Nora Tahallah

Alain Brunelle

Sabine De

La Porte

Olivier Laprévote
email: olivier.laprevote@icsn.cnrs-gif.fr

Lipid mapping in human dystrophic muscle by cluster-time-offlight secondary ion mass spectrometry imaging

Keywords: Duchenne Muscular Dystrophy, Human Skeletal Muscle Degeneration, Lipidomics, Mass Spectrometry Imaging, ToF-SIMS, Oxidative Stress, Regeneration

Lipid mapping in human dystrophic muscle by cluster-time-of-flight secondary ion mass spectrometry

imaging.

Introduction

First described by the French doctor Duchenne de Boulogne in the XIX th century, Duchenne muscular dystrophy (DMD) is a severe degenerative genetic disease. Almost exclusively male children develop this myopathy, with an estimation of 1 in 3 500 boys worldwide [START_REF] Blake | Function and genetics of dystrophin and dystrophin-related proteins in muscle[END_REF]. This X-linked recessive pathology is induced by absence of expression or by truncation of dystrophin, a 427 kDa cytoskeletal protein involved in a transmembrane noncovalent complex of more than 20 proteins. The lack of dystrophin causes a rapid progression of skeletal, smooth and cardiac muscles degeneration. Result then, already from the age of 2 or 3, bad posture, unsteady gait, difficulty in standing, walking or climbing stairs. A rolling chair is necessary around 10 to 12 years old and patients inexorably die around the age of 25-30. Dystrophin deficiency hampers the formation of an essential transmembrane complex in charge of the integrity of muscle cell walls. This results in muscle cell membranes malfunctioning and permeability, alteration of the ion drifts through the membrane (e.g. calcium), but also in oxidation stress. Diagnoses are mainly based on analyses of creatin kinase level in the serum and electromyograms associated to biopsies or genetic tests. Among the therapies used to decelerate the damages caused by the disease, are found dystrophinassociated drug design, gene and cell therapies or utrophin-associated therapies [START_REF] Voisin | L-arginine improves dystrophic phenotype in mdx mice[END_REF]. Up to date, there is unfortunately no cure for DMD yet. This last decade, research has been going on using different animal models (mice, dogs, cats, drosophila and Caenorabditis worms) to get a better understanding of this lethal pathology mechanisms. The dog model, closest to human, is rare and expensive and, although cheaper and more accessible, drosophila and Caenorabditis worms phenotypes are too different from the human one. The most spread model is the mdx (X-linked muscular dystrophy) mouse, developing an unusually high concentration of creatin and pyruvate kinases [START_REF] Bulfield | X chromosome-linked muscular dystrophy (mdx) in the mouse[END_REF]. The three advantages of the mdx mouse model are relatively low cost, good availability and relatively short lifetime.

To gain more insight on Duchenne muscular dystrophy, our group has previously carried out investigations on an mdx mouse leg muscle. Matrix-Assisted Laser Desorption Ionization-Time-Of-Flight (MALDI-TOF) profiling (4) and cluster-Time-of-Flight Secondary Ion Mass Spectrometry (cluster-ToF-SIMS) imaging [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF] have been performed on 3 weeks old mdx mice, as well as on control (nxlt) and dystrophic (xlt) mice myoblast cell cultures. These studies allowed distinguishing three regions in the mdx mouse leg: a severely damaged region, a second one degenerating and undergoing oxidative stress and phosphatidylinositol cycle deregulation and a third one that was found stable.

Concerning ToF-SIMS, its input is continuously expanding in the field of lipidomics and drugs/biomarkers mapping on biological tissues at a micrometer scale [START_REF] Brunelle | Biological tissue imaging with time-of-flight secondary ion mass spectrometry and cluster ion sources[END_REF]. ToF-SIMS imaging has the major advantage, over alternative methods, of allowing direct and simultaneous collection of mass spectra and ion images. Retrieval of molecular composition and individual compound localization on a tissue section, at micrometer scale and without the need of any prior sample treatment, makes then the analyses easier, more straightforward and the closest possible to the physiological conditions. Recent innovations, such as Bismuth (Bi 3 + ) primary ion cluster sources [START_REF] Touboul | Improvement of Biological Time-of-Flight-Secondary Ion Mass Spectrometry Imaging with a Bismuth Cluster Ion Source[END_REF], have tremendously enhanced the secondary ion emission yield of larger secondary ions. This development extended the capabilities of the method, rendering this approach particularly attractive and suitable for the investigation of biological tissues [START_REF] Nygren | A cell preparation method allowing subcellular localization of cholesterol and phosphocholine with imaging TOF-SIMS[END_REF][START_REF] Ostrowski | Mass Spectrometric Imaging of Highly Curved Membranes During Tetrahymena Mating[END_REF][START_REF] Touboul | Tissue molecular ion imaging by gold cluster ion bombardment[END_REF][START_REF] Sjövall | Mass spectrometric imaging of lipids in brain tissue[END_REF][START_REF] Monroe | Vitamin E imaging and localization in the neuronal membrane[END_REF][START_REF] Nygren | Imaging TOF-SIMS of rat kidney prepared by high-pressure freezing[END_REF][START_REF] Nygren | Localization of cholesterol in rat cerebellum with imaging TOF-SIMS -Effect of tissue preparation[END_REF][START_REF] Mas | Lipid cartography of atherosclerotic plaque by cluster-TOF-SIMS imaging[END_REF][START_REF] Touboul | MALDI-TOF and cluster-TOF-SIMS imaging of Fabry disease biomarkers[END_REF].

In order to validate the coupling of the ToF-SIMS imaging methodology, together with the mdx mouse model, as a suitable approach for research on the human pathology, we here report on first results obtained on human striated muscle sections and compare them to the previous outcomes on the mdx mouse. Therefore cluster-ToF-SIMS imaging has been used to map and characterize the molecular distribution in human control and dystrophic muscle tissue sections. Mass spectra revealed differences in the distribution of fatty acids, phospholipids, diand triglycerides. Samples were analyzed in both positive and negative ion modes. The

Materials and Methods

Surgery residues of human paravertebral striated muscles, from male control and DMDaffected 12 to 14 years old children were provided by the "Banque de Tissus pour la Recherche" (BTR, Institut de Myologie Hôpital de la Pitié-Salpêtrière, 83 boulevard de l Hôpital, 75013 Paris) after approval of the project.

Histological staining

10 µm thick sections were cut at -20°C in a CM3050-S cryostat (Leica Mycrosystèmes SA, Rueil-Malmaison, France) and immediately deposited onto a glass plate then stored at -80°C for histological staining. The dystrophic appearance of the human paravertebral striated muscle was studied with Masson's trichrome staining (Sigma kit No. HT15) to visualize muscle fibers (in red), connective tissue and collagen (colored by the aniline blue).

Histological images were taken with a DMRXA2 microscope (Leica Mycrosystèmes SA, Rueil-Malmaison, France).

Sample preparation for cluster-ToF-SIMS Imaging

Cryostat sections (20 µm) were prepared, placed on glass plates and briefly stored at -80°C.

Immediately before analyses, samples were dried under a few millibars pressure for 30 minutes, without any further treatment. Then, images were recorded with an Olympus BX51 microscope (Rungis, France) equipped with 1.25x to 50x lenses and a Color View I camera (Soft Imaging System) monitored by the Olympus DP Soft software.

ToF-SIMS data acquisition and processing

All experiments were performed on a commercial ToF-SIMS IV (ION-TOF GmbH, Münster, Germany) reflectron-ToF mass spectrometer located at the French "Institut de Chimie des Substances Naturelles" in Gif-sur-Yvette, France. The instrument was fitted with a bismuth (Bi 3 + ) cluster ion source at a 45° incidence angle. The primary ion dose density was 2 x 10 11 ions/cm 2 for each ion mode, with a 150 µs cycle time and a 0.27 pA measured pulsed current.

The ion column focusing mode ensured both a 1-2 µm beam focus and a high mass resolution, which is a prerequisite for accurate mass measurements and assignments. A low-energy electron flood gun neutralized the surface during analysis. The secondary ions were first extracted with a 2 keV kinetic energy, had an effective ion flight path of ~2m and were then post-accelerated to 10 keV before hitting the detector surface. The sample surface was directly observable through an integrated video camera. Positive and negative secondary ion images, obtained with IonImage software, had a field of view of 500 x 500 µm 2 (256 x 256 pixels). Images were compressed during data processing to 128 x 128 pixels to increase the contrast, leading to a final lateral resolution of 4 µm. For more clarity, images were also averaged and rescaled. To ensure the correct localization on each spot, a thorough examination was performed for each compound: an image was generated for each peak of the isotopic distribution of the compound (i.e. for the 12 C atoms containing peak, for the 13 C 1 containing peak and for the peak containing two 13 C). These images were then matched and summed to obtain the final compound image. However, in cases like PC34:2 (see section isotopic correction for details), only the first two images were summed. Subsequently, Regions Of Interest (ROIs) were manually selected and the corresponding mass spectra were extracted to obtain local information. These selected regions may have different areas.

Therefore, and for a proper comparison, the spectrum corresponding to each ROI needed to be normalized against the area of the smallest one. In the following, the selected cells, intercellular and damaged areas are characteristic of all the different types of cells, intercellular and damaged areas contained in the analyzed sections.

Calibration and mass assignments

The method was the same as in reference [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF]. The very low initial secondary ions kinetic energy [START_REF] Veryovkin | On the trends in kinetic energies of secondary ions produced by polyatomic ion bombardment[END_REF] allows a linear relationship between the time of flight and the square root of m/z over the whole mass range. Therefore, an internal calibration is possible, using the signals of selected. Due to the lack of an efficient MS-MS method which could be associated to TOF-SIMS imaging, structure attributions or assignments of ion peaks were made according to the instrument resolution and accuracy, the valence rule, the biological characteristics of the tissue, as already mentioned in details in reference [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF]. Many mass assignments have also been confirmed or at least reinforced with the help of the literature. Details will be given as one goes along in the next sections.

