Grégoire Rota

Grégoire Rota Graziosi
email: gregoire.rota_graziosi@u-clermont1.fr

Economic integration and political fragmentation

Keywords: Economic integration and political fragmentation economic integration, public good, secession, vote. JEL : D2, H2, H7

published or not. The documents may come

Introduction

In recent times independantist movements have appeared in some countries such as in Spain, Belgium or Canada and further regional autonomy has been demanded for example in Scotland or Catalogna. Nation-State seems today to be threatened. At the same time markets became worldwide. These two trends have brought some authors to study the relationship between economic integration and political fragmentation. Theoretical literature appears divided on this issue. On one hand, some authors in particular in political science as Young (1995) consider that different degrees of economic integration are compatible with different institutional rules as voluntary harmonization, consultative organisms, international negociations, legislative or executive instances... This author concludes then that there is no relation between economic and political integration. On the other hand, economic literature has recognized early the link between these two processes.

For instance, List conceived two centuries ago customs union as a first step towards the building of the German State. 1More recently, a literature has been built on the original analysis by [START_REF] Buchanan | Secession and the Limits of Taxation: Towards a Theory of Internal Exit[END_REF] of the possibility of 'internal exit' through secession. Unfortunately, the effects of economic integration on the cost of secession are not studied in this seminal article, since the authors consider incorrectly individual incomes as exogenous. 2 Following the works of Alesina and Spolaore (1997) or [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF] on political fragmentation, 3 Alesina, Spolaore, and Wacziarg (2000), [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] and [START_REF] Leite-Monteiro | Economic integration and fiscal devolution[END_REF] focus explicitly on the relationship between economic integration and political integration. On one hand, [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] assume a production economy where every agent owns a specific input to produce the final good.

The cost of heterogeneity increases with the size of the country. For these authors, inter-national trade liberalization improves individual income but reduces optimal country size. [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] obtains a similar result by developing a model of coalition formation. The author establishes that the optimal number of coalitions rises with the size of the market.

These two first articles conclude unambiguously that trade openness favors secessions or at least political decentralization. On the other hand, [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF] and [START_REF] Leite-Monteiro | Economic integration and fiscal devolution[END_REF] establish a non monotone relationship between economic integration and political one. [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF] show that individuals prefer political separation into different jurisdictions for smaller or larger size of market. Focusing on incomes redistribution, [START_REF] Leite-Monteiro | Economic integration and fiscal devolution[END_REF] use a model of fiscal competition and find that the incentives of political separation are decreasing and then increasing in the capital mobility costs.

In the line of these preceding articles, we examine the interactions between economic integration and political fragmentation. However, all the above quoted authors consider as exogenous the change of economic integration, which might resut from a decrease in the capital mobility costs or from an increase in the market size. 4 The main originality of our work is to consider that economic integration and politic disintegration are both collectively determined through a simple majority vote. If we admit as [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] or [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] that economic integration involves political disintegration, we wonder here if an opposition to economic openness would emerge motivated by the fear of secession. In other words, the political consequences of economic integration might be such that autarky, or in a more moderate way some protectionist barriers, would be choosen even if it means a lower private income for a majority of agents in order to maintain the political unity of the country and to insure a more advantageous provision of national public goods.

Following [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] or [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], we use the model of Salop (1979), which conforms to the more realistic formulation of differentiation from Hotelling or Lancaster. 5 The heterogenity of the population is characterised by a circle and we consider every national government as a local public good. Following [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF], [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] or [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], individual incomes result from exchanges between traders. There are two types of market: one national and one international.

For the purpose of this paper, we examine only two polar cases: autarky, as set againts complete economic integration. In autarky the market is only national and individual revenue results from the unique domestic market. Economic integration allows agents to participate within the international market. We conclude that autarky might be prefered when the productivity on the international market is insufficiently low with respect to the productivity on the domestic one.

The paper is organized as follows: the next section describes the framework and the useful assumptions; in section 3 we define critical value of parameters for political fragmentation; section 4 examines the consequences of economic integration; section 5 concludes.

The framework

We present initially the setup of the studied game. Then, we describe our world representation, the individual incomes formalization and we deduce the individual preferences.

The Setup of the game

We consider a three-stage game. We adopt the following timing for decision making:

• stage 1: inhabitants vote for the degree of trade openness;

• stage 2: a vote in the studied peripheral region determines or not if secession occurs;

5 [START_REF] Krugman | Industrial organization and international trade[END_REF] writes : "The Hotelling-Lancaster formulation has the advantage of greater realism, and leads to somewhat more plausible formulation of the nature of the gains from trade. However, it is quite difficult to work with. The Spence-Dixit-Stiglitz approach, by contrast, while less convincing, lends itself quite easily to modelling."

• stage 3: inhabitants of the relevant jurisdiction choose the location of the unique public good. This sequence of decision allows us to go a step further than the analysis of [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF]. By applying backward induction, we are then able to consider the political consequences of trade openness on the country unity. Following [START_REF] Buchanan | Secession and the Limits of Taxation: Towards a Theory of Internal Exit[END_REF] or [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF], we admit in the second step regional autodetermination principle: a region can become unilaterally independant. The game ends by the vote of the location of the unique public good in the relevant jurisdiction.

Democracy is here direct. There is neither information asymmetry between those who govern and those who are governed nor lobbying groups which would influence rulers.

World representation

We represent population heterogeneity by a circle with a unitary radius. For every agent there is a corresponding angle, denoted by x (x ∈ [0, 2π]). By assumption, individuals are uniformly distributed along the circle, their total population is 2π. The world is shared in n countries (indexed by i). Each country (i) is composed of m i regions. Following [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF], we assume a fixed partition of the world, borders are historically determined. In others words, national borders (b i-1 and b i for the country i, i = 1, ..., n) and regional borders (a ij-1 and a ij for region j of country i, j = 1, ..., m i ) are exogeneous. These assumptions constrast with [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] in which secession is choosen by unanimity. We suppose: b 0 = a 10 = b n = a nmn = 0.

Insert Figure 1 For tractability, we limit countries size to π. There exist at least two countries in the world, moreover we restrict our analysis to peripheric regions. This kind of regions is defined here by an international border, shared with the "host" country. Besides their size are supposed relatively small. More accurately, we study the behaviour of country 1 and its region 1. Their respective sizes are denoted by: s 1 = b for country 1 and s 11 = a for region 1, with by assumption a < b 2 . In the rest of the paper, we will focus only on individuals located between 0 and b.

By assumption, agents are immobile. There exist no fiscal competition between the countries. In an European context, this hypothesis might be justified by the high level of mobility costs induced by imperfections on the job market or on the real-estate market (see [START_REF] Faini | An empirical puzzle: falling migration and growing unemployment differentials among Italian regions[END_REF] for an empirical study on this issue). Another justification is cultural particularisms which reduce interregional mobility. The assumption of individual immobility corresponds to the exogeneity of individual preferences too, which is essential for the spatial model of voting (see [START_REF] Hinich | The spatial theory of voting. An introduction[END_REF]).

Individual incomes

Every agent is initialy endowed with two units of a specific good. As [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] or [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], this dotation is not productive in itself. Income results from a commercial matching between agents. There are two matches for each agent. The income of the individual x is given by:

w (b, β; x) = (1 + β) w d (b; x) + (1 -β) w e (b; x) ,
where w d (b; x) and w e (b; x) are respectively the individual incomes on national and international market. The parameter β measures the level of country economic integration: β = 1 involves autarky; β = 0 means complete economic integration. The degree of openness or of protectionism (β) results from a majority vote at the first step of the game.

We do not consider international negociations on β. Moreover, there is no trade retaliations among the new created countries in case of secession. Thus, in autarky individuals trade only on the domestic (or national) market. With complete economic integration one match is held in the national market, the other in the international market.

