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ISBN 978-952-60-5710-1 ISBN 978-952-60-5711-8 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 ISSN 1799-4942 (pdf) Aalto University School of Science Department of Biomedical Engineering and Computational Science

BUSINESS + ECONOMY ART + DESIGN + ARCHITECTURE SCIENCE + TECHNOLOGY CROSSOVER DOCTORAL DISSERTATIONS

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Modern biomagnetic devices measure the magnetic field generated by the human brain at hundreds of locations around the head. The measured field can be efficiently described by a multipole expansion. The expansion has several applications, such as interference suppression and compensation of subject movements. In this work, we clarify the requirements of the multipole-based approach. We also propose new sensor array geometries that would significantly enhance the performance of the method. The feasibility of the ideas is demonstrated by a novel prototype device. The significance of the results is better interference suppression and increased robustness, which is especially important in clinical applications of biomagnetism.

Jussi Nurm

inen T

he magnetostatic m

ultipole expansion in biomagnetism

: applications and implications

Aalto

Unive

rsity

Department of Biomedical Engineering and Computational Science

The magnetostatic multipole expansion in biomagnetism: applications and implications

Jussi Nurminen

DOCTORAL DISSERTATIONS

Aalto University publication series DOCTORAL DISSERTATIONS 79/2014


Jussi Nurminen

A doctoral dissertation completed for the degree of Doctor of Science (Technology) to be defended, with the permission of the Aalto University School of Science, at a public examination held at auditorium F239 of the school on 10 June 2014 at 12.

Aalto University School of Science Dept. of Biomedical Engineering and Computational Science

Supervising professor Prof. Risto Ilmoniemi Thesis advisor Dr. Samu Taulu Preliminary examiners Dr. John Mosher, Cleveland Clinic, Ohio, USA Prof. Ville Kolehmainen, University of Eastern Finland Opponent Prof. Jens Haueisen, Technische Universität Ilmenau, Germany

Aalto University publication series DOCTORAL DISSERTATIONS 79/2014 © Jussi Nurminen ISBN 978-952-60-5710-1 ISBN 978-952-60-5711-8 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 (printed) ISSN 1799-4942 (pdf) http://urn.fi/URN:ISBN:978-952-60-5711-8 Unigrafia Oy Helsinki 2014 Finland

Abstract Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi

Author Jussi Nurminen Name of the doctoral dissertation The magnetostatic multipole expansion in biomagnetism: applications and implications Publisher School of Science Unit Dept. of Biomedical Engineering and Computational Science

Series Aalto University publication series DOCTORAL DISSERTATIONS 79/2014

Field of research Biomedical engineering

Manuscript submitted 11 November 2013 Date of the defence 10 June 2014

Permission to publish granted (date) 19 May 2014 Language English

Monograph Article dissertation (summary + original articles)

Abstract Magnetoencephalography (MEG) is a noninvasive functional imaging method, based on

measuring neuronally generated magnetic fields outside the head. MEG has a very high temporal resolution, making it possible to track brain events on the millisecond level. Its main advantage over electroencephalography (EEG) is the simplicity of inverse modelling, enabling more accurate source localization with fewer assumptions. Previous research has shown that the extracranial magnetic fields can be completely described in terms of a magnetostatic multipole expansion derived from Laplace's equation. In recent years, the multipole expansion has found several applications in MEG, including efficient interference suppression and movement compensation; however, this approach places rather high requirements on the MEG device, demanding comprehensive spatial sampling and accurate calibration. In this thesis, we clarify the requirements the multipole-based approach places on the MEG device, in terms of calibration accuracy and other parameters. We propose novel sensor array geometries that would significantly enhance the performance of the method. We also demonstrate the effectiveness of this model by building a prototype device with diverse sensor orientations. Our results indicate improved efficiency in interference suppression as well as enhanced signal-to-noise ratio. The new developments also increase the robustness of MEG, which is particularly significant for clinical applications. The results are directly applicable to other biomagnetic measurements, such as magnetocardiography.

Keywords Magnetoencephalography, MEG, brain imaging, signal space separation, SSS, biomagnetic instrumentation, sensor arrays, multichannel arrays

ISBN (printed) 978-952-60-5710-1 ISBN (pdf) 978-952-60-5711-8

ISSN-L 1799-4934 ISSN (printed) 1799-4934 ISSN (pdf) 1799-4942

Location of publisher Helsinki Location of printing Helsinki Year 2014

Pages 126 urn http://urn.fi/URN:ISBN:978-952-60-5711-8

Tiivistelmä Aalto-yliopisto, PL 11000, 00076 Aalto www.aalto.fi

Tekijä Jussi Nurminen Väitöskirjan nimi Magnetostaattinen multipolimenetelmä biomagnetismissa: sovelluksia ja johtopäätöksiä Julkaisija Perustieteiden korkeakoulu Yksikkö Lääketieteellisen tekniikan ja laskennallisen tieteen laitos

Sarja Aalto University publication series DOCTORAL DISSERTATIONS 79/2014

Tutkimusala Lääketieteellinen tekniikka

Käsikirjoituksen pvm 11.11.2013 Väitöspäivä 10.06.2014

Julkaisuluvan myöntämispäivä 19.05.2014 Kieli Englanti

Monografia Yhdistelmäväitöskirja (yhteenveto-osa + erillisartikkelit)

Tiivistelmä Magnetoenkefalografia (MEG) on ei-invasiivinen funktionaalinen kuvantamismenetelmä,

jossa mitataan aivojen synnyttämiä magneettikenttiä pään ulkopuolelta. MEG:n etuja ovat korkea aikaresoluutio esim. funktionaaliseen magneettiresonanssikuvaukseen verrattuna, ja toisaalta yksinkertaisempi ja tarkempi lähdemallinnus EEG:hen verrattuna. Aiemmin on osoitettu, että pään ulkoinen magneettikenttä voidaan täysin kuvata magnetostaattisen multipolikehitelmän avulla. Nykyaikaiset MEG-laitteet mittaavat kentän riittävän yksityiskohtaisesti, jotta kehitelmä voidaan määrittää luotettavasti. Multipolikehitelmää on viime vuosina sovellettu MEG:n häiriönpoistoon ja liikekorjaukseen. Tämä lähestymistapa asettaa kuitenkin korkeat vaatimukset käytettävälle MEG-laitteelle. Tässä työssä keskityttiin tutkimaan MEG-anturiston vaikutusta multipolimenetelmän käyttöön. Selvensimme vaatimuksia, joita menetelmä asettaa MEG-anturistolle kalibraation ja muiden parametrien suhteen. Etsimme myös uusia anturistogeometrioita, jotka parantavat menetelmän suorituskykyä. Lisäksi rakensimme prototyyppilaitteen, jolla testattiin työssä esitettyjä ideoita. Tulosten käytännöllinen merkitys on tehokkaampi ulkoisten häiriöiden vaimennus ja parempi signaali-kohinasuhde. Lisäksi uudet tulokset auttavat parantamaan MEG-mittausten luotettavuutta, millä on merkitystä etenkin kliinisessä käytössä. Tulokset ovat suoraan sovellettavissa myös muissa biomagneettisissa mittauksissa, esim. magnetokardiografiassa.

Avainsanat Magnetoenkefalografia, MEG, aivokuvantaminen, biomagneettiset anturistot, monikanava-anturistot

ISBN (painettu) 978-952-60-5710-1 ISBN (pdf) 978-952-60-5711-8

ISSN-L 1799-4934 ISSN (painettu) 1799-4934 ISSN (pdf) 1799-4942

Julkaisupaikka Helsinki Painopaikka Helsinki Vuosi 2014

Sivumäärä 126 urn http://urn.fi/URN:ISBN:978-952-60-5711-8

Preface

This thesis work was mostly carried out in the BioMag Laboratory of Helsinki

University Central Hospital. I thank the former head of the laboratory, Pro-

fessor Risto Ilmoniemi, who inspired me to take my first steps on the path of

science and acted as my supervising professor during the thesis work. Later

BioMag has been led by Jyrki Mäkelä, who supported my work in many ways

throughout the years, and always had encouraging words to offer. Thanks to

Jyrki for being such a great boss. I also gratefully acknowledge Juha Montonen,

who patiently helped with all kinds of technical and material issues, and Suvi

Heikkilä and Pirjo Kari for their good work in keeping the laboratory running.

A large part of this work is a continuation of my Master’s thesis done in Pro-

fessor Yoshio Okada’s laboratory in Albuquerque, USA. I greatly appreciate

Yoshio’s efforts in putting new ideas into practice, as well as his contributions

as my coauthor.

Despite his busy schedule, my instructor Samu Taulu always found time to

answer my questions about magnetostatic multipoles and other puzzling con-

cepts. Obviously, Samu’s work also provided the foundation for this thesis. In

addition to Samu, I thank my other Elekta coauthors Juha Simola, Liisa Helle,

Jukka Nenonen, Dubravko Kicic and Lauri Parkkonen for their excellent work,

and particularly Antti Ahonen for his scientific contributions and support in

the tangential sensors project.

In BioMag, I have enjoyed the company of many collegues and friends through-

out the years. For all their scientific and especially social contributions, I want

to thank Jarkko Luoma, Andrey Zhdanov, Tuuli Lehti, Pantelis Lioumis, Katja

Airaksinen, Juha Wilenius, Päivi Nevalainen, Niko Mäkelä, Elina Mäkelä,

Ritva Paetau, Juha Heiskala, Simo Monto, Ville Mäntynen, Alexis Bosseler,

Bei Wang, Rozaliya Bikmullina, Ville Mäkinen, Ilkka Nissilä, Heidi Wikström,

Elina Pihko, Matias Palva, Satu Palva, Essi Rossi, Seppo Kähkönen, Leena

Lauronen, Anne-Mari Vitikainen and Dubravko Kicic, not to mention the peo-

1

Preface

ple I forgot to mention. I particularly appreciate Andrey for his ability to im-

mediately answer any technical question, and Jarkko for reading a draft of this

thesis and giving several useful comments.

I want to give a special acknowledgement to Professor Riitta Hari for encour-

aging me to finish this thesis, and generously letting me devote work hours

towards it during my time at the Brain Research Unit of O.V. Lounasmaa labo-

ratory.

I thank my examiners John Mosher and Ville Kolehmainen for suffering

through my work and providing very valuable comments, and my friend Christo-

pher Knight for proofreading the thesis and educating me on the intricacies of

the English language. Jenny and Antti Wihuri Foundation and Instrumentar-

ium Science Foundation are acknowledged for their financial support.

Finally, I’m grateful to my family for giving me everything I needed and being

there during all these years. I also thank my wonderful daughter Iiris for being

one of my best teachers.

