Matthieu Drevelle
email: matthieu.drevelle@malix.univ-paris1.fr

Global modelisation and local characteristics of French periurban spatial organization

Keywords: Periurbanisation, urban spatial structure, spatial analysis, GIS, France

This paper aims at finding global rules about periurban spatial organization in France according to three main indexes: intensity of periurbanisation, range of periurban commutes and concentration of periurban flows. Comparing all French agglomerations with more than 5.000 jobs, we have highlighted links between the population of these agglomerations and our three indexes. That is why this research proposes statistical models to determinate periurban spatial organization according to agglomeration size. Moreover, observing the spatial repartition of residuals, we have highlighted some local characteristics in this organization. Using geographical information systems, we have tried, in this paper, to explain these local characteristics by geographic facts and historic resilience.

INTRODUCTION

No one can deny that the face of our cities has been transformed by periurbanisation during the last decades. Most of the urban studies community agrees to say that today's cities cannot be reduced to the historic core or to the morphological agglomeration. In fact, our cities can be compared to an archipelago [START_REF] Beaucire | Dynamique spatiale de l'agglomération nantaise[END_REF] composed of a central continent (the historical core of the agglomeration) and a lot of islands (the periurban areas) which are attracted by the continent. Nevertheless, the boundaries of this "urban archipelago" are hard to define because we can use many different definitions. This study does not aim at giving a new definition of cities: "it's too big, we have many chance to make mistakes" (Perec, 1974), its purpose is to answer the two following questions:

-Does periurbanisation spatial characteristics (intensity, range and concentration) are linked with the size of agglomeration? Can we find a "global rule" for periurban spatial organization? -Can we observe local characteristics in the spatial organization of periurbanisation? And how can we explain these characteristics by history and geography? The periurbanisation process is well studied in geography; many recent works focus on new urban and suburban structures in the world [START_REF] Huriot | Modéliser la suburbanisation. Succès et limites de la microéconomie urbaine[END_REF]Bertaud, Malpezzi, 2004). Moreover, a lot of monographic works have been done on specific periurban areas, such as Paris' metropolitan area [START_REF] Berger | Les périurbains de Paris. De la ville dense à la métropole éclatée ?[END_REF], Rennes [START_REF] Baudelle | Les conséquences d'un choix résidentiel périurbain sur la mobilité : pratiques et représentations des ménages[END_REF] or Toulouse (Rougé, 2009). Commuting is also a well studied subject with some major comparative works [START_REF] Newman | Sustainability and Cities: Overcoming Automobile Dependence[END_REF]. Nevertheless, in France, only a few researches have tried to compare a great number of agglomerations. Le Jeannic (1997) has done a very interesting research on the demographical evolution of periurban population and on the mobility between periurban areas and core agglomerations for all the French agglomerations. But his work is now quite old (proposing an analysis on the 1962-1990 period) and it does not really deal with periurban spatial organization. [START_REF] Wiel | Forme et intensité de la périurbanisation et aptitude à la canaliser[END_REF] has calculated some indexes of periurban spatial organization for many cities, but his work as not been updated and does not concern all the French agglomerations. A new approach is to study periurban morphology at a fine scale and to characterize the form of all periurban islands [START_REF] Emangard | Diversité des formes de la périurbanisation en France[END_REF] but this method requires a priori the analysis of spatial structure at a more large scale. In this paper, we have studied periurban spatial organization for all the 354 French agglomerations with more than 5.000 jobs, using the methods developed by quantitative geography (Haggett, 1973;[START_REF] Pumain | L'analyse spatiale. 1. Localisations dans l'espace[END_REF] and GIS spatial analysis. We have also tried to explain our results with thematic and historical geography [START_REF] Braudel | L'identité de la France[END_REF][START_REF] Planhol | Géographie historique de la France[END_REF].

METHODOLOGY

Define periurban spatial characteristics in France requires to qualify periurbanisation. As we said previously the concept of 'periurbanisation' has multiple definitions. In this research, we do not choose to use INSEE's (French Institute of statistics) official definition of "periurban area" which is old (1999) and inappropriate to our work because it gives a periurban attribute to places and not to people. In fact, this official definition is based on a threshold of municipalities' workers working in the agglomeration (40%); so many periurban commuters are excluded of the definition (if the threshold of 40% is not reached). Our own definition of periurbanisation is based on people's behavior: is considered as a periurban everyone who works in an agglomeration and lives outside of this agglomeration within a range of 100km (which is the official maximum distance of "daily mobility"). With this definition, the influence area of an agglomeration is not necessary a continuous area and a municipality could be in the influence area of many agglomerations. Our definition of agglomeration is the INSEE's one (named "pole urbain" and defined as morphological agglomeration with more than 5000 jobs). Because of this morphological definition, the "pole urbain" contains the historical center and most of the suburban subcenters. Assuming a monocentic urban structure in French metropolitan areas [START_REF] Huriot | Modéliser la suburbanisation. Succès et limites de la microéconomie urbaine[END_REF], our work doesn't focus on outlying subcenters which represent a very low part of metropolitan jobs [START_REF] Aguilera | Urban sprawl, polycentrism and commuting[END_REF]. All our quantitative work and our definition of periurban is based on an individualized database of daily commuting in France in 2007 (RGP INSEE 2007). Methodology of this paper is divided in three steps:

Building indexes to define periurbanisation

Our analysis is based on three indexes: the intensity, the range and the concentration of periurbanisation. The intensity index (I) aims at quantifying the proportion of periurban people for each agglomeration. It is calculated by dividing the number of people working in the agglomeration and living outside by the total job number in the agglomeration. The range index (R) is used to know how far periurban people are living from their job and by extension how big is the influence area of the agglomeration. This index is the median (R median ), the first (R Q1 ) and the third quartile (R Q3 ) of periurban commutes length. Commutes lengths are calculated for each periurban people as the Euclidian distance between the place of residence's townhouse and the place of work's townhouse. The concentration index (C) aims at showing how the periurbanisation is concentrated in a few number of periurban towns, or how it is dispersed in a lot of periurban villages. This index is based on the weight of commuters' flows: we have calculated for each agglomeration a Gini coefficient [START_REF] Gini | Concentration and dependency ratios[END_REF] (which is an inequality index) on these weighted flows. The nearer the index is to 1, the more unequal the flows are, so the more concentrated in a few municipalities periurbanisation is. On the contrary, the nearer the index is to 0, the more equal the flows are, so the more dispersed in all the attracted municipalities periurbanisation is. These indexes have been calculated for all the agglomerations in order to make regressions and to highlight global rules for periurban spatial organization.

Observing local and spatial characteristics

As our statistical model based on agglomeration size is not perfect, we have decided to do a spatial analysis on the previous regressions' residuals. We have classified the residuals of the different regressions in different classes and plotted them on a map. Observing the repartition of agglomeration with high or low residuals could give keys to highlight local characteristics of periurban spatial structure. In order to complete our local analysis, we have also calculated a Kernel density [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF][START_REF] Parzen | On estimation of a probability density function and mode[END_REF] on the residuals with a range of 100km. Smoothing is often used in geography to transform punctual data into a continuous area [START_REF] Grasland | Lissage cartographique et animation spatio-temporelle: quelques réfléxions théoriques et méthodologiques[END_REF]. The range of 100km has been chosen in order to fit with the "daily mobility" definition. This step aims at identifying and to observing areas with high intensity of positive or negative residuals.

Trying to explain local characteristics

A last step in this local analysis is to compare our quantitative results with the analysis of thematic geography and historical geography. In fact, we assume the postulate that the actual organization of cities can be explained by the resilience of territory organization. In this work, this step is only based on map observation and comparison, but a further analysis in a future research could use a statistical approach such as Geographically Weighted Regression (Charlton, Fortheringham, 2009).

RESULTS

Range, intensity and dispersion of periurbanisation are correlated to city's population

Our first research hypothesis is that intensity, range and concentration of periurbanisation are correlated to the population of agglomeration. This hypothesis is checked by our statistical work for the 354 French agglomerations. Indeed, we get good results for our three statistical models with a R² around 0,5. Some of these results are patently obvious: the median range of periurban commutes increases with core agglomeration size. One interesting information is that the median range, but also the first and the third quartiles, are correlated with the logarithm of agglomeration population (see table 1) and that the interquartile coefficient is globally around 2,5. These regressions allow to model the repartition of periurban population around the core agglomeration: it allows us to draw three circles around the centre of city containing respectively 25%, 50% and 75% of the periurban commuters.

