Mohammed Nabil

El Korso

Rémy Boyer

Alexandre Renaux

Sylvie Marcos

A GLRT-BASED FRAMEWORK FOR THE MULTIDIMENSIONAL STATISTICAL RESOLUTION LIMIT

Keywords:

Recently, a criterion for Multidimensional Statistical Resolution Limit (MSRL) evaluation, which is defined as the minimal separation to resolve two closely spaced signals depending on several parameters, was empirically proposed in [1] but without a statistical analysis. In this paper, we fill this lack by demonstrating that this MSRL criterion is asymptotically equivalent (upon to a translator factor) to a UMP (Uniformly Most Powerful) test among all invariant statistical tests. This result is an extension of a previous work on mono-dimensional SRL (i.e., when the signals only depend on one parameter). As an illustrative example, the 3-D harmonic retrieval case for wireless channel sounding is treated to show the good agreement of the proposed result.

INTRODUCTION

Characterizing the ability of resolving closely spaced signals is an important step to quantify performance in array signal processing. The concept of Statistical Resolution Limit (SRL), i.e., the minimum distance between two closely spaced signals that allows a correct resolvability, is rising in several applications (especially in parameter estimation problems such as radar, sonar, spectral estimation [START_REF] Vantrees | Detection, Estimation and Modulation Theory[END_REF], etc.)

It is important to note that, in several estimation problems, the signals are parameterized by more than one parameters of interest per signal, for example, in the context of, near-field source localization (bearing, elevation and range), polarized source localization (DOA and the polarization state parameters) and more generally in communication applications. However, the SRL has only been defined/derived in the mono-dimensional case [START_REF] Lee | The Cramér-Rao bound on frequency estimates of signals closely spaced in frequency[END_REF][START_REF] Shahram | Imaging below the diffraction limit: A statistical analysis[END_REF][START_REF] Smith | Statistical resolution limits and the complexified Cramér Rao bound[END_REF][START_REF] Liu | Statistical angular resolution limit for point sources[END_REF][START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF] (i.e., for only one parameter of interest per signal.) This is way we recently have proposed an intuitive extension of the SRL for the multidimensional case [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF], called the Multidimensional Statistical Resolution Limit (MSRL). This criterion is based on the extension of the so-called Smith criterion [START_REF] Smith | Statistical resolution limits and the complexified Cramér Rao bound[END_REF] (i.e., based on a Cramér-Rao Bound (CRB) approach) using the k-norm distance.

Nevertheless, no analysis on the MSRL (which was proposed in [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF]) was done to check its behavior. Thus, the aim of this paper is to fill this lack. First, we link the concept of the MSRL to a binary hypothesis test which is a slight extension of [START_REF] Shahram | Imaging below the diffraction limit: A statistical analysis[END_REF].Second, we prove that the MSRL criterion [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF] is asymptotically equivalent to a Uniformly Most Powerful (UMP) test among all invariant statistical tests (which is, in the asymptotic case, considered as the strongest statement of optimality that one could expect to obtain [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF]). Note that, while the aforementioned equivalence is derived in the context of 1-norm distance, the same conclusion can be extended for the MSRL based on other k-norm distances (k ≥ 2 with k integer). Finally, we illustrate our result in the case of a 3-D harmonic retrieval model for wireless channel sounding.

PROBLEM SETUP AND BACKGROUND

Let Ω, F , y, Y, A, P ξ , ξ ⊆ R 2P +Q be the statistical experiment generated by a random vector y taking values on the measurable space {Y, A} . An observation y (ω) is the realization of the random vector y where ω takes its value on the measurable space {Ω, F } . The distribution of y is assumed to belong to a family of probability measures P ξ on Y and parameterized by the vector ξ ⊆ R 2P +Q . It is also assumed that P ξ is absolutely continuous w.r.t. a σ-finite positive measure µ on Y, such that the Radon-Nikodym derivative (i.e., the likelihood function) p (y|ξ) ∆ = dP ξ (y) /dµ (y) , ∀ξ ⊆ R 2P +Q exists. More precisely, the observation model is assumed to be structured as follows

y = f (ξ 1 ) + f (ξ 2 ) + n, (1)
where, from a signal processing point of view, y ∈ R N and n ∈ R N denote the noisy received data2 and the additive noise with a known probability density function (pdf). The noiseless received data are assumed to be the sum of two signals f (ξ m ), m = 1, 2 each one modelled from the same deterministic known waveform f (.) parameterized by a set of deterministic unknown vectors ξ m ⊆ R P +qm , m = 1, 2 in which q1 + q2 = Q. The function f (.) is assumed to be measurable and the model ( 1) is assumed to be parameter identifiable (i.e., the Fisher information matrix considered through this paper is non singular). All the parameters are collected in

ξ = ξ T 1 ξ T 2 T ⊆ R 2P +Q
. With a proper rearrangement3 of ξ one obtains ξ = ω T ρ T T where ω ⊆ R 2P and ρ ⊆ R Q denote, respectively, the parameter vector of interest, and the unwanted or nuisance parameters vector. This means that we consider P parameters of interest for each signal. To the best of our knowledge, the result of the literature [START_REF] Smith | Statistical resolution limits and the complexified Cramér Rao bound[END_REF] on the SRL have been only proposed in the case of one parameter of interest per signal. Nevertheless, the problem of deriving the so-called Multidimensional Statistical Resolution Limit (MSRL) based on the Cramér-Rao Bound (CRB) in the case of more than one parameter of interest per signal was recently studied in [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF].

The assumptions used for the MSRL derivation are the following:

• A1. The element of the ω are of the same nature, i.e., the parameters of interest have the same unit measurement. If no, please refer to [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF]Remark 2] to overcome this assumption.

• A2. Each parameter of interest w.r.t. to the first signal, ω

1 , can be as close as possible to the parameter of interest w.r.t. to the second signal ω (p) 2 , but not equal. This is not really a restrictive assumptions, since in most applications, having two or more identical parameters of interest is a zero probability event [9, p74]. Nevertheless, in the case where it exists p such that ω

(p) 1 = ω (p)
2 , please refer to [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF]Subsection 3.1] to overcome this assumption.

Under these assumptions, the intuitive MSRL criterion is defined in [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF] as follows:

Criterion 1 The MSRL, denoted by δ, for the model [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF] in the case of two signals and P parameters of interest per signal is given as the implicit solution of the following equation

δ 2 = CRB (δ) (2)
where

δ = P p=1 δp (3)
in which the so-called local SRLs are given by δp ω

(p) 2 -ω (p) 1
and where CRB (δ) denotes the CRB w.r.t. δ for the observation model [START_REF] El Korso | Statistical resolution limit for multiple signals and parameters of interest[END_REF].

In the following the latter MSRL criterion is shown to be asymptotically equivalent (upon to a translator factor) to a UMP test among all invariant statistical tests.

ANALYSIS OF THE MSRL

Hypothesis test formulation

Resolving two closely spaced sources, with respect to their parameter of interest, can be formulated as a binary hypothesis test [START_REF] Liu | Statistical angular resolution limit for point sources[END_REF][START_REF] Amar | Fundamental limitations on the resolution of deterministic signals[END_REF][START_REF] Shahram | On the resolvability of sinusoids with nearby frequencies in the presence of noise[END_REF]. Let us consider the hypothesis H0 which represents the case where the two emitted signal sources are combined into one signal (i.e., ∀p ∈ [1 . . . P ], ω

(p) 1 = ω (p)
2 ), whereas the hypothesis H1 embodies the situation where the two signals are resolvable (i.e., ∃p ∈ [1 . . . P ], such that ω

(p) 1 = ω (p)
2 ). Consequently, one can formulate the hypothesis test as a simple one-sided binary hypothesis test as follows:

H0 : δ detection = 0 H1 : δ detection > 0 (4)
where δ detection denotes a distance between two sets containing the parameters of interest. Let us denote these sets as C1 and C2 where Cq = ω

(1)

q , ω (2)
q , . . . , ω (P ) q

, q = 1, 2. Consequently, δ detection can be defined as

δ detection P p=1 ω (p) 2 -ω (p) 1 .
(

) 5
Since the separation term δ detection is unknown, it is impossible to design an optimal detector in the Neyman-Pearson sense. Alternatively, the Generalized Likelihood Ratio Test (GLRT) [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF][START_REF] Kay | Fundamentals of Statistical Signal Processing : Detection Theory[END_REF] is a well known approach appropriate to solve such a problem. The GLRT is expressed as:

LG(y) = max δ detection ,ρ 1 p(y|δ detection , ρ 1 , H1) maxρ 0 p(y|ρ 0 , H0) = p(y| δdetection , ρ1 , H1) p(y|ρ 0 , H0)

H 1 ≷ H 0 ς (6)
where δdetection , ρ1 and ρ0 denote the Maximum Likelihood Estimator (MLE) of δ detection under H1, the MLE of ρ under H1 and the MLE of ρ under H0. ς denotes the test threshold. We rewrite (6) to obtain

TG(y) = Ln LG(y) H 1 ≷ H 0 ς = Lnς , (7)
in which Ln denotes the natural logarithm.