H + ,

Calculation of lipid intensity ratios with isotopic correction

In this report, the intensity refers to the area of the ion peak in the corresponding mass spectrum. Large mass lipids, e.g. phosphatidylcholines (PCs), contain an important number of carbons. Therefore, the positive ion mode PC34:2 signal is composed of 60.7 % monoisotopic peak (m/z 758.6), 29.5 % of 13 C 1 containing ions (m/z 759.6) and 8 % with two 13 C (m/z 760.6). This latter peak contributes thus to the intensity of the PC34:1 monoisotopic peak at m/z 760.6. Besides, the emission yields and detection efficiencies for compounds of the same family and close molecular weight are considered equivalent [START_REF] Gilmore | Ion detection efficiency in SIMS: dependencies on energy, mass and composition for microchannel plates used in mass spectrometry[END_REF]. Hence, when calculating phosphatidylcholines intensity ratios the isotopic contribution correction has been applied. For the sake of precision, the calculation of the fatty acids intensity ratios, in negative ion mode, was also performed taking into account this same correction.

Ratios definition

R 1 = I m/z

Results

We investigated human striated muscle originating from surgery residues of one control (12 years old) and two distinct DMD-affected (13 and 14 years old) children. Four cryosections from the control and two sections from each DMD-affected muscle were selected. On each section, 2 to 6 zones were subjected to ToF-SIMS imaging. These 33 ToF-SIMS analyzed zones (13 control and 20 dystrophic) will be referred to as spots, in the following, for more clarity.

Fig. 1A shows the microscope optical image of a section from a control 12 years old child surgery residue. In green is delimited spot 1, one of the areas subjected to ToF-SIMS analyses. Individual, compact and regular myofibers (muscle cells) are visible, all sheathed by a membrane in whole fascicles. Fig. 1B displays the microscope optical image of a section from a 14 years old DMD-affected child surgery residue and in green are highlighted two of the analyzed spots (2 and 3) described in this report. Physical differences are visible between nonstructured, damaged and pseudo-healthy zones. The colors vary and fibers membrane sheaths are absent. Although the myofibers are clearly observed, they are not regularly sized and only few zones appear more compact. Finally, these two images helped us to choose the areas to be analysed by mass spectrometry imaging on the DMD-affected muscle sections.

Histology of control and DMD-affected human striated muscle sections

As a start, to probe the physiological status of the muscles and visualize connective tissue, muscle fibers and collagen, we applied to control and dystrophic human striated muscle sections a Masson's trichrome staining. The healthy muscle fibers in Fig. 2A appear (in red) regular, compact within a fascicle membrane surrounded by only a minute amount of collagen (in blue). Conversely, the DMD-affected muscle cells (Fig. 2B) are irregular, dislocated, lacking fascicle membrane and surrounded by an important amount of collagen and connective tissue within the numerous damaged areas.

ToF-SIMS spectrum from a control muscle section

A control human striated muscle has first been analyzed by ToF-SIMS imaging (e.g. spot 1, Fig. 1A). The total mass spectrum of spot 1 recorded in positive ion mode is displayed in Fig. by TOF-SIMS Post Source Decay measurements directly onto a rat brain section [START_REF] Touboul | Structural analysis of secondary ions by post-source decay in time-of-flight secondary ion mass spectrometry[END_REF]. The cholesterol ions, at m/z 369 and m/z 385, in positive ion mode have been already mentioned in many papers such as [START_REF] Touboul | Tissue molecular ion imaging by gold cluster ion bombardment[END_REF], [START_REF] Sjövall | Mass spectrometric imaging of lipids in brain tissue[END_REF] and [START_REF] Börner | Distribution of cholesterol and galactosylceramide in rat cerebellar white matter[END_REF]. Positive ions of vitamin E have been identified in previous imaging experiments [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF][START_REF] Monroe | Vitamin E imaging and localization in the neuronal membrane[END_REF]. Series of positive ions are observed in the m/z 500-620 range. These ions are likely to be fragments of diglycerides (DGs), triglycerides (TGs), PCs, or phosphatidylinositols (PIs). These ions are labeled with a generic name "DG type fragment". Their ion images (data not shown) reveal a general colocalization with triglycerides, suggesting that these "DG type fragment" ions mainly originate from fragmentation of diglycerides and triglycerides. PC positive ions in the region m/z 750-810 have already been observed and characterized in ToF-SIMS imaging experiments by several authors [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF][START_REF] Touboul | Tissue molecular ion imaging by gold cluster ion bombardment[END_REF][START_REF] Sjövall | Mass spectrometric imaging of lipids in brain tissue[END_REF]. In particular, the structure of these ions has already been studied directly from mouse muscular cell cultures by MALDI-TOF tandem mass spectrometry (4). Finally the ion detected at m/z 723.5 might tentatively be attributed to a sodiated sphingomyelin (SM) ion.

As the mouse samples, the human muscle sections also exhibit intact PC ions as the dominant species in the region m/z [650-820], principally PC34:2 and PC34:1 (m/z 758.6 and 760.6, respectively) and their respective sodium adducts at m/z 780.6 and 782.6 and potassium adducts at m/z 796.6 and 798.6. It must be mentioned that PC34:2 and PC34:1 ions are not the only PCs ions which are detected. Several other PC ion species appear are also present in the spectra, as it was the case for the previous studies on model mice [START_REF] Touboul | Changes of phospholipid composition within the dystrophic muscle by matrix-assisted laser desorption/ionization mass spectrometry and mass spectrometry imaging[END_REF][START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF]. PC fragments appear in the region m/z [80-300].

ToF-SIMS imaging of phosphatidylcholines

Fig. 4A depicts, from left to right, the microscope image of spot 1, the images of the m/z 758 and 760 ions over the analyzed surface as well as the selected ROIs. Indeed, ToF-SIMS imaging gives the opportunity to select particular regions of interest (ROIs) on the analyzed surface and to extract, from each of them, the corresponding spectrum. Therefore, to obtain a more precise information, we have selected different ROIs within spot 1 (Fig. 4A, right),

driven by the different distributions on the ion images. In red, green and blue are selected cells and in pink an intercellular space. From each ROI a mass spectrum was extracted, as shown in Fig. 4B, and the corresponding PC34:2/PC34:1 (758.6/760.6) intensity ratio was calculated. For the sake of clarity, this ratio is referred to as R 1 in the following. Most of the selected regions led to a R 1 ratio value of around 2 (Fig. 4C). By contrast, a unique cell (selected in green) gave R 1 at 0.80. The ion images (Fig. 4A) pointed as well to this cell as the only area where the m/z 758 ion intensity is clearly lower than the 760's. If the present results are to be compared with those obtained from the mdx mouse model [START_REF] Touboul | Changes of phospholipid composition within the dystrophic muscle by matrix-assisted laser desorption/ionization mass spectrometry and mass spectrometry imaging[END_REF], this means that this might well be a regenerating cell.

We have then analyzed the different DMD-affected muscle sections and the results are illustrated by two typical examples in Figs. 5 and6. A first pseudo-healthy region, containing almost exclusively highly uniform and compact cells, drew our attention (Fig. 5). We focused on this spot (numbered 2 in Fig. 1B) to start the comparison between the control and the DMD-affected samples. The value of ratio R 1 , for the whole spot 2, was 1.27 (see Table I:

DMD section 1), in accordance again with the mdx mouse model. However, the mass spectra (Fig. 5B) corresponding to the selected ROIs (Fig. 5A, right), and the R 1 values of the different ROIs reported in Fig. 5C showed that the cells were not fully identical, in terms of PCs composition, over this zone. Indeed, the small cells (blue ROI) exhibited a R 1 ratio value of 1.26. However, the larger cells split in two: some cells (in red) generated a ratio value of 1.21, similarly to the small ones, while others (in green) gave an R 1 ratio value of 1.42. The large cells delimited by the green ROI then presented a PCs composition similar to that of the healthy tissue, whereas the intercellular space (R 1 =0.71) behaved like a regeneration zone.

The big cells in the red ROI and the small cells in the blue ROI appeared to be in an intermediate state.

The second characteristic region of the DMD-affected human muscle is typified by spot 3 (Fig. 1B). The R 1 global ratio value for this spot was determined to be 1.13 (Table I: DMD section 1), suggesting an m/z 758.6 ion slightly more abundant than the m/z 760.6 ion. Fig. 6A depicts, from left to right, the microscope image, the images of the ions at m/z 758 and 760 and the selected ROIs over spot 3. For more detailed information, we chose ROIs corresponding to three morphologically different regions (Fig. 6A, right): relatively ordered cells were selected in red, a damaged area in green and some adipocytes in blue. Fig. 6B shows the mass spectra of each of these selected ROIs (m/z [650-820]), while Fig. 6C displays the corresponding R 1 values (1.04 within the cells and 0.83 in the damaged region). In the mass spectrum of the blue ROI (adipocytes), the signal-to-noise ratio was not sufficient to determine the R 1 ratio. Thus, in the DMD spot 3, the relatively ordered cells exhibit a similar behavior to that of the control zones in the mdx mouse model. On the other hand, the damaged area generated a R 1 value of 0.83, corresponding to that of the intercellular region of the DMD spot 2 and to the green cell of the control spot 1. Similarly to the destructured (regenerating) zone in the mdx mouse model, these ratio values were lower than 1.This could denote an intensive regenerating activity and not degeneration, as could be expected from the color, aspect and absence of fascicles on the microscope image of spot 3. Besides, when compared to spot 2, the lower R 1 ratio produced by the cells of spot 3 indicates also that these cell PCs have incorporated less C18:2, as observed previously in DMD-affected human muscles [START_REF] Pearce | Muscle lipids in Duchenne muscular dystrophy[END_REF][START_REF] Kunze | Fatty acid pattern of lipids in normal and dystrophic human muscle[END_REF]. This also means that spot 2 is in a more stable state than spot 3 and confirms the pseudo-healthy nature of spot 2.

The reproducibility of these results was verified on different sections of the same (control and dystrophic) surgery residues and their reliability was validated on sections of a second dystrophin deficient child surgery residue. Table I summarizes the global R 1 ratio calculations performed on all control and dystrophic muscle sections. The control samples provided an average ratio value of 1.85 (with a ± 0.31 standard deviation), higher than the one found for the healthy and control areas of the mouse model ( 4). For the dystrophic samples, the R 1 average value was calculated to be 1.15 (standard deviation ± 0.17), rather comparable to that found for healthy and control areas of mdx mouse. The data shown here matches the results of intensity ratios between m/z 758.6 and 760.6 for healthy and dystrophic cells on day 1 of culture (myoblastes) in Reference 4.

