# Study of the Influence of the Orientation of a 50-Hz Magnetic Field on Fetal Exposure Using Polynomial Chaos Decomposition

^{1}

^{2}

^{*}

## Abstract

**:**

**B**) orientation on fetal exposure at 50 Hz by polynomial chaos (PC). A PC expansion of induced electric field (

**E**) in each fetal tissue at different gestational ages (GA) was built as a function of

**B**orientation. Maximum

**E**in each fetal tissue and at each GA was estimated for different exposure configurations and compared with the limits of the International Commission of Non-Ionising Radiation Protection (ICNIRP) Guidelines 2010. PC theory resulted in an efficient tool to build accurate approximations of

**E**in each fetal tissue.

**B**orientation strongly influenced

**E**, with a variability across tissues from 10% to 43% with respect to the mean value. However, varying

**B**orientation, maximum

**E**in each fetal tissue was below the limits of ICNIRP 2010 at all GAs.

## 1. Introduction

_{i}collected in the input random vector

**X**, whose changes and consequent effect on the system output Y are the goal of the investigation.

**B**-field orientation at 50 Hz has been estimated on the fetal exposure at different stages of pregnancy. So far, several studies in the literature have analyzed the fetal exposure at 50 Hz by means of deterministic dosimetry with appropriate anatomical modelling [20,21,22,23,24]. In [20], models of the embryo and fetus at 8, 13, 26 and 38 weeks of pregnancy were combined with an anatomically realistic adult female model [25]. The whole-body fetus and the fetal brain and skeleton (the only organs available) were represented as spherical or cylindrical shapes. The induced current densities and the electric fields were estimated in the fetus from applied electric and magnetic fields, separately, at 50 Hz. The magnetic field was oriented to obtain a front-to-back, lateral and top-to-bottom exposure (B

_{front}, B

_{lat}and B

_{top}in the following indicated as “classical” orientations) on the pregnant body. In [21], a 30-week pregnant woman model was developed, in which only the fetal skeleton and soft tissues could be distinguished. Induced electric current densities in the fetal soft tissues and in the fetal CNS tissues have been calculated for an exposure to 100-μT homogeneous magnetic fields at 50 Hz in the three classical exposure scenarios and for a vertically-oriented 5-kV/m electric field. In [22], a model of a pregnant woman at 30 weeks of pregnancy was developed in which no organs could be distinguished in the fetus. The exposure to a homogeneous magnetic field in the front and lateral exposure (B

_{front}and B

_{lat}) and to a vertical electric field at 50 Hz has been analyzed, separately. In [23], seven pregnant female models have been analyzed, calculating the induced current densities in the fetal brain and body from applied magnetic fields at 50 Hz. Finally, in [24], the fetal exposure at 3, 7 and 9 months of gestational age (GA) to uniform magnetic fields (MF) at 50 Hz has been studied by means of advanced numerical models of pregnant women. In this work, the induced electric fields and electric current densities have been assessed in each fetal tissue at each GA (15, 17 and 26 tissues at 3, 7 and 9 months GA, respectively), when the

**B**-field is oriented to obtain the previously-mentioned classical exposure conditions on the pregnant body. Different from the studies previously described, in which there was a lack of accuracy at the level of tissues and organs in the fetal models, in [24], a complete description of the fetal exposure is given through a tissue-specific analysis; however, also in this work, only three exposure conditions have been taken into account. Overall, this previous literature gave an almost exhaustive picture of the fetal exposure only for uniform magnetic fields oriented in three classical orthogonal directions. However, the direction of the field could be different from these ones in a realistic exposure scenario, and the analysis of only these classical orientations could be insufficient to find the worst-case exposure condition for each specific fetal tissue. To tackle this issue, in this study, the variability of fetal exposure due to

**B**orientation has been investigated in terms of the assessment of the induced electric field (

**E**) in each fetal tissue at 3, 7 and 9 months GA. In order to overcome the computational effort of the deterministic dosimetry, an analytical approximation of

