Larry Bodgi

Aurélien Canet

Laurent Pujo-Menjouet

Annick Lesne

Jean-Marc Victor

Dr Nicolas Foray

Mathematical models of radiation action on living cells: from the target theory to the modern approaches. An historical and critical review

Keywords:

Cell survival is conventionally defined as the capability of irradiated cells to produce colonies. It is quantified by the clonogenic assays that consist in determining the number of colonies resulting from a known number of cells. Since the discovery of X-rays, several mathematical models were proposed to describe the survival curves, notably from the target theory; The Linear-Quadratic (LQ) model, which is to date the most frequently used model in radiobiology and radiotherapy, dominates all the other models by its robustness and simplicity. Its usefulness is particularly important because the ratio of the values of the α and β adjustable parameters on which it is based predict the occurrence of post-irradiation tissular reactions. However, the biological interpretation of these parameters is still unknown.

Throughout this review, we proposed to revisit and discuss historically, mathematically and biologically, the different models of the radiation action by providing clues for resolving the enigma of the LQ model.

Introduction

Cellular radiosensitivity was conventionally defined as the capability of irradiated cells to produce daughter cells (i.e. colonies). It is quantified by the clonogenic assays that consist in determining the number of colonies resulting from a given number of cells irradiated at a given dose. Generally plotted on a semi-log scale, the survival fraction obeys a decreasing exponential-like law with or without shoulder [START_REF] Puck | Action of x-rays on mammalian cells[END_REF]. Several mathematical models were proposed to describe cell survival curves [START_REF] Curtis | Mechanistic models[END_REF]. Interestingly, the hypotheses on which they are based reflect the conceptual advances in our understanding of the radiation response (Fig. 1):

-between the 20's and the 50's, the most extensively used cell survival models were directly derived from the target theory, such as the single target single hit, n-targets single-hit and n-hits n-targets models, including the so-called (n,D 0 ) model [START_REF] Elkind | The radiobiology of cultured mammalian cells. Gordon and Breach[END_REF].

-between the 50's and the 80's, the (n,D 0 ) model was intensely used. However, the evidence that the initial slope of the survival curve was not nil has significantly decreased its interest. In the early 80's, the linear-quadratic (LQ) model was preferred because of its very good fitting qualities, but the empiric nature of its α and β parameters encouraged the authors to develop other models [START_REF] Chadwick | A molecular theory of cell survival[END_REF].

-between the 80's and the 90's, more sophisticated models were proposed by introducing the notion of DNA damage repair but without leading to formulas simpler than the LQ model, nor providing a clear mechanistic explanation to the radiationinduced phenomena. It is notably the case of the Repair-MisRepair (RMR) [START_REF] Tobias | The repair-misrepair model in radiobiology: comparison to other models[END_REF], the Lethal-Potential Lethal (LPL) [START_REF] Curtis | Lethal and potentially lethal lesions induced by radiation--a unified repair model[END_REF], and the saturated repair models [START_REF] Goodhead | Saturable repair models of radiation action in mammalian cells[END_REF].

-since the 90's, while there is a lower infatuation towards the biostatistical models describing cell survival, radiobiologists started focusing on the description of the DNA damage repair kinetics linked to cell survival [START_REF] Bodgi | A single formula to describe radiationinduced protein relocalization: towards a mathematical definition of individual radiosensitivity[END_REF][START_REF] Cucinotta | Kinetics of DSB rejoining and formation of simple chromosome exchange aberrations[END_REF][START_REF] Foray | The repair rate of radiation-induced DNA damage: a stochastic interpretation based on the gamma function[END_REF][START_REF] Gastaldo | Induction and repair rate of DNA damage: a unified model for describing effects of external and internal irradiation and contamination with heavy metals[END_REF][START_REF] Iliakis | The role of DNA double strand breaks in ionizing radiation-induced killing of eukaryotic cells[END_REF][START_REF] Neumaier | Evidence for formation of DNA repair centers and dose-response nonlinearity in human cells[END_REF][START_REF] Radivoyevitch | Recent data obtained by pulsed-field gel electrophoresis suggest two types of double-strand breaks[END_REF][START_REF] Sontag | A discrete cell survival model including repair after high dose-rate of ionizing radiation[END_REF].

The target theory and its major related cell survival models

The genesis of the target theory

Funded by physicists, the target theory is based on two major principles:

-"1) radiation is considered to be random projectiles;

-2) the components of the cell are considered as the targets to be bombarded by these projectiles" [START_REF] Summers | Physics and genes : From Einstein to Delbrück[END_REF].

The target theory has been firstly applied by J.A. Crowther in 1924 through an analysis of data from an experiment on chick embryo cells exposed to soft X-rays that was performed by [START_REF] Strangeways | The immediate changes observed in tissue cells after exposure to soft X-rays while growing in vitro[END_REF]Oakley in 1923 (Crowther, 1924;[START_REF] Strangeways | The immediate changes observed in tissue cells after exposure to soft X-rays while growing in vitro[END_REF]. In this case, the sensitive targets were hypothesized to be mitosis cells [START_REF] Crowther | Some considerations relative to the action of X-rays on tissue cells[END_REF]. In 1929, F.

Holweck 1 and A. Lacassagne obtained survival curves from bacillus irradiated by UV, X-rays or alpha-particles (Holweck and Lacassagne, 1929 ;Holweck and Lacassagne, 1930). Marie Curie herself analyzed the data and all these authors proposed the basis of the so-called quantum radiobiology: "to destroy a bacillus is it necessary that its sensitive zone absorbs a 1 Born in 1889, Fernand Holweck became assistant of Marie Curie in 1912. During the first World War, he helped P. Langevin in his works for detecting submarines by ultrasonic waves. Holweck developed a number of instruments like the most powerful vacuum-producing, a gravimetric pendulum, the first x-ray tube with successive stages of acceleration. Through his collaboration with Dr. A. Lacassagne, Holweck rediscovered, independently of previous work by Crowther, the quantic interpretation of the biological action of radiation on microorganisms. During the second World War, he was actively engaged in defense work but was arrested, tortured and murdered by the Gestapo on December 14th, 1941, in a Paris prison. minimal number s of quantas" [START_REF] Curie | Sur l'étude des courbes de probabilité relatives à l'action des rayons X sur les bacilles Comptes[END_REF]. From all these pioneer applications of the target theory, three important comments concerning hits and targets must be done:

-1) The probability density function (pdf) that was systematically applied to describe hits inside sensitive cellular targets was a Poisson law.

-2) The actual nature of the sensitive cellular targets was not consensual: they can be sub-populations of certain cells or some part of the nucleus.

-3) The survival of irradiated cell was considered as the only result of the absence of any hit on sensitive cells.

The basic ballistic models

2.2.1

The single-target single hit-model. It is the simplest application of the target theory.

