Magali Kitzmann

Louise Lantier

Sophie Hébrard

Jacques Mercier

Marc Foretz

Celine Aguer
email: caguer@uottawa.ca

Abnormal metabolism flexibility in response to high palmitate concentrations in myotubes derived from obese type 2 diabetic patients

Keywords: lipid overflow, palmitate beta-oxidation, mitochondria, acetyl-CoA carboxylase, AMP activated protein kinase T2D, type 2 diabetes, IMCL, intramuscular lipid, ACC, acetyl-CoA carboxylase, AMP activated protein kinase, AMPK, AICAR, AMP-mimetic

Insulin resistance in type 2 diabetes (T2D) is associated with intramuscular lipid (IMCL) accumulation. To determine whether impaired lipid oxidation is involved in IMCL accumulation, we measured expression of genes involved in mitochondrial oxidative metabolism or biogenesis, mitochondrial content and palmitate beta-oxidation before and after palmitate overload (600 µM for 16h), in myotubes derived from healthy subjects and obese T2D patients. Mitochondrial gene expression, content and network were not different between groups. Basal palmitate beta-oxidation was not affected in T2D myotubes, whereas after 16h of palmitate pre-treatment, T2D myotubes in contrast to control myotubes, showed an inability to increase palmitate beta-oxidation (p<0.05).

Interestingly, acetyl-CoA carboxylase (ACC) phosphorylation was increased with a tendency for statistical significance after palmitate pre-treatment in control myotubes (p=0.06) but not in T2D myotubes which can explain their inability to increase palmitate beta-oxidation after palmitate overload. To determine whether the activation of the AMP activated protein kinase (AMPK)-ACC pathway was able to decrease lipid content in T2D myotubes, cells were treated with AICAR and metformin. These AMPK activators had no effect on ACC and AMPK phosphorylation in T2D myotubes as well as on lipid content, whereas AICAR, but not metformin, increased AMPK phosphorylation in control myotubes. Interestingly, metformin treatment and mitochondrial inhibition by antimycin induced increased lipid content in control myotubes. We conclude that T2D myotubes display an impaired capacity to respond to metabolic stimuli.

shown to be related to diverse metabolic diseases including dyslipidemia, cardiovascular disease, insulin resistance and type 2 diabetes (T2D). In healthy people, skeletal muscle accounts ~80% of postprandial glucose disposal. However, during T2D development, skeletal muscle becomes primary significant site of insulin resistance [START_REF] Defronzo | The effect of insulin on the disposal of intravenous glucose. Results from indirect calorimetry and hepatic and femoral venous catheterization[END_REF][START_REF] Shulman | Quantitation of muscle glycogen synthesis in normal subjects and subjects with non-insulin-dependent diabetes by 13C nuclear magnetic resonance spectroscopy[END_REF]. Several studies have demonstrated a strong negative relationship between ectopic lipids accumulation in skeletal muscle (intramyocellular lipid (IMCL)) and whole body insulin resistance [START_REF] Kriketos | Multiple indexes of lipid availability are independently related to whole body insulin action in healthy humans[END_REF][START_REF] Pan | Skeletal muscle triglyceride levels are inversely related to insulin action[END_REF][START_REF] Perseghin | Intramyocellular triglyceride content is a determinant of in vivo insulin resistance in humans: a 1H-13C nuclear magnetic resonance spectroscopy assessment in offspring of type 2 diabetic parents[END_REF]. However, the precise mechanisms leading to IMCL accumulation during T2D development are still under investigation. During the last decade, several studies have focused on the implication of mitochondrial dysfunction in insulin resistance development and IMCL accumulation [START_REF] Kelley | Dysfunction of mitochondria in human skeletal muscle in type 2 diabetes[END_REF][START_REF] Mogensen | Mitochondrial respiration is decreased in skeletal muscle of patients with type 2 diabetes[END_REF][START_REF] Phielix | Lower intrinsic ADP-stimulated mitochondrial respiration underlies in vivo mitochondrial dysfunction in muscle of male type 2 diabetic patients[END_REF][START_REF] Petersen | Impaired mitochondrial activity in the insulin-resistant offspring of patients with type 2 diabetes[END_REF][START_REF] Befroy | Impaired mitochondrial substrate oxidation in muscle of insulin-resistant offspring of type 2 diabetic patients[END_REF]. Mitochondrial oxidative phosphorylation was shown to be reduced in the skeletal muscle of type 2 diabetic patients [START_REF] Kelley | Dysfunction of mitochondria in human skeletal muscle in type 2 diabetes[END_REF][START_REF] Mogensen | Mitochondrial respiration is decreased in skeletal muscle of patients with type 2 diabetes[END_REF][START_REF] Phielix | Lower intrinsic ADP-stimulated mitochondrial respiration underlies in vivo mitochondrial dysfunction in muscle of male type 2 diabetic patients[END_REF] and lean insulin-resistant offspring of type 2 diabetic parents [START_REF] Petersen | Impaired mitochondrial activity in the insulin-resistant offspring of patients with type 2 diabetes[END_REF][START_REF] Befroy | Impaired mitochondrial substrate oxidation in muscle of insulin-resistant offspring of type 2 diabetic patients[END_REF]. However, more recent studies have shown that mitochondrial function is not altered in T2D skeletal muscle when normalised for mitochondrial content [START_REF] Boushel | Patients with type 2 diabetes have normal mitochondrial function in skeletal muscle[END_REF] or when diabetic and non-diabetic subjects are matched for body composition [START_REF] De Feyter | Early or advanced stage type 2 diabetes is not accompanied by in vivo skeletal muscle mitochondrial dysfunction[END_REF]. Another hypothesis to explain increased lipid content in skeletal muscle of diabetic patients is an increased fatty-acid (FA) uptake due to increased membrane expression of the major muscle FA transporter FAT/CD36 [START_REF] Bonen | Triacylglycerol accumulation in human obesity and type 2 diabetes is associated with increased rates of skeletal muscle fatty acid transport and increased sarcolemmal FAT/CD36[END_REF].

Primary human myotubes isolated from type 2 diabetic patients have been shown to express metabolic characteristics that approximate the in vivo phenotype of the donor subject [START_REF] Bouzakri | Reduced activation of phosphatidylinositol-3 kinase and increased serine 636 phosphorylation of insulin receptor substrate-1 in primary culture of skeletal muscle cells from patients with type 2 diabetes[END_REF][START_REF] Nikoulina | Impaired muscle glycogen synthase in type 2 diabetes is associated with diminished phosphatidylinositol 3-kinase activation[END_REF][START_REF] Gaster | The diabetic phenotype is conserved in myotubes established from diabetic subjects: evidence for primary defects in glucose transport and glycogen synthase activity[END_REF][START_REF] Gaster | The reduced insulin-mediated glucose oxidation in skeletal muscle from type 2 diabetic subjects may be of genetic origin--evidence from cultured myotubes[END_REF][START_REF] Henry | Glycogen synthase activity is reduced in cultured skeletal muscle cells of non-insulin-dependent diabetes mellitus subjects. Biochemical and molecular mechanisms[END_REF][START_REF] Hulver | Elevated stearoyl-CoA desaturase-1 expression in skeletal muscle contributes to abnormal fatty acid partitioning in obese humans[END_REF]. This in vitro muscle system provides an attractive model in which lipid accumulation can be evaluated apart from the systemic influences of the in vivo environment. We have recently shown in this model that an increased membrane expression of FAT/CD36 could explain ~50% of the increase in lipid content in myotubes derived from obese T2D patients [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF]. Thus, other mechanisms may be involved in increased IMCL content in myotubes derived from obese T2D patients.

Reduced lipid oxidation, particularly oxidation of saturated FA [START_REF] Gaster | Reduced lipid oxidation in skeletal muscle from type 2 diabetic subjects may be of genetic origin: evidence from cultured myotubes[END_REF][START_REF] Gaster | Differential utilization of saturated palmitate and unsaturated oleate: evidence from cultured myotubes[END_REF], as well as

A C C E P T E D M A N U S C R I P T

reduced citrate synthase activity [START_REF] Ortenblad | Reduced insulin-mediated citrate synthase activity in cultured skeletal muscle cells from patients with type 2 diabetes: evidence for an intrinsic oxidative enzyme defect[END_REF], have been shown to be maintained in myotubes established from obese T2D patients. More recently, it has been shown that myotubes derived from obese T2D patients present a decrease in intramyocellular lipid oxidation compared to control myotubes and that this decrease in lipid oxidation was not the result of lower mitochondrial mass [START_REF] Corpeleijn | Oxidation of intramyocellular lipids is dependent on mitochondrial function and the availability of extracellular fatty acids[END_REF]. However, the cause of impaired mitochondrial function per se during T2D is not clearly understood and still under investigation.

