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3D MULTI-FREQUENCY CONDUCTIVITY IMAGING

VIA CONTACTLESS MEASUREMENTS

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

MIDDLE EAST TECHNICAL UNIVERSITY

BY

KORAY ÖZDAL ÖZKAN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR

THE DEGREE OF DOCTOR OF PHILOSOPHY

IN

ELECTRICAL AND ELECTRONICS ENGINEERING

JANUARY 2013

iii

Approval of the thesis



submitted by KORAY ÖZDAL ÖZKAN in partial fulfillment of the requirements for the degree

of Doctor of Philosophy in Electrical and Electronics Engineering, Middle East Technical

University by,

Prof. Dr. Canan Özgen

Dean, Graduate School of Natural and Applied Sciences ______________

Prof. Dr. Gönül Turhan Sayan

Head of Department, Electrical and Electronics Engineering ______________

Prof. Dr. Nevzat Güneri Gençer

Supervisor, Electrical and Electronics Engineering, METU ______________

Examining Committee Members:

Prof. Dr. B. Murat Eyüboğlu

Electrical and Electronics Engineering Dept., METU ______________

Prof. Dr. Nevzat Güneri Gençer


Prof. Dr. Osman Eroğul

Biomedical Engineering Center, GMMA ______________

Prof. Dr. Kemal Leblebicioğlu


Assoc. Prof. Dr. Yeşim Serinağaoğlu Doğrusöz


Date: 22.01.2013

iv

I hereby declare that all information in this document has been obtained and presented in

accordance with academic rules and ethical conduct. I also declare that, as required by these

rules and conduct, I have fully cited and referenced all material and results that are not

original to this work.

Koray Özdal ÖZKAN

v

ABSTRACT



ÖZKAN, Koray Özdal

Ph.D., Department of Electrical and Electronics Engineering

Supervisor: Prof. Dr. Nevzat Güneri Gençer

January 2013, 117 Pages

In this study, 2D and 3D multi-frequency conductivity imaging has been performed. The study

composed of two parts, namely, theoretical studies and experimental studies. In theoretical studies,

sensitivity analysis of circular coil sensors has been performed. In addition to this, 3D inverse

problem solution has been performed. The inverse problem is solved by employing the Steepest-

Descent Method. In hardware studies, three multi-frequency data acquisition systems (DAS),

namely, DAS with CM-2251 Data Acquisition Card, DAS with Single Coil Sensor and DAS with

Array Coil Sensor have been realized to image electrical conductivity of biological tissues. The

systems differ in their sensor type and data collection hardware.

In a magnetic induction imaging system it is not straightforward to make quantitative statements

about the relationships between the resolution, accuracy, conductivity contrast, and noise.

However, knowing these relationships is essential in designing effective imaging systems. In this

study, a theoretical work is conducted to reveal the relationships between these parameters. For

this purpose, a simple detection system is analyzed that uses spatially uniform (sinusoidally

varying) magnetic fields for magnetic-induction. A circular coil is used for magnetic field

measurement. A thin cylinder with a concentric inhomogeneity is used as a conductive body. An

analytical expression is developed that relates coil and body parameters to the measurements. A

vi

set of six rules is found that reveal the relationships between resolution, accuracy, conductivity

contrast, and noise. The results are interpreted by numerical examples.

The DAS with CM-2251 Data Acquisition Card employs differential coil sensor consisting two

differentially connected receiver coils and a transmitter coil. In this setup, the Lock-in amplifier

and the Digital Multimeter are replaced with the CM-2251 Data Acquisition Card. The system is

capable of operating at a multi-frequency range between 20-60 kHz. The DAS with Single Coil

Sensor employs a single coil as a sensor. It is determined that the impedance change will be

maximum in a single coil sensor, and this yields a result that the single coil sensor is the most

efficient sensor in the sense of sensitivity. The system is capable of operating at a multi-frequency

range between 10-100 kHz. The DAS with Array coil Sensor employs an array sensor consisting

of a 1x4 array of differential coils. The system is capable of operating between 10-100 kHz. Main

advantage of the system is the time. By utilizing a 1x4 array sensor, the time required for

collecting data decreases by 4 times considering a system that uses a single sensor. The

experiments were performed and data were collected by a user interface program developed for

this purpose. The user interface program was based on Agilent VEE and MATLAB. The

sensitivity, i.e., the response of the systems to conductivity variations, was tested at each operating

frequency by using resistive ring phantoms. The results are consistent with the theory stating that

the measured signals are linearly proportional with the square of frequency. The SNR of the array

coil system was calculated at each operating frequency. It was observed that the SNR of the

system increases as the frequency increases, as expected. Spatial resolution of the array coil

system was tested at each operating frequency by using agar phantoms. The results show that the

resolving power of the system to distinguish image details increases as the frequency increases.

2D conductivity distributions of objects prepared by agar phantoms were reconstructed by

employing Steepest-Descent algorithm. The geometries and locations of the reconstructed images

matched with those of real images. 3D conductivity distribution of objects was also reconstructed.

The results show the potential of the methodology for clinical applications.

Keywords: Medical Imaging, Non-invasive Imaging, Non-destructive testing, Conductivity

imaging, Electrical impedance imaging, Magnetic induction tomography, Sensor array.

vii

ÖZ

DOKULARIN ELEKTRİKSEL İLETKENLİKLERİNİN

DOKUNMASIZ YÖNTEMLERLE 3 BOYUTLU ÇOK

FREKANSLI GÖRÜNTÜLENMESİ

ÖZKAN, Koray Özdal

Doktora, Elektrik-Elektronik Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Nevzat Güneri Gençer

Ocak 2013, 117 Sayfa

Bu çalışma teorik ve donanım olmak üzere iki kısımdan oluşmaktadır. Teorik çalışmalarda

dairesel bobinli algılayıcıların duyarlılık analizleri gerçekleştirilmiştir. Buna ilave olarak 3 boyutlu

geri problem çözümü gerçekleştirilmiştir. Geri problem çözümünde Steepest-Descent metodu

kullanılmıştır. Donanım çalışmaları kapsamında, dokuların elektriksel iletkenliğini görüntülemek

amacıyla üç adet çok frekanslı veri işleme sistemi gerçekleştirilmiştir. Bunlar, CM-2251 Veri

İşleme Kartı kullanan sistem, Tek Algılayıcılı sistem ve Algılayıcı Dizilimli sistem olarak

isimlendirilmiştir. Sistemler algılayıcı tipleri ve veri toplama donanımları noktalarında

ayrışmaktadırlar.

Bir manyetik indükleme-manyetik ölçme sisteminde, çözünürlük, doğruluk, iletkenlik kontrastı ve

gürültü arasındaki bağlantılarla ilgili somut yorumlar yapılması çok da kolay değildir. Bununla

birlikte, bu bağlantıların bilinmesi, etkin görüntüleme sistemlerinin tasarlanması için önemlidir.

Bu çalışmada, söz konusu bağlantıların açığa çıkartılması için teorik çalışmalar yapılmıştır. Bu

doğrultuda, manyetik indüklenme için uzamsal olarak değişmeyen (uniform) bir manyetik alan

kullanan basit bir algılama sistemi analiz edilmiştir. Manyetik alan ölçümü için dairesel bobin

kullanılmıştır. İletken obje olarak konsentrik anlamda homojen olmayan yapı barındıran silindirik

bir obje kullanılmıştır. Ölçümler ile bobin ve cisim parametreleri arasındaki ilişkiyi gösteren

viii

analitik ifadeler geliştirilmiştir. Çözünürlük, doğruluk, iletkenlik kontrastı ve gürültü arasındaki

bağıntıları veren altı kural seti bulunmuştur. Sonuçlar sayısal örneklerle yorumlanmıştır.

CM-2251 Veri İşleme Kartı kullanan sistem bir verici ve iki farksal bağlı alıcı bobinden oluşan bir

farksal algılayıcı kullanmaktadır. Bu düzenekte faza kilitli yükselteç ve dijital multimetre yerine

CM-2251 Veri İşleme Kartı kullanılmaktadır. Sistem 20kHz-60kHz frekans aralığında

çalışabilmektedir. Tek algılayıcılı sistem algılayıcı olarak tek bir bobin kullanmaktadır. Tek

bobinli algılayıcıda empedans değişiminin maksimum olduğu gösterilmiştir; bu durum tek

algılayıcılı sistemin duyarlılık anlamında en başarılı algılayıcı olduğu sonucunu doğurmaktadır.

Sistem 10kHz-100 kHz frekans aralığında çalışabilmektedir. Algılayıcı Dizilimli sistem 1x4

boyutunda farksal bobinlerden oluşan algılayıcı dizilimi kullanmaktadır. Sistem 10kHz ile 100kHz

frekans aralığında çalışmaktadır. Sistemin en önemli avantajı hızlı olmasıdır. 1x4 boyutunda

algılayıcı dizilimi kullanıldığında, veri toplamak için gerekli sure, bir algılayıcı kullanılan sisteme

göre dört kat daha az olacaktır. Bu amaçla hazırlanan kullanıcı arayüzü kullanılarak deneyler

yapılmış ve veri toplanmıştır. Kullanıcı arayüzü Agilent VEE ve MATLAB tabanlıdır. İletkenlik

değişimlerine sistemin tepkisi olarak tanımlanan duyarlılık direnç fantomları kullanılarak her bir

çalışma frekansı için belirlenmiştir. Sonuçların, ölçülen sinyalin frekansın karesi ile orantılı

olması gerektiğini söyleyen teori ile uyumlu olduğu gözlemlenmiştir. Her bir çalışma frekansı için

sistemin sinyal-gürültü oranı hesaplanmıştır. Beklenildiği gibi, çalışma frekansı arttığında sistemin

sinyal-gürültü oranının arttığı gözlemlenmiştir. Sistemin uzamsal çözünürlüğü her bir çalışma

frekası için agar fantomları kullanılarak belirlenmiştir. Sonuçlar frekans arttıkça sistemin

çözünürlüğünün arttığı gözlemlenmiştir. Agar fantomlarla oluşturulan cisimlerin iki boyutlu

görüntüleri Steepest–Descent algoritması kullanılarak elde edilmiştir. Görüntülerin konumları ve

şekilleri gerçek cisimlerinkiler ile uyumlu olduğu gözlemlenmiştir. Aynı zamanda cisimlerin üç

boyutlu görüntüleri elde edilmiştir. Sonuçlar metodun klinik uygulamalar için potansiyelini ortaya

koymaktadır.

Anahtar Sözcükler: Tıbbi görüntüleme, Girişimsel olmayan görüntüleme, Manyetik indükleme,

Elektriksel empedans görüntüleme, Algılayıcı dizilimi

ix

To Dr. Esra EROĞLU ÖZKAN

And in memories of

Refik ARISAN

Zahide ARISAN

Ali İhsan ARISAN

x

ACKNOWLEDGEMENTS

I am deeply grateful to Prof. Dr. Nevzat G. GENÇER for his invaluable support, guidance and

endless patience throughout this study. No doubt that Prof. Dr. Gençer is one of the most

influential people in my life.