Directly derived from the hypotheses of Crowther and Marie Curie, it was highlighted by Douglas Lea2 at the end of 50's through its book "Actions of radiation action on living cells" [START_REF] Lea | Actions of radiations on living cells[END_REF]. The single-target single hit-model dominates with both its simplicity and robustness all the approaches leading to cell survival. It is based on the hypothesis that a single impact in the sensitive part is enough to kill the cell. By considering the Poisson probability to hit k times a target:

𝑃 𝑘 = ! ! !! 𝑒 !! (1)
The probability of no impact is therefore:

𝑃 0 = 𝑒 !! (2)
We consider that m is proportional to the dose. Hence, if the survival is directly linked to the no impact probability we have:

𝑆(𝐷) = 𝑒 !!" or 𝑆(𝐷) = 𝑒 ! ! ! ! (3a,b)
Where D 0 is the mean lethal dose for which the mean number of lethal events per cell is equal to 1. At D 0 , the fraction of cell survival is equal to 1/e or 37% (0.367879 exactly) (Fig. 2A).

2.2.2

The n-target single-hit model. This model was based on the hypothesis that one cell contains n identical targets. The inactivation of one target was considered to be a sublethal event, and the accumulation of these sublethal events will lead to cell death, which will occur once the n targets are hit. Hence, by considering the probabilities of the single-target singlehit model, the probability that one target is hit once obeys the Poisson pdf:

𝑝 1 = 1 -𝑒 ! ! ! ! (4a)
Therefore, the probability that n targets from the same cell are hit once is:

𝑝 𝑛 = 1 -𝑒 ! ! ! ! ! (4b)
Thus, the survival fraction is given by the following formula:

𝑆 𝐷 = 1 -1 -𝑒 ! (4c)
The curve is described by the final slope D 0 and a parameter that defines the width of the shoulder (n or D q ). The n-target single-hit model was therefore called the (n, D 0 ) model [START_REF] Elkind | The radiobiology of cultured mammalian cells. Gordon and Breach[END_REF]) (Fig. 2B).

Criticisms of the ballistic models.

The great majority of cell survival curves obtained from mammalian cells do not obey both the single-target single hit and the (n,Do) models : to the notable exception of the hyper-radiosensitive cells that show exponential curves, all the other human cases are characterized by a shoulder and an initial part that is not nil (fig. 2A and 2B). These last observations are in clear discrepancy with the description showed by the (n,Do) model. To overcome this problem, the two-component model was proposed by considering an additional single-target component, which allows the initial slope value to be fixed at a given dose D 1 . The resulting survival equation becomes:

𝑆 𝐷 = 𝑒 !! ! ! 1 -1 -𝑒 !! ! ! ! (5)
However, although the two-component model is able to predict in an acceptable way the cell survival at low doses, it still has the default that the decrease in cell survival for a dose between 0 and D q occurs linearly, which has not been demonstrated experimentally [START_REF] Joiner | Basic clinical radiobiology[END_REF]. Even though the use of a multi-target instead of a single-target component would be able to solve this drawback, the general survival formula would become too complicated, and therefore not really useful to compare survival curves and explain the radiation response mechanisms [START_REF] Steel | Basic Clinical Radiobiology[END_REF].

To date, despite all the efforts in introducing some modifications and in addition to the problems evoked above, the models directly derived from the target theory appear to be unable to describe the phenomenon of hypersensitivity to low-dose that is characterized by a V-shaped part in the 1-400 mGy range and that is in clear discrepancy with the target theory [START_REF] Marples | Low-dose hyper-radiosensitivity: past, present, and future[END_REF]) (Fig. 2D)

The linear-quadratic model and its variants

2.3.1

The linear-quadratic model. In 1972, Kellerer and Rossi introduced the linear-quadratic (LQ) model in which a lethal event is supposed to be caused by one hit due to one particle track (the linear component αD) or to two particle tracks (the quadratic component βD 2 ) [START_REF] Kellerer | The theory of dual radiation action[END_REF][START_REF] Kellerer | A generalized formulation of dual radiation action[END_REF]. However, the probability that two particles tracks overlap is nil at biologically-relevant doses [START_REF] Goodhead | The initial physical damage produced by ionizing radiations[END_REF]. In 1973,

Chadwick and Leenhouts proposed that αD reflects unrepairable (i.e. directly lethal) DNA double-strand-breaks (DSB) and βD 2 reflects the combination of two sublethal DNA singlestrand breaks (SSB) [START_REF] Chadwick | A molecular theory of cell survival[END_REF]. Again, at biologically-relevant doses, IR-induced SSB are not close enough to produce DSB [START_REF] Goodhead | The initial physical damage produced by ionizing radiations[END_REF]. To date, while the LQ model still generates numerous debates, inherent bio-molecular mechanisms remain misknown [START_REF] Brenner | The use of the linear-quadratic model in clinical radiation oncology can be defended on the basis of empirical evidence and theoretical argument[END_REF][START_REF] Brenner | We forget at our peril the lessons built into the alpha/beta model[END_REF]. Despite of its empirical nature, the LQ model is considered as the best fitting model to describe survival (fig. 2C) [START_REF] Fertil | A benchmark of cell survival models using survival curves for human cells after completion of repair of potentially lethal damage[END_REF], and of great interest in radiation oncology through the link existing between the α/β ratio and the nature of radiotherapy-induced tissue (early or late) reactions [START_REF] Barendsen | Dose fractionation, dose rate and iso-effect relationships for normal tissue responses[END_REF][START_REF] Brenner | We forget at our peril the lessons built into the alpha/beta model[END_REF][START_REF] Dale | The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy[END_REF][START_REF] Williams | A review of alpha/beta ratios for experimental tumors: implications for clinical studies of altered fractionation[END_REF].

𝑆(𝐷) = 𝑒 !!"!!! ! (6)

The LQ model variants to describe the high-dose effects. While the LQ model

provides good fit for survival curves at biologically relevant doses, the accuracy of the survival description was found to be limited for higher or repeated doses. Douglas and Fowler therefore proposed the three-lambda model that consisted in a superimposition of three exponential terms [START_REF] Douglas | The effect of multiple small doses of x rays on skin reactions in the mouse and a basic interpretation[END_REF]. In their model the survival equation was:

𝑆 𝐷 = 𝑒 !! ! !!! ! ! ! !! !!! ! ! ! !! ! ! ! (7)
The linear-quadratic-cubic model was also proposed to describe the response to higher doses by adding another cubic term to the polynomial function of the LQ model [START_REF] Joiner | Models of radiation cell killing[END_REF][START_REF] Tobias | The repair-misrepair model in radiobiology: comparison to other models[END_REF]:

𝑆(𝐷) = 𝑒 !!"!!! ! !!! ! (8)

Criticisms of the LQ models and its variants.

While the usefulness of the three-lambda and the linear-quadratic-cubic models is quite relative since very high single doses are not really relevant for radiobiologists, the paradox of the LQ model is that it provides an actual robustness for fitting data while it remains an empirical model [START_REF] Fertil | A benchmark of cell survival models using survival curves for human cells after completion of repair of potentially lethal damage[END_REF].