Among the different treatments for type 2 diabetes, it is important to note that some of them act through the same target. Indeed, physical activity, metformin and thiazolidinediones activate AMP activated protein kinase (AMPK). AMPK is a major metabolic sensor which is activated by an increase in the ratio of AMP/ATP in order to restore energy status of the cell by stimulating ATP-producing pathways (glucose uptake, fatty acid oxidation, and mitochondrial biogenesis) [START_REF] Suwa | Effects of chronic AICAR treatment on fiber composition, enzyme activity, UCP3, and PGC-1 in rat muscles[END_REF][START_REF] Jager | AMP-activated protein kinase (AMPK) action in skeletal muscle via direct phosphorylation of PGC-1alpha[END_REF][START_REF] Koistinen | 5-amino-imidazole carboxamide riboside increases glucose transport and cell-surface GLUT4 content in skeletal muscle from subjects with type 2 diabetes[END_REF] and inhibiting ATPconsuming pathways (fatty acid synthesis, glycogen synthesis, and protein synthesis) [START_REF] Foretz | AMP-activated protein kinase inhibits the glucose-activated expression of fatty acid synthase gene in rat hepatocytes[END_REF][START_REF] Foretz | Short-term overexpression of a constitutively active form of AMP-activated protein kinase in the liver leads to mild hypoglycemia and fatty liver[END_REF][START_REF] Woods | Characterization of the role of AMP-activated protein kinase in the regulation of glucose-activated gene expression using constitutively active and dominant negative forms of the kinase[END_REF]. Interestingly, some studies have shown that the activation of AMPK induces an increase in mitochondrial function and lipid oxidation leading to decreased IMCL content [START_REF] Bandyopadhyay | Increased malonyl-CoA levels in muscle from obese and type 2 diabetic subjects lead to decreased fatty acid oxidation and increased lipogenesis; thiazolidinedione treatment reverses these defects[END_REF][START_REF] Muoio | AMP-activated kinase reciprocally regulates triacylglycerol synthesis and fatty acid oxidation in liver and muscle: evidence that sn-glycerol-3-phosphate acyltransferase is a novel target[END_REF][START_REF] Barnes | The 5'-AMP-activated protein kinase gamma3 isoform has a key role in carbohydrate and lipid metabolism in glycolytic skeletal muscle[END_REF]. This effect could be the result of an increase in long-chain FA entrance and oxidation in mitochondria due to the reduction in ACC (acetyl-CoA carboxylase) activity, a well known target of AMPK [START_REF] Merrill | AICA riboside increases AMP-activated protein kinase, fatty acid oxidation, and glucose uptake in rat muscle[END_REF][START_REF] Awan | Malonyl-CoA metabolism in cardiac myocytes and its relevance to the control of fatty acid oxidation[END_REF]. Taken together these results show that AMPK activation may be an excellent target in order to decrease lipid content in the skeletal muscle of type 2 diabetes patients.

In order to gain further insight into the mechanisms underlying elevated lipid accumulation in T2D primary muscle cells, the aim of our study was to determine whether lipid oxidation is affected in basal condition or after palmitate overload in diabetic myotubes compared to control myotubes and to better understand the mechanisms leading to these defects. We also tested whether AMPK activation could modulate lipid content in these myotubes.

A C C E P T E D

M A N U S C R I P T ACCEPTED MANUSCRIPT 6

Materials and methods

Primary human satellite cell culture

Skeletal muscle biopsy of the vastus lateralis was performed according to the percutaneous Bergström technique after local anesthesia (xylocaine) [START_REF] Thomas | Relationships between maximal muscle oxidative capacity and blood lactate removal after supramaximal exercise and fatigue indexes in humans[END_REF][START_REF] Thomas | Monocarboxylate transporters, blood lactate removal after supramaximal exercise, and fatigue indexes in humans[END_REF] [START_REF] Gaster | Reduced lipid oxidation in skeletal muscle from type 2 diabetic subjects may be of genetic origin: evidence from cultured myotubes[END_REF][START_REF] Kitzmann | The muscle regulatory factors MyoD and myf-5 undergo distinct cell cyclespecific expression in muscle cells[END_REF]. Cell culture of primary human satellite cells was performed as previously described [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF]. Cultures were maintained in a growth medium composed of (DMEM, 10% FBS and 1% Ultroser G) and when myoblasts reached confluence, medium was changed (growth medium minus Ultroser G) and the differentiation process occurred until fusion and terminal differentiation into contractile myotubes (8 days). The experiments were performed on passages 2 to 4, after 8 days of differentiation.

Myotube treatments

Treatments performed on differentiated myotubes are detailed in the figure legends and were done in triplicate for each of the independent cell cultures. The following reagents were purchased from Sigma-Aldrich (Saint Quentin Fallavier, France): L-Glutamine, DMEM, antimycin A, AMP-mimetic 5-aminoimidazole-4-carboxamide-1-β-Dribofuranoside (AICAR), metformin and palmitate. Fetal bovine serum was from Hyclone (Brebières, France). Palmitate was dissolved in chloroform (0.6 mol/l), diluted to 600 µM in DMEM containing 10% of FBS (differentiation medium) and added to the myotubes as previously described [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF][START_REF] Aguer | Lipid content and response to insulin are not invariably linked in human muscle cells[END_REF].

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 7

Lipid accumulation

Lipid accumulation in myotubes was visualized by oil red O (Sigma, Saint-Quentin Fallavier, France) staining as previously described [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF][START_REF] Aguer | Lipid content and response to insulin are not invariably linked in human muscle cells[END_REF][START_REF] Gaster | Triacylglycerol accumulation is not primarily affected in myotubes established from type 2 diabetic subjects[END_REF]. Oil red O staining was carried for one hour followed by 2 or 3 washes with distilled water. Lipid droplets were then visualized by light microscopy. In order to quantify lipid accumulation in myotubes, oil red O was extracted using isopropanol. The absorbance value was measured using a spectrophotometer set at 490 nm and blanked to untreated cells [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF][START_REF] Aguer | Lipid content and response to insulin are not invariably linked in human muscle cells[END_REF][START_REF] Gaster | Triacylglycerol accumulation is not primarily affected in myotubes established from type 2 diabetic subjects[END_REF].

Real-time RT-PCR analysis

Total RNA was isolated with Trizol (Invitrogen, Cergy-Pontoise, France) and singlestrand cDNA was synthesized from 2.5 µg of total RNA with random hexamer primers and Superscript II (Invitrogen, Cergy-Pontoise, France). Real-time RT-PCR reactions were carried out with a LightCycler reaction kit (Eurogenetec, Angers, France) in a final volume of 20 µl containing 125 ng of reverse-transcribed total RNA, 500 nM of primers [START_REF] Civitarese | Role of adiponectin in human skeletal muscle bioenergetics[END_REF], 10 µl of 2x PCR mix and 0.5 µl of SYBR Green. The reactions were carried out in capillaries in a LightCycler instrument (Roche, Rosny-sous-bois, France) with 40 cycles.

We determined the relative amounts of the mRNAs studied by means of the second derivative maximum method, with LightCycler analysis software version 3.5 and 18S

RNA as the invariant control for all studies.

Western blots

Cellular extracts were quantified and lysed in Laemmli buffer. 30-50 µg of total proteins were transferred to nitro-cellulose membranes (Schleicher and Schuell, Bioscience, Dassel, Germany). Western blots were performed as described [START_REF] Kitzmann | The muscle regulatory factors MyoD and myf-5 undergo distinct cell cyclespecific expression in muscle cells[END_REF]. The following primary antibodies were used: phospho-Ser222-ACC, ACC, phospho-Thr172-AMPK and AMPK (Cell signaling, Paris, France), non-glycosylated protein component of mitochondria (Ab2, Interchim, Montlucon, France); and anti-α-tubulin (Sigma Saint-Quentin Fallavier, France) was used as a loading control. The secondary antibodies were anti-rabbit and anti-mouse antibodies coupled to horseradish peroxidase (GE Healthcare, Orsay, France). Proteins were visualized using an enhanced luminescent reagent (Tebu-

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8
Bio), and exposed on film (GE Healthcare, Orsay, France). Expression of proteins was quantified by density analysis using ImageJ Launcher Software.

Citrate synthase activity

Cellular extracts were homogenized in a solution containing: 10 mM Tris HCl (pH 7.

Immunofluorescence

Differentiated satellite cells were fixed with 4% paraformaldehyde in PBS for 10min, permeabilized with 0.5% triton in PBS for 2min, saturated with 0.5% BSA in PBS for 10min and then incubated with primary antibodies against a non-glycosylated protein component of mitochondria (Ab2, Interchim, Montlucon, France). Secondary antibody was Alexa Fluor 596 anti-mouse (Invitrogen, Molecular Probes, France).

Palmitate beta-oxidation

Cells were cultured in 96-well plates and differentiated as described above. Differentiated satellite cells were exposed or not to cold palmitate (0.6 mM) for the last 16h of

Statistical analysis

Statistical analyses were performed using Statview 5.0. Data are means ± SEM.

