Paula Pérez-López

Eva Ternon

Sara González-García

Grégory Genta-Jouve

Gumersindo Feijoo

Olivier P Thomas

Mª Teresa Moreira

Environmental solutions for the sustainable production of bioactive natural products from the marine sponge Crambe crambe

Keywords: Sponge cultivation, crambescins, crambescidins, antitumor, Life Cycle Assessment, Crambe crambe, Life Cycle Inventory

Crambe crambe is a Mediterranean marine sponge known to produce original natural substances belonging to two families of guanidine alkaloids, namely crambescins and crambescidins, which exhibit cytotoxic and antiviral activities. These compounds are therefore considered as potential anticancer drugs. The present study focuses on the environmental assessment of a novel in vivo process for the production of pure crambescin and crambescidin using sponge specimens cultured in aquarium.

The assessment was performed following the ISO 14040 standard and extended from the production of the different mass and energy flows to the system to the growth of the sponge in indoor aquarium and further periodic extraction and purification of the bioactive compounds. According to the results, the two stages that have a remarkable contribution to all impact categories are the purification of the bioactive molecules followed by the maintenance of the sponge culture in the aquarium. Among the involved activities, the production of the chemicals (particularly methanol) together with the electricity requirements (especially due to the aquarium lighting) are responsible for up to 90% of the impact in most of the assessed categories. However, the contributions of other stages to the environmental burdens, such as the collection of sponges, considerably depend on the assumptions made during the inventory stage. The simulation of alternative scenarios has led to propose improvement alternatives that may allow significant reductions ranging from 20% to 70%, mainly thanks to the reduction of electricity requirements as well as the partial reuse of methanol.

Introduction

The largest portion of our planet's surface is covered by water and the biological diversity found in marine ecosystems make seas and oceans one of the most promising sources of natural resources for the future [START_REF] Larsen | Phenotypic taxonomy and metabolite profiling in microbial drug discovery[END_REF][START_REF] Leal | Trends in the discovery of new marine natural products from invertebrates over the last two decades -Where and what are we bioprospecting?[END_REF]. Among aquatic organisms, sponges are one of the most diverse invertebrates not only due to the number of species but also to the variety of morphological characters [START_REF] Hooper | Systema Porifera. A Guide to the Classification of Sponges[END_REF]. Indeed, between 7,000 and 8,000 different species have already been described, and at least twice that number is thought to exist [START_REF] Hooper | Biogeography of Indo-west Pacific sponges: Microcionidae, Raspailiidae, Axinellidae[END_REF][START_REF] Thakur | NW Spain): Comparative analysis of two major fishing methods[END_REF]. This diversity associated to the fact that these sessile invertebrates produce a large array of secondary metabolites make sponges a good target for the search of high value added molecules [START_REF] Leal | Trends in the discovery of new marine natural products from invertebrates over the last two decades -Where and what are we bioprospecting?[END_REF]. According to Sipkema et al. (2005a), 5,300 natural products have been isolated from marine sponges worldwide, an amount that increases annually [START_REF] Blunt | Marine natural products[END_REF]2012). Many of these molecules (e.g. halichondrin B, avarol, crambescidins) have shown high biological activities that make them valuable products for medical drugs development due to their anti-inflamatory, antitumor, immunosuppressive or neurosuppressive, antiviral or antibiotic properties, among others (Bergman et al., 2011b;[START_REF] Bondu | Additional bioactive guanidine alkaloids from the Mediterranean sponge Crambe crambe[END_REF][START_REF] Newman | Marine Natural products and related compounds in clinical and advanced preclinical trials[END_REF]Sipkema et al., 2005a). Eribulin mesylate is the first drug derived from a sponge natural product that entered the market in 2011 as an anticancer agent [START_REF] Huyck | Eribulin mesylate[END_REF].

Despite the great potential of bioactive compounds from marine origin and particularly from sponges, steady production is a key limiting factor that may hinder the development of commercial processes [START_REF] Murray | Sustainable production of biologically active molecules of marine based origin[END_REF]. As bioactive compounds of marine origin are present in small quantities in the producer organisms, fresh material is required in large amounts. Wild harvest only satisfies the demand partially and arises as an unsuitable production route (Bergman et al., 2011b;[START_REF] Osinga | Cultivation of Marine Sponges[END_REF][START_REF] Pomponi | The oceans and human health: The discovery and development of marine-derived drugs[END_REF]. Therefore, unless feasible alternatives to harvesting from the natural environment are developed, many of these target molecules will remain unexploited [START_REF] Murray | Sustainable production of biologically active molecules of marine based origin[END_REF]. For this reason, the current challenge is to develop efficient culture techniques for small to medium-scale production schemes [START_REF] Schippers | Chapter six -Cultivation of Sponges, Sponge Cells and Symbionts: Achievements and Future Prospects[END_REF].

The selection of the most appropriate culture technique can depend on the nature of the target compound and its concentration within the sponge. Thus, if the organism presents a high concentration of the desired metabolite, the cultivation of adult specimens would be the best choice, while in vitro cell cultures may constitute a more suitable method for products found in low concentrations [START_REF] Schippers | Chapter six -Cultivation of Sponges, Sponge Cells and Symbionts: Achievements and Future Prospects[END_REF]Sipkema et al., 2005b). However, in vitro cultivation systems have been found difficult to maintain in a long-term operation [START_REF] Müller | Sustainable production of bioactive compounds by sponges-cell culture and gene cluster approach: a review[END_REF][START_REF] Rinkevich | Cell cultures from marine invertebrates: obstacles, new approaches and recent improvements[END_REF].

Alternatively, aquaculture has been widely proposed as a technique to supply sponge materials, not only for the production of natural bath sponges, but more recently also for biotechnological purposes [START_REF] Duckworth | Farming sponges to supply bioactive metabolites and bath sponges: a review[END_REF][START_REF] Munro | The discovery and development of marine compounds with pharmaceutical potential[END_REF][START_REF] Osinga | Cultivation of Marine Sponges[END_REF][START_REF] Pronzato | Mediterranean commercial sponges: over 5000 years of natural history and cultural heritage[END_REF]. Cultivation of sponges can be performed either in situ or ex situ (Bergman et al., 2011b;[START_REF] Louden | An assessment of the aquaculture potential of the tropical sponges Rhopaloeides odorabile and Coscinoderma sp[END_REF]Sipkema et al., 2005b). Sea-based culture systems (in situ systems) consist of the construction of a sponge field where small cuttings (explants) from a parent are strung on a support for cultivation in the sea, so as to keep the organisms in their natural environment [START_REF] Schippers | Chapter six -Cultivation of Sponges, Sponge Cells and Symbionts: Achievements and Future Prospects[END_REF]. The main drawbacks of this alternative are the numerous risks to which sponges are exposed, including biological factors such as predation and fouling, but also diseases or adverse weather conditions [START_REF] Schippers | Chapter six -Cultivation of Sponges, Sponge Cells and Symbionts: Achievements and Future Prospects[END_REF][START_REF] Webster | A spongin-boring -proteobacterium is the etiological agent of disease in the Great Barrier Reef sponge Rhopaloeides odorabile[END_REF]. These risks are turned into very fluctuant survival rates that strongly depend not only on the considered species but also on the location of the sponge field, the season and the aquaculture method (Bergman et al., 2011a(Bergman et al., , 2011b;;[START_REF] De Caralt | In Situ Aquaculture Methods for Dysidea avara (Demospongiae, Porifera) in the Northwestern Mediterranean[END_REF], 2007;[START_REF] Voogd | An assessment of sponge mariculture potential in the Spermonde Archipelago, Indonesia[END_REF][START_REF] Ledda | Mariculture for bacterial and organic waste removal: a field study of sponge filtering activity in experimental farming[END_REF][START_REF] Louden | An assessment of the aquaculture potential of the tropical sponges Rhopaloeides odorabile and Coscinoderma sp[END_REF][START_REF] Osinga | Sponge aquaculture trials in the East-Mediterranean Sea: New approaches to earlier ideas[END_REF].

In order to circumvent these difficulties, the ex situ cultivation of sponges in closed or semi-enclosed systems such as aquarium has been proposed as an alternative strategy [START_REF] Mohamed | Changes in bacterial communities of the marine sponge Mycale laxissima on transfer into aquaculture[END_REF][START_REF] Osinga | Progress towards a controlled culture of the marine sponge Pseudosuberites andrewsi in a bioreactor[END_REF]. Even if this approach avoid seasonality effects and allow controlled conditions, the observed growth rates in aquarium are significantly lower than those of mariculture. The limited progress made in the cultivation of sponges under controlled conditions is due to the scarce knowledge on the optimal environmental conditions and ecological needs required by sponges to develop properly in a non-natural system [START_REF] Carballo | Culture of explants from the sponge Mycale cecilia to obtain bioactive mycalazal-type metabolites[END_REF]. In this regard, [START_REF] Schippers | Chapter six -Cultivation of Sponges, Sponge Cells and Symbionts: Achievements and Future Prospects[END_REF] suggest that ex situ cultivation should be performed in a semi-continuous mode instead of a batch operation, by regularly harvesting a small fraction of the culture.

This study focuses on the environmental assessment of the production of a bioactive fraction constituted by crambescins and crambescidins from the marine sponge Crambe crambe [START_REF] Schmidt | Die Spongien Des Adriatischen Meeres[END_REF], a red encrusting sponge that is widely found in the Western Mediterranean Sea as well as in the Macaronesian archipelagos [START_REF] Duran | Phylogeographical history of the sponge Crambe crambe (Porifera, Poecilosclerida): range expansion and recent invasion of the Macaronesian islands from the Mediterranean Sea[END_REF]. Both families of guanidine alkaloids have already revealed significant cytotoxic activities, and they are considered as potential anticancer drugs [START_REF] Bondu | Additional bioactive guanidine alkaloids from the Mediterranean sponge Crambe crambe[END_REF][START_REF] Laville | Bioactive guanidine alkaloids from two caribbean marine sponges[END_REF][START_REF] Martín | Differential effects of crambescins and crambescidin 816 in voltage-gated sodium, potassium and calcium channels in neurons[END_REF]. The production process consists of the periodic extraction of these biocompounds keeping the organisms alive. This alternative may allow a steady and prolonged production of antitumoral compounds as a basis for a commercial application of these biomolecules.