I would like to utter my gratitude to Prof. Dr. Murat EYUBOGLU and Prof. Dr. Osman EROĞUL

for their patience and understanding. I would like to express my special gratitudes to Prof. Dr.

Kemal LEBLEBİCİOĞLU, Assoc. Prof. Dr. Yeşim SERİNAĞAOĞLU DOĞRUSÖZ, Prof. Dr.

Gürsevil TURAN, Prof. Dr. İsmet ERKMEN, Prof. Dr. Aydan ERKMEN and Prof. Dr. Aydın

ALATAN.

I would like to utter my special thanks to my brother Dr. İ. Evrim ÇOLAK for his endless support

and encouragement during the last decade of my life.

I would like to thank to my laboratory friends Balkar ERDOĞAN, Feza CARLAK, Can Barış

TOP, Mürsel KARADAŞ, Azadeh KAMALİ and Reyhan ZENGİN. I would like to express my

special thanks to Balkar ERDOĞAN for his invaluable technical supports and discussions.

Invaluable contributions of Reyhan ZENGİN to this thesis are appreciated.

I would like to thank to Tümer DOĞAN for his supports and encouragements. And my colleagues,

Kenan ÖZCAN, Bektaş ARALIOĞLU, Erdal AKBULUT, Salih Eren BALCI and Cüneyt

KARACA for their support and friendship in ASELSAN.

I would like to thank to Onur Kaan BALCI, Belma BALCI and Salih Eren BALCI for their lovely

friendship. I would like to express my special thanks to Mustafa KÖSE for his encouragements.

This thesis was supported by The Scientific & Technological Research Council of Turkey

(TUBİTAK) (r-1001 Project, Project no: 106E170). The author was supported by The Scientific &

Technological Research Council of Turkey (TUBİTAK) from September 2006 to February 2011.

I would like to thank to TUBITAK for the supports.

xi

I would like to thank to my mother Hikmet ÖZKAN and my family for their endless support

throughout this thesis. The encouragements and motivations of my father Hacı Mustafa ÖZKAN,

my mother-in-love Ayşegül EROĞLU and my aunt Ayten ERBAŞ are appreciated.

I would like to utter my special thanks and respects to my father-in-love Faruk EROĞLU. During

those exhausting days, I felt that he was always with me and believed in me.

And my beloved wife Dr. Esra EROĞLU ÖZKAN…

She stood by me in everything. She has done so much for me. Over and over and over again, she

has sacrificed herself for me literally. And now it’s my turn… And all I am trying to say is that

Esracan, you are the best of me.

xii

TABLE OF CONTENTS

ABSTRACT ..................................................................................................................................... v

ÖZ .............................................................................................................................................. vii

ACKNOWLEDGEMENTS .............................................................................................................. x

TABLE OF CONTENTS ............................................................................................................... xii

LIST OF TABLES ........................................................................................................................ xvi

LIST OF FIGURES ...................................................................................................................... xvii

CHAPTERS

1 INTRODUCTION ......................................................................................................................... 1

1.1 Medical Imaging ................................................................................................................ 1

1.2 Electrical Impedance Imaging ........................................................................................... 2

1.3 Electrical Conductivity Imaging via Contactless Measurements ....................................... 4

1.4 Multi-frequency studies ...................................................................................................... 9

1.5 Patent Applications for Magnetic Induction Tomography ............................................... 10

1.6 Motivation and the Scope of the Thesis ............................................................................ 12

2 THEORY ..................................................................................................................................... 15

2.1 Forward Problem ............................................................................................................. 16

2.1.1 General Formulation ........................................................................................................ 16

2.1.2 Single-Coil Sensor ............................................................................................................ 19

2.2 Inverse Problem ............................................................................................................... 20

3 SENSITIVITY ANALYSIS OF CIRCULAR COIL SENSORS ................................................ 23

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3.1 Modelling the Imaging System ......................................................................................... 23

3.1.1 Introduction ...................................................................................................................... 23

3.1.2 Analytical Model .............................................................................................................. 25

3.1.2.1 Mutual Inductance Between Coaxial Coils ................................................................... 27

3.1.2.2 Current Flowing in a Cylindrical Body ........................................................................ 27

3.1.3 Numerical Model .............................................................................................................. 29

3.1.4 Comparison of the Analytical Model and the Numerical Model ...................................... 29

3.2 Sensitivity Analysis of The Imaging System by using The Analytical System Model ........ 32

3.2.1.1 The Relationship Between Sensitivity, Conductivity Contrast, Spatial Resolution And

Noise ............................................................................................................................. 32

3.2.1.2 Summary and Comments............................................................................................... 36

3.2.2 The Sensitivity of the Imaging System with Impedance Analysis of the Sensor ................ 37

3.2.3 The Sensitivity of The Imaging System with Signal-to-Noise Ratio (SNR) Analysis ......... 41

3.2.3.1 SNR Analysis of the Single coil Sensor ......................................................................... 41

4 HARDWARE STUDIES ............................................................................................................. 45

4.1 Principle of Data Acquisition ........................................................................................... 45

4.2 Sensor Design ................................................................................................................... 47

4.3 Data Acquisition Systems ................................................................................................. 50

4.3.1 Data Acquisition System with CM-2251 Data Acquisition Card...................................... 50

4.3.1.1 Experimental Setup and Operation principle ............................................................... 50

4.3.1.2 Sensitivity to Conductivity Variations ........................................................................... 51

4.3.2 Data Acquisition System with Single Coil Sensor ............................................................ 52



4.3.3 Data Acquisition with Array Coil Sensor ......................................................................... 56


4.3.3.1.1 Characteristics of the Coils Composing the Sensor Array ................................. 60

xiv

4.3.3.1.2 The Controller Card .......................................................................................... 67


5 SINGLE FREQUENCY STUDIES ............................................................................................. 81

5.1 Introduction ...................................................................................................................... 81

5.1.1 Inverse Problem Solution and Comparison of the Solution Methods: ............................. 81

5.1.2 Characteristics of the Imaging system ............................................................................. 87

5.2 Multi-Frequency Array-Coil System ................................................................................ 87

5.2.1 Inverse Problem Solution for Sensor Array: .................................................................... 88

5.2.2 Image Reconstruction at Single Frequency ...................................................................... 89

5.2.2.1 1D Scanning (Movement) ............................................................................................. 89

5.2.2.2 2D Scanning (Movement) ............................................................................................. 97

5.2.2.3 System Performance ................................................................................................... 102

5.2.2.3.1 Signal to Noise Ratio ....................................................................................... 102

5.2.2.3.2 Spatial Resolution ............................................................................................ 102

6 MULTI-FREQUENCY STUDIES ............................................................................................ 111

6.1 Introduction .................................................................................................................... 111

6.2 Image Reconstruction at Multi-Frequency ..................................................................... 111

6.3 System Performance ....................................................................................................... 112

6.4 Summary and Comments ................................................................................................ 119

7 3D IMAGE RECONSTRUCTION ........................................................................................... 121

7.1 Introduction .................................................................................................................... 121

7.2 3D Inverse Problem Solution ......................................................................................... 121

7.3 3D Image Reconstruction ............................................................................................... 123

7.4 3D Imaging Performance ............................................................................................... 129

7.5 Summary and Comments ................................................................................................ 131

8 CONCLUSION AND DISCUSSION ....................................................................................... 133

xv

8.1 Summary ......................................................................................................................... 133

8.2 Discussion ...................................................................................................................... 134

8.2.1 Interrelationships between the image quality measures and affecting factors for magnetic

induction imaging. ...................................................................................................... 134

8.2.2 Image Reconstruction within the Biological Tissue Range ............................................ 135

8.2.3 Main Contributions of the Study ..................................................................................... 138

REFERENCES ............................................................................................................................. 143

APPENDICES

A MAGNETIC FIELD MEASUREMENT USING RESISTIVE RING EXPERIMENT ......... 147

B THERMAL NOISE ................................................................................................................. 151

B.1 Equivalent Noise Bandwidth ............................................................................................ 151

B.2 Noise In IC Operational Amplifiers ................................................................................. 153

B.3 Addition of Noise Voltages ............................................................................................... 154

C DETAILS ABOUT EXPERIMENTAL PROCEDURE ......................................................... 157

D CHARACTERIZATIONS OF THE DATA ACQUISITION SYSTEMS .............................. 159

D.1 Characterization of the Data Acquisition System with CM-2251 Data Acquisition Card

.................................................................................................................................... 159

D.2 Characterization of the Data Acquisition System with Single Coil Sensor .................... 159

D.3 Characterization of the Data Acquisition System with Array Coil Sensor ..................... 160

E AGAR PHANTOM PREPERATION ..................................................................................... 161

E.1 Equipment and Materials Needed .................................................................................... 161

E.2 Preparation ...................................................................................................................... 161

xvi

LIST OF TABLES

TABLES

Table 3-1: Maximum non-linearity (NL) error between the analytical and numerical results. ...... 32 Table 3-2: Pairs of interactions between variables ......................................................................... 33 Table 4-1: Comparison of the Sensitivities at Different Operating frequencies. ............................ 52 Table 4-2: Electrical conductivities of several tissues. ................................................................... 56 Table 4-3: Mechanical Properties of the Coils Composing the Sensor array. ................................ 60 Table 4-4: Resonance Frequencies of the Coils Composing the Sensor array (in MHz). ............... 61 Table 4-5: Capacitors employed in series with the transmitter coils to cancel out the inductance of

the transmitter coil at operating frequencies. ........................................................................ 61 Table 4-6: The slopes of the sensitivities at different operating frequencies (mV/mho). ............... 78 Table 5-1: Comparison of the Inverse Problem Solution algorithms ............................................. 83 Table 5-2: SNR values of the coils comprising the sensor array. ................................................. 102 Table 6-1: SNR of the multi-frequency system at different operating frequencies. ..................... 112 Table 6-2: FWHM of the multi-frequency system at different operating frequencies. ................. 113 Table A-1: Resistor values (and corresponding 1/(resistor values)) used in the resistive ring

experiments. ........................................................................................................................ 147 Table B-1: Ratio of the Noise Bandwidth B to the 3-dB Bandwidth f0 ........................................ 152 Table D-1: Technical Specifications of the Data Acquisition System with CM-2251 Data

