domain: sde sdv sde.be

text:
Christine Paillard
email: christine.paillard@univ-brest.fr.

AN ECOLOGICAL APPROACH TO UNDERSTANDING HOST-PATHOGEN-ENVIRONMENT INTERACTIONS: THE CASE OF BROWN RING DISEASE IN CLAMS

Keywords: microparasite, Vibrio, bivalve, transmission, clinical signs, shell disease, environment, temperature

Brown Ring Disease (BRD) is used as a case study to describe an ecological approach to understanding host-pathogen interactions. BRD is caused by a bacterial microparasite, Vibrio tapetis, which elicits the inhibition of the calcification process. Epidemiological studies of BRD are facilitated by the use of characteristic clinic signs on the inner surface of the valves. The phenology and dynamics of BRD are then described by coupling epidemiological, ecological and physiological data. Spatial distribution of BRD prevalence in natural clam populations and V. tapetis load in the sediment allows the investigation of the transmission mode of this pathogen. Experimental testing of different transmission modes has demonstrated the predominant role of feces and pseudofeces in V. tapetis transmission and diffusion processes. Some other hosts can harbor V. tapetis, including reservoir species such as cockles but also vertebrate predators such as fishes, and cross-species transmission could potentially play a major role in dispersion of the pathogen during its life cycle. Environmental factors such as temperature, salinity and nutrients strongly modulate the coevolution between Vibrio and bivalve.

INTRODUCTION

Taking an ecological approach to the investigation of host-pathogen interactions primarily requires a natural environment approach known as epidemiology. Investigating the distribution of the disease in space and time allows analysis of the factors (environmental, genetic…) concurrent with its onset and development. In combination with laboratory experiments, this approach can be used to assess the role of key factors on host-pathogen interactions. The development of a numerical model of BRD allows knowledge to be synthesized in a simple manner (etiology, diagnostic tools, pathogen characterization, mechanisms of host-pathogen interaction, epidemiology), to prioritize different stages and to point to gaps and flaws of the three component system, Environment, Host and Pathogen (HPE) [START_REF] Paillard | A theoretical individual-based model of Brown Ring Disease in Manila clams, Venerupis philippinarum[END_REF]. The HPE system can be reconsidered taking into account the entire microbial community: on one hand, the pathogen community and the co-infection processes, and on the other hand, the microbiota that the host harbors [START_REF] Desriac | Exploring the hologenome concept in marine bivalvia: hemolymph microbiota as a pertinent source of probiotics for aquaculture[END_REF][START_REF] Rosenberg | The role of microorganisms in coral health, disease and evolution[END_REF][START_REF] Seabloom | The community ecology of pathogens: coinfection, coexistence and community composition[END_REF]. This ecological approach is highly interdisciplinary, drawing on techniques from fields including genetics, molecular biology, genomics, immunology, epidemiology and modeling (Figure 1).

EPIDEMIOLOGICAL APPROACH

Epidemiology is the study of diseases and health factors (environmental, genetic) in a population. The originality of epidemiology is its focus on the population rather than the individual. Quantification is paramount in epidemiology and quires to be precise and rigorous to determine the epidemiological indicators for each type of disease. These indicators will most often be expressed in relative terms compared to the population in which it is observed.

Brown Ring Disease (BRD) provides a useful case study to describe the uses of epidemiological parameters, because the disease diagnostic is based mostly on macroscopic signs. In fact, BRD is due to a bacterial microparasite, Vibrio tapetis, which elicits shell growth abnormalities by inhibiting calcification process and inducing the deposit of brown conchiolin between the mantle and the inner surface of the valves [START_REF] Paillard | The brown ring disease symptom in the Manila clam, Ruditapes philippinarum: establishment of a classification system[END_REF]. This defense process, known as nacrezation or pearl formation [START_REF] Cheng | Internal defense mechanisms[END_REF] embeds parasites inside the shell matrix. This shell repair process is very powerful against macroparasites, such as trematodes, but in the case of microparasites such as bacteria with high generation rates, this mechanism is less efficient. However, depending on the HPE interaction modulation factors, it has been demonstrated that conditions favorable to the host such as nutrition and high temperature could enhance BRD recovery [START_REF] Bris | Immune responses of phenoloxidase and superoxide dismutase in the Manila clam Venerupis philippinarum challenged with Vibrio tapetis -Part I: Spatiotemporal evolution of enzymes' activities post-infection[END_REF][START_REF] Paillard | Temperature effects on Brown Ring Disease susceptibility and defense-related activities in clams Ruditapes philippinarum[END_REF][START_REF] Plana | Variations in the biochimical composition of the juveniles of Manila clam, Ruditapes philippinarum, infected by a Vibrio sp[END_REF].