ToF-SIMS imaging of cholesterol and sphingomyelin

A remarkable feature in the m/z [650-820] region, is the presence of three signals that we tentatively attributed to the sphingomyelin species, SM34:1 (d18:1/16:0) and SM34:2 (d18:1/16:1 or d18:2/16:0). The mass spectrum of a DMD sample (Fig. 7A, lower spectrum)

showed indeed peaks corresponding to the protonated molecule (m/z 703.5, [SM34:1+H] + ), to the sodium adduct (m/z 725.5, [SM34:1+Na] + ) and to a fragment possibly corresponding to the choline headgroup elimination from the sodiated lipid (m/z 668.4). These ions have been previously described in the literature for membrane sphingolipids [START_REF] Wiseman | Mass spectrometric profiling of intact biological tissue by using desorption electrospray ionization[END_REF], which have been reported to be involved in the activation of ion channels and functional rafts [START_REF] Zhang | Ceramide, a putative second messenger for nerve growth factor, modulates the TTX-resistant Na(+) current and delayed rectifier K(+) current in rat sensory neurons[END_REF][START_REF] Titievsky | Sphingosine inhibits voltage-operated calcium channels in GH4C1 cells[END_REF] and to be colocalized with them [START_REF] Nygren | Localization of cholesterol, phosphocholine and galactosylceramide in rat cerebellar cortex with imaging TOF-SIMS equipped with a bismuth cluster ion source[END_REF]. Sphingomyelins are now known to colocalize with cholesterol and form rafts in plasma membranes of cultured cells [START_REF] Slotte | Cholesterol-sphingomyelin interactions in cells--effects on lipid metabolism[END_REF][START_REF] Ohvo-Rekila | Cholesterol interactions with phospholipids in membranes[END_REF] and PCs and cholesterol have been found to interact in membrane bilayers [START_REF] Ohvo-Rekila | Cholesterol interactions with phospholipids in membranes[END_REF][START_REF] Ramstedt | Membrane properties of sphingomyelins[END_REF]. The ion images of spot 3 revealed indeed a maximum of cholesterol species (Fig. 8A) and SMs species in the damaged region together with PC34:2 and PC34:1 (Fig. 8B). For the pseudo-healthy spot 2, a maximal accumulation of SM and cholesterol species was found in the intercellular region. The other analyzed spots confirmed their colocalization in intercellular or damaged areas. It is noteworthy that, in spot 2, some SM species were also present in the small cells unlike in the large ones. For the control spot 1, practically no sphingomyelin ion was detected except the m/z 723.5 species, tentatively assigned to the sodiated SM34:2, that gave some signal all over the spot surface (data not shown), together with ions at m/z 758.6 and 760.6. In this sample, cholesterol only accumulated in intercellular spaces or adipocytes but not in cells.

ToF-SIMS spectra of fatty acids

The mass spectra recorded in the negative ion mode (enlargement m/z [240-320]) of control and dystrophic muscles are displayed in Fig. 7B. Distributions appear distant of 28 mass units (u) with peaks separated by 2 u, characteristic of fatty acid carboxylate ions with different chain lengths bearing a variable number of double bonds [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF]. In control and DMD-affected samples the same series of fatty acid ions were present. However, their intensities varied according to the nature of the sample.

Table II summarizes the global fatty acid abundance ratios R 2 (C18:0/C18:1) for each analyzed spot of the control and dystrophic muscles, calculated as described in the Materials and Methods section. The control sample spots exhibited an average ratio (0.77 ± 0.16) about 1.5 times higher than the dystrophic ones (0.54 ± 0.14). These values significantly depended on the relative abundance of the cells in the spot area. A more thorough examination of the different ROIs ratios within each DMD-affected spot showed that the cells presented R 2 values of 0.6-0.9, i.e. higher than the average value. In other words, for the DMD-affected sections, the higher global ratios (~0.7) corresponded to pseudo-healthy regions and to spots containing healthier cells. The lower ratios were obtained from the most damaged spots (~0.3). The DMD intercellular or damaged regions gave R 2 values of 0.4-0.5, versus ~0.8 for the control intercellular regions. The adipocytes in the dystrophic spot 3 exhibited a ~0.4 R 2 value as well.

An additional striking feature is the generally high level of C18:2 in the control human muscle (Fig. 7B). In Table III, an average value of 0.98 ± 0.18 has been calculated for the R 3 (C18:1/C18:2) ratios in the control samples, while it reached 1.96 ± 0.44 for the dystrophic muscles. Hence, this ratio might also be an indicator of the DMD disease, since the control average R 3 is twice smaller than the dystrophic one. Moreover, the most structured spots in the DMD tissue sections also exhibited a relatively low C18:1/C18:2 ratio value (1.09 and 1.20, spots 1 of DMD sections 3 and 4, respectively), corroborating the healthier state of the cells in these areas.

ToF-SIMS imaging of phosphatidylinositols and triglycerides

In the negative ion mode, Touboul et al. originate not only from free fatty acids, but also from TGs fragmentations, as previously observed by ESI-MS [START_REF] Hsu | Characterization of phosphatidylinositol, phosphatidylinositol-4-phosphate, and phosphatidylinositol-4,5-bisphosphate by electrospray ionization tandem mass spectrometry: a mechanistic study[END_REF][START_REF] Stroobant | Fragmentation of conjugate bases of esters derived from multifunctional alcohols including triacylglycerols[END_REF], CF-LSIMS [START_REF] Li | Continuous flow liquid secondary ion mass spectrometric characterization of phospholipid molecular species[END_REF] and FAB-MS [START_REF] Murphy | Mass Spectrometry of Lipids (Handbook of Lipid Research[END_REF].

For the human samples, the PI fragment ions were present in all mass spectra, together with ions at m/z 259.0 and 299.0 (two other PI characteristic fragments) [START_REF] Ostrowski | Secondary ion MS imaging of lipids in picoliter vials with a buckminsterfullerene ion source[END_REF], and with the C20:4 ion at m/z 303.2 (Fig. 7B). The individual mass spectra of ROIs revealed a maximal abundance of these ions within the cells, pointing out the presence of PIs, in particular PI38:4 (m/z 885.6) which contains a C20:4 fatty acyl chain [START_REF] Sjövall | Mass spectrometric imaging of lipids in brain tissue[END_REF][START_REF] Wenk | Phosphoinositide profiling in complex lipid mixtures using electrospray ionization mass spectrometry[END_REF]. Phosphatidylinositols and triglycerides ions are observed together in the mass spectrum region m/z [760-920] (Fig. 7C), with characteristic series differing by 28 u (two methylene groups) and/or 2 u (one double bond). Their assignments are based first on the fact that TG ions are only detected in adipose or DMD-damaged tissues, secondly on the localization similitude between PI38:4 on the one hand, and PI34:0, PI36:2, PI36:1 and the PI fragments, on the other hand. Finally the mass calculation can differentiate a PI ion from a TG one, since the decimal part of a PI ion mass must be beteween".5" and ".6", while the decimal part of a TG ion is between ".7" or ".8".

Therefore the ions at m/z 831.8 and 857.8 can only be TG50:1 and TG52:2, respectively. The ion at m/z 797.7 remains as far as we know unidentified. The relative intensities of these ions varied and, in general, the more the cells were structured, the less the triglycerides were present, in favor of the PIs. To obtain more details, we have examined different ROIs to be able to distinguish the different corresponding distributions. As an example, Fig. 9A displays the mass spectra obtained in the negative ion mode of the three ROIs selected in spot 3 (cells in red, damaged area in green and adipocytes in blue). These spectra were spectacularly different. The images of the major ion peaks (Fig. 9B) led us to distinguish, on a molecular basis, the three regions of interest. Additionally, the presence of some of these species in the damaged area intimates that myofibers develop into adipocytes, accordingly to what has been previously observed in the degenerating areas of the mdx mouse model and in human dystrophic muscle tissue [START_REF] Bonsett | Oil globules' in Duchenne muscular dystrophy--history, demonstration, and metabolic significance[END_REF]. By contrast, the peaks at m/z 835.6, 837.6 and 863.6 (Fig. 9A, red spectrum) can only originate from the phosphatidylinositols PI34:1, PI34:0 and PI36:1, respectively, but not from TGs. The colocalization of the peak at m/z 885.5 with these latter ions led us to assign this ion peak to the PI38:4 species and not the TG54:2 molecule, and the peak at m/z 861.6 to PI36:2 and not to TG52:0. All images corresponding to these PI38:4 and PI36:2 ions were perfectly superimposed with the localization of the cells (red ROI). Finally, three ion peaks at m/z 795.7, 797.7 and 799.7 were detected selectively in the damaged surface. Their absence in the adipocytes and in the cells suggested that they did stem neither from triglycerides nor from PIs.

Discussion

Dystrophin deficiency generates malfunctioning in the muscle cell membranes, as the transmembrane complex is largely destabilized, deregulating functional ion channels and oxidation reaction cycles [START_REF] Blake | Function and genetics of dystrophin and dystrophin-related proteins in muscle[END_REF]. Consequently, the absence of dystrophin and of its interactions with the membrane lipids induces membrane degeneration and muscle apoptosis or necrosis.

Cell membrane dysfunctioning results in modifications of membrane lipid unsaturations as a consequence of disturbances in the fatty acids metabolism [START_REF] Kunze | Fatty acid pattern of lipids in normal and dystrophic human muscle[END_REF]. Since phospholipids are key components of muscle cell walls, the characterization of the degeneration/regeneration process, through the characterization of phospholipid composition and localization, becomes essential.

Glycerophosphatidylcholines (PCs) are known to be a major component of muscle cell membrane phospholipids. These lipids bear a quaternary ammonium moiety granting a fixed positive charge which leads to higher signals than the other lipids present in the tissue, thereby facilitating their detection by mass spectrometry. MALDI-TOF profiling on the mdx mouse model has shown, in positive ion mode, the presence of PCs as most abundant species in the region m/z 700-850, in particular PC34:2 and PC34:1 at m/z 758.6 and 760.6, respectively (4). In this lipid nomenclature (37), 34 represents the total number of carbons contained in the fatty acyl chains, with a total of 2 and 1 unsaturations, respectively. Under MALDI conditions, the intensity ratio of the ions at m/z 758.6 over 760.6 was found to be bigger than 1 in the pseudo-healthy area of the mdx mouse muscle and smaller than 1 in its destructured area. Healthy and dystrophic mdx mouse cultured cells exhibited a ratio value greater than 1 at the stage of myoblasts and smaller than 1 after cell fusion in myotubes. This suggested that the destructured areas in the mdx muscle were actually undergoing a regeneration process rather than degeneration. Likewise, ToF-SIMS spectra on mdx mouse leg tissue sections also revealed PCs as most abundant species in the region m/z 700-850 [START_REF] Touboul | Lipid imaging by gold cluster time-of-flight secondary ion mass spectrometry: application to Duchenne muscular dystrophy[END_REF].