The evaluation of the process was performed according to a Life Cycle Assessment (LCA) approach.

LCA standardized methodology was used to assess the environmental impacts of the previously described novel process from a cradle to gate perspective (ISO 14040, 2006). Although other production processes involving marine organisms, such as microalgae or macroalgae, have been already addressed through a life cycle viewpoint [START_REF] Aresta | Utilization of macro-algae for enhanced CO 2 fixation and biofuels production: Development of a computing software for an LCA study[END_REF][START_REF] Brentner | Combinatorial life cycle assessment to inform process design of industrial production of algal biodiesel[END_REF][START_REF] Campbell | Life cycle assessment of biodiesel production from microalgae in ponds[END_REF][START_REF] Clarens | Environmental life cycle comparison of algae to other bioenergy feedstocks[END_REF][START_REF] Lardon | Life-cycle assessment of biodiesel production from microalgae[END_REF]Pérez-López et al., 2013), there are not available LCA studies focused specifically on the production of high value added molecules from sponges. To the best of our knowledge, this study develops for the first time a detailed life cycle inventory (LCI) and quantification of the environmental impacts associated with the production of bioactive compounds by sponges. Moreover, this paper presents a novel method to obtain the product while maintaining the organism alive. This approach prevents from the unsustainable exploitation of sponges in natural environments, where the growth of new individuals to replace those used to extract the target compounds would take such a long time that their production would be unfeasible.

Goal and scope definition

Objectives

The main goal of this study was to identify the environmental impacts associated with the sustainable production of two potential antitumor molecules, specifically crambescin and crambescidin, from the Mediterranean sponge Crambe crambe. The production process has been developed in the Institut de Chimie de Nice at the University the Nice Sophie-Antipolis (France). Furthermore, once the major hot spots (or most problematic issues) were determined, alternative scenarios were simulated and evaluated from an environmental point of view in order to suggest feasible improvement measures that reduce impacts to obtain a more sustainable process.

The study takes into account the production of the different mass and energy flows to the system, as well as the growth of the sponge in indoor aquarium and further periodic extraction and purification of the bioactive compounds. Although only crambescidins have been patented for their cytotoxic and antiviral activities [START_REF] Rinehart | Crambescidins: New antiviral and cytotoxic compounds from the sponge Crambe crambe[END_REF], recent studies suggest that also crambescins may have interesting biological properties [START_REF] Bondu | Additional bioactive guanidine alkaloids from the Mediterranean sponge Crambe crambe[END_REF][START_REF] Martín | Differential effects of crambescins and crambescidin 816 in voltage-gated sodium, potassium and calcium channels in neurons[END_REF]. Therefore, both families of guanidine alkaloids were considered as target products.

Functional unit

The functional unit is a key parameter that provides the reference to which the inputs and outputs of the product system can be related (ISO 14040, 2006). In this case, the selected functional unit was 100 mg of total bioactive fraction, including 50 mg of pure crambescin A1 and 50 mg crambescidin 816, which corresponds to the production during one year of operation for the base scenario. It should be pointed out that both products are obtained as pure compounds and they could be directly applied for pharmaceutical purposes. The total economic value of this production is estimated in roughly 7,000 €, according to a price of 70 €/mg for a similar biocompound: halichondrin B (Sipkema et al., 2005b).

Description of the system under study

The system boundaries for the assessment of the production of crambescins and crambescidins by Crambe crambe are shown in Figure 1. The stages or subsystems of the process included within the system boundaries are further described below, considering the extraction frequency and yield of the base scenario. , which also served as water supply for other units in the facilities. Once decanted, seawater was transferred to the aquarium at a flow rate of 2 L/min.

iii) Maintenance in aquarium

Ten individuals on their substrates, with an approximate surface of 50 cm 2 each, were transferred in each aquarium, which was illuminated by conventional fluorescent lamps. Since the aquariums were fed with seawater, operational conditions changed depending on the period of the year. The temperature was kept below 20ºC during summer with a control system, whereas it fell to 10ºC in winter. Although other parameters also fluctuated depending on the season, they exhibited values around 36.8 kg/dm 3 for salinity, 8 for pH and 6.5-7 mg/L for oxygen demand. The detailed data can be accessed on the website http://somlit.epoc.u-bordeaux1.fr/fr/spip.php?rubrique48. After two days of acclimatization in these aquariums (Figure 2), which could be observed by the presence of open canals and oscules on the surface of the sponge, the specimens were ready for extraction.

Figure 2 Aquariums for the indoor cultivation of Crambe crambe iv) Extraction

Each individual was transferred alive into a closed plastic box (700 mL) with 475 mL seawater and 25 mL of ethanol 96%. Mechanical stress was applied using a -snail fork‖ and scratching with 5 cm 2 intervals, avoiding canals. Half of the volume was collected for filtration in a second closed plastic box and the stressed sponge was replaced in the aquarium for 7 days of recovering. After this period, canals and oscules were opened again in the same way as before stressing the sponge.

v) Purification of the bioactive compounds

The resulting 250 mL solution from the previous stage was filtered with a peristaltic pump through 0.22 µm filter in order to remove all insoluble particles at 50 mL/min.

After the filtration stage, the bioactive compounds were extracted from the seawater and purified by HPLC using water and methanol as solvents. The obtained fractions contained around 0.1 mg of pure crambescin A1 and 0.1 mg of crambescidin 816 (0.2 mg of bioactive compounds obtained from each individual). Both extraction and purification were repeated weekly on the same specimens. Despite the periodical scratching of the sponge surface, individuals placed in the aquarium that were not covering all the substrate were observed to grow at a rate of c.a. 20% a year (area measurement) with or without milking. No comparison was done with culture in the sea but this result evidenced that the sponges placed in aquarium were in relatively good health.

Inventory analysis, data quality and simplifications

The LCI data for the foreground system including chemicals, water and electricity consumptions, consisted of average data obtained by on-site measurements. Water and air emissions were calculated on the basis that the chemicals which are not consumed during the process are directly discharged. The global inventory of the process is shown in Table 1.

Table 1 Global inventory for the production of pure crambescin and crambescidin from Crambe crambe in indoor aquariums (functional unit: 100 mg bioactive fraction, consisting of 50 mg pure crambescin and 50 mg pure crambescidin). Concerning the background system, the corresponding inventory data for the production of all the inputs to the system were taken from Ecoinvent database. These inputs include the production of the different chemicals required for the extraction and purification stages, the electricity used in the different production stages, as well as the materials for the equipment (vessel and scuba equipment, water supply system, aquarium, fluorescent tubes, electronic devices) and waste disposal.

INPUTS from TECHNOSPHERE Materials

In the case of the vessel, a shared use of the boat was considered for the base scenario. 2010), considering manufacturers' specifications. For paint and anti-fouling emitted to marine environment, a loss of two thirds of the total amount used was considered [START_REF] Hospido | Life cycle environmental impacts of Spanish tuna fisheries[END_REF]. Solid waste was assumed to be disposed of in sanitary or inert landfills Regarding water supply, the design of the pumping and decanting system was estimated from mass balances. As the output from the decanting tank was shared with other aquariums, the corresponding amount of material was calculated from the ratio between the flow to Crambe crambe aquarium and the total flow to the decanting tank. The quantification of the polymethyl metacrylate of the aquarium was also calculated according to the dimensions of the tank and the density of the material, considering a wall thickness of 4 mm. As the inventory is associated with a hypothetical facility placed in shore, transport of equipments and chemicals was considered negligible. A detailed description of the corresponding database reports considered is shown in Table 2. The materials needed for the lab ware, as well as for the vessel and scuba equipment, were estimated as average values from manufacturers' specifications. For the equipment, different life spans were considered, according to the assumptions that are specified in Table 3. In this study, two target pure products can be distinguished: crambescin and crambescidin. As both compounds have shown comparable activities, they would have similar market prices; accordingly, mass allocation was considered. Each biocompound corresponds to 50% of the total bioactive fraction, so the environmental burdens associated with them would be half of the total impacts.

However, other fractions of crambescins and crambescidins may be obtained as by-products.

Although these fractions have been neglected in the present study further research could provide additional information about their potential use. In this case, a fraction of the environmental impacts would be allocated to these by-products and, therefore, the environmental burdens for the main target compounds would decrease with respect to the results here described.

Life Cycle Assessment of crambescin and crambescidin production by Crambe crambe

The environmental profile of the described system was assessed by performing classification and characterization stages of the LCA methodology (ISO 14040, 2006). Normalization and weighting were not conducted as these optional (and, to some extent, subjective) elements were not considered to provide additional, robust information for the objectives of the study. The characterization factors reported by the Centre of Environmental Science of Leiden University (CML 2001 method) were used [START_REF] Guinée | Life Cycle Assessment. An operational guide to the ISO standards[END_REF]. The impact potentials (or impact categories) evaluated according to the CML method were: abiotic depletion (ADP), acidification (AP), eutrophication (EP), global warming (GWP), ozone layer depletion (ODP), human toxicity (HTP), freshwater aquatic ecotoxicity (FEP), marine aquatic ecotoxicity (MEP), terrestrial ecotoxicity (TEP) and photochemical oxidants formation (POFP).

The software SimaPro 7.3 was used for the computational implementation of the inventories [START_REF] Goedkoop | Introduction to LCA with SimaPro 7[END_REF].

Identification of hot spots

The characterization results associated with the environmental impacts caused by C. crambe process in the addressed impact categories are detailed in Table 4. As shown in Figure 3, most of the environmental impacts are dominated by the purification stage, with contributions ranging from 34.0% (for TEP) to nearly 90% (87.6% for ADP and 89.9% for ODP). The maintenance in aquarium is also a significant stage, especially in terms of toxicity potentials, which present values between 40.9% and 48.7%. Among the secondary subsystems, water supply is the only stage that has a relevant contribution related to toxicity potentials, with impacts ranging from 14.4% (MEP) to 21.7% (HTP).

