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The effect of volume conductor modeling on the estimation of

cardiac vectors in fetal magnetocardiography

Rong Tao1, Elena-Anda Popescu2, William B. Drake3, David N. Jackson4,§, and Mihai

Popescu2,5,*

1Bioengineering Department, University of Kansas, Lawrence, Kansas, 66045, USA

2Hoglund Brain Imaging Center, University of Kansas Medical Center, Kansas City, KS 66160,

USA

3Section of Pediatric Cardiology, Children’s Mercy Hospitals and Clinics, Kansas City, MO 64108,

USA

4Department of Obstetrics and Gynecology, University of Kansas Medical Center, Kansas City,

KS 66160, USA

5Department of Molecular and Integrative Physiology, University of Kansas Medical Center,

Kansas City, KS 66160, USA

Abstract

Previous studies based on fetal magnetocardiographic (fMCG) recordings used simplified volume

conductor models to estimate the fetal cardiac vector as an unequivocal measure of the cardiac

source strength. However, the effect of simplified volume conductor modeling on the accuracy of

the fMCG inverse solution remains largely unknown. Aiming to determine the sensitivity of the

source estimators to the details of the volume conductor model, we performed simulations using

fetal-maternal anatomical information from ultrasound images obtained in 20 pregnant women in

various stages of pregnancy. The magnetic field produced by a cardiac source model was

computed using the boundary element method for a piecewise homogeneous volume conductor

with three nested compartments (fetal body, amniotic fluid and maternal abdomen) of different

electrical conductivities. For late gestation, we also considered the case of a fourth highly

insulating layer of vernix caseosa covering the fetus. The errors introduced for simplified volume

conductors were assessed by comparing the reconstruction results obtained with realistic versus

spherically symmetric models. Our study demonstrates a significant effect of simplified volume

conductor modeling, resulting mainly in an underestimation of the cardiac vector magnitude and

low goodness-of-fit. These findings are confirmed by the analysis of real fMCG data recorded in

mid-gestation.

Keywords

fetal magnetocardiography; cardiac vector; boundary element method; volume conductor

1. INTRODUCTION

Fetal magnetocardiography (fMCG) has emerged as an attractive technique for in-utero

assessment of cardiac electrophysiology, especially for its significant potential in assessing

*Corresponding author (mpopescu@kumc.edu).

§Current Address: Division of Maternal-Fetal Medicine, University of Nevada School of Medicine, Las Vegas, NV 89102, USA

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Published in final edited form as:

Physiol Meas. 2012 April ; 33(4): 651–665. doi:10.1088/0967-3334/33/4/651.

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pathophysiological conditions (Leeuwen et al. 2000, Wakai et al. 1998, Hamada et al. 1999,

Wakai et al. 2003, Cuneo et al. 2003, Comani et al. 2004). The major advantage of fMCG

over fetal electrocardiography (fECG) is given by its notably superior signal quality, as the

magnetic field is considerably less affected by tissues with low electrical conductivity

(Cuffin 1978), which can drastically diminish the amplitude of the fECG signals. As a result,

fECG is rarely recorded successfully in obese individuals, and it is considerably

compromised by the formation of the electrically insulating vernix caseosa in late

pregnancy. In contrast, the magnetic recordings allow high resolution measurements from

the second trimester of gestation to birth.

The fECG and fMCG signal morphology is influenced by the fetus position relative to the

sensing system, and by the geometry and electrical conductivity of fetal and maternal tissues

surrounding the fetal heart. These factors modulate the recorded cardiac cycle waveforms

and may prevent a straightforward comparison of the signals amplitude across subjects. One

approach to this problem is to examine the fetal cardiac activity in source space rather than

sensor space. Early fECG studies investigated the role of volume conductor on the

estimation of fetal cardiac vectors and fetal vector loops (FVL) (Oostendorp et al. 1989b).

Before 28 weeks of gestation, the electric potential distribution at the maternal abdomen was

quite well approximated by homogeneous conductors and a dipole model of the fetal heart.

FVLs obtained with this approach were similar to those of newborns (Ellison and Restieaux

1972), but the source strength was significantly underestimated when compared to predicted

data from animal studies (Nelson et al. 1975). The vector magnitude estimates were

improved by refining the volume conductors to account for the fetal body and amniotic

fluid. After 28 weeks, however, the fECG amplitude drops significantly, and the volume

conductor requires additional compartments to model the insulating vernix. Also, the

possible presence of non-uniformities or holes in the vernix may play a significant role on

the amplitude and distribution of the electric potential measured by fECG.