**E**in each fetal tissue has been developed by making use of the PC theory. Then, the PC expansions of

**E**built for each tissue have been used to estimate the variability of

**E**under different exposure conditions. Furthermore, the maximum induced electric field, found in each fetal tissue among all

**B**orientations analyzed, has been identified and compared to the basic restrictions of the International Commission of Non-Ionising Radiation Protection (ICNIRP) Guidelines 2010 [3] for the general public at 50 Hz. Finally, an analysis of the distribution of

**B**field orientations, which induce high electric fields (≥70% of the maximum

**E**) in the fetus whole-body at each GA, has been also carried out.

## 2. Material and Methods

#### 2.1. Polynomial Chaos Expansion of the System Output Y

**X**influences the system output Y. In detail, the system output is approximated by a suitable finite-dimensional polynomial basis

**Ψ(X)**of size P [26] as follows in (1):

_{j}are the coefficients of the PC expansion, ${\psi}_{j}(X)$ are the polynomials of the basis

**Ψ(X)**and e is the truncation error. Following the approach described in [13] and applied in [26], Y can be modelled as in (1), under the hypothesis that Y has a finite variance (E [Y

^{2}] < ∞). Furthermore, the input parameters

**X**are supposed to be independent variables characterized by the joint probability density function (pdf) f

_{X}, expressed as in (2):

**X**and ${f}_{{x}_{i}}$ is the probability density function associated with each input random parameter x

_{i}. This pdf f

_{X}is necessary to define the polynomial basis

**Ψ(X)**used to build the PC expansion (1). Indeed, considering the independence of the input parameters x

_{i}stated above, each polynomial $\psi \left(X\right)$ belonging to

**Ψ(X)**can be represented as in (3):

_{i}and ${\alpha}_{j}$ represents the maximum degree of the polynomials in ${\pi}_{{\alpha}_{j}}$. $\alpha =\left\{{\alpha}_{1},\dots .,{\alpha}_{K}\right\}$ is the vector of the degrees ${\alpha}_{j}$. Only the combinations of the ${\alpha}_{j}$, such that $\left|\alpha \right|=|{\alpha}_{1}+\dots +{\alpha}_{K}|\le p$, where p is the maximum accepted degree of the polynomial $\psi \left(X\right)$, are suitable to be used to build the polynomials $\psi \left(X\right)$. The choice of the maximum degree p is arbitrary [11,27] and depends on the desired accuracy of the PC expansion. Table 1 shows the correspondence between some classic pdfs and their corresponding orthogonal polynomials [15].

**Table 1.**Correspondence between classic probability density functions (pdfs) and families of orthogonal polynomials.

Probability Density Function | Support * | Polynomial π |
---|---|---|

Gaussian | $\mathbb{R}$ | Hermite |

Uniform | (−1,1) | Legendre |

Gamma | (0,+∞) | Laguerre |

Chebyshev | (−1,1) | Chebyshev |

Beta | (−1,1) | Jacobi |

**Ψ(X)**is indicated as P in (1) and is a function of p and K:

**Ψ(X)**has been built.

**Figure 1.**Schematic of the complete polynomial chaos (PC) procedure with a zoom of the “estimation of a

_{j}by LAR” block on the left. LAR, least angle regression.