Materials of equipments Electricity

Waste treatment

Figure 4 Relative contribution per involved activity of the production of pure crambescin and crambescidin in the base scenario.

Analysis of major contributors

Production of chemicals

As the production of chemicals is the main issue related to the environmental impacts of the production of crambescin and crambescidin, the breakdown of the contributions of these processes in all the assessed categories is shown in Figure 5. According to the graph, the production of methanol needed for the purification is the main responsible for the environmental impacts, with more than 80% of the contributions to all the categories. The reason behind this remarkable contribution is not the unitary environmental impact of methanol, but the large amount that is consumed in the process, which is two orders of magnitude above other involved chemicals with higher impacts per mass unit, such as acetonitrile or trifluoroacetic acid.

Electricity requirements

The second hot spot found in the analysis corresponds to the production of electricity in the different stages of the process. This process especially contributes to EP and toxicity categories. The impacts to EP are mainly due to the emissions of phosphate and nitrogen oxides whereas the contributions to toxicity categories are mostly related to the emissions of metals to air and water. Particularly, HTP is highly affected by emissions of selenium, arsenic and chromium VI, while the major responsible for FEP and MEP are emissions of nickel, vanadium and beryllium. Finally, the environmental impacts to TEP principally come from emissions of mercury derived from the use of coal for electricity generation and chromium VI from the distribution network.

In order to identify the stages with higher electricity requirements, the contributions are depicted in Figure 6. Nearly three fourths of the electricity consumption comes from lighting during the maintenance of C. crambe in the aquarium. This finding is consistent with the experience from previous works, which point out the importance of artificial lighting in the total energy cost of the cultivation of other marine organisms [START_REF] Das | Incremental energy supply for microalgae culture in a photobioreactor[END_REF][START_REF] Pulz | Photobioreactors: Design and performance with respect to light energy input[END_REF].

Therefore, the optimization in terms of electricity consumption should be focused on the reduction of lighting. Among the secondary stages, water supply has the highest consumption, with 62% due to water pumping. This result suggests that a recirculation in the seawater supply may help to reduce the environmental impacts associated with this stage.

Pumping

3.52%

Figure 6 Relative contribution of the electricity requirements per stage to the total environmental impact of the production of pure crambescin and crambescidin in the base scenario.

Effect of vessel operations

According to the results, the collection of the sponges from the environment is a minor contributor to all impact categories. This contribution is mainly related to the vessel operations, including fuel consumption but also material inputs for vessel construction. Despite the limited effect found for this subsystem, the results are based on the assumption that the boat is also used for fishing. Thus, only a slight fraction of the total environmental impacts associated with vessel operations were allocated to C. crambe process.

Nevertheless, energy and material inputs in fishing vessels can affect the environmental profile significantly, not only due to fuel consumption but also derived from other materials, such as antifouling agents or paints [START_REF] Hospido | Life cycle environmental impacts of Spanish tuna fisheries[END_REF]Vázquez-Rowe et al., 2010). Moreover, the assumptions considered to allocate the impacts from the vessel may considerably affect the global results. For this reason, a sensitivity analysis is shown in Figure 7.

Three alternative situations were compared to the base scenario. In the first of them (Sc 1), all the impacts associated with the vessel operations were allocated to the production process of crambescins and crambescidins, assuming that the boat used for the collection was a recreational vessel with no additional function (in terms of other material products obtained). However, it may be argued that a recreational use is associated with an immaterial function that should be taken into account. For this reason, the second scenario (Sc 2) allocates the impact of the collection of sponges according to the ratio between the number of hours associated with this process (2 h per year) and the total number of sailed hours within the year, assuming 2 h sailed per week with 52 weeks per year.

The third scenario (Sc 3) is based on the findings of previous works, which suggest that the inputs to vessel construction and maintenance have limited contributions to the total impacts of seafood products [START_REF] Hospido | Life cycle environmental impacts of Spanish tuna fisheries[END_REF]. In this case, a fishing vessel is again considered, and building materials are excluded from the system boundaries. Figure 7 shows that the assumptions considered to determine the impacts from the vessel considerably influence the global environmental profile of the production of crambescins and crambescidins. Indeed, when considering that the vessel use is only associated with sponge collection (Sc 1), the contributions to most impact categories vary between 1.5 and 3.5 times those of the base scenario. Despite these remarkable differences, it should be pointed out that this is the most unlikely scenario, as the collection stage requires the vessel use for a very short period of time. Therefore, a combined use of the boat for other purposes, such as fishing or collection of other marine specimens for product exploitation, is expected. Regarding the other analyzed scenarios, the effect of vessel operations seems rather restricted, with deviations lower than 4.5% in all the impact categories. Thus, Sc 2 results in impact increases between 0.4% (ADP) and 4.4% (HTP), whereas reductions observed in Sc 3 range from 0.03% (ADP) to 0.3% (HTP) with respect to the base scenario.

Simulation of improvement scenarios

Solvent reuse

The production of the chemicals required for the purification stage was the principal contributor to the environmental impacts in five categories (ADP, AP, GWP, ODP and POFP). More than 85% of the mentioned contributions were specifically derived from the production of methanol, due to the large use of this solvent for the purification. For this reason, an alternative scenario is suggested, consisting of the reuse of 50% of the methanol required for obtaining of pure crambescin and crambescidin.

Although this assumption was not based on experimental work, several authors have already checked the feasibility of reusing methanol to extract other similar alkaloids [START_REF] Blaicher | Lupin protein concentrates by extraction with aqueous alcohols[END_REF]Harkrader and Jones, 1998).

According to Figure 8, the reuse of methanol constitutes a promising option to improve the environmental profile of the studied process. The evaluated scenario presents remarkable reductions in terms of ADP (42.4%), AP (22.6%), GWP (32.4%), ODP (43.8%) and POFP (33.7%). The effect on other categories, such as HTP (6.5%) and TEP (11.0%) is relatively limited, though the performance in all the considered categories is better than the base scenario.

Electricity optimization

The production of electricity required throughout the whole C. crambe process was identified as a major concern in six of the ten impact categories under assessment (AP, EP, HTP, FEP, MEP and TEP). The main reason is the dependence on non-renewable sources due to the use of electricity that is directly taken from the French grid, which is characterized by a limited need of fossil fuels but a high reliance on nuclear energy (Dones et al., 2007). Hence, two possible scenarios are evaluated, concerning the use of solar and wind energy as alternative sources to the electricity taken from the grid.

In addition, the artificial lighting of the aquarium was identified as the main hot spot associated with electricity requirements, with 75% of the total electric consumption. This is due to the use of fluorescent lamps, which were switched on 24 h/day. However, the necessity of light for the sponge growth is lower than for other marine organisms such as microalgae or macroalgae [START_REF] González-Rivero | Impacts of macroalgal competition and parrotfish predation on the growth of a common bioeroding sponge[END_REF][START_REF] Ogbonna | Light requirement and photosynthetic cell cultivation -Development of processes for efficient light utilization in photobioreactors[END_REF][START_REF] Yeh | Effect of light supply and carbon source on cell growth and cellular composition of a newly isolated microalga Chlorella vulgaris ESP-31[END_REF]. Furthermore, in the evaluated process, the main goal is not maximizing the biomass production but maintaining the sponge in such healthy conditions that allow the periodical extraction of compounds from the specimens. Therefore, an alternative regime with less lighting seems a feasible strategy to reduce the total electricity consumption of the system. Thus, two additional improvement options were proposed: 16:8 regime scenario and scenario with no lighting. Furthermore, an additional scenario was evaluated, regarding the substitution of conventional fluorescent tubes by light-emitting diodes (LEDs). As well as having a longer life span (about three times higher) than fluorescent lamps, LEDs are also more efficient and can result in a 50% decrease in energy consumption [START_REF] Chen | Cultivation, photobioreactor design and harvesting of microalgae for biodiesel production: A critical review[END_REF].

According to Figure 9, all the proposed alternatives show remarkable reductions in the environmental impact for most categories, except from the solar scenario in ODP which had a higher contribution mainly due to the production of materials for the solar panels. As expected, the improvements were especially significant for toxicity categories, which were more affected by the electricity requirements, but also for AP and EP. In the solar scenario, the reductions ranged from 2.5% (ADP) and 3.1%

(POFP) up to 15% for HTP, FEP and MEP, and even 45.7% for TEP. The wind scenario showed the largest reductions, with more than 35% of improvement in six of the categories (AP, EP, HTP, FEP, MEP and TEP). The scenario with no lighting had the second best performance, with reductions between 30% and 50% for the same categories. However, it should be highlighted that this scenario is based on the assumption that sponge can be maintained in the same conditions (comparable growth rate and equivalent amount of bioactive compounds obtained by extraction) as in the base scenario without lighting. As the verification of this assumption would require further research, the 16:8 regime scenario seems a more feasible strategy to be applied in the short-term. Despite the more restricted improvement, the 16:8 regime scenario still showed significant reductions, ranging from 10% to 16%

for those categories that are affected by the use of electricity within the process. Finally, the substitution of conventional fluorescent tubes by LEDs allowed reductions between 3.5% and 24.5%.

Figure 9 Sensitivity analysis of the environmental performance considering four improved alternatives for the reduction of electricity requirements in the production of pure crambescin and crambescidin.

Recirculation effect

As indicated in the previous section, water supply constitutes a secondary contributor that may have a relevant effect in some categories, due to the electricity consumption of the pumping system. Indeed, continuous pumping of water has already been identified as a significant issue in the cultivation of other marine organisms such as microalgae [START_REF] Lam | Microalgae biofuels: A critical review of issues, problems and the way forward[END_REF][START_REF] Xu | Assessment of a dry and a wet route for the production of biofuels from microalgae: Energy balance analysis[END_REF].

