Cramer-Rao Bound for Dipole Source Localization in Infants Using

Realistic Geometry

Aleksandar Jeremic

, D. Nikolic

, G. Djuricic

, N. Milcanovic

and Z. Jokovic

Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada

University Children’s Hospital, Faculty of Medicine, University of Belgrade, Serbia

Department of Radiology, University Children’s Hospital, Belgrade, School of Medicine, University of Belgrade, Serbia

Keywords:

Source Localization, Electroencephalography, Inverse Models.

Abstract:

Source localization of electrical activity in newborn infants is important from two standpoints. From an aca-

demic standpoint such insights can enable better understanding of brain development and from clinical stand-

point localization of electrical activity can identify regions of the brain with higher than usual activity and pos-

sibly improve possible treatment outcomes. The electrical activity and the corresponding electroencephalog-

raphy (EEG) measurements are dependant on electrical properties of brain and skull tissue i.e. corresponding

conductivities and geometry. In this paper we investigate effects of realistic geometry in newborn infants by

accounting for soft spots (fontanels) that are present in newborn infants. These structures have larger conduc-

tivity than regular bone tissue and hence the estimation accuracy can potentially be improved by optimally

positioning EEG sensors on the surface of the skull. We generate forward model using realistic geometry and

ﬁnite-element model generated by COMSOL. We utilize simpliﬁed source model consisting of single dipole

source and calculate corresponding Cramer-Rao bound as a function of source intensity and locations.

1 INTRODUCTION

Neonatal convulsions are one of the most common

emergency neurological events in the early period af-

ter birth with the frequency of 1.5 to 3 in 1000 live

births (Volpe, 2001). Consequently, neonatal inten-

sive care units (NICU) continuously monitor electri-

cal activity of preterm infants for both short-term and

long-term interventions and/or treatments (Shellhaas

and Clancy, 2007) These techniques commonly uti-

lize only detection algorithms whose main purpose

is to detect events in electroencephalography (EEG)

recordings. In addition to those, estimation tech-

niques can potentially provide insight into the brain

development and indicate regions of higher convul-

sion rate. The estimation of electrical activity of the

brain in adults has been a subject of considerable

research interest in adults (Asadzadeh et al., 2020).

Most of the existing solutions utilize combination of

EEG (excellent temporal resolution and poor spatial

resolution) as a source of electrical activity informa-

tion and magnetic resonance imaging (MRI, excel-

lent spatial resolution and poor temporal resolution)

as a source of geometry information and combine

them in so called inverse models that are then used

in order to estimate the unknown parameters (usu-

ally some type of constrained spatial source models

such as distributed dipoles). In infants, however, ac-

curately describing the anatomy of the head remains

a challenge due to the complexity of the infant skull

from the electromagnetic point of view. The most sig-

niﬁcant anatomical difference with respect to adult

anatomy in addition to volume is the existence of

fontanels. soft tissue between incompletely formed

cranial bones (Cornette et al., 2002).

To this purpose in this paper we investigate the ef-

fect of the fontanelle structure on the estimation accu-

racy by evaluating Cramer-Rao lower bound (CRLB)

for a realistic geometry of the infant brain that is the

lowest attainable variance that can be achieved using

unbiased estimators. The effect of fontanels on EEG

ﬁeld has been studied in several recently published

reports e.g. (Gargiulo and Belﬁore, 2015) using for-

ward models. On the other hand, source localization

requires inverse models and consequently estimation

of the source parameters such as location. The CRLB

is a commonly used indicator of how far any proposed

inverse/estimation solution is from the theoretically

best possible performance. To this purpose our re-

sults can be used for benchmarking subsequent ma-

Jeremic, A., Nikolic, D., Djuricic, G., Milcanovic, N. and Jokovic, Z.

Cramer-Rao Bound for Dipole Source Localization in Infants Using Realistic Geometry.

DOI: 10.5220/0012470200003657

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2024) - Volume 1, pages 807-810

ISBN: 978-989-758-688-0; ISSN: 2184-4305

807

chine learning (semi-supervised and/or unsupervised)

solutions once the sufﬁciently large training datasets

are obtained.