We assume a transaction cost increasing in the distance between two commmercial partners. An agent (x) knows exaclty her location on the circle but she ignores the place of her commercial partner, denoted by v. Thus, the net gain of their transactions is uncertain. Each agent is only aware on which market they trade. So, she knows if her commercial partner, denoted by v, leaves her country

(v ∈ [0, b]) or if she is a foreigner (v / ∈ [0, b]
). The individual income on the domestic market is given by:

w d (b; x) = ω d b (1 -E [d (x, v) |v ∈ [0, b]]) , (1)
where ω d evaluates the productivity of the domestic market. A similar argument defines individual income on the international market, denoted by w e (b; x). The size of this market is 2πb and ω e is its productivity. We have:

w e (b; x) = ω e (2π -b) (1 -E [d (x, v) |v / ∈ [0, b]]) , (2)
Moreover, we assume that:

ω e > ω d > 0. (3)
For a given distance between two trading partners, international exchanges are more productive that domestic ones. This hypothesis creates an individual incitation to economic integration. We consider a quadratic normalized distance

(d(x, v) = ¡ x-v π ¢ 2
). Since we assume a uniform distribution along the circle, the expected value of distance on the two markets (domestic and interntional) yields to: 6

E [d (x, v) |v ∈ [0, b]] = b Z 0 1 b µ x -v π ¶ 2 dv, E [d (x, v) |v / ∈ [0, b]] = p (v ∈ [-π + x, 0]) E [d (x, v) |v ∈ [-π + x, 0]] +p (v ∈ [b, x + π]) E [d (x, v) |v ∈ [b, x + π]] ,
where

p (v ∈ [-π + x, 0]) = π-x 2π-b and p (v ∈ [b, x + π]) = π+x-b 2π-b . In Appendix A.1, we
6 Given the assumptions of our model, we notice that the individual income is symmetric with respect to b 2 :

∀ε ∈ • 0, b 2 ¸, w µ b, β; b 2 -ε ¶ = w µ b, β; b 2 + ε ¶ (4)
establish that domestic individual revenues are increasing with the size of the country (see relation ( 17)), while international ones are decreasing (see expression ( 18)). We highlight an advantage to large countries when tariff barriers are high and a greater incentive to openness when the country is small. Our assumptions yield to the same result as [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF].

However, in contrast to [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF], [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] or [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], our incomes formulation is such that optimal distance between two traders is nil. For an identical productivity of domestic and international exchanges, people prefers to trade with their immediate neighbours than with more distant agents. Inhabitants leaving close to the center of the country (b/2) have then the highest domestic market incomes, whilst their international market incomes are lowest. Our assumption induces an incitative for inhabitants in peripheral regions to support economic integration.

The variations of the whole individual revenue (w (b, β; x)) with respect to the size of the country (b) and its openness are then given by:

∂w (b, β; x) ∂b = Ã 1 - (b -x) 2 π 2 ! £ (1 + β) ω d -(1 -β) ω e ¤ . ( 5
)
We deduce that:

∂w (b, β; x) ∂b > 0 ⇔ β > ω e -ω d ω e + ω d or equivalently ω d ω e > 1 -β 1 + β = ω c 1 (β) . (6)
Under condition (3), the size of the country increases individual income if and only if openness is sufficiently low (or β sufficiently high). Moreover, we note that under autarky, the income is obviously increasing in the size of the country

³ ∂w(b,1;x) ∂b > 0 ´. If
we consider the variation of individual income with respect to openness (β), we have:

∂w(b,β;x) ∂β = w d (b; x) -w e (b; x).
Given the assumptions of our framework, it appears that each individual would be in favor of free trade (β = 0) as soon as international incomes exceed domestic ones. However, this result does not take into account the consequences of economic integration over political unity, and then over the national policies. More-over, we notice that:

∂ 2 w(b,β;x) ∂b∂β = ³ 1 -(b-x) 2 π 2 ´¡ω d + ω e ¢ > 0.
More protectionnist is the country, more important will be the effect of its size on individual incomes. In other words, openness and country size appear here as strategic complements. Finally, under our assumptions, we observe that economic integration reduces individual incomes inequalities. 7 Incomes are even equal when economic integration is complete (β = 0) and the productivity parameters identical (ω d = ω e ).

Preferences

As Alesina and Spolaore (1997) and [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF], we suppose that private and public goods are substitute. All individuals have the same preferences summarized in the utility function of agent (x):

υ (c p , c b ; x) = c p (x) + c b (x)
, where c p (x) is private consumption and c b (x) public consumption. Individuals consume all their revenues. Private consumption is then equal to available income:

c p (x) = C p (t, b, w; x) = (1 -t)w (b, β; x) , (7)
where t is the uniform tax rate.

Every country is represented by an unique public good called government, a term which we identify a group of administrative, judicial and economic services. Following Tiebout (1956), we suppose the public good locally produced in a single place, the capital (y ∈ [0, b]). As Alesina and Spolaore (1997), we assume the individual utility of the public good as decreasing with the distance from their ideal point, their place on the circle. Thus, public consumption is given by:

c b (x) = C b (y, g, b; x) = g (1 -d (x, y)) ,
where g is the maximum individual utility of the public good, when d (x, y) = 0. As [START_REF] Buchanan | Secession and the Limits of Taxation: Towards a Theory of Internal Exit[END_REF] or Alesina and Spolaore (1997), public good quantity is fixed.

Its cost is constant, equal to K. This assumption involves some scale economies in the furniture of public good, which yields to a centripetal force in opposition to the centrifugal force, due to the population heterogenity. Moreover, to keep the model workable we assume: K = bg. Balanced budget constraint yields to: K = btw (b, β), where w (b, β) is the average income, equal to: 8

w (b, β) ≡ 1 b b Z 0 £ (1 + β) w d (b; x) + (1 -β) w e (b; x) ¤ dx. ( 8
)
We deduce that the tax rate is equal to:

t (b, β) = g w (b, β) . ( 9
)
Differentiating the tax rate with respect to the country size (b) yields to:

∂t (b, β) ∂b = - g w (b, β) 2 µ ∂w (b, β) ∂b ¶ .
We establish that the tax rate decreases with the size of the country if the relative productivity of international exchanges is sufficiently high (see expression (24) in Appendix A.1):

∂t (b, β) ∂b < 0 ⇔ ∂w (b, β) ∂b > 0 ⇔ ω d ω e > 1 -β 1 + β ≡ ω c 1 (β) , (10)
and

∂t (b, β) ∂β > 0 ⇐⇒ ω d ω e > 8π 3 6bπ 2 -b 3 -1 = ω c 2 (b) . ( 11
)
The tax burden increases in the degree of trade barriers (β) if and only if the relative productivity of international exchanges is insufficiently low. 9

8 We notice that the mean income is always increasing in the trade openness (decreasing in β). 9 Owing to the tax rate (t) belongs to the intervall [0, 1], we have an additional condition on the level of g, which we will assume respected in the rest of the paper: g < g (b, β) = bw (b, β) .

Equilibrium determination

We turn now to the resolution of our game. Applying backward inducion, we first determine the location of the public good, then we establish if secession occurs or not and we end by determining the voted degree of trade openness.

Public good location

Applying backward induction, we consider now the third stage of the game: the location of the public good, in other terms the capital place, denoted by y. Under condition (9), the indirect utility function is given by:

V (y, b, g, β; x) = (1 -t (b, β)) w(b, β; x) + g (1 -d (x, y))
The choice of the capital for individual x is then given by the following maximization program:

y * (x) ≡ arg max y∈[0,b]
V (y, b, g, β; x). The First Order Condition (FOC) involves:

y * (x) = x.
The concavity of V (.) with respect to y allows us to determine the Condorcet winner (y M ) by applying the Median Voter Theorem. 10 We deduce that: y M = b 2 . After substituting y by its equilibrium value, the indirect utility function, denoted by U (b, g, β; x), is then given by:

U (b, g, β; x) = (1 -t (b, β)) w(b, β; x) + g µ 1 -d µ x, b 2 ¶ ¶ (12)
The population heterogeneity is double. Indeed, agents differentiate not only by their private incomes (as in [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF]), but also by their appreciation of the public good (as in Alesina and Spolaore (1997)). However, following [START_REF] Bolton | Economic theories of the break-up and integration of nations[END_REF], our formulation reduces population heterogeneity to one dimension: the individual place on the circle. 10 The conditions of unidimensionality and unimodality are respected.