Charts 1 & 2: Median range and intensity of periurbanisation according to agglomeration size

The two other statistical regressions give less intuitive results and permit to precise our model of periurban people's spatial repartition. The periurban intensity model gives us the percentage of periurban commuters. It is a power regression (or log-log regression) and it gives a surprising result: the bigger the city is, the lower the intensity of periurbanisation is. This phenomenon, traducing a strong residential attractiveness of big agglomerations, might be explained by the greater number of services offered to people living in big agglomerations. Moreover, because of congestion, these central services are harder to access to periurban people. On the contrary, in smaller agglomerations, central area can be reached quicker, so it might be more attractive to live in periurban area to benefit lower housing price and natural land. Nevertheless, this intensity index is a relative number: if the percentage of periurban is lower in big agglomerations, their number is greater. As a matter of fact, we can identify a linear relation between the population of core city and the number of periurban people with a very good R² (0,92 with all the cities and 0,78 if we exclude Paris)(see table 1). The last regression (on concentration index, based on Gini's coefficient) also gives interesting results. We can observe that periurbanisation is more concentrated in big agglomerations than in smaller ones. In other words, the commuters' flows are more concentrated in a few number of municipalities in big agglomeration and more equally distributed in the smaller ones (even if the number of municipalities attracted by an agglomeration grows exponentially with the size of the agglomeration)(see table 1). Nevertheless, if a logarithmic regression gives good results for agglomerations until 1.000.000 inhabitants (see chart 3), we can observe that the Gini's coefficient is never higher than 0,85. In fact, it seems to exist a threshold around 1.000.000 inhabitants. After that, we can see that the Gini's coefficient seems to stabilize or slowly decline (even if we have only one very big agglomeration in our study). That's why we can propose a second model for concentration using logarithms and polynomial regression (chart 4). This second model has a better R² and seems closer to the reality, so we can conclude that after one or two million inhabitants in the core agglomeration, the number of attracted municipalities is so huge that the concentration index cannot continue to grow.

Charts 3 & 4: Concentration of periurbanisation according to agglomeration size with logarithmic regression and polynomial regression

As a first conclusion, it is possible to model periurban spatial structure of French agglomerations according to their population. Indeed, our statistical models are able to determine the intensity of periurbanisation, the number of periurban commuters and their spatial localization: distance to the core agglomeration and level of concentration in a few number of municipalities (see table 1). Nevertheless, even if the precision of our models is quite good (with a R² around 0,5), it seems to be very interesting to see the spatial repartition of residuals. It will lead us to know if some local characteristics can influence our model.

Index

Formula (x=population of the core agglomeration)

Adjusted R²

Range

Behind global rules: the local characteristics of periurbanisation in France

If the statistical models have shown global rules of periurban spatial organization, the cartography of residuals is useful to highlight local characteristics independently of agglomerations size. The results of this step of our research are quite surprising because a spatial structure of residuals clearly appears for the difference indexes, but each index has its own specific spatial organization. These organizations appear in the following maps where we have plotted all our agglomerations with a specific color representing residuals. Agglomerations who are represented in the previous graphs (part 3.1) between the two lines of "+ 1 standard deviation" and "-1 standard deviation" are plotted in white because we estimate that they are explained by the model. Agglomerations with a negative residual bigger than one standard deviation are plotted in blue and these with a positive residual higher than one standard deviation are plotted red. A smoothing function (using Kernel density) permits to observe regional areas with negative or positive residuals (see map1).

Map 1: Localisation of residuals of the statistical models of periurbanisation

The first results of this analysis is that intensity of periurbanisation is greater in the North-West of France (especially in Picardie, Normandie and Bretagne), in the east (North of Alsace) and around Lyon. On the contrary the intensity of periurbanisation is lower in a diagonal starting from North-East to South-West (known in France as the "empty diagonal") and in the South-East (the Alps and the East part of Mediterranean coast). The repartition of residuals for the concentration model also shows a clear spatial organization. Periurbanisation is more concentrated along a diagonal running from west (Bretagne) to South-East (mediterrean coast) and more dispersed in North, East and South-West. The residuals of range model have a less apparent spatial structure. However, we can see that the median range of periurban commutes is lower in the Nort-East border and in the valley of Rhine and Rhone.

Explaining local characteristics of periurbanisation: geographic facts or historical resilience?

One of our hypotheses to explain this phenomenon is that periurban spatial organization is linked to historical resilience. In other word, we think that historical or geo-historical facts (such as distance between cities, boundaries of municipalities or rural population before rural-urban migration) could explain a part of today's periurban spatial organization. The first spatial repartition we want to explain is the concentration index. This index is based on commuters flows between municipalities and a problem of scale appears. In fact, French municipalities have different areas, so a high concentration index could just be caused by a territory with wide municipalities. To cancel this effect, we have done the map of municipalities' density in France and we have observed that areas with a high density of municipalities are areas with low periurban concentration (see Map 2). So we have decided to add this parameter (density of municipalities) in our previous statistical model and we have obtained a very good result (R²=0,743, see the formula in Map 6). Moreover the residuals map of this new model reveals a new spatial organization highlighting a greater concentration of periurbanisation in the Rhone valley, in the west and the north-east of France and in the north-west of Paris (see map 3). This analysis prevents us against the danger of using non regular geographical entities and shows how geo-historical facts could affect statistical index. In fact the boundaries of municipalities have not been created by chance: they have been created during the French revolution of 1789 but are based on old parishes (which size was depending of agrarian system). Area of municipality does not appear as a significant element in the spatial repartition of range (R²=0,002) except in Arcachon (on the South-West coast) where it clearly appears that the high range of periurban commutes is due to the extreme size of municipalities around the agglomeration (more than 10 times the average area of French municipalities). Nonetheless we have another geographical hypothesis to explain residuals: the urban system and concurrence between cities. This hypothesis is based on the Reilly law: if an agglomeration is far from all the others, it can extend its influence area wildly, on the contrary if an agglomeration is in a dense urban system, its influence area enters in concurrence with other agglomerations. We have not been able to link with a good correlation coefficient the range of periurbanisation and the density of the urban system. But the comparison between the map of density of urban system and the map of median range model's residuals reveals that some areas, where range of periurbanisation is lower than the model (North of France, East and Rhone valley), are also areas with a very dense urban system: the only exception is Normandy (see Map 4). The last residuals repartition to explain is the intensity index. For us, the differences in the spatial repartition of periurbanisation intensity might be explained by "rural tradition" of territories, which could influence the wish of people to live out of agglomerations. In other words, territories with a high density of rural population before the "rural exodus" may have conserved a rural tradition, so more people would like to settle in the country land. To prove our hypothesis, we have used data from the French census of 1861 to know the density of rural population before rural-urban migration. Results are quite conclusive, we observe a very good similarity between the residuals map of intensity model and the map of density of rural population in 1861 (see map 5). Of course this comparison is not enough to prove that "rural tradition" is a cause of high periurban intensity but it shows the resilience of historic facts in nowadays urban organization.

CONCLUSION

Periurban spatial organization in France can be explained by core agglomeration population: the intensity of periurbanisation decreases with the number of inhabitants in the agglomeration although the range of commuting and the concentration of periurban people increase with agglomeration size. We have also highlighted some local or regional characteristics. So, as a final result of our research, we are able to present a global statistical model of periurban spatial structure and a synthesis map which shows the local particularities of this structure (see map 6). Moreover, we have shown that a part of these local characteristics seems to be explained by geographical facts and historical resilience, such as urban system density or importance of rural population during the 19 th century. So it may be interesting to do further research on resilience in periurban spatial organization.

Map 2 :

 2 Density of municipalities compared to the residuals of the concentration model. Map 3: Location of residuals of our new concentration statistical model (population of agglomeration + density of municipalities)

Map 4 :

 4 Density of municipalities compared to the residuals of the concentration model.

Map 5 :

 5 Density of rural people in 1861 compared to the residuals of the intensity model.

Map 6 :

 6 Synthesis map of local characteristics of periurban spatial organization in France

Table 1 :

 1 Synthesis of the statistical models of spatial organization of periurbanisation

ACKNOWLEDGMENTS

I would like to thank Francis BEAUCIRE, my PhD director, and all the PhD students of the "tempête dans les transports" brain storming team for their precious advices and their help. Thanks to Fred, Marion and Alice for the review.


G Hakhumyan

C Leroy

Y Pashayan-Leroy

D Sarkisyan

M Auzinsh

Keywords: Zeeman Hamiltonian, atomic transition intensity, frequency shift, submicron thin vapor

We have implemented the so-called λ-Zeeman technique (LZT) to investigate individual hyperfine transitions between Zeeman sublevels of the Rb atoms in a strong external magnetic field B in the range of 2500 -5000 G (recently it was established that LZT is very convenient for the range of 10 -2500 G).