The models based on the sublesions hypothesis and their variants

The repair-misrepair model

Proposed by Tobias in 1985, the repair-misrepair (RMR) model describes the evolution of the U(t) function, that reflects the mean number of lesions before any repair activation [START_REF] Tobias | The repair-misrepair model in radiobiology: comparison to other models[END_REF]. The yield of induction of these lesions, U 0 , was considered proportional to the dose D:

𝑈 ! = 𝛿𝐷 (9)
The repair states R are defined as being the resultant transformation of the U lesions after the repair process. The author considered that the evolution of the U lesions can be described by the following differential equation:

!" !" = -λU t -kU ! (t) (10)
With λ the linear self-repair coefficient, considered to be the good repair pathway, and κ the coefficient for cooperative repair, involving the interaction of pairs of U lesions, that the author considered to be the misrepair. By integrating the equation we have:

U(t) = ! ! ! !!! !! ! ! (!!! !!! ) ! ! (11)
Two R-states were therefore defined: R L (t), the total number of self-repairs, which are the non-lethal lesions, and R Q (t), the total number of quadratic misrepairs, which are considered to be the lethal lesions:

R ! = λU t dt ! ! (12a) R ! = kU ! (t)dt ! ! (12b)
By considering that no new lesion is created during the repair process, we have:

U ! = U t + R ! t + R ! (t) (13)
when t→ ∞, equation ( 13) becomes:

R ! t → ∞ = U ! -R ! t → ∞ (14)
If we consider the repair ratio ε=λ/κ, and by applying Poisson pdf, the survival equation becomes:

S = e !! ! !→! = e !! ! 1 + ! ! ! ! (15)
By considering that the linear repair is not a perfect process, the author introduced the parameter ϕ that defines the probability that self-repair steps are all perfect eurepairs.

Furthermore, by considering that the repair time is limited at a time T, the survival equation becomes (fig 2E):

S !! e !! ! 1 + !!! ! ! ! (16)

The lethal-potentially lethal model

Curtis developed in 1986 the Lethal-Potentially Lethal model (LPL) model that takes the repair process into account [START_REF] Curtis | Lethal and potentially lethal lesions induced by radiation--a unified repair model[END_REF]. He proposed a classification of the radio-induced lesions: lesions "that are unrepairable and are therefore lethal", and "potentially lethal lesions for which the repair process is activated". Thus, two differential equations are necessary to describe the repair kinetics:

!! !" !" = -ε !" n !" t -ε !!" n !" t ! (17a) !! ! !" = ε !!" n !" t ! (17b)
with n PL the number of potentially lethal lesions, n L the number of lethal lesions, ε PL the constant per unit of time repair rate and ε 2PL the constant per unit of time rate of interaction between 2 potentially lethal lesions, a process that Curtis called the binary misrepair (fig. 2F).

The solutions to equations ( 16) are:

n !" (t) = ! !" ! !! !" ! ! !!(! !" /!)(!!! !! !" ! ! ) (18a) n ! (t) = N ! + ! !" !! ! !" ! !!! !! !" ! ! !!(! !" /!) !!! !! !" ! ! -ϵln 1 + (N !" /ϵ) 1 -e !! !" ! ! (18b)
where N !" = n !" (T), N ! = n ! (T) , ϵ=ϵ PL/ ϵ 2PL, T is the irradiation duration and t r is the available repair time

In order to predict the survival at a time t=T+t r , time after which the repair process is ineffective, Curtis considered that the total number of lesions per cell is the sum of lethal and potentially lethal lesions. In other words, he hypothesizes that after a certain time, all the potentially lethal lesions become lethal. Hence, the total number of lesions n !"! is

n !"! T + t ! = n ! T + t ! + n !" T + t ! (19)
The survival equation becomes:

S = e !! !"! !!! ! = e !! !"! 1 + N !" /ε(1 -e !! !" ! ! ) ! (20)
with N TOT =N L +N PL and ε = ε !" ε !"# (fig. 6)

The saturable repair model

In 1985, Dudley T. Goodhead proposed a new model, the saturable repair model that was based on the hypothesis that the efficiency of the repair system decreases with dose, and that this decrease is caused by the saturation of the repair kinetics [START_REF] Goodhead | Saturable repair models of radiation action in mammalian cells[END_REF]. He therefore considered the following repair rate for the induced lesions:

!" !" = -𝑘𝑐𝑛 ( 21
)
Where n is the number of unrepaired lesions, c the number of repair molecules or enzymes and k is a proportionality coefficient. By considering that 𝑑𝑐 = 𝑑𝑛 and that T is the time available for repair, the residual number of lesions after repair becomes:

𝑛 ! = ! ! !! ! !! ! ! ! ! ! !" ! ! !! ! (22)
Hence the survival equation:

𝑆 𝐷 = 𝑒 ! ! ! !! ! !! ! ! ! ! ! !" ! ! !! ! (23)
By considering the repair process as saturable, this model does not require the notion of sublesions like the lethal and potentially lethal/sublethal ones (fig. 2G). It was therefore presented as an alternative to the RMR and LPL models [START_REF] Goodhead | Saturable repair models of radiation action in mammalian cells[END_REF].

Criticisms of the sublesion and saturable models

The notion of the lethal and potentially lethal lesions is directly derived from the attempts to interpret the LQ model. Particularly, as evoked above, the quadratic component was systematically explained by a duality of tracks, single-strand breaks or cooperative lesions.

However, the actual nature of the sublesions still remains undefined. Besides, this idea is related to the direct/indirect effect hypothesis that suggests that some damage are directly induced by the impact of the physical particles and some other by the chemical radicals produced by such impact. In fact, it has been clearly shown that DNA damage are induced simulataneously and that radicals attack was not a 2-time-phase phenomenon (Foray et al., 1996a;[START_REF] Foray | Repair of radiation-induced DNA double-strand breaks in human fibroblasts is consistent with a continuous spectrum of repair probability[END_REF][START_REF] Kysela | Relative contributions of levels of initial DNA damage and repair of double strand breaks to the ionizing radiation-sensitive phenotype of the Chinese hamster cell mutant, XR-V15B. Part I. X-rays[END_REF]. Furthermore, both RMR and LPL models considered the misrepaired lesions as lethal lesions, which is in clear discrepancy with cytogenetics and new advances in radiobiology. Indeed, it is more accepted that misrepaired lesions are more likely involved in genomic instability and cell transformation rather than cell death and radiosensitivity [START_REF] Jeggo | DNA double-strand breaks: their cellular and clinical impact?[END_REF].

Unlike the models that are directly derived from the target theory, the sublesions (LPL and RMR) are based on the notion of DNA damage repair. Although it is suggested that these lesions might DSB, they are not formally identified as the damage of interest.

Furthermore, their repair rate per unit of time described in the LPL and RMR models was hypothesized to be constant. Such an assumption is in discrepancy with the multiphasic shape of the DSB repair kinetics which would reflect the existence of a continuous spectrum of DSB of differing reparabilities rather than a limited number of DSB subcategories (Foray et al., 1996a;[START_REF] Foray | The repair rate of radiation-induced DNA damage: a stochastic interpretation based on the gamma function[END_REF][START_REF] Foray | Repair of radiation-induced DNA double-strand breaks in human fibroblasts is consistent with a continuous spectrum of repair probability[END_REF]). An experimental proof of the continuous nature of DSB repair kinetics is given by the shape of the DSB repair curves obtained after a given dose followed by a period of time (some min to some hours): in these conditions, DSB repair curves are never mono-or bi-phasic but systematically continuously decreasing (Foray et al., 1996a;[START_REF] Foray | The repair rate of radiation-induced DNA damage: a stochastic interpretation based on the gamma function[END_REF][START_REF] Foray | Repair of radiation-induced DNA double-strand breaks in human fibroblasts is consistent with a continuous spectrum of repair probability[END_REF] which contradicts the hypothesis of the LPL and RMR models.