Political fragmentation or not

We now describe the second stage of the game, in which voters choose regional independance or not. We apply the same argument as in the preceding section when region 1 ([0, a]) becomes independant.11 With independance the capital moves from b/2 to a/2 in the center of the region 1, so y * = a 2 . The indirect utility of an inhabitant (x) of the region 1 turns into:

U (a, g, β; x) = (1 -t (a, β)) w(a, β; x) + g ³ 1 -d ³ x, a 2
´´.

For tractability, we assume that the rest of country 1 remains close after the secession of region 1. So the new capital locates in a+b 2 and the size of this new country is ba. The utility function of individual x located between a and b is then given by:

U (b -a, g, β; x) = (1 -t (b -a, β)) w(b -a, β; x) + g µ 1 -d µ x, a + b 2 ¶ ¶ .
In order to determine when the secession occurs, we define a function, denoted by D (a, b, g, β; x), which is the difference in utility terms between the two political status (separation and union) for individual x. We have:

D (a, b, g, β; x) = U (a, g, β; x) -U (b, g, β; x) .
An appropriate decomposition of D (a, b, g, β; x) allows us to capture three distinct effects of secession:

D (a, b, g, β; x) = (1 -t (b, β)) (w (a, β; x) -w (b, β; x)) | {z } E I (a,b,β;x) + w (a, β; x) (t (b, β) -t (a, β)) | {z } E F (a,b,β;x) +g µ d µ x, b 2 ¶ -d ³ x, a 2 ´ ¶ | {z } E P (a,b;x)
Indeed, we distinguish:

• An income effect, denoted by E I (a, b, β; x): individual domestic incomes increase with the size of the country, whilst individual international incomes decrease. The result remains ambiguous and depends on ω e ω d (≷ ω c 1 (β)). More formally, we know from (6) that:

if β > ω e -ω d ω e + ω d or equivalently if ω d ω e > ω c 1 (β) , then E I (a, b, β; x) < 0. ( 13
)
• A fiscal effect (E F (a, b, β; x)): the assumption of a fixed quantity of public good involves a rising of the taxes when the fiscal base contracts with secession. Expression (10) yields to:

if β > ω e -ω d ω e + ω d or equivalently if ω d ω e > ω c 1 (β) , then E F (a, b, β; x) < 0. (14)
• A political effect (E P (a, b; x)): this outcome results from the move of the capital and depends on the location of the individual (x). We remark that this effect is not affected by the openness of the country (β). We have:

x ∈ £ 0, min © a+b 4 , a ª¤ , E P (a, b; x) > 0, x ∈ £ min © a+b 4 , a ª , max © a+b 4 , a ª¤ , E P (a, b; x) 6 0. ( 15
)
If a < b/3, 12 secession provide to all inhabitants of region 1 a policy more satisfactory

¡ E P (a, b; x) > 0 ¢ .
By the first effect, we emphasize that political fragmentation has an impact on private individual incomes. This link was ignored by [START_REF] Buchanan | Secession and the Limits of Taxation: Towards a Theory of Internal Exit[END_REF]. However, while in the corrected version of [START_REF] Buchanan | Secession and the Limits of Taxation: Towards a Theory of Internal Exit[END_REF] 13 secession always involves a decrease in private income as in [START_REF] Bolton | The Break-Up of Nations: A Political Economy Analysis[END_REF], we have here an ambiguous 12 For a < b/3, we have: a < a+b 4 or equivalently d [START_REF] Rota Graziosi | The Theory of 'Internal Exit', a Comment on Buchanan and Faith[END_REF]. relation between political fragmentation and individual incomes due to the effects of international trade. The second and third effects yield to a similar trade-off as developped by Alesina and Spolaore (1997). The heterogenity of the population in the consumption of public good acts as a centrifugal force, while the scale economies in the provision of this public good involves a centripetal force. The sum of these three effects induces then the regional majority to choose the independance or not.

¡ a, a 2 ¢ < d ¡ a, b 2 ¢ . 13 See Rota
In order to highlight the combinaison of the three effects, we focus on the two polar cases: autarky (β = 1) and complete integration (β = 0). In autarky, agents based near the border are not only the poorest, but they benefit the least from the public good. We remark that the assumption of an uniform tax involves a positive relationship between individual income and fiscal obligations, thus a negative link between taxes and the distance from the capital. This effect mitigates the centrifugal force due to the heterogeneity of the population. We deduce a first result Lemma 1 No secession occurs in autarky (D (a, b, g, 1; x) < 0).

Proof: see Appendix A.2.
In autarky (β = 1) political fragmentation always involves a reduction of the individual income (E I (a, b, β; x) < 0). Combined with the rise of the taxation (E F (a, b, β; x) < 0) this outcome exceeds the possible advantage of separation, i.e. a closer government (E P (a, b; x) > 0). Thus autarky insures here political stability.

We consider now the situation in complete economic integration (β = 0). From ( 13), ( 14) and ( 15), we have the following lemma:

Lemma 2 Free trade involves secession.

Proof: since we have assumed ω e > ω d , conditions ( 13) and ( 14) yield to E I (a, b, 0; x) > 0 and E F (a, b, 0; x) > 0. For a majority at least of inhabitants in region 1 ( £ 0, a 2 ¤ ), we observe that the political effect is positive: E P (a, b, 0; x) > 0. Since the two others effects are favorable too, independance is unilaterally wished by a majority of region 1.

A sufficient condition under which economic integration favors policital fragmentation is that the productivity of international exchanges are greater than this of domestic ones (ω e > ω d ). As Alesina and Spolaore (1997), [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] and [START_REF] Casella | The Role of Market Size in the Formation of Jurisdictions[END_REF] we show that economic integration reduces the cost of the regional independance and allows secession.

Which degree of trade openness

We now turn to the last stage of the game by considering the choice of trade openness.

The maximization program of each individual utility level with respect to trade openness is given by:

∀x ∈ [0, a] , β * (x) ≡ arg max β∈[0,1] (max {U (b, g, β; x) , U (a, g, β; x)}) ∀x ∈ [a, b] , β * (x) ≡ arg max β∈[0,1] (max {U (b, g, β; x) , U (b -a, g, β; x)}) (16)
The discontinuity of the objective functions does not allow us to establish an explicit solution of this program. We restrict ourselves to determine sufficient conditions that yield to unambiguous conclusions. We focus then on corner solutions of β (= 0, 1), which corresponds to free trade or autarky. If the FOC of ( 16) is strictly negative (positive), then individual located at x will be in favor of free trade (autarky), even if free trade involves political fragmentation. We obtain the following proposition :

Proposition 1 Under our assumptions, we observe that: (i) a majority in country 1 will choose economic integration, which involves political fragmentation, if the following sufficient condition holds:

ω d ω e < ω c 3 (b) ,
(ii) a majority in country 1 will choose autarky, and then political union remains, if the following sufficient condition holds:

ω d ω e > ω c 2 (a) . Proof: see Appendix A.3.
Economic integration is not systematically choosen. Indeed, the relative productivity of the international trade must be sufficiently high (ω e > ω d /ω c 3 (b)). We do not examine individual preferences when the ratio ω d /ω e varies between ω c 3 (b) and ω c 2 (a). However, we illustrate our results in the following Figure.

Insert Figure 2 In this Figure, we represent the utility levels of four critical inhabitants, respectively located at x = 0, a, a+b 2 and b, when trade openness (β) varies from 0 to 0.4. Since the individual utility level depends on the occurence of secession, the curves are kinked in β.

Indeed, there exists a critical value of β c (x), for which individual x prefers secession to political unity. If the country is sufficiently large (b = 4π/5) as in the first graphic of Figure 2, we notice that all curves are decreasing in β. This illustrates the situation where inhabitants vote for free trade and secession occurs. In the second graphic of Figure 2, we highlight a possible conflict among country inhabitants. Indeed, while individuals located close to the borders (x = 0 or b) wish free trade in order to reduce the cost of secession, others inhabitants located close to country center (x = b/2 or (a + b) /2) support autarky.

It appears then that the initial conditions concerning trade openness matter.

The second set of graphics represents the utility levels of inhabitants for different value of trade openness.

