Linda Dominique

Fabiola Bambara

Marie Sawadogo
email: marie.sawadogo@2ie-edu.org

Daniel Roy
email: daniel.roy@univ-lorraine.fr

Didier Anciaux
email: didier.anciaux@univ-lorraine.fr

Joel Blin
email: joel.blin@cirad.fr

Salifou Koucka Ouiminga

Joël Blin

Biofuel from " Balanites aegyptiaca "

Keywords: biofuel, Balanites aegyptiaca, biomass supply chain, pre-processing, optimization

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

Introduction

Access to modern energy services remains limited in sub-Saharan Africa. Indeed, 590 million people in this part of the world still remain without access to electricity in 2017 [START_REF]Organisation for Economic Co-operation and Development[END_REF]. More than 80% of the population without electricity live in rural areas and rely on the traditional use of solid biomass (wood fuel and charcoal) for energy generation [START_REF]Organisation for Economic Co-operation and Development[END_REF]. In West Africa Sahelian countries like Burkina Faso, Mali and Niger, the use of traditional biomass account for more than 90% of primary energy consumption, used mainly for cooking. Consequently, the wood resources are pressured and over harvested, contributing to forest degradation [START_REF] Sulaiman | Wood fuel consumption, institutional quality, and forest degradation in sub-Saharan Africa: Evidence from a dynamic panel framework[END_REF]. Another consequence of the low access to energy is the socio-economic underdevelopment. Evidently, energy has been reported to be an essential development component and its consumption is an indicator of the well-being of a population [START_REF] Pereira | The challenge of energy poverty: Brazilian case study[END_REF].

Based on this energy context, any solution for the development of the energy sector in West Africa as well as in the rest of sub-Saharan Africa that aims to substitute or replace wood fuel is welcome. This could explain the interest of many West African countries in bioenergy [START_REF] Amigun | Biofuels and sustainability in Africa[END_REF][START_REF] Gasparatos | Biofuels in Africa: Impacts on Ecosystem Services, Biodiversity and Human Well-Being[END_REF] and particularly in straight vegetable oil (SVO). Evidently, SVO extracted from oilseed crops could be blended with diesel or be used directly as fuel in diesel engines when the quality standards of the oil are respected [START_REF] Blin | Characteristics of vegetable oils for use as fuel in stationary diesel engines: Towards specifications for a standard in West Africa[END_REF]. SVO could also be used in hybrid energy systems consisting of solar photovoltaic and diesel generators like those described by [START_REF] Adaramola | Multipurpose renewable energy resources based hybrid energy system for remote community in northern Ghana[END_REF][START_REF] Franco | A review of sustainable energy access and technologies for healthcare facilities in the Global South[END_REF]. They can therefore be used for electrification at the village or country level.

For SVO production in West Africa, especially in West African arid and semi-arid areas where the soil and climatic environment are not suitable to produce the plants commonly used for biofuel production (like jatropha, soybean or rapeseed), Balanites aegyptiaca (B. aegyptiaca) could be an alternative feedstock. B. aegyptiaca, which has already been targeted for the production of oilseed-based biofuels [START_REF] Bart | 6-Emerging new energy crops for biodiesel production[END_REF][START_REF] Patel | 11-Biodiesel from Plant Oils[END_REF] is a drought-resistant tree that can grow under varied ecological conditions (e.g., 100 mm to 1000 mm of rain per year) [START_REF] Ndoye | Reproductive biology in Balanites aegyptiaca (L.) Del., a semi-arid forest tree[END_REF]. The tree produces fruits that each weigh 5 to 8 g. The fruit consists of an epicarp (5 to 9% of its weight), an edible mesocarp or pulp (28 to 33% of its weight), and an endocarp (49 to 54% of its weight). The endocarp or hull surrounds the seed (8 to 12% of the weight of the fruit) [START_REF] Chapagain | Variation in diosgenin level in seed kernels among different provenances of Balanites aegyptiaca Del (Zygophyllaceae) and its correlation with oil content[END_REF][START_REF] Okia | Physico-chemical characteristics and fatty acid profile of desert date kernel oil[END_REF][START_REF] Chapagain | Characterization of Desert Date (Balanites Aegyptiaca) Saponins and Their Biological Activities; Ben-Gurion University of the Negev[END_REF]. The seed contains between 46% and 55% of its weight in vegetable oil [START_REF] Varshney | Trees: Propagation and Conservation: Biotechnological Approaches for Propagation of a Multipurpose Tree, Balanites Aegyptiaca Del[END_REF]. B. aegyptiaca is particularly abundant in arid and semi-arid regions, mainly in sub-Saharan and Sahelian regions such as the Sahelian region of Africa, the Middle East, and South Asia [START_REF] Chapagain | Variation in diosgenin level in seed kernels among different provenances of Balanites aegyptiaca Del (Zygophyllaceae) and its correlation with oil content[END_REF]. In these regions, the plant is one of the main plant species [START_REF] Boffa | Les Parcs Agroforestiers en Afrique Subsaharienne[END_REF] and the trees grow wild with no irrigation, fertilization, or care. These characteristics of wild non-cultivated products, also called non timber forest product (NTFP), described by [START_REF] Angelsen | Exploring the forest-poverty link: Key concepts, issues and research implications[END_REF] in [START_REF] Pouliot | Rural People's Reliance on Forests and the Non-Forest Environment in West Africa: Evidence from Ghana and Burkina Faso[END_REF] make B. aegyptiaca seeds an attractive biomass for SVO production.

In addition to the attractive characteristics described above, some authors who investigated the process of B. aegyptiaca oil transeterification for biodiesel production [START_REF] Yunus | Kinetics study of balanites aegyptiaca oil transesterification for the production of biodiesel[END_REF][START_REF] Dass | Production of Biodiesel Oil from Desert Dates (Balanites aegyptiaca) Seeds Oil Using a Hetrogeneous Catalyst Produced from Mahogany (Khaya senegalensis) Fruit Shells[END_REF] and studied the use of B. aegyptiaca as a fuel for diesel engines [START_REF] Naik | Assessment of toxic content in Balanites aegyptiaca seed cake and use of enzymatic-catalysed biodiesel in diesel engine[END_REF][START_REF] Naik | Experimental evaluation of a diesel engine fueled with Balanites aegyptiaca (L.) Del biodiesel blends[END_REF][START_REF] Sunil Naik | Effects of EGR on performance and emissions of a diesel engine fuelled with balanites aegyptiaca/diesel blends[END_REF] showed that B. aegyptiaca biodiesel can be used in a diesel engine as an alternate diesel. Some of these authors also concluded that B. aegyptiaca is a promising feedstock for oilseed-based biofuel production [START_REF] Naik | Assessment of toxic content in Balanites aegyptiaca seed cake and use of enzymatic-catalysed biodiesel in diesel engine[END_REF]. However, despite these interesting results, none of these authors has assessed the supply of B. aegyptiaca feedstock, which is one of the essential elements in the large-scale production of biofuels.

Indeed, as it is a wild plant, the operations involved in the B. aegyptiaca seed supply chain, which include collection, pre-processing, storage, and transport, can be complex and expensive due to the scattered and irregular distribution of the trees. This is why one of the main obstacles to the commercial exploitation of B. aegyptiaca fruits or nuts (the endocarp containing the seed) is the difficulty of obtaining adequate and regular supplies of the fruits [START_REF] Council | Lost Crops of Africa: Volume III: Fruits[END_REF]. This is a common feature of all bioenergy production systems to such an extent that in recent years, it is widely accepted that the complex and multi-faceted supply chains of bioenergy production systems call for careful consideration [START_REF] Kumar | Role of biomass supply chain management in sustainable bioenergy production[END_REF][START_REF] Rentizelas | 2-Biomass supply chains[END_REF]. Thus, for a profitable and competitive exploitation of B. aegyptiaca for biofuel production, it is important to pay particular attention to the feedstock supply chain. This ensures that the biofuel production unit will be supplied with feedstock of the required quality, quantity, within appropriate periods and at acceptable costs. In addition, a clever feedstock supply chain involving the population may become an income source as well as a growth factor.

The aim of this study was thus to develop a mathematical model for determining the optimal supply network configuration that could minimize the costs of wild oilseed biomass supply. The model has been applied to the B. aegyptiaca seed supply and has been developed by factoring in West Africa's climate, which is characterized by the weak development of infrastructures dedicated to feedstock supply. Depending on the choices of the decision makers in the supply chain, the model could be used to guarantee the profitability of the biofuel production chain while ensuring that all the stakeholders receive a fair remuneration for their work. To the best of our knowledge, this paper is one of the first to study the supply of wild biomass growing in arid climates for biofuel production. It is also one of the first to assess the costs of supplying B. aegyptiaca feedstock and to propose a model for an optimal supply for biofuel production in a West African context. The optimal supply network configuration not only presents a way to organize feedstock harvesting, it also delivers the biofuel production unit in a cost-effective way that reduces both harvesting and logistics (pre-processing, storage, handling and transport) costs.

The remainder of this paper comprises a description of the mathematical model, its implementation on a case study in Burkina Faso, a description of the obtained results, and a discussion of their implications. After this discussion, future directions are provided.

Methodology

The model proposed in this paper is a mixed integer linear program (MILP) for decisions related to the configuration of a wild oilseed crop supply chain. These decisions concern the location and the allocation of biomass harvesting sites and of the biomass collection points. They also concern the location of the pre-processing operations and the quantities of feedstock to be transported, handled, pre-processed, and stored.

The Problem and Assumptions

The system to be optimized is the B. aegyptiaca seed supply chain for SVO production. The supply network comprises biomass harvesting sites, biomass collection points, one SVO production unit, and infrastructures needed to harvest, transport, store, and pre-process B. aegyptiaca fruits. The sites mentioned above are linked by a road network. Figure 1

Methodology

The model proposed in this paper is a mixed integer linear program (MILP) for decisions related to the configuration of a wild oilseed crop supply chain. These decisions concern the location and the allocation of biomass harvesting sites and of the biomass collection points. They also concern the location of the pre-processing operations and the quantities of feedstock to be transported, handled, pre-processed, and stored.

The Problem and Assumptions

The system to be optimized is the B. aegyptiaca seed supply chain for SVO production. The supply network comprises biomass harvesting sites, biomass collection points, one SVO production unit, and infrastructures needed to harvest, transport, store, and pre-process B. aegyptiaca fruits. The sites mentioned above are linked by a road network. Figure 1 The SVO production unit is located in a Sahelian region in which B. aegyptiaca trees are abundant and where there is a periodic demand for biofuel. This periodic demand for SVO defines the minimum processing capacity of the SVO production unit.

Several biomass harvesting sites are scattered throughout the region around the SVO production unit, and it is assumed that these biomass harvesting sites are able to supply the demand for feedstock of the biofuel production unit.

To facilitate the delivery of the seeds to the biofuel production unit (mainly the labor needed to load, unload, and weigh the feedstock), all the harvested feedstock must pass through a collection point. Indeed, collection points make it possible to bring together large amounts of feedstock, and due to geographical constraints, several potential collection points are selected in advance. Each collection point is characterized by a set of means for biomass transportation, a maximum capacity, and a minimum demand for feedstock. The harvested fruits must be pre-processed. B. aegyptiaca preprocessing is a set of post-harvest operations, which consist of depulping, de-hulling, cleaning, drying, and optional crushing. It can be performed to reduce the quantity (volume or weight) of the material to be transported, to prevent deterioration of the feedstock quality during its transportation and storage, to facilitate and improve SVO extraction. There are two possible locations for preprocessing: at the biomass collection points or at the biofuel production unit. Depending on the location of pre-processing operations, the biomass transported and stored can be the non-pretreated biomass (fruits) or the pretreated biomass (seeds). When pre-processing takes place at the collection points, fruits are transported between the biomass harvesting sites and the collection points, and seeds are transported between the collection points and the biofuel production unit. When pre- The SVO production unit is located in a Sahelian region in which B. aegyptiaca trees are abundant and where there is a periodic demand for biofuel. This periodic demand for SVO defines the minimum processing capacity of the SVO production unit.