Asymptotic equivalence of the GLRT

To find the analytical expression of TG(y) in ( 7) is generally intractable. This is mainly due to the fact that the derivation of δdetection is a highly non linear optimization problem [START_REF] Ottersten | Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing[END_REF] (aside from the linear parameter model [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF][START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]). Consequently, in the following, and as in [START_REF] Liu | Statistical angular resolution limit for point sources[END_REF], we consider the asymptotic case (in terms of number of samples). In [11, eq (6C.1)] it has been proved that, for a large number of snapshots, the statistic TG(y) follows a chi-squared pdf under H0 and H1 given by

TG(y) ∼ χ 2 1
under H0 χ where χ 2 1 and χ 2 1 (κ (P f a , P d )) denote the central chi-square and the noncentral chi-square pdf with one degree of freedom. P f a and P d are, respectively, the probability of false alarm and the probability of detection w.r.t. hypothesis test (4). Whereas, assuming that CRB(δ detection ) exist, the the noncentral parameter κ (P f a , P d ) is given by [11, p.239]

κ (P f a , P d ) = δ 2 detection (CRB(δ detection )) -1 . (9)
On the other hand, one can notice that the noncentral parameter κ (P f a , P d ) can be determined exclusively by the choice of P f a and P d [START_REF] Liu | Statistical angular resolution limit for point sources[END_REF][START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF] as the solution of

Q -1 χ 2 1 (P f a ) = Q -1 χ 2 1 (κ (Pfa,Pd)) (P d ), (10)
in which Q -1 χ 2 1 and Q -1 χ 2
1 (κ (P f a ,P d )) are the inverse of the right tail of the χ 2 1 and χ 2 1 (κ (P f a , P d )) pdf. Finally, ( 9) and (10) leads to

δ detection = κ(P f a , P d ) CRB(δ detection ) (11)
where κ(P f a , P d ) = κ (P f a , P d ) is the so-called translation factor which is determined for a given probability of false alarm and probability of detection (see Fig. 1).

Fig. 1. The translator factor κ vs. the probability of detection P d and P f a . One can notice that increasing P d or decreasing P f a has the effect to increase the value of the translator factor κ. This is expected since increasing P d or decreasing P f a leads to a more selective decision [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF][START_REF] Kay | Fundamentals of Statistical Signal Processing : Detection Theory[END_REF].

Remark 1 It is worth noting that, the hypothesis test ( 4) is a binary one-sided test and that the MLE used is an unconstrained estimator. Thus, one can deduce that the GLRT, used to derive the asymptotic SRL, is [START_REF] Liu | Statistical angular resolution limit for point sources[END_REF][START_REF] Kay | Fundamentals of Statistical Signal Processing : Detection Theory[END_REF]: i) the asymptotically uniformly most powerful test among all invariant statistical tests, and ii) has an asymptotic Constant False-Alarm Rate (CFAR). This is, in the asymptotic case, considered as the strongest statement of optimality that one could hope to obtain [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF].

Finally, from ( 2) and ( 11), one can state the following result:

Result 1 The asymptotic MSRL based on the empirical extension of the Smith criterion given in (2) is equivalent (upon to a translation factor) to the binary one-sided hypothesis test given in (4). Consequently, it is equivalent to an asymptotically uniformly most powerful test among all invariant statistical tests.

Remark 2 Consequently, one should note that the MSRL based on the Smith criterion is exactly equal to the MSRL based on the detection approach for all values of P f a and P d such that κ(P f a , P d ) = 1 (cf, Fig. 1).

NUMERICAL EXAMPLE

In this Section we present a numerical example of the MSRL applied to the 3-D Harmonic retrieval model for wireless channel sounding [START_REF] Gershman | Space-time processing for MIMO communications[END_REF][START_REF] Mokios | On 3-D harmonic retrieval for wireless channel sounding[END_REF]. First we briefly introduce the considered model, then, we numerically derive its MSRL.

Model setup

The observation model can be written as [START_REF] Gershman | Space-time processing for MIMO communications[END_REF] [

Y(t)] k,l,m = [X (t)] k,l,m + [N (t)] k,l,m , t = 1, . . . , T,
for k = 1 . . . K, l = 1 . . . L, m = 1 . . . M , in which K, L and M denote the number of acquired data samples per channel, the number of receive antenna sensors and the number of transmit antenna sensors, respectively. Y(t), X (t) and N (t) denote the noisy observation multiway array, the noiseless observation multiway array and the noise multiway array at the t th snapshot, respectively. Whereas, the noiseless observation multiway array is given by [START_REF] Mokios | On 3-D harmonic retrieval for wireless channel sounding[END_REF] [X (t)] k,l,m = 2 q sq(t)e jω (1) q k e jω (2) q l e jω (3) q m , (

where the so-called electrical angles are given by

ω (1) q -2π K τq, ω (2)
q -2πdR λ cos(φq), ω

q -2πdT λ cos(θq), and τq, φq, θq denote delay, direction of arrival, and direction of departure, of the the q th multipath, respectively. dR, dT and λ are the inter-element spacings of the transmit and receive array and the carrier wavelength, respectively. sq(t) is the q th complex amplitude path [START_REF] Gershman | Space-time processing for MIMO communications[END_REF]. It can be proved that the 3-D harmonic retrieval model for wireless channel sounding follows a PARAFAC (PARallel FACtor) decomposition [START_REF] Harshman | Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis[END_REF] given by

X (t) = 2 q=1 sq(t) a(ω (1) q ) • a(ω (2) q ) • a(ω (3) q ) , (13) where a(ω (p) q )
i = e jω (p) q (i-1) and • denotes the multiway array outer-product [START_REF] Golub | Matrix Computations[END_REF]. After vectorization, the full noise free observation vector is given by x = vec T (X (1)) . . . vec T (X (L))

T .

In the same way, we define y, the noisy observation vector, and n, the noise vector, by the concatenation of the proper multiway array's entries.

MSRL derivation

First we derive the CRB for the 3-D Harmonic retrieval model. Then, we use the change of variable formula [13, p 45] to deduce CRB(δ). Assuming i.i.d. complex circular white Gaussian noise with zeromean and unknown variance σ 2 I, the joint pdf of y for a given unknown deterministic parameter vector ξ is

p( y| ξ) = T t=1 p( vec (Y(t))| ξ) = 1 (πσ 2 ) T LM K e -1 σ 2 (y-x) H (y-x) . (14)
The unknown parameter vector is given by

ξ = ω (1) 1 ω (2) 1 ω (3) 1 ω (1) 1 ω (2) 1 ω (3) 1 σ 2 T ,
where the parameters of interest are ω

1 , ω

1 , ω

2 , ω

(representing the direction of arrival and the direction of departure of each sources). After some calculus, the Fisher information matrix of the noisy observations y for orthogonal known amplitudes, is given by

FIM(ξ) = 2 σ 2 Fω 0 0 × , (15)
where,

Fω = T KM L(∆ ⊗ G), (16)
in which

∆ = T t=1 s1(t) 2 0 0 T t=1 s2(t) 2 and G =   (2K-1)(K-1) 6 (K-1)(L-1) 2 (K-1)(M -1) 2 (L-1)(K-1) 2 (2L-1)(L-1) 6 (L-1)(M -1) 2 (M -1)(K-1) 2 (M -1)(L-1) 2 (2M -1)(M -1) 6
  .

Now one can apply Criterion 1. From ( 15), one deduces numerically CRB(ξ) = (FIM(ξ)) -1 . Then, applying the change of variable formula [13, p 45], one obtains

CRB(δ) = 2 p=1 2 q=1 CRB(ω (p) q ) + 2 q=1 CRB(ω (1) q , ω (2) q ).
Finally, solving numerically the implicit equation δ 2 = CRB(δ) gives the desired MSRL as reported in Fig. 2.

From Fig. 2 one can notice that the numerical MSRL based on Criterion 1 is in good agreement with the MSRL derived using the hypothesis test approach. One can notice that, for P d = 0.37 and P f a = 0.1 the MSRL based on Criterion 1 is exactly equal to the MSRL based on the hypothesis test derived in the asymptotic case. From the case P d = 0.49 and P f a = 0.3 and P d = 0.32 and P f a = 0.1, one can notice the influence of the multiplicative factor κ on the MSRL.

CONCLUSION

In this paper, we have analyzed the multidimensional statistical resolution limit based on the empirical extension of the Smith criterion. More precisely, it has been demonstrated that the empirical MSRL criterion based on the 1-norm distance is asymptotically equivalent (upon to a translator factor) to a uniformly most powerful test which is (in the asymptotic case) considered as the strongest statement of optimality that one could expect to obtain.

2 1

 2 (κ (P f a , P d )) under H1[START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF]

Fig. 2 .

 2 Fig. 2. MSRL vs. σ 2 for L = 100: one can notice that the MSRL based on the Criterion 1 is in good agreement with the MSRL based on the hypothesis test approach (which uses an asymptotically uniformly most powerful test among all invariant statistical tests).