Insert Figure 3 The gray curve is the utility level when the country remains united, while the black one represents the utility level in case of regional secession. We consider four critical values of β (respectively 0, 0.32, 0.335 and 1). The top-left graphic is an illustration of Lemma 1 and corresponds to the autarky case (β = 1). We observe that the black curve is always below the gray one: all country inhabitants prefer political union to secession under autarky. Similarly, the top-right graphic provides an example of Lemma 2. Under complete economic integration (β = 0), the black curves are always above the gray one, which means that all inhabitants prefer secession under free trade. The two last graphics (β = 0.32 or 0.335) show a potential conflict among country inhabitants. Two distinct groups emerge: one in favor of political unity is composed by individuals leaving close to the center; the other which supports secession leaves in the peripheral areas. If this last group has the majority in the region, then political separation occurs.

Conclusion

The objective of this paper was to examine the interaction between economic integration and political fragmentation. As Alesina and Spolaore (1997), [START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] or [START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], we have established that the openness of trade barriers allows secession. But owing to this relation, autarky is sometimes preferred to economic integration by a majority of voters, in order to keep the political unity of the country.

A Appendix

A.1 Differentiations and analytical expressions

Expression ( 1) is equal to:

w d (b; x) = ω d b 3π 2 ¡ 3π 2 -b 2 + 3bx -3x 2 ¢ .
From (1), we deduce that:

∂w d (b; x) ∂b = ω d µ 1 - b 2 2π 2 ¶ > 0. (17)
Expression ( 2) is equivalent to:

w e (b; x) = ω e b ¡ 4π 3 -3π 2 b + b 3 -3b 2 x + 3bx 2 ¢ 6π 2 (2π -b) .
It yields to:

∂w e (b; x) ∂b = -ω e µ 1 - b 2 2π 2 ¶ < 0. ( 18
)
Individual income is then given by:

w (b, β; x) = b ¡ 3π 2 -b 2 + 3x (b -x) ¢ (1 + β) ω d + ¡ 4π 3 + b 3 -3b ¡ π 2 -x 2 + bx ¢¢ (1 -β) ω e 3π 2 .
We have:

∂w (b, β; x) ∂β = b ¡ 3π 2 -b 2 + 3x (b -x) ¢ ω d - ¡ 4π 3 + b 3 -3b ¡ π 2 -x 2 + bx ¢¢ ω e 3π 2 , (19)
and

∂ 2 w (b, β; x) ∂β∂x = b (b -2x) ¡ ω d + ω e ¢ π 2 ≷ 0 ⇔ x ≶ b 2 . ( 20
)
which allows us to establish that:

∂w ¡ b, β; b 2 ¢ ∂β > ∂w (b, β; x) ∂β > ∂w (b, β; b) ∂β = ∂w (b, β; 0) ∂β . ( 21
)
The differentiations of the mean individual income with respect to the size and the trade openness are given by: ∂w (b,

β) ∂b = • 1 - b 2 2π 2 ¸£(1 + β) ω d -(1 -β) ω e ¤ , (22)
and ∂w (b, β) ∂β = -

8π 3 ω e + ¡ b 3 -6bπ 2 ¢ ¡ ω d + ω e ¢ 6π 2 . ( 23
)
The tax rate given by ( 9) is also equivalent to:

t (b, β) = 6π 2 g ω d (1 + β) b (6π 2 -b 2 ) + ω e (1 -β) (8π 3 -6π 2 b + b 3 ) We have: ∂t (b, β) ∂b = - 18gπ 2 ¡ 2π 2 -b ¢ 2 £ (1 + β) ω d -(1 -β) ω e ¤ b 2 [-b (6π 2 -b 2 ) (1 + β) ω d -(b 3 -6bπ 2 + 8π 3 ) (1 -β) ω e ] 2 , (24)
and ∂t (b,

β) ∂β = 6π 2 g £ 8π 3 ω e -b ¡ 6π 2 -b 2 ¢ ¡ ω d + ω e ¢¤ [-b (6π 2 -b 2 ) (1 + β) ω d -(b 3 -6bπ 2 + 8π 3 ) (1 -β) ω e ] 2 , (25)
from which we deduce relations ( 11) and ( 11). We note that

∂E I (a, b, β; x) ∂x = - (b -a) (a + b -2x) π 2 (1 -t (b, β)) £ (1 + β) ω d -(1 -β) ω e ¤ , ( 26
) ∂E F (a, b, β; x) ∂x = - a (a -2x) π 2 (t (a, β) -t (b, β)) £ (1 + β) ω d -(1 -β) ω e ¤ , ( 27
) ∂E P (a, b; x) ∂x = g (a -b) π 2 < 0. (28)

A.2 Secession in autarky

We have: E F (a, b, 1; x) < 0 and E I (a, b, 1; x) < 0.

Since we focus on a small peripheral region of country 1, we have assumed a < b/2. Thus we have: 26), ( 27) and ( 28) we deduce that

a + b -2x > 0, ∀x ∈ [0, a]. From (
∂E I (a, b, 1; x) ∂x = - 2 (b -a) (a + b -2x) π 2 (1 -t (b, 1)) ω d < 0, ∂E F (a, b, 1; x) ∂x = - 2a (a -2x) π 2 (t (a, 1) -t (b, 1)) ω d > 0, ∂E P (a, b; x) ∂x = g (a -b) π 2 < 0. Thus, ∀x ∈ [0, a] D (a, b, g, 1; x) = E I (a, b, 1; x) + E F (a, b, 1; x) + E P (a, b; x) < E I (a, b, 1; 0) + E F (a, b, 1; a) + E P (a, b; 0) < E F (a, b, 1; a) + E P (a, b; 0)
where

E F (a, b, 1; a) = 2a ¡ 3π 2 -a 2 ¢ g µ 1 b (6π 2 -b 2 ) - 1 a (6π 2 -a 2 ) ¶ E P (a, b; 0) = g 4π 2 ¡ b 2 -a 2 ¢ . E F (a, b, 1; a) 6 g b 2 µ 3π 2 - b 2 4 ¶ Ã 1 b (6π 2 -b 2 ) - 1 b 2 ¡ 6π 2 -b 2 4 ¢ ! = - g 2 ³ 3π 2 -b 2 4 ´¡6π 2 -3 4 b 2 ¢ ¡ 6π 2 -b 2 4 ¢ (6π 2 -b 2 ) 6 - g 4 6π 2 -3 4 b 2 6π 2 -b 2 E P µ b 2 , b; 0 ¶ 6 E P (a, b; 0) 6 E P (0, b; 0) = g 4π 2 b 2 6 g 4
Thus, we deduce:

E F (a, b, 1; a) + E P (a, b; 0) 6 - g 16 µ b 2 6π 2 -b 2 ¶ < 0 ⇔ D (a, b, g, 1; x) < 0.

A.3 Trade openness

Since the utility function is discontinuous due to the possible secession, we are not able to solve explicitly the maximization progam ( 16). We restrict our analysis to define only sufficient conditions for free trade or for autarky. We consider first the case where economic integration does not involve secession. Differentiating U (b, g, β; x) with respect to β yields to:

∂U (b, g, β; x) ∂β = - ∂t (b, β) ∂β w (b, β; x) + (1 -t (b, β)) ∂w (b, β; x) ∂β ,
The symmetry of w (b, β; x) with respect to b 2 involves that: for

x ∈ [0, b], ∂U µ b, g, β; b 2 ¶ /∂β > ∂U (b, g, β; x) /∂β > ∂U (b, g, β; 0) /∂β.
First, we focus on the sign of ∂t (b, β) /∂β, ∂t (a, β) /∂β and ∂t (ba, β) /∂β. From (25), we have:

∂t (b, β) ∂β > 0 ⇐⇒ ω d ω e > 8π 3 b (6π 2 -b 2 ) -1 = ω c 2 (b) , ∂t (a, β) ∂β > 0 ⇐⇒ ω d ω e > 8π 3 a (6π 2 -a 2 ) -1 = ω c 2 (a) , ∂t (b -a, β) ∂β > 0 ⇐⇒ ω d ω e > 8π 3 (b -a) h 6π 2 -(b -a) 2 i -1 = ω c 2 (b -a) .
Since the peripheral region is by assumption small (a < b/2), we deduce that:

ω c 2 (b) < ω c 2 (b -a) < ω c 2 (a) .
We consider now the sign of ∂w (.) /∂β in the different political situations. Since trade openness has to be approved by a simple majority and not by all inabitants of the country, we consider the critical individual located in b/4 (or equivalently in 3b/4). Moreover, we define only sufficient conditions. A majority of country inhabitants will be then in favor of free trade (respectively of autarky) if: ∂t (b, β) /∂β > 0 and ∂w (b, β; b/4) /∂β < 0 (respectively ∂t (b, β) /∂β < 0 and ∂w (b, β; b/4) /∂β > 0). If the political unity remains, whatever is the trade openness, expression (19) yields to: We note that: ω c 2 (b) > ω c 3 (b). We deduce that:

  Salop, S. (1979): "Monopolistic competition with outside goods," Economic Journal, 10(1), 141-156. Tiebout, C. M. (1956): "A pure theory of local public economics," Journal of Political Economy, 64(5), 416-424. Young, R. (1995): La sécession du Québec et l'avenir du Canada. Les Presses de l'Université Laval, Québec.