The volume conductor non-homogeneities affect differently the electric potential (measured

by fECG) versus the magnetic field (measured by fMCG). One consequence is that the

electric potential on the maternal abdomen vanishes in the presence of insulating layers,

while the magnetic field does not. In addition, plane-parallel layered volume conductors, or

those with nested compartments approaching spherical symmetry can be approximated by

simplified homogeneous models in biomagnetic applications, overcoming the need to know

the electrical conductivity or to account for the presence and precise geometry of

compartments with different conductivities. Assuming that magnetic fields are less

influenced by non-homogeneities in the conductivity of the maternal abdomen and fetal

body compartments, more recent fMCG studies (Leeuwen et al. 2004, Horigome et al. 2001,

Popescu et al. 2006) used reconstruction techniques to estimate the fetal cardiac vectors by

considering either half-space or spherically symmetric volume conductors. These simplified

models allow using closed-form analytical solutions of the forward field (Sarvas 1987,

Ilmoniemi et al. 1985), and do not necessarily require imaging of the fetal-maternal unit,

which could represent substantial practical benefits. Although each of these studies

acknowledged the potential limitations of the strategy, the results of these efforts showed

promise for characterizing the strength of the cardiac vector with fetal growth ( Leeuwen et

al. 2004), or to diagnose prenatal hypertrophy ( Horigome et al. 2001). On the other hand,

data from computational studies indicates that the influence of non-homogeneities in fetal-

maternal anatomy on the forward solution of the magnetic field can be substantial (Stinstra

and Peters 2002). The extent to which these modulations affect the inverse fMCG solution

when simplified conductors are employed remains however uncertain.

One way to address this issue is to conduct a systematic investigation looking at how the

simplified volume conductor modeling is directly reflected in the accuracy of the fMCG

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inverse solution. For this purpose, we used realistic approximations of the volume conductor

derived from 3D ultrasound images of the fetal-maternal anatomy. Computer simulations

were carried out to characterize middle and late gestation by considering three-compartment

(i.e. fetal body, amniotic fluid and maternal abdomen) volume conductors, with a fourth

compartment added to account for an uniform layer of vernix caseosa in late gestation. More

complex models, e.g. including holes in the vernix have been also considered by studies

using forward magnetic field simulations (Stinstra and Peters 2002). For the purpose of our

study, however, we focus on only two cases with 3 or 4 nested compartments separated by

closed surfaces, which can in principle favor the use of simplified spherical models. The

impact of more complex modeling of the vernix layer will be discussed in the last section, in

light of the present findings. The geometry of the boundary surfaces for compartments of

different conductivity varies significantly between subjects and gestational ages (GA). In

addition, the accuracy of the inverse biomagnetic solution may also depend on such factors

like the source model, choice of source space and optimization algorithm. Thus, we

determined that the use of e.g. a single or just a few setups to characterize the effect of

volume conductor modeling is insufficient, since the way the findings generalize would

remain unknown. To overcome these issues, we use ultrasound data from a relatively large

number of subjects, and we evaluate the accuracy of the inverse solution for two

optimization schemes, in scenarios that incorporate different volume conductor models, and

none or minimal information on the approximate source location. This strategy allows to

cross-validate the results and to demonstrate the effect of volume conductor modeling on the

accuracy of the inverse fMCG solution. Finally, a comparison of the inverse solutions

obtained with realistic approximations of the volume conductor versus spherically

symmetric models is conducted using real fMCG data recorded in mid-gestation.

2. METHODS

2.1. Cardiac vector estimation in fMCG

The cardiac muscle depolarization involves spatially propagating waves that can be

described by a dense distribution of current sources (Malmivuo and Plonsey 1995). When

measurements are performed at relatively large distances from the source distribution, the

magnetic field appears to be generated by an equivalent current dipole (ECD), such that the

ECD strength and orientation approximate at any time the vector summation of all

simultaneously active current sources. This represents the magnetic equivalent of the electric

cardiac vector concept applied in adult ECG. Under this hypothesis, initial studies have used

the ECD model to fit the fMCG data at the peak of the QRS complex (Leeuwen et al. 2004,

Horigome et al. 2001). Since the ventricular depolarization generates fMCG signals that

differ substantially across sensors (figure 1), one potential difficulty is to select the QRS

time point that corresponds to a unique and well defined phase of ventricular depolarization,

e.g. the instant of maximum net cardiac current or the maximum cardiac vector magnitude

(MCVM). Using a single (e.g. most sensitive) channel or the mean global field (MGF)

across channels to determine the time point of highest signal amplitude remains vulnerable

to uncertainties introduced by the time-varying behavior of the cardiac vector orientation.

