J Lamon

M R'mili

Investigation of flaw strength distributions from tensile force-strain curves of fiber tows

The flaw strength data are determined using tensile tests on various fiber tow types including, SiC, carbon, glass, basalt and alumina. The plots of p-quintile vs. flaw strength data derived from experimental force-strain curves exhibit linearity. This indicates unambiguously that flaw strength is a Gaussian variate. The statistical distribution of data was also calculated using the Weibull distribution function. Normal and Weibull cumulative distributions of flaw strengths are found to compare fairly well. The significance of normal distribution function and of the p-quintile vs. flaw strength relation for the characterisation of fibre flaw strength is discussed. The p-quintile vs. flaw strength relation is shown to provide theoretically a material characteristic. The normal distribution function is used to construct reference empirical distributions of flaw strengths that allow the evaluation of Weibull plot and Maximum Likelihood Estimation methods as functions of sample size and composition.

1/ Introduction

Fibers control the ultimate failure of composite materials. For this reason, features of fiber fracture warrant much consideration. Like most brittle materials, ceramic fibers contain populations of microstructural flaws that act as stress concentrators. The flaws are generally distributed randomly, and they exhibit wide variability in severity, as a result of variability in shape, nature, size and location with respect to the stress-state. As a consequence, stressinduced fracture is a stochastic event, and fracture stress is a random variable.

. The fundamental approaches recognize the flaws as physical entities [4-10] while the severity of fracture-

2

inducing flaws is measured either using flaw size or flaw strength. Elemental strength and flaw size are related by fracture toughness expression. In the so-called elemental strength approach, flaw strength is defined using an elemental strength that is the critical local stress that causes extension of a flaw. A general equation of failure probability-flaw strength relation is [START_REF] Argon | Mechanical Behavior of Materials[END_REF][START_REF] Lamon | Ceramics reliability: statistical analysis of multiaxial failure using the Weibull approach and the Multiaxial Elemental Strength Model[END_REF][START_REF] Lamon | Statistical Analysis of Bending Strengths for Brittle Solids: A Multiaxial Fracture Problem[END_REF][START_REF] Lamon | Statistical approaches to failure for ceramic reliability assessment[END_REF][START_REF] Lamon | Brittle fracture and damage of brittle materials and compsites: statisticalprobabilistic approaches[END_REF]. [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] where g(S) is the flaw density function and S is the elemental strength.

In a fiber that contains a population of randomly distributed flaws, the weakest flaw of the population, induces fracture according to the weakest-link theory. Under uniform uniaxial tension, the elemental strength of the weakest flaw coincides with the applied stress on fiber at failure in the direction parallel to fiber axis.

The distribution of flaw strengths is a key issue for fracture analysis and characterization of brittle materials. The power law distribution of extreme values (often referred to as the Weibull distribution) is a versatile distribution that is very sensitive to the statistical parameters. The estimated parameters of power law generally exhibit variation. Much attention was put on this issue by researchers [START_REF] Gong | A new probability index for estimating Weibull modulus for ceramics with least square method[END_REF][START_REF] Barnett | Probability plotting methods and order statistics[END_REF][START_REF] Watson | An examination of statistical theories for fibrous materials in the light of experimental data[END_REF][START_REF] Paramonov | A family of weakest link models for fiber strength distribution Composites[END_REF][START_REF] Phani | A new modified Weibull distribution function for the evaluation of the strength of silicon carbide and alumina fibres[END_REF][START_REF] Amaniampong | Statistical variability in the strength and failure strain of aramid and polyester yarns[END_REF][START_REF] Bergman | On the estimation of the Weibull modulus[END_REF][18]. Furthermore, authors raised questions about the validity of the Weibull distribution function for ceramic strength. It is difficult to decide on the basis of small sample sizes whether the data follow a Weibull distribution or not [START_REF] Lu | Fracture statistics of brittle materials: Weibull or normal distribution[END_REF]20]. It was also observed on limited sets of flexural failure stresses (15 data) that the fitted Weibull and Normal distributions behave quite similarly [20]. A few authors considered the normal distribution function for the failure of fibers: cotton or polyester [START_REF] Peirce | Tensile Tests for Cotton Yarns-"The Weakest Link[END_REF][START_REF] Epstein | Application of the theory of extreme values in fracture problems[END_REF][START_REF] Sakai | Effect of yarn length on tensile strength and its distribution[END_REF], glass [START_REF] Foray | Statistical flaw strength distributions for glass fibres: Correlation between bundle test and AFM-derived flaw size density functions[END_REF], SiC [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF] and flax [START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF].

The rigourous and unambiguous method of evaluation of the pertinence of a distribution function requires the fit of the expected cumulative distribution function to the empirical distribution of strength data. Construction of the empirical distribution is an issue since an unbiased estimator of probability is required. This difficulty has been questioned in the literature [11-18, 25, 26]. The present paper proposes a solution based on plot of p-quintile vs.

strength. Linearity of this plot demonstrates that normal distribution function characterizes flaw strengths. This approach is close to the quintile-quintile method applied to normal distributions. Then, normal strength distribution function can afford a reference distribution to evaluate the pertinence of Weibull distribution.

The objective of the present paper is to investigate the flaw strength distributions of various fiber types along these lines. Various types of fiber tows reinforcing ceramic matrix composites were tested: SiC, glass, carbon, basalt, alumina. Emphasis was placed on Nicalon SiC fiber to exemplify some significant results. Flaw strength for a filament under tension parallel to axis is the stress at failure. Fiber tows may contain more than thousand filaments, so that tensile tests on fiber tows generate substantial sets of filament failure data. A fiber tow is regarded as a population of parallel, independent and identical single filaments (radius Rf, length ), which exhibit brittle fracture and statistical distribution of strengths. In the following, filament designates a single filament.

When filaments are broken, the surviving filaments carry equally the applied load, so that the tensile force on tow is:

F = σ Sf (Nt-Ni) (2)
Where σ is the stress on surviving filaments. Sf is filament cross-sectional area, Nt is the initial number of filaments carrying the load, Ni is the number of broken filaments.

Under monotonous loading at constant strain rate, the filaments break according to ascending order of strengths, one after one when the stress increment is such that σ reaches a filament strength (σ = σf). The force increases to a maximum, and then, decreases gradually to 0 as Ni approaches Nt [25 -30]. This failure mode is observed when filaments share the load equally, when the filaments are independent, and when the strain rate is sufficiently slow so that stress increase does not make several filaments break. The force-strain relation is convenient to characterize the tensile behavior of a tow [25 -30]. It is derived from equation (2):

F(ε ) =Ef ε Sf (Nt-Ni) (3)
Where ε = σ/Ef is the strain, Ef is filament Young's modulus.

The ratio Ni/Nt is the fraction of filaments failed at strain εi. It represents the probability of failure (denoted P(εi)) for the filament having rank Ni and strength σ fi = σ i = E f ε i .

The force-strain relation is derived from equation (3) as:

F(ε) = Nt Sf Ef ε [1-P(ε)] (4)
Equation ( 4) reduces to:

F(ε) = k0 ε [1-P(ε)] (5)
where k0 is the initial tow stiffness, given by the slope of the initial elastic domain of the force -strain curve. In the elastic domain when filaments do not fail, P(ε)=0, and F(ε)=k0 ε.