Several biomass harvesting sites are scattered throughout the region around the SVO production unit, and it is assumed that these biomass harvesting sites are able to supply the demand for feedstock of the biofuel production unit.

To facilitate the delivery of the seeds to the biofuel production unit (mainly the labor needed to load, unload, and weigh the feedstock), all the harvested feedstock must pass through a collection point. Indeed, collection points make it possible to bring together large amounts of feedstock, and due to geographical constraints, several potential collection points are selected in advance. Each collection point is characterized by a set of means for biomass transportation, a maximum capacity, and a minimum demand for feedstock. The harvested fruits must be pre-processed. B. aegyptiaca pre-processing is a set of post-harvest operations, which consist of depulping, de-hulling, cleaning, drying, and optional crushing. It can be performed to reduce the quantity (volume or weight) of the material to be transported, to prevent deterioration of the feedstock quality during its transportation and storage, to facilitate and improve SVO extraction. There are two possible locations for pre-processing: at the biomass collection points or at the biofuel production unit. Depending on the location of pre-processing operations, the biomass transported and stored can be the non-pretreated biomass (fruits) or the pretreated biomass (seeds). When pre-processing takes place at the collection points, fruits are transported between the biomass harvesting sites and the collection points, and seeds are transported between the collection points and the biofuel production unit. When pre-processing takes place at the biofuel production unit, fruits are transported through the whole supply network.

Mathematical Formulation

The mathematical formulation of the problem consists of an objective function to minimize (1) and a set of constraints ( 2) to [START_REF] Dass | Production of Biodiesel Oil from Desert Dates (Balanites aegyptiaca) Seeds Oil Using a Hetrogeneous Catalyst Produced from Mahogany (Khaya senegalensis) Fruit Shells[END_REF]. The objective function is made up of feedstock harvesting, transport, pre-processing, storage, and handling (loading and unloading) costs. All these costs take the variable and fixed costs of the operation into account. The elements of the mathematical model are described below (parameters and equations), Table 1 (sets) and in Table 2 (variables).

Q i,j
Quantity of feedstock to be transported from a biomass harvesting site i to a collection point j R+ Q j,p Quantity of feedstock collected at a collection point j if the location of the pre-processing p is chosen. R+

Q j,k,p
Quantity of feedstock to be transported from a collection point j to the biofuel production unit k if the location of pre-processing p is chosen. R+

Parameters

Feedstock demand parameters: Q k is the annual demand for seed of the biofuel production unit. This demand includes feedstock losses that may occur during the supply. α is the remaining fraction of feedstock after pre-processing. Q k,p is the quantity of feedstock expected at the biofuel production unit depending on the location of the pre-processing.

Qmin j is the minimum quantity of feedstock required at a collection point. Qmax j is the maximum capacity of a collection point. Feedstock harvesting and collection parameters: ST i is the total surface area of a biomass harvesting site.

Rend i is the territorial seed yield of a biomass harvesting site. The territorial seed yield is the ratio of the total potential seed production at the biomass harvesting site to the surface area of the biomass harvesting site.

ϑ is the fruits picking rate. Ch i is the unitary harvesting cost. W is the daily pay for harvesting workers. Hr is the prescribed number of work hours per day. NT i is the number of B. aegyptiaca trees at each biomass harvesting site. Prod is the annual amount of fruits produced by a B. aegyptiaca tree. f is the maximum number of collection points that can set up. a i,j defines the potential allocation of a biomass harvesting site i to a collection point j. "at the BPU" is the location of the pre-processing at the biofuel production unit. "at the CP" is the location of the pre-processing at the collection points. Feedstock pre-processing handling and storage parameters: Hand p is the unitary handling cost depending on the pre-processing location. Stor p is the unitary storage cost depending on the pre-processing location. Cf p is the total pre-processing fixed costs depending on the location of pre-processing. Cv p is the total pre-processing operating costs depending on the location of pre-processing. Transportation parameters: d i,j is the distance between a biomass harvesting site i and a collection point j. d j,k is the distance between a collection point and the biofuel production unit.

x i,m is a binary parameter that defines if the means of transport m is available in a the collection point j.

τ is a tortuosity factor. The tortuosity factor refers to the ratio of the actual travel distance by road to the shortest distance as the crow flies [START_REF] Sultana | Development of tortuosity factor for assessment of lignocellulosic biomass delivery cost to a biorefinery[END_REF].

C m is the unit transport cost for the means of transport m and C truck is the unit transport cost of the truck.

Pr m and P truck are the fixed costs of the means of transport m and of the truck.

Equations

Min ∑ i Ch i •S i + ∑ i,j,p,m Q i,j .d i,j .τ.C m .x i,m + ∑ i,m Pr m .x i,m .y i + ∑ j,k,p Q j,k,p .d j,k .τ.C truck +∑ j Pr truck .y j + ∑ p,j,k Hand p .Q j,k,p + ∑ p,j,k Stor p .Q j,k,p + ∑ p C f p .y p + ∑ p Cv p .y p (1)
S i ≤ ST i .y i ∀ i (2) Q k,p = Q k → p = "at the BPU" ∀ p, k (3)
Q k,p = Q k .α → p = "at the CP" ∀ p, k (4)
∑ i,j Q i,j .a i,j ≥ ∑ k Q k (5) ∑ j Q i,j ≤ S i .Rend i ∀ i (6) Q i,j ≤ S i .Rend i .y i ∀ i, j (7)
∑ i,j Q i,j .a i,j ≤ ST i .Rend i .y j (8)

∑ j Q i,j .a i,j ≥ y i ∀ i (9)
∑ i Q i,j ≤ Qmax j .y j ∀ j (10)

∑ i Q i,j ≥ Qmin j .y j ∀ j (11) Q j,p = ∑ i Q i,j .a i,j → p = "at the BPU " ∀ j, p (12)
Q j,p = ∑ i Q i,j .a i,j .α → p = "at the CP" ∀ j, p

∑ k Q j,k,p = Q j,p ∀ j, p (13)
∑ k,p Q j,k,p ≥ y j ∀ j (15) ∑ j,k Q j,k,p = ∑ k Q k,p .y p ∀ p (16) ∑ j y j ≤ f (14)
∑ p y p = 1 ( 18)

Ch i = (Rend i .W)/(ϑ.Hr) ( 19
)
Rend i = Prod.NT i /ST i (20)
Equation ( 2) ensures that the surface area used by each selected biomass harvesting site does not exceed the available surface area of each biomass harvesting site. Equations ( 3) and ( 4) calculate the quantity of feedstock required at the biofuel production unit depending on the location of pre-processing. Equation ( 5) ensures that the sum of the quantities of feedstock harvested and transported between the chosen biomass harvesting sites and the selected collection points satisfy the quantity of feedstock required by the biofuel production unit. Equations ( 6) to [START_REF] Bart | 6-Emerging new energy crops for biodiesel production[END_REF] ensure that the quantities of feedstock transported from the selected biomass harvesting sites to the selected collection points do not exceed the quantities of feedstock that can be harvested at each biomass harvesting site. Equation [START_REF] Patel | 11-Biodiesel from Plant Oils[END_REF] ensures that the maximum capacity of a selected collection point is not exceeded. Equation [START_REF] Ndoye | Reproductive biology in Balanites aegyptiaca (L.) Del., a semi-arid forest tree[END_REF] ensures that the minimum demand of each selected collection point is satisfied. Equations ( 12) and ( 13) calculate the quantity of feedstock to be transported between each selected collection point and the biofuel production unit depending on the choice of the location of pre-processing. Equations ( 14) to [START_REF] Boffa | Les Parcs Agroforestiers en Afrique Subsaharienne[END_REF] ensure that the quantities of feedstock transported between the selected collection points and the biofuel production unit satisfy the demand of the biofuel production unit depending on the location of pre-processing. Equation [START_REF] Angelsen | Exploring the forest-poverty link: Key concepts, issues and research implications[END_REF] ensures that the number of chosen collection points does not exceed the maximum number of collection points needed. Equation [START_REF] Pouliot | Rural People's Reliance on Forests and the Non-Forest Environment in West Africa: Evidence from Ghana and Burkina Faso[END_REF] ensures that only one pre-processing location is chosen. Equation [START_REF] Yunus | Kinetics study of balanites aegyptiaca oil transesterification for the production of biodiesel[END_REF] calculates the unitary costs of harvesting at a biomass harvesting site. Equation [START_REF] Dass | Production of Biodiesel Oil from Desert Dates (Balanites aegyptiaca) Seeds Oil Using a Hetrogeneous Catalyst Produced from Mahogany (Khaya senegalensis) Fruit Shells[END_REF] calculates the territorial seed yield of a biomass harvesting site.

Case Study

The model was implemented on a theoretical case study of B. aegyptiaca seed supply chain. Only the number and location of the sites in the supply network are theoretical. They have been randomized into value ranges reflecting reality in Burkina Faso. Costs and other data, etc. reflect the reality. Therefore, calculated costs reflect the real costs that could be observed in Burkina Faso. The randomization of the locations and the number of sites does not prevent the results of the model from reflecting reality in Burkina Faso. Indeed, for a given real case, there is no guarantee that the supply network configuration will follow a particular logic. Yet our model must be able to treat a supply network, whatever its configuration.

The considered biofuel production unit is in a region where an electricity company has a SVO demand of about 66,500 L/year. This SVO must feed diesel engines coupled to alternators for producing electricity. 66,500 L of SVO corresponds to a seed demand 200 tons, if it is considered that B. aegyptiaca seeds have an oil content of 46% and that the press used has an extraction efficiency of 65%. The oil density considered is 900 kg/m 3 . The need for fruits is 2000 tons, if it is considered that seeds represent 10% of the dry fruit weight.

Around the biofuel production unit several B. aegyptiaca fruit harvesting sites and several potential collection points have been identified. For a potential collection point to be selected, it must be able to collect at least 10 tons of biomass, which represent the load capacity of the trucks usually used for the transport of goods in Burkina Faso. The maximum capacity of each collection point is unlimited.

A fruit productivity per tree of 52 kg was retained for all the biomass harvesting sites [START_REF] Chapagain | Desert date (Balanites aegyptiaca) as an arid lands sustainable bioresource for biodiesel[END_REF]. A B. aegyptiaca tree density ranging from 25 trees/ha to 40 trees/ha was randomly distributed to each of the biomass harvesting sites. According to [START_REF] Hall | Ecology of a key African multipurpose tree species, Balanites aegyptiaca (Balanitaceae): The state-of-knowledge[END_REF], a density ranging from 25 to 50 trees per ha is acceptable for a region where B. aegyptiaca is well represented.

The harvesting cost per hectare was calculated based on the corresponding labor cost. It was consequently calculated based on a fruit picking rate of 9.24 kg/h found by [START_REF] Sawadogo | Optimization of Balanites aegyptiaca seeds supply chain for biofuel production in West Africa sahelian regions[END_REF] for a tree density of 20 trees/ha and a fruit yield of 545.43 kg/ha. A man-day is assumed to be 8 h. The minimum daily wage for farm workers is 1363.91 XOF/day in Burkina Faso (XOF is West African CFA francs). Fruit harvesting is done by hand or using worn household containers such as buckets. Therefore, harvesting equipment purchasing costs are not included in the feedstock cost price calculation.