Statistical analyses were performed using Student's t test for unpaired and paired comparison, or ANOVA with Fisher's PLSD post-hoc test for multiple comparisons.

P<0.05 was considered to be significant.

A C C E P T E D M A N U S C R I P T

Results

As previously shown by our team, lipid content in the presence of palmitate is increased in myotubes derived from obese T2D patients as compared to myotubes derived from healthy control subjects [START_REF] Aguer | Intramyocellular lipid accumulation is associated with permanent relocation ex vivo and in vitro of fatty acid translocase (FAT)/CD36 in obese patients[END_REF]. Here, we have confirmed this finding (Fig S1A andB, (p<0.01)).

Mitochondrial content and network are normal in T2D myotubes

We examined several mitochondrial parameters between control and T2D myotubes in order to determine whether mitochondrial content is decreased in T2D myotubes.

We first measured the mRNA levels of genes involved in mitochondrial oxidative metabolism (citrate synthase (CS) and 3-hydroxy-acyl-CoA-dehydrogenase (HAD)), of a gene known to mediate the transcriptional control of mitochondrial biogenesis (nuclear respiratory factor 1 (NRF1)) and of a gene involved mitochondrial lipid uptake (carnitine palmitoyltransferase 1 (CPT1)). The expression of all of these genes was not significantly different between control and T2D myotubes (Fig. 1A).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 11

We determined by western blot the expression of a non-glycosylated protein component of mitochondria (Ab2) which has been previously used by others as a marker of mitochondria in subcellular fractions [START_REF] Gonzalez | N-acetylcysteine, coenzyme Q10 and superoxide dismutase mimetic prevent mitochondrial cell dysfunction and cell death induced by d-galactosamine in primary culture of human hepatocytes[END_REF] (Fig. 1B). As a second marker of mitochondrial content, we measured the activity of citrate synthase ( Fig. 1C). As shown in figure 1B and C, the expression of the non-glycosylated protein component of mitochondria was variable between subjects independently of type 2 diabetes, and citrate synthase activity was not significantly different between control and T2D myotubes. Furthermore, we have performed immunofluorescence experiments to check the specificity of the antibody used in western blot as well as to determine whether the mitochondrial network was different between the two groups (Fig. 1D). As shown in Figure 1D, mitochondria Ab-2 antibody showed a specific mitochondrial "spaghetti"-like staining pattern in the cytoplasm of control as well as in T2D myotubes, without obvious differences in morphology between groups.

Taken together, results presented in figure 1 show that neither mitochondrial content nor the mitochondrial network are affected in differentiated primary muscle cells derived from obese T2D patients.

Palmitate pre-treatment reveals mitochondrial dysfunction and an abnormal response at the level of ACC phosphorylation in T2D differentiated cells

A reduction in fatty acid oxidation with no difference in mitochondrial content has been observed in T2D myotubes [START_REF] Corpeleijn | Oxidation of intramyocellular lipids is dependent on mitochondrial function and the availability of extracellular fatty acids[END_REF]. We measured palmitate beta-oxidation in fully differentiated myotubes without palmitate pre-treatment (-) or with 16h of palmitate pretreatment (+, cold palmitate, 600 µM). As shown in figure 2A, palmitate beta-oxidation was not significantly different in T2D differentiated cells compared to control cells in the basal state, but was significantly lower in T2D myotubes compared to control myotubes after palmitate overload (p<0.05). There was a trend for an increased palmitate oxidation after palmitate pre-treatment compared to basal condition in control myotubes (p=0.09),

but not in T2D myotubes (P=0.37).

We have next monitored ACC phosphorylation (expressed as a ratio of phosphorylated to total protein) (Fig. 2B andC) in 4 cultures of control and 4 cultures of T2D differentiated

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

12 muscle cells. In the basal state, ACC phosphorylation was not different between groups (Fig. 2B andC).

We next examined the impact of palmitate pre-treatment (600 µM, 16h) on ACC phosphorylation (Fig. 2D andE). Palmitate pre-treatment was able to increase ACC phosphorylation in control myotubes with a tendency of statistical significance (p=0.06)

but not in T2D myotubes (Fig. 2D andE).

Figure 2 shows that T2D myotubes are not able to increase palmitate beta-oxidation and ACC phosphorylation to the same degree as control myotubes in response to high concentrations of palmitate pre-treatment. These results suggest that the impairment in palmitate beta-oxidation in response to high level of palmitate pre-treatment in T2D myotubes (Fig. 2A) could be explained by an abnormal response to palmitate at the level of ACC phosphorylation (Fig. 2D andE).

AICAR or metformin treatments are not able to decrease lipid content in T2D myotubes

To test whether AMPK activation was able to decrease lipid content in T2D myotubes, fully differentiated myotubes were treated with 2 different AMPK activators: AICAR (5aminoimidazole-4-carboxamide-β-D-ribofuranoside) and metformin (Fig. 3). AICAR treatment for 16h showed a significantly increased AMPK phosphorylation only in control myotubes (p<0.05) but not in T2D myotubes, due to a greater variability in the response to AICAR treatment in our T2D cells (Fig. 3 A andB). However, the phosphorylation of ACC was not increased after AICAR treatment in the 2 groups of cells (Fig. 3A andC). Metformin treatment for 16h did not increase either AMPK or ACC phosphorylation in control and T2D myotubes (Fig. 3A, B andC). As a consequence, no decrease in lipid accumulation was observed in T2D myotubes after metformin and AICAR treatments (Fig. 3D, left panel). AICAR treatment also had no effect on lipid content in control myotubes (Fig. 3D, left panel). Surprisingly, long term metformin treatment induced a significant increase in lipid content in control myotubes but not in myotubes from T2D patients (Fig. 3D, right panel).

A C C E P T E D M A N U S C R I P T

Mitochondrial inhibition alters lipid content in control myotubes

AICAR activates AMPK by increasing the intracellular pool of ZMP, an AMPK analogue (for review see [START_REF] Towler | AMP-activated protein kinase in metabolic control and insulin signaling[END_REF]). Metformin activates AMPK indirectly by inhibiting complex I of the respiratory chain [START_REF] El-Mir | Dimethylbiguanide inhibits cell respiration via an indirect effect targeted on the respiratory chain complex I[END_REF][START_REF] Brunmair | Thiazolidinediones, like metformin, inhibit respiratory complex I: a common mechanism contributing to their antidiabetic actions?[END_REF].Thus we sought to examine the effect of antimycin A, a mitochondrial inhibitor (complex III), before challenging cells with a high concentration of palmitate (600 µM for 16h) in order to evaluate if the metformin effect on lipid accumulation could be attributable to mitochondrial inhibition. As shown in figure 4 A and B, after treatment of the cells with antimycin A, palmitate-induced lipid accumulation in control myotubes was significantly increased by 22% (p<0.05).

Interestingly, no increase in lipid content was found in T2D myotubes after antimycin A treatment which is in accordance with the results shown in Fig. 4 after metformin treatment.

Figure 4 shows that mitochondrial inhibition is sufficient to increase lipid content in control myotubes to the same level as in T2D myotubes.

A C C E P T E D M A N U S C R I P T

Discussion

Mitochondrial involvement in intramuscular lipid accumulation and T2D development has been widely studied over the last 10 years. Numerous studies have shown a mitochondrial dysfunction in skeletal muscle of T2D patients [START_REF] Kelley | Dysfunction of mitochondria in human skeletal muscle in type 2 diabetes[END_REF][START_REF] Mogensen | Mitochondrial respiration is decreased in skeletal muscle of patients with type 2 diabetes[END_REF][START_REF] Phielix | Lower intrinsic ADP-stimulated mitochondrial respiration underlies in vivo mitochondrial dysfunction in muscle of male type 2 diabetic patients[END_REF] as well as in muscle from persons at risk for developing T2D [START_REF] Petersen | Impaired mitochondrial activity in the insulin-resistant offspring of patients with type 2 diabetes[END_REF][START_REF] Befroy | Impaired mitochondrial substrate oxidation in muscle of insulin-resistant offspring of type 2 diabetic patients[END_REF]. However, these results have been refuted in more recent studies showing that mitochondrial function was not altered in T2D skeletal muscle after normalization by mitochondria content [START_REF] Boushel | Patients with type 2 diabetes have normal mitochondrial function in skeletal muscle[END_REF] or when T2D and control subjects were matched for obesity [START_REF] De Feyter | Early or advanced stage type 2 diabetes is not accompanied by in vivo skeletal muscle mitochondrial dysfunction[END_REF]. Thus, the involvement of mitochondrial dysfunction in T2D is still under investigation. In the present study, we have shown that baseline palmitate oxidation and mitochondrial content were similar between primary human satellite cells derived from healthy control subjects and moderately obese T2D patients. A defect in palmitate beta-oxidation was only revealed when T2D myotubes were challenged with high concentrations of palmitate, and this impaired response to palmitate overload could be the result of a lack of inhibition of ACC activity by palmitate. Thus a mitochondrial dysfunction in T2D myotubes is only revealed upon challenge with saturated fatty acid overload. This result is in accordance with another study showing that myotubes from patients with a history of type 2 diabetes were unable to adapt to a hyperglycaemic-hyperinsulinaemic challenge [START_REF] Costford | Increased susceptibility to oxidative damage in post-diabetic human myotubes[END_REF]. Thus, it seems that response to nutrient challenge is impaired in T2D myotubes.