  2 -7b 2 ) -1 ≡ ω c 3 (b) .

  Figure 1. World representation

  four specific degree of openness (b = π/2, a = π/6, ω d = 1, ω e = 2, g = 3).

Zollverein was established in 1834, monetary union came in 1857, and the German empire was instituted in 1871.

See Rota Graziosi (2006) for a corrected version of their framework.

Alesina and Spolaore (1997) extend their initial model in order to take into account the effects of economic integration. They assume a productivity factor, human capital, which is increasing in the size of the country and in the openess of its frontiers. When economic integration is achieved, individual incomes depend exclusively on a worldwide level of human capital, reducing then economic advantage of large nations (see also the chapter in[START_REF] Alesina | On the Number and Size of Nations[END_REF] on this issue).

In their working paper version,[START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] study the endogeneity of economic integration, by assuming that trade barriers are a function of country size in a world of symmetric countries. In[START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], the size of markets which captures the degree of economic integration results from individual decisions. Neither in[START_REF] Alesina | Economic Integration and Political Disintegration[END_REF] nor in[START_REF] Casella | Public Goods in Trade on the Formation of Markets and Jurisdictions[END_REF], economic integration is decided by vote.

This result can be related to the empirical works of[START_REF] Lindert | Does Globalization Make the World More Unequal?[END_REF].

The condition of non-negativity of the tax rate changes into: g < g (a, β) = aw (a, β) .


Cyril Voyant
email: cyrilvoyant@hotmail.com

Ph.D Rudy Roustit

M Sc

Jennifer Tatje

Katia Biffi

Jérome Briançon

Céline Marcovici

Therapeutic potential of atmospheric neutrons

Keywords: NBCT, neutron, Boron, atmospheric, BSH, BPA, glioblastoma

Abstract.

Background

Glioblastoma multiform (GBM) is the most common and most aggressive type of primary brain tumour in humans. It has a very poor prognosis despite multi-modality treatments consisting of open craniotomy with surgical resection, followed by chemotherapy and/or radiotherapy. Recently, a new treatment has been proposed

-Boron Neutron Capture Therapy (BNCT) -which exploits the interaction between Boron-10 atoms (introduced by vector molecules) and low energy neutrons produced by giant accelerators or nuclear reactors.

Methods

The objective of the present study is to compute the deposited dose using a natural source of neutrons (atmospheric neutrons). For this purpose, Monte Carlo computer simulations were carried out to estimate the dosimetric effects of a natural source of neutrons in matter, to establish if atmospheric neutrons interact with vector molecules containing Boron-10.

Results

The doses produced (an average of  Gy in a 1g tumour) are not sufficient for therapeutic treatment of in situ tumours. However, the non-localised yet specific dosimetric properties of 10B vector molecules could prove interesting for the treatment of micro-metastases or like (neo) adjuvant treatment. On a cellular scale, the deposited dose is approximately 0.5Gy / neutron impact.

Conclusion

It has been shown that BNCT may be used with a natural source of neutrons, and may potentially be useful for the treatment of micro-metastases. The atmospheric neutron flux is much lower than that used utilized during standard NBCT. However the purpose of the proposed study is not to replace the ordinary NBCT but to test if naturally-occurring atmospheric neutrons, considered an ionizing pollution at the Earth's surface, can be used in the treatment of a disease such as cancer. To finalize this study, it is necessary to quantify the biological effects of the physically-deposited dose, taking into account the characteristics of the incident particles (alpha particle and Lithium atom) and radio-induced effects (by-stander and low dose effect). One of the aims of the presented paper is to propose to experimental teams (which would be interesting by study the phenomena) a simple way to calculate the dose deposition (allometric fit of free path, transmission factor of brain…) [START_REF] Dorn | Boron neutron capture therapy (BNCT): A radiation oncology perspective[END_REF][START_REF] Haritz | Clinical phase-I study of NA2B12H11SH (BSH) in patients with malignant glioma as precondition for boron neutron capture therapy (BNCT)[END_REF][START_REF] Yamamoto | Current clinical results of the Tsukuba BNCT trial[END_REF]. This method, which is well adapted for intra-cranial cancer treatments, is simple and well-designed in concept but complex and difficult in its execution. It is based on the capacity of thermal neutrons to induce a reaction with Boron-10 nuclei, forming a compound nucleus (excited Boron-11 on Equ. 1 and 2) which then promptly disintegrates to Lithium-7 and an alpha particle (Fig. 1) [START_REF] Hattori | Study on the compounds containing 19F and 10B atoms in a single molecule for the application to MRI and BNCT[END_REF][START_REF] Morand | A method to build an analytic model of the 10B(n,[alpha])7Li reaction rate space distribution for boron neutron capture therapy (BNCT)[END_REF][START_REF] Nakagawa | Clinical results of BNCT for malignant brain tumors in children[END_REF].

MeV

MeV Li He n B       Equ 2
This technique is advantageous since the radiation damage occurs over a short range and thus normal tissues can be spared. The path of the reaction products is shown in Fig. 2. The NBCT is a well recognized treatment for GBM, particularly due to its efficiency, but is unfortunately very difficult to access as only a few radiotherapy units can use a proton accelerator [START_REF] Kim | Optimized therapeutic neutron beam for accelerator-based BNCT by analyzing the neutron angular distribution from 7Li(p,n)7Be reaction[END_REF][START_REF] Laramore | Boron neutron capture therapy (BNCT) for high-grade gliomas of the brain: A cautionary note[END_REF]. In France, this service is not available to patients.

An alternative to this would be to find a new source of neutrons that is easily exploitable. Because free neutrons are unstable (mean lifetime of about 15 minutes), they can only be obtained from nuclear disintegrations, nuclear reactions, or high-energy reactions such as in cosmic radiation showers or collision accelerators. Cosmic radiation interacting with the Earth's atmosphere continuously generates neutrons [START_REF] Boella | The atmospheric and leakage flux of neutrons produced in the atmosphere by cosmic ray interactions[END_REF][START_REF] Dudkin | Differential neutron energy spectra measured on spacecraft in low earth orbit[END_REF][START_REF] Goldhagen | Recent results from measurements of the energy spectrum of cosmicray induced neutrons aboard an ER-2 airplane and on the ground[END_REF][START_REF] Koi | Attenuation of neutrons in the atmosphere and a thick carbon target[END_REF][START_REF] Kovaltsov | A new 3D numerical model of cosmogenic nuclide 10Be production in the atmosphere[END_REF][START_REF] Leray | Effects of atmospheric neutrons on devices, at sea level and in avionics embedded systems[END_REF].

The cosmic rays (essentially 85% of Hydrogen and 12.5% of Helium) penetrate the magnetic fields of the solar system and the Earth, and as they reach the Earth's atmosphere, they collide with atomic nuclei in the air (78% Nitrogen, 21% Oxygen and 1% Argon) to create cascades of secondary radiation of every kind. The intensity and energy distribution of the different particles that make up atmospheric cosmic radiation vary with 3 essential parameters: altitude, location in the geomagnetic field (correlated to latitude, it deflects low-momentum charged particles back into space), and time in the sun's magnetic activity cycle [START_REF] Mavromichalaki | Applications and usage of the real-time Neutron Monitor Database[END_REF][START_REF] Phillips | An improved approach to calculating low-energy cosmic-ray neutron fluxes near the land/atmosphere interface[END_REF][START_REF] Plainaki | Modeling the solar cosmic ray event of 13 December 2006 using ground level neutron monitor data[END_REF]. Table 1 shows the implication of the first two (altitude and latitude). Atmospheric shielding, correlated to altitude, is determined by the thickness of the air mass above, called atmospheric depth.