To model the electrical activity of the brain we use

two dipole structure: low power dipoles that model

the background noise/activity of the brain and sin-

gle high power dipole whose location we aim to esti-

mate. We reiterate that our main goal is not to develop

the accurate model of electrical activity but rather to

show the accuracy dependance on the modelling of

fontanelles. To this purpose we believe that a sim-

pliﬁed model is a good preliminary approach to in-

vestigate numerically to which extent the fontanelle

structure affects the estimation accuracy. We calcu-

late the corresponding ﬁeld using AC and Medical

Imaging toolboxes in COMSOL software as well as

3D slicer for the infant brain segmentation. Using the

model predicted values we estimate the corresponding

parameters using numerical optimization techniques

discussed in Section 2 and calculate the correspond-

ing CRLB . In Section 3 we present our results for

various parameters. In Section 4 we present conclu-

sions and directions for future research.

2 MATHEMATICAL MODEL

We model the electrical ﬁeld on the surface of the

head using a volume conductor approach (Malmuvio,

1995) in which the electrical activity in the cortex

is modelled using generic current density represen-

tation J(⃗r,t. The electrical ﬁeld is then obtained by

solving Maxwell equation and the corresponding so-

lution is represented by well known Geselowitz equa-

tion (Gulrajani, 1998) that using a piecewise homo-

geneous head model consisting of the multiple closed

surfaces (skull, brain, etc.) Due to the fact that the

geometry is inherently irregular the solution of these

equations can only be obtained by using a numerical

method such as ﬁnite-element method. We use real-

istic geometry of the 9 months old infant obtained at

The University Children Hospital, University of Bel-

grade, Serbia. MRI images consisted of 110 axial MR

slices with 256x256 size and ﬁeld of view of 240 mm.

The segmentation and meshing was done using soft-

ware packages Slicer and Meshlabs that were then im-

ported as STL ﬁles in COMSOL ﬁnite-element solver.

Since our main goal is to investigate the effect of

fontanel on the accuracy of inverse model we propose

to use simpliﬁed forward model i.e. scalp EEG gen-

erator. In (Gargiulo and Belﬁore, 2015) the authors

utilized large scale computational model using large

number of dipoles to simulate EEG signal measured

on the scalp. The inverse models have to rely on much

Figure 1: MRI of Infant Head.

Figure 2: Fontanel Structure.

smaller number of parameters due to the fact the num-

ber of EEG electrodes that can be placed on infant

heads is limited due to the small area. Furthermore,

EEG signals from spatially close electrodes is known

to be highly correlated and thus of limited use in in-

verse EEG models.

To this purpose we propose to use the following

model: a) the regular brain activity is modelled us-

ing 256 Gaussian dipoles placed in the cortical layer

that represent background noise in the EEG signal

measured on the scalp distributed in the circular pat-

tern under the centre of fontanel and b) single cur-

rent dipole model with high power current dipole de-

scribed by three parameters (J

Using the ﬁnite-element solver in we calculated

the corresponding EM ﬁeld so that the measurement

model is then given by

i j

⃗

f (⃗r

⃗

θ) + e

i j

(1)

where y

i j

represents electric potential on the scalp

measured on the ith EEG sensor at timet

, f repre-

sents the solution of FE solver at location⃗r

and time

and e

i j

represents the measurements noise/residual

model error. The parameters of the model

⃗

θ are de-

ﬁned by dipole moment and location that are treated

as unknown parameters. As a preliminary approach

we assume that the measurement noise is zero-mean

Gaussian and spatiotemporally uncorrelated. We then

estimate the unknown parameters using maximum

likelihood estimation which in this scenario results

BIOSIGNALS 2024 - 17th International Conference on Bio-inspired Systems and Signal Processing

808

Figure 3: Random instance of dipole moments.

in least-squares estimate by minimizing the error be-

tween measured (simulated and model predicted val-

ues).

⃗

= argmin = ∥⃗y −

⃗

f ∥ (2)

where ⃗y is the lumped vector of all the measurements

and

⃗

f is lumped vector of all the model predicted val-

ues. The measurements consist of n spatial measure-

ments and m temporal measurements assuming that

the dipole source is not changing with time.