However, the poor contrast of the ToF-SIMS images of intact PCs did not provide spatial information, thus the composition modification revealed by MALDI-TOF could not be confirmed. Earlier studies [START_REF] Pearce | Muscle lipids in Duchenne muscular dystrophy[END_REF][START_REF] Kunze | Fatty acid pattern of lipids in normal and dystrophic human muscle[END_REF] have also shown that membrane PCs of the human dystrophic muscle incorporated less C18:2 fatty acyl chains than the healthy one, but more C18:1 and C18:0.

Although the experiments have been performed with different samples and methods, i.e.

mouse by MALDI-TOF and human by ToF-SIMS, the analyses can be qualitatively compared. Indeed, the control samples exhibited generally a R 1 ratio almost twice bigger than the DMD-affected samples and the regenerating areas generated lower ratios. Thus, these ratios followed the same trends as in the mdx mouse model, where ratio values were bigger than 1 for the control zones but smaller than 1 for the destructured (regenerating) areas. As an example, Fig. 8 depicts the images of selected ions and their accumulation over spot 3. These images reveal that, in the DMD-affected muscle, the typical PCs fragments colocalize with the parent species PC34:2 and PC34:1 (Fig. 8B) concentrated mainly in the dystrophic damaged region, as mentioned above. The specific localization and the absence of these phosphocholine species in the adipocytes and between the cells, where the low PC concentration is known, confirmed that our sample treatment did not induce any significant molecular delocalization.

Phosphatidylcholines, Cholesterol and Sphingomyelin colocalization

Cholesterol is a versatile essential membrane component involved in steroid hormones production and cell signaling [START_REF] Simons | How cells handle cholesterol[END_REF], membrane structure and function [START_REF] Simons | Functional rafts in cell membranes[END_REF] and synapse formation [START_REF] Pfrieger | Role of cholesterol in synapse formation and function[END_REF]. PC fragments and cholesterol have been found to have a complementary distribution in the rat cerebellum [START_REF] Nygren | Localization of cholesterol, phosphocholine and galactosylceramide in rat cerebellar cortex with imaging TOF-SIMS equipped with a bismuth cluster ion source[END_REF] and in capillary blood leukocytes [START_REF] Sjövall | Imaging of membrane lipids in single cells by imprint-imaging time-of-flight secondary ion mass spectrometry[END_REF]. Cholesteryl sulfate ions have been identified in TOF-SIMS imaging of human skin and renal human samples from a patient suffering from a genetic disease [START_REF] Touboul | MALDI-TOF and cluster-TOF-SIMS imaging of Fabry disease biomarkers[END_REF]. In line, the results obtained by ToF-SIMS imaging of the human control sample showed a complementary distribution of cholesterol, found in intercellular or adipocytes regions (data not shown), and PCs, mainly found within the cells. In contrast, cholesterol, cholesteryl sulfate (Fig. 8A) and intact PCs (Fig. 8B) colocalized in the damaged or intercellular regions of the DMD-affected samples.

This has not been explicitly observed with the mouse samples. Hence, the colocalization might originate from a perturbation of the lipid regulation mechanisms, directly or indirectly inducing, in the human dystrophic muscle, a modification of the PCs accumulation.

The three sphingomyelin ion peaks presented in Fig. 7A appeared in all positive ion mass spectra of the DMD-affected samples, while they were quasi absent in the mass spectra of healthy samples. Negative ion mass spectra showed that both control and DMD-affected muscles contained specific SM34:1 fragment ions (data not shown). However, it was substantially more abundant in the dystrophic samples, and increased with the extent of the damage, than in the control ones. These observations suggest that DMD-affected cells might develop a high sphingomyelinase activity induced by a disorder in the regulation of membrane sphingolipid composition. In healthy cells, after stress or apoptotic stimuli, sphingomyelinase is activated. This diminishes the level of sphingomyelin converted into ceramide (a signaling molecule involved in cell differentiation, proliferation and apoptosis)

and the dysfunctioning of ceramide-mediated apoptosis pathway induces high levels of sphingomyelin [START_REF] Kolesnick | Regulation of ceramide production and apoptosis[END_REF]. Ceramides (SM derivatives) are vital signaling substances for tumors, forming SM-rich vesicles, and responsible for their proliferation and survival [START_REF] Kim | Extracellular membrane vesicles from tumor cells promote angiogenesis via sphingomyelin[END_REF]. Besides, it has been previously shown that phospholipase C activity increases in dystrophic mouse muscle together with an increase of sphingomyelin and triglycerides [START_REF] Pearce | Muscle lipids in Duchenne muscular dystrophy[END_REF][START_REF] Kwok | Phospholipid composition and metabolism in mouse muscular dystrophy[END_REF][START_REF] Banerjee | Altered lipid composition of adipose tissue in human muscular dystrophy[END_REF]. This enzyme converts phosphatidylcholine species into diglycerides and phosphocholine, both precursors of sphingomyelin and triglyceride biosynthesis [START_REF] Ullman | The enzymatic formation of sphingomyelin from ceramide and lecithin in mouse liver[END_REF][START_REF] Kanfer | Phospholipase C catalyzed formation of sphinogomyelin--14C from lecithin and N-(-14C)-oleoyl-sphingosine[END_REF]. In line with this hypothesis, in the human dystrophic muscle, the observed increase of sphingomyelin species, in negative ion mode, goes along with a decrease of phosphatidylcholines species, in favor of diglycerides type fragments (emerging from di-and/or triglycerides), in positive ion mode (data not shown). The above cited sphingomyelin species have been observed in cryopreserved human spermatozoa [START_REF] Schiller | Lipid analysis of human spermatozoa and seminal plasma by MALDI-TOF mass spectrometry and NMR spectroscopy -effects of freezing and thawing[END_REF] and Low and High Density Lipoprotein (50). Wiseman et al. [START_REF] Wiseman | Mass spectrometric profiling of intact biological tissue by using desorption electrospray ionization[END_REF] have found higher levels of SM in cancerous human-liver adenocarcinoma than in non-tumorous tissue.

Cholesterol and SMs colocalization in the dystrophic muscle are in agreement with the literature reports on human samples ( 27) and support our peak assignments. Yet, this is at odds with the mdx mouse data where cholesterol mainly accumulated in the degenerating region together with vitamin E. Additionally, control samples displayed, in positive ion mode, cholesterol peaks with a much lower intensity than vitamin E signals (Fig. 3), whereas DMDaffected sections exhibited a similar or higher intensity for cholesterol signal than for vitamin E (data not shown). This is also consistent with previous findings on dystrophic muscles [START_REF] Young | Electrolyte and lipid composition of skeletal and cardiac muscle in mice with hereditary muscular dystrophy[END_REF][START_REF] Owens | Lipids of dystrophic and normal mouse muscle: whole tissue and particulate fractions[END_REF][START_REF] Logan | Cholesterol alterations in young dystrophic mice[END_REF]. Cholesterol, PCs and SMs colocalization, associated with a much higher proportion of cholesterol, might, therefore, suggest a tentative regulation and/or regeneration behavior of these highly damaged regions in the human DMD-affected muscles.

Fatty acyl chains composition changes

ToF-SIMS analyses of the mdx mouse muscle (5) revealed a calculated C18:0/C18:1 ratio value 6 to 7 times higher in the regenerating area than in the degenerating and control ones. In human samples, this ratio was only 1.5 times higher in the control spots than in the dystrophic ones (0.77 and 0.54, respectively). Besides, in studies on mouse lipid fractions, a C18:2/C18:1 ratio value of 1.3 has been reported for both healthy and predominantly nondestructured dystrophic myofilaments [START_REF] Owens | Lipids of dystrophic and normal mouse muscle: whole tissue and particulate fractions[END_REF]. As mentioned earlier, this is most probably due to the significantly low amount of destructured fibers, adipose and connective tissues in the dystrophic mouse muscle [START_REF] Blake | Function and genetics of dystrophin and dystrophin-related proteins in muscle[END_REF]. Similarly, a lower incorporation of C18:2 in phosphatidylcholines has been observed in human dystrophic muscles, compared to healthy ones, together with an increased level of C18:1 and C18:0 [START_REF] Pearce | Muscle lipids in Duchenne muscular dystrophy[END_REF][START_REF] Kunze | Fatty acid pattern of lipids in normal and dystrophic human muscle[END_REF]. The variation of fatty acid composition between the different areas revealed by the present study can be interpreted as an attempt to stabilize the dystrophic cells, since incorporation of saturated fatty acids in cell wall phospholipids reduces membrane flexibility. By contrast to the cells, the more damaged zones of the DMD tissues showed an increased relative abundance of monounsaturated fatty acids (C18:1) as deduced from lower R 2 and higher R 3 ratio values. Concerning the arachidonic acid carboxylate ions (C20:4), the distribution was found maximal in the degenerating zone. This fatty acid is, among others, a prostaglandin precursor that can accumulate in dystrophin deficient muscles during inflammation.

Phosphatidylinositols

The reduced relative abundance of the PI36:2 and PI36:1 compounds (m/z 861.6 and 863.6, respectively), with regards to the control tissue (Fig. 7C), might already hint at an increase of phospholipase activity in the DMD-affected muscle cells. PIs cycle rate is known to be reduced in dystrophic muscle cells, inducing a high local phospholipases A 2 , C and D activity [START_REF] Rounds | Stimulated turnover of phosphatidylinositol and phosphatidate in normal and Duchennedystrophic human skin fibroblasts[END_REF][START_REF] Liberona | Differences in both inositol 1,4,5-trisphosphate mass and inositol 1,4,5-trisphosphate receptors between normal and dystrophic skeletal muscle cell lines[END_REF]. As mentioned above, the decrease of PIs is accompanied by an increase of triglycerides and sphingomyelin, supporting the hypothesis of the enhanced phospholipase C activity. This increased activity is thought to be induced by the high concentration of intracellular calcium.