At high altitude, geomagnetic latitude has a small effect on the shape of the neutron spectrum and a very large effect on the neutron radiative flux. In Fig. 3, representing the results found by Roesler in 1998 [START_REF] Roesler | Neutron spectra in the atmosphere from interactions of primary cosmic rays[END_REF], we can see how atmospheric depth has a strong influence on neutron flux. It decreases by a factor of 100 between high altitude (20 km corresponding to an atmospheric depth of 50 g.cm -2 ) and sea level (corresponding to an atmospheric depth of 1000 g.cm -2 ).

Latitude is important as there is an increase by a factor of about 10 between the minimum intensity (at the Equator) and the maximum (at the Polar Regions).

The last parameter linked to the flux of atmospheric neutrons is solar variation. There are periodic components in these variations, mainly the 11-year solar cycle (or sunspot cycle), but also the solar magnetic activity cycle. It modulates the flux of high-energy galactic cosmic rays entering the solar system. As a consequence, the cosmic ray flux in the inner solar system (or neutron flux in the atmosphere) is correlated with the overall level of solar activity.

The atmospheric neutron flux is much lower than that used during standard NBCT (10 Vs 10 9 n/s/cm²).

However this flux is maybe small but these characteristics are certainly compatible for a clinical use: free source for an ecological approach, fractionated dose to allow normal cells to recover, localized dose deposition (only in tumor cells), and a disease no (or little) cured with conventional tools. BPA is an analog of the essential amino acid phenylalanine, and is actively taken up in cells not only as an amino acid analog for protein synthesis, but also as a tyrosine analog for melanogenesis. Because of this nature, BPA has been clinically used for the treatment of malignant cells. To characterize these molecules, it's essential to quantify the nuclear reaction rate produced by the interaction with atmospheric neutrons. In this first study, we are not considering the concentration and the biological distribution of these molecules in the organism. We hypothesise that neutrons interact with carrier molecules and simulate the different processes to establish which reactions are most probable. The estimation is done for all atmospheric neutrons (the full spectra is used). The cross sections were produced using ENDF/B library (available on http://t2.lanl.gov). Table 2 shows that the two molecules used in the simulation are appropriated for the neutron interaction: the alpha-producing reaction reveals to be the most frequent one (94% for BSH and 88% for BPA).

radiative capture (n,) (n,p) (n,n) BSH BPA 1% 1% 94% 88% ~0% ~0% 5% 11%
Table 2: nuclear reactions distribution after impact between neutron and BPA or BSH

Analytic dose expression

Before analytically calculating the absorbed dose into the tissues [START_REF] Clark | Novel carboranylporphyrins for application in boron neutron capture therapy (BNCT) of tumors[END_REF][START_REF] Montgomery | The intensity of neutrons of thermal energy in the atmosphere at sea level[END_REF] resulting from a succession of interactions between atmospheric neutrons and Boron nuclei, it is necessary to assume that the dose is comparable to kerma (transferred energy equals deposited energy) as the alpha and Lithium particles have a strong LET along with a very short pathway into matter. Equ.

            Equ 5 n ij
E is in fact equivalent to the Q-value of the reaction (default or excess of mass during the nuclear reaction).

In the case of radiative capture reaction, the value is nil as the dose will not be laid-on into the volume V. This method of calculation is relevant only in relatively simple cases. If we consider the lowering of the incident flux or a complex and heterogeneous substance, this method is impractical. Therefore, we have chosen to use a hybrid or mixed method which combines both an analytic model and Monte-Carlo type algorithms to quantify the doses in use.

Modelling methodology

The characteristics of the matter (skull, brain and tumour) are essential to simulate the absorbed dose. We have chosen to use a detailed brain and skull composition as seen in Table 3.

Element

Table 3 brain and skull characteristics

The tumour is considered similar to the brain (same chemical composition) with /g of Boron10 (average value found in the literature), representing 0.002% of a tumour weighing 1g. The first stage of the simulation determines the cumulative distribution of neutron energy in the air. This means calculating the function F=F(E), described in Equ. 6, enabling to allocate a random number between 0 and 1 to a neutron incident energy. There are 14.7 neutrons/s/cm² in this particular case.

     max min min ) ( E E i i E E i i n n E F Equ 6
Once the incident energy has been allocated, the simulation requires estimating the interaction inside the brain and the skull. The tools used are the mean free paths and the transmission (attenuation) factors.

Attenuation factor and neutron capture

This parameter allows to quantify the number of neutrons (only capture type) interacting in the brain and the skull. The probability of neutron capture reactions is defined in Equ. 7, showing the neutrons have merged to form a heavier nucleus.

) , ( . 1 . .. . , j E A N x P j a j E    Equ 7
The elastic and inelastic reactions (n,n) are not considered in the expression of the total capture cross sections because the neutrons continue to travel after impact. The transmission factor is calculated from the precedent probability factor by the expression T=1- j j E P , . So, a coefficient of zero means the neutron is stopped in the matter and a coefficient of 1 means there is no capture at all. In the intermediate case, when the coefficient ranges between 0 and 1, a random number between 0 and 1 is compared to the transmission factor T. If this number is less than T, we consider there is no neutron capture, otherwise the neutron is captured and stopped in the matter.

Mean free path

The mean free path of a particle is the average distance covered by the neutron between successive impacts.

Alternatively, it is the distance at which the intensity of particles drops by 1/e. This coefficient is calculated with the following formula:

  i a i i E N A L ) , ( .

..   Equ 8

In the case of mean free path for capture, the ) , ( i E  is related to the reaction (n, ( n,p) and (n,, and for the elastic scattering ) , ( i E  , it is only related to the elastic reaction.

Elastic interaction

We consider the energy loss is possible only with the elastic diffusion. All the capture reactions attenuate the beam but don't modify the energy of the incident particle. In the case of elastic collisions between neutrons and other particles, the energy lost is calculated in Equ. 9:

rec n rec n rec rec m m m m E E  2 2 0 cos ) ( 4 .
 

Equ 9

When E 0 <10MeV the reaction is isotropic and the cosine term is replaced by the factor ½. If the collision is done with a Hydrogen nucleus, the expression of the energy loss by the neutron is E 0 /2.

Results

Air stage

The cumulative distribution function related to the incident neutron energy is represented in Fig. 6.

Figure 6: cumulative distribution function related to incident energy neutron

This figure shows that the energy of atmospheric neutrons is very low. A random assortment from this function reveals that 50% of neutrons have energy of less than 1 MeV, and 80% energy of less than 10 MeV.

Skull stage

The thickness of the skull is about 0.5cm, which generates a weak attenuation, as shown in Fig. 7. The transmission factor varies between 99.5% and 100%, meaning that very few neutrons are absorbed by the skull. The allometric fit of the results is described by T s =0.99997-0.00162.E(eV) -0.45609 (R²=0.995). This manipulation is done by the Origin Pro software. The second observation is related to the elastic interaction in the skull. The manipulation shows that the probability for a neutron to undergo an elastic interaction is less than 0.5%. In fact, the neutron flux is unchanged by crossing the skull.

Brain stage

Without Boron (as in the healthy brain), neutrons have a higher probability of undergoing an elastic scattering process than a capture process. 4 shows the difference between those two modes. Capture reactions are rare, while those related to Elastic collisions can be very frequent (every 1cm for energies below 1KeV). Neutrons with energies higher than 50MeV were not considered in this study because they are only in few amounts and more complex reactions begin to interfere. The allometric fit for the mean free path between two elastic collisions is given by L b (cm)=0.81147+6.048.10 -5 .E(eV) 0.78671 (R²=0.999). The chosen method consists in determining whether, with this distance, the neutron will be captured or will continue on its path. If it's not mitigated, we consider it will lose some of its energy during the elastic collision. This method is repeated until the neutrons cross the brain and reaches the tumor.