When the filaments have an identical section Sf , k 0 = N t S f E f . When they have different

sections k 0 = S fi i=1 i=N t ∑ E f = N t S f E f .
Where S is the average filament diameter. Values of failure probability P(ε) are estimated from F and ε independently of the number of data [START_REF]Determination of distribution of tensile strengths and of tensile strains to failure of filaments within a multifilament tow at ambient temperature[END_REF][START_REF]Fine ceramics (advanced ceramics, advanced technical ceramics) -Reinforcement of ceramic composites -Determination of distribution of tensile strength and tensile strain to failure of filaments within a multifilament tow at ambient temperature[END_REF], according to equation (6) derived from equation [START_REF] Argon | Mechanical Behavior of Materials[END_REF]:

P(ε) = 1- F (ε) k 0 ε (6)
It is worth pointing out that this is superiority over the method of construction of Weibull plot using a probability estimator [10 -17] depending on sample size.

Flaw strength is characterized by both filament tensile failure stress and strain. In this paper, flaw strength and failure strain are employed indifferently to designate flaw strength.

2.2/ Analysis of flaw strengths using the normal distribution

The method that is proposed to demonstrate that filament flaw strength is a Gaussian variable is derived from the Henry's diagram and the quintile-quintile diagram applied to normal distribution. The Henry's diagram is a graphical method for comparing a Gaussian distribution to a set of data. When X is a Gaussian variable, with µ = mean and s = standard deviation, and N is a variable of the standard normal distribution, it comes:

P X < x ( ) = P X -µ s < x -µ s       = P(N < z) = Φ(z) (7)
Where P(.) is probability,

z = x -µ s (8), Φ is the cumulative standard normal distribution of variable z .
The cumulative distribution function of the standard normal distribution is given by the equation:

Φ(z) = 1 2π exp(- t 2 2 -∞ z ∫ )dt (9)
When linearity of relation z(x) (equation ( 8)) is observed for a set of xi data, one may assume that the xi data are occurrences of the same Gaussian variable. Then, the plot of p-quintile zp vs. xp indicates whether X is a Gaussian variable. p is the value of cumulative probability: [START_REF] Lamon | Brittle fracture and damage of brittle materials and compsites: statisticalprobabilistic approaches[END_REF] The diagram is constructed as follows:

The p-quintile zp is derived from the cumulative standard normal distribution function Φ

z p = Φ -1 p ( ) (11)
The equation of the p-quintile zp diagram is:

P( X < x i ) = p z p = Φ -1 p ( ) = x p -µ s ( 12
)
Note that xp = xi. zp can be extracted easily from Φ using a computer or tables that are available in text books. For a set of failure strain data, x is denoted ε. The relation to be satisfied by strength data is:

z p = ε p -µ s ( 13
)
In the present paper, p was taken to be the P(ε) value derived from the force -strain curve using equation ( 6). As indicated above, this value is independent of sample size.

The equation for the normal distribution PN (Ε < ε) of positive strain values is:

for ε > 0 (14) (15)
f(ε) and PN = 0 when ε ≤ 0, as the flaws cannot grow under compression. f(ε) is the density of probability, Ε and ε are failure strains, µ is the mean and s is the standard deviation.

Estimates of µ and s are obtained from the slope 1/s and the intercept µ/s of p-quintile vs failure strain diagram (equation ( 13)).

2.3/ Analysis of flaw strengths using the Weibull model

Similarly, the expression of the Weibull distribution is defined for ε > 0:

P w (E < ε) = 1-exp - ε ε l         m         ( 16
)
where m is the shape parameter (or Weibull modulus) and εl is the scale factor.

f (ε ) = 1 S 2π exp - (ε -µ) 2 2S 2       ∫ = ≤ ε ε ε ε 0 ) ( ) ( d f E P N
An expression identical to [START_REF] Amaniampong | Statistical variability in the strength and failure strain of aramid and polyester yarns[END_REF] in terms of strengths is obtained by replacing ε by σ. The corresponding scale factor is σl = Ef εl.

The statistical parameters were estimated using the mean and variance expressions derived from the first moment of the Weibull distribution:

s µ = Γ 1+ 2 m ( ) Γ 2 1+ 1 m ( ) -1           ≅ 1.2 m ( 17
)
ε l = µ Γ(1+ 1 m ) (18)
Where Γ(.) is the Gamma function.

For comparison purposes, two alternative methods were used for the Nicalon fiber: the MLE method (Maximum Likelihood Estimation) and the linear regression analysis of Weibull plot.

The parameters estimated using the MLE method are claimed to statistically approach the true values of the population more efficiently than other parameter estimation techniques as the size of the sample increases. They are derived from equations [START_REF] Lu | Fracture statistics of brittle materials: Weibull or normal distribution[END_REF] and (20)

(σ R j ) m Lnσ R j j=1 n ∑ (σ R j ) m j=1 n ∑ - 1 n Lnσ R j j=1 j ∑ - 1 m = 0 ( 19
)
σ 0 = 1 n σ R j ( ) m j-1 n ∑         1 m (20)
where σ j R is the j th strength in the set of data, n is the number of data.

Construction of Weibull plot requires values of probability P that are associated to strength data. For this purpose, the strength data are ranked in ascending order. Then, the values of probability are calculated using a probability estimator. Various estimators have been devised

in the literature [START_REF] Gong | A new probability index for estimating Weibull modulus for ceramics with least square method[END_REF][START_REF] Barnett | Probability plotting methods and order statistics[END_REF][START_REF] Watson | An examination of statistical theories for fibrous materials in the light of experimental data[END_REF][START_REF] Paramonov | A family of weakest link models for fiber strength distribution Composites[END_REF][START_REF] Phani | A new modified Weibull distribution function for the evaluation of the strength of silicon carbide and alumina fibres[END_REF][START_REF] Amaniampong | Statistical variability in the strength and failure strain of aramid and polyester yarns[END_REF][START_REF] Bergman | On the estimation of the Weibull modulus[END_REF][18][START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF]. The estimator Pj = (j -0.5)/n is recommended for limited sample sizes (j is the rank of filament strain to failure). Linear regression was applied to the plot of experimental data in terms of Ln(-Ln(1-P)) vs. Lnσ owing to the linearized equation of Weibull distribution function (Weibull plot). Statistical parameters are derived from the slope and the intercept. As mentioned in the introduction, this method is questionable [11-18, 25, 26].

2.4/ Experimental

The bundle test specimens contained more than 1000 filaments. An estimate of the number of filaments was determined from the initial slope of the force-strain curve, assuming a constant filament section. The main filament characteristics are summarized in table 1. Test specimens were prepared according to the protocol described in previous papers [START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF][START_REF] R'mili | Estimation of Weibull parameters from loose bundle tests[END_REF][START_REF] R'mili | Statistical fracture of E-glass fibres using a bundle tensile test and acoustic emission monitoring[END_REF][START_REF]Determination of distribution of tensile strengths and of tensile strains to failure of filaments within a multifilament tow at ambient temperature[END_REF]. Care was taken during specimen preparation and tests to avoid specific draw-backs, such as fiber slack [START_REF] Phoenix | Probabilistic strength analysis of fibre bundle structures[END_REF] and friction [START_REF] Hill | Weibull statistics of fibre bundle failure using mechanical and acoustic emission testing: the influence of interfibre friction[END_REF]. An example of fiber tow specimen is shown on figure 1. The tensile tests were carried out at room temperature under monotonous loading (displacement rate = 2 μm/s) on a servo-pneumatic testing machine. Test specimen elongation was measured using a contact extensometer (with a ± 2.5 mm elongation displacement transducer) that was clamped to the specimen using two 4-mm-long thermoretractable rings.