Conclusion

In summary, cluster-ToF-SIMS imaging allowed us to directly and relatively rapidly probe intact biological tissue sections with a complete preservation of the sample molecular and structural integrity, a simultaneous analysis of different morphological regions at a micrometer scale and a high molecular specificity and sensitivity without any prior preparation procedure. The specific localization of the different compounds in the positive ion mode imaging and its confirmation in the negative ion mode demonstrated the absence of any molecular delocalization at the micrometer scale on the samples. Moreover, the association of the mass spectra with images was highly useful, providing, on the one hand, molecular composition even when the image contrast was low and, on the other hand, the spatial localization even with a low signal-to-noise ratio. ToF-SIMS imaging thus gives a considerable amount of information on the local molecular composition within tissues or organs, and can be recognized as a powerful approach for localized lipidomics studies. The method emerges as a promising and highly valuable indicator of the disturbance of regulation processes, like alteration of phospholipid incorporation or lipases dysfunctioning, paving the way for substantial assistance in drug-design studies or specific inhibitors development.

Figure captions:

  negative ion mode spectra displayed signals in the region m/z 200-350, corresponding to the palmitic (C16:0, m/z 255.2), palmitoleic (C16:1, m/z 253.2), stearic (C18:0, m/z 283.2), oleic (C18:1, m/z 281.2), linoleic (C18:2, m/z 279.2) and arachidonic (C20:4, m/z 303.2) fatty acid carboxylate (FA) ions. Between m/z 350 and 450, signals originated from cholesterol and αtocopherol (vitamin E). Phospholipids and triglycerides were found between m/z 600 and m/z 900. The DMD-affected muscles displayed different distribution of these ions in the dystrophic cells and in the severely damaged areas. Positive ion mode spectra confirmed the variation in intensity of cholesterol and vitamin E in the different areas. The proportion of phosphatidylcholines also varied with the extent of degradation of the tissue. The ion images allowed differentiating the regions where accumulation of these different compounds occurred either in dystrophic cells, severely damaged areas or adipocytes.

3 (

 3 mass range m/z [80-820]). Choline and phosphatidylcholine fragment ions at m/z 86.1, 104.1, 125.0, 184.1 have already been identified by many authors and their structure are given in reference (4), while the phosphatidylcholine fragment ion at m/z 224.1 has been identified

( 5 )

 5 localized PIs fragments, at m/z 223.0 and 241.0, together with TGs and fatty acid carboxylate ions, in the degenerating area of the mdx mouse muscle sections. Their observation suggested that the fatty acid carboxylate ions could

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2:

Figure 3 :

 3 Figure 3:

Figure 4 :

 4 Figure 4:

Figure 5 :

 5 Figure 5:

Figure 6 :

 6 Figure 6:

Figure 7 :

 7 Figure 7:

Figure 8 :

 8 Figure 8:

Figure 9 :

 9 Figure 9:

  Figure 1

  Figure7

Figure 9

 9 Figure9

  The mass spectrum of the blue ROI showed the

	characteristic signals of triglyceride ions (m/z 803.8, 805.8, 827.8, 829.8, 831.8, 833.8, 853.8,
	855.8, 857.8, 859.8, 881.8 and 883.8), confirming the lipid storing function of the adipocytes.

Table I :

 I R 1 ratios calculated for each control and dystrophic spot. R 1 = I m/z 758 / I m/z 760 : ratio of the intensity of the PC34:2 protonated molecule signal at m/z 758.6 over the corrected intensity of the PC34:1 protonated molecule signal at m/z 760.6. Data originate from positive ion mode ToF-SIMS analyses on four human striated muscle sections from a control and on two sections from two distinct DMD-affected children. The spots correspond to areas over which ToF-SIMS imaging mass spectrometry experiments were carried out.

	Sample	Section number	Spot number	Spot Ratio R 1	Section average value	Standard deviation	Sample average value	Standard deviation
			1	1.13
		1	2 3	1.27 1.13	1.00	± 0.11
			4	0.99
	DMD		1	0.84
	Child 1		2	1.04
		2	3 4	0.90 1.14	1.13	± 0.11
			5	1.08
			6 1	1.02 1.54			1.15	± 0.17
			2	1.42
		3	3	1.07	1.32	± 0.19
			4	1.37
	DMD		5	1.20
	Child 2		1	1.25
			2	1.23
		4	3	0.94	1.16	± 0.13
			4	1.18
			5	1.19
			1	1.89
			2	1.82
		1	3	1.73	1.96	± 0.22
			4	2.22
			5	2.16
	Control	2	1 2	2.06 2.28	2.17	± 0.16	1.85	± 0.31
			1	1.79
		3	2	1.89	1.63	± 0.36
			3	1.22
			1	1.99
		4	2	1.50	1.65	± 0.29
			3	1.47

Table II :

 II R 2 ratios calculated for each control and dystrophic spot. R 2 = I m/z 283 / I m/z 281 : ratio of the corrected intensity of the C18:0 fatty acid carboxylate signal at m/z 283.2 over the intensity of the C18:1 fatty acid carboxylate signal at m/z 281.2. Data originate from negative ion mode ToF-SIMS analyses on four human striated muscle sections from a control and on two sections from two distinct DMD-affected children. The spots correspond to areas over which ToF-SIMS imaging mass spectrometry experiments were carried out.

	Sample	Section number	Spot number a	Spot Ratio R 2 b	Section average value	Standard deviation	Sample average value	Standard deviation
			1	0.41
		1	2 3	0.66 0.30	0.42	± 0.17
			4	0.29
	DMD		1	0.44
	Child 1		2	0.73
		2	3 4	0.58 0.66	0.62	± 0.12
			5	0.54
			6 1	0.76 0.70			0.54	± 0.14
			2	0.42
		3	3	0.45	0.50	± 0.14
			4	0.57
	DMD		5	0.35
	Child 2		1	0.73
			2	0.52
		4	3	0.50	0.59	± 0.09
			4	0.52
			5	0.60
			1	1.09
			2	0.65
		1	3	0.75	0.75	± 0.12
			4	0.77
			5	0.50
	Control	2	1 2	0.63 0.63	0.63	± 0.00	0.77	± 0.16
			1	0.88
		3	2	0.92	0.88	± 0.04
			3	0.84
			1	0.72
		4	2	0.91	0.77	± 0.12
			3	0.69

Table III :

 III R 3 ratios calculated for each control and dystrophic spot. R 3 = I m/z 281 / I m/z 279 : ratio of the corrected intensity of the C18:1 fatty acid carboxylate signal at m/z 281.2 over the intensity of the C18:2 fatty acid carboxylate signal at m/z 279.2. Data originate from negative ion mode ToF-SIMS analyses on four human striated muscle sections from a control and on two sections from two distinct DMD-affected children. The spots correspond to areas over which ToF-SIMS imaging mass spectrometry experiments were carried out.

	Sample	Section number	Spot number	Spot Ratio R 3	Section average value	Standard deviation	Sample average value	Standard deviation
			1	2.67
		1	2 3	1.74 2.41	2.29	± 0.39
			4	2.33
	DMD		1	2.74
	Child 1		2	1.86
		2	3 4	2.46 1.77	2.06	± 0.46
			5	2.03
			6 1	1.52 1.09			1.96	± 0.45
			2	1.62
		3	3	2.15	1.76	± 0.46
			4	1.71
	DMD		5	2.22
	Child 2		1	1.20
			2	1.93
		4	3	2.25	1.79	± 0.43
			4	1.50
			5	2.08
			1	0.74
			2	1.18
		1	3	1.03	0.99	± 0.18
			4	0.88
			5	1.14
	Control	2	1 2	1.00 0.85	0.93	± 0.11	0.98	± 0.18
			1	0.78
		3	2	0.81	0.96	± 0.29
			3	1.30
			1	0.92
		4	2	0.90	1.01	± 0.17
			3	1.21

Acknowledgements

We thank the French "Banque de Tissus pour la Recherche" (BTR, Institut de Myologie) for providing surgery residues of human paravertebral striated muscles. The BTR is a partner of the EuroBioBank network funded by the EC under the Fifth Framework Programme (QLRI-CT-2002-02769). NT is indebted to the Institut de Chimie des Substances Naturelles (CNRS) for a post-doctoral grant. This work was supported by the European Union (contract LSHG-CT-2005-518194 COMPUTIS).


Keywords: Voltage source inverter, 4-leg 2-level inverter, voltage control, PWM, control allocation, optimization, simplex algorithm, median voltage injection

HAL is

I. INTRODUCTION

IXED high switching-frequency pulse-width modulation (PWM) is an essential class of modulation methods for power converters [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Holtz | Pulsewidth Modulation for Electronic Power Conversion[END_REF]. For the widely-studied 3-leg 2-level inverter, two popular PWM methods are: (1) carrier-based pulse-width modulation (CBPWM) [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF]- [START_REF] Bowes | New sinusoidal pulsewidth-modulated invertor[END_REF], using lowfrequency modulating signals and a high-frequency carrier wave, and (2) space vector theory, using a 3-phase geometric vector representation of the inverter and leading to the wellestablished space vector modulation (SVM) techniques [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Holtz | Pulsewidth Modulation for Electronic Power Conversion[END_REF], [START_REF] Holtz | High-Speed Drive System with Ultrasonic MOSFET PWM Inverter and Single-Chip Microprocessor Control[END_REF], [START_REF] Van Der Broeck | Analysis and Realization of a Pulsewidth Modulator Based on Voltage Space Vectors[END_REF]. The main theoretical differences between CBPWM and space vector theory are related to the way the degree of freedom left in the inverter is exploited [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Holmes | The significance of zero space vector placement for carrier-based PWM schemes[END_REF]- [START_REF] Li | Analysis and Simplification of Three-Dimensional Space Vector PWM for Three-Phase Four-Leg Inverters[END_REF]. Typically, this degree of freedom is used to increase the inverter linearity range [START_REF] Holtz | High-Speed Drive System with Ultrasonic MOSFET PWM Inverter and Single-Chip Microprocessor Control[END_REF]- [START_REF] Bowes | The relationship between space-vector modulation and regular-sampled PWM[END_REF], [START_REF] Buja | Improvement of Pulse Width Modulation Techniques[END_REF], [START_REF] Houldsworth | The Use of Harmonic Distortion to Increase the Output Voltage of a Three-Phase PWM Inverter[END_REF], to reduce switching losses [START_REF] Ogasawara | A novel PWM scheme of voltage source inverters based on space vector theory[END_REF], [START_REF] Hava | A high-performance generalized discontinuous PWM algorithm[END_REF], or to mitigate current harmonics using suboptimal modulating solutions [START_REF] Kolar | Minimizing the current harmonics RMS value of three-phase PWM converter systems by optimal and suboptimal transition between continuous and discontinuous modulation[END_REF]- [START_REF] Holtz | Optimal pulsewidth modulation for AC servos and low-cost industrial drives[END_REF].