Atoms are confined in a nanometric thin cell (NTC) with the thickness L = λ, where λ is the resonant wavelength 794 nm for Rb D 1 line. Narrow velocity selective optical pumping (VSOP) resonances in the transmission spectrum of the NTC are split into several components in a magnetic field with the frequency positions and transition probabilities depending on the B-field. Possible applications are described, such as magnetometers with nanometric local spatial resolution and tunable atomic frequency references.

Introduction

A number of optical and magneto-optical processes running at interaction of a narrow-band laser radiation with atomic vapors are employed in laser technology, metrology, designing of high-sensitivity magnetometers, problems of quantum communications, information storage etc [1,[START_REF] Budker | AtomicPhysics[END_REF]. As known, the energy levels of atoms undergo frequency shifts and changes in their transition probabilities in an external magnetic field B. The related effects were studied for hyperfine (hf) atomic transitions in optical domain for the transmission spectra obtained with an ordinary cm-size cell containing Rb and Cs vapor in [START_REF] Tremblay | [END_REF]. However, because of Doppler broadening (hundreds of MHz), it was possible to partially separate different hf transitions only for B > 1500 G. Note that even for these large B values the lines of the 85 Rb and the 87 Rb are strongly overlapped, and pure isotopes should be used to avoid complicated spectra. In order to eliminate the Doppler broadening, the well-known saturation absorption (SA) technique was implemented to study the Rb hf transitions [4,5]. However, in this case the complexity of the Zeeman spectra in a magnetic field arises primarily from the presence of strong crossover resonances, which are also split into many components.

That is why, the SA technique is applicable only for B < 100 G. Another significant disadvantage is the fact that the SA is strongly nonlinear and, therefore, the peak amplitudes of the decreased absorption do not correspond to probabilities of atomic transitions at frequencies of which these peaks are formed. This additionally complicates the processing of the spectra. The crossover resonances can be eliminated with selective reflection spectroscopy [6], but to correctly determine the hf transition position, the spectra must undergo further non-trivial processing. It was demonstrated in [7,8] that the use of resonance fluorescence spectra of a NTC filled with Rb atomic vapor and having a thickness of L = 0.5λ (where λ = 794 nm is the wavelength of laser radiation whose frequency is resonant to the atomic transition of the D 1 line of Rb) allows one to separate and to study the atomic transitions between the levels of the hyperfine structure of the D 1 line of the 87 Rb atoms in magnetic fields with B = 10 -200 G. The achieved high sub-Doppler spectral resolution is caused by the effect of a strong narrowing of the fluorescence spectrum of a NTC with the atomic vapor column thickness L = λ/2. With the proper choice of the laser intensity and NTC temperature, it is possible to achieve an eightfold narrowing of the spectrum compared to that in ordinary 1 -10 cm-long cells (for which the Doppler width is about 500 MHz). The method was called "half-lambda Zeeman technique" (HLZT). In [9,10], the D 2 lines of Rb and Cs atoms were studied using HLZT in magnetic fields of about 50 G. Recently it has been shown that the use of the HLZT allows one to perform detailed quantitative measurements of both frequency characteristics and probabilities of large number of atomic transitions between levels of the hyperfine structure of the Rb D 1 line in magnetic fields ranging within 10 -2500 G [11]. In [12,13] it is described a technique based on the use of spectrally-narrow (close to natural line-width) velocity selective optical pumping (VSOP) resonances peaks appearing at laser intensities ∼ 10 mW/cm 2 exactly at the positions of atomic transitions in the transmission spectrum of the NTC with the Rb vapor column thickness of L = λ (λ being the wavelength of laser radiation resonant with Rb D 1 or D 2 atomic lines, 794 or 780 nm). Each VSOP resonance is split in an external magnetic field into several Zeeman components, the number of which depends on the quantum numbers F of the lower and upper levels. The amplitudes of these components and their frequency positions depend unambiguously on B-field. Hence, with the use of LZT it is also possible to study not only the frequency shift of any individual hf Zeeman transition, but also the modification in transition probability. The efficiency of LZT has been proved in the region of 1 -2500 G [13]. Among the most interesting results obtained through HLZT and LZT implementation are vanishing of some of atomic transitions for specific B-field strength, and the appearance of ∆F = 2 transitions, which are forbidden for B = 0 [13]. Moreover, the line intensity of a forbidden transition for some value of magnetic field can be higher than that for strong atomic transitions when B = 0. It is important to note that LZT could have several advantages as compared with the HLZT: in particular, the spectral width of VSOP resonances used in LZT is at least 4 times narrower than that used in HLZT, resulting in much higher resolution; also the laser power required for LZT is ∼ 10 times lower than that needed for HLZT; and finally, recording of resonant transmission spectra does not require sensitive detectors as in the case of fluorescence spectra. The aim of the work is to demonstrate that LZT can also be successfully used for even higher external magnetic fields up to 5000 G. Possible applications of LZT for diagnostics and mapping of large magnetic gradients and for making widely tunable compact frequency references are addressed too.

Experiment

Nanometric-thin cell

The first design of the NTC (called extremely thin cell) consisting of windows and a vertical side-arm (a metal reservoir), was presented in [14]. Later, this design was somewhat modified and a typical example of recent version is presented in [START_REF] Sarkisyan | Optical processes in micro-and nanometric thin cells containing atomic vapor[END_REF]. The photograph of the NTC with smoothly variable thickness wedged in the vertical direction is shown in Fig. 1. The wedged gap in this case was formed using a platinum spacer strip of 2 µm thickness. The presented NTC has garnet windows of 0.8 mm thickness (the use of thin wafers in some cases is more convenient), and in order to increase the wafers thickness at the bottom to fit the side-arm (∅ = 2 mm), two additional garnets plates are glued to the main wafers. The NTC is filled with a natural mixture of the 85 Rb (72.2%) and 87 Rb (27.8%). The region of L ≈ λ ≈ 800 nm is indicated.

The temperature limit of the NTC operation is 400 • C. The NTC operated with a specially designed oven with two ports for laser beam transmission. The source temperature of the atoms of the NTC was 120 • C, corresponding to the vapor density N = 2 • 10 13 cm -3 , but the windows were maintained at a temperature that was 20 • C higher.

Experimental setup

The sketch of the experimental setup is shown in Fig. 2. The circularly polarized beam of extended cavity diode laser (ECDL, λ = 794 nm, P L = 30 mW, γ L < 1 MHz) resonant with the 87 Rb D 1 transition frequency, was directed onto the Rb NTC (2 ) with the vapor column thickness L = λ = 794 nm at nearly normal incidence. The NTC was inserted in a special oven with two openings. The transmission signal was detected by a photodiode (4 ) and was recorded by Tektronix TDS 2014B digital four-channel storage oscilloscope (5 ). A Glan prism was used to purify initial linear radiation polarization of the laser radiation; to produce a circular polarization, a λ/4 plate (1 ) was utilized. Magnetic field was directed along the laser radiation propagation direction k (B k). About 50% of the pump power was branched by a beam splitter to an auxiliary Rb NTC (2 ′ ). The fluorescence spectrum of the latter at L = λ/2 was used as a frequency reference for B = 0. Moderate longitudinal magnetic field (B < 250 G) was applied to the NTC by a system of Helmholtz coils (not shown in Fig. 2). The B-field strength was measured by a calibrated Hall gauge. It is important to note that the use of the NTC allows one to apply very strong magnetic fields using widely available strong permanent ring magnets (PRM): in spite of strong inhomogeneity of magnetic field (in our case it can reach 150 G/mm), the variation of B-field inside atomic vapor column is ∼ 0.1 G, i.e. by several orders less than the applied B value, thanks to small thickness of the NTC (L = 794 nm).

Experimental results and discussion

The allowed transitions between magnetic sublevels of hf states for the 85 Rb and 87 Rb, D 1 line in the case of σ + (left circular) polarized excitation and selection rules ∆m F = +1 are depicted in Fig. 3 (LZM works well also for σ -excitation). In [13] the maximum attainable strength of B-field was 2500 G. In order to increase the attainable strength of the external magnetic field applied to the atomic vapor contained in NTC, the permanent ring magnets embracing the cell should be as close to each other as possible. The main limitation for the distance between PRMs is imposed by the longitudinal dimension of the cell oven (see Fig. 2). Earlier, in [13] the longitudinal dimension of the oven was 4 cm resulting in the maximum attainable strength of 2500 G. A new oven was developed specially for high B-field applications, with the longitudinal dimension of ≈ 2 cm. This allows us to produce the attainable strength of B-field > 2500 G.