In the present study, we used primary muscle cells in culture to study mitochondrial fatty acid oxidation apart from the systemic influences of the in vivo environment.

Interestingly, under basal conditions (no palmitate pre-treatment), no evidence for differential expression of genes involved in mitochondrial oxidative metabolism or in genes involved in mitochondrial biogenesis was found between myotubes derived from lean healthy subjects and obese type 2 diabetic patients, confirming results previously described [START_REF] Frederiksen | Transcriptional profiling of myotubes from patients with type 2 diabetes: no evidence for a primary defect in oxidative phosphorylation genes[END_REF]. Furthermore, we found no differences in mitochondrial content (estimated by citrate synthase activity or expression of a mitochondrial protein) or in the mitochondrial network between the two populations of cells. These results differ from previous published findings showing an intrinsic defect in citrate synthase activity [START_REF] Ortenblad | Reduced insulin-mediated citrate synthase activity in cultured skeletal muscle cells from patients with type 2 diabetes: evidence for an intrinsic oxidative enzyme defect[END_REF] but are in accordance with a recent study [START_REF] Corpeleijn | Oxidation of intramyocellular lipids is dependent on mitochondrial function and the availability of extracellular fatty acids[END_REF].

A C C E P T E D M A N U S C R I P T

Under basal conditions, we found no significant difference in palmitate beta-oxidation in cells derived from obese type 2 diabetic patients compared to those derived from control subjects. This result may be due to a lack of power in consequence of the small n size.

However, only one out of 3 T2D myotubes show smaller beta-oxidation values than the control cells. Interestingly, after palmitate pre-treatment for 16h, palmitate beta-oxidation was significantly decreased in T2D myotubes compared to control cells. Palmitate pretreatment also revealed a greater increase in lipid content in T2D myotubes as compared to control cells, whereas no difference in lipid accumulation was found between the two groups of myotubes in basal conditions (not shown). Thus, control myotubes were able to increase palmitate beta-oxidation after palmitate pre-treatment allowing an only moderate lipid accumulation whereas T2D myotubes showed a smaller response to high palmitate pre-treatment at the level of palmitate beta-oxidation (compared Fig. 2A with Fig. 2B) leading to greater lipid accumulation. The stimulation of fatty acid betaoxidation by high concentrations of palmitate in control myotubes is in accordance with another study [START_REF] Gaster | The dynamic of lipid oxidation in human myotubes[END_REF]. Moreover, direct inhibition of the electron transport chain by antimycin A and metformin was able to increase palmitate-induced lipid accumulation in control myotubes showing that mitochondrial inhibition can in fact lead to increased lipid content in control myotubes. In contrast, in satellite cells derived from obese T2D patients, neither antimycin A nor metformin treatment lead to an increase in lipid content in T2D myotubes indicating that mitochondrial function was already defective in T2D myotubes or that the lipid accumulation capacity in T2D myotubes was already at its maximum without these treatments.

To obtain mechanistic insights into the modified response to palmitate pre-treatment in T2D myotubes, we monitored changes in P-ACC. Active (dephosphorylated) ACC inhibits fatty acid oxidation by increasing the availability of malonyl-CoA which, in turn, inhibits carnitine palmitoyl transferase 1 (CPT1) and the entrance of long-chain fatty acid in mitochondria. Phosphorylation of ACC inhibits its activity leading to increased fatty acid uptake by mitochondria and, subsequently, increased fatty acid oxidation [START_REF] Park | Phosphorylation-activity relationships of AMPK and acetyl-CoA carboxylase in muscle[END_REF][START_REF] Winder | Inactivation of acetyl-CoA carboxylase and activation of AMP-activated protein kinase in muscle during exercise[END_REF]. In the basal state, no difference in P-ACC or ACC expression was observed between groups. After palmitate pre-treatment, the phosphorylation of ACC tend to be increased in control myotubes which is in accordance with the increase in palmitate beta-oxidation

A C C E P T E D M A N U S C R I P T
is those myotubes. Interestingly, in T2D myotubes, no modification of the phosphorylation of ACC was observed after palmitate overload. Thus a lack of inhibition of ACC could explain why palmitate beta-oxidation was not increase in T2D myotubes in response to palmitate overload. Interestingly lack of a significant increase in AMPK phosphorylation was also found in T2D myotubes following AICAR treatment. This result, taken together with the absence of ACC inhibition by palmitate treatment in T2D myotubes suggests that T2D myotubes display a defective response to different stimuli at the level of the AMPK-ACC pathway. These preliminary but really intriguing results need further investigation to better understand the mechanisms leading to this defect in T2D myotubes. Unfortunately, due to the inability of T2D myotubes to activate the AMPK-ACC pathway, we were not able to reduce lipid content in T2D myotubes. Other experiments are necessary to determine whether AMPK is a good target in order to decrease lipid content in T2D myotubes.

Some methodological aspects of the study should be acknowledged. ). In the present study, we focused on response to palmitate overload since it is well known that insulin resistance development in response to high FA level is the consequence of high saturated FA level (i.e. palmitate) and not to high unsaturated FA level (i.e. oleate) [START_REF] Schmitz-Peiffer | Ceramide generation is sufficient to account for the inhibition of the insulin-stimulated PKB pathway in C2C12 skeletal muscle cells pretreated with palmitate[END_REF][START_REF] Perdomo | Increased beta-oxidation in muscle cells enhances insulin-stimulated glucose metabolism and protects against fatty acid-induced insulin resistance despite intramyocellular lipid accumulation[END_REF][START_REF] Pickersgill | Key role for ceramides in mediating insulin resistance in human muscle cells[END_REF][START_REF] Yuzefovych | Different effects of oleate versus palmitate on mitochondrial function, apoptosis and insulin signaling in L6 skeletal muscle cells: role of oxidative stress[END_REF]. W acknowledge that the use of 600 µM of palmitate is not physiological since, in vivo, circulating free FA are a mix of saturated and unsaturated FA. The best experiment would be to use a cocktail of FA (i.e. oleate + palmitate) since it has been shown that the adverse effect of palmitate on cellular functions can be rescued with the addition of oleate [START_REF] Pickersgill | Key role for ceramides in mediating insulin resistance in human muscle cells[END_REF][START_REF] Yuzefovych | Different effects of oleate versus palmitate on mitochondrial function, apoptosis and insulin signaling in L6 skeletal muscle cells: role of oxidative stress[END_REF]. However, our results are in accordance with a recent study using only oleate as FA supply [START_REF] Corpeleijn | Oxidation of intramyocellular lipids is dependent on mitochondrial function and the availability of extracellular fatty acids[END_REF].

In summary, this study shows that skeletal muscle cells derived from moderately obese

  HAD, 3-hydroxy-acyl-CoA-dehydrogenase; NRF1, nuclear respiratory factor 1; CPT1, carnitine palmitoyltransferase 1 A C C E P T E D M A N U S C R I P T 1. Introduction During overweight and obesity development, a redistribution of adipose tissue occurs with increased accumulation of visceral fat. This ectopic lipid accumulation has been

  EDTA, 75 mM saccharose and 225 mM mannitol. Citrate synthase activity was measured with 0.5 mM oxaloacetate, 0.3 mM acetyl-CoA, 0.1 mM of 5, 5'-Dithiobis 2nitro-benzoic acid (DTNB), 100 mM Tris HCL (pH=8.0) and 0.1% Triton 100X. Enzyme activity was monitored by recording the changes in absorbance at 412 nm over 2min at 37°C, and normalized to protein content.

  differentiation and then exposed to differentiation medium (DMEM + 10% Foetal Bovin Serum) supplemented with 1% bovine serum albumin, 50 µM palmitate and 9.5 µM (0,09 µCi) [1-14 C]palmitate. Identical incubations were conducted on parallel plates that contained no cells. Palmitate beta-oxidation was determined by measuring production of 14 C-labeled acid-soluble metabolites (ASM), a measure of tricarboxylic acid cycle intermediates and acetyl esters. After incubation for 30 min at 37°C, reactions were terminated by aspiration of the media and addition of 100 µl of HClO 4 at 5% for 15min at room temperature. The ASM was assayed in supernatants of the acid precipitate. Radioactivity of ASM was determined by liquid scintillation counting by use of 4.5 ml of liquid scintillation cocktail (Optiphase 'Hisafe' 3, Perkin Elmer) in scintillation vials. For A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 9 protein determination, identical incubations were conducted on parallel plates with the same number of cells.

