in which the PPO/Nap, PBLG/DOX, and PEG aggregate to form t h e c o r e , s h e l l , a n d c o r o n a , respectively. The release of DOX and Nap is pH-and thermosensitive and these dual-drug-loaded micelles were shown t o b e b i o c o m p a t i b l e a t n o r m a l physiological conditions. [START_REF] Chen | Polypeptide-based "smart" micelles for dual-drug delivery: A combination study of experiments and simulations[END_REF] In an elegant study, DPD simulations were used to investigate the selfassembly process of core-polyethylene glycol-lipid shell (CPLS) nanoparticles and their potential as drug delivery vehicles. The spherical core was functionalized on surface by PEG chains with the free ends covalently bonded with lipid molecules (lipid heads). The simulations showed the formation of a complete lipid bilayer shell at the surface of the PEGylated particle core upon addition of free lipids, leading to the formation of a CPLS nanoparticle. In order to evaluate the potential of C P L S n a n o p a r t i c l e s a s d r u g c a r r i e r s , t h e chapter-in-silico-self-assembly-revised-v1-ACS_modified1.doc Printed 06/08/y 16 internalization of nanoparticles (unloaded or loaded with hydrophilic molecules) into liposomes (lipid vesicles) was studied through DPD simulations. During this process, the rearrangement of the lipid bilayer around nanoparticles was evidenced, and this represents the most important step for the nanoparticle internalization via endocytosis into diseased cells. Overall, this work highlights the significant potential of CPLS nanoparticles as drug carriers with high efficacy and non-toxicity, as their surface chemistry is identical to liposome yet their stability is superior. [START_REF] Shen | Selfassembly of core-polyethylene glycol-lipid shell (CPLS) nanoparticles and their potential as drug delivery vehicles[END_REF]

Figure 1 .

 1 Figure 1. Initial disordered state and seven representative well-organized diphenylalanine peptide assemblies generated in MD simulations. For each nanostructure, a side view (top row), a cross-sectional view through its center of mass and the axis parallel to its principal axis (middle row), and a top view (bottom row) are shown. The main-chain beads are in red, and the side-chain beads are in white. There are three families of structures: vesicles with different shapes: (a) toroid, (b) ellipsoid, (c) pot-shaped, (d) discoid, and (e) spherical vesicles; nanotube (f); and bilayer (g). For clarity, a close-up of one dipeptide together with its atomistic structure is shown in the top-left corner. In the atomistic structure, carbon atoms are in cyan, oxygen atoms in red, hydrogen atom in white, and nitrogen atom in blue. Reproduced from reference 18 .

  Similar results were obtained from the investigation of d o x o r u b i c i n (DOX) l o a d i n g a n d p Hcontrolled release from pH-sensitive 4AS-PCL-b-PDEAE-MA-b-PPEGMA 46 and (PCL) 2 (PDEA-b-PPEGMA) 2

Figure 4 .

 4 Figure 4. CGMD simulation snapshots of a binary AB+CB polymeric micelle interaction with DPPC lipid vesicle are shown. Polymeric patches in the micelle are shown in blue along with hydrophobic (cyan) and hydrophilic (yellow) polymers. One of the CG beads in the lipid headgroup is shown in bead representation, while the rest of the lipid is drawn in stick representation. Water is not shown for clarity. The initially separated polymeric micelle (at 0 ns) gradually penetrates the lipid vesicle (400 ns). The cross-sectional view of the final snapshot is shown in part H. Reproduced from reference 54 .

57

 57

Figure 6 .

 6 Figure 6. (a) Final snapshots from the 10 μs molecular dynamics simulations of liposomes (red and green beads correspond to hydrophilic entities and gray beads to hydrophobic ones) loaded with hypericin molecules (in yellow). (b) Snapshots with some lipids removed in order to illustrate the binding of hypericin; the hydrophilic parts of hypericin are shown in red and the hydrophobic parts in yellow. Reproduced from reference 57 .

Acknowledgments

Our laboratory is a member of the Laboratory of Excellence in Research on Medication and Innovative Therapeutics (LERMIT) supported by a grant from the French National Research Agency (ANR-10-LABX-33).