As it was shown [13] LZT is based on the use of spectrally-narrow VSOP resonances appearing at laser intensities ∼ 10 mW/cm 2 in the transmission spectrum of the NTC with the thickness L = λ. The VSOP peaks of reduced absorption are exactly at the atomic transitions, and these VSOPs are split into several MHz/G versus magnetic field strength. It is important to note that LZT allows one to check whether the VSOP resonance (i.e. peak of reduced absorption) is a real one or has an artificial/noise nature. For this purpose one should simply increase the side-arm temperature by additional 20 -30 degrees in order to provide larger absorption in the transmission spectrum. The real VSOP must be located exactly at the bottom of the absorption at the position of the corresponding atomic transition shifted in the magnetic field, as it is shown on the upper curve of Fig. 4 (the side-arm temperature is 140 • C). As it is mentioned the strong magnetic field is produced by two PRMs (∅ = 30 mm), with holes (∅ = 3 mm) to allow radiation to pass, placed on opposite sides of the nanocell oven and separated by a distance varied between 20 mm and 35 mm (see Fig. 2). To control the magnetic field value, one of the magnets was mounted on a micrometric translation stage for longitudinal displacement. The NTC transmission spectra for the thickness L = λ, for the 85 Rb and 87 Rb, D 1 line in the case of σ + excitation at B = 3870 G (the upper curve) are presented in Fig. 5. The labels 1 -9 denote the corresponding transitions between the magnetic sublevels shown in Fig. 3. The lower grey curve presents the fluorescence spectrum of the NTC of L = λ/2, which is the reference spectrum for the case B = 0. A new VSOP with label 9 is seen in the spectrum (the corresponding atomic transition is shown in Fig. 3), while it was absent for the case of smaller B-field shown in Fig. 4. As it is seen from Fig. 5 the most convenient is still the VSOP peak number 1 for magnetic field measurement.

F g =2 87 Rb, D 1 , σ + F e =2 F e =1 F g =1 +2 -2 +2 -2 -1 -1 0 0 +1 +1 -1 +1 0 -1 +1 0 1 2 3 1' 2' (a) F g =2 85 Rb, D 1 , σ + F e =3 F e =2 F g =3 +2 -2 -1 +1 0 +3 -3 +2 -2 -1 0 +1 +2 -2 -1 +1 0 +3 -3 +2 -2 -1 0 +1 4
The following control experiment was carried out: one of PRMs was set on the table with the micrometer step. In the magnetic field ∼ 4000 G, one PRM was shifted toward the other by the displacement of PRMs frequency reference based on a nanocell and PRMs, widely tunable over a range of several gigahertz onto high-frequency wing of transition of 87 Rb atom by simple displacement of the magnet.

Theoretical model and discussions

In this work the main interest is to study the behavior of the 87 Rb, F g = 1 → F e = 2 transitions, D 1 line in the case of σ + excitation, since from the application point of view they are most interesting.

Note, that for σ -excitation the probability of the atomic transitions rapidly decreases with magnetic field strength. Also, the probability of the atomic transitions 87 Rb, F g = 1 → F e = 1 in the case of σ + excitation rapidly decreases with magnetic field strength. Theoretical model describes how to provide the calculations of separated transition's frequencies and amplitude modification undergo external magnetic field [START_REF] Tremblay | [END_REF]6,7,8,9,10,[START_REF] Auzinsh | Optically polarized atoms: understanding light-atom interactions[END_REF]. We adopt a matrix representation in the coupled basis, that is, the basis of the unperturbated atomic state vectors |(n = 3), L, J, F, m F to evaluate the matrix elements of the Hamiltonian describing our system. In this basis, the diagonal matrix elements are given by

F, m F |H|F, m F = E 0 (F ) + µ B g F m F B Z , (1)
where E 0 (F ) is the initial energy of the sublevel |(n = 3), L, J, F, m F ≡ |F, m F and g F is the effective Landé factor. The off-diagonal matrix elements are non-zero for levels verifying the selection rules ∆L = 0, ∆J = 0, ∆F = ±1, ∆m F = 0, The diagonalization of the Hamiltonian matrix allows one to find the eigenvectors and the eigenvalues, that is to determine the eigenvalues corresponding to the energies of Zeeman sublevels and the new states vectors which can be expressed in terms of the initial unperturbed atomic state vectors,

F -1, m F |H|F, m F = F, m F |H|F -1, m F = -µ B Bz 2 (g J -g I ) × [(J+I+1) 2 -F 2 ][F 2 -(J-I) 2 ] F 1/2 F 2 -m 2 F F (2F +1)(2F -1) 1/2 . (2)
Ψ(F ′ e , m Fe ) = Fe=I+Je Fe=I-Je α e F ′ e Fe (B z , m Fe )|F e , m Fe (3)
and

Ψ(F ′ g , m Fg ) = Fg=I+Jg Fg=I-Jg α g F ′ g Fg (B z , m Fg )|F g , m Fg . (4)
The state vectors |F e , m e and |F g , m g are the unperturbated state vectors, respectively, for the excited and the ground states. The coefficients α e F ′ e Fe (B z , m Fe ) and α g F ′ g Fg (B z , m Fg ) are mixing coefficients, respectively, for the excited and the ground states; they depend on the field strength and magnetic quantum numbers m e or m g . Diagonalization of the Hamiltonian matrix for 87 Rb, D 1 line, in case of σ + polarization of exciting radiation, allows one to obtain the shift of position of energy levels in presence of external magnetic field. Fig. 7 shows the frequency shift of components 1, 2 and 3 (see diagram in Fig. 3(a)) relative to their initial position at B = 0. The probability of a transition, induced by the interaction of the atomic electric dipole and the oscillating laser electric field is proportional to the spontaneous emission rate of the associated transition A eg , that is, to the square of the transfer coefficients modified by the presence of the magnetic

field 8π 2 3ε 0 λ 3 eg | e |D q | g | 2 = A eg ∝ a 2 Ψ(F ′ e , m Fe ); Ψ(F ′ g , m Fg ); q , (5)
where D q denote the standard components of the electric dipole moment: with q = -1, 0, 1. In Eq. ( 5), the transfer coefficients are expressed as

D. e = q D q e q , (6)
a Ψ(F ′ e , m Fe ); Ψ(F ′ g , m Fg ); q = FeFg α e F ′ e Fe (B z , m Fe )a F e , m Fe ; F g , m Fg ; q α g F ′ g Fg (B z , m Fg ), (7)
where the unperturbated transfer coefficients have the following definition

a F e , m Fe ; F g , m Fg ; q = (-1) 1+I+Je+Fe+Fg-m Fe × √ 2J e + 1 √ 2F e + 1 2F g + 1   F e 1 F g -m Fe q m Fg      F e 1 F g J g I J e    , (8)
the parenthesis and curly brackets denote, respectively, the 3j and 6j symbols, g and e point respectively ground and excited states. atomic line intensity) for the case of σ + excitation versus magnetic field for (theory) versus B. However, in the experiment it is more convenient to measure the ratio of the VSOPs amplitudes A1, A2 and A3 of transitions 1, 2 and 3 versus B (shown in Fig. 8(b)), since the absolute value of the VSOP amplitude depends on laser intensity, NTC temperature, etc. Note that for B ≈ 0 the ratios A1 : A2 : A3 = 6 : 3 : 1 , while for large B, these ratios become A1 ≈ A2 ≈ A3. Also, as it is seen for B up to 5000 G the VSOP amplitude A1 is increasing, which makes it convenient to use VSOP with label 1 for a magnetic field measurement. It is obvious that a NTC with an oven can also be fixed on a table with a micrometer step so that the displacements of this system allow one to map strongly inhomogeneous magnetic fields.

For a more successful mapping, the dimensions of the NTC with oven could, in principle, be decreased further; i.e., a conducting and optically transparent deposited layer can replace the oven and the window thickness can be reduced to 100 µm with the reduction of the transverse dimensions of the NTC down a few millimeters. It should be noted that, with regard to sensitivity, the magnetometer based on LZT is far below magnetometers based on coherent processes [1,[START_REF] Budker | AtomicPhysics[END_REF], but has advantages in the measurement of strong and gradient magnetic fields. Note also that the spectral resolution achieved by LZT may also be realized with the use of well-collimated atomic beams (by means of 3 -4 m vacuum pipes where the beam will be formed) or plants for cooling of atoms. However, employment of the atomic-beam technology or atom cooling is very complicated and expensive problem, whereas LZT requires only available cheap diode lasers and a nanocell filled with alkali metal.