T2D patients exhibited decreasedFig. 1 .

 1 FIGURE LEGENDFig. 1. Mitochondrial gene expression, content and network in control and T2D myotubes. A. Expression of genes encoding proteins involved in mitochondrial oxidative metabolism (citrate synthase, CS and 3-hydroxy-acyl-CoA-dehydrogenase, in transcriptional control of mitochondrial biogenesis (nuclear respiratory factor 1, NRF1) and in lipid metabolism (carnitine palmitoyltransferase 1, CPT1) was evaluated by quantitative real-time PCR in control (n=4) and T2D differentiated satellite cells (n=4). Each value indicates the amount of mRNA expressed relative to 18S RNA. Each experiment was performed in triplicate for each of the 8 independent cell cultures. Data are presented as means ±SEM normalized to mRNA level in control myotubes. B. Representative western blot analysis of a non glycosylated protein component of mitochondria (Ab2) expression in myotubes established from control and T2D myotubes. α-Tubulin was used as a loading control. C. Citrate synthase activity was determined in control (n=3) and T2D differentiated satellite cells (n=3) under basal conditions and expressed relative to protein content. Experiments were performed in duplicate for each of the 6 independent cell cultures. Data are means ± SEM. D. Representative immunofluorescence microscopy of control and T2D myotubes after 8 days of differentiation, incubated with an antibody against a non glycosylated protein component of mitochondria (Ab2) visualized using a monoclonal secondary antibody conjugated to Alexa 546. Scale bar represents 30 µm.

Fig. 2 .

 2 Fig. 2. Effect of palmitate pre-treatment on palmitate beta-oxidation and phosphorylated acetyl-CoA carboxylase expression in control and T2D myotubes. A. Basal palmitate beta-oxidation and palmitate beta-oxidation after 16h of cold palmitate pre-treatment (600 µM) in control (n=3) and T2D differentiated satellite cells (n=3) expressed relative to protein content. Experiments were performed in triplicate for each of the 6 independent cell cultures. Data are means ± SEM. *, P<0.05, T2D cells versus control cells. B. Representative western blot analysis of P-ACC in differentiated satellite cells established from 4 control subjects and 4 type 2 diabetic (T2D) patients in the basal state (no palmitate pre-treatment). ACC protein expression was used as a loading control. C. Quantification by density analysis of western blot shown in (A). Data are means ±

Fig. 3 .

 3 Fig. 3. Effect of AMPK activation on lipid accumulation in control and T2D myotubes. A. Representative western blot analysis of P-ACC and P-AMPK in differentiated satellite cells established from a control subjects and a type 2 diabetic (T2D) patients. AMPK and ACC protein expressions were used as loading controls. B. Quantification by density analysis of P-AMPK (left panel) and P-ACC (right panel) in myotubes with (+) or without (-) AICAR (500 µM) or metformin (2 mM) treatment for 16h, from 4 cultures established from control subjects and 5 cultures established from type 2 diabetic patients (T2D). Data are presented normalized to AMPK and ACC expression, respectively. Data are shown as mean ±SEM. *p < 0.05, basal versus AICAR for control cells. C. Quantification of lipid content in control (n=3) and T2D differentiated satellite cells (n=3-4) with (+) or without (-) AICAR (500 µM, left panel) or metformin (2 mM, right panel) treatment (30 min) followed by palmitate treatment (600 µM for 16h). Data are means ±SEM. Each point was assayed in triplicate for each of the 7 independent cell cultures. *, P<0.05, T2D cells versus control cells. #, P<0.05, metformin and palmitate treatment versus palmitate alone.

Fig. 4 .

 4 Fig. 4. Effect of mitochondrial inhibition on lipid accumulation in control and T2D myotubes. A. Representative phase contrast microscopy of control and T2D myotubes after 16h of palmitate (600 µM) treatment with (+) or without (-) previous treatment with antimycin A (1 µM for 30min). Scale represents 30 µm. B. Quantification of lipid content in control (n=4) and T2D differentiated satellite cells (n=5) with (+) or without (-) antimycin A stimulation (1 µM for 30min) followed by palmitate treatment (600 µM for

Fig. S1 .

 S1 Fig. S1. Effect of palmitate treatment in control and T2D myotubes. A. Representative phase contrast microscopy of control and T2D myotubes after 16h of palmitate (600 µM) treatment. Scale represents 30 µm. B. Quantification of lipid content in control (n=4) and T2D differentiated satellite cells (n=5) after palmitate treatment (600 µM for 16h). Data are means ±SEM. Each point was assayed in triplicate for each of the 9 independent cell cultures. **, p<0.01 palmitate-treated T2D cells versus control cells.

Acklowledgements

This work was supported by grants from Association Française contre les Myopathies (AFM, MNM2 2005, n°11330), Institut National de Santé et de la Recherche Médicale (INSERM, ERI25), Languedoc Roussillon Region, University of Montpellier 1 (EA4202) and CHU of Montpellier. We are grateful to the Centre National de Recherche Scientifique (CNRS) for its support to M. Kitzmann and to AFM and EMO International for their fellowships to C. Aguer. We thank Erin L. Seifert for helpful discussion.


T T Truong

M K Nguyen

On new V-line Radon transforms in R 2 and their inversion

Radon transforms on piecewise smooth curves in R 2 are rather unfamiliar and have not been so far widely investigated. In this paper we consider three types of Radon transforms defined on a pair of half-lines in the shape a V-letter, with a fixed axis direction. These three Radon transforms arise from recently suggested tomographic procedures. Our main result consists in obtaining their analytic inverse formulas, which may serve as mathematical foundation for new imaging systems in engineering and physics.

Introduction

Radon transforms in the plane have been extensively studied in the past. Besides the cases of the straight line [START_REF] Radon | Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten Ber[END_REF], the circle intersecting a fixed point, Radon transforms on other classes of curves have been also treated (see among other works e.g. [START_REF] Cormack | The Radon transform on a family of curves in the plane[END_REF][START_REF] Quinto | Radon transforms on curves in the plane[END_REF][START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF]). The universality of Radon transform is known in imaging science ranging from medicine [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF][START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF], seismic [START_REF] Chapman | Generalized Radon transforms and slant stacks[END_REF], astrophysics [START_REF] Bracewell | Strip integration in radio astronomy[END_REF], non-destructive industrial testing [START_REF] Kershaw | The Determination of the Density Distribution of a Gas Flowing in a Pipe from Mean Density Measurements[END_REF], etc. It also plays a crucial role in pure mathematics [START_REF] John | Planes Waves and Spherical Waves Applied to Partial Differential Equations[END_REF][START_REF] Gelfand | Integral geometry and its relations to the theory of group representation Russian Math[END_REF][START_REF] Militello | Radon transform as a set of probability distributions[END_REF][START_REF] Saygili | Trkalian fields and Radon transformation[END_REF] and recently has become increasingly a powerful investigation tool in numerous fields of physics, e.g. [START_REF] Manuel | Construction of the Fermi surface from positron-annihilation methods[END_REF][START_REF] Behringer | A demonstration of ocean acoustic tomography[END_REF][START_REF] Mane | Radon reconstruction in longitudinal space[END_REF][START_REF] Asorey | Radon transform on the cylinder and tomography of a particle on the circle[END_REF][START_REF] Florescu | Single scattering optical tomography[END_REF].

In 1997 R Basko introduced a Radon transform on a pair of half-lines forming a Vletter in an attempt to model image formation in the so-called one-dimensional Compton camera [START_REF] Basko | Analytical reconstruction formula for the one-dimensional Compton camera[END_REF]. The axis of this V-line swings around a point of the plane such that its vertex lies on a straight line (representing a "scattering" detector) and the opening angle of the two half-lines (representing a Compton scattering angle) is an imaging data variable. No other cases of Radon transforms on piecewise lines (or curves) seem to have been discussed in the literature until we have put forward recently the idea of V-line Radon transforms with a fixed axis direction [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF]. An akin concept of broken ray Radon transform has just now appeared [START_REF] Florescu | Inversion formulas for the broken-ray Radon transform[END_REF].

In this work we consider the class of V-line Radon transforms with a fixed symmetry axis direction. Such Radon transforms may be of theoretical interest in integral geometry in the sense of Gel'fand [START_REF] Gelfand | Integral geometry and its relations to the theory of group representation Russian Math[END_REF]. Three types of V-line Radon transforms with fixed axis direction will be treated. The first one is mathematically the most important and arises from a coupled transmission-reflection tomographic process while the two others are its † To whom correspondence should be addressed (truong@u-cergy.fr) variants corresponding respectively to a collimated one-dimensional Compton camera and to scattered radiation emission imaging using a one-dimensional gamma camera. For all three cases an exact inverse formula is established. These results are of theoretical nature: they support the feasibility of three new tomographies. An illustration of these inversion formulas in two-dimensional medical and industrial imaging can be found in [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF]. The paper closes on a short conclusion with some forthcoming topics of investigation.