In addition, the ECD model is predicated on the assumption of a small size of the fetal heart

and a relatively large source-to-sensor distance. The validity of this assumption relates to the

spatial resolvability of dipolar sources and depends on factors such as the (unknown) source

parameters, or the (known) sensors distribution and configuration. In general, the number of

large singular values of the spatio-temporal data matrix indicates that a single fixed dipole

may not necessarily explain well the whole QRS data in fMCG recordings. Thus, in our

study we evaluate two alternative approaches, which offer more flexible frameworks for

fitting the fMCG data on the whole QRS interval: a rotating dipole and a multiple-dipole

model. A rotating dipole may be viewed as three collocated fixed dipoles with independent

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time series, such that its orientation varies with time. The multi-dipole model does not

require individual dipoles to be co-located. In this case, the cardiac vector is obtained from

the vector summation of all individual dipoles. Estimating the cardiac vector at any instant

in time allows for a direct identification of the MCVM latency on the vector magnitude

waveforms. Furthermore, the time-course of the vector magnitude would in principle enable

other cardiac measurements of interest such as the waves duration or time-amplitude

integrals.

2.2. Ultrasound recordings

Using realistic approximations of the volume conductors in fMCG requires the 3D

segmentation of the boundary surfaces of several fetal-maternal anatomical compartments

with different electrical conductivities (Stinstra and Peters 2002). The 3D images of the

fetus and maternal abdominal tissues must be acquired immediately before or after the

fMCG recording (to minimize the risk of fetal repositioning), limiting the use of MRI. For

the current study we used the so-called free-hand 3D ultrasound, in which a 3D digitizer is

attached to an ultrasound probe to record the positions and orientations of the probe

simultaneously with the B-scans, for subsequent co-registration and reconstruction of the

volumetric data. To achieve this task, we used the Stradwin 3D Ultrasound Acquisition and

Visualization software (Cambridge University, UK) with a GE Logiq-P5 ultrasound system

and a TrakSTAR positioning digitizer system (Ascension Technology Corp., USA) attached

to the ultrasound probe. The calibration of the probe-digitizer system was performed with an

ultrasound phantom. For the purpose of our simulation studies, 3D images of the fetus and

maternal abdominal tissues were acquired from 20 pregnant women. Informed consent was

obtained from each subject before participation in the experiment. The study was approved

by the Institutional Review Board of the University of Kansas Medical Center. Data from 10

subjects were recorded between 22 and 28 weeks of gestation, while the remaining 10

subjects were recorded between 32 and 36 weeks. For each case, series of images were

acquired by scanning the maternal abdominal surface from side to side and upper to lower

abdomen for a total number of ~ 5000–6000 frames.

Co-registration of the ultrasound images with the fMCG sensor array was accomplished

using three fiducial markers placed at non-collinear locations on the maternal abdomen

(right side, left side, and sternum). Each marker’s location was recorded by positioning the

center of the ultrasound probe at that location and recording the corresponding frames.

These frames were used to identify the markers locations on the volumetric image, but were

excluded from subsequent processing. The markers positions relative to the fMCG sensors

were determined in the biomagnetometer by localizing three coils placed at those locations.

2.3. Processing of volumetric ultrasound images

We developed a standard approach to model the fetal body compartment of the volume

conductor. The fetal head and trunk were approximated for each subject with a sphere and

an ellipsoid, respectively. This methodology relied on the manual identification of two

fiduciary points, and on performing two standard biometric ultrasound measurements: the

abdominal circumference (AC) and the head circumference (HC). Based on these

measurements, two parameters used to model the fetal body were derived: (1) the extent of

the small axis of the ellipsoid used to model the fetal trunk (using AC), and (2) the radius of

a sphere used to model the fetal head (using HC). The fiduciary points used to model the

fetal body were the fetal head center and the fetal coccyx, which were manually identified

on the recorded B-scans. The long axis of the ellipsoid (fetal trunk) was defined from the

head center to the coccyx, with the ellipsoid center set at the middle of this line. The fetal

body was subsequently modeled as the merged volume of the ellipsoid and sphere

determined as described above. The amniotic sac (AS) was approximated by an ellipsoid

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with size and position determined from manually identified fiduciary points. These points

defined the ends of the long axis of the sac ellipsoid in longitudinal cross-sections through

the fetal head and spinal cord, and the amniotic sac center (middle of the long axis), and

were used to estimate the length of the short axis of the ellipsoid (passing through the AS

center).