With regard to the saturable repair model, the major assumption is the saturation of the repair enzymes pool. Up to date, such an hypothesis was not verified. While some tens of DSB are induced per Gy, the average yield of each protein ranges from 10 4 to 10 8 molecules, it appears unlikely that the repair enzymes pool can be saturated. Some radiosensitive syndromes are not necessarily caused by a decreased of DSB repair kinetics, inasmuch as these syndromes are caused by mutations of cytoplasmic proteins like Huntington's disease, neurofibromatosis or Usher's syndrome [START_REF] Deschavanne | A review of human cell radiosensitivity in vitro[END_REF][START_REF] Ferlazzo | Mutations of the Huntington's Disease Protein Impact on the ATM-Dependent Signaling and Repair Pathways of the Radiation-Induced DNA Double-Strand Breaks: Corrective Effect of Statins and Bisphosphonates[END_REF]. Furthermore, that the DSB repair rate is not necessarily dependent of dose rate and the irradiation at low dose-rate for long times (some days) does lead to the absence of repair (Foray et al., 1996b).

Altogether, like the models derived from the target theory, the RMR, LPL and saturable repair models do not solve two important radiobiological questions, at least:

-the very documented hypersensitivity to low doses [START_REF] Joiner | Low-dose hypersensitivity: current status and possible mechanisms[END_REF] -the fact that some radiosensitive syndromes are not necessarily associated with DNA damage repair defect [START_REF] Deschavanne | A review of human cell radiosensitivity in vitro[END_REF][START_REF] Ferlazzo | Mutations of the Huntington's Disease Protein Impact on the ATM-Dependent Signaling and Repair Pathways of the Radiation-Induced DNA Double-Strand Breaks: Corrective Effect of Statins and Bisphosphonates[END_REF].

Modern approaches

The target theory must be reconsidered to explain radiosensitivity of mammalians

The models deriving from the target theory must consider that:

-(1) physical hits obey a Poisson pdf;

-(2) cell survival is due to the absence of hits in the sensitive areas of the irradiated cells;

-(3) since all the number of hits is proportional to the dose and since all the hits are lethal, survival is a simple exponential function of the dose.

Such hypotheses are relevant for micro-organisms but are not for mammalian cells. Indeed, by considering that the relevance of the above hypothesis (1) and that about 40 DSB are produced per human cell per Gy, the probability of an absence of impact is lower than 10 -17 .

Conversely, different DSB assays like pulsed-field gel electrophoresis and γH2AX

immunofluorescence show that the yield of induced DSB per mammalian cell does not obey a

Poisson pdf but rather a Gauss pdf [START_REF] Noda | Unrepairable DNA double-strand breaks that are generated by ionising radiation determine the fate of normal human cells[END_REF][START_REF] Rothkamm | Evidence for a lack of DNA double-strand break repair in human cells exposed to very low x-ray doses[END_REF]. Hence, the hypotheses (2) and (3) deriving from the target theory must therefore be reconsidered for mammalian cells: cells likely survive because all their DNA damage are repaired rather than because cells are not targeted by IR [START_REF] Lea | Actions of radiations on living cells[END_REF][START_REF] Sutherland | Repair dependent radiation survival : a stochastic model with Euler gamma function solutions[END_REF] (fig. 3). Furthermore, it must be reminded that, in addition to the 40 DSB induced per cells, X-and gamma-rays also induce simultaneously 1000 SSB and 10000 base damage per Gy (these numbers of are divided by more than 100 in the case of the bacteria or yeasts): it is therefore surprising that from the pioneer works of Crowther and Marie Curie (i.e. the 1930's), only few radiobiologists (D.E. Lea was one of them [START_REF] Lea | Actions of radiations on living cells[END_REF])) discussed about the relevance of the target theory for other species than micro-organisms.

The moderate radiosensitivity must be considered when testing survival models

The great majority of mammalian cells show a non-negligible initial slope and a shoulder when cell survival is plotted against dose, which is in clear discrepancy with target theory and especially (n,Do) model [START_REF] Malaise | Initial slope of radiation survival curves is characteristic of the origin of primary and established cultures of human tumor cells and fibroblasts[END_REF]. The only cell lines that can show exponential survival curve are the most hyper-radiosensitive ones such as those mutated for ATM, LIG4

or DNA-PK proteins [START_REF] Iliakis | The role of DNA double strand breaks in ionizing radiation-induced killing of eukaryotic cells[END_REF][START_REF] Joubert | DNA double-strand break repair defects in syndromes associated with acute radiation response: at least two different assays to predict intrinsic radiosensitivity[END_REF]. In the frame of the LQ model, the maximal shoulder is obtained by the cells that show a maximal β LQ parameter, which corresponds to a moderate radiosensitivity (average α). The cell lines showing the most moderate radiosensitivity are therefore a good tool to exclude a number of irrelevant cell survival models. Interestingly, as evoked above, some genetic syndromes associated with moderate sensitivity are not caused by mutations of proteins directly involved in DSB repair but are rather cytoplasmic and have no function in DNA damage repair. This is notably the case of the Huntington's disease, neurofibromatosis or Usher's syndrome [START_REF] Deschavanne | A review of human cell radiosensitivity in vitro[END_REF][START_REF] Ferlazzo | Mutations of the Huntington's Disease Protein Impact on the ATM-Dependent Signaling and Repair Pathways of the Radiation-Induced DNA Double-Strand Breaks: Corrective Effect of Statins and Bisphosphonates[END_REF]. In complete contradiction with the target theory, such syndromes provide clues that cytoplasm proteins may impact on radiation response and that considering both nuclear targets and DNA repair is not sufficient to explain all the range of human radiosensitivity.

The radiation-induced nucleo-shuttling of ATM: a solution to some enigmas?