The development of clinical signs for BRD using a classification system to assess conchiolin deposition, shell repair and phase of the disease has already been established (Figure 2) [START_REF] Paillard | Review of brown ring disease, a vibriosis affecting clams[END_REF][START_REF] Paillard | The brown ring disease symptom in the Manila clam, Ruditapes philippinarum: establishment of a classification system[END_REF]. Epidemiological Indicators Can Be Used to Describe and Quantify Brown Ring Disease (BRD)

Prevalence of BRD (BRD%): total number of individuals infected with BRD clinical signs as a percentage of the total number of sampled individuals (n≥100), at a given moment. The prevalence provides static information about the frequency of the disease, but not its development (increase, decrease or increase of new cases, such as BRD intensity). It corresponds to the balance of a disease at a given time. The prevalence is not to be confused with the incidence, which is the rate at which newly infected BRD individuals appear in a specific population during a given period.

Prevalence of conchiolin disease stages (CDS%): number of individuals with a specific stage of brown ring relative to the total number of sick animals, during a specific time period or at a given time.

Prevalence of shell repair stages (SRS%): The number of individuals at shell specific stage of repair as a percentage of the total number of animals either currently sick or previously sick (and now recovered) during a specific time period or at a given time.

Another type of indicator is the pathogen load (number of cells or CFU/ml) in clams or other hosts or in the surrounding environment (sediment, seawater). Pathogen concentration is measured using different techniques previously described (bacteriological analysis, ELISA or SSP-PCR, qPCR) [22, Bidault et al., 2015].

Considering all epidemiological indicators together, BRD in clams can be considered in a phenological way, taking into account not only the changes in the clinical signs, but also the pathogen load and mortality of clams that is not necessarily related to the presence of BRD signs. Figure 3 summarizes these results and integrates them to form a more complete vision of the vibriosis, taking into account changes in pathogen load and BRD microscopic clinical signs.

Six Phases of Vibriosis in Clams Can Be Identified Over Time

Six phases of vibriosis in clams can be identified over time.

 Phase 0 is characterized by the absence of both BRD clinical signs (BRD-) and the pathogen after detection. It therefore represents either a healthy individual, or a clam beginning BRD incubation, with a pathogen load below the limit detection of V. tapetis by qPCR (1.125 x 10 1 bacteria/ml) [START_REF] Bidault | Development of a Taqman real time PCR assay for rapid detection and quantification of Vibrio tapetis in extrapallial fluids of clams[END_REF]]  Phase 0.5 is characterized by an absence of BRD clinical signs (BRD-), but by obligatory positive detection V. tapetis. V. tapetis load can be measured in the extrapallialfluids (Figure 3, cases B or C), and/or in the hemolymph and tissues (Figure 3, cases A or B). It is a phase characterized by an absence of external BRD clinical signs and the presence of internal alterations such as increased hemocyte concentration a few hours after the inoculation, both in the mantle cavity and in the adductor muscle [START_REF] Allam | Pathogenicity of Vibrio tapetis, the etiological agent of brown ring disease in clams[END_REF]. For clams which develop shell repair defense mechanisms, this phase is only temporary. For the others (cases A and B), invasion of V. tapetis within tissues will lead to mortality. Recovery process in green.

Phases 1 to 3 are all characterized by clinical signs (Figure 2).

 Phase 1 consists of the appearance and development of the conchiolin deposit (CD).

V. tapetis concentration in extrapallial fluids tends to increase with CD stages.  Phase 2 or the intermediate repair phase of BRD. The concentration of V. tapetis reaches maximum values.  Phase 2.5 or the healing phase or "recovery". The concentration of V. tapetis starts to decrease slightly, reaching values similar to beginning of Phase 1. If there is no relapse, recovery will occur soon after.  Phase 3 characterizes a recovered animal. A scar is visible on the shell and the concentration of V. tapetis is below the limit detection of V. tapetis [START_REF] Paillard | A theoretical individual-based model of Brown Ring Disease in Manila clams, Venerupis philippinarum[END_REF]. The absence of V. tapetis must be confirmed using qPCR method [START_REF] Bidault | Development of a Taqman real time PCR assay for rapid detection and quantification of Vibrio tapetis in extrapallial fluids of clams[END_REF] during this phase. BRD associated mortalities occur generally in clams in phase 0.5 (cases A and B) when V. tapetis has colonized tissues, but also in phase 1 and 2, when the clams exhibit strong development of brown ring disease (phase 1 case B) or when the shell repair process is not completely finished (Phase 2, cases B and C) (Figure 2). Using these indicators, Figure 3 represents a synthesis of BRD dynamics coupling prevalence of disease phase and V. tapetis load at individual level. It is interesting to note that mortalities can also occur without BRD signs (if the bacteria is able to penetrate within tissues and multiply in the hemolymph) [START_REF] Allam | Pathogenicity of Vibrio tapetis, the etiological agent of brown ring disease in clams[END_REF] and at later stages of BRD development (in advanced stages of phase 2) [START_REF] Paillard | A theoretical individual-based model of Brown Ring Disease in Manila clams, Venerupis philippinarum[END_REF].