Helsinki, May 2014,

Jussi Nurminen

2

Contents

Preface 1

Contents 3

List of Publications 5

Author’s Contribution 7

1. Introduction 9

2. Generation and measurement of neuromagnetic fields 11

2.1 Origin of neuromagnetic signals . . . . . . . . . . . . . . . . . . . . . 11

2.2 Instrumentation for magnetoencephalography . . . . . . . . . . . . 14

2.2.1 Multichannel systems . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Suppression of magnetic interference . . . . . . . . . . . . . . . . . . 16

2.4 Data acquisition and postprocessing . . . . . . . . . . . . . . . . . . 17

2.5 Modelling of MEG sources . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Computation of sensor-level signals . . . . . . . . . . . . . . . . . . . 22

3. Multipole representation of MEG data and its applications 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Multipole expansion of magnetostatic fields . . . . . . . . . . . . . . 25

3.3 Estimation of the multipole coefficients . . . . . . . . . . . . . . . . 29

3.4 Applications of the multipole expansion . . . . . . . . . . . . . . . . 32

3.4.1 Signal space separation . . . . . . . . . . . . . . . . . . . . . . 32

3.4.2 Extension to the temporal domain . . . . . . . . . . . . . . . 33

3.4.3 Transformation of data . . . . . . . . . . . . . . . . . . . . . . 34

3.4.4 Movement compensation and DC recordings . . . . . . . . . 34

3.4.5 Fine calibration of the sensor array . . . . . . . . . . . . . . 35

3.4.6 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . 35

3

Contents

4. Sensor array and the multipole representation 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Geometry of the sensor array . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Calibration, imbalance and parametrization . . . . . . . . . . . . . 39

4.4 Evaluation of sensor arrays . . . . . . . . . . . . . . . . . . . . . . . 39

5. Summary of results 43

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Publication I: effects of calibration, imbalance and parametrization 43

5.3 Publication II: effects of symmetry and novel sensor array designs 44

5.4 Publication III: validation of tSSS and movement compensation . 44

5.5 Publication IV: adding tangential sensors to an existing array . . 45

5.6 Symmetry reduction by tilting sensors . . . . . . . . . . . . . . . . . 48

5.7 Alternatives to the spherical multipole basis . . . . . . . . . . . . . 50

6. Conclusions 55

References 57

Errata for publications 63

Publications 65

4

List of Publications

This thesis consists of an overview and of the following publications which are

referred to in the text by their Roman numerals.

I Nurminen J, Taulu S, Okada Y. Effects of sensor calibration, balancing and

parametrization on the signal space separation method. Physics in Medicine

and Biology, 53,1975–1987, 2008.

II Nurminen J, Taulu S, Okada Y. Improving the performance of the signal

space separation method by comprehensive spatial sampling. Physics in Medicine

and Biology, 55,1491–1503, 2010.

III Nenonen J, Nurminen J, Kicic D, Bikmullina R, Lioumis P, Jousmäki V,

Taulu S, Parkkonen L, Putaala M, Kähkönen S. Validation of head movement

correction and spatiotemporal signal space separation in magnetoencephalog-

raphy. Clinical Neurophysiology, 123,2180–2191, 2012.

IV Nurminen J, Taulu S, Nenonen J, Helle L, Simola J, Ahonen A. Improving

MEG performance with additional tangential sensors. IEEE Transactions on

Biomedical Engineering, 123,2180–2191, 2013.

5

List of Publications

6

Author’s Contribution

Sections 5.6 and 5.7 of this thesis describe results of previously unpublished

studies. These were conceived and executed solely by the author.

Publication I: “Effects of sensor calibration, balancing andparametrization on the signal space separation method”

I designed and implemented all the simulations with input from co-authors,

and wrote the manuscript.

Publication II: “Improving the performance of the signal spaceseparation method by comprehensive spatial sampling”

I developed the mathematical argument on the resolution of the basis vectors,

and designed the array geometries tested in the paper. I designed and imple-

mented all the simulations, with input from co-authors. I was the main author

of the manuscript.

Publication III: “Validation of head movement correction andspatiotemporal signal space separation inmagnetoencephalography”

I actively participated in all phases of data analysis and writing of the manuscript

and acted as the corresponding author.

7

Author’s Contribution

Publication IV: “Improving MEG performance with additionaltangential sensors”

The second, fifth and sixth authors originally conceived the idea of additional

tangential sensors. I designed and executed the simulation studies that were

the basis for the prototype device. The other authors constructed the prototype

and made the test measurements. I actively participated in the data analysis

and wrote the manuscript.

8

1. Introduction

Despite decades of research, the operation of the human brain remains largely

a mystery. Nevertheless, functional imaging methods, such as fMRI, PET, and

EEG, have contributed to our knowledge of brain function. The topic of this the-

sis is one such method known as magnetoencephalography (MEG). MEG mea-

sures the magnetic fields generated by neuronal currents and therefore enables

temporally direct tracking of brain activity. Methods such as fMRI and PET, on

the contrary, only measure metabolic correlates. MEG has many similarities

to EEG. Magnetic fields, however, are not as sensitive to the conductance prop-

erties of the head as are electric fields, making source localization with MEG

simpler and more accurate. In addition to basic research, MEG also has clinical

applications, for example in preoperative mapping of epileptic foci.

Because the magnetic fields generated by neurons are extremely weak, MEG

instrumentation is technically demanding. MEG was made practical only with

the invention of the superconducting quantum interference device (SQUID) in

the early 1970s. Since then, instrumentation has considerably matured, with

current whole-head devices employing hundreds of channels and recording ac-

tivity over most of the cortex simultaneously; however, noticeable shortcom-

ings remain. For example, suppression of ever-present magnetic interference

remains a problem in MEG recordings.

This thesis is mainly concerned with improvements in MEG instrumentation.

Current research shows that describing the measured magnetic field in terms

of a magnetostatic multipole expansion offers several advantages. This new ap-

proach makes novel demands on MEG instrumentation and also reveals ways

to improve the sensor geometry for optimal characterization of the measured

fields.

The layout of this summary is as follows: Chapter 2 provides a concise intro-

duction to the areas of neuromagnetism and magnetoencephalography relevant

to this thesis. Chapter 3 introduces the multipole-based approach to MEG in

9

Introduction

detail. Chapter 4 examines the relationship between the biomagnetic sensor ar-

ray and the multipole-based approach. Chapter 5 summarizes the publications

and other results obtained in this work. Some conclusions and future directions

are presented in Chapter 6.

10

2. Generation and measurement ofneuromagnetic fields

2.1 Origin of neuromagnetic signals

The human brain is a massively complex biological signal processing network.

Communication in the brain involves electrical and chemical signaling between

cells. According to basic physics, the related electric currents will give rise

to electric and magnetic fields, which in favorable conditions can be measured

outside of the head, forming the basis for noninvasive electrophysiological mea-

surements of brain function.

The topmost layer of the brain is known as the cerebral cortex. It is an intri-

cately folded sheet of tissue with a total area of about 2500 cm2 and thickness

of 2–4 mm. The cerebral cortex contains the facilities for processing and in-

tegrating sensory information and executing motor functions. It also plays a

key role in attention, thought and language. From the point of view of MEG,

the superficial location of the cortical sensory areas is advantageous, as they

consequently produce the strongest magnetic fields outside the head.

Information processing in the brain is thought to mostly involve neural cells,

or neurons. Neurons constitute around 10% of the human brain; the other ma-

jor type, glial cells, constitute around 90%. Originally glial cells were thought to

merely provide scaffolding for neurons, but later their significance in the brain’s

information processing has become elucidated (Fields and Stevens-Graham,

2002). Nevertheless, the generation of magnetic and electric fields observable

outside the brain involves neurons.

The neuron can be described as a nonlinear computational unit operating

on electric potentials. The potentials arrive via dendrites, where the outputs

of other neurons connect via synapses, giving rise to postsynaptic potentials.

They proceed toward the cell body or soma, as illustrated in figure 2.1 (left). If

the voltage at the soma exceeds a certain threshold, the neuron fires an action

11

Generation and measurement of neuromagnetic fields

potential, which then travels via the axon towards other neurons.

Figure 2.1. Left: a schematic of a neuron. The dashed arrows show the propagation of electricpotentials. Right: actual neurons in a rat cortex (drawing by Ramon y Cajal). Afraction of the neurons have been stained. The perpendicular orientation of theneurons can be seen.

The action potential has a fixed magnitude and a short duration of a few mil-

liseconds. In contrast, the postsynaptic potentials have a variable magnitude

and last for tens of milliseconds. Action potentials involve two nearby cur-

rent flows with opposing directions, which results in a magnetic field decaying

as 1/r3 with distance r. Postsynaptic potentials involve a single current flow

across the cellular membrane and into the interior of the cell, or in the oppo-

site direction; the corresponding field decays as 1/r2. Furthermore, the longer

duration of the PSPs favors temporal summation. Due to these differences, the

extracortical magnetic fields are mostly due to postsynaptic currents (Hämäläi-

nen et al., 1993).

Figure 2.1 (right) shows a Golgi stained slice of the rat cortex. The apical

dendrites are directed towards the surface and oriented perpendicularly to it.

Correspondingly, activation of a cortical patch leads to an average current flow

perpendicular to the surface of the cortex. According to detailed simulations

of cortical cells (Murakami and Okada, 2006), ten thousand nearby pyramidal

neurons firing in synchrony may suffice for a response detectable with MEG. In

practice, several tens of thousand of simultaneously active cells are probably re-

quired for a detectable signal, due to imperfect spatial and temporal synchrony

(Parkkonen, 2009). Cancellation of magnetic fields due to the larger scale corti-

cal anatomy also affects the MEG signal. For example, simultaneous activation

of cortex on the opposing walls of a sulcus will cause significant cancellation;

12


see (Ahlfors et al., 2010b) and figure 2.2.

Figure 2.2. Folding of the cortex. Currents near gyri (1) produce weak magnetic fields, sincethey are oriented approximately radially with respect to the head geometry. Cur-rents in sulci (2 and 3) are the strongest contributors to the MEG signal.

In a spherical conductor, radially oriented currents produce no magnetic field

outside. The head is nonspherical and therefore cannot have strictly radially

oriented currents, but nevertheless it follows that the strongest MEG signals

are produced by currents oriented tangentially to the head surface. As the post-

synaptic currents on average flow perpendicular to the cortical surface, this

means that the MEG signal is most sensitive to sources in sulci (figure 2.2).

This is in contrast to EEG, which is more sensitive to radial sources. In MEG,

the source orientation has a larger effect on the strength of the measured mag-

netic field (Ahlfors et al., 2010a; Goldenholz et al., 2009; Hillebrand and Barnes,

2002). The depth of a focal source also strongly affects the magnetic field out-

side the head; with increasing sensor-source distance r, the magnitude of the

detected field decreases approximately as 1/r2. Nevertheless, responses from

deeper structures such the brain stem have been measured with careful exper-

imental setups, e.g. in (Parkkonen et al., 2009).

The currents immediately related to neuronal activity flow inside the cells,

and are customarily known as primary currents. Looking at the situation on a

larger scale, where cellular-level details are ignored, the related changes in the

electric field will also drive ohmic volume currents throughout the intracranial

medium. The total current is then a sum of these:

J(r)= J p(r)+ Jv(r)= J p(r)+σ(r)E(r), (2.1)

where J p and Jv are the primary and volume currents, respectively, and σ(r)

is the macroscopic conductivity. Generally the goal is to estimate J p from the

measurements, which then indicates the location of the neuronal activity; how-

ever, the volume currents also contribute to the magnetic field and need to be

13


taken into account. The problem of source estimation from neuromagnetic mea-

surements will be discussed in section 2.5.

2.2 Instrumentation for magnetoencephalography

One of the fundamental challenges in MEG is the extremely small amplitude of

the neuronal magnetic fields. Consequently, a correspondingly sensitive mag-

netic field sensor is needed.