Tania S Vergara Gomez

Marc Dubois

Kaizad Rustomji

Elodie Georget

Tryfon Antonakakis

Stanislas Rapacchi

Frank Kober

Stefan Enoch

Redha Abdeddaim

Passive RF shimming with fractal metasurfaces in a 7T head coil

B 1

+ inhomogeneities in the temporal lobes of the brain are common while using a birdcage coil at 7T. Different approaches have effectively targeted this problem, at the cost of decreasing the B 1 + field in other important areas or being uncomfortable to the patient. Here, we present a new approach based on two resonant fractal metasurfaces. These fractal resonators improve simultaneously the B 1 + field in each temporal lobe without affecting the B 1 + field in other parts of the brain.

Introduction

Passive RF shimming approaches such as dielectric pads [1][2][3] and metamaterials 4,[START_REF] Dubois | Proc. Intl. Soc. Mag. Reson. Med[END_REF] have been used to reduce the B 1 + inhomogeneity observed in birdcage head coils working at 7T. While these approaches locally improve B 1 + amplitude in regions of interest like temporal lobes, such improvements come at the cost of reducing the RF field in other ROIs like the cerebellum or being limited to use a single element. In this work, we present two passive RF shimming resonators for improving the B 1 + field amplitude in each temporal lobe without diminishing the transmit efficiency of the birdcage coil elsewhere.

Material and Methods

The resonant fractal metasurfaces are copper structures printed on a 0.4-mm-thick FR-4 substrate. The design of each resonant fractal was based on 4 th -order Hilbert fractal 6 metallic strips as shown in Figure 1. Each fractal was placed on each side of the head, within the receive array. The fractal on the right side being slightly higher than the left one. The fractals were tested with a 1Tx/32Rx proton head coil (Nova Medical, Wilmington, MA, USA) in a 7T Magnetom MRI scanner (Siemens Healthineers, Erlangen, Germany) on a head specific anthropomorphic phantom (SPEAG, Zurich, Switzerland) filled with a liquid of permittivity ɛ= 45.3 and conductivity σ= 0.87 S/m.

We measured Flip Angle (FA) maps and amplitude maps using the XFL MRI sequence [START_REF] Amadon | Proc. Intl. Soc. Mag. Reson. Med[END_REF] in sagittal and axial orientation with TR= 20 s, TE= 1.4 ms, FA= 8°, Shinnar-Leroux minimum-phase saturation pulse= 90°, FOV= 256 mm, matrix= 64x64 and 12 slices of thickness=8 mm (17 slices in axial orientation). A mask extracted from GRE images was applied to the FA maps. The relative FA and amplitude signal gain expressed in percentage were computed for a quantitative analysis of the fractals' effect. All images were acquired using the same reference voltage of 260V.

Results

Flip angle maps in sagittal and axial orientations of the birdcage coil, alone (top row) and with the fractals (central row), are shown in Figure 2. The effect of the fractals can be seen at different locations in the head through the three axial slices denoted with dash lines. The first one is located at 12 cm from the top of the head. The space between the two axial slices is 24 mm. A relative FA gain map is shown in the bottom row of Figure 2. The green rectangles next to the axial slices represent schematically the position of the fractals with respect to the phantom. We can see a significant increase of the FA while using the fractal. It corresponds to a 30% gain around the area were the fractals were placed and 10% on average in the ROIs that denote the temporal lobe regions. Moreover, from the sagittal slice, no loss is observed in other regions of the brain such as the cerebellum.

Figure 3 presents GRE signal maps (unprocessed data from the XFL sequence) of the birdcage coil alone (top row) and with the fractals (central row). All GRE images were acquired with the same input power of 260V. Similarly, the last row of Figure 3 shows the relative gain between the fractals over the reference signal. We can see that the 30% enhancement obtained in the FA maps is reflected on the GRE images while the average gain in the ROIs seems to increase slightly and low loss is observed in the remaining brain region.