Acquisition Card ................................................................................................................. 159 Table D-2: Technical Specifications of the Data Acquisition System with Single Coil Sensor ... 160 Table D-3: Technical Specifications of the Data Acquisition System with Array Coil Sensor .... 160

xvii

LIST OF FIGURES

FIGURES

Figure 1-1: The ACEIT measurement system. In the figure, a time-varying current is injected to

the subject via the electrode # 10 and the electrode # 11, and the resultant voltage is

measured using the electrode # 8 and the electrode # 9 .......................................................... 3 Figure 1-2: The ICEIT measurement system. In the figure, time-varying magnetic fields, generated

by sinusoidal current carrying wires encircling the conductive body, are applied to induce

currents in the body and the voltage measurements are performed using the electrode # 8 and

the electrode # 9. ..................................................................................................................... 3 Figure 1-3: Measurement methodologies. (a) Single coil sensor, (b) two-coil sensor, (c)

differential coil sensor ............................................................................................................. 8 Figure 2-1: Data collection in the contactless conductivity imaging system (the magnetic-induction

magnetic-measurement system) with a differential coil sensor. ............................................ 15 Figure 2-2: Magnetic vector potential at point P created by current carrying loop. Here I s the

current flowing through the loop of radius a, r is the position variable defined as the distance

between the center of the coil at the point P. ......................................................................... 18 Figure 3-1: Contactless measurement system. Here, σa and σb are the conductivity of the

inhomogeneity and the tissue, respectively. ra and rb are the radius of the inhomogeneity and

the tissue, respectively. The radius of the coil, rc, is same as that of the tissue. h is the

distance between the tissue (and thus the inhomogeneity) and the coil. hm is the height of the

tissue (and the height of the inhomogeneity). The magnetic field B0 is assumed to be

uniform over the tissue (and over the inhomogeneity). ......................................................... 24 Figure 3-2 : (a)Two concentric coils representing the inhomogeneity and rest of the conductive

body. The receiver coil is also shown. Here ra, rb and rc are the radius of the inhomogeneity,

effective radius of the external conducting region and radius of the receiver coil,

respectively. (b) Circuit model of the contactless measurement system. Ic, Ia, Ib are the

currents flowing through the coil, inhomogeneity, and external part the conductive body,

respectively. Mca is the mutual inductance between the receiver coil and the internal coil that

models the concentric inhomogeneity, Mcb is the mutual inductance between the receiver

coil and the coil that models the external region of the body and Mab is the mutual

inductance between the two coils that models the two concentric regions of the conducting

body. ...................................................................................................................................... 26 Figure 3-3: Geometry for the calculation of mutual inductance between the two loops. Here a and

b are the radius of the coils and h is the distance between the coils. ..................................... 27 Figure 3-4: The conductive ring. A z-directed magnetic field applied to a conductive ring of inner

radius rb and outer radius ra. Height of the ring is indicated as hm. ....................................... 28 Figure 3-5: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS

simulations, respectively (β=0.1). The nonlinearity error is 1.38% of the full scale. ............ 30 Figure 3-6: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS



simulations, respectively (β=0.4). The nonlinearity error is 17.9% of the full scale. ............ 31 Figure 3-9: Relationship between sensitivity (S), contrast (α) and resolution (β) for particular

values of spatial resolution (The plots were drawn for h=1 mm and rc=10 mm in Figure

3-1.). ...................................................................................................................................... 33 Figure 3-10: Relationship between sensitivity (S), contrast (α) and resolution (β) for particular

values of α (The plots were drawn for h=1 mm and rc=10 mm in Figure 3-1.). .................... 34

xviii

Figure 3-11: Relationship between sensitivity (S), contrast (α) and resolution (β) for particular

values of sensitivity (The plots were drawn for h=1 mm and rc=10 mm in Figure 3-1.). ..... 35 Figure 3-12: Coaxial Coil System [77]. Here h1 and h2 are the vertical distance between the

perturbation and the transmitter and receiver coils, respectively, r is the horizontal distance

between the center of the coils and the perturbation. (with the courtesy of Prof. Dr. N.G.

GENCER) ............................................................................................................................. 38 Figure 3-13: The Sensitivity (Eq. (3-17)) variations with respect to r, for h=1 cm and f=50 kHz. 40 Figure 3-14: The Sensitivity (Eq. (3-17)) variations with respect to h, for r=1 cm and f=50 kHz. 40 Figure 3-15: The Sensitivity (Eq. (3-17)) variations with respect to f, for r=1 cm and h=1 cm. .... 41 Figure 3-16: Sensor-Perturbation geometry for SNR calculations [77]. Here h is the vertical

distance between the perturbation and the receiver coil, r is the horizontal distance between

the center of the receiver coil and the perturbation and a is the radius of the coil. (With the

courtesy of Prof. Dr. N.G. GENCER) ................................................................................... 42 Figure 3-17 : The SNR (Eq. (3-21)) variations with respect to r, for h=1 cm and f=50 kHz.......... 43 Figure 3-18: The SNR (Eq. (3-21)) variations with respect to h, for r=1 cm and f=50 kHz........... 43 Figure 3-19: The SNR (Eq. (3-21)) variations with respect to f, for h=1 cm and r=1 cm. ............. 44 Figure 4-1: The block diagram of the Low-Frequency Electrical Conductivity Imaging Data

Acquisition System. .............................................................................................................. 46 Figure 4-2: The block diagram of the circuit which performs Phase Sensitive Detection. ............. 46 Figure 4-3: The sensitivity versus radius of the coils. As the distance between the sensor and the

object increases the sensitivity decreases. Thus, the optimum radius of the coil would be the

point where the distance from the sensor to the object and the radius of the coils composing

the sensor are the same. This corresponds to 9mm in our design. ........................................ 48 Figure 4-4: Representation of the geometry for the two neighboring sensors and a perturbation. . 49 Figure 4-5: Sensitivity versus distance of two neighboring coils shown in Figure 4-4. The

perturbation is placed at the 5th

cm of the x-axis. The coils are placed 1 cm above the

perturbation. It is determined that as the distance between the coils increases the total

sensitivity decreases, while the perturbation stays at the same position. As a conclusion, the

sensitivity is determined to be maximum when the coils are almost touched to each other,

while the perturbation is placed at the intersection point of the coils (x=5cm) which yields an

intersection of two maxima of pink curve and green curve. ................................................. 49 Figure 4-6: Multi-frequency data acquisition system with CM-2251 Data Acquisition Card. ....... 51 Figure 4-7: Theoretical and measured sensitivities of the system at operating frequencies. The

sensitivity plots are normalized. The figure reveals that the sensitivity is proportional to the

square of the frequency, as expected. .................................................................................... 52 Figure 4-8: Multi-frequency data acquisition system with single coil sensor. ................................ 54 Figure 4-9: PSD output in volts as a function of conductivity. ....................................................... 56 Figure 4-10: The 1x4 array coil sensor. Each coil is constructed as a Brook’s coil, which makes

the impedance thus the sensitivity of the coils maximum.(The figure on the left is taken from

http://info.ee.surrey.ac.uk/Workshop/advice/coils/air_coils.html) ........................................ 58 Figure 4-11: The block diagram representation of the data acquisition system which comprises a

1x4 array coil sensor, relays, a controller, necessary instruments and a PC. ........................ 59 Figure 4-12: Impedance of Coil #1 as a function of frequency ...................................................... 61 Figure 4-13: Impedance of Coil #2 as a function of frequency ...................................................... 63 Figure 4-14: Impedance of Coil #3 as a function of frequency ...................................................... 64 Figure 4-15: Impedance of Coil #4 as a function of frequency ...................................................... 66 Figure 4-16: Relay Card for the Receiver coils: a) PCB layout of the card, b) Photograph of the

card ........................................................................................................................................ 68 Figure 4-17: Relay Card for the Transmitter coils: a) PCB layout of the card, b) Photograph of the

card ........................................................................................................................................ 69 Figure 4-18: The analog multiplexer and the relay driver card. The card is composed of a

multiplexer, transistors, resistors and capacitances: a) PCB layout of the card, b)

Photograph of the card (The transmitter and receiver coils are driven with two independent

cards.) .................................................................................................................................... 70

xix

Figure 4-19: The controller and controller-to-PC communication card. The controller-to-PC

communication is performed via the serial port (RS-232 protocol): a) PCB layout of the

card, b) Photograph of the card. ............................................................................................ 71 Figure 4-20: The Sensitivity of Coil #1 at operating frequency of ................................................ 72 Figure 4-21: The Sensitivity of Coil #2 at operating frequency of ................................................. 74 Figure 4-22: The Sensitivity of Coil #3 at operating frequency of ................................................. 75 Figure 4-23: The Sensitivity of Coil #3 at operating frequency of ................................................. 77 Figure 4-24: Theoretical and measured sensitivities of the coils at different operating frequencies.

The sensitivity plots are normalized. The figures reveal that the sensitivity is proportional to

the square of the frequency, as it is expected. (a) Coil #1, (b) Coil #2, (c) Coil #3 and (d)

Coil #4. .................................................................................................................................. 79 Figure 5-1: The inverse problem solution with the Steepest Descent Method (a) Conductivity

distribution (b) Error versus number of iterations. ................................................................ 84 Figure 5-2: The inverse problem solution with the Newton Rapson (a) Conductivity distribution

(b) Error function. ................................................................................................................. 85 Figure 5-3: The inverse problem solution with the Conjugate-Gradient (a) Conductivity

distribution (b) Error function. .............................................................................................. 86 Figure 5-4: Field Profile and reconstructed image of a cylindrical agar phantom with a radius of

7.5 mm and a height of 20 mm placed below the intersection of two neighboring coils: (a)

field profile, (b) reconstructed conductivity distribution. ...................................................... 90 Figure 5-5: Field Profile and reconstructed image of a cylindrical agar phantom with a radius of

7.5 mm and a height of 20 mm placed below the center of the 2nd

coil of the sensor array: (a)

field profile, (b) reconstructed conductivity distribution. ...................................................... 91 Figure 5-6: Field Profile and reconstructed image of two cylindrical agar phantoms with a radius

of 7.5 mm and a height of 20 mm, placed below the center of the 1st coil of the sensor array:

(a) field profile, (b) reconstructed conductivity distribution. ................................................ 92 Figure 5-7: Field Profile and reconstructed image of two cylindrical agar phantoms with a radius

of 7.5 mm and a height of 20 mm, placed below the center of the 2nd

coil of the sensor array:


of 7.5 mm and a height of 20 mm, placed below the center of the 3rd



of 7.5 mm and a height of 20 mm, placed below the center of the 4th





(a) field profile, (b) reconstructed conductivity distribution. ................................................ 96 Figure 5-11: Field Profile and reconstructed image of a cylindrical agar phantom with a radius of

7.5 mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15 mm, a

length of 30 mm and a height of 20 mm placed below the sensor array: (a) field profile, (b)

reconstructed conductivity distribution. ................................................................................ 98 Figure 5-12: Field Profile and reconstructed image of a cylindrical agar phantom with a radius of

7.5 mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15 mm, a

length of 30 mm and a height of 20 mm placed below the sensor array: (a) field profile, (b)

reconstructed conductivity distribution. ................................................................................ 99 Figure 5-13: Field Profile and reconstructed image of two cylindrical agar phantoms with a radius

of 7.5 mm and a height of 20 mm, placed below the 3rd

coil of the sensor array: (a) field

profile, (b) reconstructed conductivity distribution. ............................................................ 100 Figure 5-14: Field Profile and reconstructed image of two cylindrical agar phantoms with a radius

of 7.5 mm and a height of 20 mm, placed below the intersection of two neighbor coils: (a)

field profile, (b) reconstructed conductivity distribution. .................................................... 101 Figure 5-15: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm radius

placed below the center of the 1st coil: (a) field profile, (b) reconstructed conductivity

xx

distribution, (c) FWHM calculation by using the signal spread along axial direction (1st coil).