_{j}and the corresponding polynomials $\psi \left(X\right)$ in

**Ψ(X)**. In the literature, several methods have been proposed to perform this optimization [26,28,29,30,31]. Among them, here, the least angle regression algorithm (LAR), introduced by Efron and colleagues [32] and adapted to PC theory in [26], has been applied. LAR relates to the classic model-selection method known as forward selection [33]. The algorithm consists of selecting the most suitable polynomials from the chosen basis

**Ψ(X)**and then of calculating the coefficients a

_{j}by least-square regression with the aim to optimize the PC approximation of Y with respect to a series of N observations Y

_{o}= {y

_{o}

^{(1)}, y

_{o}

^{(2)},…,y

_{o}

^{(N)}} of Y. In this study, the N observations have been obtained by deterministic dosimetry applied to a random subset of N input vectors X

_{o}= {x

_{o}

^{(1)}, x

_{o}

^{(2)},…,x

_{o}

^{(N)}}, which is the “experimental design” block in the schematic of Figure 1. Once the observations Y

_{o}have been calculated, the LAR algorithm generates a collection of PC expansions, in which the first PC expansion includes a single polynomial of the basis

**Ψ(X)**, the second one includes two polynomials, and so on, until m polynomials have been included, with m = min(P, N-1). Among these m PC expansions, the optimal one Y’ (see Figure 1) is then chosen as the minimum of the m leave-one-out cross-validation (LOOCV) errors. The LOOCV approach was proposed by [34,35] and applied to assess the accuracy of PC expansions obtained by LAR in several studies, such as [17,26]. One should note that in the optimization phase, the LAR procedure could select as the best solution Y’ a PC expansion with a number of coefficients a

_{j}less than P. Hence, in the following, the number of coefficients of Y’ will be indicated as Q. In this study, a home-made Python script based on the open TURNS package [36] has been used for the implementation of the above-explained procedure.

#### 2.2. Validation of the PC Expansion

**Y**

_{val}, different from the set

**Y**

_{o}previously used to build Y’.

**Y**

_{val}has been always estimated by deterministic dosimetry on an experimental design

**X**

_{val}, randomly selected and different from

**X**

_{o}, and has size S = N/2. The residual error is the percentage mean square error (pMSE) defined as in (5):

#### 2.3. Application of the PC Theory to the Analysis of the Fetal Exposure Varying B-Field Orientation

#### 2.3.1. Definition of the Input Random Vector **X** and the System Output Y

**B**-field orientation at 50 Hz on the fetal exposure at different stages of pregnancy by means of the polynomial chaos theory. In this study, the input random vector

**X**is made by K = 2 parameters: the spherical angles theta (θ) and phi (φ), which characterize the

**B**-field orientation. These two variables are independent and supposed to be uniformly distributed over (0,180°) and (−180°, 180°), respectively. Hence, according to Table 1, each polynomial $\psi \left(X\right)$ belonging to the basis

**Ψ(X)**is made of Legendre polynomials, orthogonal with respect to the uniform distribution.

**E**averaged over a 2 × 2 × 2 mm

^{3}cube in each fetal tissue (E

_{99th}). Indeed, as the ICNIRP Guidelines 2010 [3] suggest, the induced

**E**has to be calculated as a vector average in a small contiguous tissue volume of 2 × 2 × 2 mm

^{3}in each specific fetal tissue, and the relevant value to be compared with the basic restriction is the 99th percentile value of the root mean square of

**E**in each tissue. Therefore, this is the metric modelled by polynomial chaos.

#### 2.3.2 The Observation Set **Y**_{o}

_{99th}for each fetal tissue. In this study, these observations have been obtained by deterministic dosimetry applied to an input vector X

_{o}, made of N pairs of angles [θ, φ], generated through a quasi-Monte Carlo method [37], based on Sobol’s function [38] and representing N different orientations of the

**B**-field.

**Figure 2.**On the left: pregnant woman model at 9 months gestational age (GA) with the Cartesian reference system x,y,z and the spherical coordinates θ, φ. The fetus image is also reported to present its position in the pregnant woman’s womb. On the right: table with the fetal tissues at each GA.