Discussion

We based the design of the metasurface pads on a Hilbert fractal with the aim to obtain compact, thin and flexible resonant structures. Due to their complex design, the fractals are quite sensitive to the location and orientation with respect to the birdcage coil. The optimized positions were found to respect the original asymmetry of the B1+ field amplitude in the reference flip angle maps. The presence of fractals shows a maximum +30% gain of B 1 + amplitude for both temporal lobes, for the same input voltage which is interpreted as a large increase in transmit efficiency (Figure 2). Moreover, they maintain the B 1 + amplitude almost intact in the remaining brain volume. This effect is particularly interesting for the cerebellum region which usually suffers when high dielectric constant pads or metamaterials are inserted in the birdcage coil. In our case, most of the loss seems to be directed towards the lower part of the head away from the brain ROI where the birdcage coil stops. This can be seen in the sagittal slices in Figures 2 and3. Moreover, as the relative gain observed in the FA maps in Figure 2 is conserved and slightly improved in the GRE images, it indicates that the fractals do not interfere with the receive array.

Conclusion

We designed metasurface pads inspired by a Hilbert-fractal design to reduce the transmit field inhomogeneity artefacts observed in head imaging with high-field MRI. We found that resonant fractal metasurfaces improve the B 1 + amplitude up to a maximum 30% and 10 % on average in the two temporal lobes without compromising the B1+ amplitude in the cerebellum. Their compact and flexible design allows them to fit easily between the phantom and the receive array of the head coil. Hence our approach promises better performance and patient comfort over conventional dielectric pads or metamaterials.

Figures

Figure 1 .

 1 Figure 1. Design (top row) and position (bottom row) of the Hilbert fractal-based metamaterial pads.The fractal on the right side is slightly higher than the left one.

Figure 2 .

 2 Figure 2. Experimental FA maps of birdcage coil (top) and birdcage coil with resonant fractals (middle) in sagittal and axial orientations. The bottom row shows the relative gain between fractals and reference map. The schematic position of the fractals is represented with green rectangles. The dashed lines on the sagittal slice show the location of the axial slices. We defined the temporal lobe ROIs from the reference scan (top row). The values below the bottom row are the mean values obtained inside each ROI.

Figure 3 .

 3 Figure 3. XFL amplitude signal maps of birdcage coil (top) and birdcage coil with resonant fractals (middle) in sagittal and axial orientations. The bottom row shows the relative gain between fractals and reference map. The schematic position of the fractals is represented with green rectangles. The dashed lines on the sagittal slice show the location of the axial slices. We defined the temporal lobe ROIs from the reference scan (top row). The values below the bottom row are the mean values obtain inside each ROI.

Acknowledgements

This work has received funding from the European Union Horizon 2020 Research and Innovation program under Grant Agreement No. 736937.


Tania S Vergara Gomez
email: tania-del-socorro.vergara-gomez@etu.univ-amu.fr

M Dubois

K Rustomji

E Georget

T Antonakakis

S Rapacchi

F Kober

S Enoch

R Abdeddaim

Hilbert Fractal Inspired Dipoles for B1+ Field Control in Ultra-High Field MRI

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

I. INTRODUCTION

The constructive and destructive interferences between wavelengths at ultra-high magnetic field (7 Teslas,300MHz) in clinical magnetic resonance imaging (MRI) lead to head coils' B 1 + magnetic field inhomogeneities in the temporal lobe regions. Approaches like high-permittivity dielectric pads [START_REF] Yang | Manipulation of image intensity distribution at 7.0 T: passive B1+ shimming and focusing with dielectric materials[END_REF]- [START_REF] Neves | Compressed perovskite aqueous mixtures near their phase transitions show very high permittivities: New prospects for high-field MRI dielectric shimming[END_REF] and coupled-wire metamaterials [START_REF] Dubois | Kerker Effect in Ultrahigh-Field Magnetic Resonance Imaging[END_REF]- [START_REF] Dubois | Enhancement of transmit and receive efficiencies with hybridized meta-atom in 7T head coil[END_REF] have been employed to mitigate these effects. They present a powerful and controllable improvements of the B 1 + field inhomogeneities, but there remain some challenges to overcome. The dielectric pads are based on the principle of displacement currents and are mainly composed of highpermittivity materials like calcium titanate (CaTiO 3 ) or barium titanate (BaTiO 3 ) mixed with water [START_REF] Yang | Manipulation of image intensity distribution at 7.0 T: passive B1+ shimming and focusing with dielectric materials[END_REF]- [START_REF] Neves | Compressed perovskite aqueous mixtures near their phase transitions show very high permittivities: New prospects for high-field MRI dielectric shimming[END_REF]. Their dimensions and permittivity can be optimized for a specific target. BaTiO 3 pads with 250 relative permittivity and 10x10x1cm 3 volume are examples of pads use for head imaging at 7T. They are place directly in contact with the patient's head. Since these pads are bulky, they are limited by the patients' morphology and the head coils' space which can be reduced if they are equipped with a receive coil array. This adds constraints in the set up and reduction of the patient comfort. On the other hand, the metamaterial approach is based on the mode hybridization principle, using long resonant dipoles coupled into a specific geometry [START_REF] Dubois | Kerker Effect in Ultrahigh-Field Magnetic Resonance Imaging[END_REF]- [START_REF] Dubois | Enhancement of transmit and receive efficiencies with hybridized meta-atom in 7T head coil[END_REF]. Designs such as the Hybridized Meta-Atom (HMA) provide stronger enhancement than dielectric pads since it is possible to control the coil-resonator coupling strength by adapting the length of the wires and the position with respect to the patient's head, giving more degrees of freedom. They can be placed away from the patient's head, but sometimes it can be hard to fit the resonator in the small and tight free space left in commercial MRI coils. HMA have strongly improved the B 1 + field at a single region of interest like one temporal lobe. However, maintaining such performance while targeting multiple areas by using several structures may be difficult.