Thus strain measurement was direct and unpolluted by load train deformations. The rings were located close to the grips in order to avoid possible bending introduced by the extensometer. The inner distance between the rings defined the gauge length.

The samples were first loaded up to 5% of the ultimate load, and then the extensometer was placed and adjusted. Lubricant oil was sprayed on filaments to avoid friction.

Acoustic emission was monitored during the test on SiC fiber tow in order to detect fibre fractures [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF][START_REF] R'mili | Statistical fracture of E-glass fibres using a bundle tensile test and acoustic emission monitoring[END_REF]. Two resonant PZT transducers (Acoustic Emission type μ 80) were placed at specimen ends, in order to locate fracture origins. Only those events located in the gauge length with amplitude > 60 dB (corresponding to fibre failure) were kept. 1: Characteristics of the fiber tows that were tested.

3/ Results

3.1/ Tensile behavior of tows

Figure 3 shows a typical tensile force-strain curve obtained on a SiC Nicalon tow, together with locations of acoustic emission events in the gauge length [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF]. The curve displays the conventional features of bundle tensile behavior, i.e., initial elastic deformations (at strains <0.5%), and then nonlinear deformations as a result of individual fibre breaks as indicated by acoustic emission events. Beyond maximum, the force on tow decreases progressively to 0, and the density of AE event sources looks homogeneous, which suggests that filament interactions probably did not operate. Note that the curve exhibits initial linear deformations, which indicates that all the filaments carried the load, and thus, that slacks were not present.

Tensile behavior for tows of the other fiber types displayed similar features (figure 4). The proportional limits were between 0.3 and 2%. The estimates of the initial numbers of continuous filaments in tows given in table 1 were derived from k0, assuming identical filaments. The experimental force-strain curves looked smooth. They did not exhibit stepwise decrease in load at a constant strain [START_REF] Chi | Determination of single fiber strength distribution from fiber bundle testings[END_REF][START_REF] Wilson | Statistical tensile strength of Nextel TM 610 and Nextel TM 720 fibers[END_REF], which suggests that sudden increase in load when a single filament breaks did not cause breakage of a large amount of filaments and that expected equal load sharing was satifactory. For all the other fibers of this paper, the zp(εp) plots fit straight lines (figures 6 -9). The excellent goodness of fit that was observed is confirmed by the high values of coefficients of linear regression R 2 > 0.99 for most fibers (table 2) except T300 (R 2 = 0.97). This point is discussed in a subsequent section. -the experimental cumulative distribution derived from the force strain curve using equation ( 6),

-the normal cumulative distribution functions (cdf) calculated using the mean and standard deviation estimates derived from the zp (εp) plot (table 3), -the Weibull cdf calculated using the Weibull parameters estimated using the 1 st moment method (equations ( 17) and (18); table 3). 4). As a consequence, it can be reasonably considered that sound estimates of Weibull parameters were obtained using the 1 st moment expression. The two alternative methods (linear regression on Weibull plot and MLE) underestimated the Weibull modulus (table 3).

The normal, Weibull and experimental flaw strengths distributions coincided also for the other fibers of this paper. The high values of correlation coefficients reported in table 4 assess the goodness-of-fit.

3.3.2/ Prediction of force-strain curves

Calculation of force-strain curves for the statistical parameters given in table 3 provided further assessment of the previous results. The tensile curves that were predicted using equation ( 5) and normal and Weibull expressions for probability P(ε) show both good agreement with the experimental results for most fibers.

This result was expected from the linearity of zp(εp) plots and the comparison of cdf derived from tensile curves with those that were computed. The Nicalon force-strain curve calculated for the values of parameters estimated using MLE method did not fit properly the experimental curve (figure 11a). Some discrepancy was observed on a few fibers (Almax, T300 and AS4C), as load decrease beyond maximum force looked steeper on a range of strains of the experimental curves when compared to the predicted curves (figures 11 e, g h). The corresponding zp(εp) plots displayed some deviation to linearity in this range of strains, indicating that strains were smaller than predictions. This discrepancy may be attributed to either the presence of a bi-modal population of flaws, or to the presence of extrinsic effects like fiber interfriction or fractures of small groups of fibers. given applied strain whatever the volume of specimen (figure 12). The total distribution is obtained when tow contains a large enough number of flaws. This implies a critical tow size above which the total flaw population is characterized. Peirce and Sakai raised this issue of minimum length [START_REF] Peirce | Tensile Tests for Cotton Yarns-"The Weakest Link[END_REF][START_REF] Sakai | Effect of yarn length on tensile strength and its distribution[END_REF].

Below the critical size of tow, subsets of the total distribution are obtained (figure 12). They pertain to the reference diagram whatever sample size and specimen size, since a flaw is characterized by a unique strength and a unique zp value. When bigger flaws are present in a longer tow, weaker flaw strengths are added to the data range obtained on the smaller tows, whereas a large part of flaw strengths are similar to those pertinent to the smaller filaments, and some higher flaw strengths present in the smaller filaments are eliminated. Conversely when tow size is decreased, data are added at the high flaw strength extreme whereas data are subtracted at the low strength extreme. The sets of flaw strength data are translated parallel to the zp(εp) line (figure 12). This implies that zp(long filament) < zp(short filament), and that p(long filament) < p(short filament). 16)) is acceptable only for the range on strengths in which the data lie which questions the validity of fiber strength data extrapolation [START_REF] Watson | An examination of statistical theories for fibrous materials in the light of experimental data[END_REF][START_REF] Phoenix | Statistical analysis of flaw strength spectra of high modulus fibers, Composite Reliability[END_REF]. In [START_REF] Phoenix | Statistical analysis of flaw strength spectra of high modulus fibers, Composite Reliability[END_REF] m estimates were found to depend on fiber length [START_REF] Phoenix | Statistical analysis of flaw strength spectra of high modulus fibers, Composite Reliability[END_REF]. This effect has been observed on several other unpublished results.

6.2/ Normal distribution of flaw strengths

The normal distribution indicates the probability of occurrence of a characteristic in a population of infinite size. Therefore, the normal distribution may be regarded as a natural distribution for flaw strengths in a tow, owing to the large number of filaments in the tested tows. Normal distributions are appropriate in the following conditions:

-There is a strong tendency for the variable to take a central value;

-Positive and negative deviations from this central value are equally likely;

-The frequency of deviations falls off rapidly as the deviations become larger These conditions suggest the features of flaw strength distributions. In particular, they indicate that there is a strong tendency for flaw severity to take a central value. Features of multifialement tows such as the large amount of filaments with elongated shape, together with the fabrication process of fibers may be the origin of favorable microstructure.

Figure 12: Schematic diagram showing the location of flaw strength data sets for various specimen volumes (V1>V2>V3).

6.3/ Implication of flaw strength distributions extracted from tow tensile behavior

Effects of inherent flaws.

In the presence of a large number of filaments in tow (table 1) the question arises whether the flaw strength distribution that was derived from experimental data is representative of the total population of flaws present in the material.