This project is funded by region Île de France and Digiteo Research Park.

Let us note that the study where the observation are complex can be handled by the real model (1) by stacking the real and imaginary parts of the observation vector.

To avoid a complication of notation, ξ denote both the vector parameter before and after the rearrangement. In the following we will only use the vector parameter after the rearrangement.


Thomas ; Harder
email: thomas.harder@charite.de

Charite

Andreas Plagemann

Anja Harder

Birth weight and risk of neuroblastoma: a meta-analysis

Keywords: neuroblastoma, perinatal epidemiology, meta-analysis, infancy, birth weight Birth weight, neuroblastoma, children, infants, meta-analysis

Background

Neuroblastoma is the most common solid tumor in infancy but its etiology is largely unknown. Prenatal factors might play a key role in its pathogenesis.

Previous studies investigated whether birth weight is associated with risk of neuroblastoma, with conflictive results. We conducted a meta-analysis to quantitatively summarize the published evidence.

Methods

Results from ten case-control studies and one cohort study (1966 -December 2008) were included, involving a total of 3,004 children with neuroblastoma. We constructed random-effects and fixed-effects models, performed "pool-first" analyses, assessed heterogeneity and publication bias and performed sensitivity and influence analyses.

Results

High birth weight (>4,000g) was associated with increased risk of neuroblastoma (odds ratio (OR): 1.19 ; 95% confidence interval (95%CI): 1.04-1.36). Results

for high birth weight were highly homogenous (I²=0%). Low birth weight (<2,500g) was also related to increased risk of neuroblastoma (OR: 1.24; 95%CI: 1.0-1.55), but results were more heterogeneous (I²=30%). No evidence for particularly influencial studies or for publication bias was found. However, sensitivity analysis indicated the presence of bias in studies on the association with low birth weight. Above 2,500g each 1000g-increase in birth weight was associated with a 13% (95% CI: 3-25%) increase in risk of neuroblastoma.

Conclusions

This meta-analysis shows that high birth weight is highly reproducibly associated with increased risk of neuroblastoma. The association with low birth weight was found to be less robust and deserves further studies.

Key messages

The etiology of neuroblastoma is largely unknown, but prenatal factors have been suggested

to play a key role in its pathogenesis.

This meta-analysis shows that high birth weight is consistently associated with an increased risk of neuroblastoma across studies.

Low birth weight is also associated with increased neuroblastoma risk, but the results are more heterogenous. an incidence rate of about 40 per million infants in the first year of life, thereby being the most common malignancy in infancy. [START_REF] Brodeur | Neuroblastoma[END_REF] It has been termed a "clinical enigma" of cancer research because of its hard-to-predict biological behaviour, spanning from spontaneous regression to very rapid metastatic spread, thereby making diagnosis, therapy and prognosis difficult. [START_REF] Brodeur | Neuroblastoma: biological insights into a clinical enigma[END_REF] Efforts to establish measures of secondary prevention like newborn screening for vanillylilmandelic acid in urine gave controversial results. [START_REF] Schilling | Neuroblastoma screening at one year of age[END_REF][START_REF] Hiyama | Effectiveness of screening for neuroblastoma at 6 months of age: a retrospective population-based cohort study[END_REF] Importantly, no clear-cut risk factor for neuroblastoma has been identified to far. Because of its appearance very early in life it has been suggested that risk factors which act prenatally could play a role in the pathogenesis of neuroblastoma. However, conflictive data have been accumulated over the past years on such factors like exposure to maternal smoking, [START_REF] Kramer | Medical and drug risk factors associated with neuroblastoma: a case-control study[END_REF] maternal medication [START_REF] Cook | Maternal medication use and neuroblastoma in offspring[END_REF] or vitamins during pregnancy, [START_REF] Olshan | Maternal vitamin use and reduced risk of neuroblastoma[END_REF] possibly arising from the relatively small number of studies performed so far, or from the difficulties to correctly obtain the respective exposures during pregnancies.

An alternative attempt is to focus on indicators of exposures which can be easily obtained in large samples with little bias and are established markers of the intrauterine environment.

Birth weight can be considered to be such a marker. It is routinely obtained at birth by trained personnel with little systematic error and is generally collected in medical records or registries. [START_REF] Reichman | Validation of birth certificate data. A study of women in New Jersey's HealthStart program[END_REF] Studies suggest that environmental factors such as maternal nutrition and maternal diseases are important for birth weight. [START_REF] Lunde | Genetic and environmental influenecs on birth weight, birth length, head circumference, and gestational age by use of population-based parent-offspring data[END_REF][START_REF] Brooks | Birth weight: nature or nurture?[END_REF] In the past, a number of investigators have studied birth weight in infants and children with neuroblastoma. [START_REF] Buck | Perinatal factors and risk of neuroblastoma[END_REF][START_REF] Schuz | Risk factors for neuroblastoma at different stages of disease. Results from a population-based case-control study in Germany[END_REF][START_REF] Uruyama | Birth characteristics and risk of neuroblastoma in young children[END_REF] A recent narrative review came to the conclusion that there might be an association of neuroblastoma risk with low birth

F o r R e v i e w O n l y
weight. [START_REF] Heck | The epidemiology of neuroblastoma: a review[END_REF] Here, we aimed to quantify the association and performed a meta-analysis on the relation between birth weight and risk of neuroblastoma.

Methods

We performed a comprehensive literature search according to the MOOSE guidelines for meta-analyses of observational studies, [START_REF] Stroup | Meta-analysis of observational studies in epidemiology: a proposal for reporting[END_REF] including the databases MEDLINE (1966 -December 2008) and EMBASE (1989 -December 2008) to identify studies that investigated the association between birth weight and risk of neuroblastoma, using the terms "birth weight" and "neuroblastoma". The entire search was performed in the fulltext option, without language restrictions. Furthermore, we manually searched all references cited in original studies and reviews identified.

To be eligible, a study had to fulfil the following criteria, defined a priori: 1) It had to be an original report on birth weight and risk of neuroblastoma, and 2) odds ratios (OR) and 95% confidence intervals (95% CI, or data with which to calculate them) for risk of neuroblastoma in at least two strata of birth weight had to have been presented. Alternatively, an odds ratio and 95% CI for the change in neuroblastoma risk per unit change in birth weight had to have been reported. If only data for risk ratio were reported (cohort studies), they were used for pooling since under the rare-disease assumption OR and risk ratio are nearly equal. [START_REF] Porta | A dictionary of epidemiology[END_REF] The course of the systematic literature review is illustrated in Figure 1.

From all studies included, data were abstracted in duplicate, using a standardized form. Three different meta-analytic approaches were used: 1) A birth weight cutoff of 4,000 g (high birth weight) [START_REF] Guyer | Annual summary of vital statistics -1998[END_REF] was used to compare risks of neuroblastoma above and below this value (dichotomous comparison).

2) The dichotomous approach was repeated for a birth weight cutoff of 2,500 g (low birth weight). [START_REF] Guyer | Annual summary of vital statistics -1998[END_REF] 3) The pool-first method [START_REF] Greenland | Methods for trend estimation from summarized doseresponse data, with applications to meta-analysis[END_REF] was used to combine regression coefficients obtained from the studies for trend analysis.

For dichotomous comparisons, we extracted data on numbers of subjects with and without neuroblastoma above or below the cutoff value and calculated corresponding crude ORs and 95% CIs. We constructed both fixed-effects and random-effects models to estimate the pooled ORs for risk of neuroblastoma above versus below the respective cutoff value across all studies.

To studies that provided data for more than two categories of birth weight, we applied the 'pool-first method' to quantify the dose-response relationship between birth weight and risk of neuroblastoma. This method, as introduced by Greenland and Longnecker, [START_REF] Greenland | Methods for trend estimation from summarized doseresponse data, with applications to meta-analysis[END_REF] allows the meta-analytic pooling of regression data, but accounts for the fact that estimates used to calculate single regression coefficients are correlated to each other within one study. This problem is solved by firstly pooling the data within each study ("pool first") to obtain respective regression coefficients for each study, followed by pooling these regression coefficients. The resulting pooled regression coefficient can be interpreted like a regression coefficient from a single study. Visual inspection of the plots of eleven out of the twelve individual studies revealed a u-shaped relation between birth weight and risk of neuroblastoma. For each of these studies a study-specific linear and a quadratic regression coefficient and corresponding 95% CIs were calculated using a log-quadratic model. Following the pool-first approach, [START_REF] Greenland | Methods for trend estimation from summarized doseresponse data, with applications to meta-analysis[END_REF] the resulting linear and quadratic regression coefficients and their 95% CIs were pooled using a random-effects model. After having performed the sensitivity analyses (see below), we repeated the pool-first analysis, now applying a linear model restricted to the birth weight spectrum above 2,500g and to studies that used registries to obtain birth weight data.

Heterogeneity of study results was assessed by calculating the I², as proposed by Higgins et al. [START_REF] Higgins | Measuring inconsistency in metaanalyses[END_REF] I², ranging from 0-100%, is a direct measure of inconsistency of study results in a metaanalysis. Robustness of the pooled estimate was checked by influence analysis, using a random-effects model: each of the study estimates was individually omitted from the data set, followed in each case by re-calculation of the pooled estimate of the remaining studies.