............................................................................................................................................ 105 Figure 5-16: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm radius

placed below the center of the 2nd

coil: (a) field profile, (b) reconstructed conductivity

distribution, (c) FWHM calculation by using the signal spread along axial direction (2nd

coil). .................................................................................................................................... 106 Figure 5-17: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm radius

placed below the center of the 3rd


distribution, (c) FWHM calculation by using the signal spread along axial direction (3rd

coil). .................................................................................................................................... 108 Figure 5-18: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm radius

placed below the center of the 4th


distribution, (c) FWHM calculation by using the signal spread along axial direction (4th

coil). .................................................................................................................................... 109 Figure 6-1: SNR of the system as a function of frequency ........................................................... 113 Figure 6-2: FWHM calculation by using the signal spread along axial direction (4

th coil): a)

FWHM=31mm at 50 kHz, (b) FWHM=25mm at 75 kHz, (c) FWHM=16mm at 100 kHz 114 Figure 6-3 Field Profile and reconstructed image of a cylindrical agar phantom with a radius of 7.5

mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15 mm, a length

of 45 mm and a height of 20 mm placed below the sensor array: ....................................... 116 Figure 6-4 Field Profile and reconstructed image of a cylindrical agar phantom with a radius of 7.5

mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15 mm, a length

of 45 mm and a height of 20 mm placed below the sensor array: ....................................... 117 Figure 6-5 Field Profile measurements and reconstructed image of a cylindrical agar phantom with

a radius of 7.5 mm and a height of 20 mm, and a bar shaped agar phantom with a width of

15 mm, a length of 45 mm and a height of 20 mm placed below the sensor array: (a) field

profile, (b) reconstructed conductivity distribution, at 100 kHz. ........................................ 118 Figure 7-1: 3D visualization of the medium and the inhomogeneity to be imaged. ..................... 122 Figure 7-2: 3D inverse problem solution: the medium is divided into voxels. ............................. 123 Figure 7-3: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a

height of 27 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction................... 125 Figure 7-4: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a bar

shaped agar phantom with a width of 15 mm and a length of 30 mm. The height of the

objects is 20 mm. (a) 2D Image Reconstruction, (b) 3D Image Reconstruction (XZ-

crosssection) ........................................................................................................................ 126 Figure 7-5: Reconstructed images of two cylindrical agar phantoms with a radius of 7.5 mm and a

height of 20 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction (XZ-

crosssection) ........................................................................................................................ 127 Figure 7-6: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a

height of 20 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction................... 128 Figure 7-7: Response of the system (each sensor) against distance from conductive object. (a)

Sensor #1, (b) Sensor #2, (c) Sensor #3, (d) Sensor #4 ....................................................... 129 Figure A-1: Magnetic field measurement using resistive ring experiment setup ......................... 148 Figure A-2: Receiver coil output voltage versus conductivity plot obtained by using resistive ring.

Mean value of the difference data is plotted as a function of: (a) resistor values, (b)

1/(resistor values) (100 sample are taken for each measurement). ...................................... 148 Figure B-1: Ideal bandwidth of low-pass and band-pass circuit elements .................................... 152 Figure B-2: Actual response and equivalent noise bandwidth for low-pass circuit. ..................... 152 Figure B-3: Typical op-amp circuit with an absolute gain A = Rf/R1 ........................................... 153 Figure B-4: Typical op-amp circuit (Figure B-3) with the equivalent noise voltage and current

sources included; (a) Circuit of Figure B-3 with noise sources added; (b) Circuit of Figure

B-4(a) with noise sources combined at one terminal for the case Rs1 = Rs2 = Rs. ............. 155

1

CHAPTER 1

INTRODUCTION

1.1 Medical Imaging

Medical imaging is employed in clinical evaluations to diagnose specific diseases by displaying

the distribution of a physical property in human body. Alternatively, it may be used to understand

working mechanisms of various organs. Most of these techniques also have industrial applications.

Mathematically speaking, medical imaging usually deals with the solution of inverse problems.

This means that the cause (i.e., the properties of the living tissue) is inferred from the effect. The

effect is the response probed by various means. In the case of ultrasonography, the probe is an

ultrasonic detector; whereas for radiography, the probe is an X-ray detector.

Radiographs, Fluoroscopy, Computed Tomography (CT), Magnetic Resonance Imaging (MRI)

and Ultrasound are the well-known modern imaging modalities [1]. Recently proposed/developed

medical imaging modalities are as follows: Diffuse Optical Tomography (DOT) [2], Elastography

[3], Electrical Impedance Tomography (EIT) [4], Nuclear Medicine [5], Optoacoustic Imaging [6],

Positron Emission Tomography (PET) [7], and Ophthalmology [8].

In this thesis, the results of a new imaging modality which uses magnetic means to image

electrical conductivity of body tissues are presented. The technique is closely related to its

predecessors that were proposed about two decades ago to image electrical impedance of body

tissues. For completeness, prior to reporting the results of the new approach, some of these

previous methods will be reviewed.

2

1.2 Electrical Impedance Imaging

Electrical Impedance Imaging (EII) attempts to image the conductivity and permittivity

distribution of the tissue by electrical measurements. There are mainly three types of EII

depending on how the current is applied and measurements are taken. The earliest method is to

apply currents to body and measure the voltage on the body. This method is called Applied

Current EII (ACEII). The second method is the Induced Current EII (ICEII) in which the currents

are induced by time-varying magnetic fields from outside the body and voltage is measured on the

body [9], [10], [11], [12-14]). The last method is called Contactless EII (CEII) . In this method,

current is induced by time-varying magnetic fields from outside the body and secondary magnetic

field caused by these curents are sensed by a sensor outside the body.[14], [15], [16], [17], [18].

ACEII and ICEII are discuessed in the remaining of this section, while Contactless EII will be

discussed in the following section.

The EII is usually implemented as Electrical Impedance Tomography (EIT) in the literature. In

EIT, currents can be introduced into the subject either by using electrodes or by using magnetic

induction. In applied-current electrical impedance tomography (ACEIT), time-varying currents

with constant amplitude are injected to the subject via the surface electrodes, and the resultant

voltages are measured using the same electrodes [19], [20], [21], [22] (Figure 1-1). The

performance of this method is affected negatively by the limitations on the electrodes and high

resistive tissues like bones [19], [23], [24], [20].

In induced-current electrical impedance tomography (ICEIT), time-varying magnetic fields,

generated by sinusoidal current carrying wires encircling the conducting body, are applied to

induce currents in the body and voltage measurements are performed using the surface electrodes

[11], [9], [10], [12], [13], [25] (Figure 1-2). This method is advantageous over ACEIT since

screening effect of high resistive tissues are eliminated and it is flexible in terms of induced

currents. However, voltage is again measured by large number of electrodes attached to body.

Some notable features of ICEIT compared to ACEIT are the following:

The electrodes are used for a single function, i.e., for voltage measurements.

Consequently, the voltage sensing electronics can be optimized,

Internal current level is not limited by the current density at the injection electrodes (it is

not limited to the safe current density on the skin where the injection electrodes are

attached). It can be enhanced by applying different magnetic field patterns to increase the

signal-to-noise ratio in the measurements,

3

4

12

15

14

2

13

3

5

6

7

109

11

116

Electrodes

Applied

Current

σ

Measured

VoltageIejwt

σ

V

8

+

-

Figure 1-1: The ACEIT measurement system. In the figure, a time-varying current is injected to

the subject via the electrode # 10 and the electrode # 11, and the resultant voltage is measured

using the electrode # 8 and the electrode # 9

s1

-

9+

Iejwt

4

12

15

14

2

13

3

5

6

7

810

11

116

Electrodes

σ

V

-

Measured Voltage

Figure 1-2: The ICEIT measurement system. In the figure, time-varying magnetic fields,

generated by sinusoidal current carrying wires encircling the conductive body, are applied to

induce currents in the body and the voltage measurements are performed using the electrode # 8

and the electrode # 9.

4

For a given number of electrodes, the number of independent measurements can be

increased by introducing spatially independent magnetic field patterns. In principle, it is

possible to manipulate the applied magnetic fields and thus the induced current

distribution to examine a particular part of the region in detail.

One disadvantage of induced-current approach is the induced EMF imposed in the measurement

cables. However, this effect can be minimized by orienting the cables appropriately with respect to

the applied field and keeping them rigidly fixed during the measurements.

A number of applications were reported for EIT, namely, monitoring lung and thorax function

[26], heart imaging [27], [28], [29], detection of cancer in the skin and breast [30], [31], [32], [33],

[34] and location of epileptic foci [35]. A review of these experimental applications can be found

in [36].

In geophysics, the exploitation of akin ideas dates back to 1930s. A similar technique is employed

to locate resistivity anomalies using electrodes on the surface of the earth or in bore holes [37]. In

industrial process monitoring, arrays of electrodes are used, for example, to monitor mixtures of

conductive fluids in vessels or pipes, or to measure the degree of salt content in sea water. The

method is also applicable for nondestructive testing [38], [39].

The credit for the invention of EIT as a medical imaging technique is usually attributed to John G.

Webster in around 1978 [40], although the first practical realization of a medical EIT system was

due to David C. Barber and Brian H. Brown [19].

1.3 Electrical Conductivity Imaging via Contactless

Measurements

Figure 1-3 shows different measurement strategies of an alternative method: A transmitter coil is

driven by a sinusoidal current to generate time varying magnetic fields. When a conducting body

is brought to the vicinity of the coils, eddy currents are induced in the body as a function of the

body's impedance distribution. A secondary magnetic field is created due to the induced currents

and the resulting electromotive force can be measured via a receiver coil by means of different

approaches.

Compared to the EIT techniques mentioned above, this method has the following advantages:

5

i. There is no physical contact between the body and the measurement system,

ii. Currents can be coupled into the body avoiding the screening effects of the superficial

insulating layers,

iii. The number of measurements can be increased by simply shifting the transmitter and

receiver coil array.

The measurement technique is known as Induction Logging in geophysics and used for the

purpose of geophysical inspection [37]. In process tomography applications, the technique is

named as Mutual Inductance Tomography or Electromagnetic Inductance Tomography [35], [38].

Inspired from the process tomography applications, the method was also proposed for imaging

body tissues, and it was termed as Electromagnetic Imaging and Mutual Inductance Imaging [27],

though the proposed method was basically a tomography system.