When the flaw strength distribution obtained on a tow is representative of the total population of inherent flaws, owing to the presence of lower and higher bounds (figure 12) the

V 1 V 2 V 3 Strain Quintile z p Lower bound V 3 >V 2 >V 1
Upper bound probability is 0 that weaker or stronger flaws are present in other identical tows. As a consequence, the force-strain curves would coincide. No variation would be observed on statistical parameters nor on the critical fiber at maximum force. It is worth noting that very limited variation was observed on the statistical parameters estimated on several SiC [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF] or flax [START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF] fiber tows. In [START_REF] Calard | Failure of fibres bundles[END_REF] it was shown using the Nt -dependent coefficient of variation [START_REF] Daniels | The statistical theory of the strength of bundles of threads I[END_REF][START_REF] Mccartney | Statistical theory of the strength of fiber bundles[END_REF][START_REF] Gurvich | Strength size effect of laminated composites[END_REF], that the forces at maximum should not be scattered when the total number of filaments is large.

Furthermore, the probability of finding a more severe flaw on a longer tow would be 0 also, owing to the presence of lower bound.

If the flaw strength distribution obtained on a tow were not representative of the total flaw population, the tensile curves and the statistical parameters would show variation. Size effects on strength distribution and statistical parameters would be effective [START_REF] Phoenix | Statistical analysis of flaw strength spectra of high modulus fibers, Composite Reliability[END_REF]. Furthermore S.L.

Phoenix showed that the Weibull modulus increases with decreasing fiber length. He attributed this effect to limited domain of stress data produced so that the proper cumulative flaw strength spectrum is generally not characterized properly. One must consider a critical tow size above which the total flaw population is characterized.

Proper characterization of the total distribution depends on spatial distribution of critical flaws with respect to gauge length. It can be anticipated that a low flaw density will require long filaments, whereas shorter filaments will be appropriate in the presence of a high flaw density.

Obviously, a large number of filaments is preferred. Then, the determination of the minimum number of filaments is not straightforward. It depends first on the spatial density of flaws versus gauge length. Second, it must be such that the fiber tow contains the weakest and the strongest flaws of the total distribution. The chances of presence of these extreme flaws increase with the number of filaments in tow.

6.4/ Implications: Evaluation of Weibull plot and MLE methods of estimation of Weibull parameters

The linearity of p-quintile vs. strain diagram demonstrated that the normal cumulative distribution function characterizes flaw strengths. Then, the Weibull distribution was found to fit the normal distribution. Thus, it is inferred that normal cdf can be used as a reference to evaluate conventional methods of determination of Weibull parameters that are claimed more or less biased.

Evaluation of Weibull plot based method

Comparison of Weibull plot with the Ln(-Ln(1-P)) vs. Lnε plot of normal cumulative distribution is shown on figure 13 for Nicalon fiber [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF]. The Weibull plot deviates from the reference normal distribution and also from linearity despite the presence of a unimodal flaw population, and although the Weibull distribution function was shown to fit quite well the normal distribution (figure 10). This effect was reproduced on the other fibers of this paper. It indicates that Weibull plot can fail to indicate whether Weibull distribution characterizes strength. As mentioned earlier, this effect is attributed to probability estimator.

Representativeness of subsets of data: evaluation of Weibull plot and MLE methods

Subsets of 20 and 30 data were extracted randomly (five draws per sample size) from the experimental set of strains for Nicalon fiber tow. Such sample sizes are generally recommended for the estimation of Weibull parameters. The probability estimator Pj = (j -0.5)/n was used for the construction of Weibull plot. On figure 13, the Weibull plots show also deviation from the reference normal distribution (Ln(-Ln(1-P) vs.Lnε) and from linearity. This indicates that these sets of data are not characterized by the Weibull function and that they do not represent the total distribution of flaw strength pertinent to Nicalon fiber. Furthermore, results for subsets having the same size produced by several draws lead to Weibull modulus estimates that displayed significant variation: 4.75 < m < 7.18 for data subset size n = 20 and 4.55 < m < 6.05 for n = 30 [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF]. Detailed results on Nicalon fiber and flax fiber are reported in [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF][START_REF] Lamon | Investigation of statistical distributions of fracture strengths for flax fibre using the tow-based approach[END_REF]. Thus, it appears that the composition of batches of test specimens may yield to subsets of data that are disjoint and not representative of the flaw strength distribution and that are not characterized by the Weibull function. This agrees with the above-mentioned statement that the Weibull equation ( 16) is acceptable only for the range on strengths in which the data lie [START_REF] Phoenix | Statistical analysis of flaw strength spectra of high modulus fibers, Composite Reliability[END_REF]. These results demonstrate that the estimation of Weibull parameters does depend not only on the sample size, but also on the composition of the sample in terms of flaw strengths. When the flaw strength distributions in samples are not representative of the total distribution pertinent to a fiber type, they vary with sample. The variation is reflected by the above-mentioned variability of statistical parameters. The MLE method was applied to two subsets of 30 data selected from the Nicalon flaw strength data with a view to obtain even and uneven patterns of dispersion of data, i.e. an 5) differ from the value reported in table 2 for Nicalon (m = 5.2, εl = 1.24). Furthermore, they display a significant variation (table 5) when compared with results from fiber tow behavior (5.23 < m < 5.43, 1.20 < εl < 1.25) [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF]. Then, they depend on sample size, and also on homogeneity of data distribution. These results agree with the findings of the previous section.

6.5/ Influence of extrinsic factors

The contribution of extrinsic flaws, such as slacks, filament interactions, random or local load sharing may affect the tensile behavior of fiber tows. In [START_REF] Calard | Failure of fibres bundles[END_REF] the influence of various extrinsic factors (such as load sharing, non-uniform loading) on tow strength was examined. It was shown that imperfect load sharing induces a drop in tow strength and an enhanced scatter in data. In the present paper, the experimental force-strain curves looked smooth.

Step-wise decreases in load at a constant strain were not observed, which suggests that simultaneous breakage of several filaments did not take place and that the expected equal load sharing was satisfactory. Moreover, the linear deformation from the origin of force-strain curves suggested that slack filaments were not present in tows. The plot of zp(εp) derived from tests on Nicalon fiber tow (figure 5) was found to agree with that obtained from tests on single filaments (figure 14). This suggests that results were not polluted by extrinsic factors. Furthermore, a very small variation was observed on the statistical parameters estimated on different Nicalon tow specimens [START_REF] R'mili | Flaw strength distributions and statistical parameters for ceramic fibres: the Normal distribution[END_REF] and the Weibull modulus estimates were consistent with results available in the literature for single filaments of Nicalon (m=5.2 (table 3) against m = 5.5 in [START_REF] Lissart | Statistical analysis of failure of SiC fibers in the presence of bimodal flaw populations[END_REF]).

However, the zp(εp) diagrams evidenced deviation from linearity for Alumina Almax and carbon T300 and AS4C at strains when the force decrease looks steeper (figures 7,8, 11e, 11f, 11g). But, the load decrease was not as steep as the load drops induced by the simultaneous failure of filaments. The strains being smaller than those expected from the trend shown by the linear domain of zp(εp) it may be argued that the tows were not free to deform because of interfiber friction. This effect was evidenced on glass fiber tows using lubricants having different viscosity [START_REF] R'mili | Statistical fracture of E-glass fibres using a bundle tensile test and acoustic emission monitoring[END_REF]. The presence of a bimodal flaw population may be another cause.

These issues warrant further consideration.