Four different subgroup analyses were performed to identify potential sources of heterogeneity between study results. First, we calculated separate estimates for all studies from North America (U.S. and Canada) and for studies from Europe. Second, we stratified all studies according to method of obtaining birth weight (registry/certificate vs interview).

Third, subgroup estimates were calculated according to method of obtaining case diagnosis (registry vs others). Fourth, we stratified all studies according to source of controls (registry/certificate vs others). For all analyses, a random-effects model was used.

Publication bias was assessed by inspection of the funnel plot, and formal testing for funnel plot asymmetry using Begg's test and Egger's test.

All calculations were performed using STATA 8 (Stata corporation, College Station, TX, USA).

Results

During the systematic review, we identified 13 studies which provided data on birth weight and subsequent risk of neuroblastoma. One study [START_REF] Yeazel | High birth weight and risk of specific childhood cancers: a report from the Children's Cancer Group[END_REF] had to be excluded since only an adjusted OR was reported for neuroblastoma after high birth weight (OR: 1.40; 95% CI: 0.90-2.20; adjusted for maternal age, birth order, weeks of gestation and sex) and an unadjusted OR could not be calculated from the data. Another study [START_REF] Schuz | Risk factors for neuroblastoma at different stages of disease. Results from a population-based case-control study in Germany[END_REF] had to be excluded since case overlap to a previous study [START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF] was substantial. Thereby, a total of eleven studies could be used for meta-analysis (Figure 1). Study size ranged from 417 to 2,127,452 participants. In seven of the studies, cases were derived from cancer registries, while in the remaining four studies they came from other sources.

Ten studies provided data for calculation of OR (95% CI) of risk of neuroblastoma in subjects with high birth weight (> 4,000 g), compared to those below this cutoff value. In Figure 2, the forest plot with ORs and 95% CIs and the pooled estimate for risk of neuroblastoma in subjects with high birth weight are shown. High birth weight was associated with increased risk of neuroblastoma. This effect measure was identical when using the random-effects model as well as the fixed-effects model (OR: 1.19; 95% CI: To further investigate the association with both high and low birth weight, we performed sensitivity analysis. Regarding the association with high birth weight, region-specific pooled estimates showed no large differences (Table 2). The association with low birth weight was more pronounced in studies from Europe than in those performed in the U.S. or Canada. The method of obtaining data on birth weight did not influence the strength of the association with high birth weight. By contrast, it had a strong effect on the association with low birth weight:

Studies which used interview-based data showed a more than 60% increased risk, whereas those which relied on data from registries had a much smaller estimate. While the method of obtaining the case diagnosis had no large effect on neither the estimate for high birth weight, nor on the one for low birth weight, the estimate for low birth weight was strongly influenced by source of controls, with registry-based studies showing smaller effects (Table 2).

Since the sensitivity analysis thereby indicated a considerable bias for low birth weight, we repeated the "pool-first" analysis, now using only birth weight data above 2,500g which came from studies that used registries to obtain birth weight. These data from the individual studies indicated a linear positive relation between birth weight and risk of neuroblastoma above 2,500g. Using these data, we therefore calculated linear regression coefficients for the individual studies. Pooling of these linear regression coefficients revealed that above 2,500g, each 1000g-increase in birth weight was associated with a 13% (95% CI: 3-25%) increase in risk of neuroblastoma.

Discussion

A number of studies and meta-analyses have shown that birth weight is associated with risk of different types of tumors in infants and children, like leukaemia, [START_REF] Hjalgrim | Birth weight as a risk factor for childhood leukemia: a meta-analysis of 18 epidemiologic studies[END_REF] Wilms' tumor [START_REF] Schuz | High birth weight and other risk factors for Wilms' tumour: results of a population-based case-control study[END_REF] and brain tumors. [START_REF] Harder | Birth weight and subsequent risk of childhood primary brain tumors: a meta-analysis[END_REF] Regarding neuroblastoma, however, the evidence appeared to be inconclusive yet.

This meta-analysis, on first view, indicates that birth weight might be related to followed by an increased risk of developing neuroblastoma in later life. However, our sensitivity analysis gave clear indication for bias in the case of low birth weight, with studies using interview data reporting stronger associations than those that used registries as data sources.

Remarkably, the results of this meta-analysis are in some contrast to the conclusions drawn by authors of the individual studies: In only one of the publications [START_REF] Uruyama | Birth characteristics and risk of neuroblastoma in young children[END_REF] it was explicitely concluded that high birth weight is a risk factor for neuroblastoma, whereas a further three groups concluded that low birth weight is associated with neuroblastoma. [START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF][START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF][START_REF] Johnson | Neuroblastoma: Case-control analysis of birth characteristics[END_REF] In only two studies, [START_REF] Munzer | Birth-related characteristics, congenital malformation, maternal reproductive history and neuroblastoma: The ESCALE study (SFCE)[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF] the authors concluded that a u-shaped relation exists, as described here. In four reports it is stated that no association exists with birth weight, [START_REF] Chow | Maternal and perinatal characteristics in relation to neuroblastoma[END_REF][START_REF] Johnson | Perinatal characteristics and risk of neuroblastoma[END_REF][START_REF] Neglia | Prenatal and perinatal risk factors for neuroblastoma. A case-control study[END_REF][START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] whereas the authors of one study did not comment on this issue. [START_REF] Buck | Perinatal factors and risk of neuroblastoma[END_REF] At least in some cases, the authors of individual studies might have overlooked an association with high birth weight, either since it was not statistically significant, or since the choice of the reference stratum for birth weight might have "masked" the result (see also Table 1, last column for the original study results as they were calculated by the authors of the studies).

In addition to the problem of measurement error of birth weight which cannot be fully disregarded, studies on associations between birth weight and later outcome are prone to confounding in multiple ways. Adjustment for confounders has been performed in the studies included here mainly by matching. However, matching was generally restricted to only two variables, that are, age and gender. In four of the studies, [START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF][START_REF] Chow | Maternal and perinatal characteristics in relation to neuroblastoma[END_REF][START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF] additional adjustments were made, mainly for socioeconomic status and ethnicity. Adjusted estimates, however, did not differ largely from the unadjusted results. In only one study, [START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF] the possibility that the associations observed might also be related to the degree of maturity of the child at birth, such a relation between gestational age and neuroblastoma risk was only present in two of the studies, [START_REF] Chow | Maternal and perinatal characteristics in relation to neuroblastoma[END_REF][START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] but not in the remaining six [START_REF] Buck | Perinatal factors and risk of neuroblastoma[END_REF][START_REF] Uruyama | Birth characteristics and risk of neuroblastoma in young children[END_REF][START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF][START_REF] Munzer | Birth-related characteristics, congenital malformation, maternal reproductive history and neuroblastoma: The ESCALE study (SFCE)[END_REF][START_REF] Neglia | Prenatal and perinatal risk factors for neuroblastoma. A case-control study[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF] which investigated this variable.

Although the relation between high birth weight and risk of neuroblastoma was only weak, we observed an unusually high degree of homogeneity between the single study results.

According to I², a meanwhile established indicator of between-study-heterogeneity, 19 0% of the variation across studies was due to heterogeneity. While, on the one hand, this might indicate a highly constant association between high birth weight and neuroblastoma risk, it might, on the other hand, also raise questions regarding other potential reasons for this extreme between-study homogeneity. Recently, Ioannidis et al. have suggested a number of such potential alternative reasons for this phenomenon. [START_REF] Ioannidis | Extreme between-study homogeneity in metaanalyses could offer useful insights[END_REF] Among them, only the issue of correlated data resulting from studies that came from identical authors or overlapping patient groups is relevant in the context of this meta-analysis. We therefore checked whether there was any overlap in study region or year of diagnosis between the studies and included the relevant information in Table 1. However, we did not find indication that a considerable overlap between any of the study populations could explain our findings. Moreover, influence analysis showed that the effect measure of the relation between high birth weight and risk of neuroblastoma was very robust. The association with high birth weight was also present in the largest study, which is also the only cohort study on this issue, [START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] supporting the validity of the estimate. Furthermore, in two out of three studies which reported estimates for birth weight > 4,500g, [START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF][START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] risk of neuroblastoma was further increased in this category of "very high" birth weight. participation is known to be skewed towards higher socioeconomic status which itself is related to low birth weight. [START_REF] Kramer | Investigating socio-economic disparities in preterm birth: evidence for selective study participation and selction bias[END_REF] This interpretation is further supported by the result of stratification by source of controls. For low birth weight, studies which used methods that required active participation (like telephone interviews) gave higher estimates than those which used less biased sources (like registries) to recruit controls. However, in the three casecontrol studies which reported response proportions of cases and controls they did not differ greatly in this parameter (cases: 73%, controls: 72%; 23 cases: 69%, controls: 71%; 26 cases: 80%, controls: 62-69% [START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF] ). Moreover, there are other possible explanations for the observed differences. One might be an under-representation of low-birth-weight children in the control group as discussed by Schuz and Forman, [START_REF] Schuz | Birthweight by gestational age and childhood cancer[END_REF] which could have caused the results at least in the German study. [START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF] Since low education and income are associated with increased risk of low birth weight, as mentioned above, to the extent that the controls have fewer low birth weight infants, an increased OR would result from response bias.