The usage of magnetic induction-magnetic measurement technique to measure the conductivity of

the biological tissues was first proposed by Tarjan and McFee in 1968 [14]. The method was used

to determine the average conductivity variation of human torso and head at an operating frequency

of 100 kHz employing a differential coil sensor of 13 cm diameter. Single-point measurements

following conductivity fluctuations in the human heart are reported [14]. Almost 25 years later,

Al-Zeibak and Saunders proposed the use of the same method to produce tomographic images of

conducting bodies immersed in saline solutions [41]. A small drive-coil (excited at an operating

frequency of 2MHz) and a distant pickup-coil is used to scan saline phantoms. The images were

reconstructed by employing computational algorithms developed for X-ray CT. The method was

named as Mutual Induction Tomography (MIT). They reported that fat and fat-free tissues could

be distinguished and the internal and external geometry of simple objects can be determined.

Around the same period, Netz et al. utilized the same technique with miniaturized coils (25-mm

diameter) at an operating frequency of 100 kHz [42]. They reported that the calibration

measurements using equal volumes of NaCl solutions of different concentrations show linear

dependence on the electrolyte content of the solutions.

Korzhenevskii and Cherepenin (1997) presented a theoretical study of a two-coil arrangement and

proposed the direct measurement of phase angle for detecting the eddy currents in the conducting

bodies. They applied the filtered back-projection algorithm to the simulation resulted to obtain

images [43]. Almost three years later, Korjenevsky et al. (2000) reported the implementation of

this method [16]. The system employed 16 electronically switched excitation and detection coil

units arranged in a circle. The carrier frequency of the system was 20 MHz and this was down-

converted to 20 kHz for processing. The demonstration of imaging of the cylindrical objects, with

positive and negative conductivity contrast, in a saline bath is given in the same study. The

6

objects, each with diameter corresponding to 29 % of the array diameter, were clearly resolved

using a filtered-back projection algorithm. Korjenevsky and Sapetsky (2000) showed that reduced

spatial distortion could be achieved when the images were reconstructed by a neural network for

some simple distributions of conductivity [44].

A review of magnetic impedance tomography studies can be found in [45]. There is considerable

interest in applying this technique to medical imaging in the last decade:

Scharfetter et al. (2001) implemented a new system that operates at relatively low frequencies (20-

370 kHz). The measurement system consists of a solenoid transmitter, a planar gradiometer which

functions as a receiver, and a sensitive phase detector [17]. Sensitivity maps of the system are

investigated [46], [47] and a fast computation algorithm of sensitivity map is developed [48]. They

adopted an existing inverse problem solution [15], which they called as 3D inverse eddy current

solution, to reconstruct images [49].

Watson et al. (2004) implemented a new sensor, which in principle, cancels the primary magnetic

field by a single receiver, mounted such that no magnetic flux runs through it, instead of using an

additional back-off coil [50]. The numerical models and solutions obtained by Morris et al. (2001)

[51] are used for simulation studies. The above mentioned sensors are compared with the axial

gradiometers [52]. The frequency is selected between 1-10 Mhz. The implementation of this

system was demonstrated by Igney et al. (2005) using an array of coils [53].

Independent from the earlier studies, the feasibility of magnetic induction-magnetic measurements

method was also explored for sub-surface imaging by testing the safety conditions at 50 kHz [54],

[55], [15]. Gencer and Tek calculated the pick-up voltages for a miniaturized coil configuration

over a uniformly conductive semi-infinite region. Transmitter and receiver coils of radii 10 mm

were placed coaxially above the half space (the transmitter and receiver coil distances to the

sample are 6 cm and 1 cm, respectively). The secondary voltages were about 10 μV while the

induced currents were well below the safety limit (1.6 mA/cm2 at 50 kHz). In that study, the

effects of displacement currents and the propagation effects were also discussed. It was observed

that, for certain tissues, like heart muscle, kidney, liver and lung, the displacement currents can be

assumed negligible for frequencies below 100 kHz. For a survey distance of 20 cm, the % error

between the magnitude of the propagation term and unity becomes significant as the frequency

increases (7.64% at 100 kHz). Consequently, Gencer et al. chose lower frequencies (<100 kHz) to

overcome the difficulties arising from both displacement currents and wave propagation delays

[55]. They also discussed the safety considerations of the system [15]. To apply the method to

realistic body geometries, the forward problem was analyzed by using Finite Element Method

7

(FEM). The inverse problem solutions were obtained using the pseudo-inverse of the sensitivity

matrix [15]. In the latter studies, two data acquisition systems were developed and implemented

[56], [18], [57]. The performance of these systems was investigated using saline solution filled

glass tubes and biological tissues. After those studies, an improved data acquisition system

operating at 14.1 kHz was developed by Colak and Gencer [57]. Recently, a novel data acquisition

system operating at multi-frequency was developed by Ozkan and Gencer [58]. These four

implementations will be shortly reviewed next.

In the first implementation, two different measurement methodologies were proposed [59]. The

first approach employs a single-coil (Figure 1-3(a)). In this model, single sensor functions serves

as both the transmitter and the receiver. When the conductive object is placed close to the sensor,

the impedance of the sensor will change. An impedance-to-frequency converter is used to convert

these impedance variations to the square waveform whose frequency is a function of the body

conductivity. This frequency output is directly connected to a parallel port of a PC. The

dependence of the output frequency to the concentration of saline solutions (of concentration 10

gr/l to 100 gr/l).was investigated. As a figure of merit for sensitivity, 100 Hz frequency change at

the output for every 1.9 mS/cm conductivity variation was reported at the operating frequency of

1.4 MHz. The details of the measurement system and further information can be found in [59].

The second configuration in [59] employs two coils (Figure 1-3(b)). In this approach, there are

two different channels for the sensor: The first coil serves as the reference, and the other is used

for the measurement. Output of these two channels is fed to a differential amplifier. After

performing phase sensitive detection (PSD) to the signal at the output of the differential amplifier,

the analog output of PSD block is digitized with an analog-to-digital converter (ADC). The ADC

output is then fed to the PC. The system operates at 15 KHz. The SNR of the system was reported

as 39 dB while the spatial resolution was determined as 9 mm.

In this second implementation, a differential coil sensor which was utilized by a PC controlled

scanning system was used to image conduction phantoms (Figure 1-3(c)). The data acquisition

system was constructed using a PC controlled lock-in amplifier instrument. The operating

frequency of the system was 11.6 kHz. The SNR of the system was reported as 34 dB. The

conductivity images are reconstructed using the sensitivity matrix approach. The spatial resolution

was determined as 12.35 mm for agar phantoms and 17 mm for isolated conducting phantoms.

The sensitivity of the system is measured as 21.47 mV/(S/m) while the linearity is 7.2% of the full

scale. In addition, the measured field profile of a biological tissue was reported for the first time in

the literature. Thus, the potential of this methodology for clinical applications was shown [18],

[60]). The details of the measurement system and further information can be found in [61].

8

Eddy Currents

σ

Secondary Flux

Impedance to Frequency

Converter

Sensor of

single coil

(a)

Eddy Currents

σ

Transmitter 1

Receiver 1

Transmitter 2

Receiver 2

Differential Amplifier

which can handle common

mode signals of 200 Vp

Sensor Auxiliary Coil

Secondary

Flux

–

+

(b)

σ

Receiver 1

Transmitter

Receiver 2

Eddy CurrentsSecondary Flux

-

V

+

Primary Flux

(c)

Figure 1-3: Measurement methodologies. (a) Single coil sensor, (b) two-coil sensor, (c)

differential coil sensor

9

In the third implementation, the former data acquisition system [61] was improved to obtain

measurements with a faster scanning speed. Besides, a data acquisition card was realized to

eliminate the use of the Lock-in instrument in the phase sensitive measurements, thus achieving a

portable system. The performance of the system was investigated with a novel test method

employing resistor rings. The minimum conductivity value that can be distinguished was

determined as 2.7 S/m. The field profiles of the agar objects with different geometries were

reported. The details of the measurement system and further information can be found in [57].

In the fourth implementation, a multi-frequency data acquisition system was developed [58].

Operating frequencies of the system were chosen to be within 30-90 kHz. Field profiles and

reconstructed conductivity distributions of the agar phantoms were obtained at different

frequencies. The performance of the system was investigated by employing resistor rings. The

normalized sensitivity of the system was 18.2mV/Mho at 30 kHz, 50.7 mV/Mho at 50 kHz, 73.1

mV/Mho at 60 kHz and 171.2 mV/Mho at 90 kHz. The spatial resolution of the system was found

as 19.8 mm at 30 kHz, 10.8 mm at 60 kHz and 9 mm at 90 kHz. The results were in consistence

with the theory stating that the measured signal is proportional to the square of the frequency.

1.4 Multi-frequency studies

Previous researchers studied on the adopted measurement approach which concentrated on single

frequency measurements [59], [61], [57]. However, electrical properties, namely, conductivity and

permittivity, of biological tissues may vary with frequency [20]. In another words, the electrical

properties of biological tissues are strictly dependent on the operating frequency [62]. As

frequency increases, the cell membrane permits the current to pass inside the intracellular fluid.

This is accompanied with an increase in conductivity values. This phenomenon is known as fi-

relaxation or Maxwell-Wagner dispersion [62]. At this region range, the permittivity decreases as

the frequency increases. Since the change of conductivity as a function of frequency variation

differs in different tissues, multi-frequency studies should be investigated. Thus, tissues that

cannot be distinguished at a particular frequency can be resolved at another frequency. This can,

i.e., operating at multi-frequency, for example, enable the detection of different diseased tissues.

The feasibility of multi-frequency studies working in the fi-dispersion region was investigated

experimentally on biological tissues, and a conductivity spectrum of a potato was determined by

Scharfetter et al. [46]. This study showed the feasibility of a spectroscopic system as well as the

feasibility of on-line monitoring of brain oedema. Ozkan and Gencer performed multi-frequency

experiments with agar phantoms at the operating frequencies between 30 kHz to 90 kHz [58].

10

Field profiles and reconstructed conductivity distributions of the objects were obtained at different

frequencies. The results were in consistence with the theory stating that the measured signal is

proportional to the square of the frequency.

In this study, a new data acquisition system is realized to perform electrical conductivity imaging

of biological tissues via contactless measurements. Performance of the system is tested with a

novel test method which employs conductive rings representing the conductive objects. The

operating frequency of the system is between 10 kHz and 100 kHz. In the experiments, agar

phantoms, i.e. biological tissue equivalent phantoms, are used, thus, the induced currents are

allowed to flow between the object and the conductive medium where the object is immersed in.

Images of agar phantoms at different operating frequencies are presented. The results show the

feasibility of the contactless, multi-frequency conductivity imaging of the biological tissues.

In the literature, 3D and 2D multi-frequency scanning images of an agar object by using sensor

array have not been reported yet. To the best of the authors’s knowledge, this study is the first to

provide with scanning results which are obtained with the multi-frequency contactless

conductivity imaging technique.