However, the zp(εp) diagram is not affected when the linear part only is considered, because, according to equation [START_REF] Batdorf | A statistical theory for the fracture of brittle structures subjected to nonuniform polyaxial stresses[END_REF], the values of P(ε) are not affected when data are removed. The strains > 1.6 % for AS4C and those < 0.7 % and > 1.1% for T300 were eliminated from the data sets. Table 3 gives the corrected parameters.

7/ Conclusions

Flaw strength distributions were extracted from the tensile behavior of fiber tows of various types: carbon, SiC, glass, basalt and alumina. Several features such as steady force decrease beyond maximum, spatial distribution of accoustic emission sources, limited variation in statistical parameters, consistency of Weibull modulus with literature, suggested that the testing conditions were well controlled in order to reach independent and successive breakage of filaments under constant strain rate.

Linear p-quintile vs. strain relation was obtained on all the tested fiber tows. This demontrates that flaw strength is a Gaussian variate. Then, the Weibull cumulative distribution function was found to characterize flaw strength as well. must be considered. This issue will require further investigation of flaw strength strength distributions obtained at various specimen sizes.

Normal distribution for flaw strength indicates that there is a strong tendency for the flaw severity to take a central value. This may be related to the large number of filaments in tested specimens, but also to the fabrication process, the microstructure and the elongated shape of fibers.

The normal distribution provides a sound reference for assessing the Weibull distribution and the statistical parameters. It allowed assessment of the estimation of parameters using the first moment of Weibull distribution. It also confirmed the limits of the method of estimation based on the construction of Weibull plot using a probability estimator, and of the MLE method that led to sample size dependent estimates. It appeared that subsets of the total distribution of flaw strengths generated on small sample sizes are not representative of the total distribution, they can be disjoint, and they may not be characterized by Weibull function even when the reference distribution is Weibull-type. This questions the issue of failure predictions using the Weibull model.

It seems reasonable to anticipate that the results of this paper can be extended to most brittle fibers.

1 /

 1 Generation of flaw strength data.

Figure 1 :

 1 Figure 1: Example of tow specimen with ends stuck in tubes for gripping in the testing machine jaws and thermoretractable clamping rings.

Figure 2 :

 2 Figure 2: Tow test set up with acoustic emission sensors.

Figure 3 :

 3 Figure 3: Load-strain curve and location of AE events along specimen axis for a Nicalon fibre

Figure 5 :

 5 Figure 5: Plot of zp(εp) for SiC Nicalon filaments.

Figure 6 :

 6 Figure 6: Plot of zp(εp) for glass and basalt filaments.

3. 3 /

 3 Figure 10 compares the following cumulative flaw strength distributions determined on the

Figure 7 :

 7 Figure 7: Plot of zp(εp) for alumina filaments.

Figure 8 :

 8 Figure 8: Plot of zp(εp) for carbon filaments.

Figure 9 :

 9 Figure 9: Comparison of zp(εp) plots.

Figure 10

 10 Figure 10 typifies on Nicalon fiber tows the excellent fit of the experimental distribution of

.

Figure 10 :Figure 11 :

 1011 Figure 10: Fit of experimental cumulative distribution by Weibull and normal distributions (Nicalon fiber tow).

6 / Discussion 6 . 1 /

 661 Uniqueness of zp(εp) diagramFlaw strength is defined as the elemental strength, i.e. the critical value of the stress that operates on a volume element containing a flaw. When a uniaxial uniform stress-state prevails, the elemental strength is equal to the remote stress at failure. This is the theoretical situation during a tensile test on filaments. The strain at failure is an appropriate characteristic of flaw strength for an elastic brittle filament. It is in proportion to failure stress according to Hooke's law: σ p = E f ε p Flaw strength, by definition, is independent of filament dimensions, whatever σp or εp are considered. It depends on flaw criticality characterized by flaw size and shape[START_REF] Lamon | Brittle fracture and damage of brittle materials and compsites: statisticalprobabilistic approaches[END_REF].Therefore, the main findings of this paper on flaw strength (i.e. linearity of zp(εp) plots and equivalence of normal and Weibull distribution functions) do not depend theoretically on filament dimensions. It can be noted that they were obtained on different gauge lengths: 115mm for Nicalon tows and 65mm for the other fiber types.The total population of flaws is characterized by a unique zp(εp) diagram, and a flaw is characterized by unique zp, and strength εp. Owing to equation[START_REF] Batdorf | A statistical theory for the fracture of brittle structures subjected to nonuniform polyaxial stresses[END_REF], zp does not depend on sample size. The zp(εp) diagram is bounded by the strengths of the biggest and the smallest flaws and the corresponding values of zp (figure12). The zp(εp) diagram for the total distribution is thus an invariant for a single population of flaw strengths. It is a material characteristic. It allows one to determine the probability of presence of a critical flaw for a

Figure 12

 12 Figure 12 illustrates the process of size effect by evolution of location of data sets on the

Figure 13 :

 13 Figure 13 : Comparison of various Weibull plots of strains-to-failure for Nicalon fiber tow: (1) total sets of data, normal distribution (P=PN) and empirical data (P=j/N) ; (2) subsets of 20 and 30 empirical data (P=(i-0.5)/n), (3) Truncated normal distribution (P=Pt).

  homogeneous distribution and an inhomogeneous distribution with clusters of data. The Weibull parameter estimates (table

Figure 14 :

 14 Figure 14: Plots of zp(εp) for Nicalon SiC filaments determined from tests on single filaments

Table 2 :

 2 Table 2 also reports the number of filaments and the number of data. Note that the number of data is smaller than the number of filaments. This difference does not have any incidence on the values of p and zp since p=P does not depend on the number of data as it is derived from F/ε according to equation (6), as pointed out in section 2.1. Sizes of data sets used for the construction of zp(εp) plots, and coefficients of correlation. The numbers of filaments are also given.

		Number of	Number	R 2
		filaments	of data
	SiC Nicalon	1013	402	0.998
	E-Glass	1952	686	0.99
	T30 Glass	1952	1223	0.99
	Basalt	2870	1256	0.996
	Alumina Almax	1337	155	0.99
	Carbon T300	5259	704	0.97
	Carbon T800H	12071	656	0.99
	Carbon AS4C	3267	197	0.988

The zp(εp) plots differentiate three groups of filaments according to mean and standard deviation values:

-large mean and standard deviation: glass and basalt filaments (figures 6 and 9) -intermediate mean and standard deviation: SiC Nicalon and C filaments (figures 7 and 9) -low mean and standard deviation: alumina fibers (figures 8 and 9).

Table 3 :

 3 Statistical parameters of Weibull and Normal distributions extracted from the force-

						method
	Fibre	m	ει (%)	µ (%)	s (%)	for Weibull
	SiC Nicalon 5.2	1.24	1.14	0.26	1 st moment
		4.86	1.20			Weibull plot
		4.37	1.25			MLE
	GlassT30	6.89	2.20	2.05	0.36	1 st moment
	Glass E	5,12	2.14	1.98	0.46	1 st moment
	Basalt	5.57	2.72	2.51	0.54	1 st moment
	Alumina	9,68	0.46	0.44	0.055	1 st moment
	ALMAX
	C T300	3.43	1.34	1.20	0.42	1 st moment
	C T300	3.20	1.37	1.22	0.46	1 st moment
	(corrected)
	C T800H	6,19	1.56	1.44	0.28	1 st moment
	AS4C	7.6	1.77	1.67	0.26	1 st moment
	AS4C	6.14	1.89	1.76	0.34	1 st moment
	(corrected)
	strain curves.