During recent years, it has been suggested that neuroblastoma in infancy and in later childhood represent different entities. [START_REF] London | Evidence for an age cutoff greater than 365 days for neuroblastoma risk group stratification in the Children's Oncology Group[END_REF] We therefore additionally checked whether the strength of the association between high birth weight and risk of neuroblastoma differs by age of manifestation. In four of the studies, [START_REF] Uruyama | Birth characteristics and risk of neuroblastoma in young children[END_REF][START_REF] Munzer | Birth-related characteristics, congenital malformation, maternal reproductive history and neuroblastoma: The ESCALE study (SFCE)[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF][START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] Since it cannot be assumed that birth weight or fetal growth per se is a causal factor leading to later disease risks, as recently again emphasized by Basso, [START_REF] Basso | Birth weight is forever[END_REF] the key question concerns the mechanisms underlying associations between high birth weight and risk of neuroblastoma.

Among the factors which stimulate prenatal weight gain maternal diabetes mellitus might be the most prevalent. It is well established that maternal diabetes leads to excess weight gain in utero and, consequently, to increased birth weight, [START_REF]Hyperglycemia and adverse pregnancy outcomes[END_REF] as even shown in a randomized trial. [START_REF] Crowther | Effect of treatment of gestational diabetes mellitus on pregnancy outcomes[END_REF] Remarkably, one of the studies included in this meta-analysis showed that maternal diabetes, in particular gestational diabetes, is associated with an increased risk of neuroblastoma in the offspring, [START_REF] Chow | Maternal and perinatal characteristics in relation to neuroblastoma[END_REF] whereas, however, another study did not observe an association. [START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF] On the population level, maternal diabetes during pregnancy, particularly gestational diabetes which was estimated to affect up to 14% of pregnant women, [START_REF] Jovanovic | Gestational diabetes mellitus[END_REF] is likely to be the most important risk factor for high birth weight, leading to a 2-3 fold increased risk. [START_REF] Landon | A multicenter, randomized trial of treatment for mild gestational diabetes[END_REF] Similar applies to maternal overweight during pregnancy. However, it is obvious that one might only speculate whether this proportion is large enough to explain the 19% increase in risk of neuroblastoma after high birth weight.

During recent years some studies have found an increase in the incidence of neuroblastoma in the general population: In Europe, the age-standardized incidence rate for infants increased from 35.4 to 57.8 per million. [START_REF] Spix | Neuroblastoma incidence and survival in European children (1978-1997): report from the Automated Childhood Cancer Information System project[END_REF] It has been discussed that these increases cannot be completely attributed to intensified screening. [START_REF] Spix | Neuroblastoma incidence and survival in European children (1978-1997): report from the Automated Childhood Cancer Information System project[END_REF][START_REF] Honjo | Neuroblastoma trends in Osaka, Japan, and Great Britain 1970-1994, in relation to screening[END_REF] If the association between high birth weight and neuroblastoma risk exists, it might offer a provocative explanation for this trend (Switzerland) to 25% increase (Sweden) over the last 20 years in European countries. [START_REF] Rooth | Increase in birthweight: a unique biological event and an obstetrical problem[END_REF][START_REF] Bergmann | Secular trends in neonatal macrosomia in Berlin: influences of potential determinants[END_REF] However, as in the case of maternal diabetes discussed above, it remains speculative whether this trend is strong enough to explain the trend over time in the incidence of neuroblastoma.

Taken together, this meta-analysis shows that high birth weight is a risk factor for neuroblastoma. Previous observations that low birth weight is related to increased risk of neuroblastoma 14 might be distorted by bias. However, on the basis of the currently published literature it cannot be excluded that low birth weight is also a risk factor for neuroblastoma, themoreso as data exist showing that rapid early weight gain, which is frequent in low-birthweight babies, is also associated with increased risk of neuroblastoma. [START_REF] Suminoe | Rapid somatic growth after birth in children with neuroblastoma: a survey of 1718 patients with childhood cancer in Kyushu-Okinawa district[END_REF]

F o r R e v i e w O n l y

Figure 2

Odds ratios for neuroblastoma in subjects with high birth weight (>4,000 g), as compared to a birth weight of <4,000 g (random-effects model).

Figure 3

Odds ratios for neuroblastoma in subjects with low birth weight (<2,500 g), as compared to a birth weight of >2,500 g (random-effects model).

  Birth weight and risk of neuroblastoma: a meta-analysis Journal: International Journal of Epidemiology Manuscript ID: IJE-2009-07-0608.R2 Manuscript Type: Original Article Date Submitted by the Author

  solid tumor arising from primitive sympathetic neural precursor cells, has

  a u-shaped manner. High birth weight as well as low birth weight are

  that the first notion that low birth weight is also associated with increased risk of neuroblastoma might be biased by maternal recall because interview-based studies gave stronger estimates than studies in which birth weight data came from more objective sources like birth registries and birth records. One might speculate that mothers of children with neuroblastoma and a relatively low birth weight might recall their babies' birth weight even lower than it was. Alternatively, selection bias might have occurred. Interview

  birth weight, as well as the incidence of high birth weight has been increasing continuously in the United States and Europe in recent years, ranging from 6.9%

Figure 1

 1 Figure 1Flow diagram of study selection process.

F

  ratio (95% CI) for risk of neuroblastoma

Table 1

 1 [START_REF] Buck | Perinatal factors and risk of neuroblastoma[END_REF][START_REF] Uruyama | Birth characteristics and risk of neuroblastoma in young children[END_REF][START_REF] Schuz | Association of childhood cancer with factors related to pregnancy and birth[END_REF][START_REF] Chow | Maternal and perinatal characteristics in relation to neuroblastoma[END_REF][START_REF] Hamrick | Association of pregnancy history and birth characteristics with neuroblastoma: a report from the Children's Cancer Group and the Pediatric Oncology Group[END_REF][START_REF] Johnson | Neuroblastoma: Case-control analysis of birth characteristics[END_REF][START_REF] Johnson | Perinatal characteristics and risk of neuroblastoma[END_REF][START_REF] Munzer | Birth-related characteristics, congenital malformation, maternal reproductive history and neuroblastoma: The ESCALE study (SFCE)[END_REF][START_REF] Neglia | Prenatal and perinatal risk factors for neuroblastoma. A case-control study[END_REF][START_REF] Mclaughlin | Perinatal risk factors for neuroblastoma[END_REF][START_REF] Bjorge | Birth and parental characteristics and risk of neuroblastoma in a population-based Norwegian cohort study[END_REF] Study characteristics of included reports are shown in

. Studies involved a total of 2,159,927 individuals, of whom 3,004 suffered from neuroblastoma. One study was a cohort study. The remaining eleven studies had a casecontrol design with matching ratios ranging from 1:1 to 1:10. The first study was published in 1985, while the most recent appeared in 2008. Studies were performed in either the United States, Canada, France, Norway or Germany. Age of participants ranged from 0 to 18 years.

  No indication for publication bias was found, as investigated by visual inspection of the funnel plots (not shown), and Begg's test (p = 0.28) and Egger's test (p = 0.50).

	1.40). Ten studies gave data for calculation of OR (95% CI) of risk of neuroblastoma in subjects
	with low birth weight (< 2,500 g), compared to those above this cutoff value. Figure 3 shows
	the forest plot with ORs and 95% CIs and the pooled estimate for risk of neuroblastoma after
	low birth weight. Low birth weight was related to increased risk of neuroblastoma by random-
	F o r effects model (OR: 1.24; 95% CI: 1.0-1.55) as well as by using the fixed-effects model (OR: F o r
	R R
	e v e v
	i e i e
	w w
	O n l O n l
	y To quantitatively describe this u-shaped relation between birth weight and risk of y
	neuroblastoma, we first checked whether the individual studies fulfilled the assumption of a
	quadratic model. Since all studies except of one 29 did so, we pooled the linear and quadratic
	regression coefficients from the remaining eleven single studies obtained by the pool-first
	method. Both the linear regression coefficient (0.52; 95% CI: 0.28-0.96) as well as the
	9 10

1.04-1.36)

. According to I², 0% of the total variation across studies was due to heterogeneity. Influence analysis (randomeffects model) showed that the pooled estimate was very robust: Omission of individual study estimates led to pooled ORs ranging from 1.17 (95% CI: 1.01-1.36) to 1.20 (95% CI: 1.04-1.24; 95% CI: 1.04-1.50). According to I², 30% of the total variation across studies was due to heterogeneity. Omission of individual studies revealed that no particular study had a major influence on the pooled estimate, with pooled ORs ranging from 1.16 (95% CI: 0.97-1.39) to 1.30 (95% CI: 1.04-1.61).