In the case of C. crambe process, this contribution is mostly associated with the seawater collection and pumping from the sea to the decanting tank. Therefore, an alternative scenario where 50%

seawater was recycled to the aquarium instead of its direct discharge to the sea was assessed.

However, the results indicate that the improvement achieved with this measure would be rather limited, with reductions of impact between 0.7% and 4.3%. The highest reductions are found in the toxicity categories, which were those with a significant contribution of electricity. The improvements related to these categories would range from 3.5% for HTP to 4.3% for TEP.

Improved waste treatment

Although the environmental burdens associated with waste treatment are rather slight in comparison with other subsystems of the process, an alternative option was proposed, regarding the final disposal of the waste. In the present study, the assumption that the materials of the equipment and infrastructure were finally sent to landfill was considered. Nevertheless, previous LCA studies of related processes proposed other approaches, such as sending these materials to recycling [START_REF] Collet | Life-cycle assessment of microalgae culture coupled to biogas production[END_REF]. In this case, the final disposal of steel, plastic materials and concrete to landfill was substituted by the recycling of these materials. However, the improvement observed with this measure is very limited and the highest reductions were between 1% and 2.4% for the categories of EP, FEP and MEP.

Best performance vs most feasible scenario

Several of the simulated scenarios can be simultaneously applied, allowing higher reductions of impact. Therefore, the compatible improvement alternatives were combined in two hypothetical scenarios:

-Best performance scenario. In this case, the maintenance in aquarium with no lighting is considered together with the wind electricity supply for the other electricity requirements.

Additional, a methanol re-use of 50% is taken into account, along with a 50% seawater recirculation and a recycling scenario.

-Most feasible scenario. Despite having the lowest environmental burdens, the best performance scenario is based on the assumptions that the reduction of lighting and the reuse of the solvent neither affect the yield of the process nor the purity of the two produced compounds. As further research should be needed to prove the accuracy of these assumptions, another alternative scenario that seems more feasible in a short period of time was proposed. In this case, a 16:8 regime was considered jointly with a solar electricity supply and the use of LEDs, as well as a methanol re-use of 25%, was assumed. Seawater recirculation and recycling of materials were also considered.

The improvements that may be achieved by the combined implementation of the alternative scenarios are shown in Figure 10. According to these results, the environmental profile of the production of pure crambescin and crambescidin can be significantly enhanced, with reductions of impact for the best performance scenario between 52 and 78% depending on the category. Even if a more conservative approach is considered, the most feasible scenario in a short term period would allow improvements up to 40% for the categories of AP, EP, HTP, FEP and MEP, and as high as 65% for TEP.

Conclusions

The principal aim of this study was to evaluate the environmental performance of a novel process for the production of high value added biomolecules from a marine sponge. The life cycle approach was used to identify the main contributors to the assessed impact categories, allowing the proposal of several improvement scenarios that were simulated and analyzed through LCA methodology.

The use of large amounts of methanol during the purification stage and the high electricity requirements essentially due to the continuous illumination of the aquarium were found as the two major environmental concerns of the base scenario. The environmental burdens of other stages, such as the collection of sponges, considerably depend on the assumptions made during the inventory analysis stage.

Among the proposed improved alternatives, the most promising scenarios were those related to electricity optimization, including the reduction of electricity requirements together with the use of LEDs and the substitution of electricity from the grid by renewable sources, as well as the partial reuse of methanol. The combined implementation of the evaluated options may allow impact reductions in the short term ranging between 20% and 65% depending on the considered category. Further environmental improvements up to 70% could be achieved according to the best potential scenario.

The results of this paper should be taken into account, not only due to the importance of LCA as a tool to develop improved production systems but also due to the novelty of the proposed process. This production process avoids the over-exploitation of the marine environment, as it replaces the need of wild harvesting by the maintenance of specimens in a closed aquarium. Moreover, the process is currently under a scale up phase and will be able to deliver enough sponge natural products to be amenable to commercialization. Therefore, this alternative may allow to overcome existing bottlenecks regarding the marine-based biotechnology and specifically in the field of sponges cultivation.

Figure 1

 1 Figure 1 System boundaries and process chain of the production of pure crambescin and crambescidin from Crambe crambe in indoor aquariums.

Figure 3

 3 Figure 3 Relative contribution per stage of the production of pure crambescin and crambescidin in the base scenario.

Figure 4

 4 Figure 4 presents the relative contribution of the different activities that are involved in the process.

Figure 5

 5 Figure 5 Relative contribution of the production of the different chemicals involved in the production of pure crambescin and crambescidin in the base scenario.

Figure 7

 7 Figure 7 Effect of vessel operations in the environmental profile of the production of pure crambescin and crambescidin.

Figure 8

 8 Figure 8 Effect of methanol reuse in the environmental performance of the production of pure crambescin and crambescidin.

Figure 10

 10 Figure 10Comparative environmental profiles of the base scenario, the best potential scenario and the most feasible scenario.

  Thus, 1600 hours of annual operation were assumed, corresponding to 200 days of operation for 8 h/day. The amount of materials associated with the collection of sponges itself was estimated considering that this stage only requires 2 h of sailing within the whole year. The effect of this assumption will be further discussed in following sections. Emissions from fuel combustion were determined as shown in the

EMEP/EEA air pollutant emission inventory guidebook of 2009 (EMEP/EEA, 2009). Chemicals related to vessel operations (i.e. paint, anti-fouling paint, marine lubricant oil) were inventoried according to Vázquez-Rowe et al. (

Table 2

 2 Summary of data sources.

	Energy	Electricity (French electricity profile) Diesel	Ecoinvent database (Dones et al. 2007) Ecoinvent database (Jungbluth 2007)
	Chemicals related to vessel operation	Anti-fouling Boat paint Marine lubricant oil	Vázquez-Rowe et al. (2010)
		Deionized water	Ecoinvent database (Althaus et al. 2007)
		Tap water
	Solvents	Methanol Ethanol	Ecoinvent database (Sutter 2007)
		Acetonitrile
		Trifluoroacetic acid †
	Air for scuba	Compressed air	Ecoinvent database (Steiner and Frischnecht
	equipment		2007)
		Glass fibre reinforced plastic, polyester	Ecoinvent database (Kellenberger et al 2007)
		resin
		Concrete
		Steel	Ecoinvent database (Classen et al 2007)
	Materials	Synthetic rubber	Ecoinvent database (Hischier 2007)
		PVC
		Polymethyl methacrylate
		Polypropylene
		Lamp (60 W)	Ecoinvent database (Hischier et al. 2007)
	Waste treatment	Inert landfill Sanitary landfill	Ecoinvent database (Doka 2007)
	† Assimilated to acetic acid

Table 3

 3 Life spans and assumptions for materials' quantification.

	Equipment	Component	Life span	Assumptions
		Hull (polyester)	30 years	Calculated material increased by 25% to account for
				vessel repairs and maintenance (Hospido and Tyedmers,
				2005).
		Diesel engine (steel)	15 years	Average weight estimated from manufacturers. Estimated
	Vessel			weight increased by 50% to account for vessel repairs and
				maintenance (Hospido and Tyedmers, 2005).
				Life span estimated from EMEP/EEA (2009).
		Anti-fouling	1 year	2 coats per year assumed, according to manufacturers.
		Paint	1 year	1 coat per year assumed, according to manufacturers.
	Scuba	Diving cylinder (steel) Diving equipment (neoprene)	15 years 10 years	10 uses/year, with 1 use related to C. crambe process. 10 uses/year, with 1 use related to C. crambe process.
		Pumps for water supply to	20 years	Designed for 10 m 3 decanting tank with 2.5 m 3 /h flow rate.
		decanting tank (steel)		14% of water pumped associated with C. crambe
		Pipes for water supply to	20 years	cultivation.
		decanting tank (PVC)
	Water supply	Decanting tank (concrete)	20 years
		Pumps for water supply to	20 years	Designed to feed three aquaria with 2 L/min flow rate
		aquarium (steel)		each.
		Pipes for water supply to	20 years
		aquarium (PVC)
	Cooler	Polypropylene (PP)	20 years	According to manufacturers' specifications.
		Polymethyl metacrylate (PMMA)	10 years	Weight calculated from on-site direct measurement of
	Aquarium	tank		dimensions.
		Lights	30000 h	According to manufacturers' specifications.
	Filtration and purification system	Plastic boxes, polypropylene (PP) Pump (steel) Separation columns	20 years 20 years 20 years	According to manufacturers' specifications.

Table 4

 4 Impact

	Impact category	Unit		Value
	Abiotic Depletion (ADP)	kg Sb eq	17.27
	Acidification (AP)	kg SO2 eq	3.44
	Eutrophication (EP)	kg PO4 -3	eq	1.23
	Global Warming (GWP)	kg CO2 eq	967.54
	Ozone Layer Depletion (ODP)	g CFC-11 eq	0.131
	Human Toxicity (HTP)	kg 1,4-DB eq	640.36
	Freshwater aquatic Ecotoxicity (FEP)	kg 1,4-DB eq	260.47
	Marine aquatic Ecotoxicity (MEP)	kg 1,4-DB eq	171.02
	Terrestrial Ecotoxicity (TEP)	g 1,4-DB eq	62.48
	Photochemical Oxidants Formation (POFP)	g C2H4 eq	319.45

assessment results (characterization step) associated with the base scenario of the production of pure crambescin and crambescidin from Crambe crambe in indoor aquariums (FU: 100 mg bioactive fraction).

Acknowledgements This project has been developed within the framework of the BAMMBO Project (Project reference: FP7 KBBE-2010-4 -265896) and has been partially financed by the Xunta de Galicia (Project Reference GRC 2010/37). P. Pérez-López would like to express her gratitude to the Spanish Ministry of Education for her research fellowship (AP2012-1605). We also acknowledge the assistance of D. Luquet for the recollection of sponges and the Observatoire d'Océanologie de Villefranche sur Mer (G. Gorsky) for hosting the aquariums.