The prevalence of BRD phase could represent a pertinent integrative indicator to determine health status of clam population.

Spatial Distribution of BRD and V. tapetis in the Field

Spatial distribution of disease (prevalence, pathogen load…) is also very useful in epidemiological studies. Spatial distribution of BRD prevalence was assessed according to the host density and density of pathogens in sediment (performed by ELISA) [START_REF] Nöel | Development of a colonyblot ELISA assay using monoclonal antibodies to identify Vibrio P1 responsible for "brown ring disease" in the clam Tapes philippinarum[END_REF] during field sampling carried out during spring in the Gulf of Morbihan (Figures 4a and4b). The distribution of BRD prevalence among burrowing and rising clams did not show any relationship with either host densities or pathogen densities in clams and in the sediment. However, a significant relationship between the density of the clams and the average concentration of V. tapetis in thesediment was demonstrated. Areas with very high densities of clams were also those that harbored the highest densities of V. tapetis in sediments. Simulations of Lagrangiantrajectories for different numbers of particles or "larvae"using the hydrodynamic model of the Gulf of Morbihan [START_REF] Janin | Qualité des eaux du Golfe de Morbihan -Utilisation d'un modèle hydrodynamique[END_REF] have confirmed that the areas with high densities of clams in the center of Truscat sandbank (cf. Figure 4a, purple squares) were those subject to low current swirling waters, favoring the accumulation and deposition of fine particles [START_REF] Guillou | Simulations de la dérive lagrangienne des larves[END_REF]. Larval transport and the spread of bacteria could also follow the same pattern of dispersion within the Truscat sandbank, resulting in the strong correlation between clam density and pathogen load.

DISEASE TRANSMISSION MODES

Considering the results of spatial distribution of BRD and V. tapetis in the natural population of clams, an experiment was performed to investigate transmission of BRD. Three disease transmission modes were considered: by water, via contact, and by feces and pseudofeces. To determine the role of water in transmission, the animals were placed just under the surface, on top of a grid that allowed their feces to sink to the bottom of the tanks. Clams were not in contact with feces or with other clams. For testing the role of contact transmission, the bivalves were placed in the same conditions as above; however, they were kept in continuous contact with each other. To simulate the process of infection by feces and pseudo-feces, clams were fixed to the bottom of the tank, without contact with each other. Feces and pseudo-feces were resuspended by stirring every two days. V. tapetis contamination was performed by placing V. tapetis inoculated individuals (10% of the total number of clams) in each aquarium. Each transmission mode was investigated at two different densities: 100 and 400 clams/m 2 . For each transmission mode and for each clam density, the effect of water change was also tested (no water change, or water change every two days). For this experiment, because disease-free clams were not available in the wild or from hatcheries, a treatment bath of chloramphenicol (8 mg/l for 48 hours) was performed on the clams prior to the start of the experiment. A subsequent thermic treatment (clams were subjected to a temperature change of 4°C every 24 hours, starting at the experimental temperature of 15°C, rising to a maximum of 27°C, and falling again to 15°C) and another antibiotic bath of 48 hours were applied to limit the development of opportunistic bacteria. The animals were fed every two days. After one month, the animals were sacrificed and the pathogenic load in the pallial and extrapallial fluids of clams determined by the ELISA method [START_REF] Nöel | Development of a colonyblot ELISA assay using monoclonal antibodies to identify Vibrio P1 responsible for "brown ring disease" in the clam Tapes philippinarum[END_REF]. The results are summarized in Figure 5.

Whether water was renewed or not, the concentration of V. tapetis in fluids of clams placed in tank bottom in contact with feces and pseudofeces was significantly higher than those placed in open water or in intimate contact between each other. We were able to demonstrate that the concentration of V. tapetis in diseased clam feces represented 80 to 90% of the total heterotrophic microflora. When clams were at high density, water exchange allowed a reduction in the pathogen load (factor 10). Transmission through the water only appeared to be effective if the density of the host was high (at least 400 clams/m 2 ) and if the water was not renewed. These are the conditions under which clams live in the Gulf of Morbihan (muddy sediments and densities of 100-600 clams/m2) (Figure 4a). However, within the trays on which the animals were placed in intimate contact, the distribution of hosts with high pathogen loads was independent of the location of the animals inoculated with V. tapetis. In conclusion, two possible transmission paths have been demonstrated: by water and by feces. Whatever the density of clams or water renewal treatment, transmission through the feces seems crucial. Vertical transmission of V. tapetis was also considered, but to date, no experiments have been conducted to test this hypothesis.