Energy

Figure 8: transmission factor for the brain

Fig. 8 shows the results for the transmission of neutrons during the passage through the brain. The analytic expression of this curve can be given by a standard allometric equation: T s =0.99943-0.003735.E(eV) -0.49955 (R²=0.999). We can observe that neutrons with energy higher than 0.1eV will not be altered, and will continue on their path. During the simulations, we used the analytic expression of the transmission coefficients or the free average distance, which greatly reduce computation time (42s instead of 15 hours for 6,000 neutrons).

Tumour stage

As we mentioned earlier, we consider that the tumour composition is equivalent to that of the brain comprising of Boron. The mean free paths of elastic collisions are almost equivalent to those shown in the previous section.

However, the transmission factor will be changed. Collisions like (n, will be predominant at low energy, amending the first part of the transmission curve of the tumour (Fig. 9). The procedure is the same as that established in the case of the brain.

Model validation

To validate the model and the method of calculation, we have made use of the calculation code SIEVERT (System of information and evaluation of flight exposure to cosmic radiation in transports, as available on www.sievert-system.org), which is a tool developed by IRSN. This tool allows quantifying the dose received by aircrew during a flight [START_REF] Vukovic | Cosmic radiation dose in aircraft -a neutron track etch detector[END_REF], which is associated with the presence of neutrons at that altitude. Thus, we could compare the results given by our methodology with those established by the IRSN. In this manipulation, the numerical phantom does not contain Boron or tumour. We only simulated a flight and calculated the dose deposited in a cylindrical model representing a person. The code SIEVERT cuts airspace in 265,000 stitches according to altitude, latitude and longitude and then assigns them a dose rate (recalculated every month). For a flight from Paris to New York, the software simulates an effective dose of 6,1Sv/h. From our side, we have made large approximations with the available data. The spectrum used corresponds to the one applied during the previous manipulation, for a similar latitude but at different solar period. The simulation target is a cylinder 20cm x 1.80m with a mass of 56.5Kg. Its composition is the same as that previously described for the brain. As before, there are very few capture reactions. Furthermore, the only reactions of interest to us are the elastic reactions (n,n), the deposit dose caused by the recoil nucleus. The average energy transferred to the cylinder is calculated by

    ) ( ). ( . 1 E w E E m D r i f
. The result of 4.37Sv/h implies a difference of 30% with the reference SIEVERT. However, the numerous approximations make this result interesting and in accordance with the requirement we have set ourselves, namely to only work on the order of magnitude so as to validate the interaction of atmospheric neutrons with the nuclei of Boron-10.

Dosimetric aspect

The study of El Massaoui [START_REF] El Moussaoui | Monte Carlo calculation for the development of a BNCT neutron source (1 eV-10 KeV) using MCNP code[END_REF] shows that pre-filtering the neutron beam allows to ameliorate the registered depth dose, the incident energy of neutrons must be reduced to between 1eV and 10keV. To obtain this value, the Moroccan team simulated different "neutron decelerator" and noted the results. Inspired by this study, we studied the effects of a decelerator formed by water on the incidental spectra of neutrons. By interposing 10cm (or 20cm with less impact) of water between the beam and the test patient, we considerably increased the proportion of neutrons having an energy included between 1eV and 10keV. The chosen simulation conditions are the attenuator of 10cm of water, a crane thickness of 0.5cm and a tumour of 1cm containing 20µg of Boron-10. The simulation is made with 6,000 incident neutrons. In addition to changing the spectrum, the attenuator induced a neutron loss of 5% (capture reaction), but these are neutrons that would not have been able to reach the tumour because they mostly had a very low energy (Fig. 10). 5.6% of incident neutrons interact with the tumour, implying that the average dose to the tumour is in the order of 1Gy for 1hour of irradiation (3,000 interactions).

We see that the value of the mean absorbed dose within the tumour is very low, so the use of this technique as a sole therapy at a tumour site seems inappropriate. However, given the irradiation heterogeneity and low runs of secondary particles (alpha and lithium), this dose does not present any "physical issue", for better understanding we enter the field of micro dosimetry. The deposit is important but on a very small volume. Indeed, at the cell level represented by a cube of 1m 3 , where the dose will fill after a shock to the Boron-10, the absorbed dose is this time about 0.5Gy / shock. Typically, this level of radiation induces important DNA base damage (i.e. for 1Gy per cell, there are 2,000 oxidized and reduced bases and 250 abasic sites, 1,000 single and 40 double strand breaks, and 150 clustered lesions) [START_REF] Spotheim-Maurizot | Radiation chemistry from basics to applications in material and life science[END_REF]. The non-localized, but specific, irradiation suggests this methodology as a possible source of treatment of micro-metastases (a form of metastasis in which secondary tumours are too minuscule to be clinically detected). Adjuvant atmospheric NBCT would focus on destroying these micrometastases. In addition, we must not forget that the exposure time and the concentration of Boron-10 (20 g fixed for the study, but could be increased to at least 50 g) are directly proportional to the number of interactions. If one of these parameters increases, the number of interactions within the tumour will also increase. These values must also be contrasted with the different radiobiological effects inherent in the nature of secondary particles depositing dose (alpha and Lithium). A relatively weak dose rate but with a well-targeted dose within a single cell will lead to radiobiological effects. The first is related to the low dose rate effect [START_REF] Hegyesi | Low Dose Radiation Induced Transcriptional Alterations in Directly Irradiated and Bystander Fibroblast Cells[END_REF],

whereby the main effects are repair, repopulation and re-oxygenation, which increase the radiosensitivity of the cells. This process is equivalent to the process observed during brachytherapy. The second aspect is related to the deleterious effects, such as mutagenesis, observed in cells that are not directly hit, a process called the bystander effect [START_REF] Mothersill | Bystander Effects Induced by Exposure to Sublethal Radiation and Heavy Metals in Atlantic Salmon[END_REF][START_REF] Spotheim-Maurizot | Radiation chemistry from basics to applications in material and life science[END_REF]. Another effect to consider is where mutations are induced in cells that are hit only in the cytoplasm and not in the nucleus where DNA is located. Normal cells (without Boron) do not undergo great dose deposition, at least no more than during a standard flight. For these healthy cells, the ratio (concentration of boron) with malignant cells is about 10, revealing the low toxicity of Boron injection at altitude in healthy cells.

Figure 1 :

 1 Figure 1: schematic of 10 B(n,  Li capture reaction in a cancer cell (left), and total cross-section (right)

Figure 2 :

 2 Figure 2: penetration depth of alpha particle (1.47MeV) in the muscle at left. On the right, the zoom of the last micrometer. Red point are, the elastic scattering on hydrogenous nucleus, and green points the scattering of the secondary hydrogenous particle. (SRIM© calculator)

Figure 3 :

 3 Figure 3: the depth dependence of the neutron flux, calculates for three different conditions. Polar region (minimum and maximum) and equatorial region.

3 and 4

 4 stand for the expression of the average onlaid dose after a succession of interactions between neutrons and tissues. It becomes possible to define a mass coefficient of absorption thus noted ) (

3

 3

[ 3 ]

 3 … The initial conditions are presented in Fig. 4.

Figure 4 : 5 .