Sara Cadoni

Emilie Chouzenoux

Jean-Christophe Pesquet

Caroline Chaux
email: caroline.chaux@univ-amu.fr

A BLOCK PARALLEL MAJORIZE-MINIMIZE MEMORY GRADIENT ALGORITHM

Keywords: Majorization-Minimization, Blockalternating optimization, Parallel algorithm, Memory gradient technique, 3D deconvolution, Depth-varying blur

In the field of 3D image recovery, huge amounts of data need to be processed. Parallel optimization methods are then of main interest since they allow to overcome memory limitation issues, while benefiting from the intrinsic acceleration provided by recent multicore computing architectures. In this context, we propose a Block Parallel Majorize-Minimize Memory Gradient (BP3MG) algorithm for solving large scale optimization problems. This algorithm combines a block coordinate strategy with an efficient parallel update. The proposed method is applied to a 3D microscopy image restoration problem involving a depth-variant blur, where it is shown to lead to significant computational time savings with respect to a sequential approach.

INTRODUCTION

In many inverse problems encountered in image processing, one has to generate an image estimate x ∈ R N by minimizing an appropriate cost function F , which has the following composite form:

(∀x ∈ R N ) F (x) = S s=1 f s (L s x) (1)
where, for every s ∈ {1, . . . , S}, L s ∈ R Ps×N , P s ∈ N * , and f s is a function from R Ps to R. In the context of maximum a posteriori estimation, L 1 is a degradation operator (for instance, a convolution operator) related to the acquisition process and f 1 measures the fidelity between the data and the observation model in accordance with the noise statistics. The remaining terms (f s ) 2 s S act as regularization functions incorporating a priori information on the sought solution, or more generally on a linear transform of it, thanks to matrices (L s ) 2 s S . Iterative optimization methods must generally be employed to find a minimizer of (1). In the case of large scale image recovery problems, a major challenge is to design an optimization algorithm able to deliver reliable numerical solutions in a reasonable time.

When all the involved functions (f s ) 1 s S are differentiable on R N (but non necessarily convex), state-of-the art smooth optimization methods are nonlinear conjugate gradient (NLCG) and low memory BFGS (L-BFGS) algorithms [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF][START_REF] Hager | A survey of nonlinear conjugate gradient methods[END_REF][START_REF] Chen | Large-scale L-BFGS using MapReduce[END_REF]. The Majorize-Minimize Memory Gradient (3MG) algorithm [START_REF] Chouzenoux | A Majorize-Minimize strategy for subspace optimization applied to image restoration[END_REF][START_REF] Chouzenoux | A stochastic majorize-minimize subspace algorithm for online penalized least squares estimation[END_REF] can be viewed as a special instance of NLCG with optimal stepsize and optimal conjugacy parameter and, when more than one previous direction is employed in the subspace, as a particular L-BFGS algorithm with optimal combination of memory directions. The main advantage of 3MG algorithm is its high efficiency since it relies on a Majorize-Minimize (MM) approach, combined with a subspace acceleration technique. At each iteration, a quadratic majorizing approximation of (1) at the current iterate is constructed, and the next iterate results from its minimization within a two-dimensional subspace spanned by the current gradient and the previous direction. In addition, the 3MG algorithm enjoys nice convergence properties which have been investigated in [START_REF] Chouzenoux | A Majorize-Minimize strategy for subspace optimization applied to image restoration[END_REF] in the convex case, and in [START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF] in the non-convex case. Comparisons with graph-cut based discrete optimization methods, and primal-dual proximal algorithms on a number of image restoration problems have also shown the good performance of 3MG in terms of practical convergence speed [START_REF] Chouzenoux | A majorize-minimize subspace approach for ℓ 2 -ℓ 0 image regularization[END_REF][START_REF] Chouzenoux | A majorize-minimize memory gradient algorithm applied to X-ray tomography[END_REF][START_REF] Florescu | A majorize-minimize memory gradient method for complex-valued inverse problems[END_REF][START_REF] Chouzenoux | Approche variationnelle pour la déconvolution rapide de données 3D en microscopie biphotonique[END_REF]. However, when the size of the problem becomes increasingly large, as it may happen in 3D image processing or video processing, running this kind of algorithm becomes difficult, due to memory limitation issues.

An efficient way to overcome difficulties related to memory requirements in optimization algorithm consists in adopting a parallel block alternating minimization approach [START_REF] Repetti | A parallel block-coordinate approach for primal-dual splitting with arbitrary random block selection[END_REF][START_REF] Konečný | Federated optimization: Distributed optimization beyond the datacenter[END_REF][START_REF] Onose | Scalable splitting algorithms for big-data interferometric imaging in the SKA era[END_REF]. The target vector is split into non-overlapping blocks of reduced dimension and, at each iteration of the algorithm, few blocks are selected, and updated in a parallel manner according to a given optimization scheme. This strategy allows to control the required memory, since the number of blocks can be set by the user to the number of available processors.

Several works in the context of image restoration, have shown that MM algorithms benefit from a high potential of parallelization [START_REF] Sotthivirat | Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms[END_REF][START_REF] Legendre | Primal-dual interior-point optimization based on majorization-minimization for edge preserving spectral unmixing[END_REF]. These studies are based on the use of majorant functions having a particular block separable structure that leads inherently to parallel update schemes which may be of great interest when the optimization is performed on multicore systems or GPGPU boards. The usage of block alternating techniques in MM strategies has also been investigated (see, for instance, [START_REF] Erdogan | Monotonic algorithms for transmission tomography[END_REF][START_REF] Hong | A unified algorithmic framework for block-structured optimization involving big data: With applications in machine learning and signal processing[END_REF]). However, up to our knowledge, the resulting methods do not present any parallel structure, as only one block must be updated per iteration, so that the gain in terms of memory usage is usually obtained at the expense of a substantial decrease of the convergence rate [START_REF] Sotthivirat | Image recovery using partitioned-separable paraboloidal surrogate coordinate ascent algorithms[END_REF].

In this work we propose a new Block Parallel 3MG (BP3MG) algorithm for smooth, possibly nonconvex, optimization problems involving massive datasets. The proposed method has two key benefits: it takes advantage of recent technological advances in parallel computing on multicore architectures and it allows the computation load to be tailored to the number of available processors. We show that these new features result in a highly accelerated method for solving a 3D deblurring problem with depth-variant blur.

The paper is organized as follows: in Section 2 we introduce the proposed BP3MG algorithm. In particular we describe the block subspace and the block separable majorant function used for its derivation. Section 3 describes a parallel implementation of this method for the restoration of 3D microscopy images and shows its very good performance. Some conclusions are drawn in Section 4.

PROPOSED METHOD

Notation

In our block-coordinate approach, the vector of unknowns

x = (x n ) 1≤n≤N ∈ R N is split into J vectors x (j) of reduced dimension N j = 0 with j ∈ {1, . . . , J} (N 1 +• • •+N J = N ).
We will also need to define subsets S ⊂ {1, . . . , J} gathering such block indices as illustrated in Fig. 1. The vector of variables associated with a set S is then denoted by x (S) = x (j) j∈S and its components are (x p ) p∈S where S ⊂ {1, . . . , N } with cardinality |S| = j∈S N j .

For every s ∈ {1, . . . , S}, the matrix L s in (1) can be decomposed as

L (1) s |. . .| L (J) s
, where, for every j ∈ {1, . . . , J}, L

(j) s ∈ R Ps×Nj . Finally, we denote by L (S) s ∈ R Ps×|S| , the columnwise concatenation of matrices (L (j) s ) j∈S .
A similar notation is used for the (rowwise) block decomposition of the gradient ∇F (x) at some given vector x ∈ R N .

Block separable quadratic majorant function

The majorize-minimize approaches [START_REF] Jacobson | An expanded theoretical treatment of iteration-dependent majorize-minimize algorithms[END_REF][START_REF] Hunter | A tutorial on MM algorithms[END_REF][START_REF] Zhang | Surrogate maximization/minimization algorithms and extensions[END_REF] often rely on the use of a surrogate function which constitutes a quadratic tangent majorant functions for Criterion [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF]. Let x ∈ R N x = x (1) x (j) x (J)

x (S) = and, for every x ∈ R N , let A(x) ∈ R N ×N be a symmetric positive matrix. The quadratic function

(∀x ∈ R N ) Q(x, x) = F ( x) + ∇F ( x) ⊤ (x -x) + 1 2 (x -x) ⊤ A( x)(x -x),
is said to be a tangent majorant for F at x, if

(∀x ∈ R N ) Q(x, x) ≥ F (x) and Q( x, x) = F ( x). (2)
In the sequel we will assume the existence of matrices fulfilling (2) of the form

(∀x ∈ R N ) A(x) = S s=1 L ⊤ s Diag {ω s (L s x)} L s , (3)
where, for every s ∈ {1, . . . , S}, L s is a P s × N real-valued matrix and ω s is a function from R Ps to ]0, +∞[ Ps whose expression depends on the properties of the potential f s [5, Tab. I]. For instance, if f s is β s -Lipschitz differentiable with β s ∈]0, +∞[, we can choose ω s (•) = β s 1 Ps , where 1 Ps is a vector with length P s whose components are all equal to one.

In order to derive our block-coordinate algorithm, we need to define, for every nonempty set of blocks S ∈ {1, . . . , J}, a quadratic majorant function of the restriction of F to these blocks. According to [20, Rem. 2.4], for every x ∈ R N , a possible choice is

(∀v ∈ R |S| ) Q (S) A (S) (v, x) = F ( x) + ∇F (S) ( x) ⊤ (v -x (S) ) + 1 2 (v -x (S) ) ⊤ A (S) ( x)(v -x (S) ), (4)
where

(∀x ∈ R N ) A (S) (x) = ([A(x)] p,p ) p∈S . (5)
However, in practice, the matrices defined in (5) do not necessarily have a block diagonal structure so that Q (S)

A (S) (•, x) is not separable with respect to each block j ∈ S. Therefore, its minimization cannot be performed efficiently in a parallel manner. The following lemma defining block diagonal matrices majorizing (5) will be the cornerstone of the parallel structure of our algorithm: Lemma 1. Let S ⊂ {1, . . . , J}. For every j ∈ S, for every x ∈ R N , let matrix B (j) (x) ∈ R Nj ×Nj be given by

B (j) (x) = S s=1 (L (j) s ) ⊤ Diag {b s (L s x)} L (j) s , (6)
where, for every s ∈ {1, . . . , S} and p ∈ {1, . . . , P s }, the p-th component of vector b

s (L s x) is 1 [b s (L s x)] p = [ω s (L s x)] p [|L (S) s |1 |S| ] p /[|L (j) s |1 Nj ] p . (7) Then, for every x ∈ R N , A (S) (x) B (S) (x)
, where B (S) (x) = BDiag B (j) (x) j∈S (i.e. the block diagonal matrix with |S| blocks equal to B (j) (x), j ∈ S).