In the frame of our collection of skin biopsies from radiotherapy patients showing adverse tissue reactions, we have accumulated from 2003 some hundreds fibroblast cell lines whose radiation response has been investigated with the major DSB repair biomarkers (Granzotto et al., submitted). Interestingly, it appeared that the phosphorylated forms of the ATM protein translocate from cytoplasm to nucleus in response to radiation. For the patients showing moderate radiosensitivity, such radiation-induced nucleo-shuttling of ATM was delayed, probably because the ATM proteins are sequestrated by mutated proteins in cytoplasm [START_REF] Ferlazzo | Mutations of the Huntington's Disease Protein Impact on the ATM-Dependent Signaling and Repair Pathways of the Radiation-Induced DNA Double-Strand Breaks: Corrective Effect of Statins and Bisphosphonates[END_REF]. Since ATM protein is known to phosphorylate the γH2AX variant histone that is considered as the recognition step of DSB by the preponderant DSB repair pathway in humans, the non-homologous end-joining (NHEJ), two categories of lethal DSB can be defined: 1) the recognized by non-repaired DSB; 2) the non-recognized therefore nonrepaired DSB. We have shown that these DSB categories called α-type and β-type, increase with dose or with the square of the dose, respectively (Bodgi and Foray, submitted). Such findings support therefore that the LQ model, that provides the best cell survival data fits, provides also the best relevance with molecular mechanisms in response to radiation. It is also noteworthy that the theory of the nucleo-shuttling of ATM and the LQ model are the most compatible with the hypersensitivity to low-dose phenomenon [START_REF] Bodgi | A single formula to describe radiationinduced protein relocalization: towards a mathematical definition of individual radiosensitivity[END_REF][START_REF] Colin | MRE11 and H2AX biomarkers in the response to low-dose exposure: Balance between individual susceptibility to radiosensitivity and to genomic instability[END_REF][START_REF] Joiner | Low-dose hypersensitivity: current status and possible mechanisms[END_REF][START_REF] Thomas | Impact of doserate on the low-dose hyper-radiosensitivity and induced radioresistance (HRS/IRR) response[END_REF] (Bodgi and Foray, submitted).

Conclusions

To date, it appears to date that most of the biostatistical models of cell survival are not relevant to describe the radiation response of mammalians. However, the re-analysis of the princeps papers provides strong evidence that the general theory and pdf on which these models are based were built from micro-organisms data whose size and characteristics are clearly different from the mammalians case. It is therefore not surprising that the LQ model, based on a very permissive 2 nd degree polynomial function provides the best cell survival data fits. Nevertheless, the biological interpretation of the LQ parameters remained unsolved since the 1970's. Today, by taking into account the radiation-induced nucleo-shuttling of ATM that is very far from the target theory, coherent explanations of the descriptive power of the LQ model and of some misknown radiobiological phenomena can be proposed.

Figure legends:

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1 : Historical synopsis related to the cell survival models and their variants

Figure 4 :

 4 Figure 4 : Schematic illustration of our new model of the ATM nucleo-shuttling

Douglas E. Lea (MA, PhD) was born in 1910 in Great Britain. During his career, he worked as a physicist at England's Strangeways Laboratory and as a reader in the Department of Radiobiology in the Department of Radiotherapeutics at Cambridge University. The majority of his work dealt with the effects of radiation on cells. Lea died in an accident inCambridge, England, on June 16, 1947.


Zaher Al Barakeh

Philippe Breuil

Nathalie Redon

Christophe Pijolat

Nadine Locoge

Jean-Paul Viricelle

Development of a normalized multi-sensors system for low cost on-line atmospheric pollution detection

Keywords: Air pollution detection, semi-conductor gas sensors, sensors array, calibration and normalization procedure, fuzzy logic, Neural Network calibration. Sensors and Actuators B: Chemical, Elsevier

Introduction

Atmospheric pollution monitoring is usually done by a network of measuring stations. These stations are based on high performance industrial gas analyzers placed in fixed monitoring sites. Measurements from these stations are used to define air quality indexes which depend on some regulated gases (NO 2 , SO 2 , O 3 and/or particles concentrations (PM10).

For example, in France, nearly eight hundred permanent stations throughout the territory contribute to foresee short -and mid-term air quality forecast. The measurements done by these stations are based on high performance industrial gas analyzers, which offer an excellent selectivity and precision towards targeted gases. However their high price and heavy weight make their use complicated [START_REF] Kamionka | Calibration of a multivariate gas sensing device for atmospheric pollution measurement[END_REF], especially for some application where selectivity and precision are not required [START_REF] Yamazoe | New perspectives of gas sensor technology[END_REF] like pollution meshing [START_REF] Ma | Air Pollution Monitoring and Mining Based on Sensor Grid in London[END_REF] or source detection [START_REF] Ködderitzsch | Sensor array based measurement technique for fast-responding cigarette smoke analysis[END_REF].

A complementary method for air pollution monitoring lies in the use of gas sensors arrays based on semiconductor gas sensors [START_REF] Ma | Air Pollution Monitoring and Mining Based on Sensor Grid in London[END_REF][START_REF] Pijolat | Gas detection for automotive pollution control[END_REF][START_REF] Rivière | Development of tin oxide material by screenprinting technology for micro-machined gas sensorsSensors and Actuators B[END_REF][START_REF] Kamionka | Atmospheric pollution measurement with a multimaterials sensing device[END_REF]. Semiconductor gas sensors have been optimized for more than thirty years [START_REF] Watson | The tin oxide gas sensor and its applications[END_REF], giving them a high sensitivity for certain gas detection, a fast response time of up to several seconds, a small size, low cost (few euros) compared to other types of sensors, and ease of use in electronic systems which make them attractive cheap consumers devices [START_REF] Kamionka | Atmospheric pollution measurement with a multimaterials sensing device[END_REF] [START_REF] Zampolli | An electronic nose based on solid state sensor arrays for low-cost indoor air quality monitoring applications[END_REF].

Even if semiconductor gas sensors show a low selectivity [START_REF] Padilla | Drift compensation of gas sensor array data by Orthogonal Signal Correction[END_REF][START_REF] Tomic | Standardization methods for handling instrument related signal shift in gas-sensor array measurement data[END_REF][START_REF] Korotcenkov | Instability of metal oxide-based conductometric gas sensors and approaches to stability improvement (short survey)[END_REF][START_REF] Romain | Long term stability of metal oxide-based gas sensors for e-nose environmental applications: An overview[END_REF][START_REF] Vito | NO 2 and NO x urban pollution monitoring with on field calibrated electronic nose by automatic bayesian regulation Sensors and Actuators B[END_REF], reproducibility and repeatability [START_REF] Romain | Long term stability of metal oxide-based gas sensors for e-nose environmental applications: An overview[END_REF][START_REF] Tomic | Recalibration of a gas-sensor array system related to sensor replacement[END_REF] to specific gases in a complex mixture, each type of sensor reacts differently to each family of gas species [START_REF] Vito | NO 2 and NO x urban pollution monitoring with on field calibrated electronic nose by automatic bayesian regulation Sensors and Actuators B[END_REF]. So, with adequate pattern recognition algorithms (multivariate analyses such as neural networks), this drawback is used as an advantage to identify an overall signature of pollution [START_REF] Shaffer | A comparison study of chemical sensor array pattern recognition algorithms[END_REF][START_REF] Sommer | Neural networks and abductice networks for chemical sensor signals: a case comparision Sensors and Actuators B[END_REF][START_REF] Natale | Pattern recognition in gas sensing: well-stated techniques and advances Sensors and Actuators B[END_REF][START_REF] Hines | Electronic noses: a review of signal processing techniques[END_REF][START_REF] Szecowka | On reliability of neural netwok sensitivity analysis applied for sensors array optimisation[END_REF].