Given these findings of increased risk of neuroblastoma at both ends of the birth weight spectrum, we repeated the dichotomous comparisons, now using 'normal birth weight' (2500-4000g) as reference for all studies that gave data on both low and high birth weight (n = 10 studies). Pooled estimates after high birth weight (OR: 1.21; 95% CI: 1.05-1.39) as well as low birth weight remained increased (OR: 1.23; 95% CI: 0.98-1.55). quadratic regression coefficient (1.10; 95% CI: 1.01-1.20) were different from 1.0.

  neuroblastoma which manifest in infancy is particularly vulnerable to prenatal influences. Moreover, one has to mention that in none of the studies stage of disease was related to birth weight, leaving the question open whether there is a relation between prenatally acting risk factors and particularly aggressive forms of neuroblastoma.

	specific subtype of
	F o r
	R
	e v
	i e
	w
	O n l
	y
	14

results were stratified by age. Interestingly, in three of them the association with high birth weight was stronger in infants below one year of age, as compared to older children. Further studies are needed to investigate whether a

Table 1 (

 1 continued)Characteristics of studies on the relation between birth weight and risk of neuroblastoma

	Further studies on

** Values calculated from the respective years of diagnosis and age at diagnosis. *** Risk ratio was used for calculations.

Table 2

 2 Birth weight and risk of neuroblastoma: sensitivity analysis (random-effects model)

	Study characteristic	Category	High birth weight	Low birth weight
			(OR (95% CI))	(OR (95% CI))
	Geographic area	North America	1.17 (1.0-1.38)	1.16 (0.95-1.41)
		Europe	1.21 (0.95-1.56)	1.43 (0.68-2.99)
	Source of data for	Registry	1.18 (1.01-1.39)	1.11 (0.90-1.37)
	birth weight
		Interview	1.20 (0.93-1.53)	1.63 (1.06-2.49)
	Source of case	Registry	1.23 (1.05-1.44)	1.18 (0.87-1.60)
	diagnosis
		Other *	1.07 (0.83-1.39)	1.36 (0.95-1.94)
	Source of controls	Registry/certificate	1.20 (1.0-1.43)	1.14 (0.93-1.41)
		Other **	1.20 (0.93-1.53)	1.63 (1.06-2.49)
	* included: death certificates, records, surveillance
	** included: population, random digit dialing

82-2.55) 2.59 (1.41-4.76) 1.04 (0.66-1.64) 1.24 (1.0-1.55) Bjorge et al., 2008 (29) 0.53 (0.17-1.65) Johnson et al., 2008 (25) 1.26 (0.66-2.41) McLaughlin et al., 2008 (28) 1.24 (0.90-1.69)

	Study	OR (95% CI)
		0.88 (0.38-2.04)
		0.73 (0.37-1.46)
		1.24 (0.76-2.03)
		2.08 (0.86-5.02)
		1.45 (0.
		0.1	1	10

Acknowledgements

The authors thank Swantje Scholz for assistance in study identification.

Funding

This study was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft; grant number PL 241/5-1).

Conflict of interest none


Ashish ; Kamat
email: ashish.kamat@volkswagen.de

Andreas Huth

Olaf Klein

Stephan Scholl

Chronoamperometric Investigations of the Electrode-Electrolyte Interface of a Commercial High Temperature PEM Fuel Cell

Keywords: Diffusive Mass Transport, Gas Diffusion Electrode (GDE), Interface, Mass Transport, Mathematical Simulation, MEA, Performance Chronoamperometry, Cottrell, Diffusion Limiting Current, PBI, Phosphoric Acid Wiley-VCH Fuel Cells

come   L'archive ouverte pluridisciplinaire

Introduction

Polymer electrolyte membrane (PEM) fuel cells are one of the most promising candidates for powering transportation systems. Fuel cell driven cars are already being introduced on the road and undergoing final stages of testing and development [1--4]. The high temperature PBI / phosphoric acid PEM fuel cell has some operational advantages (no humidification, higher CO tolerance) over the low temperature Nafion® based PEM fuel cell. Nevertheless, in comparison, the HT-PEM fuel cell has lower cell current densities. This is largely due to the presence of phosphoric acid in the electrode-electrolyte interface. Limited permeability of oxygen in phosphoric acid medium and the influence of the adsorbed phosphate species on oxygen reduction reaction (ORR) kinetics are identified as the limiting factors [5].

The electrode-electrolyte interface behavior in a commercial HT-PEM fuel cell largely determines its performance. The attempt to describe this interface is often done via a three phase boundary model, where it is assumed that there exists a physical contact area between the solid Pt/C electrode particles, the liquid electrolyte and the gas phase [5]. The hydrogen oxidation reaction (HOR) is faster than the ORR and hence the anode side over-potentials are often safely neglected for modeling purposes [6]. On the cathode side, the oxygen permeability in phosphoric acid is at least an order of magnitude lower than in Nafion®/water electrolyte. Mass transport investigations in commercial cells are rarely carried out due to larger inaccuracy of the diffusion and solubility parameters. Most research focuses on small well-defined electrodes, which can be useful in precisely determining the mass transport parameters, but cannot clearly describe the complex processes in a commercial cell [7--10]. Mass transport investigations via chronoamperometry are often reported in literature as well as their analysis using Cottrell or Cottrell-like equations [11,12]. This approach is revisited to arrive at useful techniques to better understand the processes occurring at the electrode-electrolyte interface of a commercial fuel cell.

Theoretical Background

Chronoamperometry is an established method where the test cell is subjected to a voltage step change and the transient response from the cell current is recorded. In order to study the diffusion processes, the transient current in the mass transport limitation region of the polarization curve is of higher relevance. To accomplish this, the cell voltage is stepped down from open circuit voltage (OCV) to lower than 0.2 V. The corresponding transient cell current can be described by the following equation,

cap d cell i i i + = (1)
where i cell is the total cell current, i d is the diffusion limiting current and i cap is the capacitive current. This study uses the chronoamperometric experiments and the Cottrell experiment and equation to assemble the experimental setup and analyze the experimental results [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF]. The Cottrell equation for diffusion limiting current is

F o r P e e r R e v i e w 2 / 1 2 / 1 * 2 / 1 2 2 ) ( t C nFAD t i O O d π = (2)
where n is the number of electrons transferred in one electrochemical reaction step, F is the Faraday constant, A is the active electrode area, D is the diffusion coefficient of oxygen in electrolyte, C* is the initial bulk concentration of oxygen in electrolyte and t is the time. This equation is valid for a planar electrode in an unstirred electrolyte solution. This transient current when plotted against the inverse square root of time (Cottrell plot) depicts a straight line heading towards the origin.

In case of a spherical electrode, the Cottrell equation can be extended to

δ π * 2 / 1 2 / 1 * 2 / 1 2 2 2 2 ) ( O O O O d C nFAD t C nFAD t i + = (3)
where the time independent second term includes "δ" which is the radius of the spherical electrode. Its Cottrell plot is a straight line intercepting the cell current axis.

For short times (t <0.1s) the capacitive current is a considerable contribution to the total cell current and can be described by the equation

) exp( ) ( d u u cap C R t R E t i - = (4)
where E is the magnitude of the cell voltage step, R u is the uncompensated resistance, C d is the double layer capacitance and t is the time. At times greater than 0.1 seconds, this contribution however is almost negligible. This is largely due to the exponential decay in the capacitive current with a cell time constant (product of uncompensated resistance and double layer capacitance) [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF]. Measured current data at time intervals approximately 10 times larger than the cell time constant are considered to be reliable measurement points for the diffusion limiting cell current i d .

Experimental

The experimental setup consists of test cells mounted on fuel cell test benches C100 from FuelCon AG. The fuel cell hardware comprises of two metal alloy blocks that provide connections for fuel flow inlets and outlets, temperature and voltage sensors as well as the current collector plates. The inner side of this block holds a graphite plate with square shaped flow field in the center exhibiting flow channels of serpentine design. The total area of the flow field approximates 10 cm². The Pt/C electrode paste is prepared in house and printed on the GDL material and has a combined thickness of 300 µm. Both the anode and cathode are identically prepared with estimated platinum loading of 1.5 mg cm -². The dry PBI membrane is doped with 85% phosphoric acid and has an impregnated thickness of around 50 µm. This membrane is sandwiched between the electrodes. For structural integrity of the MEA, two extra components are used. A thin film of about 25 µm made of insulating Kapton material is used to hold the membrane in place. The MEA is then set between two polytetrafluroethylene (PTFE) sealings. These films are about 200 µm thick and also provide electrical insulation along with impermeability to reactant gases. The fuel cell hardware blocks on either side of the MEA are screwed together. The test cell is then mounted on the test bench, which provides operational support.