The first measurements of the brain’s magnetic field were done by David Co-

hen in 1968 using a million-turn induction coil with a ferrite core (Cohen, 1968).

Due to the small signal-to-noise ratio, the brain signal was detectable only after

averaging thousands of alpha-rhythm1 cycles, with the averaging triggered by

the EEG signal. The feasibility of recording neuronal magnetic fields was thus

demonstrated, but it was clear that a different kind of sensor would be needed

to make the method practical.

The theoretical foundations for a new kind of magnetic field sensor had been

laid already in 1962, when Brian Josephson predicted the tunneling of super-

current carriers (Cooper pairs) over weak links (Josephson, 1974). This led to

the invention of the DC (direct current) SQUID (Jaklevic et al., 1964), and later,

the radiofrequency (RF) SQUID. The DC SQUID is the preferred type in mod-

ern neuromagnetic measurements, due to its higher sensitivity (Clarke, 1989).

It is a superconducting ring interrupted by two weak links, known as Josephson

junctions. Due to interference of the Cooper pair wavefunctions on the two sides

of the ring, the DC SQUID develops a voltage that is periodically dependent on

the applied magnetic flux. The period is one flux quantum Φ0 = h/2e ≈ 2.07 fWb.

A bias current is applied to keep the junctions resistive for all flux values, and

thus allow a voltage to develop. For a comprehensive analysis of DC SQUID

operation, see (Ryhänen et al., 1989). A schematic of the DC SQUID is shown

in figure 2.3.

In principle, changes in magnetic flux through the SQUID could be measured

by counting the periods the voltage goes through. A more practical way, how-

ever, is to use a feedback loop to keep the applied flux constant and instead

track the feedback current. The current is related to the feedback flux and thus

to the applied flux. This technique is known as a flux-locked loop. An operating

point is chosen where the voltage response to flux is near its maximum, allow-

ing tracking of flux changes much smaller than one flux quantum; thus, the

SQUID is operated as an extremely sensitive flux-to-voltage transducer. The

1An ubiquitous type of neuronal oscillation with a frequency of about 10 Hz.

14


Figure 2.3. a) Schematic of the DC SQUID. Crosses indicate the Josephson junctions. Ib is thebias current. b) Voltage response of the biased SQUID to an applied magnetic field.The period is one flux quantum Φ0. The dashed line indicates a suitable operatingpoint for the flux-locked loop, where the voltage response to changes in the magneticflux is maximum.

periodic response means that absolute flux values cannot be deduced from the

SQUID output, only their changes.

To reduce noise, it is desirable to minimize the area of the SQUID (Clarke,

1989). The small area, however, weakens its direct coupling to the external

magnetic field; therefore, the magnetic field is generally coupled to the SQUID

using a superconducting flux transformer. The transformer consists of a pickup

coil and an input coil, which are connected in series. An external flux generates

a shielding current in the pickup coil. This current in turn generates a magnetic

field in the input coil, which is coupled inductively to the SQUID. Since the flux

coupling is based on shielding current, the frequency response extends down to

DC. Different pickup coils are illustrated in figure 2.4.

Figure 2.4. Some common pickup coil geometries. a) Magnetometer. b) Axial gradiometer. c)Planar gradiometer.

A magnetometer simply measures flux. In a gradiometer, the loops of the

coil are wound in opposite orientations, leading to cancellation of homogeneous

fields. Therefore, the sensor approximately measures the spatial gradient of

the magnetic field. Spatial gradients decay faster as a function of distance

than the magnetic field, leading to cancellation of fields from faraway inter-

ference sources. A problem with gradiometers is that to obtain cancellation

of homogeneous fields, the areas (or area-turn products) of the loops need to

15


be equal. With wire-wound gradiometers, an imbalance on the order of 1% is

typical. Planar gradiometers can be manufactured with more precise thin-film

technologies, with attainable imbalance on the order of 0.1% (Publication I).

In principle, higher order gradiometers can be constructed, but their physical

realization is cumbersome and prone to errors.

The sensors typically employ low-temperature superconductors, which re-

quire cooling well below the transition temperature. This is accomplished by

suspending the sensors in liquid helium, at a temperature of 4.2 K, in a vacuum-

insulated vessel called the dewar.

2.2.1 Multichannel systems

The earliest MEG recordings were made with single-channel instruments, which

required movement of the sensor around the head to obtain topographic infor-

mation. The first multichannel instruments with 4–5 channels were introduced

in the early 1980s; they recorded over an area with a diameter of few cm. By

1990, several systems with tens of channels had been produced (Hämäläinen

et al., 1993). The first system providing whole-head coverage with 122 planar

gradiometer channels was operational in 1992 (Ahonen et al., 1993b). A typical

modern whole-head system has 200–400 channels in a helmet-like arrange-

ment. The helmet is generally designed to fit the majority of adult population,

but cannot accommodate all head sizes optimally. This is especially problematic

for pediatric measurements.

With multiple channels, crosstalk between the channels becomes an issue.

Crosstalk refers to the fact that the shielding current circulating in the flux

transformer of one channel will also distort the flux measured by its neigh-

bouring channels. The feedback wiring of the SQUIDs can exacerbate the phe-

nomenon; however, crosstalk can be corrected by determining the mutual in-

ductances between coils. Finally, the spacing, distribution and orientation of

the sensors in a multichannel array is a complex optimization problem. Since

the array performs spatial sampling of the magnetic field, considerations of

sampling theory become fundamentally important. A comprehensive analysis

has been published in (Ahonen et al., 1993a).

2.3 Suppression of magnetic interference

Suppression of interference is crucial in MEG. To reduce external interference,

the measurements are almost always performed in a magnetically shielded

16


room (MSR). The two basic principles of magnetic shielding are flux diversion

(also known as ferromagnetic shielding) and eddy-current shielding. Flux di-

version refers to use of materials with high magnetic permeability, such as μ-

metal, in the MSR construction. They act as a preferred path for magnetic flux

lines, diverting flux along the shielded room walls and away from the interior.

Eddy-current shielding is based on materials with high conductivity, such as

aluminium. In these materials, induced currents are generated that oppose

the imposed flux. Shielded rooms typically combine ferromagnetic and eddy-

current shielding in several layers, to obtain good shielding at a wide range of

frequencies. Shielding factors range from 50–110 dB for 1–100 Hz, which are

the typical frequencies of interest in neuromagnetism. Passive shielding using

the above principles can be combined with active shielding systems, which at-

tempt to measure the external interference and generate compensating fields

using coils placed around the MSR (Hämäläinen et al., 1993; Andrä and Nowak,

2007).

As noted above, gradiometer sensors provide intrisic shielding against exter-

nal interference by suppressing homogeneous fields. Their effectiveness is lim-

ited by interference from nearby sources, which is not spatially homogeneous.

A further limiting factor is the imperfect manufacturing of the gradiometers

which manifests as imbalance, misaligned loops, and complexities created by

the wiring. These problems are especially prominent in wire-wound gradiome-

ters.

Another method is the use of reference sensors. These sensors are placed near

the main sensor array, but far enough from the interior of the helmet not to

detect brain activity. Thus, based on the reference signals, the interference sig-

nals can be estimated and subtracted from the signals of the main sensor array.

The subtraction coefficients may be chosen either to optimize SNR or to create

synthetic gradiometers. Compared to first-order gradiometers, synthetic third-

order gradiometers can reduce low-frequency environmental noise by about a

factor of 20 (Vrba and Robinson, 2001). For nearby interference sources whose

fields are less homogeneous, the signal extrapolation implicit in the reference

sensor approach will be less accurate, reducing the shielding performance.

2.4 Data acquisition and postprocessing

To obtain the magnetic field data, the voltage outputs of the sensors are low-

pass filtered, digitally sampled and converted to the appropriate units, either

T for magnetometers or T/m for planar gradiometers; axial gradiometers are

17


traditionally converted only to T.

Before further analysis, some pre-processing is generally applied to the data.

The purpose is to increase the signal-to-noise ratio and extract the information

that is most relevant for further analysis. Filtering methods may be roughly

divided into two classes: time domain and spatial domain. The former only use

the time course of the signal (or its frequency domain representation); the latter

take advantage of the spatial information intrisic in multichannel recordings.

A MEG measurement may be considered as a N ×T matrix, where N is the

number of channels and T is the number of acquired temporal samples. At

typical sampling frequencies of 1–5 kHz and for recording times of one hour or

longer, several gigabytes of storage may be required. As a first step, downsam-

pling may be used to reduce the size of data, if the full frequency response is

not required.

Frequency-domain filtering is often desirable to remove high-frequency or

low-frequency noise, to restrict the analysis to a certain frequency range, or

to suppress narrowband interference such as line-frequency noise. It is often

important to consider both the frequency and time domain characteristics of the

filters; a filter which has a satisfactory frequency response may still cause unac-

ceptable phenomena in the time domain, such as excessive ringing or temporal

delays. In offline filtering, forward-backward (also known as zero-phase) filter-

ing may be utilized to cancel delays and phase distortion (Mitra, 2001). Other

temporal domain methods include various techniques based on statistics, such

as independent component analysis (Vigário et al., 2000).

In an evoked-response paradigm, stimuli are presented to the subject and

triggers are recorded with the magnetic data to indicate stimulus onsets. If the

brain evoked response to the stimulus may be considered constant, epochs time-

locked to the stimulus onsets can be averaged to increase the signal-to-noise

ratio. Here, “noise” is defined as any other signal than the evoked response.

This model of superposed evoked responses may not be completely realistic,

as the stimulus may also modify the underlying (non-stimulus-related) brain

activity to some degree. Assuming a nonchanging, time-locked response that

is independent of the noise, averaging of M responses improves signal-to-noise

ratio by a factor of M.2

In contrast to purely temporal filtering, spatial filtering uses the knowledge

of spatial properties of the magnetic field that multichannel recordings provide.

2This applies to signal-to-noise ratio in terms of power, which is the usual definition.Note that Publication I, Publication II and section 4.3 use SNR defined in terms ofamplitude, not power.

18


In spatial methods, it is useful to describe the data as vectors in N-dimensional

signal space. One tradional approach is signal space projection (SSP) (Uusitalo

and Ilmoniemi, 1997; Tesche et al., 1995), based on orthogonal projections in

the signal space. SSP is a purely mathematical projection method, which does

not take the specific properties of MEG signals into account. The next chapter

is devoted to a different approach to spatial filtering, based on the physical

properties of magnetic fields.

2.5 Modelling of MEG sources

After postprocessing, the MEG data may be analyzed directly on the sensor

level, to examine the time course and frequency content of the brain responses.

The spatial distribution of the signals over the sensors also gives a rough in-

dication of the activated brain areas. Nevertheless, it often would be desirable

to localize the generating primary currents as accurately as possible, using all

available information. This is known as the MEG inverse problem. As a prereq-

uisite, it requires solution of the forward problem, i.e. the computation of the

magnetic field corresponding to the specified current distribution.

The relationship between electromagnetic fields and their generators, e.g. cur-

rents or charges, is given by Maxwell’s equations. In the analysis of MEG sig-

nals, the equations can be simplified by invoking the quasistatic approxima-

tion, in which field propagation delays, inductance and capacitive effects are

ignored. The approximation is justified, since these time-varying phenomena

have only a small effect on the electric and magnetic fields in the low-frequency

regime3 relevant in biomagnetism (Plonsey and Heppner, 1967; Hämäläinen

et al., 1993). The quasistatic Maxwell’s equations are

∇·E = ρ/ε0, (2.2)

∇×E = 0, (2.3)

∇·B = 0, (2.4)

∇×B =μ0J, (2.5)

where ρ is charge density, J is current density, and E and B are the electric

and magnetic fields, respectively. μ0 is the permeability of vacuum.