 45 Figure 4: numerical phantom simulation

Figure 5 :

 5 Figure 5 : schematic model used

Figure 7 :

 7 Figure 7: transmission factor for the skull (thickness 0.5cm)

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: schematic of 10 B(n,  Li capture reaction in a cancer cell (left), and total cross-section (right) versus

Figure 4 :Figure 6 :

 46 Figure 4: numerical phantom simulation Figure 5 schematic model used Figure 6: cumulative distribution function related to incident energy neutron

Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 78910 Figure 7: transmission factor for the skull (thickness 0.5cm) Figure 8: transmission factor for the brain Figure 9: transmission factor for the tumour Figure 10: neutron spectra after interaction with attenuator (top, 5.12% of neutron stopped), brain (middle, 21.12% of neutron stopped) and tumor (bottom, 26.72% neutron stopped)

  Deflexion angle in the system of the center of mass (rad)

			NOMENCLATURE
	D ij	Dose deposited by the reaction i and the nucleus j (Gy)	L s,b,t	Mean free path (m)
	N o	Total number of neutron (NU)	T s,b,t	Transmission factor for skull, brain and tumour (NU)
	E	Energy variable (J)	Depth	Tumour depth in brain (m)
	t		Time variable (s)	F(E)	Cumulative distribution function for neutron related to energy (NU)
	S N j V	Surface variable (m) Number of nuclei j (NU) Volume (m 3 )	n i N a A j	Number of neutron with energy i (NU) Avogadro constant (mol -1 ) Atomic mass number for nucleus j (atomic mass unit, u)
	 	cross section for reaction i and nucleus j (m²)	j	(m²)
	E	n ij	Energy deposited after the reaction i on the nucleus j (J)	P E,j	Capture probability on nucleus j with the energy E (NU)
	x	Depth (m)	m rec	Mass of recoil nucleus during elastic collision (kg)
	m	Mass (kg)	m n	Neutron mass = 1.0087u
		Differential neutron flux (J -1 s -1 m -2 ).
		Density (g.cm -3 )	E 0	Energy of incident particle (J)
	 en		Attenuation coefficient (J.kg -1 )	w r (E)	Weighting factor designed to reflect the different radiosensitivity of the tissues (NU)
	E a,s,b	Neutron energy in air, skull and brain (J)	E i,f	Initial and final neutron energy (J)

rec 

1 Introduction

 1 In 2007, cancer caused about 13% of all human deaths(7.6 million). Some of most invasive malignant tumours are breast cancer, colorectal cancer, lung cancer, and stomach and liver cancer. Brain tumours are not very common as they account for only 1.4% of all cancers in United States. Patients with benign gliomas may survive for many years, but in most cases of glioblastoma multiforme (GBM), survival is limited to a few months after diagnosis without treatment[START_REF] Clark | Novel carboranylporphyrins for application in boron neutron capture therapy (BNCT) of tumors[END_REF][START_REF] Laramore | Boron neutron capture therapy (BNCT) for high-grade gliomas of the brain: A cautionary note[END_REF]. Despite being the most prevalent form of primary brain tumour, GBM occurs in only 2-3 cases per 100,000 people in Europe and North America. It is the most common and most aggressive type of primary brain tumour in humans. The usual multimodality treatment consists of open craniotomy with

surgical resection of as much of the tumour as possible, followed by concurrent or sequential chemotherapy, antiangiogenic therapy with bevacizumab, gamma knife radiosurgery, standard radiotherapy, and symptomatic care with corticosteroids. Another therapeutic approach is based on the Boron Neutron Capture Therapy (BNCT), which was proposed in 1936 by Dr. Gordon Lecher only 4 years after the discovery of the neutron

  Both the alpha particle and the Lithium ion produce closely-spaced ionizations in the immediate vicinity of the reaction, with a range of approximately 4 and 8 µm, for Lithium-7 and alpha particle respectively, or roughly the diameter of one cell (10 µm).

	10 5	B		4 2  n 1 0	He 	4 2 Li		79 . 2	MeV	%) 6 (	Equ 1
	10		1		4	4
									48 . 0 (	)	31 . 2	%) 94 (
	5		0	2	2

Geographic Location Atmospheric Depth (g.cm -2 ) Altitude (km) Neutron fluence rate (cm -2 .s -1 )

  The purpose of the proposed study is not to replace the ordinary NBCT but to test if naturally-occurring atmospheric neutrons, considered an ionizing pollution at the Earth's surface, can be used in the treatment of a disease such as cancer. Before to begin the calculus, we know that atmospheric neutron are not candidate to be used like a global and single therapy (case of standard BNCT), however It's interesting to study the ability of atmospherics neutrons to be used like an

	adjuvant or neo-adjuvant treatment
	2 Materials and methods
	2.1	BSH (Na 19°N,127°W	53.5	20.3	1.24
		54°N,117°W	56	20.0	9.8
		56°N,121°W	101	16.2	9.6
		38°N,122°W	201	11.9	3.4
		17°N,76°W	1030	0	0.0127

2 B 12 H 12 S 1 ) and BPA(C 9 H 12 O 4 BN)

In clinical neutron capture therapy 10 B compounds such as BPA and BSH have been widely used as shortrange alpha particle-producing agents

[START_REF] Ghaneolhosseini | Synthesis of novel boronated acridines-and spermidines as possible agents for BNCT[END_REF][START_REF] Gottumukkala | Synthesis and cellular studies of an octa-anionic 5,10,15,20-tetra[3,5-(nidocarboranylmethyl)phenyl]porphyrin (H2OCP) for application in BNCT[END_REF][START_REF] Heber | Biodistribution of GB-10 (Na210B10H10) compound for boron neutron capture therapy (BNCT) in an experimental model of oral cancer in the hamster cheek pouch[END_REF][START_REF] Jonnalagadda | Synthesis of [alpha]-carboranyl-[alpha]-acyloxy-amides as potential BNCT agents[END_REF][START_REF] Lechtenberg | Synthesis of a (B12H11S)2-containing glucuronoside as potential prodrug for BNCT[END_REF][START_REF] Narayanasamy | Hydrophilically enhanced 3-carboranyl thymidine analogues (3CTAs) for boron neutron capture therapy (BNCT) of cancer[END_REF]

. It is believed that BSH, which has been clinically used in brain tumours, can pass through the disrupted blood-brain barrier and thus selectively accumulate in brain tumour tissue.

Table 1 : relation between neutron flux, altitude and geographic location

 1

Table 4 : mean free path in the brain, for the atmospheric neutrons. The average is related to the neutron energy distribution Table

 4

		L capture (cm)	L elastic (cm)
	0.025eV<…<0.5eV	119	0.811
	0.05eV<…<1keV	2360	0.815
	1keV<…<500keV	32600	1.54
	500keV<…<50Mev	5460	8.58
	0.025eV<…<50MeV	12800	4.53

Table legends NomenclatureTable 1 :

 legends1 relation between neutron flux, altitude and geographic location

Table 2 :

 2 nuclear reactions distribution after impact between neutron and BPA or BSH

Table 3 :

 3 brain and skull characteristics Table4: mean free path in the brain, for the atmospheric neutrons. The average is related to the neutron energy distribution

Acknowledgments

The authors thank Franglo-Traduction and Samantha Warren for contributions during the drafting.

Conclusion

The present study shows that atmospheric neutrons can interact with Boron-10 found on molecular targets (BSH and BPA). These results were clearly predictable beforehand, but the code developed allows to quantify simply the dose after the penetration in brain. However, the low flux of neutrons doesn't allow us to assume that it's possible to treat visible tumours (large cell colony) but maybe the micrometastasis. To validate this hypothesis, the presented study must be completed with laboratory experiments to quantify the real effect of this radiation on the B-10 and biological cells. It's the intersection of several disciplines (microdosimetry, nuclear physics, bystander effect, low dose effect…). The aim of the presented paper is to propose to experimental teams (which would be interesting by study the phenomena) a simple way to calculate the dose deposition (allometric fit of free path, transmission factor of brain…). Moreover, the spectrum used as a basis for this study is a medium spectrum; obviously, according to location and altitude, the flux will vary. Likewise, according to the injection protocol, the Boron concentration within GBM as well as the exposure duration will proportionally increase along with the on-laid dose. Sun cycle phases will equally modify the on-laid dose into the tumour. Considering the parameter values before maximising the atmospheric neutron output, we'll be able to increase the absorbed dose up to a hundred times. Although there are a lot of parameters which influence the dose rate, this radiation is not out of any possibility of control. Previous calculus (don't presented here) have shown that the use of dosimeter may allow to quantify the dose deposited in the tissue by neutron on Boron. The detectors studied (but all neutron detector are certainly valid) were a rare-earth-doped alkaline-earth sulfide with 2% of boron (SrS:Ce,Sm:B), they are luminescent materials (Optically Stimulated Luminescence). The first results suggest that the dose in the tissue can be calculus by the dosimeter measure.. In brief, further studies based on radioinduced effects occurring after the interaction between neutrons into GBM would make it possible to establish whether an aircraft may change into a radiotherapy room.