The C100 test bench has connections for gas supply, heating rods, temperature sensors as well as the cell current and voltage sensors. The complete overview of the operational test cell parameters is observed and controlled from the software interface. Temperatures, pressures, gas flow rates, cell voltage and cell current can be varied by manual inputs from this interface. It is also possible to run user defined scripts for long term trials. Along with hydrogen and air as reactant gases, nitrogen is also supplied as an inert gas, often used for purging remnants of the reactant gases during shut down or start-up phases. The gas flow rates and absolute pressures can be maintained separately for the anode and the cathode. Standard operating conditions are set at 160°C temperature, 2.5 bar absolute pressure and anode and cathode flow rates at 0.1 and 0.3 liters per minute respectively.

The electrode-electrolyte interface remains to be the main focus for any improvement in performance of the HT-PEM fuel cell. Figure 1 shows the schematic depiction of the membrane in contact with the electrode (cathode), where both the two phase as well as the three phase boundary reaction zones are formed due to the electrolyte reaching into or partially flooding the porous electrode. The three phase boundary reaction zone is shown to be the contact area where the three phases namely the liquid electrolyte (I), the solid catalyst particle (II) and the air in the gaseous phase (III) are present. Chronoamperometric analyses on fuel cells with such an undefined electrode-electrolyte interface are hardly useful without separating the two phase boundary reaction zone from the three phase boundary reaction zone. For achieving this separation, certain modifications are made by changing the extent of the electrolyte flooding of the electrode, thus resulting into three distinct states of the electrode. These states are depicted in Figure 2 The "Regular Cell" has a standard electrolyte loading in the membrane, which assures optimal performance at standard operating conditions. The "Flooded Cell" membrane has 10% more phosphoric acid solution in comparison and the electrodes are separately doped with the additional electrolyte. The "Dry Cell" electrolyte loading in the membrane is approximately 20% less than that in the "Regular cell". This is done to assure minimal flooding of the electrode and thereby create a condition opposite to the "Flooded Cell". Chronoamperometric tests on these cells are performed with the potentiostat PGSTAT20 from AUTOLAB®. This potentiostat however is incapable of recording cell currents higher than 2 A. The regular cell with a 10 cm² active surface area, under standard operating conditions when subjected to a cell voltage step down from OCV to 0.1 V, produces an instantaneous cell current overshoot to around 100 A before stabilizing at around 30 A. To be able to get accurate readings from this potentiostat, certain adjustments are made. An additional 20 A current booster (BSTR 20A also from AUTOLAB®) is used in connection to the potentiostat and the electrodes are cut down to 1 cm².

The sequence of the chronoamperometric experiment is described below and is repeated for the three test cells.

1. The cell is initially kept close to the OCV before making a step change to lower voltage.

2. The step down in voltage is made from OCV to 0.1V and the transient current response is recorded every 10 milliseconds. 3. The whole procedure is carried out at 160°C, 120°C and 80°C. 4. Polarization curves for the above temperatures are recorded.

Results and Discussion

Polarization curves

The steady state behavior of the three modified electrodes can be seen from their I/E curves at 160°C as shown in Figure 3. The curves for the dry and the regular cell are almost identical. The electrode modifications have no affect on the steady state behavior for these two types. The flooded cell on the other hand shows poor performance throughout. If referred to Figure 2 again, the lower cell current in the flooded cell can be attributed to the absence of the three phase boundary. Since both the dry and the regular cell are assumed to have the same geometric area of the three phase boundary region, the steady state currents are identical over the entire cell voltage range.

As the temperatures are lowered to 120°C (Figure 4), the I/E curves for the same cells present a slightly different picture. The flooded cell shows a similar curve as at 160°C but the dry cell curve is now quite distinct in comparison to the regular cell. At higher cell voltage and low cell current also known as the activation region, the dry cell shows poor performance. This continues to be the case in the quasi-linear ohmic region, but in the mass transport limitation region the regular cell current drops quite early in comparison. The regular cell current drops below the dry cell at 0.4V. The boiling point of water at 2.5 bar absolute pressure is around 129°C. This can make the removal of product water at 120°C rather slow when compared to that at 160°C. Since the regular cell is assumed to have a part of its electrode flooded with electrolyte, further flooding by product water can cause complete flooding and thus induce current losses due to oxygen transport resistances. This however is not the case for the dry cell. Since most of its electrode is expected to be void of the electrolyte, flooding by product water is not as detrimental to its performance at the same voltages.

In Figure 5 the I/E curves at 80°C show identical behavior of the dry and regular cell in the activation and the ohmic region. But the cell current drop due to mass transport losses is now observed much earlier (0.5V) for the regular cell as compared to the dry cell. This is an expected trend due to lowering of temperatures. The dry cell behavior at lower temperatures in the mass transport limitation region has the best performance. Nevertheless, for most fuel cell applications, cell voltages in the range of 0.6V -0.7V are favorable and that's where the regular cell has the better cell currents in the investigated temperature range. During the fabrication stage the partial flooding of the regular cell is difficult to quantify. It is seen clearly that this electrode-electrolyte interface plays a crucial role in the fuel cell performance. In the next section, dynamic measurements on three test cells are done via chronoamperometry. The analysis of the recorded transient currents by Cottrell equation provides a possible diagnostic method for characterizing the electrode-electrolyte interface.

Reaction Zones

The chronoamperometric measurements are carried out on the three test cells at 160°C. In Figure 6, the left graph shows the transient current recorded when a cell voltage step from OCV to 0.1V is applied. The right figure is the corresponding Cottrell plot, where the transient current after the applied voltage step is plotted against the inverse square root of time. The highest peak current is observed for the flooded cell, where maximum platinum catalyst particles are expected to be in contact with the electrolyte. This current however drops rapidly below the currents observed from the other two cells. The regular and the dry cell show almost similar behavior, which was also the case for their polarization curves at 160°C. However, it is observed here that the cell current peak after the voltage step is higher for the regular cell. This indicates that since the electrolyte in this case floods the electrode more than in the dry cell, more platinum catalyst particles are in contact with the electrolyte and therefore electrochemically active. While the steady state behavior shows no such differences for the regular and the dry cell, the dynamic behavior indicates the difference in the electrodeelectrolyte interface.

Similar plots at 120°C are shown in Figure 7. The peak current is the largest for the flooded cell and its Cottrell plot is linear. This straight line heading towards the origin is similar to a planar electrode in the Cottrell experiment. The regular and the dry cell Cottrell plots are more distinct with the peak current of the regular cell larger than that of the dry cell, but the steady state current considerably lower. The chronoamperometric and Cottrell plots at 80°C for the three cells are shown in Figure 8. The flooded test cell Cottrell plot is similar to the Cottrell experiment for a planar electrode. It is linear heading towards the origin. This describes the two phase (liquid and solid) boundary reaction zone where the oxygen permeates i.e. dissolves into the electrolyte and diffuses towards the electrode. The slight non linearity in the plot is due to the capacitive current at short times (3 > t -1/2 >10).

The dry test cell has a Cottrell plot similar to the spherical electrode with a straight line intercepting the cell current axis. The magnitude of the current for the dry cell is almost five times higher than that of the flooded cell at steady state. This can be attributed to the reaction at the three phase boundary, which requires permeation of oxygen in air, which is much faster than that in the electrolyte. The Cottrell plot decreases until t -1/2 = ~1, beyond which a slight increase and then a decrease in cell current in observed. The increase in the cell current is expected to be due to product water, which acts as an extension of the electrolyte in the electrode. This brings previously unused catalyst particles in contact with the electrolyte and therefore slightly increases the active area and also the cell current. For times longer than a couple of seconds, effects other than diffusion (mainly convection) play a significant role and therefore these times on the Cottrell plots (between 0 and 1 on the x-axis) deviate from expected behavior [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF].

The regular test cell Cottrell plot shows a larger slope at short times similar to the flooded test cell. This behavior transitions into a slope more like the dry cell at longer times. It is therefore plausible to assume that the regular test cell with sufficient electrolyte forms both, a two phase and a three phase reaction boundary and has a Cottrell plot similar to a combination of the plots of the other two cells. This can be simulated and validated as shown in the section 4.4. The following sections consider the experimental results performed at 80°C for further analysis.

Electrode Active Area

The Cottrell equation for the flooded cell, Eq. ( 2), is used to simulate the Cottrell plot while taking the active electrode area A as an unknown. The oxygen diffusion coefficient in phosphoric acid, D and initial concentration in phosphoric acid C* are obtained by other ex-situ experiments (see Table 1) and have similar magnitude to the data reported in literature [START_REF] Liu | [END_REF]. This equation is applied only to the relevant measurement data points, i.e., the intermediate measured points. The shorter times that have a large influence from cell capacitance and the times longer than 1 second are neglected for this analysis. This is depicted in Figure 9 where one sees the simulated line being linear heading towards the origin in good agreement with the measurement data at intermediate measured points. The active electrode area is the only unknown in this equation and is estimated for these measured data points to be approximately 3500 cm². As the platinum particles used in electrode preparation have an average weight specific surface area of 500 m² g -1 , the maximum possible surface area based on Pt loading of the catalyst in the cathode is estimated to be around 7000 cm² per cm geometric area. Even in case of the flooded cell, not all catalyst particles are accessible by the electrolyte. This estimation suggests that approximately 50% of the platinum surface area is not available for reaction. This can be due to loss of catalyst particles in the porous carbon support, sintering and other agglomeration effects.