The elements of primary current in the brain are typically modelled as cur-

rent dipoles. The current dipole is a mathematical idealization representing a

current which flows for an infinitesimally small distance. The current density

3Approximately <1000 Hz.

19


of the dipole may be written as

J p(r)=Qδ(r− rQ), (2.6)

where δ(r) is the Dirac delta function and rQ is the location of the current

dipole. Q is current dipole moment with units of A·m. The total current is then

J(r)= J p(r)+ Jv(r)=Qδ(r− rQ)+σ(r)E(r). (2.7)

In the quasistatic approximation, the Biot–Savart law

B(r)= μ0

4π

∫ J(r′)× (r− r′)dV ′

‖r− r′‖3 (2.8)

gives the magnetic field due to a current in a region of space. Primed coordi-

nates refer to the source region and r is the point where the field is evaluated.

Substitution of (2.7) gives

B(r)= μ0

4πQ× (r− rQ)

(r− rQ)3+ μ0

4π

∫σ(r)E(r)× (r− r′)dV ′

‖r− r′‖3 (2.9)

In an infinite homogeneous conductor (σ = constant), it can be shown that the

integral term vanishes (Sarvas, 1987), meaning that the volume currents do

not contribute to the magnetic field. Thus, in such a medium, the magnetic

field of the current dipole reduces to

B(r)= μ0

4πQ× (r− rQ)

(r− rQ)3 . (2.10)

A more realistic model is a finite conductor with σ = 0 outside, in which case

the volume currents generally do contribute to the magnetic field. The simplest

model for the head is a spherical conductor. In this case, the magnetic field

outside of the sphere (where σ = 0 and J = 0) may be obtained via a rather

simple analytical formula, without explicitly computing the volume currents

first (Sarvas, 1987). Assuming the conducting sphere to be centered at the

origin of the coordinate system, the result is

B(r)= μ0

4πFQ× rQ − (Q× rQ · r)∇F

F2 , (2.11)

where

F = a(ra+ r2 − r · rQ)

a= r− rQ

r = ‖r‖,a = ‖a‖. (2.12)

Interestingly, neither the spherically symmetric conductivity profile nor the ra-

dius of the sphere affect the magnetic field outside, simplifying the calculations.

20


Further, even in case of anisotropy, the conductance in the radial direction can

vary arbitrarily with no consequence on the magnetic field (Ilmoniemi, 1995).

This provides justification for ignoring the conductance details of the head, e.g.

the relatively well-conducting axon bundles in the white matter, and fontanels

in infants (Lew et al., 2013). Such simplifications are not possible for the elec-

tric potential, which is strongly affected by the conductance. This is a signif-

icant advantage of MEG compared to EEG, which requires detailed conductor

models for accurate inverse solutions (Hämäläinen et al., 1993).

From (2.11), the magnetic field is seen to vanish for any radially oriented

dipole, since then Q × rQ = 0. At the center of the sphere (rQ = 0) the dipole

produces no magnetic field. These are consequences of the perfect symmetry of

the sphere.

The simple spherical conductor model is often sufficiently accurate, and other

sources of error such as the signal-to-noise ratio and errors in coregistration

of coordinate systems dominate the localization accuracy. Nevertheless, the

sphere model may not always produce satisfactory results, particularly when

modelling deep sources or frontal or frontotemporal sources (Hämälainen and

Sarvas, 1989; Tarkiainen et al., 2003). A more refined model based on the ac-

tual anatomy of the head may be constructed based on the magnetic resonance

images (MRI) of the subject. This may be either a three-shell model comprising

the brain with the cerebrospinal fluid, skull and skin, or alternatively a single-

shell model which consists only of the brain and the cerebrospinal fluid. With

these models, computation of the magnetic fields is more involved than with the

sphere model, requiring first the computation of the electric potential on the

compartment boundaries. The potential is discretized on the boundaries, lead-

ing to an application of the boundary-element method (BEM) (Mosher et al.,

1999; Hämälainen and Sarvas, 1989). In recent years, automatic segmentation

tools for anatomical images have become widely available. Consequently, it has

been argued that three-shell BEM models should now be routinely used instead

of simplified ones (Stenroos et al., 2014).

Once the forward problem is solved, the inverse problem can be tackled. Un-

fortunately, when given a measured magnetic field outside a source volume,

the current distribution inside the volume cannot be determined uniquely, as

was shown by Hermann von Helmholtz already in the 19th century. For every

magnetic field, there exist an infinite number of source configurations that may

in theory have produced it. Only by placing suitable constraints on the source

distribution can a unique solution be obtained.

The simplest source model is the equivalent current dipole (ECD), commonly

21


employed in MEG studies. It attempts to explain the measured signal with

a single current dipole, whose location and orientation are adjusted in an opti-

mization procedure until the field predicted by the forward computation matches

the measured field as closely as possible. In case of a distributed activation, the

ECD then appears at the “center of gravity” of the activated area. The single

ECD is a good model for a highly focal activation, for example the early response

from the primary somatosensory cortex (Forss et al., 1994). A natural extension

is the multidipole model, where a user-specified number of current dipoles are

permitted to be active simultaneously (Hämäläinen et al., 1993; Hansen et al.,

2010; Mosher et al., 1992).

Another class of source models are distributed current estimates, where the

number of active sources is not determined a priori. These methods divide

the source region into voxels and permit a current dipole to be active at each

voxel. With suitable additional constraints, a unique total current distribution,

i.e. magnitude and orientation of current at each voxel, can then be obtained.

Commonly used versions include the minimum norm estimate (Hämäläinen

and Ilmoniemi, 1994), which assumes the minimum L2 norm for the current

distribution and the minimum current estimate (Matsuura and Okabe, 1995;

Uutela et al., 1999), which uses the L1 norm.

2.6 Computation of sensor-level signals

So far, the physical quantities have been described in terms of continuous vector

fields. Mathematically, the magnetic flux density bi measured by a sensor may

be computed by integration over the area Ai of its pickup loop:

bi = 1Ai

∫Ai

B(r) ·dA i. (2.13)

In numerical computations, the integral is approximated by summing as

bi ≈N∑

i=1wiB(pi) ·n, (2.14)

where n is the normal vector of the loop, and wi and pi are chosen according to

the loop geometry. Values are given in references such as (Stroud, 1971).

The current distribution can also be related directly to the sensor signal. The

linearity of the Biot–Savart law indicates that the output of each sensor can be

obtained by a weighted sum of current over the source volume. Therefore, there

exists a vector field Li so that

bi =∫Li · J p(r) dV , (2.15)

22


where Li is known as the lead field of the ith sensor (Sarvas, 1987; Hämäläinen

et al., 1993). It describes the sensitivity of the sensor to the currents flowing in

the source volume. In the case of a spherical conductor model, the lead fields are

always strictly tangential in orientation, since radial currents do not produce a

magnetic field outside of the conductor.

23


24

3. Multipole representation of MEG dataand its applications

3.1 Introduction

Traditionally, sensor-level MEG data is used either directly or as an input to

source modelling. (Taulu, 2008) presents a method where the data is first trans-

formed into another coordinate system. This transformation corresponds to

estimating the magnetostatic multipole expansion of the measured magnetic

field. The expansion represents the field in a device-independent form.

This approach has several advantages. The representation is optimal in the

sense that different multipole terms correspond to mutually orthogonal current

distributions. The redundancy in the sensor-level multichannel data is elimi-

nated, and the number of multipole terms corresponds closely to the actual

degrees of freedom in the measured magnetic field. Further, due to the geome-

try of the measurement, the multipole terms are naturally separated into two

sets, corresponding to sources internal and external to the sensor array. This

property enables suppression of magnetic interference.

The device-independent multipole representation has several other applica-

tions, which will be reviewed in this chapter. The issue of estimating the multi-

pole expansion from noisy data will also be discussed. To provide a foundation

for later discussion, the chapter begins with a rather detailed derivation of the

multipole expansion, mostly following (Taulu et al., 2004a, 2005; Taulu and

Kajola, 2005).

3.2 Multipole expansion of magnetostatic fields

According to (2.5), the magnetic flux density B satisfies ∇×B = μ0J. It follows

that in current-free regions, ∇×B = 0. Thus, in such regions, B can be expressed

25

Multipole representation of MEG data and its applications

as a gradient of a magnetic scalar potential ψ:

B =−∇ψ. (3.1)

Therefore

∇·B =−∇·∇ψ=−∇2ψ= 0, (3.2)

where (2.4) has been used. Thus, the scalar potential satisfies Laplace’s equa-

tion ∇2ψ= 0 in current-free regions.

In spherical coordinates, a general solution to (3.2) may be written in terms

of spherical harmonics Ylm as

ψ(r)=−∞∑

l=0

l∑m=−l

αlmYlm(θ,φ)

rl+1 −∞∑

l=0

l∑m=−l

βlmrlYlm(θ,φ), (3.3)

where r refers to the point (r,θ,φ) in the spherical coordinate system, and αlm

and βlm are coefficients. In the following, primed coordinates will be used to

refer to sources, and unprimed coordinates to the points where the field is eval-

uated.

Applying (3.1) yields an expansion for B itself:

B(r)=∞∑

l=0

l∑m=−l

αlm∇[

Ylm(θ,φ)rl+1

]+

∞∑l=0

l∑m=−l

βlm∇[rlYlm(θ,φ)

]. (3.4)

The gradient terms closely resemble the commonly defined spherical vector har-

monics (Arfken and Weber, 1995), so the series may be conveniently written

using modified vector harmonic functions νlm and ωlm combined with radial

terms (Taulu et al., 2005):

B(r)=∞∑

l=1

l∑m=−l

αlmνlm

rl+2 +∞∑

l=1

l∑m=−l

βlmrl−1ωlm. (3.5)

Equation (3.5) gives a general series expansion for any magnetic field in a

source-free region. It consists of two series with a different dependency on

r. Now consider the expansion in a source-free region of space defined by

r0 < r < r1. Decreasing r will increase the magnitude of the first series. There-

fore, it may be deduced that the first series must correspond to magnetic field

sources closer to the origin than r0. Similarly, the second series must corre-

spond to sources further from origin than r1.

Next, place the origin inside the sensor array, and define r0 and r1 so that

the sensors are wholly contained in the region r0 < r < r1 (figure 3.1). Now,

due to the argument above, the first series in (3.5) must correspond to sources

“inside” the sensor array (r′ < r0), and the second to sources “outside” the sensor

array (r′ > r1). In MEG, the brain is located inside the array and contains

the sources of interest, and field sources outside the array produce undesired

26


Figure 3.1. Geometry of the MEG measurement. The dot indicates the origin for the series ex-pansion. r0 is the distance from the origin to the nearest sensor. r1 is the max-imum origin-sensor distance. These determine the intermediate region (shadedgray) which contains the sensors and is assumed to be source-free. Interferencesources are characterized by r′ > r1 and brain sources by r′ < r0.

interference. Thus, with a suitable choice of origin, the series expansion offers

a way to separate the signals from the two types of sources.