LIFE CYCLE OF V. TAPETIS

Previously, we have shown that V. tapetis could be found in sediment, where high clam density occurs. Rodrigues et al. [START_REF] Rodrigues | Vibrio tapetis, the causative agent of Brown Ring Disease, forms biofilms with spherical components[END_REF] have demonstrated the ability of V. tapetis to form a biofilm in vitro on abiotic surfaces. Within the clams, a biofilm of V. tapetis on shell secretions constitutes the first steps of the infectious process. Then, V. tapetis multiplies in different compartments, primarily in the brown deposit, the periostracal secretion, the external mantle epithelium and the pallial and extrapallial fluids [START_REF] Allam | Effects of the pathogenic Vibrio tapetis on defence factors of susceptible and non-susceptible bivalve species: I. Haemocyte changes following in vitro challenge[END_REF][START_REF] Allam | Effects of the pathogenic Vibrio tapetis on defence factors of susceptible and non-susceptible bivalve species: II. Cellular and biochemical changes following in vivo challenge[END_REF]. In times of winter mortality or pre-spring time, V. tapetis can be also isolated in the hemolymph of moribund clams [Paillard et al., unpublished data]. V. tapetis can also be detected during the same period in the pore water of the sediment, the water surrounding moribund clams, and in the feces and pseudofeces of BRD clams after placing them in sterile vials [START_REF] Maes | Pathologie bactérienne chez deux invertébrés marins d'interêt commercial, Ruditapes philippinarum et Paracentrotus lividus[END_REF].

Hosts other than R. decussatus and R. philippinarum, have also been identified. V. tapetis has been isolated from other species of veneridae that suffer from BRD (Venerupis aurea, Tapes rhomboïdes, Dosynia exoleta), and the cockle, Cerastoderma edule, has also been identified as a potential vector species [START_REF] Maes | Effet de Vibrio P1, pathogène de Ruditapesphilippinarum sur d'autres espèces de bivalves[END_REF]. Indeed the ability to cross species barriers is a feature of V. tapetis, and can also occur from invertebrates to vertebrates. In fact, V. tapetis has been isolated from several species of fishes during their mortality events, such as flat fishes (Hypoglossus hypoglossus, Dicologoglossa cuneata, Solea solea) and the wrasse (Symphodus melops) [START_REF] Declercq | Vibrio tapetis isolated from vesicular skin lesions in Dover sole Solea solea[END_REF][START_REF] Jensen | Characterization of strains of Vibrio splendidus and Vibrio tapetis isolated from corkwing wrasse (symphodus melops) suffering vibriosis[END_REF][START_REF] Lopez | Characterization of Vibrio tapetis strains isolated from diseased cultured Wedge sole (Dicologoglossa cuneata Moreau)[END_REF]. All these fishes are likely to be predators of clams and could also potentially be considered as secondary hosts of V. tapetis. Figure 6 summarizes the life cycle of V. tapetis as a pathogen of clams.

VIBRIO BEHAVIOR MANIPULATION?

The two schematic figures presented (ecological level of Figure 3 and Figure 6) summarize our knowledge of how micro-habitats, transmission modes and seasonal change are able to influence the spread of V. tapetis. It thus appears that a predation/parasitism interaction could exacerbate mortalities; as weakened infected bivalves rise to the surface of the sediment and are more accessible to predators. In addition, the rising clams have a very high pathogenic load of vibrios, varying from 2 x 10 5 to 8 x 10 9 CFU/ml, which is on average 100 times higher than in a burrowing clam. The rising clams therefore also constitute a source of contamination and spread of the pathogen. Could this be similar to the favorisation process induced by trematode parasites in the clam Tapes aureus? [START_REF] Combes | Parasitism. The Ecology and Evolution of Intimate Interactions[END_REF]. These parasites modify the host behavior by promoting its rising in the sediment, making it more accessible to the secondary hosts (birds) [START_REF] Bartoli | parasites d'oiseaux des côtes de Camargue: systématique, biologie et écologie[END_REF]. Indeed, this same behavior was observed for heavily V. tapetis infected clams, and rising clams could even act like delayed explosive bombs in the predator! Thus, fish-eating clams, such as wrasse and flatfish may play a major role in the infectious cycle of V. tapetis, especially during periods of winter mortality.