Carts pulled by donkeys and motor tricycles are considered for transporting the feedstock between harvesting sites and collection points. Old trucks are considered for feedstock transportation between collection points and the biofuel production unit. The unit transport cost by cart and tricycle are estimated on the basis of the driver's minimum wage, the maximum load per trip, the velocity of each means of transport and fuel consumption for the tricycle. For carts, a maximum load is about 0.5 tons/trip [START_REF] Lhoste | La traction Animale[END_REF] and the maximum traveling distance covered in a day is 20 km when considering a velocity of 5 km/h [START_REF] Lhoste | La traction Animale[END_REF] and an 8-h work day. For tricycles, the maximum load is 1.1 ton and its fuel consumption is 3.5 L/100 km [START_REF] Sawadogo | Optimization of Balanites aegyptiaca seeds supply chain for biofuel production in West Africa sahelian regions[END_REF]. The maximum distance covered in a day is 160 km considering a velocity of 40 km/h and an 8-h work day. For trucks, the unit transport cost is 52 XOF/ton.km for unpaved roads [START_REF] Amassa | Manuel d'information sur le transport des produits agricoles au Mali[END_REF]. Its maximum load is about 10 tons. The costs for each means of transport are listed in Table 3. Only fixed costs arising from the transport of B. aegyptiaca fruits or seeds were calculated, as in rural areas in West Africa, the means of transport are not reserved for a single activity. In this case study, it has been considered that the means of transport are used only from October to December, i.e., three months per year, for the biofuel production unit supply. This period of three months is in the ripening period of the B. aegyptiaca fruits, which according to [START_REF] Ndoye | Reproductive biology in Balanites aegyptiaca (L.) Del., a semi-arid forest tree[END_REF], lasts from October to February.

Concerning the supply network and number, the geographical coordinates of the biomass harvesting sites were randomized over an area with a radius of 50 km. The resulting supply network was used for the whole study. The surface area value of each biomass harvesting site was set to 10 ha. All distances are Euclidean distances to which a tortuosity factor was applied. With no knowledge of the road network in the study area, the value of the tortuosity factor was set to √ 2 (it was considered that the straight-line distance is the diagonal of a square, the sides of which represent the actual path) [START_REF] Okia | Balanites Aegyptiaca: A Resource for Improving Nutrition and Income of Dryland Communities in Uganda[END_REF].

Pre-processing consists of the depulping and de-hulling of B. aegyptiaca fruits. Due to the difficult access to electricity in rural areas in Burkina Faso, and due to the large quantities of biomass to be collected at the biofuel production unit, it was assumed that pre-processing is manual when it takes place at the collection points and motorized when it takes place at the biofuel production unit. Manual pre-processing involves soaking fruits in water for three days and washing off the pulp to obtain nuts. Nuts are sun-dried for several hours. Seeds are obtained by cracking nuts with a hard, solid object on top of a hard, flat surface.

Pre-processing costs were calculated based on their operating parameters. The manual pulping and de-hulling operating parameter is 2.5 kg/day per person [START_REF] Sp/Cpsa | Etude-Diagnostic sur L'organisation, les Performances et Perspectives de la Filière Niébé Burkinabè[END_REF]. Motorized pulping operating parameters are 278.7 kg/h per two persons, 30 kWh/t and 2125 L/t (water consumption) and motorized de-hulling operating parameters are 1009.5 kg/h per two persons and 30 kWh/t [START_REF] Sawadogo | Optimization of Balanites aegyptiaca seeds supply chain for biofuel production in West Africa sahelian regions[END_REF]. Water costs 1091 XOF/m 3 (Burkina Faso Water Company in 2017) and electricity costs 114 XOF/kWh (Burkina Faso Electricity Company in 2017). All the pre-processing machines considered have a lifespan of 10 years and their investment cost is amortized using the declining balance method at a rate of 5%. Unitary costs for each type of pre-processing are described in Table 4.

At the collection points, biomass is deposited in polypropylene bags before the bags are transported and stored at the biofuel production unit. The handling for a bag full of biomass cost is 100 XOF [START_REF] Allard | Evaluation de la Rentabilité Economique de la Filière Jatropha dans la Région de Teriya Bugu (Mali)[END_REF]. The storage cost is assumed to be equal to the cost of the bags in which the biomass is bagged for transport. This storage cost amounts to 250 XOF for a bag full of biomass [START_REF] Allard | Evaluation de la Rentabilité Economique de la Filière Jatropha dans la Région de Teriya Bugu (Mali)[END_REF][START_REF] Aviara | Some Physical Properties of Balanites Aegyptiaca Nuts[END_REF]. A full bag weighs 60 kg if it contains fruit and 70 kg if it contains seeds.

Four different scenarios were designed to observe the effects of the location of pre-processing and the effects of the means of transport on the supply costs.

In scenario 1, carts pulled by donkeys are used to transport fruits from the harvesting sites to the collection points and two pre-processing locations (at the collection points and at the biofuel production unit) are possible. In scenario 2, motor tricycles are used to transport the fruits from the harvesting sites to the collection points and the pre-processing locations options are the same as in scenario 1. In scenario 3, carts pulled by donkeys are used to transport the feedstock from the biomass harvesting sites to the collection points and only one pre-processing location (at the biofuel production unit) is possible. In scenario 4, motor tricycles are used to transport the feedstock from the biomass harvesting sites to the collection points and pre-processing location options are the same as in scenario 3.

In the four scenarios, the maximum number of collection points that can be opened is constant and equal to the total number of potential collection points. After determining the scenario with the lowest feedstock cost price at the biofuel production unit gate, this scenario was run again but the value of the parameter f in Equation ( 19) was varied from 35 to 1. The objective was to observe the influence of the reduction in the maximum number of collection points that can be opened on the supply costs and on the network configuration. This reduction could occur when some constraints appear (inaccessibility of a collection point by truck for example).

Results and Discussion

The model was implemented and solved in the optimization software Xpress IVE 7.9. The experiments were performed on an Intel Core i7-5500U CPU 2.4 GHz with 8 GB RAM on a 64-bit platform. The solution was obtained in less than five minutes using the branch and bound algorithms available in the software solver.

The model determines the number of biomass harvesting sites and the optimal surface area to be harvested; the number, location and allocation of biomass harvesting sites; the number, location and allocation of collection points; the quantities of fruit to be harvested and pre-processed; the location of pre-processing if more than one location is possible; the quantities of fruit or seed to be transported, handled and stored depending on the pre-processing location; total supply costs (optimal costs of harvest, transport, pre-processing, storage, and handling).

The results of the implementation of the four scenarios are presented and discussed below. These results concern the configuration of the supply network, and the cost results produced by the different scenarios. Then, for the scenario with the lowest seed cost at the biofuel production unit gate, the results of the variation in the maximum number of collection points that can be opened are presented and discussed.

Results of the Implementation of the Four Scenarios

Supply network configuration: For each scenario, the result of the pre-processing location, total harvested surface area, the number of chosen biomass harvesting sites and number of chosen collection points are presented in Table 5. In scenarios 1 and 2, pre-processing always takes place at the collection points. The configuration of the supply network is the same in the scenarios with donkey carts (scenarios 1 and 3). The configuration of the supply network is also the same in the scenarios with tricycles (scenarios 2 and 4). These configuration for the scenarios with tricycles and carts are presented in Figure 2. The total harvested surface area, the number of chosen biomass harvesting sites and chosen collection points in the scenarios with carts is higher than in those with tricycles.

Sustainability 2018, 10, x FOR PEER REVIEW 9 of 15

Results of the Implementation of the Four Scenarios

Supply network configuration: For each scenario, the result of the pre-processing location, total harvested surface area, the number of chosen biomass harvesting sites and number of chosen collection points are presented in Table 5. In scenarios 1 and 2, pre-processing always takes place at the collection points. The configuration of the supply network is the same in the scenarios with donkey carts (scenarios 1 and 3). The configuration of the supply network is also the same in the scenarios with tricycles (scenarios 2 and 4). These configuration for the scenarios with tricycles and carts are presented in Figure 2. The total harvested surface area, the number of chosen biomass harvesting sites and chosen collection points in the scenarios with carts is higher than in those with tricycles.

The supply network in scenario 1

The supply network in scenario 2

The supply network in scenario 3

The supply network in scenario 4 Cost results in the four scenarios are presented in Figure 3. Harvesting and pre-processing costs represent the costs per kg of B. aegyptiaca fruits. Transport, handling, and storage costs represent the costs per kg of B. aegyptiaca seed.

Cost results in the four scenarios are presented in Figure 3. Harvesting and pre-processing costs represent the costs per kg of B. aegyptiaca fruits. Transport, handling, and storage costs represent the costs per kg of B. aegyptiaca seed.

The cost price of B. aegyptiaca seeds at the biofuel production unit gate fluctuates between 119.5 XOF/kg (scenario 1) and 139.5 XOF/kg (scenario 4). The lowest seed cost prices at the biofuel production unit gate as well as the lowest costs of all the other supply operations except harvesting and pre-processing are found in the scenarios where pre-processing takes place at the collection points (scenarios 1 and 2). Transport costs fluctuate between 41.4 XOF/kg in scenario 1 and 52 XOF/kg in scenario 4. Transport costs represent about 35% of seed cost price at the biofuel production unit gate in the four scenarios. Harvesting costs are the same in all the scenarios and represent about 15% of the seed costs at the biofuel production unit in all the scenarios.

Preprocessing costs represent respectively 45% and 8% of the seed cost price at the biofuel production unit gate in scenarios 1 and 2 and in scenarios 3 and 4. Storage and handling costs represent respectively 4% and 32% of the seed costs at the biofuel production unit in scenarios 1 and 2 and in scenarios 3 and 4. It can be seen that scenarios 1 and 2 pre-processing costs are four times higher, and storage and handling costs are more than 10 times higher than the same items cost in scenarios 3 and 4.

Discussion

According to the results presented above, the scenarios with donkey carts (scenarios 1 and 3) have the highest number of chosen biomass harvesting sites and the highest number of chosen collection points. These results can be explained by the short distance travelled by the cart each day (20 km) compared to the distance travelled by the tricycle per day (160 km). Indeed, when the tricycles are used, the model is not constrained by short travel distances. Thus, collection points that have the highest fruit yields and are simultaneously located closest to the biofuel production unit and biomass harvesting sites are chosen. The cost price of B. aegyptiaca seeds at the biofuel production unit gate fluctuates between 119.5 XOF/kg (scenario 1) and 139.5 XOF/kg (scenario 4). The lowest seed cost prices at the biofuel production unit gate as well as the lowest costs of all the other supply operations except harvesting and pre-processing are found in the scenarios where pre-processing takes place at the collection points (scenarios 1 and 2). Transport costs fluctuate between 41.4 XOF/kg in scenario 1 and 52 XOF/kg in scenario 4. Transport costs represent about 35% of seed cost price at the biofuel production unit gate in the four scenarios. Harvesting costs are the same in all the scenarios and represent about 15% of the seed costs at the biofuel production unit in all the scenarios.

Preprocessing costs represent respectively 45% and 8% of the seed cost price at the biofuel production unit gate in scenarios 1 and 2 and in scenarios 3 and 4. Storage and handling costs represent respectively 4% and 32% of the seed costs at the biofuel production unit in scenarios 1 and 2 and in scenarios 3 and 4. It can be seen that scenarios 1 and 2 pre-processing costs are four times higher, and storage and handling costs are more than 10 times higher than the same items cost in scenarios 3 and 4.

Discussion

According to the results presented above, the scenarios with donkey carts (scenarios 1 and 3) have the highest number of chosen biomass harvesting sites and the highest number of chosen collection points. These results can be explained by the short distance travelled by the cart each day (20 km) compared to the distance travelled by the tricycle per day (160 km). Indeed, when the tricycles are used, the model is not constrained by short travel distances. Thus, collection points that have the highest fruit yields and are simultaneously located closest to the biofuel production unit and biomass harvesting sites are chosen.

Despite the performance of the tricycle, in scenario 1 the carts that are used have the lowest seed cost price and the lowest transport cost. This can be explained by the fixed costs of the motor tricycles, which are 7.7 times higher than the carts fixed costs (Table 3).

Transport costs between biomass harvesting sites and collection points are mainly affected by the purchase price of the means of transport when motor tricycles are used. When carts pulled by animals are used, transport costs between biomass harvesting sites and collection points are mainly affected by the operating costs of transportation.