**B**-field at 50 Hz. The root mean square amplitude of the uniform

**B**-field was set to the reference levels of the ICNIRP 2010 [3] for the general public at 50 Hz, which is 200 μT.

#### 2.4. Analysis of the Fetal Exposure

_{99th}for each fetal tissue has been achieved, the statistical moments of the first and second order, i.e., the mean $\mu $ and the variance ${\sigma}^{2}$, of E

_{99th}in each fetal tissue have been analytically derived from the PC coefficients following a method proposed and validated by Blatman and colleagues in [26]. In detail, the mean and the variance are expressed as in (6) and (7):

_{99th}for each fetal tissue.

**B**-field orientation on the fetal exposure of each tissue and at each GA.

_{99th}in each tissue was computed for 10,000

**B**orientations, defined as pairs of angles [θ, φ] randomly selected, and the maximum among, 10,000 E

_{99th}(mE

_{99th}), has been compared to the limits (E

_{lim}) indicated by the basic restrictions of the ICNIRP Guidelines 2010 [3] for the general public at 50 Hz. These limits are 0.02 V/m for the central nervous system (CNS) tissues of the head and 0.4 V/m for all of the other tissues of the head and body. For convenience, the percentage ratio between mE

_{99th}and E

_{lim}in each fetal tissue (in the following identified as worst-case scenario WS%) is used to provide quantitative information about the worst exposure scenario with respect to the permitted values and is defined as:

**B**orientations, which induce E

_{99th}≥ 70% mE

_{99th}of the fetus whole-body at each GA, has been performed. The threshold of 70% was chosen, because it corresponds to an amplitude reduction of 3 dB with respect to the maximum value [44].

## 3. Results and Discussion

#### 3.1. Validation of the PC Expansion in Each Fetal Tissue

_{99th}in each fetal tissue, the validation procedure described in the Material and Methods Section has been performed, estimating the residual error, expressed as pMSE (see Equation 5), between a validation set

**Y**

_{val}and several PC expansions of E

_{99th}built from sample sets

**Y**

_{o}of increasing size N and changing the maximum degree p of the polynomials $\psi \left(X\right)$. The error pMSE was found to be lower than the error threshold τ, fixed at 0.5%, when the PC expansions were built using N = 300 observations and a degree p = 15. Therefore, this number of observations and the same degree p were used in this study to build all of the PC expansions.

#### 3.2. Estimation of the Statistical Moments

_{99th}for each fetal tissue at 3, 7 and 9 months GA estimated by PC coefficients. This graph represents the effect of the variation of

**B**orientation on E

_{99th}for each fetal tissue.

**Figure 3.**Mean value (bar) and standard deviation (whiskers) of E

_{99th}in each fetal tissue at 3, 7 and 9 months GA.

_{99th}across the

**B**orientations is greater with GA due to the increase in size of fetal organs and tissues. Moreover, considering only the common tissues at all GAs, the tissues with the highest mean E

_{99th}are the fetal skin, fat, liver, spinal cord and kidney, with a mean value up to 7.10 mV/m at nine months GA in the skin tissue. At three months GA, the highest mean E

_{99th}has been found in the fetal fat (around 2.16 mV/m), while the lowest exposed tissue is the eye-lens, with a mean E

_{99th}of about 0.68 mV/m. The standard deviation has a maximum of 0.49 mV/m in the spinal cord and a minimum of 0.15 mV/m in the eye-lens and brain. At seven months GA, the highest exposed tissue, in terms of mean E

_{99th}, is the fetal skin (5.09 mV/m). Furthermore, fetal brain, bone, spinal cord and muscle present similar mean E

_{99th}values of about 3 mV/m, whereas the fetal stomach and gallbladder, which are the lowest exposed tissues, show mean E

_{99th}values around 1.40 mV/m. The highest standard deviation is 0.73 mV/m in liver, and it is around 0.70 mV/m in skin, kidney, lung and subcutaneous adipose tissue (SAT), whereas the lowest variation of 0.19 mV/m is in the stomach tissue. At nine months GA, some tissues present similar mean E

_{99th}(e.g., brain, heart, adrenal, CSF, esophagus and thyroidal) of about 2.60 mV/m, and the highest standard deviation is up to 1.40 mV/m in kidney.