The V-line Radon transform

Short review of the standard Radon transform

Before going into the subject, we briefly go over the standard Radon transform. It maps a

L 1 (R 2 )-function f (x, y) onto R f (p, ϕ) via the integral R f (p, ϕ) = ∫ R 2 d x d y f (x, y) δ(p -x cos ϕ -y sin ϕ),
in which p is the distance from the coordinate system origin to the straight line and ϕ is the angle of the normal unit vector to the line with the Ox axis, see Fig. From the structure of the delta function kernel, it follows that

R f (p, ϕ) = Rf (-p, ϕ + π).
There exists an explicit inverse formula [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]

f (x, y) = 1 2π 2 ∫ π 0 d ϕ ( P.V. ∫ R d p R ′ f (p, ϕ) (p -x cos ϕ -y sin ϕ) ) (1)
where

R ′ f (p, ϕ) = ∂ ∂p R f (p, ϕ).
In eq. ( 1), the p-integral is a Cauchy principal value (P.V.). An alternative form in Fourier space of this inverse formula is

f (x, y) = ∫ π 0 d ϕ ∫ R d k |k| e 2i πk(x cos ϕ+y sin ϕ) ∫ R d q e -2i πqk R f (q, ϕ),
which expresses a summation image of the "ramp"-filtered Radon data [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF].

Definition

Consider a pair of half-lines meeting at a point M of abscissa ξ on the Ox-axis, each of which making an angle ω (0 < ω < π/2) with the O y-axis, see Fig. 2. Let f (x, y) be a L 1 (R 2 ) nonnegative continuous function with compact support in {R 2 |y > 0}. We call Eq. ( 2) may be put under the form of an integral transform in R 2 with a delta function kernel concentrated on the V-line

V f (ζ, τ) = ∫ ∞ 0 d r ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) , (2)
V f (ζ, τ) = ∫ R 2 d x d y K V (ζ, ω|x, y) f (x, y), with K V (ζ, ω|x, y) = ( δ((x -ζ) cos ω -y sin ω) + δ((x -ζ) cos ω + y sin ω) ) .
Thus the image of a given point source at (x, y) ∈ R 2 , is the sum of two Dirac delta distributions, which have a support in the upper (τ, ζ)-plane, consisting of two half-lines, with τ > 0, meeting at ξ = x on the ξ-axis and having a slope ± y -1 , i.e.

τ = | x -ζ y | or ω = arctan | x -ζ y |.
In a (ω, ζ)-representation, this is just the arctan-curve, which is the support of the pointsource image. Thus the support of V f (ζ, τ) is not compact in this representation.

Clearly the V-Radon transform can also be written in the language of the standard Radon transform as

V f (ζ, τ) = ∫ R d x ∫ ∞ 0 d y f (x, y) ( δ(u -x cos ω -y sin ω) + δ(u -x cos ω + y sin ω) ) ,
where u = ζ cos ω, the distance of the coordinate origin to the two half-lines forming the letter V. Note also that the unit normal vectors to these half-lines make an angle ±ω with the Ox-axis. So the result appears as the sum of two standard Radon transforms of f (x, y), except for the y-integration range.

Physical implementation

In standard Computed Tomography, a calibrated X-ray pencil emitted from a point-source S traverses a two-dimensional slice of an objet along a straight line before being absorbed by a point-like detector D. In this way the attenuation of the radiation beam is expressed as an integral of the linear attenuation function of the object to be imaged over a straight line. This is a Radon's problem which is solved by eq. ( 1).

The V-line Radon transform may be conceived as representing the measurement of a X-ray pencil attenuation along a broken path in the shape of a V-letter, which is physically realized by putting a mirror at the vertex M of the V-line as shown in Fig. 2. Measurements requiring various angles ω may be performed by varying the distance SD, but in practice one may set a series of point-sources and a series of point-detectors respectively on the left and on the right of a given axis of symmetry of the V-line. To vary the parameter ζ, one may put the object on a conveyor belt in a scanning process. Such a device is clearly a new type of X-ray scanner which could prove to be more efficient wherever the rotational motion of a standard X-ray scanner cannot be used and should be replaced by a translational motion. Such a situation would arise when tomographic images of series of objects on a conveyor belt are to be taken successively. As possible applications, one may think of luggage control in airports as well as continuous imaging of series of biological objects in medical research.

Inversion formula

We now establish a relation between Radon transform and Fourier transform which is used to obtain the inverse of the V-line Radon transform. For the standard Radon transform, such a relation tells how this transform appears in Fourier spaces. It is known in imaging science as the central-slice theorem [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF]. As V-lines of given opening angle ω can be deduced from each other by x-translations, we may introduce the x-Fourier transform f (p, y) of f (x, y) by

f (x, y) = ∫ ∞ -∞ d p e 2i πpx f (p, y).
Eq. ( 2) can be now rewritten as

V f (ζ, τ) = ∫ ∞ -∞ d p e 2i πpζ ∫ ∞ 0 dr f (p, r cos ω) 2 cos (2πpr sin ω). ( 3
)
To proceed further we set z = r cos ω. Recalling that 0 < ω < π/2, we have

z > 0 and cos ω = 1 1 + τ 2 > 0.
Then eq. ( 3) appears as

V f (ζ, τ) = 2 √ 1 + τ 2 ∫ ∞ -∞ d p e 2i πpζ ∫ ∞ 0 d z f (p, z) cos (2πpτz). (4)
Extracting the p-Fourier component V f (p, τ) of V f (ζ, τ) on the left-hand-side of eq. ( 4)

gives ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 2 ∫ ∞ 0 d z f (p, z) cos (2πpτz).
We now multiply both sides of this equation by cos(2πτv) and integrate on τ > 0. Using the identity

∫ ∞ 0 d τ cos (2πτ v) cos(2πkz τ) = 1 4 (δ(kz + v) + δ(kz -v)) , (5)
we get

∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2 ∫ ∞ 0 d z f (p, z) ( δ(pz + v) + δ(pz -v) ) ,
with (p, v) ∈ R 2 and z > 0. We now perform the τ-integration

• if v/p > 0, then only δ(v -pz) = |p| -1 δ(z -v/p) contributes, ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2|p| f (p, v/p), • if v/p < 0, then only δ(v + pz) = |p| -1 δ(z + v/p) contributes, ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2 ) = 1 2|p| f (p, -v/p),
Hence we can regroup these two cases into a single formula as

f (p, |v/p|) = 2|p| ∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp ( V f (ζ, τ) 1 + τ 2
)

.

Putting y = |v/p| > 0, we can reconstruct f (x, y) by inverse Fourier transform in p and f (x, y) is expressed as a triple integral on the V-line Radon data

f (x, y) = ∫ ∞ -∞ |p| d p e 2i πpx ∫ ∞ 0 d τ 2 cos (2πpτy) 1 + τ 2 ∫ ∞ -∞ d ζ e -2i πζp V f (ζ, τ), (6)
where we have made use of the evenness of the cos-function.

Inspection of convergence shows that exchange of integration order is possible and eq. ( 6) may now be rewritten as

f (x, y) = ∫ ∞ 0 d τ 1 + τ 2 ∫ ∞ -∞ |p| d p e 2i πpx ( e 2i πpτy + e -2i πpτy ) ∫ ∞ -∞ d ζ V f (ζ, τ) e -2i πζp . (7)
Thus analogously to the Radon transform, we can write

f (x, y) = 1 2π 2 ∫ ∞ 0 d τ 1 + τ 2 ( P.V. ∫ R d ζ ( V ′ f (ζ, τ) (ζ -x -yτ) + V ′ f (ζ, τ) (ζ -x + yτ) )) , ( 8
)
where

V ′ f (ζ, τ) is the ζ-derivative of V f (ζ, τ).

Alternative form of the inversion formula

Reintroducing u = ζ cos ω, which is the common distance from the coordinate system origin to the two branches of the V-line, and redefining f (ζ, τ) as F (u, ω), eq. ( 8) becomes

f (x, y) = 1 2π 2 ∫ π/2 0 d ω ( P.V. ∫ R d u [ V ′ F (u, ω) (u -x cos ω -y sin ω) + V ′ F (u, ω) (u -x cos ω + y sin ω) ]) , (9)
where V ′ F (u, ω) is the derivative of VF (u, ω) with respect to u. This expression is a sum of two ω-integrals which, as functions of (x, y) respectively take a constant value along the half-line of unit normal vector n = (cos ω, sin ω) and along the half-line of unit normal vector n ′ = (cos ω,sin ω). A similar feature occurs in the standard Radon transform but with a double angular range [0, π]. Thus we may view the V-line Radon transform as the sum of two half-line Radon R-transforms, as suggested by the defining equation ( 2). In imaging science equation ( 9) is called a "summation image" of back-projected Radon data [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF], the ζ-derivative of the data being back-projected on the left and on the right branches of the V-line in the ζ-integrand of eq. ( 9).