IMPACT OF ENVIRONMENTAL FACTORS ON THE HOST-PATHOGEN INTERACTIONS

In natural populations of Manila clam along the coasts of Brittany (Bay of Brest, North Finistere and Morbihan), we have shown previously that maximum prevalence occurs during the winter and beginning of spring periods, peaking in March. Animals from one natural population in the Gulf of Morbihan, ("Île aux oiseaux") were sampled during this spring period and maintained in aquaria at different temperature (4, 14, 18 and 21°C), allowing quantification of the BRD dynamic. At 14°C, the prevalence reached 90% after two months (Figure 7). At other temperatures (4, 18 and 21°C), the clams developed BRD but in much lower proportions. Environmental conditions strongly modulate the development of BRD. Experimental infections in laboratory with V. tapetis at various temperature have also shown that maximum BRD development in clams occurs at 14°C, and that at 21°C the prevalence is very low due to high recovery rates.

Temperature is also a major modulating factor impacting growth and survival of V. tapetis. Maximum growth occurs around 22°C but its survival decreases rapidly after 25°C and death is observed at 27°C [START_REF] Paillard | Epidemiological survey of the brown ring disease in clams of Atlantic coast: role of temperature in variations of prevalence[END_REF]. The death of V. tapetis at a temperature generally measured in the sediment at summer conditions could explain the low prevalence of BRD in southern sites such as the Adriatic and Mediterranean Seas. Other environmental factors such as salinity can also modulate the Vibrio-clam interaction; in fact lower salinities enhance BRD development [START_REF] Soudant | Salinity effect on the vibriosis model[END_REF]. The quantity and quality of nutrients available also have a great impact on the condition index and on recovery processes of clam [START_REF] Delaporte | Effect of a mono-specific algal diet on immune functions in two bivalve species Crassostre gigas and Ruditapes philippinarum[END_REF][START_REF] Plana | Variations in the biochimical composition of the juveniles of Manila clam, Ruditapes philippinarum, infected by a Vibrio sp[END_REF]. Figure 8 integrates these different pressures on the host-pathogen environment system. The theoretical individual-based model of Brown Ring Disease in Manila clams suggested also that the outcome of the host-parasite interaction is dependent on food supply and temperature [START_REF] Paillard | A theoretical individual-based model of Brown Ring Disease in Manila clams, Venerupis philippinarum[END_REF][START_REF] Richard | -Immune responses of phenoloxidase and superoxide dismutase in the Manila clam Venerupis philippinarum challenged with Vibrio tapetis -part II: combined effect of temperature and two Vibrio tapetis strains[END_REF]. Simulations with increase temperature (1 or 2°C) favor generally disease development, indicating that climate warming might favor the spread of BRD, although this disease is a cold water vibriosis.

CONCLUSION

Brown Ring Disease in clams, caused by Vibrio tapetis, represents a unique model to explore how a bacterium, acting as a microparasite, can intimately alter bivalve physiology and ecology. Adopting an ecological approach to studying BRD will allow the coupling of various phenotypes (characteristic external signs, pathogen load and other life traits such as physiological and immunological patterns) to better understand host-parasite coevolution; moreover, BRD phenology could also be followed along with its response to climatic phenomena. New investigations into the role of microbiota on the host-pathogen-environment interplay represent an intriguing possibility for future research into ecological approaches to mollusc diseases.

Figure 1 .

 1 Figure 1. The ecological approach of the Host-Pathogen-Environment System.

Figure 2 .

 2 Figure 2. BRD clinical signs (BRD+) that could be observed macroscopically and microscopically from Phase 1 to Phase 3 of Brown Ring Disease. In Phase 1 (cases B and C), only conchiolin deposit is observed without shell repair. In the other phases (Phase 2, 2.5 and 3), the conchiolin deposit is recovered by shell repaired layers at different recovery steps, intermediate (phase 2), almost recovery (Phase 2.5) and completely recovery (Phase 3).

Figure 3 .

 3 Figure 3. Synthesis of Brown Ring Disease (BRD) phenology, described at ecological, physiological and epidemiological levels. A: Vibrio tapetis only within tissues (purple); C: V. tapetis only in extrapallial and pallial fluids (red); B: V. tapetis in internal and external compartments (purple and red). Recovery process in green.

Figure 4a .

 4a Figure 4a. Density of Manila clams (purple squares) and V. tapetis concentrations in the surface (0-2 cm) sediment (yellow circles) in the Gulf of Morbihan, France.March campaign 1999.