_{99th}values, due to the different

**B**-field orientations, is more clearly highlighted in Figure 4 in terms of CV coefficients for the various fetal tissues at all GAs. CV has been found always higher than 10% at all GAs. Fetal spleen, gallbladder and thyroidal gland present the highest CV at 3, 7 and 9 months GA with values up to 32.37%, 25.30% and 43.25%, respectively. CVs higher than 30% and up to 35% have been also observed in the fetal gallbladder, esophagus and ovary at nine months GA.

**B**-field at three months GA than at the other stages of pregnancy, with CV up to three-times higher in the spinal cord at three months of GA than for the other two GAs. This difference among GAs could be related to computational uncertainty due to the different grid resolution at three months GA (i.e., of 0.3 mm) with respect to seven and nine months GA (i.e., 1 mm), adopted for the application of the deterministic dosimetry to estimate the observations

**Y**

_{o}necessary to build the PC expansion (see Figure 1 above). However, Liorni and colleagues [24] demonstrated that the uncertainty of the grid resolution affects the estimated E

_{99th}by no more than 5%; thus, the high CV value at three months GA is probably linked to the specific shape and small size of the involved tissue. Among the tissues that are common to all GAs, fetal bladder, eye-lens, kidney and spleen always present CV almost equal or higher than 20%, whereas for the fetal bone, the maximum CV is 22.5% at three months GA, with values of 14.8% and 12.59% at seven and nine months, respectively.

#### 3.3. Analysis of the Fetal Exposure Respect to the Limits

_{99th}(mE

_{99th}) in each fetal tissue has been found over 10,000 different

**B**orientations and compared with the exposure limits (E

_{lim}) proposed in the ICNIRP Guidelines 2010 [3]. The analysis has been carried out considering (i) the central nervous system (CNS) tissues of the head and (ii) all of the other tissues of the head and body, separately.

_{99th}is notably lower than the ICNIRP limits at all GAs.

**Figure 5.**Worst-case scenario (WS%) in all the tissues of the head (excluding the CNS) and body for the mE

_{99th}found among the 10,000 values of E

_{99th}calculated from the PC expansion, changing randomly the

**B**orientations.

**B**orientations, among the 10,000 analyzed, which induce E

_{99th}higher than 70% of mE

_{99th}in the fetus whole-body at 3, 7 and 9 months GA. In detail, in the figures, the blue circles represent the

**B**orientations for which E

_{99th}in the fetus whole-body is in the range from 70% to 79% of mE

_{99th}, the green circles in the range from 80% to 89% of mE

_{99th}and the red circles all of the orientations of the

**B**-field for which E

_{99th}is higher than 90% of mE

_{99th}.

**Figure 6.**Distribution of

**B**-field orientations on a unitary sphere represented in the pregnant woman’s reference system x,y,z (Figure 2), which determines, on the fetus whole-body at three months GA, induced electric field E

_{99th}in the ranges 70% mE

_{99th}to 80% mE

_{99th}(blue circles), 80% mE

_{99th}to 90% mE

_{99th}(green circles) and higher than 90% mE

_{99th}(red circles), respectively. These

**B**orientations have been also indicated with respect to the reference system centered on the pregnant woman adopted in this study.

**Figure 7.**Distribution of

**B**-field orientations on a unitary sphere represented in the pregnant woman’s reference system x,y,z (Figure 2), which determines, on the fetuswhole-body at seven months GA, induced electric field E

_{99th}in the ranges explained in Figure 6. The

**B**orientations have been also indicated with respect to the reference system centered on the pregnant woman adopted in this study.

**Figure 8.**Distribution of

**B**-field orientations on a unitary sphere represented in the pregnant woman’s reference system x,y,z (Figure 2), which determine, on the fetus whole-body at nine months GA, induced electric field E

_{99th}in the ranges explained in Figure 6. The

**B**orientations have been also indicated with respect to the reference system centered on the pregnant woman adopted in this study.