Extension to the range π/2 < ω < π

Up to now, we have consider L 1 0 (R 2 )-functions with compact support. By appropriate translations, their support can be shifted into the upper half-plane. But when the support of f (x, y) is extended over all R 2 , we need to collect data pertaining to the part of its support below the real axis. This can be achieved using the angular range π/2 < ω < π, for which cos ω and τ are negative. The V-line is now up-side down. In this case, we set z = -z ′ and τ = -τ ′ in eq. (2.4), to have

∫ R d p e -2i π pζ ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 2 ∫ ∞ 0 d z ′ f (p, -z ′ ) cos (2πpz ′ τ ′ ), since zτ = z ′ τ ′ , d z = -d z ′ and because of cos ω = -1/ 1 + τ ′2
. This shows that again we have to invert a cosine-Fourier transform to reconstruct f (x, y) for y < 0. Following the previous steps we obtain

∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2 ∫ ∞ 0 d z ′ f (p, -z ′ ) ( δ(pz ′ + v) + δ(pz ′ -v) ) .
with (p, v) ∈ R 2 and z ′ > 0. We now perform the z ′ -integration on the right-hand-side of this last equation.

• if v/p > 0, then only δ(v -pz ′ ) = |p| -1 δ(z ′ -v/p) contributes, ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2|p| f (p, -v/p), • if v/p < 0, then only δ(v + pz ′ ) = |p| -1 δ(z ′ + v/p) contributes, ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2 ) = 1 2|p| f (p, v/p),
Analogously to the case with 0 < ω < π/2 (see eq. ( 6)), we can write a single formula

f (p, -|v/p|) = 2|p| ∫ ∞ 0 d τ ′ cos(2πτ ′ v) ∫ R d ζ e -2i πζp ( V f (ζ, -τ ′ ) 1 + τ ′2
) .

Putting y = -|v/p| > 0, we can reconstruct f (x, y) by inverse Fourier transform in p for y < 0. Using again the evenness of the cos-function, f (x, y) is then expressed as a triple integral on the V-line Radon data

f (x, y) = ∫ ∞ -∞ |p| d p e 2i πpx ∫ ∞ 0 d τ ′ 2 cos (2πpτ ′ y) 1 + τ ′2 ∫ ∞ -∞ d ζ e -2i πζp V f (ζ, -τ ′ ).
And hence for y < 0, we obtain

f (x, y) = 1 2π 2 ∫ ∞ 0 d τ ′ 1 + τ ′2 P.V. (∫ R d ζ ( V ′ f (ζ, -τ ′ ) (x + τ ′ y -ζ) + V ′ f (ζ, -τ ′ ) (x -τ ′ y -ζ) )) . ( 10
)
To sum up, formulas [START_REF] Bracewell | Strip integration in radio astronomy[END_REF] give the reconstruction of a function f (x, y) with compact support, which can be always, by an appropriate choice of the coordinate origin, positioned in the upper half-plane (y > 0). But when the support of f (x, y) is infinitely extended and f (x, y) decreases fast at infinity, then the reconstruction formula ( 10) for (y < 0) should be used. We can put ( 8) and [START_REF] John | Planes Waves and Spherical Waves Applied to Partial Differential Equations[END_REF] together in one single formula as

f (x, y) = 1 2π 2 ∫ ∞ 0 d σ 1 + σ 2 P.V. (∫ R d ζ ( V ′ f (ζ, (sgny)σ) (x + σy -ζ) + V ′ f (ζ, (sgny)σ) (x -σy -ζ) )) ,
where (sgny) is the sign of y. This completes the proof of the inversion of the V-line Radon transform with fixed axis direction for all values of y ∈ R.

Adjoint transform

The kernel K V (ζ, ω|x, y) may be used to define an adjoint V-line Radon transform V + . For an integrable function g (ζ, ω), V + g (x, y) is given by

V + g (x, y) = ∫ π/2 0 d ω ∫ R d ζ K V (ζ, ω|x, y) g (ζ, ω).
This is also a V-line Radon transform in (ζ, τ)-space on the V-line given by eq. (2.2). After performing the ζ-integration, the following form appears

V + g (x, y) = ∫ π/2 0 d ω cos ω ( g (x -yτ, ω) + g (x + yτ, ω) ) ,
where τ = tan ω as before. We observe that g (xyτ, ω) is constant on the left half-line of equation (yy 0 )τ = (xx 0 ), whereas g (x + yτ, ω) is constant on the right half-line of equation -(yy 0 )τ = (xx 0 ). The inverse formula of eq. ( 8) has been viewed already as a "summation image" of back-projected V-line Radon data (see eq. ( 9)). Here it may be viewed now as the V-line adjoint Radon transform of the data with the identifications

∫ ∞ 0 d τ 1 + τ 2 = ∫ π/2 0 d ω,
and

g (x -yτ, ω) = P.V. 1 2π 2 ∫ R d ζ V ′ f (ζ, τ) (ζ -x + yτ) , g (x + yτ, ω) = P.V. 1 2π 2 ∫ R d ζ V ′ f (ζ, τ) (ζ -x -yτ) .
This behavior shows up also in the standard Radon transform [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF].

The V * -line Radon transform

Definition

A variant of the V-line Radon transform arises when the integration measure on the halflines is performed with a non-uniform density along the branches of the V-line. Technically this occurs in the so-called one-dimensional special Compton camera for flat radiating objects [START_REF] Morvidone | On the V-line Radon transform and its imaging applications[END_REF], which is a special version of an imaging device investigated by [START_REF] Cree | Towards direct reconstruction from a gamma camera based on Compton scattering[END_REF] in three dimensions, see Fig. 3. Here a radiation emitting object is described by its radio-activity emission density f (x, y). Emitted radiation from a point-source N will Compton scatter electric charges, under a scattering angle ω, in the scatter detector before entering a collimated absorbing detector. The amount of scattered gamma radiation detected is thus given by the V * -line Radon transform of f (x, y), over an integration measure m σ (r ), which accounts for radiation dispersion from object emission site to scattering site on the scatter detector. Upon close analysis, the integration measure is given by the two-dimensional photometric law as

m σ (r ) = 1 πσ tan -1 σ 2r ,
where σ is the linear size of scatterers. As in the previous section, we shall assume that the support of f (x, y) is strictly in the upper half-plane y > 0. To keep the discussion simple and concentrate on the nature of this transform, we have ignored the Compton kinematic factor, which includes the Klein-Nishina scattering probability density.

V * f (ζ, τ) = ∫ ∞ 0 d r m σ (r ) ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) .
For σ ̸ = 0, an inversion formula cannot be worked out at present. But for vanishingly small target size σ → 0, it has a simpler form

V * f (ζ, τ) = ∫ ∞ 0 d r 2πr ( f (ζ + r sin ω, r cos ω) + f (ζ -r sin ω, r cos ω) ) , (11)
which can be inverted with the procedure of section 2. Since the support of f is "above" the Ox-axis, the integral in equation ( 11) is well defined.

The integral kernel has the form

K V * (ζ, ω|x, y) = 1 2π y ( δ((x -ζ) cos ω -y sin ω) + δ((x -ζ) cos ω + y sin ω)
) .

It differs from the kernel K V (ζ, ω|x, y) by a factor 1/2πy due to the presence of 1/2πr in the r -integration. Its support is the same as the one given by eq. (2.2).