Conclusion

The "λ-Zeeman technique" is shown to be a convenient and robust tool for the study of individual transitions between the Zeeman sublevels of hyperfine levels in an external magnetic field of 10 -5000 G (taking into account previously obtained results of the LZT use for the range of 10 -2500 G). LZT is based on NTC resonant transmission spectrum with thickness L = λ, where λ is the resonant wavelength (794 nm) for D 1 line of the Rb. Narrow VSOP resonances (of ∼ 20 -30 MHz linewidth) in the transmission spectrum of the NTC are split into several components in a magnetic field; their frequency positions and transition probabilities depend on the B field. Examination of the VSOP resonances formed in a nanometric-thin cell allows one to obtain, identify, and investigate the atomic transitions between the Zeeman sublevels in the transmission spectrum of the 87 Rb D 1 line in the range of magnetic fields 10 -5000 G. Nanometric-thin column thicknesses (∼ 794 nm) allow of the application of permanent magnets, which facilitates significantly the creation of strong magnetic fields. The results obtained show that a nano -magnetometer in the range of 10 -5000 G with a local spatial resolution of ∼ 794 nm can be created based on a NTC and the atomic transition of the 87 Rb D 1 , F g = 1, m F = +1 → F e = 2, m F = +2. This result is important for mapping strongly inhomogeneous magnetic fields. LZT can be successfully implemented also for the case of Cs, Na, K, Li atoms. Experimental results are in a good agreement with the theoretical values.

Figure 1 :

 1 Figure 1: Photography of the NTC. Since NTC gap thickness is of order of visible light wavelength, interferometric pattern occurs. Region of L = λ = 800 nm is indicated, NTC has a sapphire side-arm.

Figure 2 :

 2 Figure 2: Sketch of the experimental setup. ECDL -diode laser, FI -Faraday isolator, 1 -λ/4 plate, 2 -NTC in the oven, 2 ′ -an auxiliary NTC and the oven, 3 -ring magnets, 4 -photodetectors, 5 -digital storage oscilloscope.

Figure 3 :

 3 Figure 3: The hfs energy level diagram of D 1 line of 87 Rb (a) and 85 Rb (b) in magnetic field and possible atomic Zeeman transitions for σ + polarized exciting laser radiation.

Figure 4 :

 4 Figure 4: NTC transmission spectra for L = λ for the transition between sublevels of hf states for the 85 Rb and 87 Rb D 1 line in the case of σ + excitation at B = 2910 G (the upper curve) and B = 2430 G (middle curve). The lower grey curve presents the reference spectrum, which is the fluorescence spectrum of the NTC of L = λ/2, for B = 0. Oblique arrows indicate the positions of the VSOPs resonance frequencies with the labels 1 -8 for B = 2430 G and B = 2910 G.

by 15 µm leading to the frequency shift of component 1 by 4 2 1100%Figure 5 :

 425 Figure 5: NTC transmission spectra for L = λ for the transition between sublevels of hf states for the 85 Rb and 87 Rb D 1 line for the case of σ + excitation at 3870 G (the upper curve). The lower grey curve presents the reference spectrum, which is the fluorescence spectrum of the NTC of L = λ/2, for B = 0.

6 Reference, 2 1Figure 6 :

 626 Figure 6: NTC transmission spectra for L = λ for the 85 Rb and 87 Rb D 1 line in the case of σ + excitation at B = 4770 G (the upper curve). The lower grey curve presents the reference spectrum, which is the fluorescence spectrum of the NTC of L = λ/2, for B = 0.

Figure 7 :

 7 Figure 7: Transition's frequency of components 1, 2 and 3 relatively to the initial position at B = 0, solid line is the theory.

Fig. 8 (Figure 8 :

 88 Figure 8: (a) The probability for the atomic transition 1, 2, 3 in the case of σ + excitation versus magnetic field (theory), (b) curve 1 shows the ratio of the amplitudes A(1)/A(3) and curve 2 shows the ratio of the amplitudes A(1)/A(2) versus B.

Acknowledgements

The authors are grateful to A. Sarkisyan for his valuable participation in fabrication of the NTC as well as to A. Papoyan and A. Sargsyan for useful discussions. Research conducted in the scope of the International Associated Laboratory IRMAS. Armenian team thanks for a research grant Opt 2428 from the Armenian National Science and Education Fund (ANSEF) based in New York, USA.


PhD Cécile Proust-Lima

PhD Jeremy Mg Taylor

PhD Solène Sécher

MD Howard Sandler

MD Larry Kestin

MD Tom Pickles

PhD Kyoungwha Bae

Roger Allison

MD FRANZCR Scott Williams
email: scott.williams@petermac.org

Confirmation of a low α/β ratio for prostate cancer treated by external beam radiation therapy monotherapy using a post-treatment repeated measures model for PSA dynamics

Keywords: Prostate cancer, Prostate-specific Antigen, Progression of disease, Radiation therapy, Alpha/beta, Radiosensitivity

Purpose: To estimate the  ratio of prostate cancer treated with external beam radiation only using a model of long term prostate specific antigen (PSA) dynamics.

Materials and methods

: Repeated measures of PSA from 5,093 patients from 6 institutions treated for localized prostate cancer by external beam radiation therapy (EBRT) without planned androgen deprivation were analysed. A biphasic linear mixed model described the post-treatment evolution of PSA, rather than a conventional model of time to biochemical recurrence. The model was adjusted for standard prognostic factors (T-stage, initial PSA and Gleason) and cohort specific effects. The radiation dose fractionation effect was estimated from the long term rate of rise of PSA.

Results: Adjusted for other factors, total dose of EBRT and sum of squared doses per fraction were associated with long term rate of change of PSA (respectively p=0.0017 and p=0.0003), an increase of each being associated with a lower rate of rise. The ratio  was estimated at 1.55Gy (95% confidence bands 0.46, 4.52Gy). This estimate was robust to adjustment of the linear mixed model.

Conclusions:

Analysing a large EBRT-only cohort along with a method that utilises all the repeated measures of PSA after end of treatment, a low and precise  was estimated. These data support the use of hypofractionation at fractional doses of 1.8-2.8Gy, but cannot presently be assumed to accurately represent higher doses per fraction.

The radiation oncology literature is permeated with suggestions that the apparent sensitivity of prostate cancer to hypofractionated radiotherapy is a valuable therapeutic target [START_REF] Duchesne | What is the alpha/beta ratio for prostate cancer? Rationale for hypofractionated high-dose-rate brachytherapy[END_REF][START_REF] Fowler | What hypofractionated protocols should be tested for prostate cancer?[END_REF][START_REF] Brenner | Fractionation and protraction for radiotherapy of prostate carcinoma[END_REF][START_REF] Dasu | Is the alpha/beta value for prostate tumours low enough to be safely used in clinical trials?[END_REF]. When quantifying the dose and dose-per-fraction (DPF) relationship with a linear-quadratic biological model, the consensus is that the derived  ratio is substantially lower in prostate cancer than in most cancers, with typical purported values between 1 and 3Gy [START_REF] Brenner | Fractionation and protraction for radiotherapy of prostate carcinoma[END_REF][START_REF] Brenner | Direct evidence that prostate tumors show high sensitivity to fractionation (low alpha/beta ratio), similar to late-responding normal tissue[END_REF][START_REF] Fowler | Is alpha/beta for prostate tumors really low?[END_REF][START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF]. While we are yet to see any prospective controlled studies to confirm these observational data, the hypothesis generated by them is that larger DPF therapies may be optimal therapy in this cancer, which has been incorporated into several clinical techniques [START_REF] Demanes | High dose rate prostate brachytherapy: the California Endocurietherapy (CET) method[END_REF][START_REF] Kupelian | Higher than standard radiation doses (> or =72Gy) with or without androgen deprivation in the treatment of localized prostate cancer[END_REF][START_REF] Martinez | Interim report of image-guided conformal highdose-rate brachytherapy for patients with unfavorable prostate cancer: the William Beaumont phase II dose-escalating trial[END_REF].