It is now necessary to describe the situation in terms of the multichannel

sensor array. The array samples the magnetic field at N discrete locations, so

one temporal sample may be represented as an N-dimensional signal vector:

φ= [b1 b2 . . .bN ]T , (3.6)

where b1 . . .bN are the signals of individual sensors. Define the multichannel

measurement function F that maps magnetic fields to N-dimensional signal

vectors. Then we may define

alm = F(νlm

rl+2

)(3.7)

blm = F(rl−1ωlm

), (3.8)

and, for a general magnetic field B,

φ= F (B) . (3.9)

27


We will refer to the basis functions in (3.8) as VSHR (vector spherical harmonic

+ radial term). The resulting alm and blm will be called basis vectors. In

practice, alm and blm are computed by numerical integration, using (2.14).

Applying F to (3.5) yields

φ=∞∑

l=0

l∑m=−l

αlmalm +∞∑

l=0

l∑m=−l

βlmblm. (3.10)

Thus, we have decomposed any measurable signal vector to a linear combina-

tion of basis vectors. At this point, the sum is infinite. Assume, however, that

the first series can be truncated at index Lint, and the second series at index

Lext. Then we can move to a matrix representation by defining the column

matrices

Sint =[a1,−1 a1,0 a1,1 . . . aLint,Lint

](3.11)

Sext =[b1,−1 b1,0 b1,1 . . . bLext,Lext

](3.12)

S= [Sint Sext] . (3.13)

The column spaces of Sint and Sext will be called the internal and external sub-

spaces, respectively. The corresponding coefficient vectors are

xint = [α1,−1 α1,0 α1,1 . . . αLint,Lint ]T (3.14)

xext = [β1,−1 β1,0 β1,1 . . . βLext,Lext ]T (3.15)

x= [α1,−1 α1,0 . . . β1,−1 . . .βLext,Lext ]T. (3.16)

With these definitions, the linear combination in (3.10) may be written as

φ= Sx. (3.17)

Normally, the measured signal vector φm is known and the coefficients x are

desired. They may be estimated e.g. by the pseudoinverse as

x= S+φm. (3.18)

Having estimated the coefficients, we may discard the xext part and compute

the signal vector that corresponds only to sources inside the sensor array:

φint = Sint xint. (3.19)

In the above discussion, we have made the assumption that the series can

be truncated at finite indices Lint and Lext. This is justified since the MEG

sensor array can measure spatial frequencies only up to a limited threshold,

and increased Lint and Lext correspond to higher spatial frequencies. The effect

of Lint on dipole signals was investigated in (Taulu et al., 2005) on the 306-

channel Elekta Neuromag® array, where Lint = 8 was found sufficient for all

28


source configurations. To limit reconstruction noise, Lext is typically limited to

3 or 4.

Another assumption concerns the source-free region r0 < r < r1. In a real

measurement, the region will contain sources, e.g. in the form of biological cur-

rents flowing in the subject’s body. The Laplace equation will still hold in any

region where currents are negligible, e.g. in the vicinity of sensors, where the

fields are evaluated. Nevertheless, due to the finite number of expansion terms,

signals arising from sources located in the region r0 < r < r1 cannot be com-

pletely modeled by either the internal or the external subspace. Thus, when the

reconstruction is performed, their contribution will appear in both subspaces.

The temporal extension of the method, discussed later in this chapter, was de-

signed to deal with this “leakage” phenomenon. It also addresses the problem

of nearby external interference sources.

To conclude, the multipole coefficients xint provide a device-independent rep-

resentation of the magnetic field, with external interference suppressed. The

sensor-level signal may be recomputed from the coefficients by applying (3.19),

or the magnetic fields may be extrapolated to other points in space by applying

a corresponding basis matrix.

Similarly to the sensor signals in (2.15), the coefficients alm can also be di-

rectly related to the current distribution using a lead field-like operator (Taulu

and Kajola, 2005):

αlm =∫

λlm · J(r) dV , (3.20)

where the multipole lead field λlm can be expressed in closed form as

λlm = i2l+1

√l

l+1rl X∗

lm, (3.21)

and X∗lm is the complex conjugate of a tangential vector spherical harmonic

(Taulu and Kajola, 2005). The lead fields of different multipole components are

also orthogonal over a spherical volume V ; that is,∫Vλlmλ∗

LM dV = 0 ∀ l = L,m = M. (3.22)

In this sense, the multipole coefficients may be regarded as “sensors” that con-

vey mutually independent information about the current distribution, unlike

physical sensors, which have non-orthogonal lead fields.

3.3 Estimation of the multipole coefficients

As elaborated in the previous section, the MEG measurement may be inter-

preted as discretization of VSHR basis functions into N-dimensional basis vec-

29


tors of the signal space:

alm = F(νlm

rl+2

)(3.23)

blm = F(rl−1ωlm

)(3.24)

S= [a1,−1 . . . aLint,Lint b1,−1 . . . bLext,Lext

]. (3.25)

Now assume that M is the number of multipole components. The truncation

orders Lint and Lext are assumed to be low enough so that N > M. Then the

relationship between the multipole components and the signal vector

φ= Sx (3.26)

is an overdetermined system of linear equations. Equation (3.26) describes

a perfect fit of multipole coefficients x to noiseless data. For noisy data, the

overdetermined system of equations is not consistent and does not have an ex-

act solution in terms of x. Thus, denote the measured signal vector by

φm =φ+φn, (3.27)

where φ is the ideal noiseless data, and φn represents additional noise not ex-

plained by the multipole model. The corresponding noisy estimate for the mul-

tipole coefficients is x. A minimum-norm solution can be sought by minimizing

the 2-norm of the residual

‖Sx−φm‖22, (3.28)

leading to the least squares solution

xLSQ = (STS)−1STφm = S+φm

= S+φ+S+φn = x+S+φn, (3.29)

where S+ = (STS)−1ST is often defined as the pseudo-inverse of S (Golub and

van Loan, 1996). The solution consists of the noiseless multipole coefficients

x and a noise term, which is affected by the pseudoinverse S+ and φn (Taulu

et al., 2005).

Via the singular value decomposition (SVD) of S, it may be shown that the

least squares solution of (3.29) is equivalent to

xLSQ =N∑

i=1

uTi φm

σivi, (3.30)

where ui and vi are left and right singular vectors of S, respectively. This

form reveals the numerical sensitivity of the pseudoinverse. Specifically, an

ill-conditioned matrix will have very small singular values σi, which strongly

30


affect the solution but cannot be computed accurately in a finite precision arith-

metic. The conditioning of the S can be characterized by its condition number,

defined as

κ=σ1/σN (3.31)

i.e. the ratio of the largest and smallest singular values. It is affected by the

geometry of the sensor array, as discussed in the next section.

A large condition number indicates that the pseudoinverse solution will be

overly sensitive to errors, and necessitates the use of regularization. The most

obvious way is to neglect the terms corresponding to to smallest σi, according

to some threshold. The regularized solution is then

xREG =K∑

i=1

uTi φm

σivi, (3.32)

where K is a suitable cutoff. This method is commonly known as truncated

singular value decomposition (TSVD). Neglecting the terms corresponding to

i > K will bias the solution by limiting it to the subspace spanned by vectors

vi, i < K .

Other regularization methods may produce better results than TSVD. It is

useful to take the structure of x into account. The measured signal can of-

ten be described in terms of limited number of multipole components, and ac-

cordingly the elements of x tend to have highly dissimilar values. Therefore,

methods that produce smooth solutions, such as classical Tikhonov regulariza-

tion with minimization of the 2-norm, are not well suited to this problem. A

method known as PP-TSVD (Hansen and Mosegaard, 1996) is able to recon-

struct piecewise continuous functions and was shown to perform well in the

multipole estimation problem (Nurminen, 2005). Alternatively, an iterative so-

lution may be used, as suggested by (Taulu, 2013). On each round of iteration,

low-order multipole components explaining most of the variance in the data

are fitted first. When fitting each new multipole component, the contribution of

other already fitted components is first subtracted from the data. The iteration

typically converges in fewer than 10 rounds.

The condition number of S is most strongly affected by the similarity of Sint

and Sext. Denote their column spaces by Sint and Sext, respectively. The

distance between Sint and Sext can be studied directly using the concept of

subspace angles, also known as canonical angles (Golub and van Loan, 1996;

Knyazev and Argentati, 2002; Taulu et al., 2005). Sint and Sext are searched

for a pair of vectors v1int ∈ Sint and v1

ext ∈ Sext that maximize θ1 =∠(v1int,v

1ext).

These form the first principal vectors, and the angle θ1 is the first principal an-

gle. Then the search is repeated in new subspaces S ′int orthogonal to v1

int and

31


S ′ext orthogonal to v1

ext, to find v2int and v2

ext. This process is repeated until the

dimension of the smaller subspace (typically Sext) is exhausted. Intuitively, the

angles describe the “minimum distance” and “maximum distance” between the

subspaces.

3.4 Applications of the multipole expansion

3.4.1 Signal space separation

As mentioned above, external interference can be suppressed simply by first

estimating the full set of multipole components, and then discarding the com-

ponents corresponding to the external subspace in the reconstruction of the

signal vector. This method is known as signal space separation (SSS). In prac-

tice, the shielding factor attainable by SSS is limited by the truncation of the

external basis at Lext and the finite calibration accuracy of the MEG device,

discussed in more detail in chapter 4.

The SSS method may be contrasted with the traditional signal space projec-

tion (SSP) (Uusitalo and Ilmoniemi, 1997). In SSP, the interference subspace is

typically determined based on the data. For example, let D be an N ×T matrix

of data, where N is the number of channels and T is the number of time points.

Suppose that D consists of empty room data, i.e. data recorded in the presence

of external interference only. The sample covariance matrix C is then

C= 1T −1

DDT. (3.33)

Denote the eigenvectors of C by yk. The M eigenvectors corresponding to

the largest eigenvalues now determine the M-dimensional subspace of domi-

nant interference. Place them into a column matrix Y = [y1 y2 . . . yM

]. Since

the eigenvectors of C are orthogonal, an orthogonal projection operator can be

formed simply as P⊥ = I−YYT, where I is the identity matrix. This operator can

then be used to project data away from the interference subspace. Projection

operators can also be defined based on a single artifact, such as a QRS complex.

Since the brain signals are generally not orthogonal to the interference sub-

space, the projection will also distort them, which has to be taken into account

in source modelling. In contrast, SSS is not based on projection, but on re-

construction in terms of the complete multipole basis. As the basis explicitly

models both the brain signals and the interference, the distortion arising from

orthogonal projection is avoided. Yet, the data-determined SSP subspace auto-

matically accounts for properties of the MEG device such as calibration errors

32


and imbalance, which adversely affect the performance of the SSS method. In

practice, SSP can achieve interference suppression by a high factor, but only for

interference signals that fall into the predetermined subspace.

3.4.2 Extension to the temporal domain

Fields of sources very close to the sensors, possibly even in the sensor region

r0 < r < r1, are not adequately represented by either the Sint or Sext basis. Thus,

in the estimation of multipole components, their contribution will appear in

both xint and xext. The imperfect calibration of the MEG device also contributes

to this mixing of signals into both internal and external subspaces. If, how-

ever, temporal information is available, the leakage can be utilized. With time-

dependent data, time-dependent multipole components xint(t) and xext(t) can

be estimated. Identifying correlated temporal waveforms of xint(t) and xext(t)

then provides a way to identify sources with waveforms that have leaked into

both subspaces. After identifying such waveforms which are necessarily arti-

factual, a temporal projection operator can be formed to suppress them. The

method is known as spatiotemporal SSS or tSSS (Taulu and Simola, 2006).