High pre-processing costs in the scenarios with pre-processing at the collection points (scenarios 1 and 2), are due to the high cost of manual pre-processing at the collection points. Despite these high pre-processing costs, pre-processing always takes place at the collection points regardless of the means of transport in scenarios where the model is not constrained to choose only one pre-processing location (scenarios 1 and 2). This is due to the fact that less amounts of biomass have to be transported, handled and stored when the pre-processing takes place at the collection points. Indeed, seeds represent only 10% of the whole B. aegyptiaca fruit weight. Therefore, when pre-processing is located at the collection points, the quantities of biomass to be transported are divided by 10.

Pre-processing is manual at the collection points because access to electricity is difficult in rural areas and because to date, mechanical pre-processing of B. aegyptiaca is not well developed. Therefore, improving B. aegyptiaca fruit pre-processing operations by developing affordable pre-processing machines could significantly reduce seeds cost prices at the biofuel production unit. E. Mamman et al. and Aviara et al. [START_REF] Mamman | Effect of Moisture Content and Loading Orientation on the Mechanical Properties of Balanites Aegyptiaca Nuts[END_REF] had already addressed this difficulty to manually perform B. aegyptiaca de-hulling. They have indeed worked on the physical properties of the B. aegyptiaca nuts in order to allow the development of machines facilitating fruit de-hulling. Another aspect concerning pre-processing is depulping, which until now consumed significant amounts of water (more than 2 m 3 /t), especially when the rainfall of the Sahelian regions is taken into consideration. The limited access to water in Sahelian climates could therefore be a brake on the exploitation of B. aegyptiaca in these regions.

To conclude the four scenarios, scenario 1 is the best because pre-processing takes place at the collection points and also because carts pulled by animals are used for feedstock transportation between biomass harvesting sites and collection points. Scenario 1 which produced the lowest total cost was run again by varying the value of the maximum number of collection points that can be opened. Scenario 1 was intentionally re-run to observe the influence (on the supply costs and on the network configuration) of the reduction in the maximum number of collection points that can be opened. The results are presented and discussed in the following section of the paper.

Results and Discussion of the Variation in the Maximum Number of Collection Points That Can Be Opened

The results of the variation of the maximum number of collection points that can be opened are presented in Table 6 and Figure 4.

These results show that when the maximum number of collection points is under 4, no solution is found. This can be explained by the fact that the organization of biomass harvesting sites and of collection points does not allow 3, 2, or 1 gathering point(s) to fulfill the demand for feedstock by the biofuel production unit.

When the maximum number of gathering points that can be opened is between 4 and 12, feedstock cost prices and transport costs decrease when the maximum number of collection points that can be opened increase (see Figure 4). This result can be explained by the fact that the model is forced to choose biomass harvesting sites, which are located at increasingly far distances when the maximum number of collection points that can be opened decreases. Indeed, the maximum distance between the farthest biomass harvesting site from a gathering point decreases from 18 km to 10.8 km when the number of collection points increases from 4 to 12. Moreover, the percentage of biomass harvesting sites at a distance equal to or greater than 11 km also decreases from 45% to 0% when the number of collection points increases from 4 to 12. This can be seen in Table 6.

When the maximum number of collection points that can be opened is equal to or greater than 12, the costs and logistics network configurations remain constant. These costs and logistics network configuration are exactly the same as the configuration of the logistics network and costs found for the maximum number of gathering points equal to 12 (optimum value found for scenario 1 in the above paragraph). This result implies that without any constraint on the number of collection points, 12 collection points are sufficient and optimal for feedstock collection. One more collection point would be useless. This confirms the model's performance in locating and choosing the optimal collection points.

It can be learned from this variation in the number of collection points that when carts pulled by animals (or any other means of transportation having a low velocity and a low load capacity) are used, a high number of collection points helps reducing feedstock cost prices. These reductions in costs are induced by the choice of biomass harvesting sites closer to collection points. When the maximum number of collection points that can be opened is equal to or greater than 12, the costs and logistics network configurations remain constant. These costs and logistics network configuration are exactly the same as the configuration of the logistics network and costs found for the maximum number of gathering points equal to 12 (optimum value found for scenario 1 in the above paragraph). This result implies that without any constraint on the number of collection points, 12 collection points are sufficient and optimal for feedstock collection. One more collection point would be useless. This confirms the model's performance in locating and choosing the optimal collection points.

It can be learned from this variation in the number of collection points that when carts pulled by animals (or any other means of transportation having a low velocity and a low load capacity) are used, a high number of collection points helps reducing feedstock cost prices. These reductions in costs are induced by the choice of biomass harvesting sites closer to collection points.

Conclusions

A mathematical model for making decisions regarding the wild non-timber forest products supply chain for biofuel production is herein described. It has been applied to the Balanites aegyptiaca seed supply chain.

Conclusions

A mathematical model for making decisions regarding the wild non-timber forest products supply chain for biofuel production is herein described. It has been applied to the Balanites aegyptiaca seed supply chain.

Based on the characteristics of the studied region and considering different means of transport and different pre-processing options, the model determines the optimal supply network configuration for feedstock supply to a biofuel production unit.

The model was implemented as a theoretical case study in the sub-Sahelian region of Burkina Faso. The results of the case study implementation show that in such a context, more than 35% of the cost price of seeds at the biofuel production unit gate is the cost of transport. Pre-processing, handling, and storage costs represent about 50% of the cost price of the seeds. In the best scenario, the cost of collecting feedstock represents more than 70% of the transport costs. Due to the high fixed costs of the motor tricycles, the most efficient way to transport biomass from the biomass harvesting sites to the collection points is by using animal drawn carts. Motor tricycles could be profitable for the transport of biomass if their purchasing cost could be reduced, as they can travel faster and have a higher load capacity than carts. Moreover, when the number of collection points is limited, tricycles could be more profitable because they enable fewer collection points than do carts.

Despite the fact that the cost of motorized pre-processing at the biofuel production unit is the lowest, pre-processing is better performed before transport. This is due to the very low weight of B. aegyptiaca seed compared to the weight of the whole fruit. Consequently, improving B. aegyptiaca fruit pre-processing operations by developing affordable mechanical pre-processing machines could significantly reduce the cost of seeds at the biofuel production unit. This would indeed simultaneously reduce biomass pre-processing and transportation costs.

A serious and controlled exploitation of B. aegyptiaca could make it a good feedstock for producing biofuel in Sahelian areas where the soil and climatic conditions are not suitable for growing the plants commonly used for biofuel production. The produced biofuels could be used for electrifying rural areas of Sahelian countries such as Burkina Faso, Mali, Niger, and Chad.

The next step in this research will be the study of Balanites aegyptiaca reverse logistics. The aim is to examine the benefits that a reverse supply chain might have on the feedstock cost prices.

  is a schematic representation of the supply network. Biomass harvesting sites are sites where B. aegyptiaca trees are well represented.

  is a schematic representation of the supply network. Biomass harvesting sites are sites where B. aegyptiaca trees are well represented.

Figure 1 .

 1 Figure 1. Schematic diagram of the supply network structure with the possible allocations of the sites.

Figure 1 .

 1 Figure 1. Schematic diagram of the supply network structure with the possible allocations of the sites.

Figure 2 .

 2 Figure 2. The supply network in scenarios 1-4.

Figure 2 .

 2 Figure 2. The supply network in scenarios 1-4.

Figure 3 .

 3 Figure 3. Cost results in the four scenarios.

Figure 3 .

 3 Figure 3. Cost results in the four scenarios.

Figure 4 .

 4 Figure 4. Variation in the cost price of the feedstock at the biofuel production unit and in the transport cost.

  collection points that can be opened Transport cost (XOF/kg) Cost price of feedstock (XOF/kg)

Figure 4 .

 4 Figure 4. Variation in the cost price of the feedstock at the biofuel production unit and in the transport cost.

Table 1 .

 1 Sets.

	Sets	Index
	Biomass harvesting sites	i
	Biomass collection points	j
	Biofuel production unit	k
	Means of transport	m
	Location of pre-processing	p

Table 2 .

 2 Variables.

	Variables	Description	Domain
	S i	Surface area of a biomass harvesting site i	R+
	y i	Selection of a biomass harvesting site i	{0,1}
	y j	Selection of a collection point j	{0,1}
	y p	Decision on the location of pre-processing	{0,1}

Table 3 .

 3 Attributes of the means of transport.

		Tricycle	Cart	Truck
	Interest rate %	0.05	0.05	0.05
	Life time (years)	5	5	10
	Price (XOF)	1,000,000	130,000	4,000,000
	Fixed cost (XOF)	57,744	7507	129,505
	Unitary cost (XOF/ton.km)	27	136	52

Table 4 .

 4 Pre-processing costs (calculated by the authors based on operating parameters).

	Type	Manual	Motorized
	Operations	Pulping and De-Hulling	Pulping	De-Hulling
	Labor (XOF/t)	54,560	1225	340
	Electricity (XOF/t)	-	3420	3420
	Water (XOF/t)	-	2320	-
	Operating costs (XOF/t)	54,560	6965	3760
	Fixed costs for one machine (XOF/year)	-	485,645	679,900

Table 5 .

 5 Results of the supply network configuration.

	Scenarios

Location of Pre-Processing Numbers of Chosen Biomass Harvesting Sites Numbers of Chosen Collection Points Total Harvested Surface Area (ha)

	Scenario 1	Collection points	119	12	1181.3
	Scenario 2	Collection points	109	6	1089.8

Table 5 .

 5 Results of the supply network configuration.

	Scenarios	Location of Pre-Processing	Numbers of Chosen Biomass Harvesting Sites	Numbers of Chosen Collection Points	Total Harvested Surface Area (ha)
	Scenario 1	Collection points	119	12	1181.3
	Scenario 2	Collection points	109	6	1089.8
	Scenario 3	Biofuel production unit	117	9	1167.2
	Scenario 4	Biofuel production unit	107	2	1079.8

Table 6 .

 6 Results of the reduction in the maximum number of collection points (GP) that can be opened.

	Sustainability 2018, 10, x FOR PEER REVIEW		12 of 15
	Number of GP	Maximum Distance	Distance > 11 km	Distance < 11 km
	4 GP	18 km	45%	55%
	5 GP	15.7 km	36%	64%
	6 GP	14.1 km	29%	71%
	7 GP	13 km	23%	77%
	8 GP	12.7 km	14%	86%
	9 GP	11.7 km	8%	92%
	10 GP	11.4 km	2%	98%
	11 GP	10.8	0%	100%
	12 GP	10.8	0%	100%

Table 6 .

 6 Results of the reduction in the maximum number of collection points (GP) that can be opened.

	Number

of GP Maximum Distance Distance > 11 km Distance < 11 km

	4 GP	18 km	45%	55%
	5 GP	15.7 km	36%	64%
	6 GP	14.1 km	29%	71%
	7 GP	13 km	23%	77%
	8 GP	12.7 km	14%	86%
	9 GP	11.7 km	8%	92%
	10 GP	11.4 km	2%	98%
	11 GP	10.8	0%	100%
	12 GP	10.8	0%	100%

Funding: This research received no external funding.

Acknowledgments: This paper is expanded and improved from the conference paper. Sawadogo, M.; Bambara, L.D.F.; Blin, J.; Anciaux, D.; Roy, D. Optimization of Balanites aegyptiaca seeds supply chain for biofuel production in West Africa sahelian regions. In Proceedings of the 2017 8th International Renewable Energy Congress (IREC), Amman, Jordan, 21-23 March 2017; pp. 1-5, doi:10.1109/IREC.2017.7926017.

Conflicts of Interest:

The authors declare no conflicts of interest.