Figure 4b .

 4b Figure 4b. BRD prevalence in burrowing (red circles) and in rising clams at the surface of the sediment (green squares)in the Gulf of Morbihan, France. March campaign 1999.

Figure 5 .

 5 Figure 5. Log 10 [V. tapetis] in the pallial and extrapallial fluids of clams for each density condition and transmission mode at day 30. Blue: transmission by water; green: contact transmission; red:

  through feces. WC: with water change every two days. NWC: No water change. Low density (100 clams/m 2 ); High density (400 clams/m 2 ).

Figure 6 .

 6 Figure 6. Life cycle of V. tapetis pathogenic to clams. The black arrows indicate dissemination, the blue arrows indicate transmission modes and the red arrows indicate mortality associated with the rising of the clams to the surface of the sediment.

Figure 7 .

 7 Figure 7. Evolution of BRD prevalence in clams harvested from natural populations (Île aux oiseaux, Gulf of Morbihan) in March 1998 and maintained in aquariums without food for two months at different temperatures (4, 14, 18 and 21°C).

Figure 8 .

 8 Figure 8. Synthesis of the Host-Pathogen-Environment system in the case of Brown Ring disease (BRD) in R. philippinarum.

Acknowledgments

I would like to thank, Romain Froquet, Susan Ford and Vanessa Thomann for the analysis of clams from field studies and laboratory experiments, Lionel Alano from Bailleron island for field survey and Sébastien Hervé for the illustrations and Ewan Harney for English corrections.


Optimization methods for phase estimation in differential-interference-contrast (DIC) microscopy Simone Rebegoldi 1 , Lola Bautista [START_REF] Bautista | IEEE 13th Int. Symp. on Biomedical Imaging[END_REF][START_REF] Fletcher | [END_REF] , Laure Blanc-Féraud [START_REF] Fletcher | [END_REF] , Marco Prato 1 , Luca Zanni 1 and Arturo Plata 2 1 Università di Modena e Reggio Emilia, [START_REF] Bautista | IEEE 13th Int. Symp. on Biomedical Imaging[END_REF] Universidad Industrial de Santander, [START_REF] Fletcher | [END_REF] Université Côte d'Azur,

The DIC phase estimation problem

The DIC image is formed by the interference of two orthogonally polarized beams that have a lateral displacement (called shear) and are phase shifted relatively one to each other. The resulting image has a 3D high contrast appearance, which can be enhanced by introducing a uniform phase difference between the beams (called bias).

Model: the DIC image formation is described by the polychromatic rotational-diversity model [1,2]

(o k,λ ℓ ) j = (h k,λ ℓ ⊗ e -iφ/λ ℓ ) j 2 + (η k,λ ℓ ) j , k = 1, . . . , K , ℓ = 1, 2, 3, j ∈ χ
• k is the index of the rotation of the specimen w.r.t. the horizontal axis, ℓ is the index denoting one of the RGB channels and j = (j 1 , j 2 ) is a 2D-index varying in the set χ = {1, . . . , M} × {1, . . . , P} • λ ℓ is the ℓ-th illumination wavelength

• o k,λ ℓ ∈ R MP is the ℓ-th color component of the k-th observed image o k = (o k,λ 1 , o k,λ 2 , o k,λ 3 ) ∈ R MP×3 • φ ∈ R MP is the unknown phase vector and e -iφ/λ ℓ ∈ C MP is defined by (e -iφ/λ ℓ ) j = e -iφ j /λ ℓ • h k,λ ℓ ∈ C MP is the discretization of the continuous DIC Point Spread Function • η k,λ ℓ ∈ R MP is the noise corrupting the data, η k,λ ℓ ∼ N (0, σ 2 I (MP) 2).
Problem: given the rotationally diverse images o 1 , . . . , o K , retrieve the phase vector φ by solving min φ∈R MP J(φ) ≡ J 0 (φ) + J TV (φ), (P)

• J 0 (φ) = 3 ℓ=1 K k=1 j∈χ (o k,λ ℓ ) j -(h k,λ ℓ ⊗ e -iφ/λ ℓ ) j 2 2
is the nonconvex least-squares term

• J TV (φ) = µ j∈χ ((Dφ) j ) 2 1 + ((Dφ) j ) 2
2 + δ 2 is the total variation (TV) functional, where µ > 0 is the regularization parameter and δ ≥ 0 is the smoothing parameter (δ = 0 → standard TV).