The tSSS provides a way to overcome the limitations of the spatial SSS method.

Interference sources very close to the sensors can be effectively suppressed, en-

abling studies of subjects previously unsuited for MEG recordings. Examples

include patients with an implanted deep brain stimulator (Airaksinen et al.,

2011, 2012; Park et al., 2009), vagal nerve stimulators (Kakisaka et al., 2012;

Carrette et al., 2011; Song et al., 2009; Tanaka et al., 2009; Jin et al., 2013),

and subjects with dental implants (Hillebrand et al., 2013). Another novel type

of study is tSSS-facilitated analysis of single-trial responses (Taulu and Hari,

2009).

TSSS depends on a few user-defined parameters. The correlation limit (CL)

is used as a threshold to determine correlated temporal waveforms and affects

how many waveforms are projected out from the data. With CL too high (close

to one) some interference may remain; with CL too low, waveforms originat-

ing from brain sources may also be suppressed. (Medvedovsky et al., 2009)

investigated the effect of CL on alpha rhythm. No suppression of alpha was

found at CL 0.8–0.98; at CL 0.6, some evidence of suppression was observed.

Fine-tuning CL provides a way to improve interference suppression in demand-

ing situations. Another parameter is the epoch length used in computing the

temporal waveforms and correlations. A typical length is approximately 10

seconds. Shorter epoch lengths require less memory, but may lead to suppres-

sion of low frequencies; low-frequency signals increasingly resemble DC as the

33


window length is shortened, and therefore their correlation tends to increase.

The epoch-based processing may lead to slight temporal discontinuities in the

processed MEG data, which should be taken into account if necessary.

3.4.3 Transformation of data

As noted before, the sensor-level signals may be reconstructed from the esti-

mated multipole coefficients x as

φ= Sx, (3.34)

where the matrix S describes the coupling of the sensor array to the magneto-

static multipoles. By computing S with a different origin for the VSHR basis

functions, the position of the sensor array relative to the subject’s head can

be altered. This is equivalent to translation of the subject’s head to a differ-

ent position within the array. The translation enables standardization of head

positions between different measurement sessions, facilitating comparison of

sensor-level data (Taulu et al., 2005; Lioumis et al., 2007). It should be noted

that while the head may be translated closer to the sensors, the original signal-

to-noise ratio of the measurement cannot be increased. Thus, large shifts in

head position may result in noisy sensor-level data (Medvedovsky et al., 2007).

The data can also be transformed to a completely different sensor array sim-

ply by using the appropriate coupling matrix. This enables direct comparison

of data recorded with different MEG systems, similarly to the approach used in

(Burghoff et al., 2000).

3.4.4 Movement compensation and DC recordings

Head movements are a major problem in magnetoencephalography, especially

in measurements of children (Wehner et al., 2008) and certain patient groups

(e.g. Parkinson’s disease, epilepsy). The possibility to virtually translate the

subject’s head to different positions also enables compensation of head move-

ments, provided that continuous information about the head position is avail-

able (Taulu et al., 2005; Taulu and Kajola, 2005). The Elekta Neuromag®MEG

instrument provides continuous head position monitoring (cHPI) by continu-

ously energizing the head position indicator coils during the measurement.

Movement compensation is performed by translating the acquired head posi-

tion to a desired reference position at every point in time.

Interestingly, subject movement can also be utilized to record steady (DC)

fields. Normally, SQUID sensors will not detect DC fields, since the zero level

of the sensors is reset at the beginning of the measurement. With subject move-

34


ment, the DC sources will be modulated to time-varying fields that can be mea-

sured. When movement compensation is performed, fields of the DC sources

are demodulated and will thus appear as DC shifts on sensor-level data (Taulu

et al., 2004b). Similarly to full-band EEG (Vanhatalo et al., 2005), DC MEG can

reveal previously obscure aspects of brain function (Burghoff et al., 2004). On

the other hand, DC magnetic fields can be challenging to interpret due to mul-

tiple potential artifact sources (Cohen, 2003), and so far this topic has received

relatively little attention.

3.4.5 Fine calibration of the sensor array

The multipole basis represents the signals that are ideally measured by the

MEG device, and thus may be utilized for calibration. If the measured signals

have sufficiently low spatial frequency to not to be affected by the truncation

of the multipole basis, the remaining discrepancy between the basis and the

signal must be due to either sensor noise or calibration error. Thus, the calibra-

tion coefficients can be adjusted by an optimization procedure until an optimal

agreement between the data and the multipole basis is achieved. Such fine-

calibration is performed for the Elekta Neuromag® at the factory, increasing

the shielding by a factor of 3–10 (Taulu and Kajola, 2005) compared to the de-

fault calibration. A related approach is described in (Chella et al., 2012).

3.4.6 Other applications

Sensor-level MEG data are inherently redundant due to spatial oversampling.

In other words, the fields measured by nearby sensor units tend to be strongly

correlated. The typical number of channels (200–400) in modern instruments

clearly exceeds the number of degrees of freedom in the data, estimated to

be around 50–100 (Uutela, 2001). The multipole expansion presents the data

in nonredundant fashion, and the number of necessary multipole components

(typically about 80 for fields originating inside of the sensor array) is much

closer to the number of actual degrees of freedom in the data.

Since the multipole coordinates are free of redundancy, they are more suitable

for numerical operations than the original sensor-level data. The multipole

coordinates can be directly used as input to source modelling (Taulu, 2008).

Distributed-source approaches based on the multipole lead fields and estima-

tion of single equivalent current dipoles are both possible. As another example,

the orthogonality of the multipole lead fields enables straightforward estima-

tion of total information from the data, providing a new index to characterize

35


neuronal activity (Nenonen et al., 2007).

36

4. Sensor array and the multipolerepresentation

4.1 Introduction

The multipole-based approach to multichannel biomagnetic data is highly de-

pendent on the properties of the sensor array. There are two main factors

to consider: the geometry of the device and the accuracy of calibration and

parametrization. In this chapter, the relevant theory and methods are dis-

cussed.

4.2 Geometry of the sensor array

The structure of the multipole matrix S is determined entirely by the geometry

of the sensor array. To generate a well-conditioned basis matrix, the array

should be able to distinguish between different VSHR functions as clearly as

possible. A critical issue is the similarity of the basis vectors alm and blm for

fixed l,m. The following analysis is based on Publication II.

The general expansion for the magnetic field of (3.5) reads

B =∞∑

l=1

l∑m=−l

αlmνlm

rl+2 +∞∑

l=1

l∑m=−l

βlmrl−1ωlm.

The VSHR terms can be expanded in terms of the spherical harmonics Ylm as

νlm(θ,φ)rl+2 = 1

rl+2

[(−l−1)Ylm(θ,φ)er + ∂Ylm(θ,φ)

∂θeθ+ im

Ylm(θ,φ)sinθ

eφ

](4.1)

and

rl−1ωlm(θ,φ)= rl−1[lYlm(θ,φ)er + ∂Ylm(θ,φ)

∂θeθ+ im

Ylm(θ,φ)sinθ

eφ

], (4.2)

where er, eθ and eφ are the radial, polar, and azimuthal unit vectors, respec-

tively. The similarity of the VSHR functions is readily apparent; they differ

only in their radial component and the r-dependent multiplier.

37

Sensor array and the multipole representation

Next we examine the situation in terms of the sensor array. Without loss of

generality, assume pointlike magnetometer sensors with the kth sensor located

at r = rk, θ = θk, φ=φk, having the normal vector nk = nr,ker +nθ,keθ+nφ,keφ.

This sensor will then measure the corresponding kth elements of the alm and

blm basis vectors as

alm,k =νlm(θk,φk)

rl+2k

· (nr,ker +nθ,keθ+nφ,keφ) (4.3)

= 1rl+2

k

[−nr,k(l+1)Ylm(θk,φk)+nθ,k

∂Ylm

∂θ

∣∣∣∣θ=θk,φ=φk

+nφ,k imYlm(θk,φk)

sinθk

]

blm,k =rl−1k ωlm(θk,φk) · (nr,ker +nθ,keθ+nφ,keφ)

=rl−1k

[nr,klYlm(θk,φk)+nθ,k

∂Ylm

∂θ

∣∣∣∣θ=θk,φ=φk

+nφ,k imYlm(θk,φk)

sinθk

].

The k dependency of these expressions determines the direction of the basis

vectors alm and blm in the signal space. Identical dependency on k would mean

that the angle between the vectors is zero. Since the rk-dependent multiplier

differs between alm,k and blm,k, it is seen that radial separation (i.e. varying rk)

between the sensors improves the resolution of the VSHR functions, regardless

of which field components are measured.

Next, assume radial sensors, i.e. nk = er. Then

alm,k =−(l+1)

rl+2k

Ylm(θk,φk) (4.4)

blm,k = lrl−1k Ylm(θk,φk).

Now, alm and blm are distinguished only by their different dependency on rk.

In this case, if all sensors are located on a spherical surface (rk = R), alm and

blm can no longer be resolved. A similar analysis shows that alm and blm be-

come parallel in the case of tangential sensors on a spherical surface. However,

inclusion of both radial and tangential sensor orientations enables resolution

between alm and blm, even on a spherical surface. This is in accordance with

the result in (Taulu et al., 2005), concerning the singularity of the multipole

basis.

In conclusion, the resolution between the internal and external VSHR func-

tions can be improved by creating radial separation between the sensors and

measuring both radial and tangential field components. This result provides

theoretical motivation for the novel sensor geometries presented in Publication

II, summarized in the next chapter.

38


4.3 Calibration, imbalance and parametrization

The numerically computed multipole basis never exactly matches the signals

recorded by the MEG device. In addition to random sensor noise, the sensors

have manufacturing errors causing gradiometer imbalance, and their positions

and orientations are known only with finite precision. The same applies to the

calibration coefficients used for voltage-to-flux conversion. The SSS shielding

factor is approximately inversely proportional to the overall calibration accu-

racy of the device (Taulu et al., 2004b).

As suggested in Publication I and (Taulu and Kajola, 2005), the mismatch be-

tween the multipole basis and a signal vector φ can be quantified by computing

the subspace angle γ between them:

γ=∠(S,φ) (4.5)

When the signal φ is represented in the multipole basis S, the upper limit for

the reconstruction SNR is then 1/tanγ. The subspace angle can be used to study

the representation of signal vectors in presence of calibration errors and imbal-

ance. Ideally, all measurable signal vectors should be accurately representable

in terms of the multipole basis.

4.4 Evaluation of sensor arrays

The performance of the multichannel sensor array may be studied from dif-

ferent points of view. Traditionally, criteria such as total information (Kemp-

painen and Ilmoniemi, 1990; Nenonen et al., 2004; Schneiderman, 2013) and

dipole localization accuracy (Ahonen et al., 1991) have been used. However,

these don’t reflect the performance of the array in the multipole-based ap-

proach, so novel figures of merit are needed.