Xavier Bertrand

Mathieu Pucheault

Laurent Chabaud

Jean-François Paquin

Synthesis of Tertiary Fluorides Through an Acid-Mediated Deoxyfluorination of Tertiary Alcohols

The combination of methanesulfonic acid and potassium bifluoride is reported for the deoxyfluorination of tertiary alcohols. Under those metal-free conditions that use readily available, cheap, and easy-to-handle reagents, a range of tertiary alcohols could be converted to the corresponding fluorides in excellent yields (average yields of 92% for 19 examples) without the formation of any side product. The application of these conditions for the fluorination of ether and ester is also demonstrated.

The importance of fluorine in many fields of chemistry, such as in pharmaceutical, agrochemical and material science has long been recognized. 1 From 2015 to 2020 between 23 and 53% of new small-molecule drugs approved by the FDA contained at least one atom of fluorine. 2 Yet, most of the functional group present were either aryl fluoride or trifluoromethylated arenes. 2a The presence, or the absence, of a fluorinated functional group can be, in part, correlated to the ease of its synthetic incorporation. [START_REF] Brown | Analysis of Past and Present Synthetic Methodologies on Medicinal Chemistry: Where Have All the New Reactions Gone?[END_REF] Tertiary alkyl fluoride has recently emerged as a potentially useful motif in medicinal chemistry (e.g., odanacatib).

A number of methods for its synthesis has been reported starting from various precursors such as alcanes, 4 carboxylic acids and derivatives, 5 tertiary halides, 6 alkenes 7 as well as others (Scheme 1a). [START_REF] Zhu | Modern Approaches for Asymmetric Construction of Carbon-Fluorine Quaternary Stereogenic Centers: Synthetic Challenges and Pharmaceutical Needs[END_REF] The direct conversion of an alcohol to a fluoride, namely the deoxyfluorination reaction, is another useful way for the incorporation of a single fluorine atom onto an organic molecule. [START_REF] Champagne | Monofluorination of Organic Compounds: 10 Years of Innovation[END_REF] Most of the reagents commonly used for the deoxyfluorination exploit the sulfur-fluorine bond. [START_REF] Ni | Good Partnership between Sulfur and Fluorine: Sulfur-Based Fluorination and Fluoroalkylation Reagents for Organic Synthesis[END_REF] Sulfur tetrafluoride was first reported by DuPont in 1958 for the conversion of carbonyl functions to the difluoromethylene moiety. 11 Since then, it has been used for the fluorination of alcohols, ketones, and carboxylic acids. [START_REF] Hasek | Fluorination reactions of sulfur tetrafluoride[END_REF] Over the years, new reagents have been developed to make this transformation easier. Notably, DAST, Deoxo-Fluor ® , XtalFluor ® reagents, Pyfluor and AlkylFluor, as well as many others, have all been used for the fluorination of oxygenated functions, such as alcohols, phenols, aldehydes, ketones, and carboxylic acids. 13,14 Although these reagents are widely used, they still present issues in terms of thermal stability, reactivity, accessibility, and storage. 15 While there have been many reports for primary and secondary alcohols; the deoxyfluorination of tertiary alcohols remains challenging as the activated intermediate tends to eliminate and generates the corresponding alkene (Scheme 1b). 13,16 Therefore, the development of an efficient method for the conversion of tertiary alkyl alcohols, which are easily accessible, to the corresponding fluoride would be of great interest. To overcome this complication, alternative approaches have been proposed. The derivatization of alcohols to oxalates, followed by radical decarboxylation and fluorination has been reported by a few groups (Scheme 1c). 17 Later on, the group of Xiao reported the direct conversion of tertiary alcohol to fluorine by an in situ generation of the corresponding bromide or iodide, followed by radical fluorination using Selectfluor (Scheme 1d). [START_REF] Zhang | Dehydroxylative Fluorination of Tertiary Alcohols[END_REF] Along those lines, we recently reported the synthesis of tertiary alkyl fluoride by hydrofluorination of alkenes. 7a This reaction was performed using a combination of a strong acid (i.e., methanesulfonic acid), and a nucleophilic source of fluorine (i.e., triethylamine trihydrofluoride). [START_REF] Franz | Ueber Trishydrofluoride Tertiaerer Amine Und Ihren Einsatz Als Fluorierungsmittel[END_REF][START_REF]For reviews on the properties and reactivity of Et3N•3HF, see: (a) McClinton, M. A. Triethylamine Tris(hydrogen fluoride): Applications in Synthesis[END_REF] We envisioned that we could apply similar conditions to tertiary alcohols, which in the presence of a strong acid could lose a molecule of water to generate the analogous carbocation. A nucleophilic source of fluoride could then yield the desired tertiary alkyl fluoride compound. Herein, we report the use of a combination of methanesulfonic acid and potassium bifluoride for the deoxyfluorination of tertiary alcohols (Scheme 1e). This method employs inexpensive and easy-to-handle reagents for the simple transformation of alcohol to fluoride without the need for prior alcohol derivatization. With these optimized conditions, the elimination product was never observed for all tertiary alcohols that were tested.

Scheme 1. Tertiary alkyl fluorides

Considering the similarities between the mechanism of the hydrofluorination of alkenes and our proposed deoxyfluorination, we started investigating the reaction on 1a using similar reaction conditions and the results are shown in Table 1. 7a Hence, with a combination of methanesulfonic acid (5 equiv) and triethylamine trihydrofluoride (5 equiv), the corresponding fluoride (2a) was obtained in a promising 64% NMR yield (Table 1, entry 1). Inorganic sources of fluoride afforded the desired product in similar yields, with KHF2 showing the best results (Table 1, entries 2-3). Using concentrated sulfuric acid significantly diminished the yield of the reaction (Table 1, entry 4). Based on these results, we selected a combination of methanesulfonic acid and KHF2 for the rest of the optimization. By screening the temperature (Table 1, entries 5-6) and the time (Table 1, entries 7-8) of the reaction, we found that running the reaction at 0 °C for one hour afforded the desired fluorinated compound in 96% yield (Table 1, entry 9). While the elimination product was formed at room temperature, only the fluorinated product was obtained at 0 °C. Efforts to diminish the number of equivalents of acid and fluoride decreased the yields significantly (Table 1, entries 10-11) with an increased variability of the outcome (a typical result is shown in the Table 1). The time of the reaction could be lowered to 5 minutes without sig-nificant loss of yield (Table 1, entry 12). To ensure a full conversion for all substrates, we still ran the reaction for one hour for the rest of the study. With these optimized conditions in hand, we went on to investigate the scope of the reaction for various tertiary alcohols with the results presented in Scheme 2. A variety of commonly used alcohol protecting groups were tested and were found to be compatible with the reaction conditions (2b-g). Tertiary alcohol derived from fatty acids were obtained in excellent yields (2h-i), with the internal disubstituted alkene of the oleic acid derivative remaining untouched, without isomerization. Alcohol bearing other functional groups such as nitrile, nitro, glyoxyl and phthalimide could also be converted to the corresponding fluoride in excellent yields (2j-m). No substitution was observed on the α-chloroester (2n) as well as on the tosylate (2o). The presence of a furan group (2p), as well as variety in the side chain (2q) were also tolerated in the reaction. A natural product derivative could also be used, with the substrate 2r being obtained in a good 71% yield. However, the presence of a pyridine on the substrate seemed to shut down the reaction (2s). Primary and secondary alcohols were also tested (2t-u), but no conversion was observed in both cases, pointing that the formation of a tertiary carbocation might be the limiting step. This reaction could be applied for the synthesis of 1-bromo-6-fluoro-6-methylheptane (2v), a key synthetic intermediate of trecetilide, a compound being developed for the treatment of atrial arrhythmia. A first synthesis was reported by Hester and coworkers in 2001 using a convergent synthesis for the introduction of the fluorinated chain. 21 In their synthesis, the secondary amine is alkylated with 1-bromo-6-fluoro-6-methylheptane (2v), synthesized in 5 steps from 5-chloropentanol, to obtain trecetilide. We therefore considered using our conditions for the synthesis of this intermediate. Starting from fairly inexpensive caprolactone, we could obtain the 7-bromo-2-methylheptan-2ol (1v) in 73% yield over two steps. [START_REF] Zhang | Dehydroxylative Fluorination of Tertiary Alcohols[END_REF] We were pleased to see that in our conditions, not only is the bromide not substituted nor eliminated, but the desired product 2v is obtained in an excellent 99% yield without the need for chromatography purification (Scheme 3).

Based on the assumed mechanism of the reaction, we envisioned that other CO bonds could also be fluorinated under these conditions. To test this hypothesis, we synthesized the corresponding trimethylsilyl ether (3a), acetate (4a) and methyl ether (5a). Upon treatment with the standard fluorination conditions, the corresponding fluoride 2a were obtained in excellent yields of 97% and 91% starting from the silyl ether (3a) and the acetate (4a) respectively, and a good yield of 65% from the methoxy (5a) (Scheme 6). This could be useful for late-stage fluorination of protected alcohols.

Scheme 6. Deoxyfluorination of tertiary alcohols derivatives

In summary, we have described a simple and effective method for the deoxyfluorination of tertiary alcohol using a combina-tion of methanesulfonic acid and potassium bifluoride. The reaction has been applied to a wide range of tertiary alcohols with excellent yields, and the corresponding elimination product has never been observed in these conditions. Being selective for tertiary alcohol, this method is complementary to the well establish deoxyfluorination of primary and secondary alcohols.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website.

Detailed experimental procedures and full spectroscopic data for all new compounds (PDF)

Scheme 2 .a

 2 Scheme 2. Scope of the deoxyfluorination of tertiary alcohols a,b

Table 1 . Optimization for the deoxyfluorination of 1a a

 1

	Entry	Fluoride	Acid	Temp. (°C)	Time (h)	Yield (%) b
	1	Et3N•3HF MsOH	20	22	64
	2	KF	MsOH	20	22	57
	3	KHF2	MsOH	20	22	70
	4	KHF2	H2SO4 c	20	22	24
	5	KHF2	MsOH	40	22	40
	6	KHF2	MsOH	0	22	81
	7	KHF2	MsOH	20	8	80
	8	KHF2	MsOH	20	1	86
	9	KHF2	MsOH	0	1	96 (93) d
	10 e	KHF2	MsOH	0	1	3329-67
	11 f	KHF2	MsOH	0	1	6
	12	KHF2	MsOH	0	5 min	90

a See the Supporting Information for the detailed experimental procedures. b Estimated by

[START_REF] Franz | Ueber Trishydrofluoride Tertiaerer Amine Und Ihren Einsatz Als Fluorierungsmittel[END_REF]

F NMR analysis of the crude mixture after workup using 2-fluoro-4-nitrotoluene as the internal standard. c Concentrated sulfuric acid (95-97%). d Isolated yield. e 3 equivalents of KHF2 and MsOH were used. f 1 equivalent of KHF2 and MsOH was used.

ACKNOWLEDGMENT

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Quebec-Nature et technologies (FRQNT), Campus France, the Université Laval, the University of Bordeaux and CNRS. This work has benefited from the facilities and expertise of the Biophysical and Structural Chemistry platform (BPCS) at IECB, CNRS UMS3033, Inserm US001, Bordeaux University.

AUTHOR INFORMATION

Corresponding Author *E-mail: laurent.chabaud@u-bordeaux.fr and jean-francois.paquin@chm.ulaval.ca ORCID Jean-Francois Paquin: 0000-0003-2412-3083 Laurent Chabaud: 0000-0002-2590-8707

Notes

The authors declare no competing financial interest.


Par

Fang GUO Title : Beam selection optimization for non-coplanar intensity-modulated radiation therapy Keywords : IMRT, deep learning, simulated annealing, dose prediction Abstract : Over the decades, one of the popular methods in cancer treatment is Intensity Modulated Radiotherapy (IMRT). The goal of IMRT is to conform the radiation dose to the target and to avoid or reduce exposure to healthy tissue to limit the side effects of treatment. Non-coplanar IMRT selects the beam direction in 2-dimensional angle space. Despite the development of using IMRT in cancer treatment, determining the trajectory and the optimal set of beam orientations is still challenging due to the large beam angle sampling space. Common methods limit the search space for the optimal beam orientations due to the large size and the combinatorial nature of the problem leading to local solutions, while the possibility of collision detection and trajectory is also limited.