1.5 Patent Applications for Magnetic Induction Tomography

A number of patent applications related with the magnetic induction tomography have been filled

or granted especially during the last decade. These applications are summarized here:

a) Magnetic Induction Tomography System and Method, Igney, 2007

In this invention an MIT system and a method is presented. A high resolution MIT

technique was provided without increasing the number of coils [63].

b) Method and Device For Calibrating A Magnetic Induction Tomography System,

Yan, 2009

In this invention a method and device for calibrating the offset of an imaging system

(MIT system) was developed. The invention provides a reduction in the imaging

interferences caused by the reference object during monitoring [64].

c) Magnetic Induction Tomography with Two Reference Signals, Watson, 2009

The proposed apparatus in this invention comprising 1) an excitation signal generator, 2)

a primary excitation coil, 3) a primary receiver coil, 4) a signal distribution arranged to

receive the detection signal from the primary receiver coil, 5) a passive reference

11

detector, 6) an active reference signal generator for generating an active reference signal,

7) an active reference source for receiving the active reference signal from the active

reference signal generator [65].

d) Method and Device For Magnetic Induction Tomography, Philips Elect., 2010, July

This invention introduces an apparatus comprising a coil arrangement with at least one

transmitting coil and at least one measurement coil. It also comprises motion sensing

means to sense a relative motion between the object and coil arrangement and generating

a trigger signal as the relative motion occurs [66].

e) Method and Device For Magnetic Induction Tomography, Philips Elect., 2010

This invention provides a device consisting of a plurality of transmitting and

measurement coils. A first pair of transmitting coils are selected and excited among the

transmitter coils, which minimizes the primary magnetic field at the location of

measurement coil(s). By this minimization, the dynamic range of measurement coils can

be reduced. Then, the hardware design for MIT is simplified [66].

f) Magnetic Induction Tomography Systems With Coil Configuration, Eichardt, 2010

In this invention, an MIT system with an excitation and measurement coil system was

developed. Several excitation coils were used for both generating an excitation magnetic

field and measuring the fields generated by the induced fields. The arrangement of the

measurement coils were in a volumetric geometry. Each measurement coil was oriented

transverse to the field line of the excitation magnetic field of the excitation coils. The

system provides an improvement in the image quantity of volumetric objects [67].

g) Coil Arrangement and Magnetic Induction Tomography System Comprising Such a

Coil Arrangement, Chen, 2011

This invention provides a coil arrangement comprising at least one transmitting coil and a

plurality of measurement coils. This coil arrangement results in a reduction of the signal

strength of the induced signal on the measurement coil and the signal dynamic range

[68].

h) Correction Of Phase Error In Magnetic Induction Tomography, Scharfetter, 2011

The signals measured in MIT are corrected with regard to a phase error in this invention

[69].

i) Device and Method For Magnetic Induction Tomography, Scharfetter, 2011

12

A device for MIT consists of at least one transmitter coil and at least one receiver coil.

This invention can provide high termination impedance to a transmitter coil and drive the

same coil at two or more different frequencies [70].

j) Device and Method Magnetic Induction Tomography, Scharfetter, 2011

This invention provides a method and an apparatus for MIT, where an object having

inhomogeneous electrical properties is exposed to alternating magnetic fields. This

apparatus consists of at least one transmitter and one receiver coil. It allows eliminating

artifacts due to movements of the body under investigation without causing high costs

and complicated equipment [71].

1.6 Motivation and the Scope of the Thesis

Contactless conductivity imaging is a main research topic of the METU Brain Research

Laboratories. Researchers have been conducting this research for almost 25 years. The motivation

of this study is to progress on developing a contactless conductivity imaging system for clinical

applications.

In our previous study [58, 72] a differential coil sensor was employed for field measurements. The

measurements were taken by performing rectilinear scanning. Average data collecting time

including scanning and measurements was 0.29 sec/mm2. One of the important concerns for

clinical applications is the long data collecting time. Since it is difficult to make the patient steady

during the scanning process, the data collecting time must be as short as possible. Therefore, one

of the aims of this study is to increase the data collecting speed or similarly to decrease the data

collection time.

It is well known that the conductivity of the tissues varies as the frequency varies. This

observation can be used to collect a number of data by changing the operating frequency of the

system. By doing this, the quality, i.e., the resolution, signal-to-noise, etc., of the images can be

enhanced. Another aim of this study is to perform multi-frequency measurements.

Three-dimensional (3D) imaging is one of the most important abilities that an imaging system

must possess. 3D image imaging has not been studied in the low frequency subsurface imaging

with a real data yet. Constructing the 3D sensitivity matrix and then solving the 3D image

reconstruction problem are other goals of this study.

13

In subsurface contactless imaging, the sensitivity of the sensor depends on the area of the coil,

number of turns, amplitude of the excitation current and frequency, i.e., the rate of change of the

magnetic flux through the coil, and the geometry and the conductivity of the objects. In general,

for an imaging system employing magnetic induction-magnetic measurement technique, a number

of parameters limit the sensitivity of the sensor and thus the performance of the system.

Theoretical limits to sensitivity and resolution in EIT have been investigated in [27]. However,

relation between the sensitivity of the sensor and the parameters affecting the quality of the

measurements has not been studied in MIT and Subsurface Conductivity Imaging. Finally, another

aim of this study is to develop a mathematical model of the sensor and conductive medium

comprising a conductive object and inhomogeneity. For this purpose, the sensor and the

conductive medium will be analytically and numerically modeled and the validity of the analytical

model will be verified with the numerical model. After then, four parameters, namely, sensitivity,

spatial resolution, conductivity contrast, conductivity resolution and noise will be introduced.

Finally, relationships between the sensitivity and spatial resolution, conductivity contrast,

conductivity resolution and noise will be investigated.

Briefly, the motivation of this work is based on the following facts: 1) electrical conductivities of

biological tissues are different, and 2) the electrical properties change with the operating

frequency. Thus, besides providing a valuable tool for diagnostics imaging, electrical

conductivities of biological tissues may also provide with complementary images for existing

imaging systems using other physical properties of tissues. The objectives of this study are listed

as follows:

To design and to develop a multi-frequency prototype systems with different sensors. The

system should be able to measure very small ac magnetic fields and be capable of phase

sensitive detection. The output of the system should linearly follow the conductivity

variation,

To develop a mathematical model relating the measurements of a circular coil

configuration to the conductive body parameters,

To investigate the relationships between resolution, accuracy, conductivity contrast, and

noise by using the developed mathematical model.

To design and develop a multi-frequency data acquisition system. Necessary hardware

and sensor should be designed and implemented,

To perform multi-frequency experiments. There must be a consistency between the

results and the theory,

To obtain field profiles of the phantoms,

To reconstruct 2D conductivity profiles of agar objects by employing the field profiles.

To reconstruct 3D conductivity profiles of agar objects by employing the field profiles.

15

CHAPTER 2

THEORY

The system is used to image electrical conductivity of tissues via contactless measurements. For

that purpose, sinusoidally varying currents (eddy currents) are induced in the conductive body by

means of an external magnetic field. The external field is created by a transmitter coil carrying a

sinusoidal current. A receiver coil which is placed nearby the body surface measures the magnetic

fields due to the induced eddy currents. By changing the location of the coils (transmitter/receiver

coils), i.e. by scanning the body surface, it is possible to obtain a number of measurements which,

in turn, are used to obtain the conductivity distribution of the biological object under investigation.

σ

Receiver 1

Transmitter

Receiver 2

Eddy CurrentsSecondary Flux

-

V

+

Primary Flux

Figure 2-1: Data collection in the contactless conductivity imaging system (the magnetic-

induction magnetic-measurement system) with a differential coil sensor.

Figure 2-1 shows the basic measurement principles of the electrical conductivity imaging via

contactless measurements. This modality uses magnetic excitation to induce currents inside the

body and measures the magnetic fields of the induced currents (eddy currents). As a result, the

measurement system has no physical contact with the conducting body in contrast to the other

electrical impedance imaging methods. A transmitter coil is driven by a sinusoidal current to

provide a time varying magnetic field (primary field). When a body is brought nearby these coils,

16

eddy currents are induced in the body. The distribution of the eddy currents depends on the

impedance distribution of the body. The induced currents create a secondary magnetic field and

the electromotive force induced due to the primary and the secondary fields is measured by the

receiver coil. The measurement hardware utilizes the phase sensitive detection method to

distinguish the component of the electromotive force which arises due to the impedance of the

body under investigation.

It should be noted that since the conductivities of different tissues vary as a function of frequency,

a number of data can be collected by simply changing the operating frequency of the system. In

this manner the quality, i.e., the resolution, signal-to-noise, etc., of the images can be enhanced.

The theory behind this principle can be found in [15], [72].

2.1 Forward Problem

In this imaging modality, the forward problem is defined as the calculation of the secondary

magnetic fields for a known conductivity distribution. In this subsection, the relation between the

conductivity of the object and the magnetic measurements will be presented first. After then,

maximum sensitivity case of the proposed imaging modality will be discussed.

2.1.1 General Formulation

Theoretical formulation relating conductivity to magnetic measurements is given in ([15], [18] and

[58]. However, for completeness, it will be briefly presented here. In a linear, isotropic, non-

magnetic medium, the electric field E

has two sources: Namely, the time-varying magnetic field

and the surface and volume charges. For the sinusoidal excitation, E

can be expressed as the

combination of these two sources [15]:

AjE (2-1)

Where

A , and represent the magnetic vector potential, scalar potential and radial

frequency respectively.

Figure 2.2 shows a circular loop carrying a sinusoidal current I (e jt

time variation is assumed).

The loop is of radius a, and centered at origin in the x-y plane. When there is no conductive body

17

nearby the loop, the primary magnetic vector potential

A at point P due to the current in the loop

is calculated from [55] (page 182):

2

2

22

0 22

sin2

4

4,

k

kEkKk

arra

IrA a

p

(2-2)

where K and E are the elliptic integrals of first and second kinds [61], respectively and r the

position variable is the distance between the center of the coil and the point P. Argument k of the

elliptic integral can be calculated using:

sin2

sin422

2

arra

ark

(2-3)

Under quasi-static conditions, when a conductive body is located near the coil, the scalar potential

can be calculated by solving the following differential equations [10], [73]:

ss

pA

(2-4)

pnAn

(2-5)

where pnA

is the normal component of the primary magnetic vector potential on the surface of

the conductive body whose conductivity is represented as σ.

In the above equation, the scalar potential has only imaginary component. Consequently,

E can

be expressed as:

AjE (2-6)

18

Figure 2-2: Magnetic vector potential at point P created by current carrying loop. Here I s the

current flowing through the loop of radius a, r is the position variable defined as the distance

between the center of the coil at the point P.

The induced current density in the conductive body is related to the electric field as follows:

EJ I s .