However, the difficulty of developing control strategies in-A. Bouarfa (corresponding author) and M. Fadel are with LAPLACE, Université de Toulouse, CNRS, INPT, UPS, France: 2 rue Charles Camichel, F-31071 Toulouse, France (bouarfa@laplace.univ-tlse.fr, fadel@laplace.univtlse.fr).

M. Bodson is with University of Utah, Dept. Electrical and Computer Engineering, 50 S Central Campus DR RM 2110, Salt Lake City UT 84112-9206, USA (bodson@ece.utah.edu).

creases rapidly as the number of switches grows, especially with the widely-used space-vector representation. For the 4-leg 2-level inverter, the addition of the fourth-leg raises the number of possible locations of a reference voltage vector from a 2D plane divided into 6 sectors [START_REF] Holtz | High-Speed Drive System with Ultrasonic MOSFET PWM Inverter and Single-Chip Microprocessor Control[END_REF], [START_REF] Van Der Broeck | Analysis and Realization of a Pulsewidth Modulator Based on Voltage Space Vectors[END_REF] to 24 tetrahedrons in a 3D representation [START_REF] Li | Analysis and Simplification of Three-Dimensional Space Vector PWM for Three-Phase Four-Leg Inverters[END_REF], [START_REF] Zhang | Three-Dimensional Space Vector Modulation for Four-Leg Voltage-Source Converters[END_REF]- [START_REF] Perales | Three-Dimensional Space Vector Modulation in abc Coordinates for Four-Leg Voltage Source Converters[END_REF].

Starting from these observations, the paper proposes a new control strategy different from traditional PWM control and based on on-line numerical optimization using linear programming techniques. This strategy is expected to be more generic, to be less dependent on the switch count, and to avoid difficulties encountered with a geometric approach. The control problem of the converter is formulated as an underdetermined constrained optimization problem. Such problem is remarkably similar (although not identical) to the control allocation problem studied previously in flight control and marine applications [START_REF] Bodson | Evaluation of Optimization Methods for Control Allocation[END_REF], [START_REF] Johansen | Control Allocation-A survey[END_REF]. In this paper, the focus is placed on the 4-leg 2-level inverter, where a complete characterization of the solutions is possible without the use of numerical techniques, thus enabling a good understanding of the possibilities opened by this new formulation.

A previous implementation of the control allocation approach method for the 4-leg inverter was based on the description of its active voltage vectors [START_REF] Bouarfa | A New Control Allocation Method for Power Converters and its Application to the Four-Leg Two-Level Inverter[END_REF] and the regular-sampled symmetric PWM (RSPWM) [START_REF] Bowes | New sinusoidal pulsewidth-modulated invertor[END_REF]. The correct non-zero vector sequence was computed using a numerical optimization method based on the simplex algorithm, removing the need to identify a reference tetrahedron, as was done in earlier work.

Considering switching cells as available resources subjected to ranges of operation, a different control allocation method is introduced here, reducing the size of the problem considerably. Moreover, the paper proposes an analytical solution of the optimization problem that reduces the control of the inverter to the computation of the median of a special series. Furthermore, CBPWM-equivalent formulas are given for modulation laws derived from the new control allocation problem.

As constraints are taken into account in the optimization problem, solutions resulting from control allocation methods for power converters naturally yield a maximal extension of the linearity range of the inverter [START_REF] Bouarfa | A New Control Allocation Method for Power Converters and its Application to the Four-Leg Two-Level Inverter[END_REF], [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF]. Specific choices of the algorithm's parameters produce modulation laws with different properties, all implementable with few computations. Under the general umbrella of the proposed strategy, not only An optimization formulation of converter control and its general solution for the [START_REF] Holtz | High-Speed Drive System with Ultrasonic MOSFET PWM Inverter and Single-Chip Microprocessor Control[END_REF]

A. Inverter and load

The 4-leg 2-level inverter is a well-proven solution for 3phase systems with neutral wire, for applications like distributed generation, active power filtering, common-mode active filtering or fault-tolerant operation of electric drives [START_REF] Li | Analysis and Simplification of Three-Dimensional Space Vector PWM for Three-Phase Four-Leg Inverters[END_REF], [START_REF] Zhang | Three-Dimensional Space Vector Modulation for Four-Leg Voltage-Source Converters[END_REF]- [START_REF] Perales | Three-Dimensional Space Vector Modulation in abc Coordinates for Four-Leg Voltage Source Converters[END_REF]. Fig. 1 illustrates the inverter, star-connected to a load with per-phase resistances and inductances values RK, LK, K{A,B,C,N}. The load can be unbalanced or nonlinear. The fourth-leg (N) is connected to the neutral point of the load in order to control the neutral voltage and also handle possible unbalanced currents. Here, RN and LN are neglected, and the ground reference G is the lower DC bus in order to facilitate the derivation of control equations. EDC is the DC bus voltage.

B. Switching cells

A switching cell is taken to be an association of two functional switches working at complementary binary states. Dead times are neglected. Each leg is comprised of one switching cell. For K{A,B,C,N}, the state of the switching cell K is the state of the corresponding upper switch, SK, equal to 0 when the switch is off and 1 when the switch is on.

C. Output voltages, load voltages

The 4 output voltages VKG are referred to the ground G. Regarding the star-connection, it is useful to define for each of the three first legs a relative switching state SKN = SK -SN. Then, the 3 independent load voltages, referred to the neutral point N, are given by VKN = EDC SKN, for K{A,B,C}.

D. Pulse-width modulation

The mean value DK =<SK>Ts of SK over the switching period TS defines the duty cycle of the switching cell K, DK[0,1]. Mean-value references voltages are obtained with RSPWM by determining the gating pulses SK from the duty cycles DK as illustrated on Fig. 2. The load voltage mean values are

    KN DC N K DC T KN D E D D E V C B A K S        , , ,  
The linearity range of the converter is defined here as the range of amplitude of the 3-phase sinusoidal reference voltage system for which the converter delivers a voltage system with the desired values. Because of the star-connection, with classic sinusoidal PWM (SPWM), the linearity range is restricted to EDC/2. A common modification extends this range by varying the neutral potential of the load. When the desired voltages are balanced and three-phase, the maximal output amplitude achievable linearly is 1/3•EDC, as for SVM [START_REF] Holtz | High-Speed Drive System with Ultrasonic MOSFET PWM Inverter and Single-Chip Microprocessor Control[END_REF], [START_REF] Van Der Broeck | Analysis and Realization of a Pulsewidth Modulator Based on Voltage Space Vectors[END_REF] or SPWM with 1/6 third-harmonic injection (THIPWM1/6) [START_REF] Buja | Improvement of Pulse Width Modulation Techniques[END_REF], [START_REF] Houldsworth | The Use of Harmonic Distortion to Increase the Output Voltage of a Three-Phase PWM Inverter[END_REF].

III. CONTROL ALLOCATION FOR INVERTER CONTROL

A. Control allocation methods

Control allocation methods were developed in aerospace problems as a solution for over-actuated systems subjected to constraints [START_REF] Bodson | Evaluation of Optimization Methods for Control Allocation[END_REF], [START_REF] Johansen | Control Allocation-A survey[END_REF]. It is assumed that the process to be controlled can be modeled by a constrained system of m control equations with n unknowns, with m<n:

 max min , u u u a u B      
The matrix B, of size m×n, specifies the effectiveness of the actuators. B is assumed to be of rank m (rows are linearly independent). The vector u, of length n (column), is the control vector that specifies the chosen use of actuators. The vector a, of length m (column), is the resulting output vector. Denote umin and umax (vectors) as the minimum and maximum bounds of each component of the control vector u, respectively. One wants to find a control vector u to obtain a given desired output vector ad. Considering the constraints, the problem (2) may have zero, one, or an infinity of solutions. Methods have been developed to extract a solution from the whole set of feasible solutions by formulating an optimization problem.

The objective of the 4-leg 2-level inverter control is to obtain a desired reference vector ad = Vref = (VANref, VBNref, VCNref) T of 3 load voltages on average over the switching period TS. In order to scale the voltages with respect to EDC, define the scaled reference vector as ΔDref = Vref / EDC = (DANref, DBNref, DCNref) T . Here, ΔDref is similar to the difference of duty cycles between the three first legs and the fourth one, see [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF]. Then, the system of equations of (2) depends on the chosen control variables and the converter control requirements.

B. Switching cell formulation

The control problem can be formulated as finding duty cycle commands DS to the cells such that

 1 0 ,     S S S ref D D M D                  1 1 0 0 1 0 1 0 1 0 0 1 S M    T N C B A S D D D D D   
The switching cell formulation ( 3) is a simpler alternative to the space vector formulation in [START_REF] Bouarfa | A New Control Allocation Method for Power Converters and its Application to the Four-Leg Two-Level Inverter[END_REF], introducing less control variables and a smaller matrix. In order to find a control solution to the under-determined constrained control problem (3), the following optimization problem is defined:

1 0 subject to , min 0 1           S pref S S ref S S D D D D D D M J J S    
The primary criterion Jctrl(DS) = ||MSDS -ΔDref||1 is a control error based on the control equation in [START_REF] Bowes | New sinusoidal pulsewidth-modulated invertor[END_REF]. Because the solution is often not unique, the secondary criterion Jpref(DS) = ε•|DS -DSpref| is a deviation error added to specify preferred uses of the switches, where | | denotes the application of the element-wise absolute value. The column vector DSpref = (DApref, DBpref, DCpref, DNpref) T corresponds to the preferred duty cycle values. The row vector ε = (εA, εB, εC, εN) helps to give further preference to one or more switches, which makes it possible to affect the weight of deviation errors from preferred positions in Jpref. Although the parameters εK can be real numbers, we will restrict them to be integers, with little loss of generality. This choice will result in a simple solution of the optimization problem. Finally, the small real parameter ε0 gives priority to the minimization of the control error Jctrl [START_REF] Bodson | Evaluation of Optimization Methods for Control Allocation[END_REF].

The optimal solutions are determined by the choice of the parameters DSpref and ε, which produce a large number of possible configurations. We will discuss a few choices in the next sections. The associated control diagram is shown on Fig. 3.