In this thesis we propose a novel optimization method based on simulated annealing and learning for non-coplanar IMRT. The simulated annealing is first employed to find the optimal set of beam orientations, and a pretrained neural network is called to predict the dose distribution in potential beam orientations during simulated annealing optimization. This thesis also presents the comparison of the proposed method and conventional noncoplanar IMRT using clinical data obtained from the hospital. In conclusion, the results of this work can be adapted for use at other treatment centers with equipment from different manufacturers. In the future, we aim to perform an evaluation study using methods and algorithms on equipment in order to compare the generated treatment plans to the methods used in the clinic.

  Sciences de la Vie et de la Santé Spécialité : Analyse et Traitement de l'Information et des Images Médicales

	THESE DE DOCTORAT DE Titre : Optimisation de la sélection du faisceau pour la radiothérapie à modulation d'intensité non
	coplanaire
	Mots clés : IMRT, deep learning, simulated annealing, prédiction de dose
	Résumé : Au fil des décennies, l'une des	nature combinatoire du problème conduisant à
	méthodes populaires de traitement du cancer L'UNIVERSITE est la radiothérapie à modulation d'intensité	des solutions locales, tandis que la possibilité de détection de collision et a trajectoire est
	(IMRT). L'objectif de l'IMRT est de conformer DE BRETAGNE OCCIDENTALE la dose de rayonnement à la cible et d'éviter	également limitée. Dans cette thèse, nous proposons une nouvelle méthode
	ou de réduire l'exposition aux tissus sains ECOLE DOCTORALE N° 637 pour limiter les effets secondaires du	d'optimisation basée sur le recuit simulé et l'apprentissage profond pour l'IMRT non
	traitement.	L'IMRT	non	coplanaire	coplanaire. Le recuit simulé est d'abord utilisé
	sélectionne la direction du faisceau dans un	pour trouver l'ensemble optimal d'orientations
	espace angulaire bidimensionnel. Malgré le	de faisceau, et un réseau neuronal pré-formé
	développement de l'utilisation de l'IMRT dans	est appelé pour prédire la distribution de dose
	le traitement du cancer, la détermination de	dans les orientations de faisceau potentielles
	la trajectoire et de l'ensemble optimal	lors de l'optimisation du recuit simulé. En
	d'orientations de faisceau est toujours difficile	conclusion, les résultats de ce travail peuvent
	en raison du grand espace d'échantillonnage	être adaptés pour être utilisés dans d'autres
	de l'angle de faisceau. Les méthodes Beam selection optimization for non-coplanar intensity-modulated centres de traitement avec des équipements orientations optimales du faisceau en raison radiation therapy courantes limitent l'espace de recherche des de différents fabricants.
	de la grande taille et de la
	Président :	Schick ULRIKE	PU-PH, Département Radiothérapie, CHRU de Brest
	Examinateurs : Charlotte ROBERT	Maître de Conférence, HDR, Université Paris Sud
		Issam El NAQA	Professeur, University of South Florida
		Schick ULRIKE	PU-PH, Département Radiothérapie, CHRU de Brest
	Dir. de thèse : Antoine VALERI	PU-PH, Département Urologie, CHRU de Brest
	Invité(s) :
		Julien BERT	Chercheur, CHRU de Brest, HDR, LaTIM

Thèse présentée et soutenue à Brest, le 26/06/23 Unité de recherche : LaTIM UMR 1101 Rapporteurs avant soutenance : Charlotte ROBERT Maître de Conférence, HDR, Université Paris Sud Issam El NAQA Professeur, University of South Florida Composition du Jury :


Nicolas Forcadel

Cyril Imbert

Régis Monneau

Coercive Hamilton-Jacobi equations in domains: the twin blow-ups method

In this note, we consider an evolution coercive Hamilton-Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient variables are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017) to extend their comparison principle to more general boundary conditions and to Hamiltonians that are not globally Lipschitz continuous in the time variable. Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared. We refer to our technique as the twin blow-ups method since two blow-ups are performed simultaneously, one for each variable of the doubling variable method. The Lipschitz regularity of the regularized subsolution provides a key Lipschitz inequality satisfied by the two blow-up limits, that are a priori allowed to be infinite. For expository reasons, the result is presented here in the framework of space dimension one and the general case is treated in a companion paper.

Introduction: a comparison principle

Given T > 0, we consider viscosity solutions of a standard evolutive Hamilton-Jacobi equation posed in the geometric setting of a domain Ω := (0, +∞), (1.1) u t + H(X, u x ) = 0 in (0, T ) × Ω

where X := (t, x), supplemented with the boundary condition, u t + F (X, u x ) = 0 in (0, T ) × ∂Ω and the initial condition, u(0, •) = u 0 in {0} × Ω.

Since the boundary condition can be lost when characteristics reach ∂Ω, it has to be imposed in a weak sense. When working with viscosity solutions, a classical way to handle this discrepancy is to impose that either the boundary condition or the equation is satisfied (in the viscosity sense) at the boundary, (1.2) u t + min {F, H} (X, u x ) ≤ 0 in (0, T ) × ∂Ω (for subsolutions), u t + max {F, H} (X, u x ) ≥ 0 in (0, T ) × ∂Ω (for supersolutions).

Comparison principles are strong uniqueness results for Hamilton-Jacobi equations. In the case of the previous initial boundary value problem, it is known (see [START_REF] Lions | Neumann type boundary conditions for Hamilton-Jacobi equations[END_REF][START_REF] Barles | Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications[END_REF][START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF][START_REF] Barles | An Illustrated Guide of the Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities[END_REF]) that it is difficult to treat the case when tangential coordinates, such as the time variable t, and the normal derivative u x of the solution, are strongly coupled reaching the boundary (0, T ) × ∂Ω. It is standard to make the (strong) assumption of uniform continuity in time t, uniformly in the gradient u x . Such an assumption is not satisfied by the following simple example, (1.3) u t + a(t, x)|u x | = 0 in (0, T ) × Ω, u tb(t, x)u x = 0 in (0, T ) × ∂Ω when a, b ≥ 1 are bounded Lipschitz continuous functions (here with b(t, x) = b(t, 0)).

In this note, we choose structural assumptions on H and F that encompass a large variety of examples (including (1.3)) but we do not seek for generality to avoid technicalities in proofs as much as possible. We assume that H and F are continuous in X, Lipschitz continuous and (semi-) coercive in p, with a time dependance allowing strong coupling with the gradient variable. Precisely, we assume that there exists a constant C > 0 such that, (1.4)

    
H is continuous and |H(X, 0)| ≤ C and |H(X, p) -H(X, q)| ≤ C|p -q| H(X, p) → +∞ as |p| → ∞ uniformly in X |H(s, x, p) -H(t, x, p)| ≤ C|t -s| (1 + max(0, H)(t, x, p))

(1.5)      F is continuous and |F (X, 0)| ≤ C and |F (X, p) -F (X, q)| ≤ C|p -q| F (X, p) → +∞ as p → -∞ uniformly in X |F (s, x, p) -F (t, x, p)| ≤ C|t -s| (1 + max(0, F, H)(t, x, p)) .
We make artificially appear the dependence of F on x ∈ Ω in order to unify the presentation of assumptions for H and F .

By regularizing the subsolution by tangential sup-convolution, and without regularizing the supersolution, we are able to prove the following new comparison principle.

Theorem 1.1 (A comparison principle with strong time coupling). Let T > 0, assume that (1.4) and (1.5) hold true and that u 0 is bounded and Lipschitz continuous. Let u : [0, T ) × Ω → R (resp. v) be a bounded upper semi-continuous viscosity subsolution (resp. bounded lower semi-continuous viscosity supersolution) of

(1.1)-(1.2). If u(0, •) ≤ u 0 ≤ v(0, •) in Ω, then u ≤ v in [0, T ) × Ω.
Remark 1.2. Here, we assume for simplicity that the initial data u 0 is Lipschitz continuous. With some additional (classical) technicalities, it is possible to deal with uniformly continuous u 0 's and to relax the boundedness assumption on u, v, u 0 by imposing that they grow at most linearly. Remark 1.3. Lipschitz continuity of F with respect to p is only used to get barriers in the proof of Theorem 1.1. Moreover, the monotonicity of F with respect to p is not used in the proof of the comparison principle. However it is required in order to ensure that classical solutions are viscosity solutions. This important and natural property is used when constructing viscosity solutions by Perron's method. Remark 1.4. Notice that, given (1.4), we can always define the state constraint boundary function

H -(t, x, p) := sup q≤p H(t, x, q) for (t, x) ∈ [0, T ] × ∂Ω and p ∈ R
and it satisfies (1.5). Up to our knowledge, the comparison principle was also an open problem for F = H - in the generality of this note. Remark 1.5. Notice that in Theorem 1.1, when F is nonincreasing in p, condition (1.5) iv) can be replaced by the weak continuity of the subsolution u on the boundary (0, T ) × ∂Ω, using [4, Proposition 3.12]) and replacing F by F 1 := max(F, H -).

Comparison with known results. J. Guerand [START_REF] Guerand | Effective nonlinear Neumann boundary conditions for 1d nonconvex Hamilton-Jacobi equations[END_REF] proved a comparison principle in our geometric setting in the case where H and F are independent of (t, x). She also proved a comparison principle for non-coercive Hamiltonians.

P.-L. Lions and P. Souganidis [START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF] introduced a new method to prove comparison principle for junctions with N ≥ 1 branches (or half-spaces) between bounded uniformly continuous sub/supersolutions. They use a blow-up argument that reduces the study to a 1D problem. The authors show the comparison principle in the case of Kirchoff-type boundary conditions and non-convex Hamiltonians. As far as (t, x) dependence is concerned, these authors can handle Hamiltonians that are Lipschitz in t, see [START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF]Assumption (4)].

This result is generalized by G. Barles and E. Chasseigne [3, Theorem 15.3.7, page 295] to the case of bounded semi-continuous sub/supersolutions under three different junction conditions. Even if they are presented for N = 2 branches, we present their results in our geometric setting: a junction reduced to a single branch N = 1 in 1D. The three cases are the following: F is constant in (1.2), the Neumann problem, and general nonincreasing continuous F . In the third case, the normal derivative is not coupled with the tangential coordinates in F (see also the very end of [3, §13.2.2 and condition (GA-G-FLT) p. 247]).

As explained above, we improve these results in the case where the functions H and F imply a strong coupling of the time variable with the normal derivative of the solution.

Organization of the note. In Section 2 we recall the definition of (limiting) semi-differentials and we state and prove a technical lemma relating some slopes (that we call critical) at the boundary with semi-differentials (Lemma 2.3). We deduce from this technical lemma a corollary about critical slopes of stationary semisolutions of the boundary value problem (Corollary 2.5). In Section 3, we state a barrier result (Lemma 3.1) that helps us dealing with the initial time condition; we also state a result (Lemma 3.2) about regularized subsolutions. In Section 4, the comparison principle (Theorem 1.1) is proved.

Acknowledgements. The authors thank G. Barles and E. Chasseigne for enlighting discussions during the preparation of this note. This research was funded, in whole or in part, by l'Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010. For the purpose of open access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

Boundary lemmas

Definition 2.1 ((Limiting) semi-differentials). For a set A ⊂ Ω = [0, +∞) and a point x 0 ∈ A, we define the (first order) super/subdifferential at x 0 of a function u in A as

D ± A u(x 0 ) = {p ∈ R, such that 0 ≤ ± {u(x 0 ) + p(x -x 0 ) + o(x -x 0 ) -u(x)} in A}
and the limiting (first order) super/subdiffential at the boundary x 0 = 0 of u in Ω as

D± Ω u(0) = p ∈ R, there exists a sequence p n ∈ D ± Ω u(x n ) with x n ∈ Ω and (x n , p n ) → (0, p) .
Remark 2.2. Note that if p ∈ D+ Ω u(0) and u is a subsolution of H(u x ) ≤ 0 in Ω, then H(p) ≤ 0.