Table 4 :

 4 Correlations coefficients of fit of the cumulative distributions.

	Fibre	Exp vs	Exp. vs	Weibull vs
		normal	Weibull	normal
	SiC Nicalon 0.999	0.999	0.999
	GlassT30	0.998	0.999	0.999
	Glass E	0.999	0.999	0.999
	Basalt	0.999	0.999	0.999
	Alumina	0.999	0.998	0.999
	ALMAX
	C T300	0.99	0.99	0.999
	C T800H	0.999	0.997	0.999
	AS4C	0.997	0.996	0.999

Table 5 :

 5 They illustrate the limitation of MLE method and the issue of representativeness of flaw strength distributions in fracture analysis. Weibull parameters estimated using MLE method on a large set of data and on subsets of 30 data.

	number of data	402	30	30
	m	4.37	4.07	5.23
	εl (%)	1.25	1.26	1.32
	pattern		even	uneven

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.


MD. Benoit Godbert

PharmD. Nadine Petitpain
email: n.petitpain@chru-nancy.fr

MD. Anthony Lopez

PharmD. Yann-Eric Nisse

MD Pierre Gillet

Hepatitis B reactivation and immune check point inhibitors Authors

Keywords: Immune checkpoint inhibitors, viral hepatitis, fulminant hepatitis, reactivation, case report

Background: Liver toxicity during immune checkpoint inhibitor treatment is mostly due to immune mediated hepatitis. Viral hepatitis, as well as auto-immune or metabolic hepatitis, are considered as exclusion criteria for ICI induced immune hepatitis diagnosis. However, considering the high prevalence of viral hepatitis B infection and the increasing prescription of immune checkpoint inhibitors, their use in patients with HBV chronic viral infection may be common, even more if patients are treated for hepatocellular carcinoma. Few clinical studies directly deal with the risk of HBV reactivation during ICI therapy and real-life data is currently based on five reported cases of HBV reactivation, one with fatal outcome.

In this review, we summarize the current available clinical information about HBV reactivation risk during ICI treatment, its hypothetic mechanism, and propose practical recommendations about verifying and monitoring HBV status throughout the treatment.

Introduction

Immune checkpoint inhibitors (ICI) targeting programmed cell death protein 1 (PD-1)/PD ligand 1 (PD-L1) and cytotoxic T lymphocyte antigen 4 (CTLA-4) now have an essential place in the treatment of many cancers. Liver related adverse events (AE) may occur in 2 to 30% of patients with a typical pattern of hepatocellular injury, although cholestasis or hepatocellular followed by mixed injury are also possible [START_REF] Jennings | Hepatotoxicity induced by immune checkpoint inhibitors: a comprehensive review including current and alternative management strategies[END_REF] . The mechanism is generally an immune mechanism [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF] , resulting from immune related T-cell activation, leading to secretion of T4 helper cell cytokines and cytolytic CD8 T-cell tissue infiltration. ICI-induced immune mediated hepatitis (ICI-IMH) may rarely have fatal outcome [START_REF] Wang | Fatal Toxic Effects Associated With Immune Checkpoint Inhibitors: A Systematic Review and Meta-analysis[END_REF] .

Viral hepatitis, as well as auto-immune or metabolic hepatitis are considered as exclusion criteria for the diagnosis of ICI-IMH [START_REF] De Martin | Characterization of liver injury induced by cancer immunotherapy using immune checkpoint inhibitors[END_REF][START_REF] Hsu | Workup and Management of Immune-Mediated Hepatobiliary Pancreatic Toxicities That Develop During Immune Checkpoint Inhibitor Treatment[END_REF] . However, viral hepatitis reactivation remains a real issue during ICI treatment, even more since hepatocellular carcinoma (HCC) is becoming a new area for ICIs [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] El-Khoueiry | Nivolumab in patients with advanced hepatocellular carcinoma (CheckMate 040): an open-label, non-comparative, phase 1/2 dose escalation and expansion trial[END_REF] . Therefore, it is important to have the knowledge of prior hepatitis B virus (HBV) or hepatitis C virus (HCV) exposure at the start of treatment and to assess current serologic and/or viral load status.

As we recently observed a fatal case of fulminant viral hepatitis B (HBV) reactivation occurring after an unique durvalumab infusion, an anti-PD-L1 monoclonal antibody, we aim to provide here an overview about current available clinical information from studies and real life setting about the risk of HBV reactivation during ICI treatment.

Search strategy and selection criteria

We searched on PubMed up to August 2020 all studies and case reports on viral hepatitis reactivation during ICI treatment. The following search terms were used: "viral hepatitis", "ipilimumab", "nivolumab", "pembrolizumab", "durvalumab", "atezolizumab", "avelumab", cemiplimab", "immunotherapy", "immune check point inhibitor", "reactivation". No language restriction was applied. We focused on full-text articles, although relevant abstracts were considered. In addition, further studies were identified through the accurate evaluation of the reference lists of the assessed manuscripts. Finally, the studies were included in our review on the basis of originality and relevance.

Data from studies

A few clinical studies directly deal with the risk of HBV reactivation during ICI therapy.

Zhang et al. tried to assess the reactivation rate in a cohort study of 114 patients with positive HBs antigen (HBs-Ag) who received an anti-PD-1/PD-L1 antibody for cancer. They identified six patients (5.3%) developing HBV reactivation after the beginning of ICI, with a median onset of 18 weeks (range, 3-35 weeks). None of them died. The lack of antiviral prophylaxis appeared as the only significant risk factor for HBV reactivation (OR 17.50 [95% CI, 1.95-157.07], p =0 .004) [START_REF] Zhang | Hepatitis B virus reactivation in cancer patients with positive Hepatitis B surface antigen undergoing PD-1 inhibition[END_REF] .

Pu et al. [START_REF] Pu | Safety and efficacy of immune checkpoint inhibitors in patients with HBV/HCV infection and advanced-stage cancer: A systematic review[END_REF] , in their recent systematic literature review about ICI (PD1 or CTLA4 inhibitors) use in patients with advanced-stage cancer and previous HBV and/or HCV infection, gathered the data of 186 patients including 89 HBV and 98 HCV. A large proportion of their series came from studies on patients with hepatocellular carcinoma (HCC), especially the Check-Mate 040 trial assessing nivolumab in HCC patients [START_REF] El-Khoueiry | Nivolumab in patients with advanced hepatocellular carcinoma (CheckMate 040): an open-label, non-comparative, phase 1/2 dose escalation and expansion trial[END_REF] . Three HBV patients (3.4%) and 17 HCV patients (17,3%) experienced grade 3 or 4 transaminases increase during ICI treatment.

No treatment related death was reported. Among the 89 HBV patients, 67 did not receive concomitant antiviral therapy, and most of them did not experience HBV-VL change during treatment, except two patients (2.8%) who experienced HBV-VL increasing, and seven (10.4%) who had HBV-VL decreasing. The authors finally concluded that ICIs were safe in HBV or HCV infected patients, but with possibilities of viral hepatitis reactivation and, therefore, they recommended close monitoring and, if needed, antiviral therapy before or during ICIs treatment 8 . Kothapalli et al. [START_REF] Kothapalli | Safety and efficacy of anti-PD-1 therapy for metastatic melanoma and non-small-cell lung cancer in patients with viral hepatitis: a case series[END_REF] described seven patients treated with PD-1 inhibitors (nivolumab and pembrolizumab) for metastatic melanoma or metastatic NSCLC having chronic or past HBV/HCV infection. One patient showed a grade 2 increase in ALT that returned to the normal range following treatment of his HCV infection with ledipasvir /sofosbuvir. Four patients showed a grade 1 ALT increase and two patients experienced no hepatic toxicity after a follow-up greater than 24 cycles (nivolumab). Based on their findings, the authors recommended to closely monitor patients with specialized consultations and to treat with antiviral therapy if necessary [START_REF] Kothapalli | Safety and efficacy of anti-PD-1 therapy for metastatic melanoma and non-small-cell lung cancer in patients with viral hepatitis: a case series[END_REF] .