In applying the available radiobiology knowledge to clinical protocols in prostate cancer,  ratio estimates which have displayed the tightest estimates are, quite understandably, those which clinicians have relied on. These estimates have unfailingly used external beam radiation therapy (EBRT) in combination with brachytherapy data to date. From a modelling viewpoint, this combined data is required as it enables one to either: theoretically eliminate the β component from calculations involving the low-dose-rate (LDR) brachytherapy data [START_REF] Brenner | Fractionation and protraction for radiotherapy of prostate carcinoma[END_REF][START_REF] Fowler | Is alpha/beta for prostate tumors really low?[END_REF][START_REF] Wang | How low is the alpha/beta ratio for prostate cancer?[END_REF]; or to have access to large enough DPF variation to successfully model both the  and  components by using hypofractionated highdose-rate (HDR) brachytherapy data [START_REF] Brenner | Direct evidence that prostate tumors show high sensitivity to fractionation (low alpha/beta ratio), similar to late-responding normal tissue[END_REF][START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF]. Each of these approaches can be criticized on a number of fronts, such as including yet not accounting for uncertainties associated with comparing the relatively homogenous dose of EBRT to highly inhomogenous brachytherapy dose distributions [START_REF] Brenner | Hypofractionation for prostate cancer radiotherapy--what are the issues?[END_REF].

Inherently, these combined modality data should not be extrapolated to the design of EBRT-only fractionation strategies.

A more acceptable approach would be to derive the  ratio from fractionation studies incorporating EBRT only. Unfortunately, the largest randomised study to date comparing conventional DPF and hypofractionation [START_REF] Lukka | Randomized trial comparing two fractionation schedules for patients with localized prostate cancer[END_REF] has failed to provide precise estimates [START_REF] Bentzen | The alpha/beta ratio for prostate cancer: what is it, really?[END_REF], while data from observational models is similarly imprecise when using only EBRT data [START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF]. One constant throughout all the previous studies has been the use of a conventional binary failure endpoint to assess outcome, such as a time-to-recurrence endpoint (biochemical or clinical). Yet, repeated measures of PSA are collected after therapy; the evolution of which can be described using a linear mixed model. We have previously provided a framework for evaluating prognostic and other factors that may be associated with disease progression in such a model of prostate cancer [START_REF] Proust-Lima | Determinants of change in prostate-specific antigen over time and its association with recurrence after external beam radiation therapy for prostate cancer in five large cohorts[END_REF]. By incorporating the complete PSA history, rather than reducing it to a somewhat arbitrary single point, such as biochemical recurrence, this modelling approach has the potential to greatly increase the power with which prognostic variables can be interpreted. A finding from previous work [START_REF] Proust-Lima | Determinants of change in prostate-specific antigen over time and its association with recurrence after external beam radiation therapy for prostate cancer in five large cohorts[END_REF] using this mixed modelling strategy was that the dose of radiation was associated with the pattern of post treatment PSA values, but that given the pattern of PSA the dose of radiation provided no further significant information about clinical recurrence. This provides a rationale for the approach in the current paper where we are investigating the effect of the radiation fractionation schedule on disease progression and we can do that by using just the linear mixed model approach focussing on the PSA values after treatment and not considering clinical recurrence. The present studies aim is to use this modelling strategy to attempt to describe the  ratio using data from a large number of men treated with EBRT containing little dose inhomogeneity, where this has been unable to be done prior with adequate precision.

METHODS AND MATERIALS

Patient cohorts

The analysis included all patients treated for a prostate cancer by external beam radiation therapy (EBRT) from 6 different cohorts: University of Michigan, Ann Arbor, Michigan (UM) [START_REF] Taylor | Individualized predictions of disease progression following radiation therapy for prostate cancer[END_REF], Radiation Therapy Oncology Group (RTOG 9406) [START_REF] Roach | Penile bulb dose and impotence after threedimensional conformal radiotherapy for prostate cancer on RTOG 9406: findings from a prospective, multi-institutional, phase I/II dose-escalation study[END_REF], Peter MacCallum Cancer Center, Melbourne, Australia (PMCC) [START_REF] Williams | An international multicenter study evaluating the impact of an alternative biochemical failure definition on the judgment of prostate cancer risk[END_REF], William Beaumont Hospital, Detroit, Michigan (WBH) [START_REF] Kestin | Defining biochemical cure for prostate carcinoma patients treated with external beam radiation therapy[END_REF], Royal Brisbane Hospital, Brisbane, Australia (RBH) [START_REF] Williams | Both pretreatment prostate-specific antigen level and posttreatment biochemical failure are independent predictors of overall survival after radiotherapy for prostate cancer[END_REF] and British Columbia Cancer Agency, Vancouver, Canada (Vancouver) [START_REF] Pickles | Evaluation of the Houston biochemical relapse definition in men treated with prolonged neoadjuvant and adjuvant androgen ablation and assessment of follow-up lead-time bias[END_REF]. Loco-regional and systemic disease was assessed at the discretion of the treating physician, and typically included computed tomography of the abdomen and pelvis, along with whole body bone scanning. Surgical staging of lymph nodes was done in <1% of cases. Those having planned neoadjuvant or adjuvant androgen deprivation therapy (ADT) were ineligible. All patients had clinically localized prostate cancer of clinical stage T1 to T4 and were node and metastasis negative. All data were acquired under Institutional Review Board approval at the respective institutions and at UM, where the data were aggregated and analysed.

In summary, EBRT (either 2D or 3D conformal) was targeted to the prostate or prostate plus seminal vesicles depending on clinical risk. Whole pelvic radiotherapy was not routinely performed. All cases were required to have at least one year follow-up without clinical recurrence or salvage ADT and at least two PSA measurements before end of follow-up. All PSA measures collected after EBRT until the end of follow-up (minimum time to clinical recurrence or lost to follow-up) or initiation of salvage ADT were analysed. PSA measures were logarithmically transformed using ln(PSA+0.1 ng/mL) to satisfy the normality assumptions for inclusion in the statistical models.

Statistical analysis

Distributions of prognostic factors over cohorts were compared using Kruskall-Wallis test for continuous variables and Chi-square test for categorical variables. Change over time of PSA after end of radiation therapy was described using a linear mixed model (LMM) [START_REF] Laird | Random-effects models for longitudinal data[END_REF]. Evolution of PSA was separated into 3 components as proposed by Proust-Lima et al. [START_REF] Proust-Lima | Determinants of change in prostate-specific antigen over time and its association with recurrence after external beam radiation therapy for prostate cancer in five large cohorts[END_REF] and described in figure 1: [START_REF] Duchesne | What is the alpha/beta ratio for prostate cancer? Rationale for hypofractionated high-dose-rate brachytherapy[END_REF] post-treatment level of PSA, denoted as post-treatment PSA (ptPSA) ; (2) short term evolution during the first year after end of EBRT, approximated by f 1 (t)=((1+t) -1.5 -1) and denoted as short term drop of PSA (stPSA) ; (3) long term evolution of PSA, described by a linear trend and denoted as long term rise of PSA (ltPSA).

The linear mixed model accounted for inter-individual variability through individual correlated Gaussian random-effects for each component (ptPSA, stPSA and ltPSA), as well as intra-individual correlation through correlated errors in addition to the independent Gaussian measurement errors (detailed in Appendix eI). The full model was adjusted for iPSA, T-stage and GS, and their interactions with short term drop and long term rise of PSA. The  ratio was derived from the dose and squared dose effect evaluated on long term PSA rise. The analysis also adjusted for the dose and squared dose effects on post-treatment level of PSA and short term drop of PSA. A sensitivity analysis examined a proposed parsimonious model [START_REF] Proust-Lima | Determinants of change in prostate-specific antigen over time and its association with recurrence after external beam radiation therapy for prostate cancer in five large cohorts[END_REF]. Associations were tested using the Wald test and likelihood ratio test (LRT).

For the pooled data, cohort-specific intercepts, short term drops and long term rises were included to correct for cohort heterogeneity. The  ratio was estimated as the ratio of the effect of dose and squared dose on long-term rise of PSA. Confidence bands (CB) were obtained from the profile likelihood as described in the appendix. Statistical analyses were performed with SAS statistical software version 9.1.3 (SAS Institute, Cary, NC). Statistical tests were at the two-sided =0.05 level of significance.

RESULTS

Description of cohorts

Summary variables for he 6 cohorts are described in Table 1. The median follow-up was 4.7 years, with relatively shorter time of follow-up for UM and WBH (respectively a median of 4.1 and 3.7 years) and longer time for RBH with a median of 5.3 years. The prognostic factor distribution differed significantly over cohorts (p<0.0001 for each factor). RTOG included more T-stage 1 and less T-stage 3 and 4 while PMCC and Vancouver had a converse distribution. The proportion of patients having a GS of 7 varying from 19.9% for BM to 37.4% for UM, while high grade cancers were relatively less frequent in the PMCC data (4.8%) than at WBH (9.0%). Finally, the median institutional pre-treatment PSA level ranged from 7.4 ng/mL for RTOG to 15.0 ng/mL for RBH.