Lemma 2.3 (Critical slopes and semi-differentials).

Let Ω := (0, +∞) and u, v : Ω → R ∪ {-∞, +∞} with u upper semi-continuous and v lower semicontinous such that u(0) = 0 = v(0) and u ≤ v in Ω. The critical slopes defined by

(2.1) p := lim sup Ω∋x→0 u(x) x , p := lim inf Ω∋x→0 v(x) x
satisfy the following (limiting) semi-differentials inclusions

(2.2) R ∩ p, p ⊂ D+ Ω u(0) ∩ D- Ω v(0) if p ≥ p (2.3) R ∩ p, p ⊂ D + Ω u(0) ∩ D - Ω v(0) if p ≤ p (2.4) p ∈ D+ Ω u(0) if p ∈ R p ∈ D- Ω v(0) if p ∈ R
Proof. We first notice that (2.3) is a straightforward consequence of the definition of semi-differentials. We now focus on the proof of

(2.5) R ∩ p, p ⊂ D+ Ω u(0) in case p > p
and will even show the following better result

(2.6) R ∩ q, p ⊂ D+ Ω u(0) in case p > q := lim inf Ω∋x→0 u(x) x
where the last inequality is assumed because u ≤ v implies q ≤ p. This claim is a variant of [8, Eq. ( 18)], and its proof is a variant of [START_REF] Barles | An Illustrated Guide of the Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities[END_REF]Lemma 15.3.1]. We give the details for the sake of completness. Notice that p ∈ (q, p) means

lim sup Ω∋x→0 u(x) x = p > p > q = lim inf Ω∋x→0 u(x) x
We deduce that for any ε > 0, there exists y ε ∈ (0, ε) and

z ε ∈ (0, y ε ) such that u(z ε ) x ε > p > u(y ε ) y ε Hence the function ζ(x) := u(x) -px satisfies ζ(0) = 0 > ζ(y ε ). Let M := sup [0,yε] ζ ≥ ζ(z ε ) > 0. Hence at a point x ε ∈ (0, y ε ) of maximum of ζ, we see that p ∈ D + Ω u(x ε ).
In the limit ε → 0, we recover p ∈ D+ Ω u(0). Then (2.6) follows from the fact that D+ Ω u(0) is closed. Then (2.5) holds true for u. The similar inclusion for v implies (2.2) in the special case where p > p. On the other hand, notice that (2.4) implies (2.2) in the case p = p. Hence it remains to show (2.4).

We now explain why the following fact holds true,

(2.7) p ∈ D- Ω v(0) if p ∈ R
This result is a property of the critical slope for any lower semi-continuous functions. Its proof follows exactly the lines of the proof of [6, Lemma 2.9]. A similar result holds for u and this proves (2.4). As a consequence of Lemma 2.3, we have Corollary 2.5 (Boundary viscosity inequalities). Let Ω and u, v be as in statement of Lemma 2.3. For γ = α, β, consider continuous functions H γ , F γ : R → R with H α coercive and F α semi-coercive. Assume that we have the following viscosity inequalities for some η > 0

(2.8)            H α (u x ) ≤ 0 on Ω ∩ {|u| < +∞} min {F α , H α } (u x ) ≤ 0 on {0} ∩ {|u| < +∞} H β (v x ) ≥ η on Ω ∩ {|v| < +∞} max {F β , H β } (v x ) ≥ η on {0} ∩ {|v| < +∞}
p ∈ [a, b] such that either H α (p) ≤ 0 < η ≤ (H β -H α )(p) or max(F α , H α )(p) ≤ 0 < η ≤ (F β -F α )(p)
Remark 2.6. Corollary 2.5 can very easily be extended to the case of junctions where the Hamiltonians H α 's are coercive on each branch and the junction function is semi-coercive in the sense of [6, Eq. (2.2)]. Remark 2.7. Under the assumptions of Corollary 2.5, we can show that the subsolution u is globally Lipschitz continuous in Ω.

Proof of Corollary 2.5. We sketch the proof that p ∈ R. Because H α is coercive and F α is semi-coercive, we know from [4, Lemma 3.8] that u is weakly continuous at x = 0, i.e.

(2.9) 0 = u(0) = lim sup

Ω∋x→0 + u(x).
Then the proof [6, Lemma 2.10] shows additionally that p > -∞. Now we claim that we also have p < +∞. Indeed, it can be seen by contradiction, leaving fall down above the graph of u on [0, y], some straight lines of slopes s = u(y) y for large positive s and using the equation satisfied by u. We conclude that p ∈ R ∩ (2.10)

   H α (a) ≤ 0 because a ∈ R 0 < η ≤ H β (b) if b ∈ R min {H α , F α } ≤ 0 < η ≤ max {H β , F β } on [a, b] ∩ R
where the last line follows from (2.3), and the first two lines follow from (2.4).

Intermediate claim. Now we assume by contradiction that there exists ε > 0 (small enough) such that (2.11)

   i) H β -H α < η -ε or ε < H α and ii) F β -F α < η or ε < max(F α , H α ) for all p ∈ [a, b] ∩ R Recall that the coercivity of H α means H α (±∞) := lim inf p→±∞ H α (p) = +∞. Case 1: H α (b) > ε
Here b can be finite or equal to +∞. We get

H α (b) > ε > 0 ≥ H α (a)
Therefore by continuity, there exists p ∈ (a, b) such that H α (p) = ε. Hence in the last line of (2.10), the first inequality implies that F α (p) ≤ 0. Because (2.11) i) and ii) hold true for p, we get

H β (p) < η and F β (p) < η
which leads to a contradiction in the last line of (2.10), the second inequality. Conclusion. We just proved that (2.11) does not hold true. This implies that for all ε > 0 small enough, there exists some

p ε ∈ [a, b] ∩ R such that we have at p ε i) H α ≤ ε < η -ε ≤ H β -H α or ii) max(F α , H α ) ≤ ε < η ≤ F β -F α
Because H α is coercive, we see in both cases i) or ii), that we can always extract a subsequence as ε → 0

such that p ε → p ∈ [a, b] ∩ R.
As a consequence, we get that p satisfies i) or ii) for ε = 0, which is the desired conclusion.

Barriers and tangential regularization

In the proof of the comparison principle, two standard results about the construction of barriers and regularization of subsolutions by sup-convolution are needed.

Lemma 3.1 (Barriers). Assume (1.4) and (1.5) and that the initial data u 0 is bounded and Lipschitz continuous. Assume that u (resp. v) is a bounded upper semi-continuous subsolution (resp. a bounded lower semi-continuous supersolution) of (1.1)-(1.2). Then there exists some constant λ > 0 such that the functions u ± (t, x) = u 0 (x) ± λt satisfy the following barrier properties:

• if u ≤ u 0 in {0} × Ω, then u ≤ u + in [0, T ) × Ω, • if v ≥ u 0 in {0} × Ω, then v ≥ u -in [0, T ) × Ω.
The previous lemma is a direct consequence of the definition of viscosity solutions if u 0 is C 1 . In the general case, it follows by a standard approximation procedure.

We now turn to the regularization of subsolutions with respect to tangential variables. Even if the proof is very standard, we will give below a short sketch of it. Lemma 3.2 (Tangential regularization of subsolutions by sup-convolution). Assume that H satisfies (1.4). Let u : [0, T ) × Ω → R be an upper semi-continuous subsolution of (1.1) which satisfies

|u -u 0 | ≤ C T in [0, T ) × Ω.
We extend u to t = T by u(T, x) := lim sup{u(s, y) : (s, y) → (T, x), (s, y) ∈ [0, T ) × Ω} and to R × Ω by,

u(t, x) = u(T, x) for t ≥ T, u(t, x) = u(0, x) for t ≤ 0.
Then for ν > 0, we the tangential sup-convolution

u ν (s, x) := sup t∈R u(t, x) - |t -s| 2 2ν = u( t, x) - | t -s| 2 2ν
and any such t (depending on (s,

x) ∈ R × Ω) satisfies t ∈ [s -θ ν , s + θ ν ] with θ ν := 2 √ νC T < T /2.
If I ν denotes the time interval (θ ν , Tθ ν ), then the function u ν is Lipschitz continuous with respect to t in R × Ω and with respect to x in I ν × Ω,

|∂ s u ν | L ∞ (R×Ω) ≤ θ ν ν and |∂ x u ν | L ∞ (I ν ×Ω) ≤ L ν with L ν := sup p ∈ R, min X∈[0,T ]×Ω H(X, p) ≤ θ ν ν .
Assume moreover that that u is a subsolution at the boundary (0, T ) × ∂Ω, i.e. satisfies the first line of (1.2), for some F that satisfies (1.5). Then u ν is Lipschitz continuous in space and time on I ν × Ω of Lipschitz constant L ν := max( θ ν ν , L ν ). Sketch of the proof. It is easy to check that uu ν ≤ 2C T which gives the bound on θ ν ≥ | t -s|. Moreover the time derivative of u ν is like t-s ν which gives the bound on ∂ s u ν . The PDE inequality satisfied by u ν gives naturally the bound on ∂ x u ν . Finally, when F satisfies (1.5), we see using [START_REF] Forcadel | Non-convex coercive Hamilton-Jacobi equations: Guerand's relaxation revisited[END_REF]Lemma 3.8]) that u (and then u ν ) is weakly continuous on I ν × ∂Ω, which implies the Lipschitz continuity of u ν in I ν × Ω. This ends the sketch of the proof.

Proof of the comparison principle

Before proving our comparison principle, we describe the main steps.

We first use the doubling variable technique with respect to time with a parameter ν > 0. This procedure can be interpreted as a sup-convolution in time of the subsolution u only (Step 1).

We then focus on the case where the supremum of u(t, x)v(s, x)correction/penalization is reached at some ( t, s, x) with x on the boundary of Ω. (Step 2). At this stage, the sup-convolution u ν of the subsolution u is L ν -Lipschitz continuous in space and time (thanks to the coercivity of the Hamiltonian), but the supersolution is only lower semi-continuous (Step 3).

We then consider twin blow-ups (Step 4): one at ( t, x) for u, and one at (s, x) for v (up to some correction terms on v). After blow-up, we get half-relaxed limits U 0 , V 0 that are globally defined on R t × Ω and satisfy U 0 (0, 0) = 0 = V 0 (0, 0). We show the following key Lipschitz inequality, (4.1)

U 0 (t, x) -V 0 (s, y) ≤ L ν |x -y| + b(t -s) with b := t - s ν
where the Lipschitz constant L ν is inherited from u ν .

Then, considering u, the supremum in time of the map t → U 0 (t, x)bt and v, the infimum in time of the map s → V 0 (s, y)bs (Step 5), we see that u * and v * are subsolution and supersolution of a 1D problem with moreover the key Lipschitz inequality

1 u * (x) -v * (y) ≤ L ν |x -y| with u * (0) = 0 = v * (0).
This procedure reduces the study to a 1D problem that is solved using the boundary viscosity inequalities from Corollary 2.5 (Step 6).

Proof of Theorem 1.1. The proof is split into several steps. The first two steps are standard and new ideas appear in the next steps.

Step 1: approximate supremum. Let η > 0. This parameter will be small enough but will not vary until we prove that the following quantity is non-positive,

M = sup t∈[0,T ),x∈Ω u(t, x) -v(t, x) - η T -t .
It turns out that

(4.2) M = lim ν→0 lim α→0 M ν,α . with M ν,α := sup t,s∈[0,T ), x∈Ω Ψ ν,α (t, s, x)= Ψ ν,α ( t, s, x)
and (with a careful choice of the penalization term

η T -s instead of η T -t ) Ψ ν,α (t, s, x) := u(t, x) -v(s, x) - η T -s -αg(x) - |t -s| 2 2ν with g(x) := x 2 2 .
Moreover all maximisers ( t, s, x) in the definition of M ν,α satisfy

(4.3) lim ν→0 lim α→0 | t -s| 2 ν = 0, lim ν→0 lim α→0 αg(x) = 0, lim sup ν→0 lim sup α→0 η T -s ≤ 2C T .
Step 2: Reduction to the case where the supremum is reached at the boundary.