_{99th}has been found for both the front-to-back and back-to-front exposure with respect to the pregnant woman body (Figure 6). The solid angle of the sphere under the region E

_{99th}≥ 90% mE

_{99th}is equal to 0.27 steradians (sr) with respect to the front side of the pregnant woman and symmetrically also to the back side. Furthermore, the other

**B**orientations, which determine the three ranges of values of E

_{99th}indicated above, are always located around the

**B**orientation for which E

_{99th}is maximum, and the total solid angle under these three areas analyzed is equal to 1.43 sr.

_{99th}is induced by the lateral exposure (both right-to-left and left-to-right) with respect to the pregnant woman body (Figure 7). Furthermore, in this case, the region of the sphere with E

_{99th}≥ 90% mE

_{99th}is determined by a solid angle of 0.26 sr with respect to each lateral side of the pregnant woman.

_{99th}≥ 80% mE

_{99th}have been found for

**B**orientations significantly far from the lateral exposure and closer to the top and bottom exposure (as can be observed in Figure 7). The solid angle under the sphere, which represents this last region, is 0.48 sr for the top side and symmetrical for the bottom one.

_{99th}has been found for the front-to-back and back-to-front exposure as at three months GA (Figure 8). The region of the sphere in which

**B**orientations induce E

_{99th}≥ 90% has a solid angle of 0.34 sr with respect to the front side and symmetrically for the back side of the pregnant woman. However, as previously said for the seven-month fetus, there are several orientations of the

**B**-field around the lateral exposure, which induce electric fields higher than 80% and up 89% of the maximum. The solid angle relative to this region is 0.24 sr for each lateral side of the pregnant woman.

**B**-field with respect to the pregnant woman’s body, whose morphology was unchanged. However, one should take into account that also the movements of the fetus in its mother’s womb would additionally contribute to varying the induced fields in the fetal tissues. In [24], the variability of the exposure due to the fetal posture was quantified on three pregnant woman models at three months GA, in which the fetal posture was changed among the most statistically significant at that stage of pregnancy. A detailed description of the various fetal positions during pregnancy is provided in [45]. The three months GA pregnant models were exposed to uniform

**B**-field oriented with respect to the classical directions B

_{front}, B

_{lat}and B

_{top}. In this condition, the maximum variation of E

_{99th}induced in the whole-body, due to the fetal posture, was found to be up to 18%.

## 4. Conclusions

**B**orientation was assessed, by developing an approximation of the induced electric field E

_{99th}in each fetal tissue by polynomial chaos and studying how the variability of

**B**orientation influences E

_{99th}. Indeed, there is the necessity to close the gap of knowledge about possible worst-case exposure scenarios in fetal tissues different from the ones obtained for the classical orientations of the magnetic field (i.e., B

_{front}, B

_{lat}and B

_{top}) that have so far been analyzed in the previous literature by deterministic dosimetry.

_{99th}in each fetal tissue (from the validation procedure, a maximum pMSE of 0.48% was obtained across all fetal tissues) and to perform a complete analysis of the fetal exposure at a lower computational cost than the one required by the deterministic dosimetry. Indeed, the number of observations

**Y**

_{o}(in this case N = 300), calculated by deterministic dosimetry and necessary for building the PC expansion of E

_{99th}, was much lower than the number of possible variations of the input parameters under study, lowering therefore the computational cost. Once the observations

**Y**

_{o}have been collected, the construction of the PC expansion by means of the home-made Python script and the calculation of the E

_{99th}, changing

**B**orientations, require only a few minutes.

**Y**

_{o}is a critical issue and depends on the compromise between the computational cost necessary for their estimation and the precision that can be considered acceptable for the specific situation under investigation. Indeed, there could be cases where the estimation of the observations is complex and time consuming and the vector

**Y**

_{o}has to be reduced at the expense of the accuracy of the stochastic modelling.