The non-linearity due to cell capacitance is now taken into account by adding the 2 nd term, Eq. ( 4).

) exp( )

( 2 / 1 2 / 1 * , 2 / 1 , 4 3 2 4 3 2 d u u PO H O PO H O Flooded C R t R E t C nFAD t i - + = π (6)
For the second term, the magnitude of the cell voltage step E is a known value, whereas the cell capacitance C d and uncompensated resistance R u are unknown. The two time dependent terms share no common parameters. The double layer capacitance obtained from this equation is approximately 0.0875 F and the uncompensated resistance value of 0.64 Ohms. The solution for this equation with the estimated values of C d and R u also shows good agreement with the measurement points as shown in Figure 9.

This approach can be used as a comparative analysis for estimating the maximum available catalyst area in commercial electrodes. The estimated double layer capacitance is due to the entire Pt/C area in contact with the electrolyte. However, the contribution to the cell current at the intermediate measurement points used to estimate the active platinum area has negligible influence from the capacitive current and therefore the estimated active area is expected to be the active platinum catalyst area,

Thin Film Boundary

The Cottrell plot of the dry cell can be simulated using Eq. ( 3). In this equation, the first time dependent term has the unknown active electrode area A and the second time independent term has the radius of the spherical electrode δ. While the test cell electrode is in no sense spherical, one can treat this parameter as a shape factor. Even though it's difficult to describe this theoretical radius in the context of a porous electrode, its inclusion in the second time independent constant term serves only to ascertain the magnitude of the intercept on the cell current axis. This shape factor therefore has no effect on the slope of the curve, which holds the permeability parameters and the active electrode area.

The three phase boundary reaction zone assumes the oxygen diffusion coefficient and initial concentration are that in air. This input however estimates an active electrode area around 0.6 mm², which is much smaller than the geometrically projected area (~1 cm²) of the test cell and therefore an improbable scenario. An alternative to the simple three phase boundary reaction zone model is the thin film boundary model as seen in Figure 10. Similar model approaches are widely reported in the literature for describing active catalyst regions in a Nafion based PEM fuel cell [16 -18]. In the presented case the catalyst particle are covered by a thin film of the electrolyte and the oxygen which diffuses through the air has to first dissolve in the electrolyte film in order to reach the reaction site. An important assumption here is that the thickness of the electrolyte film is extremely small and all the dissolved oxygen molecules are directly available for reaction on the catalyst surface. This is in agreement with the work done by Bjerrum et al [15]. Here, the thickness of the phosphoric acid film on electrodes was estimated to be 0.1µm order of magnitude. With this consideration, the diffusion of the dissolved oxygen molecule to the active site within the electrolyte thin film can be neglected and only the diffusion of oxygen in air is considered to be rate limiting. The initial concentration of oxygen is however that in the electrolyte itself and not in air. The diffusion term can be accordingly modified as shown here.

δ π * , , 2 / 1 2 / 1 * , 2 / 1 , 4 3 2 2 4 3 2 2
) (

PO H O Air O PO H O Air O Dry C nFAD t C nFAD t i + = (7)
In this equation, the initial concentration of oxygen in air is thereby replaced with the initial concentration of oxygen in phosphoric acid and the model estimates an active electrode area of about 3 cm². The simulation is in close agreement with the experimental data as seen for the dry cell in Figure 11. As seen in the Cottrell plots for all three cells at 80°C, the steady state current for the dry cell is five times larger than that for the flooded cell. Thus it can be concluded that a much smaller active electrode area with fast oxygen transport in the dry cell can account for a larger cell current when compared to a much larger active electrode area but slow oxygen transport in the flooded cell. A build up of such a thin film boundary in the electrode-electrolyte region can therefore determine the performance of the fuel cell.

Commercial Electrode Behavior

Even though the dry cell performs extremely well at low temperatures in the mass transport limitation region, the cell current at working voltages of 0.6V -0.7V, the regular cell performs consistently better in the tested temperature range. This goes to show that a certain degree of partial flooding is important for power generation. However, quantifying such a character of the electrode-electrolyte interface after a commercial fuel cell is assembled is not a trivial task. The Cottrell plot of the regular cell (commercial electrode) can be considered to be combination of the flooded and dry cell plots. Therefore it is possible to add the Cottrell equations for the flooded and dry cells along with an additional transition term to be able to describe the standard cell. One such extension has been presented in the works of S. Mitsushima et al [START_REF] Ota | Handbook of Fuel Cells[END_REF], which was validated for a platinum microelectrode.

) 3911 . 0 exp 273 . 0 ( ) ( 2 / 1 2 / 1 0 0 0 2 / 1 2 / 1 2 / 1 2 2 2 2 2 2 2 t D r r C nFAD r C nFAD t C nFAD t i O O O O O O O d - + + = π (8)
This equation is an extension of the Cottrell equation with the first term representing the flooded part, the second term accounting for the intercept on the current axis and the third term, which contains the exponential time dependence. The diffusion coefficient and initial solubility of the oxygen is in the electrolyte. The third term is a product of the current intercept (second term) and the exponential time dependent term with the preexponential and exponential constants (positive 0.2732 and negative 0.3911). The term r 0 in this equation is the electrode radius and is analogous to the term δ used in all previous equations. The exponential term presented in this work is rather empirical in nature, but can be used to describe the transition of the Cottrell plot from the flooded cell behavior to the dry cell behavior. In case of the standard cell, the electrode is partially flooded along with the thin film boundary. For the Cottrell plot, that would mean partial contribution from the flooded cell equation term and full contribution from the dry cell equation term assuming a similar thin film boundary area. The short time influence of the flooded cell term, which transcends into the long time influence of the dry cell term, is sufficiently connected by the exponential term of the equation. This can be written as:

) . exp ( ) ( 2 / 1 2 / 1 0 2 / 1 , 2 4 3 , 2 2 t D r Q P C nFAD i i x t i O PO H O Air O Dry Flooded cell δ + + = (9)
In this equation the unknown parameters are x, P and Q, where x is the extent of flooding of the regular cell ranging between zero for a dry cell to one for a flooded cell. The parameter P and Q are left completely free to be solved to match the experimental measurements and the result is shown in Figure 11. The estimated value of parameter x is found to be 0.6 and those of P and Q are 1.3808 and -0.5979 respectively. The empirical constants P and Q have no physical significance, but when compared with Eq. 8 for the microelectrode, one can observe the increase in magnitude for both as well as maintaining the negative sign for constant Q. The value of x signifies the contribution of the flooded cell equation to the regular cell current equation. This technique shows the estimation of flooding of the electrode by the electrolyte in a commercial fuel cell.

Conclusions and Outlook

Chronoamperometric testing and Cottrell analysis performed on commercial electrodes give valuable insights into its electrode-electrolyte interface. This interface is experimentally and analytically described using the combination of the two extreme cases of electrode flooding. Polarization curves as well as Cottrell plots of the three test electrodes show very distinct behavior. The flooded cell analysis provides a new possibility to estimate the maximum available catalyst area in a commercial HT-PEM fuel cell. The formation of the thin film boundary layer in the electrode that determines largely the output of the cell can be seen from the dry cell analysis. A certain level of "floodedness" of the electrode is necessary to maximize the cell performance under typical fuel cell operating conditions as seen from the performance of the regular cell. This approach can be a useful diagnostic method for characterizing MEAs postproduction. Finally, the estimation of maximum active area available in the electrode is a motivation for further optimization of platinum usage for important cost reductions in commercial electrodes.

  as homogenously distributed Pt/C particles with the different electrolyte flooding.

Fig. 1

 1 Fig. 1 Reaction zones: depiction of two phase and three phase boundary Fig. 2 Schematic representation of the modified electrodes Fig. 3 Polarization curves for the three electrodes at 160°C Fig. 4 Polarization curves for the three electrodes at 120°C Fig. 5 Polarization curves for the three electrodes at 80°C Fig. 6 Transient current responses at 160°C and corresponding Cottrell plot Fig. 7 Transient current responses at 120°C and corresponding Cottrell plot Fig. 8 Transient current response at 80°C and corresponding Cottrell plot Fig. 9 Cottrell plot and simulation results for flooded cell current Fig. 10 Schematic depiction of a thin film boundary reaction zone model Fig. 11 Cottrell plot and simulation results for dry cell and regular cell current

Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Fig. 11

 1234567891011 Fig. 1 Reaction zones: depiction of two phase and three phase boundary 266x120mm (96 x 96 DPI)

Table 1

 1 Modeling parameters used in and obtained from simulations (for 80°C).

	Modified Electrodes

Wiley-VCH Fuel Cells

Acknowledgements

The authors are thankful to B. Schaar at the Volkswagen AG and U. Krewer at the Max Planck Institute, Magdeburg, for their valuable contribution to the discussion. Also the authors are grateful for the support extended by O. Groeger, G. Huebner, J. Huslage and F. Seyfried at the Volkswagen AG.

List of Symbols