The proof of Lemma 1 is omitted due to the lack of space. It makes a clever use of Jensen's inequality, following techniques similar to those in [21, Sec. III] (see also [13, Sec. III-A]).

BP3MG algorithm

We are now ready to present our algorithm. At each iteration k ∈ N, a subset S k ⊂ {1, . . . , J} of block indices with cardinality C ∈ N * , is selected. C typically corresponds to the number of available processors. The corresponding blocks are then updated thanks to a local 3MG iteration, corresponding to the minimization of Q

(S k ) B (S k ) •, x k within the subspace spanned by the columns of D (S) k = BDiag{(D (j) k ) j∈S }
where, for every k ∈ N and j ∈ S k ,

D (j) k = -∇F (j) (x k ) if j / ∈ k-1 ℓ=0 S ℓ , -∇F (j) (x k ) x (j) k -x (j) k-1 ] otherwise. ( 8
)
The above condition on j simply allows us to distinguish blocks which have never been updated, from the blocks which have been updated at least once. Thanks to the blockseparable structure of the majorant functions, the C selected blocks with indices j ∈ S k can be updated in a parallel manner, leading to the following new algorithm:

Initialize x 0 ∈ R N . For k = 0, 1, 2, . . .                    Select S k ⊂ {1, . . . , J} s.t. |S k | = C Parfor j ∈ S k           Compute ∇F (j) (x k ) Compute B (j) k (x k ) using (6) Construct D (j) k using (8) u (j) k = -(D (j) k ) ⊤ B (j) k (x k )D (j) k † (D (j) k ) ⊤ ∇F (j) (x k ) x (j) k+1 = x (j) k + D (j) k u (j) k Set , for every j ∈ {1, . . . , J} \ S k , x (j) k+1 = x (j) k . Share x (j) k+1 j∈S k between all cores. ( 9
)
Hereabove, (•) † denotes the pseudo inverse operation. The selection rule for the blocks plays an important role in the convergence analysis. Actually, it suffices to assume a quasicyclic rule, where the blocks are updated in an arbitrary order, as soon as each one is updated at least once per a finite number K ∈ N * of iterations. Under this assumption, the monotonic convergence of the criterion sequence (F (x k )) k∈N to a (locally) optimal value can be established, using the same theoretical tools as in [START_REF] Chouzenoux | A block coordinate variable metric forwardbackward algorithm[END_REF].

It is important to emphasize, that, because of Expressions ( 1) and ( 6), in practice, it is not necessary to send the full vector x k+1 to all the cores, at iteration k. Actually, let j ∈ {1, . . . , J} and let J j be the index set of the components of vector x k+1 belonging to the j-th block. Only the components of x k+1 with indices in

N j = S s=1 n ∈ {1, . . . , N }|(∃p ∈ P s,j ) [L s ] p,n = 0 ,
where P s,j = {p ∈ {1, . . . , P s } |(∃j ∈ J j ) [L s ] p,j = 0}, are required to compute the gradient and the majorant matrix that are useful for the update of the j-th block. The cardinality of N j is usually very small with respect to N . For instance, if S = 1 and L 1 is a discrete gradient operator with one pixel neighborhood, then |N j | = 3.

APPLICATION TO 3D IMAGE DECONVOLUTION

Observation model

We focus on the following linear degradation model:

y = Hx + b, (10)
where

x ∈ R N represents the 3D original (unknown) image of size N = N X × N Y × N Z , y ∈ R N is the observed image,
H is a linear operator modeling a 3D convolution with depth variant blur, and b is a realization of an additive random noise. The goal is to solve the associated inverse problem, i.e. to find an estimate of x from y.

Objective function

A hybrid penalized least squares criterion involving a smooth 3D regularization term is used for the objective function F [START_REF] Chouzenoux | Approche variationnelle pour la déconvolution rapide de données 3D en microscopie biphotonique[END_REF]. Then, F takes the form (1) with

• P 1 = N , L 1 = H, f 1 = 1 2 • -y 2 , • P 2 = N , L 2 = I N , f 2 = η d 2 [xmin,xmax] N , • P 3 = 2N , L 3 = [(V X ) ⊤ (V Y ) ⊤ ] ⊤ , f 3 (L 3 x) = λ N n=1 ([V X x] n ) 2 + ([V Y x] n ) 2 + δ 2 , • P 4 = N , L 4 = V Z , f 4 = κ • 2 ,
where (η, λ, δ, κ) ∈ (0, +∞) 4 , V X ∈ R N ×N , V Y ∈ R N ×N , V Z ∈ R N ×N are discrete gradient operators along X,Y, and Z directions, d E is the distance to a set E, and (x min , x max ) ∈ R 2 are some minimal and maximal bounds on the sought intensity values. The same matrices as in [START_REF] Chouzenoux | Approche variationnelle pour la déconvolution rapide de données 3D en microscopie biphotonique[END_REF] are used for defining (3).

Implementation

The BP3MG algorithm derived in Section 2 has been implemented in MATLAB R (2015b) using the Message Passing Interface command SPMD of the Parallel Computing Toolbox TM . More precisely, the blocks correspond to the J = N Z 2D slices of the 3D volume. For a given number of active cores C, the first core is used as a master process while the C = C -1 remaining cores are used as slave processes. The master process carries out the main loop of Algorithm [START_REF] Chouzenoux | Approche variationnelle pour la déconvolution rapide de données 3D en microscopie biphotonique[END_REF]. At each iteration k ∈ N, it selects C block indices, stored in S k , and sends to each slave the required data allowing it to update the j-th block with j ∈ S k . The slave processes perform their tasks simultaneously to compute the variables x (j) k+1 j∈S k , and finally send their results to the master.

Experimental results

We considered two image restoration scenarii described by Model [START_REF] Repetti | A parallel block-coordinate approach for primal-dual splitting with arbitrary random block selection[END_REF]. The 3D microscopical images FlyBrain 2 with size N = 256 × 256 × 48 and Tube 3 with size N = 284 × 280 × 48 are considered for x. In both cases, the linear operator H models a 3D depth-varying Gaussian blur, with kernel size 5 × 5 × 11. For each depth z ∈ {1, . . . , N Z }, the blur kernel is characterized by different variance and rotation parameters (σ X (z), σ Y (z), σ Z (z), ϕ Y (z), ϕ Z (z)), following the model from [START_REF] Wirjadi | Approximate separable 3D anisotropic gauss filter[END_REF]. In practice, the values of these five parameters are chosen randomly and independently, following a uniform distribution for every z, leading to a blurred signal to noise ratio (BSNR) equal to 14.09 dB (resp. 12.58 dB). A zero-mean white Gaussian noise with standard deviation 0.02 is then added to the blurred volume, leading to a SNR equal to 13.42 dB (resp. 11.53 dB). The regularization parameters (λ, δ, κ, η), and (x min , x max ) = (0, 1) are chosen so as to maximize the SNR of the restored volume. Here, we obtain a final SNR equal to 16.98 dB (resp. 14.47 dB). Fig. 2 displays some examples of original, degraded and restored slices, for two slices of FlyBrain. In Fig. 3, we plot, for different values of the number of active cores C, the ratio between the required computation time for one core versus the actual computational time, for reaching the stopping criterion x kx k-1 10 -5 x k . The computations were performed on an Intel(R) Xeon(R) CPU E5-2670 v3 @ 2.30GHz with 24 physical cores. The result of a linear fitting illus-2 http://imagej.nih.gov/ij/plugins/3d-viewer/ 3 http://adm.irbbarcelona.org/image-j-fiji trates the great potential of parallelization of the proposed algorithm.

CONCLUSION

In this paper, we have presented a new Block Parallel Majorize-Minimize Memory Gradient algorithm for handling large-size optimization problems such as those encountered in 3D image restoration with depth-variant blur. In this context, our experimental results illustrate the high efficiency of the proposed method in terms of acceleration for multi-core architectures. Our future work will focus on the implementation in other languages that may lead to a further improved performance.

Fig. 1 .

 1 Fig. 1. Example of block subset, with Card(S) = 3. Each block has here 16 components.

Fig. 2 .

 2 Fig. 2. Original (left), degraded (middle) and restored (right) images corresponding to slices z = 6 (top) and z = 18 (bottom) of the 3D volume FlyBrain.

Fig. 3 .

 3 Fig. 3. Ratio between the computation time for one core and the computation time for C cores (crosses) with linear fitting (dotted line), for the restoration of FlyBrain (left) and Tube (right).

The notation |M | is used to denote the matrix whose elements are equal to the absolute value of those of M .

This work was supported by the CNRS Imag'In project under grant 2015 OPTIMISME.


Yoann Roth
email: yoann.roth@cea.fr

Jean-Baptiste Dor
email: jean-baptiste.dore@cea.fr

Laurent Ros
email: laurent.ros@gipsa-lab.grenoble-inp.fr

Vincent Berg
email: vincent.berg@cea.fr

EXIT Chart Optimization of Turbo-FSK: Application to Low Power Wide Area Networks

Keywords: Low Power Wide Area, LPWA, Internet of Things, PHY-layer, FSK, Turbo Code, Low Rate, Low SNR, Machine-to-Machine, EXIT Chart

Low Power Wide Area (LPWA) networks aim at connecting low cost devices with high constraints on the link budget. Focusing on the physical layer, the combination of Frequency Shift Keying (FSK) modulation and coding in the same process has been shown efficient when associated to a turbo receiver. This paper considers the optimization of Turbo-FSK parameters for the LPWA context using the EXtrinsic Information Transfer (EXIT) Chart analysis. After adapting the metric to the FSK waveforms, EXIT chart performance results are extended and compared to short packet lengths using extensive Bit Error Rate (BER) computations.