Optimization methods

Case δ > 0: problem (P) is differentiable → we use a gradient-descent method. Algorithm 1 Limited Memory Steepest Descent (LMSD) method [3] Set ρ, ω ∈ (0, 1), m > 0, α Case δ = 0: problem (P) is non differentiable → we use a proximal-gradient method. Algorithm 2 Inexact Linesearch based Algorithm (ILA) [4] Set ρ, ω ∈ (0, 1), 0 < α min ≤ α max , τ > 0, φ (0) ∈ R MP , n = 0.

n ρ i n satisfies J(φ (n) -α n ∇J(φ (n) )) ≤ J(φ (n) ) -ωα n ∇J(φ (n) ) 2 . 2. Compute the new point as φ (n+1) = φ (n) -α n ∇J(φ (n) ). 3. Update G = [G ∇J(φ (n) )] and Θ = [Θ α -1 n ].

WHILE True

1. Set α n = max min α (0) n , α max , α min , where α

(0) n is chosen as in Algorithm 1. 2. Let ψ (n) = prox α n J TV φ (n) -α n ∇J 0 (φ (n) ) = argmin φ∈R MP h (n) (φ). Compute ψ(n) such that h (n) ( ψ(n) ) -h (n) (ψ (n) ) ≤ ǫ n and 0 ≤ ǫ n ≤ -τ h (n) ( ψ(n) ). 3. Set d (n) = ψ(n) -φ (n) . 4. Compute the smallest non-negative integer i n such that λ n = ρ i n satisfies J(φ (n) + λ n d (n) ) ≤ J(φ (n) ) + ωλ n h (n) ( ψ(n) ). 5. Compute the new point as φ (n+1) = φ (n) + λ n d (n) . 6. Set n = n + 1. END

Convergence and numerical results

Convergence: Any limit point of Algorithm 1 and 2 is stationary for problem (P). Since J satisfies the Kurdyka-Lojasiewicz property, Algorithm 1 converges to a limit point; the same result can be proved for Algorithm 2 when the proximal point is computed exactly [4]. Results: for both objects, cone (top row) and cross (bottom row), K = 2 DIC images have been generated.

  > 0, φ (0) ∈ R MP , G = [ ], Θ = [ ], n = 0. WHILE True FOR l = 1, . . . , m 1. Compute the smallest non-negative integer i n such that α n = α (0)

4 .

 4 Set n = n + 1. END 6. Define the (m + 1) × m matrix Γ = diag(Θ) zeros(1, m) -zeros(1, m) diag(Θ) . 7. Compute the Cholesky factorization R T R of the m × m matrix G T G . 8. Solve the linear system R T r = G T ∇J(φ (n) ). 9. Define the m × m matrix Φ = [R, r ]ΓR -1 and its approximation Φ = diag(Φ) + tril(Φ, -1) + tril(Φ, -1) T , which is symmetric and tridiagonal. 10. Compute eigenvalues θ 1 , . . . , θ m of Φ and define α (0) n+i-1 = 1/θ i , i = 1, . . . , m. END

  True phase Noisy DIC image Rec. phaseThe parameters of the methods have been tuned as follows: ρ = 0.5, ω = 10 -4 , m = 4, α min = 10 -5 , α max = 10 2 , τ = 10 6 -1, φ (0) = 0. The methods are compared with the Polak-Ribière conjugate gradient method equipped with the strong Wolfe conditions (PR + -SW) and a linesearch based on polynomial approximation (PR-PA)[1].


Simone Rebegoldi
email: simone.rebegoldi@unimore.it

Lola Bautista
email: bautista@i3s.unice.fr

Laure Blanc-Féraud
email: laure.blanc_feraud@inria.fr

Marco Prato
email: marco.prato@unimore.it

Luca Zanni
email: luca.zanni@unimore.it

Arturo Plata
email: aplatag@yahoo.com

Accelerated gradient-based methods for phase estimation in differential-interference-contrast microscopy

In the last forty years, differential-interference-contrast (DIC) microscopy has gained popularity in biomedical research as an effective optical microscopy technique used to observe unstained transparent specimens under a transmitted-light configuration. The DIC image formation is caused by the interference of two orthogonally polarized beams, which are laterally split of a few tenths of a micrometer by a Wollaston prism, phase shifted when passing through different materials across the specimen and then successively recombined by a sliding prism. The resulting image has a three dimensional, high contrast appearance, which can be enhanced by adjusting the sliding prism along the direction of the split in order to introduce a uniform phase difference between the two beams. From the mathematical viewpoint, the DIC image acquisition is described by the following nonlinear model [START_REF] Bautista | Phase estimation in differential-interference-contrast (DIC) microscopy[END_REF][START_REF] Preza | Rotational-diversity phase estimation from differential interference contrast microscopy images[END_REF]