Total dose of EBRT and dose per fraction

A large range of total dose of EBRT was observed (50 -84.6Gy). The total dose distribution varied significantly across cohorts (p<0.0001) from a median of 63.2Gy to 78.0Gy for RBH and RTOG respectively (Table 1). The dose per fraction varied from 1.70Gy to 2.77Gy varied significantly across cohorts (p<0.0001) from a median of 1.80Gy for WBH to 2.11Gy for RBH.

 estimation

Parameter regression coefficients for dose and squared dose using the biphasic linear mixed model are given in Table 2. Total dose of EBRT was associated with post-treatment initial level of PSA (p=0.0002); a higher total dose being associated with a lower post-treatment level of PSA. Fractional dose was not associated with post-treatment PSA level (p=0.876). Neither dose nor squared dose were significantly associated with the short term drop of PSA (respectively p=0.135 and p=0.163). The total dose and dose were highly associated with long term PSA rise, with both higher total doses (p=0.0017) and doses per fraction (p=0.0003 associated with a lower long term rate of rise of PSA. This association led to an estimation of  =-0.0062/-0.0040=1.55Gy with 95%CB of 0.46Gy to 4.52Gy.

Robustness of the estimate

To evaluate whether this estimate was robust to specification of the linear mixed model, we compared the estimate of  for different specifications of the structure of correlation of the repeated measures, different adjustment for prognostic factors and different adjustment for dose on post-treatment PSA and short term drop of PSA. Using complete or parsimonious adjustment for prognostic factors did not substantially change the estimate of nor its 95% confidence band (=1.58, 95%CB=0.46; 4.69).

Accounting for the correlation between repeated measures of PSA improved markedly the fit of the data (log-likelihood improved by 3200 points). The estimate of  was slightly reduced if independent errors were assumed (=1.15, 95%CB =0.24; 3.23). Not adjusting for squared dose on posttreatment PSA or short term drop of PSA gave =2.37 (95%CB =0.83; 7.63), while when adjusting only for dose on the post-treatment level of PSA, =1.64 (95%CB =0.38; 5.96). This hence highlights the need for the appropriateness of complete adjustment as used in the principal analysis.

Variability over cohorts

The variability between cohorts due to their heterogeneity in recruitment, time period and other unknown factors was accounted for by including cohort-specific mean trajectory parameters. This greatly improved the goodness-of-fit of the model (the log-likelihood was improved by 108 points). There were significant differences between cohorts in the initial post-treatment level, probably reflecting differences between the patient populations that were not accounted for by the other prognostic factors. There were also small differences in the rate of rise of PSA. However, not including this adjustment for cohort effects did impact the  estimate (=10.6) with only the RBH cohort data changing the  ratio estimate range. Excluding this dataset led to a still low  (3.16Gy) with larger CB (95%CB=0.95; 21.20Gy) while excluding one of the 5 other cohorts led to  ratios from 1.14 to 1.53Gy. These findings are plausible given that the RBH data has the largest dose per fraction variation and hence the largest impact on the α/β ratio.

As a sensitivity analysis we adjusted for interactions between prognostic factors and cohorts, and adjusted for the year of treatment that could have explained differences over cohorts. The estimate of  remained in the same range as for the main analysis (=1.47 and =2.02 respectively). Using the complete model, we estimated the α/β ratio separately for each cohort (Table 3). Estimates of  varied greatly from cohort to cohort. However, except for RTOG and Vancouver, wide confidence bands allowed for extremely low value of  as well as infinitely large values and suggests that the  ration cannot be derived from individual cohorts.

 estimation for subgroups of prognostic factors

We evaluated whether the  was different when considering subpopulation of patients with different prognostic factors values: T1, T2 and T3-4; Gleason≤6, 7 and 8-10; iPSA≤10ng/mL or >10ng/mL. The fully-adjusted linear mixed model was then derived within each of these subgroups and the resultant dose and squared dose parameters are given in Table 4. For some of the subgroups the effect of dose and dose squared are not statistically significant. This occurred in the smaller subgroups that tended to have more advanced disease. Except for the Gleason 8-10 category, estimates of remained very low varying from -0.90 to 2.88. Among subjects with GS 8-10, was estimated close to 20 but with a lack of precision (95% confidence bands -∞; +∞).

DISCUSSION

The design of optimal radiation fractionation schedules to treat prostate cancer is an active area of clinical research presently. The consensus for a low α/β ratio in prostate cancer via a number of modelling studies [START_REF] Brenner | Fractionation and protraction for radiotherapy of prostate carcinoma[END_REF][START_REF] Brenner | Direct evidence that prostate tumors show high sensitivity to fractionation (low alpha/beta ratio), similar to late-responding normal tissue[END_REF][START_REF] Fowler | Is alpha/beta for prostate tumors really low?[END_REF][START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF] has produced a groundswell of support for schedules incorporating large doses per fraction, with considerable early implementation using high dose-rate brachytherapy in particular [START_REF] Martinez | Interim report of image-guided conformal highdose-rate brachytherapy for patients with unfavorable prostate cancer: the William Beaumont phase II dose-escalating trial[END_REF]. EBRT techniques have also been designed to meet these theoretical benefits [START_REF] Martin | Phase II trial of hypofractionated image-guided intensity-modulated radiotherapy for localized prostate adenocarcinoma[END_REF], despite requiring an assumption that combined EBRT and brachytherapy data can be directly extrapolated to the EBRT monotherapy situation. The present data support the low α/β ratio consensus, but are set apart by a lack of reliance on the inclusion of brachytherapy results to derive accurate estimates.

Drawing from summary outcome data, early clinical α/β ratio models based on EBRT and low doserate brachytherapy (LDRB) data [START_REF] Brenner | Fractionation and protraction for radiotherapy of prostate carcinoma[END_REF][START_REF] Fowler | Is alpha/beta for prostate tumors really low?[END_REF] produced α/β ratio estimates which appeared consistent at 1.5Gy with great precision. These analyses were to some extent reliant on assumptions that both the dose-rate and the dose homogeneity differences between treatment modalities did not require accounting for. Subsequent avoidance of the dose-rate assumption by deriving estimates from combined EBRT and high dose-rate brachytherapy (HDRB) results also arrived at a very similar (1.2Gy) estimate [START_REF] Brenner | Direct evidence that prostate tumors show high sensitivity to fractionation (low alpha/beta ratio), similar to late-responding normal tissue[END_REF]. Further to these studies, we focussed on replacing the previously used grouped outcome data with that of the individual patient to enable better potential control of confounding factors, using multivariable techniques [START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF]. With EBRT data alone (to avoid dose inhomogeneity), a model based on a standard binary actuarial bF endpoint was unable to estimate the α/β ratio (95%CI 1.1Gy to infinity) despite having >3000 men and >100 unique fractionation schedules to analyse.

Incorporation of the very large fractional doses from a relatively small number of HDRB boost cases was necessary to substantially improve the estimate precision to 2.6Gy (95%CI 0.9-4.8Gy). These estimates were dependent on the specification of the HDRB dose however, suggesting that dose homogeneity, and the manor in which it was accounted for, can be a potential influencing factor within a modelling system incorporating homogenous (EBRT) plus inhomogenous (brachytherapy) dose data. Our present data suggesting a low α/β ratio in patients all treated with a homogenous dose and constant fraction size are therefore both unique and reassuring in light of the previous assumptions.

The present data support a most likely α/β ratio of 1.5Gy, but suggest confidence limits which have a range of 0.5-4.5Gy. This range is similar to previous studies using HDRB data [START_REF] Brenner | Direct evidence that prostate tumors show high sensitivity to fractionation (low alpha/beta ratio), similar to late-responding normal tissue[END_REF][START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF], but wide in comparison to the range of values suggested in both the LDRB-based analyses (with values of 95%CI 0.8-2.2Gy (3) and 1.3-1.8Gy ( 6)). In employing these data to the design of clinical fractionation schedules, consideration need be given to such estimate uncertainties. Taking the present data as being the most directly applicable to the situation of EBRT monotherapy, the design of hypofractionation schedules should go beyond deriving a dose and fractionation at only an α/β ratio of 1.5Gy. Importantly, consideration should also be given to what detriment there may be to a new schedule should the value be at either end of the plausible α/β ratio range. For instance, a protocol of 36Gy in uniform 6 fractions may be suggested as approximately isoeffective with 78Gy in conventional 2Gy fractions at an α/β ratio of 1.5Gy. However, at an α/β ratio of 4Gy this isoeffect would fall to that of just 60Gy (as an equivalent in 2Gy fractions), with an attendant reduced efficacy likely. Even if the α/β ratio is 1.5Gy, some men will undoubtedly have a low α/β ratio and some a higher one, which will create a spread of outcomes which will be more apparent as the DPF increasesthose with low or average values will do as well or better than conventional fractionation, but those with a higher α/β ratio will have an over-representation of poor biologically effective doses in the hypofractionated group. In practical terms, a move to hypofractionation with EBRT with uniform large fraction sizes throughout the course would most safely be done by judiciously maintaining the overall total dose, so that if the α/β ratio is ultimately suggested to not be as low as expected, the hypofractionated patients still receive a biologically useful radiation dose.