The current flowing in the transmitter coil and the conductive object creates a magnetic flux

which is picked up by the receiver coil. Using the reciprocity theorem [13], can be calculated

using the following integrals in the corresponding volumes [15]:

bodyIRR

COILTRR

dVJAI

dVJAI

11 (2-7)

where RA

is the magnetic vector potential created by the reciprocal current RI in the receiver

coil, dVbody and dVcoil are respectively differential volume elements of the conductive body and

receiver coil, TJ

and IJ

represent the current density in the transmitter coil and the induced

current density in the conductive object, respectively. The first term on the right is the primary

flux, directly coupled from the transmitter coil. The second term represents the flux caused by the

induced currents. Then, the electromotive force v in the receiver coil can then be expressed as:

19

bodyT

R

R

R

RT dVAw

I

Awld

I

AwIjv

wjv

)()())(( s

(2-8)

As seen in the above equations, the induced voltage in the receiver coil has two components. The

imaginary quadrature component (with j coefficient) arises from the transmitter coil current while

the real part which is the in-phase component includes the conductivity information. Quadrature

component is very large compared with the in-phase component and is cancelled out by use of

differentially connected receiver coils. For perfectly matched receiver coils which have the same

electrical characteristics, placed at the same distance from the transmitter coil, the differential

connection produces the output voltage of [61]:

bodyT

R

R

R

RdVAw

I

Aw

I

Awwv )(

2

2

1

1 s

(2-9)

Here 1RA

and 2RA

are the primary magnetic vector potentials created by the reciprocal currents

1RI and 2RI of the two coils in the differential sensor, respectively. 1RA

and 2RA

can be

calculated using the same formula given in Eq. (2-2) and Eq. (2-3).

2.1.2 Single-Coil Sensor

In this thesis, two different sensor types are used, namely, the differential- coil sensor and the

single-coil sensor. The theory behind the single-coil sensor will be explained in this subsection.

The secondary flux due to the eddy currents in the body changes the inductance and thus the

impedance of the sensor. As a result, the sensitivity of the sensor can also be determined using its

electrical impedance. Assuming that the sensor has a transmitter and a receiver coil, the voltage at

the receiver coil due to the conductive object can be expressed as

20

objectT

R

RdVA

I

Awv s)(2

(2-10)

Here, RA

is the magnetic vector potential due to the reciprocal current IR in the receiver coil and

TA

is the magnetic vector potential due to the transmitter coil current IT. Since the operating

frequency is lower than 100 kHz, the term can be neglected in the equation. Accordingly, the

transfer impedance of the coil is determined by dividing the pick-up voltage to the transmitter

current IT,

object

T

T

R

R

T

dVI

A

I

Aw

I

vZ s

2

(2-11)

It can be seen that the impedance change (ΔZ) will be maximum when the magnetic vector

potential vectors are equal, which dictates nothing but the use of a single-coil for both

transmission and receiving. In such a case, the impedance of the sensor is expressed as follows:

objectdVAI

wZ s

2

2

2

(2-12)

2.2 Inverse Problem

The inverse problem is defined as calculation of the conductivity distribution from the magnetic

field measurements. Since the scalar potential distribution is a function of the unknown

conductivity distribution, the relation between the conductivity distribution and the voltage

induced in the detector coil due to the secondary magnetic field is non-linear. One method to find

an estimate of the conductivity distribution from a set of measurements is to linearize the pick-up

voltage expression (Equation 2-9) around an initial conductivity distribution. Using this approach,

it is possible to obtain a linear relation between the perturbation in conductivity and the changes in

the measurements.

For m measurements, if the conductive body is discretized into n elements (of constant

conductivity), it is possible to relate the changes in the voltage measurements to the conductivity

perturbations using the following matrix equation [15]:

21

s Sv (2-13)

where v is an m × 1 vector representing the changes in measurements and s is an n x 1

vector representing the perturbations in the element conductivities around an assumed

conductivity value. The sensitivity matrix S can be calculated as explained in detail in [15]. It

should be noted that the calculation of S requires the solution of the scalar potential distribution

for an assumed conductivity distribution. For a body of arbitrary geometry, a numerical method,

such as the FEM [15], must be employed for that purpose.

In this thesis, to obtain a fast estimate of the conductivity distribution, the effects of the ∆ϕ term

on the magnetic field measurements are simply neglected which can be done safely for the

frequencies below 100 KHz [15]. For a differential-coil sensor, this yields the following form of

Equation 2-9:

bodyT

R

R

R

R dVAwI

A

I

Awv )(

2

2

1

1

s (2-14)

Thus, one obtains a linear relation between the measurements and the conductivity distribution. In

matrix notation, the resultant equation takes the following form:

v sS' (2-15)

where S’ denotes the coefficient matrix of the simplified equations.

This approach, previously proposed by Ulker and Gencer [18], considerably simplifies the

numerical solutions. The reconstructed images using the experimental data (see CHAPTER 5)

proves that this approach provides satisfactory estimates while reducing the computation time.

23

CHAPTER 3

SENSITIVITY ANALYSIS OF

CIRCULAR COIL SENSORS

3.1 Modelling the Imaging System

3.1.1 Introduction

Subsurface Conductivity Imaging [15, 18, 54, 55, 72] or Magnetic Induction Tomography (MIT)

[51, 74] are contactless conductivity imaging modalities based on magnetic-induction magnetic-

measurement principle. In these modalities, usually circular coils are preferred for both current

inductions and magnetic field measurements.

Time varying magnetic fields generated by an excitation coil induce eddy currents in the

conductive body. These currents produce secondary magnetic fields that are measured using

receiver coils. The technique is used for different purposes, such as non-destructive testing [38],

geophysical inspections [37], process tomography [35] , and in biomedicine [14, 42, 43, 75].

Depending on the application type, different sensors with various geometries are employed.

In all these applications the quality of the image (described by two measures, namely, the spatial

resolution and conductivity resolution) is limited by a number of factors: 1) the position in the

body, 2) the contrast within the body, 3) noise in the measurements, 4) number of measurements,

and 5) the reconstruction algorithm. Knowing the relationship between these factors and the

quality measures is important in designing effective imaging systems.

24

Figure 3-1: Contactless measurement system. Here, σa and σb are the conductivity of the

inhomogeneity and the tissue, respectively. ra and rb are the radius of the inhomogeneity and the

tissue, respectively. The radius of the coil, rc, is same as that of the tissue. h is the distance

between the tissue (and thus the inhomogeneity) and the coil. hm is the height of the tissue (and the

height of the inhomogeneity). The magnetic field B0 is assumed to be uniform over the tissue (and

over the inhomogeneity).

In an earlier study [76], a theoretical work was conducted to reveal such relationships in applied

current electrical impedance imaging. Inspired from that work, in this study, a simple detection

system (Figure 3-1) is analyzed which uses the magnetic-induction and magnetic-measurement

technique. It is assumed that a circular receiver coil is above a thin cylindrical body with a

concentric inhomogeneity. The body is in a spatially uniform sinusoidally varying magnetic field.

Concentric inhomogeneity is chosen to be the most difficult detection problem using a circular

receiver coil. In that sense, the first factor in the above list (i.e., the position in the body) is not

considered. Study on the effects of inhomogeneity location can be further studied. The effects of

two other factors, i.e., the number of measurements and the reconstruction algorithm are not also

considered. However, the relationships between the two quality measures (spatial resolution and

conductivity resolution) and two factors (contrast within the body and noise in the measurements)

are studied in detail.

In this part of the study, the terminology used in [76] is adopted. The term resolution (β) is used

to mean the spatial resolution and it is defined as the smallest region (i.e., the concentric

inhomogeneity of smallest radius) in which the conductivity can be determined. It is quantified by

the ratio ra/rb. The term contrast (α) represents the conductivity contrast and is defined as α = sa

/sb. The conductivity resolution is defined as the fractional change in conductivity contrast (dα/α)

rc

ra rb

25

and is referred as accuracy throughout this text. One last term employed frequently in this study is

the noise in the measurements. The smallest change in conductivity that can be detected results in

a change in the measurement which just exceeds the noise. The fractional change in the measured

voltage (dV/V) is used to mean the noise in that measurement.

In the following subsections, first a mathematical model relating the measurements of a circular

coil configuration to the conductive body parameters is introduced. Thereafter, the relationships

between resolution, accuracy, conductivity contrast, and noise are investigated using the

developed mathematical model.

3.1.2 Analytical Model

The sensors used in this modality can be designed in different forms, like single-coil, dual-coil

(one for excitation and one for receiving) or a differential-coil (a transmitter coil and two

differentially connected receiver coils). In this work, we assume a dual-coil system and model the

system accordingly. However, the transmitter coil is assumed to be larger than the receiver coil

and positioned at a distant location yielding a uniform magnetic flux density on the receiver coil

and the conductive body. The proposed three-ring model (that represents receiver coil, body

without concentric inhomogeneity, and concentric inhomogeneity) and corresponding circuit

model are shown in Figure 3-2. The concentric inhomogeneity and rest of the conducting body are

assumed as rings with effective resistances and inductances. The pick-up voltage VC in s-domain

due to induced currents (Ia and Ib) in the coils (modeling the conducting body) can be expressed as

follows:

TCTPCPC MIsMIsV (3-1)

Here M represents the mutual inductance between the two coils and defined as 2122

12 LLkM ,

where k2 is the coefficient of coupling, i.e., the fraction of the magnetic flux of a circuit that

threads a second circuit and L1 and L2 are the self-inductances of the 1st and 2

nd coil, respectively

[38]. The self-inductance L of a loop is related to its radius rloop and is given by,

looploop rgL1

(3-2)

Where g1 is the numerical constant related to the wire radius, permeability of the air and relative

permeability of the medium. Thus, the mutual inductance Mcb between the external body coil and

the receiver coil is given as:

bccbcb rrcM (3-3)

And the mutual inductance Mca between the concentric inhomogeneity and the receiver coil is

given as:

26

accaca rrcM (3-4)

The definitions of the coefficients cac and cb

c are given in the next subsection.

(a)

(b)

Figure 3-2 : (a)Two concentric coils representing the inhomogeneity and rest of the conductive

body. The receiver coil is also shown. Here ra, rb and rc are the radius of the inhomogeneity,

effective radius of the external conducting region and radius of the receiver coil, respectively. (b)

Circuit model of the contactless measurement system. Ic, Ia, Ib are the currents flowing through the

coil, inhomogeneity, and external part the conductive body, respectively. Mca is the mutual

inductance between the receiver coil and the internal coil that models the concentric

inhomogeneity, Mcb is the mutual inductance between the receiver coil and the coil that models the

external region of the body and Mab is the mutual inductance between the two coils that models the

two concentric regions of the conducting body.

27

3.1.2.1 Mutual Inductance Between Coaxial Coils

Figure 3-3 shows the geometry used to calculate the mutual inductance between two circular coils.

The radius of the coils are a and b and the distance between the coils is h. The magnetic flux

density is assumed to be uniform over the area of the loops. The mutual inductance between the

two loops is then determined as:

2

3

22

220

2

hb

baM

(3-5)

Comparing this formula with equation (3-3) and (3-4), we can find the coefficients cbc and cac

as,

2

3

22 hr

rrkc

c

bccbcb

(3-6.a)

2

3

22 hr

rrkc

c

accaca

(3-6.b)

Figure 3-3: Geometry for the calculation of mutual inductance between the two loops. Here a and

b are the radius of the coils and h is the distance between the coils.