Using the doubling variable technique with respect to x for u and v and considering u(t, x), v(s, y) with a further penalization term of the form |x-y| 2 2δ , we can rely on barrier estimates close to t = 0 to get estimates on maximum points ( tδ , xδ , sδ , ȳδ ) → ( t, x, s, x) as δ → 0 (for some subsequence δ and some suitable limit ( t, s, x)). We deduce that in the limit, the following fact holds true,

(4.4) t, s ∈ [τ η , T -τ η ], x ∈ [0, ρ α ], t - s ν , p ∈ D1,+ t,x u( t, x), |p| ≤ L ν
where L ν is the Lipschitz constant of u ν in Lemma 3.2. Moreover, it is possible to choose τ η depending on η only and ρ α depending on α only. 2

1 Notice that without the Lν -Lipschitz inequality (4.1), we would only get u ≤ v and u(0) = 0 = v(0), which is not sufficient to conclude.

2 Up to increase λ and C T in the barrier Lemma 3.1, and to decrease η in the time penalisation term, we can assume that λT = C T and it is possible to show in this case that we can choose τη := η

4C T and ρα = 6C T α .
If x > 0, then we are in the classical case where we can conclude by writing viscosity inequalities for the sequence ( tδ , xδ , sδ , ȳδ ) and by combining them in the classical way. Classical details for this step are given in the companion paper. We are thus reduced to deal with the case where x = 0.

Step 3: the key Lipschitz estimate. Following Lemma 3.2, we extend u and consider

     U ν (s, x) = sup t∈R u(t, x) - |t -s| 2 2ν V (s, x) = v(s, x) + η T -s + αg(x)
and there exists some (possibly non unique) ts

∈ [s -θ ν , s + θ ν ] such that U ν (s, x) = u( ts , x) -| ts-s| 2 2ν . If s ∈ (θ ν , T -θ ν
), then we see that ts ∈ (0, T ) and then we also have

U ν (s, x) := sup t∈[0,T ) u(t, x) - |t -s| 2 2ν .
In particular for (s, x) = (s, x), we can choose ts = t where ts is given by Lemma 3.2 and ( t, s, x) appear in Steps 1 and 2. Now we choose ν > 0 small enough such that θ ν < τ η and we set I ν := (θ ν , Tθ ν ). Moreover we have

u(t, x) - |t -s| 2 2ν -V (s, x) = Ψ ν,α (t, s, x) ≤ Ψ ν,α ( t, s, x) = U ν (s, x) -V (s, x)
and then taking the supremum in t ∈ R, we get for all s ∈ I ν , x, y ∈ Ω

(4.5) U ν (s, y) -V (s, y) ≤ U ν (s, x) -V (s, x) U ν (s, x) -U ν (s, y) ≤ L ν |x -y|
where the second line follows from the fact that U ν is L ν -Lipschitz continuous (see Lemma 3.2). Notice also that we have

           ∂ t u + H(t, x, ∂ x u) ≤ 0 in (0, T ) × Ω ∂ t u + min(F, H)(t, x, ∂ x u) ≤ 0 in (0, T ) × ∂Ω -η (T -s) 2 + ∂ s V + H(s, x, ∂ x V -α∂ x g) ≥ 0 in (0, T ) × Ω -η (T -s) 2 + ∂ s V + max(F, H)(s, x, ∂ x V -α∂ x g) ≥ 0 in (0, T ) × ∂Ω.
Step 4: the twin blow-ups.

We then consider the following twin blow-ups with small parameter ε > 0: one blow-up for u at the point ( t, x) and one blow-up for V at the point (s, x),

(4.6) U ε (τ, ξ) := ε -1 {u( t + ετ, x + εξ) -u( t, x)} , U ε (0, 0) = 0, V ε (σ, ξ) := ε -1 {V (s + εσ, x + εξ) -V (s, x)} , V ε (0, 0) = 0.
Before passing to the limit ε → 0, they satisfy (4.7)

           ∂ τ U ε + H( t + ετ, x + εξ, ∂ ξ U ε ) ≤ 0 in I ε t × Ω ∂ τ U ε + min(F, H)( t + ετ, x + εξ, ∂ ξ U ε ) ≤ 0 in I ε t × ∂Ω -η ε + ∂ σ V ε + H(s + εσ, x + εξ, ∂ ξ V ε -α∂ x g(x + εξ)) ≥ 0 in I ε s × Ω -η ε + ∂ σ V ε + max(F, H)(s + εσ, x + εξ, ∂ ξ V ε -α∂ x g(x + εξ)) ≥ 0 in I ε s × ∂Ω with ηε (σ) := η (T -(s + εσ)) 2 and I ε r := - r ε , T - r ε for r = t, s.
Because U ν is globally L ν -Lipschitz in space and time, it is also interesting to consider the auxiliary blow-up for U ν at a point (s, x),

U ν,ε (σ, ξ) := ε -1 {U ν (s + εσ, x + εξ) -U ν (s, x)} , U ν,ε (0, 0) = 0.
This auxiliary function is indeed intimately related to U ε since we have,

(4.8) U ν,ε (σ, ξ) = sup τ ∈R {U ε (τ, ξ) -A ε (τ -σ)} , A ε (q) = bq + ε |q| 2 2ν , b := t - s ν .
Moreover from (4.5), we have for all σ ∈ I ε s and ξ, ζ ∈ Ω,

U ν,ε (σ, ζ) -V ε (σ, ζ) ≤ 0, U ν,ε (σ, ξ) -U ν,ε (σ, ζ) ≤ L ν |ξ -ζ| and then for all σ ∈ I ε s , τ ∈ R and ξ, ζ ∈ Ω, (4.9) U ν,ε (σ, ξ) -V ε (σ, ζ) ≤ L ν |ξ -ζ| U ν,ε (σ, ξ) ≥ U ε (τ, ξ) -A ε (τ -σ)
where the last line follows from (4.8). Thanks to the uniform Lipschitz estimate for U ν and the fact that U ν,ε (0, 0) = 0, we can extract a subsequence (still denoted by ε → 0) and get, U ν,ε → U ν,0 locally uniformly on compact sets of R × Ω, U ν,0 (0, 0) = 0.

We then define (along the already extracted subsequence) the following half-relaxed limits

   U 0 := lim sup ε→0 * U ε , U 0 (0, 0) ≥ 0, V 0 := lim inf ε→0 * V ε , V 0 (0, 0) ≤ 0.
Passing to the limit in (4.9), we get with A 0 (q) = bq

U ν,0 (σ, ξ) -V 0 (σ, ζ) ≤ L ν |ξ -ζ| U ν,0 (σ, ξ) ≥ U 0 (τ, ξ) -b(τ -σ)
which shows in particular that (4.10)

U 0 (τ, ξ) -V 0 (σ, ζ) ≤ L ν |ξ -ζ| + b(τ -σ), U 0 (0, 0) = 0 = V 0 (0, 0).
Moreover, passing to the limit in (4.7) thanks to the discontinuous stability of viscosity solutions, we get (4.11)

           ∂ τ U 0 + H( t, x, ∂ ξ U 0 ) ≤ 0 in (R × Ω) ∩ |U 0 | < +∞ ∂ τ U 0 + min(F, H)( t, x, ∂ ξ U 0 ) ≤ 0 in (R × ∂Ω) ∩ |U 0 | < +∞ -η + ∂ σ V 0 + H(s, x, ∂ ξ V 0 ) ≥ 0 in (R × Ω) ∩ |V 0 | < +∞ -η + ∂ σ V 0 + max(F, H)(s, x, ∂ ξ V 0 ) ≥ 0 in (R × ∂Ω) ∩ |V 0 | < +∞ with η := η (T -s) 2
. We used the fact that α∂ x g(x) = αx = 0. From (4.11), we get (again from stability) that these functions satisfy in particular for X := ( t, x) and Ȳ := (s, x)

(4.13)            b + H( X, ∂ ξ u * ) ≤ 0 in (R × Ω) ∩ {|u * | < +∞} b + min(F, H)( X, ∂ ξ u * ) ≤ 0 in (R × ∂Ω) ∩ {|u * | < +∞} -η + b + H( Ȳ , ∂ ξ v * ) ≥ 0 in (R × Ω) ∩ {|v * | < +∞} -η + b + max(F, H)( Ȳ , ∂ ξ v * ) ≥ 0 in (R × ∂Ω) ∩ {|v * | < +∞} .
Step 6: getting a contradiction from structural assumptions. We now apply Corollary 2.5. In order to do so, we consider From the assumption on the function F , see (1.5) ii), we get a similar contradiction in the second case, η ≤ F ( Ȳ , p) -F ( X, p) ≤ C| t -s| 1 + max(0, max(F, H)( X, p))

≤ C| t -s| {1 + max(0, -b)} ≤ C | t -s| 2 ν + | t -s| → 0 as ν → 0.
We conclude that M ≤ 0. Recalling that

M = sup t∈[0,T ),x∈Ω u(t, x) -v(t, x) - η T -t ≤ 0,
it is enough to let η → 0 to get u ≤ v as desired.

Definition 2 . 4 (

 24 Coercive and semi-coercive functions). Consider a function G : R → R. Then G is coercice if lim |p|→+∞ G(p) = +∞, and semi-coercive if lim p→-∞ G(p) = +∞.

For

  p, p defined in (2.1), we set a := min(p, p) and b := max(p, p). Then p ∈ [a, b] ∩ R and there exists a real number

  [a, b]. Assume first that p ≤ p. Here (2.2) shows that H α (p) ≤ 0 < η ≤ H β (p) and then η ≤ (H β -H α )(p) for all p ∈ p, p ∩ R which implies in particular the desired conclusion. If p > p, then we have [a, b] ⊂ (-∞, +∞] with a < b and

Case 2 :

 2 H α (b) ≤ ε Then b ∈ R and (2.11) i) implies for p = b that H β (b) < η, which is in contradiction with the second line of (2.10).

Step 5 :

 5 the 1D problem. We now define the following functions as supremum/infimum in time of the functions defined in R × Ω,u(ξ) := sup τ ∈R U 0 (τ, ξ)bτ , v(ξ) := inf τ ∈R V 0 (τ, ξ)bτ .From (4.10), these functions satisfy-∞ ≤ -L ν |ζ -ξ| + u(ξ) ≤ v(ζ) ≤ +∞, 0 ≤ u(0) ≤ v(0) ≤ 0and then u(0) = 0 = v(0). Because of this Lipschitz inequality, this is also the case for their semi-continuous envelopes, i.e. we have (4.12) -∞ ≤ -L ν |ζ -ξ| + u * (ξ) ≤ v * (ζ) ≤ +∞, u * (0) = 0 = v * (0).

p

  := lim sup Ω∋x→0 u * (x) x , p := lim inf Ω∋x→0 v * (x) x , a := min(p, p), b := max(p, p) and we get that there exists p ∈ [a, b]∩ R = ∅ such that either b + H( X, p) ≤ 0 < η ≤ H( Ȳ , p) -H( X, p) or b + max(F, H)( X, p) ≤ 0 < η ≤ F ( Ȳ , p) -F ( X, p).One of these facts are true along a subsequence ν → 0. In the first case, we get from the assumption on the Hamiltonian H, see (1.4) ii), that,η ≤ H( Ȳ , p) -H( X, p) ≤ C| t -s| 1 + max(0, H( X, p)) ≤ C| t -s| {1 + max(0, -b)} ≤ C | t -s| 2 ν + | t -s| → 0 as ν → 0where we have used b = ts ν in the third line, and (4.3) in the last line. Contradiction because η ≥ η/T 2 > 0.