**B**. Moreover, in these same tissues, in almost all cases, E

_{99th}was found more influenced by the

**B**-field orientation at three months GA than at the other stages of pregnancy. This could probably be due to the shape and size of the tissues and the organs at that GA.

**B**-field orientation cannot be considered negligible, considering that a variability of E

_{99th}has been estimated ranging from 10% to 43% in terms of CV across all fetal tissues and GAs. On the other side, even analyzing the maximum E

_{99th}found across 10,000

**B**orientations, the fetal exposure resulted in being largely lower than the basic restriction of the ICNIRP 2010 [3] for the general public at 50 Hz for both the CNS tissues of the head and for all of the other tissues of the head and body. Indeed, a maximum WS% of 23.50% was found in the CNS tissues of the head at nine months GA, whereas it has been always found lower than 3% in the evaluation of all of the other tissues of the head and body. Therefore, the induced electric fields were well within the ICNIRP basic restrictions in all cases for exposures at the general public reference levels, disregarding the orientation of

**B**.

**B**orientations showed that the highest induced electric field in the fetus whole-body at all GAs was found for the classical orientations B

_{front}at three and nine months GA and B

_{lat}at seven months GA, in agreement with the results already found in [24]. However, the induced E

_{99th}≥ 90% mE

_{99th}have been also found for other

**B**orientations around the classical worst-case exposure condition, which describe a region that is quite similar in size at all GAs.

**B**orientations, different from the ones that induce the maximum electric field in the fetus whole-body, which determine E

_{99th}in the range 80% to 89% of mE

_{99th}. This means that with increasing the gestational age, there is a major dependence of the fetal exposure to the orientation of the incident

**B**-field, which is able to induce a high electric field in the fetus for several orientations.

**B**-field orientation at 50 Hz on the fetal exposure by developing a PC expansion of E

_{99th}in each fetal tissue. This study has demonstrated that polynomial chaos is a powerful tool to evaluate the influence of the variation of the input parameters in a realistic exposure scenario, overcoming the problems of the high computational costs faced by deterministic dosimetry. Therefore, it could be used in the future for the analysis of the variations of other parameters that influence the human exposure to EMF.

**B**orientation significantly influences the fetal exposure in some specific tissues at all GAs. However, though there is a significant variation of exposure, the highest levels of induced electric fields in all tissues are always found within the limits of the ICNIRP 2010 [3]. Finally, several directions of

**B**, far from the exposure conditions that determine the maximum of E

_{99th}, have been found to induce a high electric field (from 80% to 89% of the maximum) especially at seven and nine months GA. Therefore, the polynomial chaos approach also permitted accurately studying the influence on the maximum exposure due to several

**B**orientations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Liorni, I.; Parazzini, M.; Fiocchi, S.; Ravazzani, P. Study of the Influence of the Orientation of a 50-Hz Magnetic Field on Fetal Exposure Using Polynomial Chaos Decomposition. *Int. J. Environ. Res. Public Health* **2015**, *12*, 5934-5953.
https://doi.org/10.3390/ijerph120605934

**AMA Style**

Liorni I, Parazzini M, Fiocchi S, Ravazzani P. Study of the Influence of the Orientation of a 50-Hz Magnetic Field on Fetal Exposure Using Polynomial Chaos Decomposition. *International Journal of Environmental Research and Public Health*. 2015; 12(6):5934-5953.
https://doi.org/10.3390/ijerph120605934

**Chicago/Turabian Style**

Liorni, Ilaria, Marta Parazzini, Serena Fiocchi, and Paolo Ravazzani. 2015. "Study of the Influence of the Orientation of a 50-Hz Magnetic Field on Fetal Exposure Using Polynomial Chaos Decomposition" *International Journal of Environmental Research and Public Health* 12, no. 6: 5934-5953.
https://doi.org/10.3390/ijerph120605934