This point constitutes the sensors array approach. Such systems can be used to perform semi-quantitative measurements [START_REF] Negri | Identification of pollutant gases and its concentrations with a multisensors array[END_REF]. However, complex situations with large numbers of gas can limit their performance [START_REF] Bourgeois | The use of sensor arrays for environmental monitoring: interests and limitations[END_REF].

Most of time, pollution detection with commercial gas sensors is usually focused on identifying one single pollutant in the atmosphere [START_REF] Kamionka | Calibration of a multivariate gas sensing device for atmospheric pollution measurement[END_REF] [7] [START_REF] Tsujita | Gas sensor network for air-pollution monitoring[END_REF] [START_REF] Shurmer | Sensitivity enhancement for gas sensing and electronic nose applications Sensors and Actuators B[END_REF]. The presence of different interfering gases in the atmosphere renders these detections difficult and vulnerable for mistakes.

Therefore, in this work, we present three identical autonomous portable systems each containing 9 different commercial semi-conductor gas sensors, one humidity sensor and a temperature sensor especially developed to address ambient air pollution problematic. First, through many field experiments, we have evaluated the capability of these multi-sensors arrays to distinguish different types of air pollution like traffic, urban or photochemical pollution, and then we have associated them to an index level, according to French air quality standards. To reach this goal, the developed systems were placed in six different air network stations. Previously, one year of data history from each station was used to establish correlations and repeated patterns between pollutants and atmospheric pollution types. Neural networks models are built from sensors responses in order to predict membership of pollution to the categories previously defined (traffic, urban or photochemical pollution). Two field campaigns, during winter and summer in order to take into account the influence of atmospheric conditions, were conducted to establish the modelling and to validate it.

The portability and robustness of the models for different sensors arrays on different stations was also evaluated.

Sensors arrays and responses corrections

2.1.Measuring System development:

Three identical sensor arrays (Fig 1 ) were developed, each containing nine commercial semiconductor gas sensors (from Figaro Sensors and SGX Sensortech, Table 1) and also humidity (Measurement Specialities HS1101) and temperature (Texas Instruments LM35) sensors. The gas sensors were selected to detect the widest gas species usually present in urban atmosphere. We focused firstly on regulated pollutants (O 3 , NO, NO 2 , CO) in order to correlate them to databases provided by the network of air quality monitoring stations. Secondly, we targeted components sensitive to VOCs because those gases can also be specific to kind of contamination like traffic or urban pollution. Finally, we completed our choice, with sensors able to detect unregulated gas like NH 3 , H 2 S, or alkanes because of their specificity to large types of pollution. Table 1 presents the sensors chosen with their commercial references, their target gases and the detection range as specified by the constructors. All these gas sensors undergo a variation in conductance when the gas concentration in their environment is changed. A dimensionless quantity called "relative conductance variation" was used to characterize the sensor response (Eq.1).

X R = (G-G 0 )/G 0 [START_REF] Kamionka | Calibration of a multivariate gas sensing device for atmospheric pollution measurement[END_REF] With:

-X R : relative conductance variation of a sensor -G: sensor's conductance -G 0 : reference conductance (base-line) measured under "reference air" (synthetic air, nitrogen /oxygen 80/20 vol.% with H 2 O, 12g/m 3 (1.5% volume) ).

It is clearly noticeable that these specified detection ranges are much higher than the usual pollutants concentrations in the atmosphere. Therefore testing these sensors under relatively low concentrations of pollutant gases was a necessity. To perform such experiments, a test bench allowing to characterize each array under defined gases was used. These gases were methane, toluene, nitrogen dioxide, carbon monoxide, ozone, ammonia and ethanol. For each gas, the concentrations were fixed by dilution in reference air described previously, thanks to massflow regulators. These concentrations were chosen according to levels that can be encountered in ambient air. Each signature is specific for a given gas at a given concentration. These sensors show a significant response toward different gases at concentrations much lower than the detection ranges specified by their constructors, tested concentrations levels can be representative of what can be found in polluted sites.

2.2.Baseline drifts correction:

The drift of the sensors signal can cause significant problems of temporal validity for the produced models. So a periodic measurement of this change is necessary to ensure proper monitoring of the baseline. Sensors baseline drift was studied by measuring and giving each month a reference conductance (G 0 ) to each sensor calculated under reference air for specific conditions of temperature and relative humidity (50% at 25°C). Fig 3 shows the evolution of sensors relative conductance variation X R (defined by eq 1) for zero air flow during 7 months of tests for the first prototype array. This figure shows a decreasing trend similar between the three different Mics 5525 sensors. However, the amplitude of the variation varies from one sensor to another. We note that responses have sharply decreased in the first months (July-November), and after this period, they were relatively stable. This confirms that a periodic measurement of these changes is necessary to ensure proper monitoring of the baseline. Therefore a calibration correction was done on a weekly basis to detect this drift in order to have new measurements of the reference G 0 , although it can probably be reduced to a monthly basis with no significant changes if sensors are aged at least 3 months before use.

2.3.Sensitivity change compensation:

Besides the drift of the baseline, semi-conductor gas sensors also tend to undergo changes in their sensitivity through time. Moreover, same sensors type can have different sensitivity to the same gases, inducing a low reproducibility between them. To homogenise a relative repeatable and reproducible responses between the 3 arrays, a normalisation procedure was elaborated. The principle is based on normalization of the relative conductance of each same type sensors under one reference gas as shown in Table 2. Four gases were chosen as representatives of pollutants in the atmosphere (CO, CH 4 , NO 2 , NH 3 ), and each sensor was normalized toward the gas for which the sensor shows highest sensitivity.

Sensor

Reference Normalization procedure for a given sensor "i" was made by dividing its relative conductance variation X R,i by its relative conductance variation X N-i under the chosen normalization gas "N" with a concentration C N :

S i = X R,i / X N,i (2)
This test can also be periodically used as a diagnostic tool to detect any sensor failure that implies its replacement. Figure 5 shows the sensors response to reference gases and the effect of this sensitivity normalization on the 3 multi-sensors arrays:

On site measurements and monitoring method

3.1.Field measurement sites and classes of pollution:

Two measurement campaigns (one in summer and one in the end of autumn) conducted on site during this work were done in six stations of the National Network of Accredited Associations for Air Quality Monitoring of France. AASQA networks are based on different types of monitoring stations placed in specifically determined position according to the EU directive (2008/50/EU). These six monitoring stations, distributed in the north of France, are representative of three different categories of majority pollutions, named urban, suburban and traffic as shown in Table 3. For each pollutant, mean and maximum value as well as standard deviation over one year history's data are reported. During these measuring campaigns, measurements were done during 5 to 14 consecutive days to ensure the registration of special events, like traffic spikes for example, and also to get data for the calculation of the neural network modelling. Previously, one year data history issued from analyzers and collected from these stations were studied in order to define a credible pollution categorization according to its source. The analysis was oriented to find a repeatable patterns or specific characteristics that can define existing pollution in a station. Thus repeated patterns in stations from similar category (urban, suburban or traffic) were studied. Then three pollution categories were defined using a fuzzy logic approach from the measurements of O 3 , NO, and NO 2 concentration in the atmosphere:

-Traffic pollution: sites near emission of pollutants, high correlation between NO and NO 2 , ozone absent (reaction with other primary pollutants),

-Urban pollution: more far from emission, NO is partially destructed.