IV. CHARACTERIZATION OF ALL FEASIBLE SOLUTIONS WHEN THE REFERENCE IS ACHIEVABLE

A. Constraint set when the reference is achievable

When the reference vector is achievable, the optimization problem can be simplified. Solutions can be determined analytically, depending on the parameters DSpref, and ε. This leads to relatively simple expressions of the modulating signals.

Due to the particular form of MS, the solutions that achieve the reference can be characterized by the single variable DN:

   N ref KN K D D D C B A K     , , ,  
If there is at least one positive and one negative voltage in the reference vector (it is the case if the reference voltage vector is three-phase), bounds DNmin and DNmax can be defined as

ref N ref N D D D D       max 1 , min max min  
and the whole set of feasible solutions that achieve the reference vector is described by constraints on DN alone:

     0 , , max min    N ctrl N N N D J D D D  
Therefore, as long as the constraint set is not empty, i.e., as long as DNmin ≤ DNmax, any signal resulting from solutions of (5) will achieve a maximal linearity range.

B. Reduction of the solution to a single decision variable

The secondary criterion to be minimized is

   pref K K N C B A K K S pref D D D J     , , , ) (   
By defining a series dmod of elements dmodK as

   ref KN pref K K D D d C B A K     mod , , ,        N C B A K K pref N N d d D d , , , mod mod mod     
and by substitution in [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF], the criterion Jpref can be now reformulated as an explicit function of DN:

   K N N C B A K K N pref d D D J mod , , , ) (       
that is a sum of distances on the real-axis from DN to the points dmodK and weighted by the parameters εK. The single decision variable DN solely represents the available functional degree of freedom. The optimization problem is considerably simplified.

C. Solution of the optimization problem

The minimization of sums of weighted absolute deviations from a unique number (here DN) to other fixed numbers (here dmodK) is a well-known mathematical problem [START_REF] Small | A Survey of Multidimensional Medians[END_REF]- [START_REF] Drezner | Facility Location: Applications and Theory[END_REF]. Here, the solution is introduced thanks to the definition of the special series dmodε, for which each dmodK appears εK times. From [START_REF] Small | A Survey of Multidimensional Medians[END_REF]- [START_REF] Drezner | Facility Location: Applications and Theory[END_REF], it can be deduced that, without considering constraints, the solution of the minimization of ( 13) reduces to the determination of the median (denoted med) of the series dmodε for all values of the design parameters.

The sum of all the εK gives the size (or number of elements) of dmodε. In practice, two distinct cases are encountered: (1) the series dmodε is of odd size: the optimal value of DN denoted DNopt is the midpoint of the series and is unique; (2) the series dmodε is of even size: DNopt is a set of values of DN belonging to the segment delimited by the two midpoints of the series.

Fig. 4 illustrates the optimal solutions depending on the nature of the size of dmodε. The criterion Jpref is a piecewise linear function due to the use of absolute values. The "even" case in Fig. 4 (a) is inconvenient because there is an infinite number of solutions, and control is not uniquely determined. In practice, a slight adjustment of the problem statement is sufficient to make the optimal solution unique, as shown in Fig. 4 (b).

Finally, the median set of the series dmodε has to be constrained by the bounds DNmin and DNmax in order to obtain feasible solutions. Therefore, optimal solutions of the constrained problem are the elements DNopt belonging to the intersection of the median set of dmodε and of the segment [DNmin, DNmax].

D. Special choices of parameters

The choice of parameters DSpref and ε has a direct and significant effect on the nature of dmodε, on the optimal set of DN, and on the resulting modulation strategy. Sets of parameters having interesting practical implications are given as examples.

The adjustment of DSpref translates into preferred duty cycle values of for each switch. The most common choices for DSpref would be DSpref {(0…0) T , (1…1) T , (0.5…0.5) T }.

The first obvious choice of ε is a vector of ones, implying no discrimination between switches. More generally, to avoid non-symmetric behavior with respect to the three phases, the first three weights should be identical. This yields the following propositions for adjusting ε: {(1…1), (1 1 1 0), (0 0 0 1)}.

V. EXAMPLES & SIMULATION RESULTS

Specific solutions are derived and illustrated for the most interesting choices of parameters DSpref and ε, as summarized in Table I. Simulations are carried out in the MATLAB Simulink environment. Evaluations of switching losses, THD on voltages and THD on currents are given later for comparison.

For the following figures showing duty cycles, the scaled reference amplitude value increases from left (a) to right (c) until the limit of 1/3 is reached. A whole fundamental period is illustrated. Reference voltages are chosen to be three-phase and balanced for simplicity, but the method also works with unbalanced references. As it will be useful for determining optimal solutions, we denote ΔDmed = med ΔDref the median of the scaled reference vector.

A. Configuration 1

Selected configuration: First of all, consider the natural choice where all preferred duty cycles are set to 0.5, which means DSpref = (0.5 0.5 0.5 0.5) T and ε = (1 1 1 1).

Determination of the median: As all the εK are identical, all the dmodK have the same occurrence in the series dmodε:

  5 . 0 , 5 . 0 , 5 . 0 , 5 . 0 mod ref CN ref BN ref AN D D D d       
The size is even. Therefore, optimal solutions DNopt belong to the segment defined by the two midpoints of dmodε. In the case where the reference system is three-phase, 0.5 is necessarily one of the two midpoints of dmodε. The second midpoint is linked to ΔDmed. Consequently,

   5 . 0 , 5 . 0 med med mod D d      
Remark: If ΔDmed is negative, the segment [0.5-ΔDmed, 0.5] has to be replaced by [0.5, 0.5-ΔDmed]. For simplicity, this trivial adjustment will be assumed to be made if necessary.

Optimal solutions: Finally, optimal solutions are any value belonging to [0.5-ΔDmed, 0.5] and saturated by [DNmin, DNmax].

Fig. 5 gives an illustration of the domain of the DNopt waveforms. The bounds DNmin and DNmax are represented by the extremal bold lines, delimiting the saturation domain. As the reference amplitude increases, the saturation domain becomes narrower. The scaled value of 1/3 is the limit where for some instants =k/3, k{0,...,6}, the two extremal bold lines meet, meaning that DNmin is equal to DNmax, see Fig. 5 (c).

In Fig. 5, the two plain midlines draw the bounds of the optimal domain of DN (hatched area). As the scaled amplitude increases, the median segment grows wider and eventually encounters the saturation domain (from (b)). The two lines labelled (1) and (2) correspond to the maximum and minimum bounds of the nonsaturated optimal solution set, respectively. Fig. 6 shows the optimal domain of the DA waveforms, determined from the optimal domain of DN by using [START_REF] Holmes | The significance of zero space vector placement for carrier-based PWM schemes[END_REF]. The fundamental wave is represented by the dashed gray midline (when visible). As for Fig. 5, the optimal domain of DA obtained if DN's is not saturated is represented by the lines labelled (1) and ( 2). The parts of any optimal saturated DN modulating signal that encounters the bounds DNmin or DNmax correspond to periods where a leg is continuously clamped to the DC bus or to the ground, as observed on Fig. 6 (b)-(c).

When DN is not saturated, a certain regular waveform of DA is identified, which stays proportional to the reference amplitude until saturation appears. Then, as a result of the saturation, the optimal domain of DA changes form compared to Fig. 6 (a) in order to achieve the correct fundamental wave. This particular property is shared by most of modulation strategies obtained from the proposed control allocation method.

TABLE I CONFIGURATION EXAMPLES OF THE CONTROL ALLOCATION METHOD n° DSpref T ε

Resulting PWM laws 1 (0.5 0.5 0.5 0.5) (1 1 1 1) undetermined 2 (0.5 0.5 0.5 0.5) (1 1 1 0) OMIPWM [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF] 3 (0.5 0.5 0.5 0.5) (0 0 0 1) ASPWM

4

(1 1 1 1) (1 1 1 1) DPWMmax [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Da Silva | Pulsewidth Modulation Strategies[END_REF], [START_REF] Dai | A FPGA-Based Generalized Pulse Width Modulator for Three-Leg Center-Split and Four-Leg Voltage Source Inverters[END_REF] ~ (0 0 0 0) (1 1 1 1) DPWMmin [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Da Silva | Pulsewidth Modulation Strategies[END_REF], [START_REF] Dai | A FPGA-Based Generalized Pulse Width Modulator for Three-Leg Center-Split and Four-Leg Voltage Source Inverters[END_REF] OMIPWM: opposite median-voltage injection PWM; ASPWM: adaptive sinus PWM; DPWM: discontinuous PWM.

B. Configuration 2

Selected configuration: With the previous configuration, the optimal solution was not unique. In order to force the series to be of odd cardinal, a first possibility is to remove the preference on the fourth duty cycle that controls the neutral potential while keeping the symmetry between the three phases. This leads to DSpref = (0.5 0.5 0.5 0.5) T and ε = (1 1 1 0).

Determination of the median: dmodN does not appear in the series dmodε and, as the other εK are identical,

  ref CN ref BN ref AN D D D d     5 . 0 , 5 . 0 , 5 . 0 mod   
Thus, the median is directly given by med dmodε = 0.5-ΔDmed.

Optimal solution: the optimal value DNopt is unique and is equal to 0.5-ΔDmed saturated by [DNmin, DNmax]. It corresponds to parts of the optimal domain given by the previous configuration and farthest from 0.5 in Fig. 5. Thanks to the analytical solution, a carrier-based equivalent method can be given. It consists in injecting the opposite of the median reference in the waveform of DN as a zero-sequence signal. This new modulation method was recently proposed in [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF] under the name of opposite median-voltage injection PWM (OMIPWM). Unlike SVM, where the zero-sequence signal always corresponds to the value of the midpoint between DNmin and DNmax (see [START_REF] Holmes | The significance of zero space vector placement for carrier-based PWM schemes[END_REF], [START_REF] Jacobina | Digital scalar pulse-width modulation: a simple approach to introduce nonsinusoidal modulating waveforms[END_REF]- [START_REF] Dai | A FPGA-Based Generalized Pulse Width Modulator for Three-Leg Center-Split and Four-Leg Voltage Source Inverters[END_REF], [START_REF] Hava | A high-performance generalized discontinuous PWM algorithm[END_REF], [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF]), DN will tend to reach the bounds DNmin and DNmax for OMIPWM. From a certain reference amplitude value, discontinuous modulation appears with increasing discontinuous periods. This feature is highlighted in Fig. 5 (b)-(c) at =/6+k/3, k{0,…,5}. OMIPWM is a new method, easy to implement, which automatically ensures the transition from a continuous to a discontinuous modulation scheme as a function of the reference amplitude [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF].