Here, we propose compact, thin and flexible resonant meander dipoles based on a Hilbert fractal. The main concept behind these meander electric dipoles is to introduce small passive resonators free of lumped elements, able to resonate at 300MHz and to fit within a maximum of 10x10cm 2 area. First, we investigated space filling structures based on Hilbert fractal [START_REF] Anguera | The fractal Hilbert monopole: A two-dimensional wire[END_REF]- [START_REF] Engheta | Metamaterials: physics and engineering explorations[END_REF]. Hilbert fractals end-feed have been used as monopoles [START_REF] Anguera | The fractal Hilbert monopole: A two-dimensional wire[END_REF]. In order to satisfy our geometrical constraints, we used two 3 rd -order Hilbert monopoles in series to form our passive meander electric dipole. Akin to dielectric pads, the Hilbert like dipoles are placed close to the patient's head. Due to their sub millimeter thickness, they do not impact the patients' comfort. Our compact and flexible structures are able to improve the B 1 + field in each temporal lobe maintaining the transmit efficiency of the head coil elsewhere.

II. MATERIAL AND METHODS

Exhaustive numerical studies were carried out on CST Microwave studio to compare the effect on the 16-legs head coil's B 1 + field while using the original 3 rd -order Hilbert fractal and our proposed meander dipole (Fig. 1). The head coil was fed by 2 ports in quadrature. A third port was placed on the structure used to check the resonance and the coupling with the head coil. This third port was later short-circuited to obtain B 1 + magnetic field maps with the structure inductively coupled to the main coil. We printed the copper meander dipoles on a 0.4-mm-thick FR-4 substrate to make them thin and flexible. They were tested using a 1Tx/32Rx proton head coil (Nova Medical, Wilmington, MA, USA) in a 7T Magnetom MRI scanner (Siemens Healthineers, Erlangen, Germany) using a specific anthropomorphic head phantom (SPEAG, Zurich, Switzerland) filled with a liquid with permittivity ɛ = 45.3 and conductivity σ = 0.87 S/m. Using the turbo-flash XFL MRI sequence [START_REF] Amadon | Slice profile corrections in the XFL (magnetization-prepared turbo-FLASH) B1-mapping sequence[END_REF], we obtained B 1 + field distribution maps in sagittal and axial orientation, with the following parameters: repetition time = 20 s, echo time = 1.4 ms, flip angle= 8°, Shinnar-Le Roux minimum-phase saturation pulse = 90°, FOV = 256 mm, matrix = 64x64 and 12 slices of 8 mm thickness (17 slices in axial orientation). All images were acquired using the same reference voltage of 260V. For a quantitative analysis, we computed relative B 1 + field difference maps expressed in percentage. + field on Y direction at the center of the phantom and Z direction at -3 cm below the center. The dash lines show the position where the slices were obtained, color scale from 0 to 0.5 T.