The ultrasound volumes were then resliced using planes oriented in the axial direction

relative to the maternal abdomen. Resliced frames along with the coordinates of the

fiduciary points were processed in Matlab to set the voxels of the fetal body and AS

compartments to predefined intensity levels. An additional dark gray layer has been added to

represent the outer maternal abdominal skin. The images were transformed to ANALYZE

format for a final processing step involving the segmentation of the different compartments

(performed with CURRY 5.0, Compumedics Neuroscan). All compartments were visually

inspected, and the AS compartment was locally adjusted using regional 3D dilation and/or

setting pass markers to improve its local shape and to ensure that the fetal body and AS

compartments did not intersect with each other. The triangularization of the boundary

surfaces was done using average triangle sides of 5 mm, 10 mm, and 20 mm for the fetal

body, AS and abdominal compartments, respectively. For late gestational data, a 5 mm

dilation of the fetus compartment was used to derive a vernix mesh with an average triangle

side of 5 mm. Source space points were created inside the middle part of the fetal trunk as a

regular grid with 3 mm average spacing and no points at less than 5 mm from the fetal body

surface. Surfaces (exemplified in figure 2) and source space points were used for further

processing in Matlab. Across subjects and models, the total number of vertices varied

between 2598 and 9828.

2.4. BEM feasibility experiments

The BEM algorithms used throughout this study and the specific implementation strategies

are described elsewhere (Ferguson et al. 1994, Mosher et al. 1999, Hämäläinen et al. 1993).

The numerical algorithms were first tested for accuracy in two feasibility studies. First

(feasibility experiment 1) we aimed to characterize the numerical errors bounds of the BEM

forward solutions. Since for a radial source in a spherically symmetric conductor the

primary and volume currents counterbalance each other’s contribution to the magnetic field,

geometries approaching a sphere in combination with radial sources can introduce larger

numerical errors. We defined a 4-layer spherically symmetric volume conductor, with radii

of 6, 6.2, 8 and 15 cm, and conductivities of 0.22, 2×10−6, 1.4, and 0.05 S/m (Oostendorp et

al. 1989a, Stinstra and Peters 2002) to replicate the approximate size and conductivity of the

fetal body, vernix, amniotic fluid and surrounding tissues, respectively. Each of the spherical

surfaces was triangulated using 642 triangles per surface. In a second approach, the thin

resistive layer was excluded leading to a 3-layer model of the volume conductor. For each

case, the analytical solution of the electric potential on the external surface, computed using

the Legendre polynomials method (Zhang 1995) with 24 terms in the truncated expansion,

and the analytical solution of the magnetic field (Sarvas 1987) for axial gradiometer sensors

were compared with the BEM forward solutions for dipoles oriented along the y and z axes,

respectively. The dipole position was varied on the x axis. The BEM solutions were

computed for both linear-collocation and linear-Galerkin implementations (Mosher et al.

1999). For the 4-layer model, the BEM solutions were computed with the Isolated Problem

Approach (IPA) (Hämäläinen and Sarvas 1989). Relative Difference Measures (RDM)

between the analytical and numerical solutions (figure 3b) indicated adequate accuracy for

the BEM linear-collocation approach. The linear-Galerkin approach provided slightly

smaller RDMs over a range of source eccentricities, but the computational time was

significantly increased. Since the linear-collocation provided accurate solutions and efficient

computational times, it was considered optimal for the current studies.

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In a second series of simulations (experiment 2) we assessed the errors that can be

introduced by variations in the thickness of the vernix. Volume conductors that were not

spherically symmetric were obtained by positioning the inner compartments in figure 3a

(representing the AS and fetus covered by a thin resistive layer) at different locations inside

the outermost spherical compartment. The thickness of the resistive layer was varied at 0.3,

1, 2, 3, 4 and 5 mm. We used the BEM solution (linear-collocation approach with IPA)

obtained for 2 mm thickness as a reference, and we computed the RDMs between this

reference and the solutions obtained for other thickness values. The results (figure 3c) show

that thickness variations of the resistive layer lead to very small differences in the magnetic

field, and agree with previous reports (Stinstra and Peters 2002) indicating that volume

currents are in this case largely confined to the fetus compartment of the volume conductor.

Hence, throughout the simulations in this study, we considered a uniform, 5 mm thick vernix

layer. This choice is motivated by the fact that the inner and outer surfaces of the thin layer

should in principle be tessellated using triangles of a size comparable to the thickness of the

layer, which increases considerably the computational burden for very thin layers.

2.5. Simulation experiments

Simulation experiments were performed for the sensor array of the CTF fetal

biomagnetometer with 83 axial gradiometers (with 5 cm distance between the pick-up coils

of each sensor). The magnetic field was simulated in two conditions. First, we defined 3-

compartment volume conductors for the whole set of 20 subjects (using both middle and late

gestation setups). We assigned conductivities of 0.22 S/m to the fetus, 1.4 S/m to the

amniotic fluid, and 0.05 S/m to the maternal abdomen (Stinstra and Peters 2002). In a

second condition, setups from late gestation (10 subjects) were also used in simulations with

4-compartment volume conductors. In this case, a fourth compartment (5 mm thick,

conductivity of 2 × 10−6 S/m) covered the fetus, to test for potential differences introduced

by the presence of vernix.