The condition number of the multipole basis and the canonical angles be-

tween the internal and external subspaces are simple to compute, but their

exact relationship to the performance of the sensor array is not clear. They are

also not sensitive to important properties of the array such as the level of gra-

diometer imbalance and calibration error. To obtain more concrete indicators

of performance, Monte Carlo simulations can be used to estimate the shield-

ing factor obtained by the spatial SSS method and the residual reconstruction

noise (Taulu et al., 2005; Taulu and Kajola, 2005). Array properties includ-

ing calibration errors, imbalance and parametrization errors can be included in

such simulations.

39


Suppose that the measured signal vector φe consists only of external inter-

ference, e.g. fields generated by sources with r > r1, where r1 is the radius of

the smallest sphere fully enclosing the sensors (section 3.2). In simulations,

φe may be the field of a magnetic dipole, which serves as a good model for a

source of interference. Estimate the multipole coefficients x from this signal

vector. The field generated by internal sources can then be reconstructed as

Sint xint, and should ideally be zero, since internal sources are not present. The

actual residual may be compared with the norm of the original signal vector φ

to obtain the SSS shielding factor

ξ= ‖φe‖‖Sint xint‖

. (4.6)

The shielding factor depends on the sensor–source distance, source position

and orientation, and the calibration of the sensor array. Thus, in simulations it

should be evaluated for several different interference sources. Multiple realiza-

tions of a randomly generated calibration error and imbalance can also be used.

An overall shielding factor can then be obtained by averaging the results. The

evaluation of shielding factor can also be performed on real data, as in (Taulu

et al., 2005) and Publication IV.

In addition to external interference, MEG data also contains spatially ran-

dom noise generated by e.g. electronics and the SQUID sensors themselves. It

is useful to know how such noise is affected by the multipole reconstruction pro-

cedure. Let φn be a signal vector consisting purely of Gaussian noise. Estimate

the corresponding multipole coefficients xn by the pseudoinverse or some other

procedure. Reconstruct the signal vector for internal sources φint,n as

φint,n = Sint xint,n, (4.7)

which represents the portion of the noise that falls into the internal subspace.

The relative reconstruction noise can then be computed as

nr = ‖φint,n‖/‖φn‖. (4.8)

Thus if nr > 1, the multipole reconstruction increases the magnitude of spa-

tially random noise. The result should be averaged over several realizations of

the random noise vector.

A single figure of merit that would capture all aspects of sensor array per-

formance does not yet exist. To evaluate array performance in Publication II

and Publication IV, we used the shielding factor, relative reconstruction noise,

the condition number of the multipole basis and the canonical angles between

the internal and external subspaces. These figures are somewhat interrelated,

40


e.g. the canonical angles seem to be closely correlated with the shielding factor,

though the exact relationship is not clear.

41


42

5. Summary of results

5.1 Introduction

This chapter provides a summary of the publications. In addition, sections 5.6

and 5.7 present previously unpublished results.

5.2 Publication I: effects of calibration, imbalance andparametrization

In Publication I, the accuracy of the multipole-based representations for var-

ious signals was studied by simulation, in the presence of parametrization,

calibration and imbalance errors. The subspace angle method in section 4.3

was used. Magnetic dipoles and current dipoles in a spherical conductor model

were used to model interference and brain sources, respectively, and their sig-

nals were compared with the corresponding SSS bases.

We found that gradiometer imbalance is the main factor affecting the repre-

sentation of the external interference signal, with imbalance around 0.5% or

larger causing a significant error. The large effect can explained by the fact

that the relative error caused by imbalance is largest for nearly homogeneous

signals. In contrast, signals originating from inside the helmet have large spa-

tial gradients and are not significantly affected by imbalance. However, if the

interference signal cannot be represented accurately by the Sext basis, the re-

construction of the internal signal will also be severely distorted.

The practical implication of the results is that wire-wound gradiometers may

be difficult to manufacture with the required precision; thin-film gradiometers

have an advantage in this respect. Compared to the effect of imbalance, calibra-

tion errors and inaccuracy of sensor parametrization caused relatively minor

errors.

43

Summary of results

5.3 Publication II: effects of symmetry and novel sensor arraydesigns

In Publication II, we considered the effect of the sensor array geometry on the

multipole basis. As elaborated in section 4.2, the resolution of different VSHR

components can be improved by creating radial separation between the sensors

and measuring different field components at the same time. Based on this

result, we created various models of two-layer sensor arrays and vector sensors.

The arrays were then evaluated by computing the condition number of the SSS

basis, shielding factor and the relative reconstruction noise.

The results indicate markedly improved performance of the signal space sep-

aration method on two-layer and vector arrays, especially combination of two

sensor layers and vector sensors on the second layer. The role of the second sen-

sor layer is not to detect brain signals, but to improve the characterization and

resolution of magnetic fields originating from internal and external sources.

Therefore, a small number of sensors (about 30–50) in the second layer is suf-

ficient for a significant improvement in performance. This was later confirmed

experimentally in Publication IV. We compared the two-layer array with a tra-

ditional reference array geometry, which did not perform equally well in the

multipole approach. The reason is apparently that a typical reference array

subtends a smaller solid angle than the full second layer of sensors, and there-

fore does not characterize the magnetic field as fully.

On a two-layer geometry with vector sensors on the second layer, a shield-

ing factor of about 1000 and relative reconstruction noise below 0.5 are attain-

able in the simulation. The corresponding numbers for the 306-channel Elekta

Neuromag® device are approximately 200 and 1.2.

5.4 Publication III: validation of tSSS and movement compensation

A validation study of the tSSS method and multipole-based movement com-

pensation was performed in Publication III, using auditory and somatosensory

responses in 20 healthy volunteers. In one condition, subjects were instructed

to continuously move their heads inside the helmet. In another condition, addi-

tional interference was generated by attaching magnetized particles to subjects’

heads. The data were then processed with tSSS and movement compensation.

Single equivalent current dipoles in a spherical conductor were used as the

source model. As the reference, we used SSS-processed data in a stationary

measurement condition.

44

Summary of results

Figure 5.1. Simulated sensor arrays used in Publication II. Top: single-layer array with 234radial sensors. Bottom: two-layer array with 78 radial sensors on the second layer.

The methods were able to reliably recover data corrupted by movement and

interference. The source locations obtained by dipole fitting differed by 5–7 mm

from the reference. Response amplitudes and latencies were not significantly

affected.

5.5 Publication IV: adding tangential sensors to an existing array

Arrays with multipole sensor layers and vector sensors appear promising based

on the results of Publication II, but designing and constructing a completely

new sensor array is a large undertaking. Alternatively, existing systems could

be modified. In Publication IV, we studied the addition of tangential sensors to

the existing 306-channel Elekta Neuromag® system. Since the sensors in the

original array are predominantly oriented in the radial or near-radial direc-

tion, the additional tangential sensors are expected to provide complementary

information about the magnetic field.

To test the concept and optimize the arrangement of the tangential sensors,

we first performed computer simulations. We started by identifying 46 locations

in the sensor array where physical constrains would allow insertion of tangen-

tial sensor adapters. In the simulations, each new sensor was then placed in

the location that provided the maximal improvement in the condition number

45

Summary of results

κ. The shielding factor and the mean canonical angle between subspaces were

then determined for the new sensor configuration; the results are shown in fig-

ure 5.2. Based on these results and practical constraints, 18 additional triple-

sensors were selected as the final configuration, as illustrated in figure 5.3.

0 5 10 15 20 25 30 35 40 45 5010

15

20

25

30

35

40

Number of added sensors

Mea

n pr

inci

pal a

ngle

bet

wee

n S

SS

sub

spac

es (

°)

magnetometerstriple−sensors

0 5 10 15 20 25 30 35 40 45 50200

250

300

350

400

450

500

Number of added sensors

Shi

eldi

ng fa

ctor

magnetometerstriple−sensors

Figure 5.2. Simulation of adding tangential sensors to a 306-channel system. Improvement ofmean principal angle between subspaces (top) and the SSS shielding factor (bottom)as a function of added sensor units.

Instead of optimizing the position of one sensor at a time, a global optimization

procedure could have produced better results. Nevertheless, as long as the ad-

ditional sensors are distributed approximately equally around the helmet, the

exact placement of sensors does not have a large effect on the performance of

the array.

For the finalized configuration, the simulations predicted 74% increase in the

46

Summary of results

Figure 5.3. Adding tangential sensors to the Elekta Neuromag® 306-channel system. The finalconfiguration with 18 tangential magnetometers and 360 channels is shown.

SSS shielding factor and 44% decrease in residual reconstruction noise. As

seen in figure 5.2, the shielding factor and mean principal angle between the

internal and external subspaces are highly correlated.

Next, a prototype device was constructed. Due to the modularity of the sys-

tem, only relatively minor changes in hardware and software were required to

accommodate the tangential sensor units. Fine calibration and crosstalk cor-

rection were performed for the new device. Afterwards, the shielding factor

was evaluated using empty-room data and artificial interference generated by

two types of coils. For further verification, evoked auditory and somatosensory

responses were recorded from volunteer subjects.

The results were largely in agreement with the simulations. The shielding

factor was improved by about 100%, while the residual noise after SSS recon-

struction decreased by about 20%. In the experiments, the evaluation of resid-

ual noise cannot be performed in exactly the same way as in simulations, which

probably explains the discrepancy.

The study practically demonstrates the benefits of comprehensively sampling

the magnetic field, validating the theory and results of Publication II. The

tangential sensors improve the resolution between the internal and external

multipole subspaces, resulting in increased shielding against interference and

reduced reconstruction noise. The subject data was processed with the temporal

47

Summary of results

extension of SSS, showing that the improved spatial sampling of the magnetic

field also has a direct benefit for the tSSS method.

A modest number of additional sensors was found to be sufficient for improv-

ing the characterization of the multipole subspaces. Also, the simulations in-

dicate that using additional single-channel magnetometer sensors instead of

the triple-sensor units would provide much of the same improvements. We

emphasize that the positioning of the additional sensors is not designed to for

detection of cortical magnetic fields, and thus their main function is different

from the tangential sensors investigated in previous studies (Haueisen et al.,

2012; Arturi et al., 2004; Di Rienzo et al., 2005; Nara et al., 2007).

5.6 Symmetry reduction by tilting sensors

As previously noted, the spherical symmetry of the sensor array is detrimental

to its performance, and the symmetry may be reduced by incorporating both

radial and tangential directions of measurement. Another way of breaking the

symmetry is using diversely oriented sensors. The simulation below shows the

effect of tilting the sensors in random directions.

A hypothetical 256-channel spherical array (radius=10 cm) with pointlike

magnetometers was used. The multipole basis had dimensions Lint = 7,Lext = 3,

with the origin at the symmetry center of the sensor array. At 0 degrees of tilt,

the sensors have radial normal vectors. Each sensor is then tilted to a random

direction by the specified angle, and the SSS basis is recomputed for the modi-

fied array. The resulting condition number and mean subspace angle are shown

in figure 5.4, as a function of the tilt angle.

At zero degrees of tilt, the condition number indicates singularity of the mul-

tipole matrix, in accordance with the results in section 4.2. Correspondingly,

the principal angles are all zero, since alm and blm are identical. Apparently,

even few degrees of tilting is enough to disrupt the symmetry, resulting in a rea-

sonable condition number and resolution between the subspaces. At 90 degrees

of tilt, the sensors become tangential and the basis becomes singular again. In-

terestingly, the resolution according to the mean subspace angle is optimal at

near-tangential orientation (about 70 degree of tilt). The results do not signifi-

cantly depend on Lint and Lext.