III. RESULTS

According to the S-parameters in Fig. 1, both S33 parameters indicate that Hilbert fractal and the meander dipole resonate around 300MHz. S31 and S32 coefficients characterize the coupling to each head coil ports. We only present S32 results as it is the dominant contribution. With the meander dipole we obtain a better coupling (+3dB) to the head coil compared to the original Hilbert fractal. The right side of Fig. 1 shows the simulated B 1 + field maps obtained from a head coil alone (reference) and coupled with the Hilbert fractal and with the meander dipole. The reference present characteristic inhomogeneities (regions with low amplitude) at the temporal region of the brain. We can see that the Hilbert fractal has an effect only on the surface of the phantom. On the other hand, the meander dipole is able to enhance B 1 + field further in the phantom and on a larger area.

Fig. 2. Position of the meander dipoles (left) and B 1 + field distribution maps of the head coil alone (top right) and coupled with the resonant meander dipoles (middle right) in sagittal and axial orientations. The bottom right row shows the relative gain between meander dipoles and reference map.

The experimental results show the effect of the meander dipoles at different locations of the phantom head (Fig. 2). A central sagittal slice and three different axial slices of B 1 + field maps are presented. The green rectangles next to the axial slices represent schematically the position of the meander dipoles with respect to the phantom. The dashed lines on the sagittal slices show the different examined locations selected through the head. The first axial slice analyzed is located at 12 cm from the top of the head and the space between two consecutive axial slices is 24 mm. At the bottom of Fig. 2, difference maps were calculated. We defined the temporal lobe ROIs from the reference scan. The values at the bottom of each figure are the mean values obtained inside each temporal lobe ROI. A significant increase of the B 1 + field can be seen when using the meander dipoles. A 30% gain was obtained around the area were the meander dipoles were placed, and 10% gain was obtained on average in the ROIs that denote the temporal lobe regions. Furthermore, the central sagittal slice and the lower axial slice illustrate that no loss is observed in other regions of the brain such as the cerebellum.

IV. DISCUSSION AND CONCLUSIONS By designing the passive meander dipoles based on a Hilbert's fractal we were able to obtain compact, thin and flexible resonant structures. We show in the first section that our passive meander dipole design outperforms the original Hilbert fractal structure. The original Hilbert fractal design mostly radiates a magnetic field normal to the fractal plane (similar to a loop). Since the main B 1 + magnetic field of the head coil is parallel to the fractal plane, this reduces the inductive coupling between them. Here, we proposed the meander dipole design based on two connected fractal patterns, obtaining a magnetic field polarization collinear to the main B 1 + field produced by the head coil. Thus, a better coupling (S32 coefficient) and a better B 1 + field enhancement can be obtained. The concept was studied experimentally on a phantom head using two meander dipoles, one on each side of the head. Due to their design, the meander dipoles are quite sensitive to the position and orientation with respect to the head coil. Therefore, some tests were needed to find the most efficient position for each meander dipole, respecting the original asymmetry given by the reference B 1 + field distribution maps. This is the reason for the asymmetric position of the meander dipoles with respect to the head. The experimental results show a strong correlation with the numerical studies. The presence of meander dipoles led to up to 30% gain in B 1 + field amplitude, with an average gain of 10% within both temporal lobes for the same input voltage (Fig. 2). This can be interpreted as a large increase in transmit efficiency. Furthermore, the field in the remaining brain volume remains almost intact. This last effect is particularly important for applications involving regions like the cerebellum, which usually exhibits field losses when other approaches are used. In our case, most of the loss seems to be directed towards the lower part of the head, close to the head coil's lower edge (Fig. 2).

The compact and flexible design of our meander dipoles make them fit easily between the phantom and the receive array of the head coil. Hence, our approach promises better performance and patient comfort over conventional approaches with dielectric pads or coupled-wire metamaterials.

Fig. 1 .

 1 Fig. 1. Design and study of original Hilbert fractal and meander dipole. Top left, structures' design and position with respect to the head and the coil. Bottom left, S-parameters showing the coupling between the structures and the head coil. Right side, Amplitude of B 1+ field on Y direction at the center of the phantom and Z direction at -3 cm below the center. The dash lines show the position where the slices were obtained, color scale from 0 to 0.5 T.

ACKNOWLEDGEMENTS

This work has received funding from the European Union Horizon 2020 Research and Innovation program under Grant Agreement No. 736937.