The general principle of our simulation approach is that the cardiac electrical activity can be

described as a current distribution of elementary current sources that summate into a few

effective dipoles, which can be reconstructed from multi-channel fMCG measurements. The

number of effective dipoles depends on factors such as the position and orientation of the

heart with respect to the sensor array as well as the heart size, and therefore can show

significant inter-subject variability. Since the number of effective dipoles is unknown a

priori, we modeled the heart using a large number of elementary currents, and we tested the

reconstruction algorithms for retrieving the data-driven or subject-dependent (small) number

of effective dipoles. Thus, to simulate the forward magnetic field, we used a fetal heart

source model that approximates the propagation of the depolarization wave front through the

ventricular walls during the QRS interval (Popescu et al. 2006). The model uses a modulated

profile cylindrical surface to seed elementary current dipoles around the cardiac axis. For

middle gestation, the radius of the largest heart circumference was R=0.75 cm, and the

ventricular dimension along the heart axis was Z=1.5 cm. For late gestation, we selected

R=1.5 cm, and Z=3.0 cm. The peak current latencies were parameterized to generate a

traveling wave progressing from the apex to the ventricles’ upper part. A total of 480 time

samples were generated with 1200 Hz sampling rate. The vector magnitude (VM) was

derived by vector summation of all currents at each time point, and the currents strengths

were scaled such that the peak VM was 853 nAm and 1706 nAm for middle and

lategestation, respectively. The heart source model was positioned at the fetal heart location

identified on ultrasound images, and the forward magnetic field was computed at the sensor

positions using the BEM linear-collocation approach. White Gaussian noise(RMS=3 fT) was

added to the data.

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2.6. Source reconstruction and evaluation of volume conductor effects

The source reconstruction results were evaluated for two algorithms: (1) multiple dipole

search using R-MUSIC (Mosher and Leahy 1999), and (2) fitting of a single rotating-dipole

using least-squares source-space scanning (RDSS) to find the location that minimizes the

relative residual deviation, ε, between the measured (B) and estimated data (Ri) on the QRS

interval:

(1)

where Li is the m×3 location-wise gain matrix, and (+) denotes the pseudo-inverse of a

matrix. Note that in a spherically symmetric volume conductor, radial currents do not

produce any magnetic field outside the volume conductor (Sarvas 1987), and in that case Li

is the m×2 matrix formed by the gain column vectors along the two local tangential

directions.

The source-space scanning approach for fitting a rotating dipole has been preferred to

alternative nonlinear minimization algorithms, since the latter can sometimes provide

erroneous results due to their vulnerability to getting trapped in local minima (Popescu et al.

2006). The R-MUSIC algorithm searches over a 3D grid to find the locations for which a

linear combination of columns of the gain matrix Li projects entirely onto the signal

subspace of the spatio-temporal data matrix. This is achieved using subspace correlation

metrics (Mosher and Leahy 1999). Once a first dipole is identified, the algorithm is repeated

to search for other sources which explain the remaining data. The rank of the signal

subspace was set to 5 for all our experiments. Also, we used a threshold value of 0.95 for the

subspace correlation to represent an adequate correlation of a source. The recursive search

stops when the number of found dipoles equals the signal subspace rank, or when no more

sources satisfying the threshold subspace correlation are found. When the algorithm stops,

the dipoles magnitudes are estimated by multiplying the data matrix by the pseudo-inverse

of the dipoles gains matrix.

The MUSIC algorithm appears suited for the problem at hand because it provides a flexible

framework to fit multiple dipole sources, and the optimal effective dipoles do not

necessarily need to be co-located. Another difference between fitting a rotating dipole

(RDSS) versus MUSIC arises from the optimization scheme and the implicit cost function.

Minimizing the residual variance in RDSS inherently assigns a greater weight to the

channels with maximal amplitude (thus selecting the solution which provides good fitting

for those channels) in contrast to MUSIC, which uses subspace correlations.