In contrast, the study of Shah et al. [START_REF] Shah | Safety and efficacy of immune checkpoint inhibitors (ICIs) in cancer patients with HIV, hepatitis B, or hepatitis C viral infection[END_REF] , including 50 patients with chronic viral infection ( 16HIV, 29 HBV and/or HCV, and five patients with HIV and HBV or HCV) treated with PD(L)-1 inhibitors (alone or in combination with anti-CTLA4 or chemotherapy) for several types of neoplasia, reported no case of viral reinfection. Two patients developed grade 3 hepatitis, both with grade ≥ 2 hepatitis at baseline before ICI initiation. This led the authors to definitely support the inclusion of patients having chronic viral infections in ICI trials or treatments [START_REF] Shah | Safety and efficacy of immune checkpoint inhibitors (ICIs) in cancer patients with HIV, hepatitis B, or hepatitis C viral infection[END_REF] .

De Martin et al. [START_REF] De Martin | Characterization of liver injury induced by cancer immunotherapy using immune checkpoint inhibitors[END_REF] identified one patient with hepatitis E (HEV) among 19 patients with grade ≥3 hepatitis in a cohort of 536 patients treated with anti-PD-1/PD-L1 or CTLA-4 immunotherapies. The authors did not specify this case corresponded to HEV reactivation.

In a study assessing nivolumab use in patients with metastatic renal carcinoma, Tsifameyeu et al. compared 22 patients with chronic HCV infection to 22 non-infected patients and found longer overall survival (OS) and progression free survival (PFS) in the HCV group. They identified five patients with elevation of ALT/AST but no concomitant change in viral load.

In patients not receiving antiviral treatment, nivolumab had a non-significant impact on the viral charge (available in 14 patients; mean change 210 IU/mL, p=0.82), whereas patients treated with the combination of ledipasvir and sofosbuvir kept an undetectable viral load throughout nivolumab treatment [START_REF] Tsimafeyeu | Nivolumab in patients with metastatic renal cell carcinoma and chronic hepatitis C virus infection[END_REF] .

Real life evidence

Four case-reports of HBV reactivation during ICI treatment have been published so far (table 1) [START_REF] Pu | Safety and efficacy of immune checkpoint inhibitors in patients with HBV/HCV infection and advanced-stage cancer: A systematic review[END_REF][START_REF] Pandey | A Rare Case of Pembrolizumab-Induced Reactivation of Hepatitis B[END_REF][START_REF] Lake | Hepatitis B reactivation in a long-term nonprogressor due to nivolumab therapy[END_REF][START_REF] Koksal | HBV-related acute hepatitis due to immune checkpoint inhibitors in a patient with malignant melanoma[END_REF][START_REF] Ragunathan | Hepatitis B Reactivation After Administration of Pembrolizumab (KEYTRUDA): A Unique Case Report[END_REF] . Three were related to ICI targeting anti-PD1 (pembrolizumab or nivolumab) and one to ipilimumab (anti-CTLA4). Two patients had no documented HBV status before the start of ICI [START_REF] Pandey | A Rare Case of Pembrolizumab-Induced Reactivation of Hepatitis B[END_REF][START_REF] Ragunathan | Hepatitis B Reactivation After Administration of Pembrolizumab (KEYTRUDA): A Unique Case Report[END_REF] , and one patient experienced a seroconversion [START_REF] Lake | Hepatitis B reactivation in a long-term nonprogressor due to nivolumab therapy[END_REF] .Three patients recovered after antiviral treatment (tenofovir (TFV)) [START_REF] Pandey | A Rare Case of Pembrolizumab-Induced Reactivation of Hepatitis B[END_REF][START_REF] Lake | Hepatitis B reactivation in a long-term nonprogressor due to nivolumab therapy[END_REF][START_REF] Koksal | HBV-related acute hepatitis due to immune checkpoint inhibitors in a patient with malignant melanoma[END_REF] , whereas one patient only received corticosteroids, with an unspecified outcome [START_REF] Ragunathan | Hepatitis B Reactivation After Administration of Pembrolizumab (KEYTRUDA): A Unique Case Report[END_REF] .

For our part, we observed the case of a 67-y-old man (figure 1), ex-smoker with history of cholecystectomy, recently diagnosed with metastatic right superior lobe bronchial adenocarcinoma (PDL1 expression 80%). After partial response to four previous chemoradiation cycles, durvalumab was started (D0) and the first infusion (900 mg) was rapidly followed by a slight elevation of transaminases. The second infusion was postponed, then then cancelled on D21 when the patient was jaundiced with altered general physical state. Liver enzymes and bilirubin were highly increased and prothrombin rate (PR) decreased (factor V unavailable). Liver ultrasonography showed enlarged spleen, liver steatosis areas, with no portal and hepatic vein thrombosis, and non-dilated bile ducts. Viral serological tests and HBV-VL revealed HBV infection with high HBV-VL Hepatitis A, HCV, HEV and HIV antibodies were all negative, while EBV, CMV and HSV1/2 antibodies were consistent with past infections. Autoimmune antibodies were negative (anti-LKM < 40, anti-smooth muscles < 80, anti-mitochondria <40 and liver cytosol (LC1) <40). No liver biopsy was performed.

TFV (245 mg/day) was immediately started, and liver enzymes and HBV-VL significantly decreased. However, liver function worsened concomitantly to infected ascites related sepsis with acute renal failure requiring continuous hemodialysis. Despite antibiotherapy and intensive care, the patient died from severe multiorgan failure 40 days after the single durvalumab infusion. This observation is probably the first ICI-induced HBV reactivation with fatal outcome reported so far.

To summarize real life data, the five reported case show that a) all types of ICI can be involved (anti-CTLA-4, -PD1, -PD-L1), b) HBV status before ICI start is often lacking, and c) liver enzymes may rapidly increase after the first ICI infusion.