Inversion formula

We follow now the steps and the same notations as before. Eq. ( 11) becomes now

∫ R d ζ e -2i πζp V * f (ζ, τ) = 1 π ∫ ∞ 0 d z ( 1 z f (p, z) ) cos (2πτpz).
We apply now on both sides of this equation the integral operator

∫ ∞ 0 d τ cos(2πτv),
and use the identity [START_REF] Barrett | The Radon Transform and its Applications in Progress in Optics[END_REF] to have

∫ ∞ 0 d τ cos(2πτv) ∫ R d ζ e -2i πζp V * f (ζ, τ) = ∫ ∞ 0 d z 2πz f (p, z) 1 2 ( δ(v + pz) + δ(v -pz) ) .
As (p, v) ∈ R 2 and z > 0, there are two types of contributions

• if p v > 0, only δ(v -pz) = |p| -1 δ(z -v/p) contributes, hence ∫ ∞ 0 d z 2πz f (p, z) 1 2|p| δ(z - v p ) = 1 4πv sgn(p) f (p, v p ) = f (p, v p ) 4π|v| , since sgnp = sgnv. • if p v < 0, only δ(v + pz) = |p| -1 δ(z + v/p) contributes, hence ∫ ∞ 0 d z 2πz f (p, z) 1 2|p| δ(z + v p ) = 1 2|p| 1 2π(-v p ) f (p, - v p ) = f (p, -v p ) 4π|v| , since |p| ( - v p ) = sgn(p) (-1) |v| sgn(v) = |v|, because p v < 0.
We can put these two cases together with y = |v/p| to have

f (p, |v/p|) = 2π|v| ∫ ∞ 0 d τ 2 cos(2πτv) ∫ R d ζ e -2i πζp V * f (ζ, τ). ( 12
)
The reconstruction formula is then for y = |v/p| > 0, thanks to the evenness of the cosfunction

f (x, y) = 2πy ∫ R d p |p| e 2i πx ∫ ∞ 0 d τ 2 cos(2πτy p) ∫ R d ζ e -2i πζp V * f (ζ, τ).
It can be rewritten also as a Barrett "summation image" as

f (x, y) = 2πy ∫ ∞ 0 d τ ∫ R d p |p| e 2i πx ( e 2i πp yτ + e -2i πp yτ ) ∫ R d ζ V * f (ζ, τ) e -2i πζp .
The p-integration is the same as in eq. ( 7) and yields

f (x, y) = y π ∫ ∞ 0 d τ ( P.V. ∫ R d ζ ( V ′ * f (ζ, τ) (ζ -x -yτ) + V ′ * f (ζ, τ) (ζ -x + yτ)
)) ,

where

V ′ * f (ζ, τ) is the ζ-derivative of V * f (ζ, τ).
This expression can also be represented as a summation image on back-projected data on a V-line in (x, y)-space, with f (ζ, τ) = F (u, ω)

f (x, y) = y π ∫ π/2 0 d ω cos ω P.V. ∫ R du [ V ′ * F (u, ω) (u -x cos ω -y sin ω) + V ′ * F (u, ω) (u -x cos ω + y sin ω) ] . (13)
Note that the apparent singularity in the angular integration at ω = π/2 does not exists thanks to the assumption on the support of f (x, y).

Extension to the range π/2 < ω < π

Repeating the approach of subsection 2.5, we obtain the complement of eq. ( 12)

f (p, -|v/p|) = 2π|v| ∫ ∞ 0 d τ ′ 2 cos(2πτ ′ v) ∫ R d ζ e -2i πζp V * f (ζ, -τ ′ ),
and the corresponding reconstruction formula for y = -|v/p| < 0

f (x, y) = 2πy ∫ R d p |p| e 2i πx ∫ ∞ 0 d τ ′ 2 cos(2πτ ′ y p) ∫ R d ζ e -2i πζp V * f (ζ, -τ ′ ), or f (x, y) = ( y π ) ∫ ∞ 0 d τ ′ ( P.V. ∫ R d ζ ( V ′ * f (ζ, -τ ′ ) (ζ -x -yτ ′ ) + V ′ * f (ζ, -τ ′ ) (ζ -x + yτ ′ ) )) . ( 14
)
Formula [START_REF] Saygili | Trkalian fields and Radon transformation[END_REF] gives the reconstruction of f (x, y) when its support is compact for y > 0 and when it is not compact one must add [START_REF] Manuel | Construction of the Fermi surface from positron-annihilation methods[END_REF]. Note these formulas differs from that of the V-transform. We can put (3.2) and ( 14) together in a single formula as

f (x, y) = ( y π ) ∫ ∞ 0 d σ ( P.V. ∫ R d ζ ( V ′ * f (ζ, (sgny)σ) (ζ -x -yσ) + V ′ * f (ζ, (sgny)σ) (ζ -x + yσ) )) ,
where (sgny) is the sign of y. This completes the inversion of the V * -line Radon transform by a factor y/π and by the form of the τ-integration. An interpretation of equation ( 14) in terms of back-projected data or in terms of its adjoint transform can also be made along the lines of the previous section.

The V ♯ -line Radon transform

A third version of the V-line Radon transform is a so-called "compounded" or integrated form of the V * -Radon transform over a given integration measure. It arises in scattered radiation emission imaging, a concept which has been extensively investigated in recent years [START_REF] Nguyen | On an integral transform and its inverse in nuclear imaging[END_REF][START_REF] Nguyen | Radon transforms on a class of cones with fixed axis direction[END_REF][START_REF] Nguyen | On a novel approach to Compton scattered emission imaging[END_REF][START_REF] Truong | On a class of generalized Radon transform and its role in imaging science[END_REF]. If the Compton camera exploits the scattering of object emitted radiation by an external scatterer in the form of a scatter detector for imaging purpose, this time it is the object itself (in fact its electric charges) which serves as natural scatterer. Thus the outgoing radiation from the object itself is registered now by a motionless collimated gamma camera (as opposed to a moving collimated linear detector as in Single Photon Emission Tomography (or SPECT)). The data is collected at various values of the scattering energies -or equivalently at various values of the scattering angles ω), -see Fig. 4. Hence the detected radiation flux density appears as the V ♯ -line Radon transform of the object radioactivity emission density f (x, y), i.e.

V ♯ f (ζ, τ) = ∫ ∞ 0 d η ( 1 πσ ′ tan -1 σ ′ 2η ) ∫ ∞ 0 d r ( 1 πσ tan -1 σ 2r ) × [ f (ζ + r sin ω, η + r cos ω) + f (ζ -r sin ω, η + r cos ω) ] , (15)
where σ ′ has the same meaning as in subsection 3.1 ‡. A successful inversion of this V ♯ -line Radon transform would allow to reconstruct the radio-activity emission density f (x, y) with a stationary linear collimated detector collecting series of "images" at different scattering angles ω. This imaging process presents a very important advantage over the existing ones: it avoids the motion of the detector around the object during data acquisition, hence does not require the heavy bulky rotational mechanics to be included.

This type of V-line transform is called "compounded" V-line Radon transform or V ♯line Radon transform, by analogy to the three-dimensional case [START_REF] Nguyen | On an integral transform and its inverse in nuclear imaging[END_REF]. Now let us define

∫ ∞ 0 d η ( 1 πσ ′ tan -1 σ ′ 2η ) f (ξ ± , η + r cos ω) = h(ξ ± , r cos ω), (16)
where ξ ± = (ζ ± r sin ω). Then we see that

V ♯ f (ζ, τ) = V * h(ζ, τ).
Hence the inversion problem is that of V * for h, followed by the extraction of f from eq. ( 16).

Eq. ( 15) may be rewritten as an integral transform with the kernel

K V ♯ (ζ, τ|x, y) = √ 1 + τ 2 ( Y (x -ζ) Y (y -τ(x -ζ)) m σ ((x -ζ) √ 1 + τ 2 ) m σ ′ (y -τ(x -ζ))+ Y (-x + ζ) Y (y + τ(x -ζ)) m σ (-(x -ζ) √ 1 + τ 2 ) m σ ′ (y + τ(x -ζ))
) ,

where Y (x) is the Heaviside unit step function. Here the kernel is not a Dirac delta function.

Section 3 shows how to reconstruct h(x, y) from the data V ♯ f (ζ, τ) for all (x, y) ∈ R 2 .

Thus given h(x, y), f is the solution of the convolution equation ∫ R d z e -2i πzq h(x, z), [START_REF] Florescu | Single scattering optical tomography[END_REF] as h(x, z) is known for all z ∈ R. As an example, in the limit σ → 0, using the Fourier table [START_REF] Lavoine | Transformation de Fourier des Pseudo-fonctions avec Tables de Nouvelles Transformées[END_REF], we have an exact expression for m σ (q) = -

( ln 2π|q| -γ + i π 2 sgnq
) , to be substituted in equation [START_REF] Florescu | Single scattering optical tomography[END_REF] to reconstruct f (x, y). We observe that the inversion of the V ♯ the product of the inversion of V ⋆ times an inverse Fourier transform.

Conclusion and perspectives

In this article we have presented for the first time three types of Radon transforms in R 2 which are defined on a piecewise continuous curve having the shape of a letter V with a fixed axis direction. The main theoretical result is their analytic inverse formulas. These transforms are instrumental in new possible tomographic processes involving scattering, coupled transmission-reflection and emission phenomena. Numerical simulations on test phantoms to ascertain the viability of these imaging processes will appear in a separate publication.

We also envisage to study Radon transforms on more general piecewise continuous curves consisting of connected pieces of arcs, e.g. arcs of circle, as well as Radon transforms on V-line with an axis swinging around a fixed point. Their inversion would open the way to new advantageous imaging processes. These topics are the object of future research.

Figure 1 .

 1 Figure 1. Parameters of the Radon transform

Figure 2 .

 2 Figure 2. Parameters of the V-line Radon transform

Figure 3 . 1 0 (R 2 +

 312 Figure 3. The V ♯ -line Radon transform and its parameters

h(x, y) = ∫ ∞ 0 dFigure 4 .

 04 Figure 4. Setup and parameters of the V ♯ -line Radon transform