Each reconstruction scheme was applied within three scenarios characterized by a different

choice of the volume conductor model and/or source space selection. First (scenario 1), we

evaluated the reconstruction strategies for the case when perfect knowledge of the volume

conductor is available and the source space is confined to the middle part of the fetal trunk

volume. This evaluation is necessary to characterize the errors introduced by the limited

number of sensors, presence of noise in the data, and intrinsic properties of the

reconstruction algorithms. Second (scenario 2), a sphere fitted to the sensors was used as a

simplified volume conductor model. This scenario assumes also that minimal information

about the fetal body position is available, such as an approximate fetal heart-to-sensors

distance (as proposed for example by Horigome et al. 2001). To account for this, the source

space was selected to coincide with the grid in scenario 1, i.e. confined to the middle part of

the fetal trunk. The gains were computed using the Sarvas equations for the forward

magnetic field (Sarvas 1987). Finally, scenario 3 assumed the same spherical volume

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conductor model, with a source space that spanned a large (14×25×18 cm) rectangular

volume in the upper hemisphere of the volume conductor, to replicate conditions in which

anatomical information is unavailable.

The performance of the two reconstruction schemes was evaluated using the estimated

cardiac vector magnitude(VM̃). For R-MUSIC, VM̃ is the net current strength derived at

each time point by vector summation of the reconstructed dipoles. Amplitude errors were

assessed by the relative error of the peak VM̃ (reVM) with respect to the known VM peak

amplitude. We use the convention that positive and negative errors indicate VM

underestimation and overestimation, respectively. The localization performance was

evaluated by the Euclidian distance between the heart’s geometrical center and the average

location of the retrieved dipoles (R-MUSIC) or the location of the best-fit rotating dipole

(RDSS). The relative residual deviation (1) was used to characterize the goodness-of-fit.

3. RESULTS

3.1. Three-Compartment Volume Conductors

Fig. 4 exemplifies the estimated VMs (panels a, b) and summarizes the results for 3-

compartment volume conductor models (panels c, d). For perfect knowledge of the volume

conductor (scenario 1), the R-MUSIC and RDDS schemes provide good VM estimates

(overall mean errors of 3.0±9.5% and 1.0±11.5%, respectively) and small localization errors

(overall mean values of 0.9±0.4 cm and 1.0±0.5 cm, respectively), irrespective of the

gestational age (early vs. late). reVM for both R-MUSIC and RDSS passed D’Augustino-

Pearson omnibus normality tests (K2s<023, ps>0.89), and subsequent two-tailed t-tests

indicated that mean reVM values were not significantly different than zero (ts<1.4,

ps>0.18). To test if any reconstruction scheme performs better for this scenario, we

conducted a 2×2 ANOVA with reVM as a dependent variable and independent factors

reconstruction algorithm(with repeated measures R-MUSIC vs. RDSS) and gestational age

(early vs. late). The test showed no significant main effects or interaction (Fs<0.32,

ps>0.58).

The use of simplified volume conductors led to relatively large reVMs (fig. 4, c) and higher

localization errors (fig. 4, d) for each reconstruction scheme and source space selection.

reVMs were positive for all subjects and tested scenarios (2 and 3), indicating a clear trend

of underestimating the true VM. Similar 2×2 ANOVAs were conducted separately for

scenarios 2 and 3, indicating a significant main effect of reconstruction algorithm on the

reVMin each case (scenario 2: F=9.6, p=0.004; scenario 3: F=5.3, p=0.03), but no

significant main effect of gestational age or interactions. The optimal reconstruction

algorithm for these scenarios (i.e. RDSS, which provided lower reVMs, as seen in fig. 4c)

has been further tested to see if the selection of the source space affects its performance.

This was done using ANOVAs with independent factors scenario (with repeated measures

scenario 2 vs. scenario 3) and gestational age (early vs. late). These tests indicated a

significant main effect of scenario( F=11.2, p=0.003) on the localization error, but no

significant main effects on reVM( Fs<1.15, ps>0.29). Thus, high reVMs are largely

determined by the mismatch between the gains in realistic and simplified models, and thus,

they are not significantly improved by confining of the source space to the fetal trunk.

The residual deviation increases when using simplified volume conductors. In scenario 1, R-

MUSIC and RDSS solutions explained well the signals (mean residual deviation of

7.6±6.0% and 5.6±3.8%%, respectively, pooled over gestational ages). For simplified

conductors however, a significant part of the signals remains unexplained: the mean residual

deviation was 48.4±30.6% in scenario 2 and 40.9±28.2% in scenario 3 for R-MUSIC, and

31.1 ±13.8% in scenario 2 and 25.8±11.3% in scenario 3 for RDSS.

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3.2. Impact of vernix caseosa

Figure 4e, f compares the results for 3- vs. 4-compartment volume conductors. Since both

models are derived from the same group of subjects in late gestation, the differences reflect

the role of the insulating layer of vernix. This layer can change the magnetic field

distribution and typically lowers the mean global field across sensors (Fig. 5).