-Photochemical pollution: high concentration of ozone, low concentrations of NO and NO 2 , typical of suburban sites, However, if the traffic, urban and photochemical pollution types are most probable on respectively traffic, urban and suburban site types, each type of pollution can still be met on each type of site.

As boundaries between categories are not so easy to define, which may be detrimental to the modelling, we have chosen a "fuzzy logic" approach. Fuzzy logic was introduced in 1965 by Lotfi Zadeh [START_REF] Zadeh | Fuzzy sets[END_REF] based on his mathematical theory of fuzzy sets. By introducing the notion of a membership function (MF) allowing a condition to be in a state other than true or false unlike boolean logic, it provides flexibility and makes possible to take into account inaccuracies and uncertainties. Indeed, a boolean logic uses a step function defined with one threshold while fuzzy logic uses a membership function, which is a sigmoid function that can be defined with two thresholds l 1 and l 2 , to calculate a "membership degree" (MD) (Fig 6 and Table 4). For a given assertion, this membership degree indicates its probability, from 0 (not probable) to 1 (very probable). The two thresholds are calculated from the median and the average values of the concentrations for one year of measurements and for the six stations. Then the membership degree of the different classes can be calculated from the membership degree of the pollutants and their equations (Table 5), following the fuzzy logic theory laws [START_REF] Zadeh | Fuzzy sets[END_REF].

3.2.Pollution index:

To complete this categorization with a quantitative information, a pollution index (PI) was created in order to describe the pollution intensity. This index is inspired from the index defined by AASQA [START_REF]Arrêté du 22 juillet 2004 relatif aux indices de la qualité de l'air[END_REF], named ATMO, which is defined as the maximum of 4 minor indexes calculated out of the concentration of 4 pollutants (SO 2 , O 3 , NO 2 , PM10), and is used to inform the public about the ambient air quality in France. In our case, the chosen pollution categories are based on only 3 minor index calculated from pollutants concentrations (NO, NO 2 and O 3 ). Each index is integer and varies from 1 to 10, 1 being the case of a no or low concentration of a pollutant and 10 being the maximum defined threshold in the European directives. The highest value among them conveys the value of the overall index. O 3 and NO 2 indexes can be calculated in the same way defined by ATMO, but a new index for NO had to be calculated (Table 6) as this gas is not included in the AASQA measurements. It was estimated in order to keep the same frequency distribution for pollution indexes of NO and NO 2 encountered in the year 2010. The three categories of pollution and the pollution index are illustrated as a function of the concentrations of pollutants on. Then our goal is to predict, from the gas sensors signals, this pollution index and the class of pollution which is the one with the highest membership degree.

3.3.Artificial Neural Network Modelling:

In order to perform model of the array sensor response, we have opted for an artificial neural network (ANN, Fig 8) for its ability to model non-linear phenomena. The input data of the ANN are the normalized responses S i from the 9 sensors forming each multi-sensor system, to which are added relative humidity, temperature, time of day and if it is a weekday or not. Output data "Y" correspond to the membership degree of the 3 categories of pollution to which we add the pollution index. The output layer is connected to the input layer via one hidden layer each consisting of 4 to 12 neural nodes. The relations between the different layers are either linear combinations of the preceding layer for input and output layers, either non-linear combination for the hidden layer. A hyperbolic tangent function was used as the non-linear function in our case. The building of the target data "Ŷ", to which the output data Y resulting from the ANN is compared during the learning step, is summarized on Fig 9 . Starting from gas concentrations measured by gas analysers, membership degree and pollution index for each three gas are calculated. Then, the overall index and the membership degree of the three pollution category constitute the target data "Ŷ". For a typical ANN-modelling, three sets of data are used: the two to five first measurement days were used for the learning, followed by one day for the validation and one day for the test. The modelling is iteratively optimised during the learning step by adjusting the coefficients of the neural networks so that the Root Mean Error of Prediction (RMSEP) comparing the Y variables calculated from the conductance of the sensors, and Ŷ calculated from measurements of gases analysers, is as low as possible for the learning set. This iterative optimisation is stopped when the RMSEP for the validation set no longer decreases. Once the coefficients of the model are optimized, they are applied to the test set, and a classification rate is calculated: it represents the percentage of right classifications compared to the categorization obtained by measurements done with AASQA analysers. Similarly, the PI prediction quality is quantified with the RMSEP.

Results of classification and prediction:

Fig 10 shows a short example of learning/classification/prediction. Results are shown over a three days period. The two first correspond to the learning period while the last one is the test period. At any time, the predicted classes corresponds to the highest membership degree of the 3 classes. Agreement between predicted class from sensor array and from analysers is observed for more than 88% of the test duration. There is also a very good correlation between predicted and calculated pollution index. The RMSEP of the pollution index is less than 0.3 during the test period. Table 7 shows the overall classification rates and RMSEP for the pollution index for the three multi-sensors arrays deployed on the six stations. Data are issued from more than 200 hours of test and 1200 hours of learning and validation acquired during the summer and autumn campaigns. All performances are close to each other and especially satisfactory since all classification rates are higher than 85%. These performances show the ability of a multi-sensors system to obtain relevant information about atmospheric pollution, using a model based on neural networks. But it is also important to study the domain of validity of the calculated models, assessing their portability from one multi-sensors array to another and also from one measuring site to another. The aim is to build a robust model. To check the first criteria, inter-array portability, various combinations of arrays were made for both learning and predictions steps on one site. Table 8 reports the results obtained in the case of measurements done on the station of Nogent. As it can be seen, the model calculated on a multi-sensors array (n°1) and applied on itself has very good performances but is useless because when this same model is applied to another multi-sensors array (n°2), it has a very low success rate, around 25%. In contrast, a model calculated on a set of two multi-sensors arrays (n°1+n°2) and applied to a multi-sensors array which data were not used in the calculation of the model parameters (n°3), has correct success rate over 70%. These results tend to demonstrate that if the number of multi-sensors arrays is large enough for learning phase, the model becomes more general, but slightly less efficient. Anyway, with only 3 multi-sensors arrays, the statistical size seems to be good enough to make a pertinent modelling.

In order to study the inter-site portability, same types of experiments were carried out by changing the site for learning and prediction with a given array (Table 9). Like for inter-array portability, a model calculated from data collected on one station (Bethune) and applied to another one (Nogent) presents a bad success rate, around 50%. However, if the parameters are calculated from a set of different stations (including urban types), and applied to a station that was not taking into account in the calculation, then the results are satisfying, always upper than 80%. So, the more sites representative of each pollution categorization are involved in the learning and validation phases, the more robust is the model.