Discussion

Hepatotoxicity of ICIs is reported to occur in 2-30% of patients and is dominated by the risk of IMH, due to the disruption of self-tolerance. ICI-IMH, generally leads to mild elevation of liver enzymes, though it can rarely be associated with life threatening hepatic injury with absence of increased values of antibodies [START_REF] Jennings | Hepatotoxicity induced by immune checkpoint inhibitors: a comprehensive review including current and alternative management strategies[END_REF][START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF] . Its management, including ICI discontinuation is adapted to the magnitude of liver damage [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] Lombardi | Review article: immune checkpoint inhibitors and the liver, from therapeutic efficacy to side effects[END_REF] and is based on corticosteroids, possibly at higher dose and shorter duration than commonly used in autoimmune hepatitis [START_REF] Riveiro-Barciela | Immune-related hepatitis related to checkpoint inhibitors: Clinical and prognostic factors[END_REF] , some patients possibly being steroid refractory [START_REF] Hsu | Workup and Management of Immune-Mediated Hepatobiliary Pancreatic Toxicities That Develop During Immune Checkpoint Inhibitor Treatment[END_REF] . Risk of hepatotoxicity increases when using multiple ICIs and in patients who develop other immune related adverse events. Other risk factors for hepatotoxicity include: underlying chronic liver disease, such as viral hepatitis, higher dosages of ICI, and anti-CTLA-4. Serial liver enzymes monitoring (at least monthly) is recommended and consultation with a liver specialist should be considered for any patient who develops grade 2 hepatotoxicity or greater. Permanent discontinuation of ICI therapy is recommended for grade ≥3 injury 2 . The lack of confirming biomarkers for the diagnosis of ICI-IMH highlights the need to exclude other causes of hepatitis prior to starting corticosteroids [START_REF] Riveiro-Barciela | Immune-related hepatitis related to checkpoint inhibitors: Clinical and prognostic factors[END_REF] .

Viral hepatitis infection or reactivation is precisely considered as one of the main exclusion criteria for ICI-IMH [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] Lombardi | Review article: immune checkpoint inhibitors and the liver, from therapeutic efficacy to side effects[END_REF] . Viral hepatitis has a high prevalence with approximately 380 million of individuals infected by HBV or HCV worldwide [START_REF] Razavi-Shearer | Global prevalence, treatment, and prevention of hepatitis B virus infection in 2016: a modelling study[END_REF] , and patients with chronic viral infections such as HBV, HCV or HIV are at increased risk of developing malignancies [START_REF] Shah | Safety and efficacy of immune checkpoint inhibitors (ICIs) in cancer patients with HIV, hepatitis B, or hepatitis C viral infection[END_REF] .

HBV reactivation can be triggered by the administration of cancer chemotherapy or immunosuppressive/biologic therapies (e.g. rituximab and ofatumumab, doxorubicin, epirubicin, corticosteroids (high dose > 20 mg prednisone daily, or moderate dose 10-20 prednisone daily), tumor necrosis factor antagonists, tyrosine kinases, bortezomib) [START_REF] Lombardi | Review article: immune checkpoint inhibitors and the liver, from therapeutic efficacy to side effects[END_REF][START_REF] Smalls | Hepatitis B Virus Reactivation: Risk Factors and Current Management Strategies[END_REF] . A global assessment in patients with solid tumors showed a prevalence of HBV reactivation at 2.1% (95% CI, 0.3-3.9%), and highlighted low HBs antibody (HBs-Ab) titer (< 10 IU/mL) and a high average daily dexamethasone dose (>1.0 mg/day) as significant high-risk factors [START_REF] Kotake | Prevalence and risk factors of hepatitis B virus reactivation in patients with solid tumors with resolved HBV infection[END_REF] .

Considering the increasing prescription of ICIs, their use in patients with chronic viral infection may be common, even more if patients are treated for HCC, and even more so since combinations of ICIs and tyrosine kinase inhibitors are now proposed [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] Lombardi | Review article: immune checkpoint inhibitors and the liver, from therapeutic efficacy to side effects[END_REF] . Nivolumab and pembrolizumab received conditional approval by the US Food and Drug Administration (FDA) as second line treatment of HCC where the immunosuppressive milieu typical of progressive chronic liver disease profoundly modulates tumor growth by facilitating tumor evasion [START_REF] Lleo | Hepatotoxicity of immune check point inhibitors: Approach and management[END_REF][START_REF] Jenne | Immune surveillance by the liver[END_REF] . Therefore, the knowledge of prior HBV and HCV exposure and current serologic status and viral load at the start of treatment are essential for the diagnosis and management of ICI hepatotoxicity, regardless the type of cancer treated . Two cases reports [START_REF] Pandey | A Rare Case of Pembrolizumab-Induced Reactivation of Hepatitis B[END_REF][START_REF] Ragunathan | Hepatitis B Reactivation After Administration of Pembrolizumab (KEYTRUDA): A Unique Case Report[END_REF] and our observation illustrate the "real-life" absence of documented HBV status before the beginning of ICI treatment. Under this circumstance, glucocorticoid treatment can worsen viral hepatitis reactivation [START_REF] Pandey | A Rare Case of Pembrolizumab-Induced Reactivation of Hepatitis B[END_REF][START_REF] Ragunathan | Hepatitis B Reactivation After Administration of Pembrolizumab (KEYTRUDA): A Unique Case Report[END_REF] .

The mechanism by which ICI treatment may induce HBV reactivation remains unclear.

Patients with chronic HBV infection are characterized by a population of exhausted T-cells, which have weak virus-specific T-cell responses during chronic HBV infection, impeding the clearance of virus and recovery from hepatitis [START_REF] Ye | T-cell exhaustion in chronic hepatitis B infection: current knowledge and clinical significance[END_REF] . The beneficial role of ICI treatments in chronic HBV (and HCV) infection is precisely supported by their action of restoring exhausted T-cells function. The occurrence of HBV reactivation could, therefore, appear as a paradoxical effect and suggests that ICI treatments may disrupt the balance of chronic HBV infection and, in some conditions, lead to its reactivation rather than to a hypothetic improvement.

To prevent HBV reactivation in HBs-Ag positive patients, initiation of prophylactic antiviral treatment is recommended irrespective of baseline HBV-VL, and HBV VL should be monitored during ICI treatment and even after [START_REF] Zhang | Hepatitis B virus reactivation in cancer patients with positive Hepatitis B surface antigen undergoing PD-1 inhibition[END_REF] . If reactivation is diagnosed during ICI treatment, antiviral treatment (TFV) should be considered very rapidly [START_REF] Zhang | Hepatitis B virus reactivation in cancer patients with positive Hepatitis B surface antigen undergoing PD-1 inhibition[END_REF][START_REF] Lombardi | Review article: immune checkpoint inhibitors and the liver, from therapeutic efficacy to side effects[END_REF] .

Conclusion

The unawareness or the lack of monitoring of HBV status in cancer patients can be seriously deleterious during immunomodulating therapies, including ICI treatments. Scientific data now clearly highlight the need for checking HBV/HCV status before treatment and for considering treating latent infection before ICI initiation. This risk is clearly labeled in SmPC of ICIs, but clear guidelines are still awaited to definitely prevent this risk, as it is already done for other immunomodulating agents.

Table 1 .

 1 Main characteristics of published cases of HBV reactivation during ICI treatmentFigure 1. Chronological sequence of the reported fatal case of HBV reactivation during durvalumab treatment.

	Ref Pat Disease	HBV status	ICI	Management Outcome	ICI after
				before ICI
	12	M	Lung	Unk	PEMBRO	CTC then	Recovered	PEMBRO
		51	cancer			TFV	in 10
							weeks
	13	M	Lung	Undetectable NIVO	TFV	Recovered No
		72	cancer
	14	M	Melanoma HBs positive IPILI (4	TFV	Recovered NIVOL
		56			cycles) then
					NIVO
	15	M	Lung	Unk	PEMBRO	CTC	unk	Unk
		51	carcinoma
	Our	M	Lung	Unk	DURV	TFV	Fatal
	case	67	cancer
	CTC, corticosteroids ; DURV, durvalumab ; IPILI, ipilimumab; NIVO, nivolumab;
	PEMBRO, pembrolizumab; TFV, tenofovir; UNK, unknown.