A 2×2 ANOVA was conducted for scenario 1, with the dependent variable reVM and

independent factors reconstruction algorithm and volume conductor model (with repeated

measures for 3- vs. 4-compartment). The test showed a significant main effect of volume

conductor model (F=6.4, p=0.02), indicating a trend for both R-MUSIC and RDSS to

retrieve less accurate solutions for the 4-compartment data. Additional insights have

revealed that the mean accuracy in this case was however affected by the presence of one

large misfit in the retrieved solutions for each scheme. In one case, RDSS retrieved a

solution farther away from the simulated heart location, associated with a high rEVM error

(−59%). Likewise, R-MUSIC has retrieved a large rEVM error in one case (−61%). Similar

ANOVAs for scenario 2 and 3 showed no significant effects or interactions (Fs<0.65,

ps>0.64). Thus, the presence of a uniform layer of vernix does not change the effect of

imperfect volume conductor modeling. ANOVAs for the localization error showed no

significant effects in any scenario (Fs<1.6, ps>0.22).

3.3. Assessment on real fMCG data

The use of realistic approximations of the volume conductor was tested in a preliminary

evaluation with real fMCG measurements collected in mid-gestation, to avoid uncertainties

about the presence of vernix. Fifteen pregnant women (gestational age 23 to 25 weeks)

undergone a continuous 4 minute fMCG recording (1200 Hz sampling rate, 0.5–200 Hz

band-pass). The signals were filtered using ICA (Mantini et al. 2006) to segregate the

contribution of the fetal cardiac source. Data from 4 subjects were discarded due to gross

fetal body movements during the recording, identified as non-stationarities of the QRS

amplitude in the sensor signals, associated with shifts of the fetal cardiac source activity

from one independent component to another. For the remaining subjects without observable

fetal body movement, the QRS peak was detected by an automatic algorithm, and the

averaged cardiac beat was estimated and used for source reconstruction. Free-hand

ultrasound images acquired immediately after the fMCG recordings were processed as

explained in Methods to derive 3-layer realistic approximations of the volume conductor.

Figure 6 exemplifies the averaged fMCG data and the reconstructed cardiac vectors using R-

MUSIC. Out of the 11 subjects, a good fitting (>90% explained data variance) was obtained

in 5 cases, a moderately good fitting (between 75% to 90% explained data variance) was

obtained in 4 cases, and a low-quality fit (<75% explained data variance) was obtained in 2

cases. The mean cardiac vector peak-magnitude was 747±259 nAm across the 5 subjects

with good fitting, and 872±319 nAm across all 9 subjects with good and moderately good

fitting. Simplified spherical models were also used for comparison: with this strategy, the

mean peak-magnitude obtained across the same 9 subjects was 227±103 nAm, and the mean

explained variance was 52±0.25%. These results allow making several observations. First,

the peak-magnitude estimators obtained with simplified volume conductors generally agree

with the ones reported by previous fMCG (Leeuwen et al. 2004, Horigome et al. 2001) and

fECG (Oostendorp et al. 1989b) studies using similar reconstruction strategies. Second, the

refinement of the volume conductor to include non-homogeneities introduced by the fetal

body and amniotic sac increases the estimated values of the vector peak-magnitude and

improves the goodness-of-fit for most subjects. Lastly, the values of the vector peak-

magnitude in these cases are closer to those predicted by studies on vertebrate animals,

which indicate an expected mean value of ~750 nAm for this range of gestational ages

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(Nelson et al. 1975, Alexander et al. 1998). Thus, although the magnetic field is apparently

less sensitive to the details of the volume conductor than the electric potential, these results

indicate that the refinement of the volume conductor model for mid-gestation to account for

inhomogeneities introduced by the fetal body and amniotic sac apparently improves the

vector magnitude estimation in fMCG in a similar manner as previously reported for fECG

(Oostendorp et al.1989b).

4. DISCUSSION

A reliable methodology for the estimation of the cardiac signal strength from fMCG data

can extend its clinical usefulness by allowing studies of fetal cardiac electrophysiology in

conditions associated with increased risk of cardiac hypertrophy. Our study used realistic

approximations of the volume conductor for a relatively large number of subjects and

demonstrated the limitations of using simplified models to achieve this objective. These

findings are similar to observations made by earlier studies using fECG (Oostendorp et al.

1989b), and challenge the view that magnetic recordings would be less sensitive to the

details of the volume conductor to such extent that they would allow the use of simplified

models in experimental applications.