One disadvantage of tilted sensors is that finite-area pickup loops would be

slightly lifted relative to the cortical surface, increasing the sensor–source dis-

tance by approximately d/2sin(φ), where d is the diameter of the pickup loop.

However, with small degrees of tilt (<10 degrees) and typical pickup loop di-

48

Summary of results

0 10 20 30 40 50 60 70 80 9010

2

104

106

108

1010

1012

1014

1016

Tilt angle (degrees)

Con

ditio

n nu

mbe

r of

SS

S b

asis

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16

Tilt angle (degrees)

Me

an

prin

cip

al a

ng

le b

etw

ee

n s

ub

sp

ace

s (

de

gre

es)

Figure 5.4. Condition number of the whole multipole basis (top) and mean subspace angle be-tween external and internal bases (bottom) as a function of sensor tilt. Each sensorwas tilted in a random direction. 0 degrees refers to radial orientation.

49

Summary of results

ameters, the lift would be smaller than 2 mm, with a rather small effect on

the SNR. Another disadvantage is that the accurate manufacturing and char-

acterization of a tilted-sensor array might be somewhat more difficult than of a

regular one.

The orientation pattern of the sensors affects the characterization of the mag-

netic field (e.g. nearby sensors should probably be oriented differently, to max-

imize the obtained information). Global optimization techniques such as sim-

ulated annealing could be used to search for optimal orientation patterns. To

speed up computations, the VSHR functions can be precomputed at the sensor

points and simply projected on the rotated sensor normals, which should pro-

vide a good approximation for small rotation angles. The condition number of

the multipole basis or the mean principal angle between internal and external

multipole subspaces are good candidates for the optimization target.

5.7 Alternatives to the spherical multipole basis

The vector spherical harmonics obtained by solving Laplace’s equation in spher-

ical coordinates are a good model for most MEG sensor array geometries. In

principle, the equation could also be solved in e.g. the prolate spheroidal coor-

dinate system, to provide an optimal fit to the sensor array.

Alternatively, instead of starting from an analytical solution to Laplace’s equa-

tion, a Monte Carlo method can be used to obtain bases for the signal sub-

space. The idea is to populate the desired region with a large number of ran-

dom sources, solve the forward problem for all sources, and then determine the

corresponding most significant directions in the signal space.

In practice, this may be accomplished using the singular value decomposition

(SVD). Assign M random sources to the selected region, compute the resulting

signal space vectors φ1 . . .φM , and place them into a column matrix:

Φ= [φ1 . . .φM

]. (5.1)

The SVD of Φ may be written as

Φ=USVT (5.2)

where the columns of U and V are called the left-singular and right-singular

vectors of Φ, respectively. S contains the corresponding singular values on its

diagonal. As a basic property of the SVD, the columns of U form an orthonor-

mal basis for the range of Φ. Further, the columns are assigned in order of

importance; the first column vectors, corresponding to the largest singular val-

ues, are the most significant signal space directions spanned by the sources. An

50

Summary of results

appropriate number of basis vectors can be chosen by examining the spectrum

of singular values. For a similar application of the SVD, see (Jerbi et al., 2002),

where field distributions produced by extended source regions were compared

with the dominant modes of the current multipole expansion of the magnetic

field.

In principle, subspaces could be estimated for arbitrary spatial regions. Here

the method is demonstrated by obtaining subspaces corresponding to spherical

regions inside and outside the sensor array. They are then compared to VSHR

subspaces with Lint = 8, Lext = 3.

Figure 5.5 shows M = 5000 dipole locations inside the 306-channel Elekta

Neuromag® sensor array. The locations were chosen inside a spherical region

(r=8.5 cm) centered at [0,2,0] cm, which is also the origin of the VSHR functions.

A randomly oriented tangential current dipole was placed at each location, and

the corresponding signal space vectors were computed to obtain Φ. A spherical

volume conductor model was used for the forward calculation. Next, SVD was

performed on Φ. Components corresponding to the 80 largest singular values

were chosen to match the dimension of the VSHR basis. The external basis

was obtained similarly, with random magnetic dipoles inside a spherical shell

(inner radius = 0.5 m, outer radius = 6 m). 15 external components were chosen,

matching the dimension of the external VSHR basis.

Figure 5.5. Randomly generated source locations for subspace estimation. View from the bot-tom of the 306-channel sensor array.

The internal basis vectors are illustrated in figure 5.6. There is a clear sim-

ilarity between the VSHR and the MC basis vectors. The condition numbers

51

Summary of results

are 280 and 323 for the VSHR and MC basis, respectively. The mean princi-

pal angles between the internal and external subspaces are 9.7° (VSHR inter-

nal/external) and 11.3° (MC internal/external).

The external VSHR and MC bases are almost identical, with a mean principal

angle of 0.7 degrees between them. In contrast, the internal subspaces clearly

differ in some dimensions. This seems to indicate that the VSHR basis has

dimensions that are absent from the MC basis. They possibly correspond to

VSHR functions that are poorly coupled to the sensor array and thus do not

become significant in the SVD-based analysis.

Next, the shielding factor was obtained according to (4.6), where randomly

oriented magnetic dipoles were used as a model for the interference sources.

They were positioned at a distance 0.5 m–3.0 m from the origin. The result is

shown in figure 5.7.

Based on these results, it seems that VSHR and MC methods offer similar

performance. However, obtaining the basis by the Monte Carlo method may

have advantages in some situations. If the the sensor array does not conform

well to the spherical geometry, some regions near the array may be poorly rep-

resented by the VSHR basis. With the MC basis, arbitrary source regions can be

defined to obtain an optimal fit to the array. Another advantage is that the SVD

procedure automatically finds components that are well coupled to the sensor

array. In contrast, some VSHR functions may be poorly coupled to the array,

increasing reconstruction noise. By evaluating the expected SNR of each ba-

sis vector, such basis vectors can be neglected in the reconstruction, as is done

in the commercial MaxFilter® implementation of the signal space separation

method.

The magnetostatic multipole coefficients constitute a device-independent har-

monic expansion (section 3.4.3). In contrast, the MC coefficients are determined

by the properties of the particular sensor array, and are thus device-dependent.

Therefore in the MC approach, it is not possible to perform e.g. translation of

head position simply by switching to a different basis. The computation times

for the MC and VSHR bases are roughly similar.

The mismatch between the computed multipole model and the measurements

was discussed earlier. In particular, it was noted that gradiometer imbalance

may cause large errors in the representation of signals. One possible way to

alleviate this problem would be to measure actual interference by e.g. moving a

coil around the sensor array. Using the SVD procedure, this data could then be

included in the basis. This “real-world” basis would automatically include the

effects of imbalance and other nonideal properties of the sensor array.

52

Summary of results

Figure 5.6. Vector spherical harmonic+radial (VSHR) and Monte Carlo (MC) internal basis vec-tors in signal space, visualized as interpolated color-coded plots on top of the sensorhelmet. The color (arbitrary scale) represents the magnitude of the basis vectors onmagnetometer channels.

53

Summary of results

0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

Distance (m)

Shi

eldi

ng fa

ctor

VSHMC

Figure 5.7. Shielding factor against external interference with the spherical harmonic (VSHR)and Monte Carlo (MC) bases.

54

6. Conclusions

The multipole expansion provides an elegant way to describe multichannel bio-

magnetic signals. It has several practical applications which have been in clini-

cal and research use already for some years, as reviewed in this thesis. New ap-

plications are still emerging, e.g. multipole-based source localization, which has

received relatively little attention so far. Additionally, data review and analysis

software that would take full advantage of the multipole-based approach is not

yet available.

The role of the multichannel biomagnetic instrument is to characterize the

magnetic field as completely as possible. Since the magnetostatic multipole

expansion fully describes the measurable magnetic fields, it is a useful tool

for investigating the properties of the sensor array. In this work, we used the

expansion to gain insight into the performance of different sensor arrays. We

were able to clarify the requirements of the multipole-based approach, and to

elucidate the limitations of traditional MEG sensor array geometries. We also

introduced new geometries based on these considerations.

The practical significance of the novel sensor arrays introduced in this work

is improved performance in methods such as SSS and tSSS. The improvement

also has a more fundamental significance, as it implies a more complete char-

acterization of the magnetic field. Therefore, the new arrays can be expected

to perform better in a variety of applications. This is especially important for

clinical magnetoencephalography, where robustness in the face of varying mea-

surement conditions is needed.

The present work indicates some possible directions for future research. First,

the Monte Carlo approach for determining signal subspaces offers an alterna-

tive or complementary method to the spherical multipole expansion. This ap-

proach could be studied further e.g. in sensor arrays with unusual geometries,

or when the source region otherwise needs to be restricted or defined in arbi-

trary fashion. Second, the preliminary simulation of diversely oriented sensors

55

Conclusions

demonstrates an alternative approach to orthogonal vector sensors. To continue

this line of investigation, optimal patterns of sensor orientations could be stud-

ied by global optimization methods such as simulated annealing. A third topic

of future study would be better figures of merit for multichannel arrays, with

the ultimate goal of characterizing the performance of the array as completely

as possible.

In conclusion, the multipole approach gives insight into the optimal multi-

channel measurement of magnetic fields, with applications for design of bio-

magnetic sensor arrays. In the present work, we have examined this approach

to sensor array optimization from a theoretical perspective and with numerical

simulations. We have also built a prototype device that successfully demon-

strates the approach. Further, a novel pediatric MEG sensor array based on

the ideas presented here is soon expected to be operational at Children’s Hos-

pital Boston, USA. The results obtained in this work would also be applicable

to other biomagnetic multichannel instruments, e.g. magnetocardiography.

56

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62

Errata for publications

Publication II

Equation (10) should read

νlm(θ,φ)rl+2 = 1

rl+2

[(−l−1)Ylm(θ,φ)er + ∂Ylm(θ,φ)

∂θeθ+ im

Ylm(θ,φ)sinθ

eφ

]

i.e. θk and φk should not appear in this equation.

63

Errata for publications

64

9HSTFMG*afhbab+

ISBN 978-952-60-5710-1 ISBN 978-952-60-5711-8 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 ISSN 1799-4942 (pdf) Aalto University School of Science Department of Biomedical Engineering and Computational Science

BUSINESS + ECONOMY ART + DESIGN + ARCHITECTURE SCIENCE + TECHNOLOGY CROSSOVER DOCTORAL DISSERTATIONS

Aalto-D

D 7

9/2

014

Modern biomagnetic devices measure the magnetic field generated by the human brain at hundreds of locations around the head. The measured field can be efficiently described by a multipole expansion. The expansion has several applications, such as interference suppression and compensation of subject movements. In this work, we clarify the requirements of the multipole-based approach. We also propose new sensor array geometries that would significantly enhance the performance of the method. The feasibility of the ideas is demonstrated by a novel prototype device. The significance of the results is better interference suppression and increased robustness, which is especially important in clinical applications of biomagnetism.

Jussi Nurm

inen T

he magnetostatic m

ultipole expansion in biomagnetism

: applications and implications

Aalto

Unive

rsity

Department of Biomedical Engineering and Computational Science


Jussi Nurminen

DOCTORAL DISSERTATIONS

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