To evaluate the performance of the proposed al-

gorithms we calculate Cramer-Rao bound which rep-

resents the lower bound on the variance of unbiased

estimators. This bound is a theoretical limit of the

lowest possible variance of the unbiased estimator

and hence it is desirable to have the smallest pos-

sible value as it is a value to which the variance of

the proposed estimator will converge if the number of

measurements is sufﬁciently high. The Cramer-Rao

bound is calculated using the Fischer information ma-

trix given by

i j

(

⃗

θ) = −E

∂lnp

(⃗y,

⃗

θ)

∂θ

∂lnp

(⃗y,

⃗

θ)

∂θ

(3)

where p(⃗y,

⃗

θ) is the probability density function of the

measurement vector calculated using FEM solver for

a given

⃗

θ. The analytical expression for the Gaus-

sian case can be obtained following (Kay, 1993) using

the Gaussian distribution with a nonlinear parametric

mean.

3 NUMERICAL RESULTS

To simulate the background EEG signal we use cir-

cular grid of 32x32 dipoles with randomly generated

dipole intensities so that in the fontanel region the

background noise dipole density has expected value

of 20µA/cm

and standard deviation of 100µA

/cm

In the outer region of the cortex the background EEG

dipoles have expected value of 10µA/cm

and stan-

dard deviation of 100µA

/cm

. For the source we

are trying to estimate/localize we use a dipole with

density of 40µA/cm

. Following the values used in

(Gargiulo and Belﬁore, 2015) we vary the conductiv-

ity of the fontanel region from 0.01 to 1.51 S/m to

evaluate its effect on the Cramer-Rao bound.

Since in our model the measurement noise is as-

sumed to be Gaussian the Fischer information matrix

and consequently CRLB depend only on the gradient

of the model predicted solution

⃗

f with respect to the

unknown parameters

⃗

θ. We calculate the correspond-

ing gradients using ﬁnite difference approximation in

the parameter space. Note that if parameters of in-

terest can be modelled as random variables the cal-

culation of CRLB would require Monte-Carlo simu-

lations. We use the multichannel EEG sensor model

consisting of 4 sensors distributed on the fontanel pe-

riphery. The distance of the source is measured with

respect to the centre of the fontanel structure as illus-

trated in Figure 4.

Figure 4: EEG sensor locations.

1.5

dist

0.5

0.0

Figure 5: Source localization CRB.

In Figure 5 we illustrate the CRB as a function

of distance of the high activity dipole source from

the centre of the fontanel and conductivity. As ex-

pected for larger conductivity values we obtain the

lower CRLB which improves our ability to estimate

the source location accurately. In Figure 6 we illus-

Cramer-Rao Bound for Dipole Source Localization in Infants Using Realistic Geometry

809

50 100 150 200 250 300

−3

−2

−1

Number of Measurements

Variances

CRB cond=0.1

LS estimate

CRB Cond=1

Figure 6: Source localization CRB as a function of time.

trate the CRLB and mean square error as a function

of number of measurements. For illustration purposes

we calculate two CRLB for different conductivity val-

ues of 0.1 and 1 and mean square error of LS estima-

tor using 1000 runs. Note that in this example we as-

sume that the dipole intensity is ﬁxed during the mea-

surement interval which may not be valid in realistic

scenario due to the fact that EEG activity epochs are

quite dynamic with respect to time. As expected the

estimator variance decreases with the number of mea-

surements and is expected to asymptotically approach

CRB.

4 CONCLUSIONS

We proposed a computational framework for calculat-

ing theoretical bound for localization of single dipole

source in an infant head using single dipole model

and realistic geometry. The proposed framework en-

ables us to calculate the lowest possible variance that

can be obtained for a given geometry. Our prelim-

inary results indicate that the ability to localize sin-

gle dipole depends on the conductivity as well as ge-

ometry of the subject as well as the conductivity of

fontanel. Therefore an effort should be place on im-

proving our ability to estimate the conductivity jointly

with source localization. In addition, an effort should

be made to investigate the accuracy of the model with

respect to signal-to-noise ratio. Although this analy-

sis is important it is left for future studies as the CRB

of power is signiﬁcantly dependent on the conductiv-

ity and thus may require improved knowledge on con-

ductivity values of the fontanels. Furthermore, based

on the aforementioned CRB studies the performance

can be signiﬁcantly increased if we position the EEG

sensors in a such way to minimize CRLB at the largest

possible number of possible source (regions of high

activity) instances. Our results indicate that an ade-

quate localization error can be achieved using inverse

EM modelling approach although it may require ad-

vanced signal processing algorithms.

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