o k,λ = h k,λ * e -i φ λ 2 + η k , (1)
where φ ∈ R n is the true phase function, k is the rotation index and λ is the value of the RGB wavelength in micrometers, o k,λ ∈ R n and h k,λ ∈ C n are respectively the observed image and the DIC Point-Spread-Function at the k-th rotation of the specimen and illumination wavelength λ, * is the convolution operator, | • | 2 is the component-wise square modulus and η k is the realization of a Gaussian random variable with zero mean and variance σ 2 . Following the rotational-diversity model proposed in [START_REF] Preza | Rotational-diversity phase estimation from differential interference contrast microscopy images[END_REF] and furtherly extended in [START_REF] Bautista | Phase estimation in differential-interference-contrast (DIC) microscopy[END_REF], one is interested in retrieving the specimen's phase function from a set of DIC intensity images acquired at different rotations of the specimen. Since this problem is highly ill-posed, one must look for a solution of the following regularized minimization problem min

φ∈R n J LS (φ; o) + J R (φ), (2)
where J LS is a least-squares term measuring the distance between the observed image o k,λ and the predicted image and J R is the regularization term. When a differentiable regularizer, such as a Tikhonov penalty term, is considered, one can address ( 2) by means of a nonlinear conjugate gradient method [START_REF] Preza | Rotational-diversity phase estimation from differential interference contrast microscopy images[END_REF], which is particularly suited for least squares problems. However, the computation of the line search parameter at each iteration may require several evaluations of both the function and its gradient in order to ensure convergence [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF], which significantly increases computational time when such evaluations are expensive, as is the case of the DIC functional. Furthermore, a conjugate gradient method can not handle the presence of a non differentiable regularizer, such as the total variational functional.

To overcome these limits, we propose to address the problem of phase estimation in DIC microscopy by means of a recently proposed proximal-gradient method [START_REF] Bonettini | Variable metric inexact line-search based methods for nonsmooth optimization[END_REF]. The key features of this approach are the use of an Armijo-like rule to determine the step size along the descent direction and the possibility of adopting a variable metric to compute the proximal point. Since providing an efficient scaling matrix for the DIC problem is a quite difficult task, the acceleration of the method relies uniquely on a clever adaptive choice of the stepsize at each iteration [START_REF] Fletcher | A limited memory steepest descent method[END_REF]. The method is shown to be efficient both when the total variational functional and a smooth approximation of it is used. In the latter case, the method shows also significant improvements in terms of efficiency and stability with respect to widely used conjugate gradient methods.

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Simone Rebegoldi
email: simone.rebegoldi@unimore.it

Lola Bautista
email: bautista@i3s.unice.fr

Marco Prato
email: marco.prato@unimore.it

Luca Zanni
email: luca.zanni@unimore.it

Laure Blanc-Féraud
email: laure.blancferaud@inria.fr

Arturo Plata
email: aplatag@yahoo.com

An efficient gradient-based method for differential-interference-contrast microscopy

Keywords: DIC microscopy, inverse problems, gradient-descent method

Differential-interference-contrast (DIC) microscopy is an optical microscopy technique widely used in biology to observe unstained transparent specimens, in which a two-dimensional image is formed from the interference of two waves that have a lateral differential displacement (shear) and are phase shifted relative one to each other. Following the rotational-diversity model proposed in [START_REF] Preza | Rotational-diversity phase estimation from differential interference contrast microscopy images[END_REF], one is interested in retrieving the specimen's phase function from a set of DIC intensity images acquired at different rotations of the specimen. This highly nonlinear, illposed problem is solved by adopting a least squares approach and thus looking for a regularized solution of a smooth nonconvex optimization problem. As already done in [START_REF] Preza | Rotational-diversity phase estimation from differential interference contrast microscopy images[END_REF], one can address the DIC problem by means of a nonlinear conjugate gradient method, which is particularly suited for least squares problems. However, the computation of the line search parameter at each iteration may require several evaluations of both the function and its gradient in order to ensure convergence [START_REF] Gilbert | Global convergence properties of conjugate gradient methods for optimization[END_REF], which significantly increases computational time when such evaluations are time-consuming, as is the case of the DIC problem. In this light we propose an efficient gradient-descent method for the estimation of a specimen's phase function from polychromatic DIC images. The method minimizes the sum of a nonlinear least-squares discrepancy measure and a smooth approximation of the total variation and exploits a recent updating rule for the choice of the step size [START_REF] Fletcher | A limited memory steepest descent method[END_REF]. Numerical simulations on two computer-generated objects show significant improvements in terms of efficiency and stability with respect to widely used conjugate gradient methods.