C. Configuration 3

Selected configuration: another possibility is to place a preference for the fourth duty cycle at 0.5, which means keeping the neutral potential nearest to half of the DC bus voltage: DSpref = (0.5 0.5 0.5 0.5) T and ε = (0 0 0 1).

Determination of the median: This time, only εN is non-null. dmodN = 0.5 is the only element of dmodε, thus, med dmodε = 0.5.

Optimal solution: DNopt is unique and corresponds to the value 0.5 saturated by [DNmin, DNmax]. Fig. 7 shows the resulting optimal waveform of DN. In fact, most of the time, the optimal solution corresponds to SPWM, as shown on Fig. 8. However, SPWM has a linearity range limited to the scaled amplitude value of 0.5. As can be deduced from Fig. 7 (b), the constant DN solution value of 0.5 starts to be saturated by the bounds DNmin and DNmax. It gives the minimal adjustment of the neutral potential needed to achieve the reference. A benefit of the optimal solution given by configuration 3 is that, unlike for SPWM, the maximal limit of 1/3 is reached. It appears that this solution has not been proposed before in the literature. The associated modulation law is called here adaptive sinus PWM (ASPWM).

D. Configuration 4

Selected configuration: it is also possible to adjust DSpref in order to specify a preferred value of duty cycle. For configuration 4, we set DSpref = (1 1 1 1) T and ε = (1 1 1 1).

Determination of the median: the median of the series med dmodε is not reduced to a unique element. It corresponds to [1-ΔDmed, 1] and it is always out of the saturation domain.

Optimal solution: With the selected vector DSpref, the optimal solution is DNmax, whatever the reference amplitude value is (of course comprised between 0 and 1/3). This solution (1)

out of frame (1) (2)
(2)

max DAopt = min DAopt (1) (2)
out of frame corresponds to the DPWMmax technique [START_REF] Holmes | Pulse Width Modulation for Power Converters: principles and practice[END_REF], [START_REF] Jacobina | Digital scalar pulse-width modulation: a simple approach to introduce nonsinusoidal modulating waveforms[END_REF]- [START_REF] Dai | A FPGA-Based Generalized Pulse Width Modulator for Three-Leg Center-Split and Four-Leg Voltage Source Inverters[END_REF], [START_REF] Hava | A high-performance generalized discontinuous PWM algorithm[END_REF].

(1) (2)
The same reasoning can be carried out with a preferred duty cycle value of 0. This choice leads to the optimal solution DNmin, which corresponds to the DPWMmin technique.

E. Comparison of performances

Performance criteria of importance to power electronic converters were evaluated in simulations to gain a better understanding of the differences between the various solutions. Simulation parameters are given in Table II.

Basic estimation of switching-loss energies over the simulation time were obtained by using simple parametric models derived from curves available on datasheets. Only general trends should be considered. Switching losses are shown in Fig. 9 (a). Two quasi-parallel lines, one higher than the other, describe the evolution of the switching losses, for SVM and for DPWMmin or DPWMmax. The trend for OMIPWM is remarkable since the curve follows the SVM curve, and then switches to the DPWM curve as a function of the amplitude [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF]. It is also the case for ASPWM, but the transition occurs for a much greater reference signal, because ASPWM is equivalent to SPWM until the scaled value of 0.5 is reached.

Fig. 9 (b) shows the evolution of the THD on the load voltage VAN. THD was evaluated using the function thd of the Signal Processing Toolbox in MATLAB. THD trends on voltages are very similar for all methods, but some differences are found for currents that highlight differences in the spectral distributions, see in Fig. 9 (c). As for switching losses, only general trends should be considered since the precise value of THD depends on the load impedances. The transition effect of OMIPWM is also observed with the THD on the currents. At low amplitudes, the THD for OMIPWM on the currents is higher than SVM, but fairly close compared to the THD for the DPWM's. As the amplitude increases, the THD on currents for OMIPWM converges to DPWM THD's, which have dropped and are closer to the SVM THD's. The same trend is found for ASPWM, but in a more mitigated manner, as ASPWM remains a continuous PWM method most of the time.

OMIPWM presents better harmonic quality at low reference amplitude and lower switching losses at high reference amplitude. It constitutes an advantageous compromise between continuous and discontinuous modulation [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF]. ASPWM has the particularity of moving the neutral potential as little as possible from half the DC voltage, and moreover of not having an effect on the neutral potential when not needed.

VI. EXTENSION TO MULTILEVEL, MULTILEG CONVERTERS

In this paper, we introduced a new control allocation method for the voltage control of the 4-leg 2-level inverter. However, the general approach is also applicable to multilevel, multileg converters, as long as one derives control equations linking duty cycles of switching cells to desired outputs. As the numerical optimization is based on the simplex algorithm, control equations must be linear (or linearized).

We already developed control allocation methods for the 3phase multilevel flying capacitor inverter [START_REF] Bouarfa | A fast active-balancing method for the 3-phase multi-level flying capacitor inverter derived from control allocation theory[END_REF] and the 3-phase modular multilevel inverter [START_REF] Bouarfa | A numerical optimization method using the simplex algorithm for control of modular multilevel converters[END_REF]. To ensure safe and efficient operation of these converters, additional issues must be addressed, like the active-balancing of quantities related to energy-storing elements, like capacitor voltages. Linear discrete control equations can be derived from differential equations of capacitor voltages and/or output currents by sampling and holding state values and by performing 1-order predictions. Then, a LP problem is formulated. The problem is updated and solved for each control period. As a result of the optimized control, the method in [START_REF] Bouarfa | A fast active-balancing method for the 3-phase multi-level flying capacitor inverter derived from control allocation theory[END_REF] led to a fast active-balancing of capacitor voltages. Finally, 4-leg inverters are already multileg inverters, but, in the same way, the approach is expected to be also applicable to converters with more than 4 legs.

VII. ABOUT THE HARDWARE IMPLEMENTATION OF THE CONTROL ALLOCATION METHOD

To achieve an efficient PWM operation, a high control frequency is needed. Typical switching periods are around few hundreds of microseconds. FPGA implementation is a solution for complying with these strong time constraints, thanks to short memory read/write access times and advantages of hardwired logic like parallelism and pipelining. Up to now, the problem of the FPGA implementation of the simplex algorithm has been rarely addressed, and the first implementation known to the authors was proposed in 2006 [START_REF] Bayliss | An FPGA implementation of the simplex algorithm[END_REF] with 18-bit data precision. It led to a gain in computing times up to 20 times those obtained with optimized commercial solvers on PC's [START_REF] Bayliss | An FPGA implementation of the simplex algorithm[END_REF].

We are currently working on an FPGA implementation of the simplex algorithm dedicated to control allocation methods for power converters, using a Terasic TR4 development board (Altera Stratix IV FPGA core), with 32-bit floating-point data representation. A first validation of our implementation was carried out with small-size arbitrary linear programs thanks to the ModelSim Altera software. For the control problem of the 4-leg 2-level inverter, we estimated computation times around 50 µs, which is appropriate for switching frequencies up to 10 kHz. Depending on the control objectives, the delay induced by the computing time should be taken into account by the control strategy. A solution is to perform a two-step prediction of the system states, and optimal control solutions will be applied only for the next control period.

VIII. CONCLUSIONS

In this paper, a new formulation of the control of a 4-leg 2level inverter was proposed. The problem was stated in terms of an optimization problem similar to the control allocation problem that was solved earlier in other applications, especial-ly for flight control. The control objective was thus formulated as the distribution of the 3 reference voltages to the available redundant actuators, which consisted of the duty cycles of the switching cells. The control allocation theory presents an alternative solution to converter control taking a step back from the known electronic or geometric viewpoints. The available degree of freedom is used in an optimized manner as the voltage constraint set is fully taken into account. A major interest of this framework is its possible application to problems with large number of switching cells [START_REF] Bouarfa | A fast active-balancing method for the 3-phase multi-level flying capacitor inverter derived from control allocation theory[END_REF], [START_REF] Bouarfa | A numerical optimization method using the simplex algorithm for control of modular multilevel converters[END_REF]. In the context of this paper, however, where a small size, 4-leg 2-level inverter was considered, the main result was the development of an approach providing a whole range of solutions giving a maximal extension of the inverter linearity range, all under the umbrella of a single mathematical formulation. Investigations on optimization parameters revealed several possible choices with some corresponding to existing solutions, and others providing new approaches with distinct advantages, like OMIPWM [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF] and ASPWM. An interesting feature of the analytic solutions obtained was that the resulting algorithms all reduced to carrier-based equivalent PWM methods. Moreover, they are applicable to the classic 3-leg 2-level inverter connected to a threephase balanced load [START_REF] Bouarfa | A new PWM method for a 3phase 4-leg inverter based on the injection of the opposite median reference voltage[END_REF].

As the control allocation method optimizes the use of available resources with a general problem formulation, the approach seems promising for implementing a fault tolerant capability. However, this feature was not studied in this paper as the 4-leg 2-level inverter is not adequate for a meaningful investigation of redundancy.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Illustration of the 4-leg 2-level inverter connected to a three-phase load with impedances ZA, ZB, ZC.

Fig. 3 .

 3 Fig. 3. Diagram of the proposed control allocation method.

Fig. 4 .

 4 Fig. 4. Illustration of the effect of parameters on the solving of the simplified optimization problem with arbitrary ΔDref. DSpref = (0.5 0.5 0.5 0.5) T .

Fig. 5 .

 5 Fig. 5. Optimal waveforms of DN with DSpref = (0.5 0.5 0.5 0.5) T , ε = (1 1 1 1). Labels (1) and (2) indicate nonsaturated lines for max, min DNopt, respectively.

Fig. 6 .

 6 Fig. 6. Optimal waveforms of DA with DSpref = (0.5 0.5 0.5 0.5) T , ε = (1 1 1 1). Labels (1) and (2) indicate nonsaturated lines for max, min DAopt, respectively.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Optimal waveforms of DN with DSpref = (0.5 0.5 0.5 0.5) T , ε = (0 0 0 1), corresponding to ASPWM.

Fig. 9 .

 9 Fig. 9. Performance evaluation in simulation for proposed configurations. (a) Switching losses. (b) THD on load voltage VAN. (c) THD on load current IA.