I. INTRODUCTION

Machine-to-Machine (M2M) type communication is expanding, and the potential market for low-powered wireless solutions in the Internet-of-Things (IoT) is expected to grow exponentially [START_REF] Rebbeck | Low-powered wireless solutions have the potential to increase the M2M market by over 3 billion connections[END_REF]. In this Low Power Wide Area (LPWA) networks, terminals are expected to be low cost, to have a low power consumption, and to be able to communicate at long range. A link budget improvement of 15-20dB in comparison to existing cellular technologies is also expected [START_REF]LTE-M -Optimizing LTE for the Internet of Things[END_REF]. These critical issues set major constraints on the physical layer, as complexity at the terminal level and sensitivity working levels must be kept minimum.

A commonly employed technique designed to reach low sensitivity levels is the well-known repetition scheme (spread spectrum). The information data is transmitted λ times in order to give a processing gain of 10 log 10 (λ) in Signal-to-Noise Ratio (SNR) at the receiver, but the spectral efficiency is divided by λ. This technique improves the sensitivity level when a fixed bandwidth is considered, but does not improve the energy efficiency. Many systems rely on this scheme and so does Direct Sequence Spread Spectrum (DSSS). The IEEE standard 802.15.4k [3] has been specifically designed for low power infrastructure monitoring using the DSSS technique. A natural choice for low data rate and low power applications is the use of M -ary orthogonal modulation. For this specific type of non-linear modulation, increasing the alphabet size M gives a gain in energy efficiency while reducing the spectral efficiency [START_REF] Proakis | Digital Communications 3rd Edition, ser. Communications and signal processing[END_REF]. This property can be extended to Orthogonal Sequence Spread Spectrum (OSSS), and is purportedly used by the LoRa technology [START_REF]LoRa Alliance[END_REF], as suggested in [START_REF] Seller | Low power long range transmitter[END_REF]. This solution aims at long range communication for the IoT.

The ultimate Shannon's limit [START_REF] Shannon | A mathematical theory of communication[END_REF] gives the maximum transmission rate with arbitrarily small bit-error probability, for a given Signal-to-Noise Ratio (SNR) and a given bandwidth. This limit defines the minimum required E b /N 0 for a reliable communication as an increasing function of the spectral efficiency, and it can be shown that this value is lowerbounded by the ultimate limit E b /N 0 = -1.59dB. The previously mentioned schemes have some significant limitations when this limit is considered. Orthogonal modulations are shown to reach the ultimate limit, but an infinite size of alphabet is required, thus reducing the spectral efficiency to 0. Alternatively, DSSS does not improve the energy efficiency, as the E b /N 0 is not affected by spreading the information, and leads to potential energy waste. Our previous work [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] already highlighted this issue, and suggested that in the LPWA context, more elaborate decoding schemes, such as turbo processing [START_REF] Berrou | Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1[END_REF], should be considered as efficient alternatives. These schemes indeed offer high performance at the expense of complexity increase at the receiver side, but low power consumption still can be ensured at the terminal side.

The use of orthogonal alphabet combined with turbo decoding was first proposed with the introduction of the Turbo-Hadamard code presented in [START_REF] Ping | Low-rate turbo-Hadamard codes[END_REF], where binary Hadamard codes are studied. In [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF], we adapted the scheme and decoder computations to M -ary Frequency-Shift-Keying (FSK) modulation, and introduced the technique in the LPWA context. FSK modulation has already been considered for low power wireless networks as its constant envelope property provides a power efficient solution. This modulation is widely used for monitoring applications, for which power efficient and low cost off-the-shelf optimized chips are available [START_REF]SX1272 from Semtech, datasheet[END_REF]. A comparison with standard LPWA solutions is done in [START_REF] Roth | A Comparison of Physical Layers for Low Power Wide Area Networks[END_REF], showing the potential of turbo processing for the IoT context. Limitations of the Turbo-Hadamard scheme have been studied in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF], using the EXtrinsic Information Transfer (EXIT) Chart analysis [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF], [15]. The emphasis has primarily been on finding the minimum pinch-off value for every possible set of parameters. Pinch-off value is defined as the E b /N 0 value for which the decoder can recover without error the information word after a certain number of iterations. The Turbo-Hadamard scheme was shown to approach Shannon's limit by only 0.29dB.

In this paper, we propose to optimize the energy efficiency of the Turbo-FSK scheme, using the EXIT Chart analysis. The computation proposed by [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF] for Turbo-Hadamard does not directly apply and adaptations are derived for the FSK case. Then, we perform simulations to find the pinch-off values for every set of parameters. The analysis assumes perfect decorrelation between the different sent versions, and this hypothesis tends to be valid when a large information block size is considered. For LPWA, short length packets are expected, we therefore propose to perform extensive Bit-Error-Rate (BER) simulation to confirm if the behavior predicted by the EXIT Charts can be valid to short packet sizes.

The paper is organized as follows. The Turbo-FSK ar- chitecture from [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] is briefly presented in Section II. EXIT Chart computation is presented in Section III, while Section IV presents the optimization process proposed. Section V concludes the paper.

II. SYSTEM MODEL

In this section, we will briefly introduce the Turbo-FSK scheme presented in [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF].

A. Transmitter

Transmitter architecture is given in Figure 1. The structure is composed of λ stages, each one encoding a differently interleaved version of the input bits. Information bits are gathered into P groups of r bits, and Convolutional-FSK encoding is applied, as described in Figure 2. For every block of r bits, a single parity bit is added. The parity bit consists of the modulo-2 sum of the r input bits and the parity bit of the previous block. The resulting r + 1 bits are then mapped to a complex codeword of the FSK alphabet, whose size is M = 2 r+1 . Thanks to the accumulation, every consecutive symbol is related to the previous one. The output of each encoder is then a set of P FSK codewords: this scheme mixes coding and modulation in the same process. A Parallelto-Serial Conversion is then done in order to transmit the FSK codewords through the channel. The normalized spectral efficiency is defined by

1 η = log 2 (M ) -1 λM (1)
and expressed in bits/s/Hz.

B. Channel

We use a discrete-time model and consider an Additive White Gaussian Noise (AWGN) channel, where the output is defined as with x is the emitted signal. n is the complex additive white Gaussian noise vector, with variance σ 2 . x, y and n are complex vectors of size M λ(P + 1).

y = x + n, (2)

C. Receiver

The receiver, depicted in Figure 3, includes a Serial-to-Parallel Conversion, to reconstruct the λ stages emitted at the first place. A soft FSK detector is used to determine the probabilities of each possible codeword. This step can be performed using the Fast Fourier Transform (FFT) algorithm. These probabilities are then fed to the decoder, which uses them as channel observation, while output of the other decoders will be used as a priori information. BCJR algorithm [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate (corresp.)[END_REF] is used to decode the trellis, and the A Posteriori Probabilities (APP) of the information bits are computed. Orthogonality of the transmitted codewords is one of the key features of this scheme, which offers a great performance gain compared to simple repetition schemes. For detailed explanations about the receiver's computations, reader is invited to refer to [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF].

III. EXIT CHART ANALYSIS

The EXIT Chart analysis was first introduced in [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF] as a way to predict the threshold E b /N 0 for which the BER drops significantly (the "waterfall" region). The analysis was performed on standard Parallel Concatenated Convolutional Codes (PCCC), where, at the receiving side, two decoders exchange information in an iterative way. During the process, the information bits are represented by Log Likelihood Ratio (LLR), defined as

L(b) = log p(b = 1) p(b = 0) , (3)
where p(b = 1) (resp. p(b = 0)) is the probability that the bit b equals 1 (resp. 0). The Turbo-Hadamard code EXIT Chart analysis has been realized in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF].

A. EXIT Chart computation for the FSK case

While the PCCC or Turbo-Hadamard code are systematic codes, the suggested architecture in [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] uses FSK waveforms as codeword. The process to compute the extrinsic information is depicted Figure 4. The observation from the channel are directly converted to codewords probabilities, represented in blue on the graph. The BCJR decodes the trellis and gives the output LLRs on the information bits L APP , that can be expressed where L A is the output of the a priori information simulation block, or the a priori LLR used by the BCJR algorithm. L E is the extrinsic information created by the algorithm, and L ch is the information from the channel. The FSK codewords being non-systematic, it is not possible to extract the LLR from the channel as it is usually done for systematic codes; hence the value L ch cannot be properly computed 2 . It is however possible to subtract the a priori LLR L A , and to refer to the results as the extrinsic information. The influence of this approximation will be discussed later.

L APP = L A + L ch + L E (4)

B. Multidimensional EXIT Chart analysis

The extension of the EXIT Chart analysis to multidimensional codes, where more than two decoders are used, has been studied in [15]. If λ decoders are considered, the representation of the exchanges of information should be represented in λ dimensions. Tracking directly the exchanges of information becomes impossible for λ > 3 (as it would require more than 3 dimensions), but in the case where all decoders are the same, a two-dimensional projection may be computed. If the projection does not intersect with the line going from (0, 0) to (1, 1), it indicates that convergence toward the maximum of mutual information is possible. The a priori information fed to one decoder is the sum of the extrinsic information of the (λ -1) other decoders. To obtain the projection, the formula of the a priori mutual information I A must be modified. Using the function J as defined in [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF], it can be expressed

I A = J √ λ -1σ A , (5)
where σ A is the standard deviation of the a priori model, chosen as Gaussian as suggested in [START_REF] Brink | Convergence behavior of iteratively decoded parallel concatenated codes[END_REF]. To compute the extrinsic mutual information I E , we use the general formula where density probability functions need to be integrated.