Conclusions

The concept of qualitative and semi-quantitative pollution detection using a gas sensor array was presented in this work. The method is based on modelling temporal responses of conductances in multi-sensors systems. Three identical sensors arrays with nine different commercial semiconductor gas sensors were constructed. Calibration and standardization techniques were elaborated to correct drifts of baseline and of sensitivity and assure an interarray homogeneity in sensors response. The study was carried out by measuring existing pollution in different polluted sites. Six sites were chosen, and the pollution type definition was made based on pollutant concentrations measured by air quality network stations. A pollution index was also introduced to describe pollution's severity. This combination of predicted pollution category and pollution index can be used to inform the public on the pollution in real time, and with a high spatial accuracy. Such multi-sensors arrays can be multiplied and used in a way to mesh a sensible zone to be monitored. Up to now, and according to our knowledge, no equivalent instrument combining multi-sensors arrays with models based on neural networks using the proposed semi-quantitative approach exists. The classification success rate for pollution categories prediction had a success rate higher than 80% with a low error in the prediction of the pollution index. Presented results show that the model can be applied to a large number of uncharacterised arrays of sensors (thanks to calibration / normalization procedure) and unknown sites if a sufficient number of reference sites and arrays for the learning and validation phases of the neural network are used. Further modelling approach could be applied in order to build equivalent categorizations for inside air quality problematic.

Fig 1 :

 1 Fig 1: Sensors array including 9 semiconductor sensors Table 1: Chosen semiconductor based gas sensors with their target gases and their specified detection range, according to constructor specifications (* TGS: Figaro Sensors, MICS: SGX Sensortech)

  Fig 2 presents sensors responses from one of the sensors arrays measured in such conditions.

Fig 2 :

 2 Fig 2 : Sensors relative conductance variation (X R ) for low concentration of pollutants

Fig 3 :

 3 Fig 3 : Relative conductance variation (X R ) under zero air variation during 7 months of measurements Baseline drift value is specific to each sensor. It is imperative that this test is always repeated under the same conditions. Drifts can take different shapes depending on the sensor's type, but sensors of same type can also have different drifts. Fig 4 shows the evolution of the baseline of Mics 5525 sensors from the 3 different arrays for another period of 8 months.

Fig 4 :

 4 Fig 4: Baseline drift registered for different Mics5525 sensors from three arrays during 8 months of observation.

Fig 5 :

 5 Fig 5: Sensors responses to normalization gases: (a) X R before normalization, (b) S after normalization .

Fig

  Fig 5a represents relative conductance of sensors under target gases before normalization, and Fig 5b shows these variations after normalization. Although differences of normalized response to a given target gas is still observable between the three sensors of same type for each array, normalization has decreased in a remarkable way the amplitude of these variations.

Fig 6 :

 6 Fig 6 : Membership function of the assertion "pollution with NO", Boolean logic and Fuzzy logic

  Fig 7.

Fig 7 :

 7 Fig 7 : Illustration of categories of pollution and pollution index

Fig 8 :

 8 Fig 8 : Structure of the artificial neural network used for the pollution categorization

Fig 9 :

 9 Fig 9 : Building of the target data Ŷ to be compared with output data "Y" from the artificial neural network

Fig 10 :

 10 Fig 10 : Short example of classification and prediction from array n° 3, Béthune, period from 10/8 to 13/8; 2 days for artificial neural network learning, 1 day for test. (a): classification from analyzers data, (b): classification from gas sensors data, (c): pollution index (PI): calculated from analyzers and predicted from sensors array

Table 2 : Reference gas and concentration chosen for each sensor for normalization

 2

		gases N C N
	MICS5525	CO	20 ppm
	MICS2710	NO 2	0.4 ppm
	MICS2611	NO 2	0.4 ppm
	TGS2600	NH 3	4 ppm
	TGS2602	NH 3	4 ppm
	TGS2620	NH 3	4 ppm
	TGS2611	CH 4	20 ppm
	MICS5521	CO	20 ppm
	MICS5914	NH 3	4 ppm

Table 3 : Statistical analysis of pollutant concentrations in six AASQA stations Station NO (µg/m 3 ) NO 2 (µg/m 3 ) O 3 (µg/m 3 ) Station Category Mean Max Std dev Mean Max Std dev Mean Max Std dev

 3

	Creil	Urban	9	243	16	24	122	17	42	235	29
	Douai	Urban	9	296	18	25	100	15	39	182	27
	Bethune	Urban	6	191	13	22	150	16	44	182	27
	Nogent	Suburban	12	254	21	26	139	17	43	236	30
	Roubaix	Traffic	32	801	46	46	301	25	---	---	---
	Valenciennes	Traffic	20	405	29	37	152	18	---	---	---

Table 5 : Determination of membership degrees for the 3 categories of pollution from the concentrations of pollutants according to Boolean or Fuzzy logic equations

 5

	pollution class	Boolean logic equation	Fuzzy logic equation
	Traffic	[𝑁𝑂] * [𝑂 3 ] ̅̅̅̅̅̅	MD(NO)*(1-MD(O 3 ))
	Urban	[𝑁𝑂 2 ] * [𝑁𝑂] ̅̅̅̅̅̅̅ * [𝑂 3 ] ̅̅̅̅̅̅	MD(NO 2 )*(1-MD(NO))*(1-
	Photochemical	[𝑂 3 ]	MD(O 3 )) MD(O 3 )

Table 6 : Minor pollution indexes (PI) as a function of pollutants concentrations for the three pollutants considered in this study (NO, NO 2 and O 3 )

 6

	Pollu	NO	NO 2	O 3
	tion	(µg/m 3 )	(µg/m 3 )	(µg/m 3 )
	index PI 1	0-9	0-29	0-29
	2	10-34	30-54	30-54
	3	35-74	55-84	55-79
	4	75-154	85-109	80-104
	5	160-239	110-134	105-129
	6	239-314	135-164	130-149
	7	315-379	165-199	150-179
	8	380-439	200-274	180-209
	9	440-499	275-399	210-239
	10	>= 500	>=400	>=240

Table 7 : Multi-sensors arrays classification success rates and pollution index prediction error (RMSEP)

 7

			Multi-sensors	Multi-sensors	Multi-sensors
			array n°1		array n°2		array n°3
	Site	Type	Classification rate	RMSEP PI	Classification rate	RMSEP PI	Classification rate	RMSEP PI
	Creil	Urban	87%	0.04	90%	0.03	89%	0.05
	Nogent	Suburban	90%	0.03	91%	0.01	89%	0.01
	Saint-Amand Suburban	89%	0.03	87%	0.03	88%	0.03
	Valenciennes	Traffic	91%	0.02	90%	0.01	89%	0.02
	Béthune	Urban	91%	0.03	98%	0.02	98%	0.03
	Douai	Urban	91%	0.01	92%	0.01	93%	0.02

Table 8 : Example of inter-arrays model portability for the urban site of Nogent Array use for model learning… Array use for Prediction done Classification Success Rate

 8

	Pollution index
	RMSEP

Table 9 : Example of inter-sites model portability

 9

			Array 1
	Model learning site	Prediction site	Classification success rate	Pollution index RMSEP
	Bethune -Urban	Bethune -Urban	91%	0.03
	Bethune -Urban	Nogent -Suburban	47%	0.15
	All stations out of Bethune (all types)	Bethune -Urban	86%	0.09