An apparent biological phenomenon suggested by the model revolves around the differing impact of radiation dose and DPF at various phases of PSA evolution. Although both phases of PSA evolution are related to iPSA, GS and T-stage, the rate of PSA fall early after EBRT appears dependent on a biological system unaffected by the amount of radiation injury. Conversely, a rising PSA profile indicative of recurrence shows a strong and independent correlation to EBRT dose and DPF. In relation to existing studies, most have looked at the nadir PSA level or the time taken to reach that point rather than the pure "halving" time. A low PSA nadir at a late time has been the most consistent favourable feature identified in a number of these studies [START_REF] Aref | The relationship between biochemical failure and time to nadir in patients treated with external beam therapy for T1-T3 prostate carcinoma[END_REF][START_REF] Ray | PSA nadir predicts biochemical and distant failures after external beam radiotherapy for prostate cancer: a multi-institutional analysis[END_REF], and would be plausible based on these data which suggest an advantage for indolent cancers which have undergone a large therapeutic outcome. Further biological insights from subset analyses of know prognostic factors were not forthcoming. For the subdivisions of T-stage, iPSA and GS analysed, there was no consistent trend to the estimated α/β ratio. Small numbers of cases in the T3 or GS 8-10 categories precluded confident estimates to be made. As with our previous analysis [START_REF] Williams | Use of Individual Fraction Size Data from 3756 Patients to Directly Determine the alpha/beta Ratio of Prostate Cancer[END_REF], plausible biological suggestions that high grade cancers may have a higher α/β ratio, secondary to a higher growth rate, continue to be unsupported.

For the first time, the  ratio of prostate cancer has been directly estimated from the effect of dose on the longitudinal change of PSA. This has certain advantages over the use of a conventional biochemical failure endpoint in this situation. By using the evolutionary pattern of all the repeated measures of PSA as an outcome, the power of the analysis is greatly increased; enabling confidently modelling of the small changes in fractionation seen in EBRT data. Sensitivity analysis has shown the α/β ratio estimates to be robust to model misspecification (α/β ratio between 1.2 and 2.4Gy depending on model), although the model incorporating all the PSA evolution phases proved to have the optimal performance. From our previous experience in both development [START_REF] Taylor | Individualized predictions of disease progression following radiation therapy for prostate cancer[END_REF] and validation (15) of this modelling system, the complexities of the model do appear justified, with a demonstrated ability to accurately predict both PSA evolution and associated clinical failure risk. However, we found that whether or not we adjusted for cohort differences changed considerably the estimate. Each cohort had a different distribution of prognostic factors and different range of fractionation schedules, especially the RBH data. The RBH data had higher doses per fraction than the other cohorts, and these data were most influential in determining the α/β ratio.

While we did perform a considerable amount of sensitivity analysis and adjusted for known prognostic factors there remains the potential that the results are biased by other confounding factors. Thus the results should be interpreted with some caution. Patients were recruited at different times over a 17 year window, changes in treatment planning and administration have occurred, and the criteria for assigning a GS may have also drifted. As observational data, the dose given to each patient was chosen partially based on that patients prognostic factors, while we adjusted for known prognostic factors other prognostic factors may have been important for some patients. As a result, adjustment for cohort variability was the only way to provide an estimate of the ratio  that was adjusted as best as possible for this heterogeneity over cohorts. As with all other analyses of this topic, adjustment for cohort or other measured factors cannot prevent bias in the estimate of  from other unknown influential factors.

In conclusion, we propose that EBRT fractionation schemae can be modelled using a low α/β ratio, for a dose per fraction range of 1.7-2.8Gy without androgen deprivation therapy. Hypofractionation beyond this level should still be regarded as unsupported by clinical evidence, as no homogenous dose data presently exist to confirm that the model holds true at more ablative fractionation schedules.

Prospective studies continue to be highly anticipated in this regard.

1 .

 1 Three prognostic factors were used in the analysis: the initial level of PSA at diagnosis (iPSA; transformed on the continuous scale to iPSA=ln(iPSA+0.1)), T-stage category (1, 2, 3-4) and Gleason score (GS; ≤6, 7, 8-10). Dose of radiation D was defined as the sum of all fractional doses d j given during the J sessions of therapy  To evaluate the  ratio, squared dose D² was defined as the sum of all squared fractional doses given during the therapy  is the ratio of the regression parameters of D and D² in accordance with the linear-quadratic biological model. The mean dose per fraction was defined as the average dose per fraction received over sessions

Figure 1 :

 1 Figure 1: Evolution of PSA after end of radiation therapy: post-treatment level of PSA, short term drop of PSA and long term rise of PSA.

Acknowledgements: This research was supported by grant CA110518 from the US National Cancer Institute.

Conflict of interest notification: no actual or potential conflicts of interest exist.


Emmanuelle Petelet-Giraud

Romain Millot

Catherine Guerrot

Philippe Négrel

Spatial variability of rainwater stable isotopic composition (δ18O and δ2H) over the French territory

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Spatial variability of rainwater stable isotopic composition (δ 18 O and δ 2 H) over the French territory

Emmanuelle Petelet-Giraud, Romain Millot, Catherine Guerrot, and Philippe Négrel BRGM, Metrology Monitoring Analysis Department, Orléans, France (e.petelet@brgm.fr) Regarding isotopic hydrogeology, one important step for characterizing an aquifer is to compare the stable isotope signature of local rainwater with that of groundwater, which should reflect the mean weighted value of recharge by the successive rain events. Thus, considering that rainwater constitutes the main input into a hydrogeological system, the knowledge of the spatial variability of rainwater isotopic composition appears to be essential for hydrogeological investigations as well as for achieving sustainable water management. Five stations over France were monitored monthly for the δ 18 O and δ 2 H atmospheric signal (Brest, Dax, Orléans, Avignon and Thonon). They constitute the French monitoring network which is part of the IAEA/IOW Global Network for Isotopes in Précipitation (GNIP ; http://www-naweb.iaea.org/napc/ih/IHS_resources_gnip.html and log to WISER) for isotopes of the water molecule. Other rain data originate from the BDISO databank, gathering isotopic data available on French groundwaters, surface waters and rainwater (http://infoterre.brgm.fr/). They often represent only a few months of monitoring and do not refer to the same periods, as they stem from dedicated studies of local aquifers by the five French laboratories involved in BDISO (BRGM in Orléans, University of Paris-Sud in Orsay, University of Avignon, Centre de Recherches Géodynamiques in Thonon-les-Bains and CEA in Saclay). Among the available data, data points were selected applying the following criteria: (1) at least one year of monitoring should be avaliable, (2) isotopic data should be associated to the rain amount, (3) when two points are close, only the longest and more recent series of measurements are chosen. A total of 44 rain chronicles were selected. Some regions are poorly documented and complementary data were gathered: (1) lakes, that can be considered as natural pluviometers under specific conditions (upper part of watershed to limit runoff, pristine environment), are used after correction for evaporation, (2) young groundwaters were used as they often represent long term rainfall reasonably well, (3) landsnail shells integrating over the lifetime of the animal. Finally, to better constrain the isotopic signature near the borders of the French territory, we selected some data from the bibliography or long term rain monitoring in the GNIP database in the neighboring countries. The contour maps of δ 18 O and δ 2 H values reflect well the main effects that are known to affect the isotopic signature of precipitations; the continental and altitude effects are clearly visible. These maps constitute unique tools to assess the stable isotopic signature of oxygen and hydrogen in aquifer recharge. Nevertheless, it is worth noting that frequently rainwater data are used which integrate only over one year of precipitations and it was evidenced that the mean annual weighted δ 18 O values may vary by 1 to 2‰. Under these conditions our maps can constitute good complements to a local rain monitoring when studying a specific aquifer.