We showed that the simplified volume conductor models lead to significant and consistent

underestimation of the cardiac vector, with a mean retrieved peak amplitude ranging from

20% to 50% across the scenarios tested in our study, irrespective of factors like the selection

of the source space or reconstruction scheme. These results, as well as our preliminary

findings with real fMCG measurements, can largely explain an apparent discrepancy

between the vector peak-magnitude reported previously by fMCG studies using simplified

models (Leeuwen et al. 2004, Horigome et al. 2001) and the values predicted by studies on

vertebrate animals (Nelsonet al. 1975).

For late gestation, the above observations were made using simulations in two limit cases,

i.e. in the complete absence or presence of a uniform layer of vernix. In real experiments, it

is likely that intermediate states (i.e. patches of vernix covering partially the fetal body) may

be encountered after 28 weeks of gestation. The presence of holes in the vernix has been

shown to change the amplitude and distribution of the electric potential on the maternal

abdomen, as well as the forward magnetic field (Oostendorp et al. 1989b, Stinstra and Peters

2002). The errors in the fMCG inverse solution for simplified models that approximate

conductors with nested compartments and uniform layers of vernix indicate that similar or

larger inaccuracies would be also obtained for such spherically-symmetric approximations

of more complex conductors that include additional inhomogeneities introduced by the

presence of holes in the vernix. Since there is no reliable way to gain evidence for the

presence of vernix in practical applications, or to determine the existence and size of its

holes, efforts to define and use realistic approximations of the volume conductors will most

likely find applications in a “window of opportunity” between ~22 weeks of gestation (when

reliable fMCG recordings can be obtained) to ~28 weeks (when uncertainties about the

presence of vernix would start to impact the modeling and its success rate).

From this perspective, fMCG measurements have several advantages compared to the early

fECG studies: (1) the availability of large arrays systems with dense sensor coverage, (2) the

availability of new computational methods for eliminating interferences from maternal

cardiac and other artifacts (e.g. ICA), which allow more accurate estimation of the averaged

fetal cardiac signals, and (3) recent advances in free-hand ultrasound recording and

processing, which can facilitate the definition of realistic approximations of the volume

conductor. Whereas the use of simplified volume conductors offers the advantage that

inverse calculations do not need to account explicitly for the effect of volume currents, the

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BEM modeling implies (1) the availability of free-hand ultrasound systems to collect feto-

maternal anatomical information, (2) additional experimental time for ultrasound recordings,

and (3) an increased computational burden. Furthermore, BEM modeling requires the 3D

segmentation of the compartments with different electrical conductivity from ultrasound

images. In this study, we used a standardized approach for modeling the different volume

conductor compartments, which is robust to inherent inter-individual variations in

ultrasound image quality, but it requires manual definition of several fiduciary points, and

local manual correction of the amniotic sac boundaries. A fully automatic segmentation of

the different compartments of the volume conductor remains contingent to additional

development of the image processing techniques. Our preliminary results obtained for real

fMCG data in mid-gestation show promise, but additional studies are necessary to address

the sensitivity of the source estimators to slight variations in the geometry of the individual

volume conductor compartments, within the range of errors that are inherent to the proposed

methodology. Definitive answers regarding the sensitivity and specificity of such an

approach for the detection of hypertrophy based on the cardiac vector strength await also

further studies in fetuses with cardiac hypertrophy confirmed by m-mode cardiac ultrasound

examination.

Acknowledgments

This work was supported in part by grant R21EB006776 from the National Institute of Biomedical Imaging and

Bioengineering. The Hoglund Brain Imaging Center at the University of Kansas Medical Center is supported by a

generous donation from Forrest and Sally Hoglund. Authors would like to thank the study coordinator JoAnn

Lierman, RNC, PhD, and the anonymous reviewers who provided valuable suggestions for improving the

manuscript.

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Figure 1.

fMCG signals exemplifying an averaged QRS complex from a 36 week old fetus.

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Figure 2.

(a) Exemplification of boundary surfaces modeling for a 3-compartment volume conductor.

(b) Top view of the outer abdominal surface co-registered with the biomagnetic sensor

array.

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Figure 3.

(a) Setup of the simulations in feasibility experiment 1. (b) RDMs for different orientations

of a dipole with strength of 300 nAm, and for the 3- and 4-shells models. Due to spherical

symmetry, RDMs for the electric potential are shown only for dipoles orientated along the y-

axis. Results are shown for the linear-collocation approach. (c) Results from feasibility

experiment 2. RDMs are shown for a four-shells volume conductor, with the center of the

third sphere shifted with 3 cm on both y and z axes with respect to the center of the

outermost sphere, and the center of the first and second spheres shifted with 1 cm on the x

axis relative to the center of the third sphere. RDMs are computed relative to the solution

obtained with 2mm thickness.

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