C. Comparison with the Turbo-Hadamard case

The EXIT Chart analysis done for the Turbo-Hadamard in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF] considers both a systematic and multidimensional code. We chose to study the impact of: (1) the implication of dealing with non-systematic codes and (2) the non-binary codewords inducing the modification of the EXIT Chart computation previously presented. For (1), we perform simulation of the EXIT Chart of the Turbo-Hadamard case for which the systematic bits are repeated. In comparison to the case where systematic bits are not repeated, the process of repeating them changes the observed channel on those bits. We refer EXIT Chart and BER performance for the Turbo-FSK scheme with parameters M = 128, λ = 4. EXIT Chart is computed for E b /N 0 = -1.12dB, with N = 100000. BER performance is computed using an information block size of N = 100000 and the MAP algorithm.

to this scheme as Turbo-Hadamard*. For (2), we compare for the same parameters the EXIT Chart of the Turbo-FSK decoder with the equivalent Turbo-Hadamard decoder. The EXIT Charts of the three different decoders are depicted Figure 5. The EXIT Charts are very close, but these non-similarities will induce small differences in the pinch-off values of each scheme.

D. Comparison with the BER performance

In order to check if the EXIT Chart analysis predicts the waterfall region, we look for the pinch-off value of a particular case and compute the Bit-Error-Rate (BER) for the same case with a large block size of information bit (N = 100000). The case M = 128 and λ = 4 is selected. The pinch-off value found is E b /N 0 = -1.12dB. The EXIT Chart for this E b /N 0 and the BER around this value are depicted Figure 6. After 100 decoder iterations, the waterfall appears around -1.1dB, very close to the value predicted by the EXIT Chart.

IV. PARAMETERS OPTIMIZATION

The Turbo-FSK scheme offers two adjustable parameters that can be optimized for better energy efficiency. Using the EXIT Chart analysis previously introduced and BER computations, this section is dedicated to the exhaustive search for sets of parameters (M, λ) as close as possible of the Shannon's limit. For all simulations, coherent detection of the FSK and perfect synchronization are assumed. MAP algorithm without any approximation is considered.

A. Optimization using the EXIT Chart analysis

As the EXIT Chart satisfyingly predicts the waterfall, this tool can be used to determine the energy efficiency of the Turbo-FSK scheme for large information block sizes. An exhaustive search was realized on a large number of sets of (M, λ). Alphabet size selected varies from M = 16 to M = 2048, and repetition factor is chosen between 2 and 6. A block size N = 100000 is used. The pinch-off values obtained are presented Table I 3 . The minimum E b /N 0 value for each alphabet size is denoted by a star * . The couple of parameters with the lowest pinch-off value is (512, 3), only 0.29dB away from Shannon's limit. For the Turbo-Hadamard case, an equivalent result is obtained in [START_REF] Wu | On the limiting performance of turbo-Hadamard codes[END_REF], despite the differences between the codes. The performance depending on the value of λ seems unpredictable and the exhaustive search enables the optimization of the parameters.

B. Optimization based on BER computations

As previously mentioned, the EXIT Chart analysis assumes perfect independence between the different interleaved information messages of the decoder and a large size of information block guarantees this property. Considering the LPWA context, small block sizes are expected. In order to analyze the influence of the block size and the interleaver, we perform extensive BER computation. For every set of parameter, we search the E b /N 0 value for which BER reach 10 -4 . Block size is set to N = 1000, and the decoder performs 10 iterations. A random interleaver is used. Values obtained are depicted Table II. For the best case, the gap to Shannon's limit is now 1.35dB. The corresponding set of parameters is (1024, 3). The performance of the best set found with EXIT Chart analysis is 3 For λ = 2, values may be overestimated due to the fact that the EXIT Charts cross the line (0, 0) → (1, 1) very close to the point (1,1). only 0.02dB away from this set. Performance improvements may be achieved through the use of an optimized interleaving function.

C. Comparison to the channel capacity

The spectral efficiency of the Turbo-FSK, as defined in (1), is different for every set of parameters. Considering the normalized spectral efficiency versus the energy efficiency, the values previously obtained can be compared to the ultimate Shannon's limit [START_REF] Shannon | A mathematical theory of communication[END_REF], defined as

E b /N 0 ≥ 2 η -1 η . (6)
The result is depicted in Figure 7, where for each alphabet size, λ values from 3 to 6 are represented. As λ is increased, the normalized spectral efficiency is lowered; the point with the higher η is λ = 3, then 4 and so on. The figure clearly shows the optimum value for the parameters, for both the EXIT Chart analysis and the BER computations for small block sizes.

The general trend obtained with the EXIT Chart seems to be valid when the block size is shortened. The performance loss incurred by the block size reduction is 1dB on average (between 0.8 and 1.25dB).

V. CONCLUSION

Within the LPWA context, the energy efficiency of the transmission scheme is a major issue. The Turbo-FSK scheme is a way to achieve very low levels of sensitivity, exploiting repetition, orthogonal modulation, and a complex decoding algorithm, while ensuring a low consumption transmitter. Optimization using both EXIT Chart analysis and BER computations was performed. The best set of parameters is shown to be only 0.29dB away from the channel capacity for large block size, and 1.35dB away for a block size of 1000 bits which is compatible given considerations for the M2M context. A highly reliable communication can be achieved with a nearlyoptimal use of the energy resource. However, reducing the sensitivity also implies to work at very low levels of Signal-to-Noise Ratios, for which detection and synchronization become problematic. Repetition in the preamble offers here again a solution, but such a simple scheme will cause severe E b /N 0 degradation (e.g a preamble as long as the data will induce a 3dB loss of E b /N 0 ). More elaborate algorithms for synchronization can be conceived to settle this issue.

Fig. 1 .Fig. 2 .

 12 Fig. 1. The Turbo-FSK transmitter.

  Fig. 3. The Turbo-FSK receiver.

Fig. 4 .

 4 Fig. 4. The Turbo-FSK EXIT extrinsic information extraction.

Fig. 5 .

 5 Fig. 5. Comparison of the different schemes for the same parameters M = 16, λ = 3. The information block size was set to N = 100000, and channel setting is E b /N 0 = 2dB.

  λ

Fig. 7 .

 7 Fig. 7. Values from Table I (EXIT Chart analysis with N = 100000) and Table II (BER simulation for N = 1000) in the plot normalized spectral efficiency versus energy efficiency, for λ values going from 3 to 6.

TABLE I .

 I PINCHOFF VALUES IN E b /N 0 FOR EVERY M AND λ TESTED. EXIT CHART ARE COMPUTED USING AN INFORMATION BLOCK SIZE N = 100000. BEST VALUE FOR EACH M IS DENOTED WITH A SYMBOL * , AND BEST COUPLE OF PARAMETERS IS IN BOLD.

TABLE II .

 II E b /N 0 VALUES FOR WHICH BER IS 10 -4 , FOR EVERY M AND λ. BER IS COMPUTED FOR AN INFORMATION BLOCK SIZE N = 1000, MAP ALGORITHM IS USED, AND 10 ITERATIONS ARE PERFORMED. BEST VALUE FOR EACH M IS DENOTED WITH A SYMBOL * , AND BEST COUPLE OF PARAMETERS IS IN BOLD.

The trellis termination has been omitted. One extra FSK codeword must be emitted in order to close the trellis, thus slightly reducing the spectral efficiency.

Decoding only the FSK modulation would still include the extrinsic information of the orthogonal code itself.


Anton Dogadov

Franck Quaine

Christine Serviere

A MODEL OF THE MIDDLE FINGER EXTENSOR MECHANISM WITH THE RETENTION APPARATUS

A model of the middle finger extensor mechanism with the retention apparatus

Anton Dogadov, Franck Quaine, Christine Serviere

Introduction

Biomechanical modeling of force transmission in fingers is important for surgical treatment planning. In recent years, there has been an increasing interest in extensor mechanism (EM) modeling, which plays an important role in both extension and flexion of the fingers [1]. It consists of tendons transmitting active forces and retention apparatus (or ligaments) transmitting passive forces. However, no studies have added the retention apparatus into account despite its important role in finger biomechanics. The aim of this study was to model the EM with the retention apparatus for the middle finger. Two postures, full flexion and full extension, were studied. Model deformation was compared with anatomical published data [2].

Methods

The model was based on the modern anatomical description of EM [3] Both elements of EM, tendons and ligaments, were modeled as elastic strings. The Minimum Potential Energy principle was used to find the equilibrium of the system. Potential energy was calculated as: PE = SE+WP+PT, where SE stands for energy of the system, WP is a work potential, PT is a penalty term, which is introduced to take into account the constraints, disabling the tendons from penetrating into the bones orgetting out of the pulleys. Model is shown in the Figure 1. Retention apparatus is presented by FI band (oblique retinacular ligament) and the band going from the F point to the symmetrical point on the ulnar side of the finger (triangular ligament).

Results

EM deformation was studied by finding the angles between the tendons (Fig. 1). The angles produced by the model were compared with the anatomical data (Fig. 2 and3).

Discussion and Conclusion

The current study found good agreement between estimated and measured angles. The Root-mean-square error of modeled data was 6.18° for extension posture and 6.17° for flexion posture. Excluding of the retention apparatus from the model leads in RMSE increase until 6.32° (in full extension) and 6.85° (in full flexion). The remaining differences from anatomical data could be explained by lack of knowledge about the model parameters values. Further work, such as model parameter estimation, needs to be done to perform better model fitting to the anatomical data.

Figure 1 :

 1 Figure 1: View of the model. Calculations were performed in MATLAB R2011b software (The MathWorks, Inc., Natick, MA, USA).

Figure 2 :

 2 Figure 2: Model-predicted and measured angles between the tendons for full extension posture.

Figure 3 :

 3 Figure 3: Model-predicted and measured angles between the tendons for full flexion posture.

Acknowledgements

The authors thank Prof. François Moutet and Dr. Mazen Alamir for their valuable comments.