3.1.2.2 Current Flowing in a Cylindrical Body

Assume a uniform magnetic flux density zo awtBtB )(cos)( is applied on a conductive ring as

shown in Figure 3-4. The currents Ia and Ib induced within the tissue and the concentric

inhomogeneity can be found as follows [58]:

b

a

r

h

28

hm

Cross-section, A

σ

Contour C

a

rb

r

ra

Surface Sz

x

y

B0 cos (wt)

a

Figure 3-4: The conductive ring. A z-directed magnetic field applied to a conductive ring of inner

radius rb and outer radius ra. Height of the ring is indicated as hm.

thBrI aaa s sin4

10

2 (3-7.a)

thBrrI abbb s sin4

10

22 (3-7.b)

Substitution of the current expressions into the voltage equation (3-1) and use of the relevant

mutual coupling expressions result in the following expression for the receiver voltage,

b

a

b

acb

b

acabbccC

r

rc

r

rcrrthBkV

s

ss

32

30

2 1sin

(3-8)

Here ck is the numerical coefficient related to the permeability and geometry of the medium.

Replacing the terms that correspond to the definitions of contrast ( ) and resolution ( ) we

obtain,

3230

2 1sin scbcabbccC ccrrthBkV (3-9)

By taking the derivative of VC with respect to α and multiplying by α/V yields

32

3

1cbca

cbcc

c

c

cc

cVV

V

VS

(3-10)

29

Equation (3-10) appears to be the key expression to investigate the relationship between spatial

resolution, contrast, noise and accuracy in this measurement system. Note that, the term on the left

hand side of this equation is the ratio of noise (dVc / Vc) to accuracy (dα /α), and the right hand side

is a function of spatial resolution (β) and contrast (α). Hereby the term on the left is denoted by S

representing sensitivity as used in [76] for impedance imaging. When the relations between

sensitivity, resolution and contrast are analyzed, it is easy to extend these relations to the four

parameters (noise, accuracy, resolution and contrast) as noted as the ultimate goal of this study.

The sensitivity S obtained with the defined model is verified with simulations. The details of the

simulation studies will be given in the next section (Section 3.1.3).

3.1.3 Numerical Model

In this part of the study we simulate the model in Figure 3-1 by using ANSYS, a commercial

simulation program based on the Finite Element Method. In this simulation, a cylindrical

perturbation with conductivity of σp is placed within a cylindrical object of conductivity of σ=0.2

S/m. A circular coil is placed above the conductive body in such a way that the plane of the

objects and the coil are parallel and that their axes are coaxial. The distance between the coil and

the object is set to 1 mm. The aim is to verify the results obtained with (3-10). For this purpose,

the conductivity of the perturbation is changed and the pick-up voltage in the receiver coil is

calculated for

σb =k. σa =0.2k where k=2,3,4,5,6,7,8,9 (3-11)

Then, the fractional change in the measurements as a response to the fractional changes in

conductivity contrast is determined.

3.1.4 Comparison of the Analytical Model and the Numerical Model

In order to verify the model explained in Section 3.1.2, the sensitivity values obtained using the

analytical approach (by employing Eq. (3-10)) are compared with the numerical results obtained

with ANSYS simulations (Section 3.1.3). Figure 3-5 to Figure 3-8 show the resultant sensitivity

versus contrast plots using both approaches for particular values of resolution. The results

obtained for h=1mm, hm=10mm and rc=10mm in Figure 3-1.

30

Figure 3-5: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS

simulations, respectively (β=0.1). The nonlinearity error is 1.38% of the full scale.



0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

= sp / s

t

Sensitiv

ity,

SSensitivity for =0.1

analytical

numerical

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

= sp / s

t

Sensitiv

ity,

S

Sensitivity for = 0.2

analytical

numerical

31





0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

= sp / s

t

Sensitiv

ity,

S

Sensitivity for =0.3

analytical

numerical

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

= sp / s

t

Sensitiv

ity,

S

Sensitivity for =0.4

analytical

numerical

32

The nonlinearity errors calculated for different resolutions are tabulated in Table 3-1. It can be

seen that the error increases as β increases. Possible sources of these errors are listed below:

Modeling errors in the analytical formulation,

Uniform B-field assumption,

Simulation errors such as the number of the mesh elements, etc.

Table 3-1: Maximum non-linearity (NL) error between the analytical and numerical results.

b

a

r

r

Maximum

error (%)

0.1 1.38

0.2 11.2

0.3 16.6

0.4 17.9

3.2 Sensitivity Analysis of The Imaging System by using The Analytical

System Model

3.2.1.1 The Relationship Between Sensitivity, Conductivity Contrast, Spatial

Resolution And Noise

In order to determine the relationship between sensitivity, conductivity contrast, spatial resolution

and noise, we appeal to Equation (3-10),

32

3

1cbca

cb

cc

cS

(3-10)

The relationship given by Equation (3-10) is shown in Figure 3-9 through Figure 3-11. Note that,

it is the same information used in each of these figures; the figures are merely differing in the

choice of the axis. The following approach is used to obtain the graphs: a feasible value is set for

one variable and then the interaction between the other variables is examined (Table 3-2).

33

Table 3-2: Pairs of interactions between variables

Figure number Sensitivity, S Conductivity Contrast, α Spatial Resolution, β

Figure 3-9 Fixed

Figure 3-10 Fixed

Figure 3-11 Fixed

Figure 3-9 shows the relationship between sensitivity (S), contrast (α) and spatial resolution (β) for

particular values of the spatial resolution. For small conductivity contrast, the logarithm of the

sensitivity is linearly related to the logarithm of the contrast. The slope of the lines β=constant is

32

2

1

1

ln

ln

cbca

ca

cc

c

d

dS

Sd

Sd

(3-12)


values of spatial resolution (The plots were drawn for h=1 mm and rc=10 mm in Figure 3-1.).

10-3

10-2

10-1

100

101

102

103

10-4

10-3

10-2

10-1

100

Sensitiv

ity,

S

Conductivity Contrast, = sa / s

b

The Relationship between S, and ; for particular values of

=0.1

=0.15

=0.05

=0.35

=0.25

=0.3

=0.4

=0.2

34

In such a case, if the fractional change in contrast is K then the fractional change in sensitivity is

almost K, i.e., improvement in sensitivity by a factor of K can be obtained by enhancing the

conductivity contrast by a factor of K. Note that this coefficient is also a function of contrast. In

general, as the contrast increases, any fractional change in contrast is less reflected to the

fractional changes in sensitivity.

Figure 3-10 shows the relationship between sensitivity (S), contrast (α) and resolution (β) for

particular values of contrast. For small radius, the logarithm of the sensitivity is linearly related to

the logarithm of the contrast. The slope of the lines α= constant is

32

2

1

2

ln

ln

cbcb

cb

cc

c

d

dS

Sd

Sd

(3-13)

The slope of the asymptote, as β approaches to zero, is 3. Thus, an improvement in sensitivity by a

factor of K balances an improvement in the spatial resolution by a factor of K

1/3.


values of α (The plots were drawn for h=1 mm and rc=10 mm in Figure 3-1.).

10-3

10-2

10-1

100

10-5

10-4

10-3

10-2

10-1

100

Sensitiv

ity,

S

Spatial Resolution, = ra / r

b

The Relationship between S, and ; for particular values of

=0.5

=100

=10

=2

=10

=10

=10

=10-5

-4

-3

-2

-1

=10

=5

35

Figure 3-11 shows the relationship between sensitivity (S), contrast (α) and resolution (β) for

particular values of sensitivity. For all sensitivity values, the logarithm of the contrast is linearly

related to the logarithm of the resolution. The slope of the lines for any S value is

2

2

1

3

ln

ln

d

d

d

d

(3-14)

The slope of the asymptote, as β approaches to zero, is -3. Thus, an extension in contrast by a

factor of K is balanced by a degradation of resolution by a factor of K 3.


values of sensitivity (The plots were drawn for h=1 mm and rc=10 mm in Figure 3-1.).

The fourth relation is given by the definition of sensitivity (Equation (3-10): An improvement in

noise by a factor of K balances an improvement in accuracy by a factor of K.

10-3

10-2

10-1

100

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Contr

ast,

Spatial Resolution, = ra / r

b

The Relationship between S, and ; for particular values of S

S=0.9

S=10

S=10

S=10

S=10

S=10

S=10

S=10

S=10

S=10-1

-2

-3

-4

-5

-6

-7

-8

-9

36

3.2.1.2 Summary and Comments

Combining the above relationships, one may obtain the following rules:

1. resolution-accuracy:

degrading accuracy by K balances improving resolution by K 1/3

.

2. resolution-noise:

improving noise by K balances improving resolution by K 1/3

.

3. accuracy-noise:

improving noise by K balances improving accuracy by K

4. resolution-contrast:

extending contrast by K balances degrading resolution by K 3

5. accuracy-contrast:

extending contrast by K balances degrading accuracy by K

6. noise-contrast:

extending contrast by K balances improving noise by K

These relationships can be used to investigate/improve the performance of the system. For

example, let us assume that we need to improve the conductivity resolution by a factor of two. It is

known that the conductivity resolution is limited by the noise. Then in accordance with the above

rules, we have to improve the noise by a factor of eight (23=8). If we wish to improve both

conductivity contrast and conductivity resolution by a factor of 2 we have to improve the noise by

a factor of 16 (2×23=16).

Figure 3-9 to Figure 3-11 can be used to determine a parameter value while the other three are

known. For example, suppose an imaging system has a noise level of 1 in 104

4101 CC VV , and a resolution of 1 in 50 02.0ba rr . The question that we would like

to answer is the following: What is the range of conductivity contrast that can be measured

with an accuracy better than 10% 1.0 ? The given noise and accuracy require a

sensitivity of 1x10-3

(=1×10-4

/0.1). Finding the intersection of the lines sensitivity=1x10-3

and

radius=0.02 on Figure 3-10 shows the contrast is approximately 5. Thus, the conductivities

within a contrast range of 5 to 1 can be imaged to an accuracy of at least 10%.

These relationships can also be used in the design of an imaging system. Following design-

evaluation procedure may be adopted for this purpose:

(i) Define a realistic resolution by considering clinical applications (for instance assume

a tumor in a breast),

37

(ii) Calculate the noise of the electronics and assume a contrast value (this assumption is

consistent for the tumor-in-breast case in (i)).

(iii) Estimate the sensitivity by using Equation (3-10) or Figure 3-9 to Figure 3-11).

(iv) Substitute the unknowns in Equation (3-10) and find the accuracy.

(v) If the accuracy is too low, trade off some resolution against accuracy (using Equation

(3-10)), or interaction 1 in Section 3.2.1.2) or make some improvement in the noise

(using Equation (3-10), or interaction 3 in Section 3.2.1.2)

For example, suppose an imaging system has a noise level of 1 in 106 6101 CC VV , and a

resolution of 1 in 20 05.0ba rr . Figure 3-9 shows that, assuming α=1, the sensitivity must be

about 0.0005. The accuracy is therefore 10-6

/0.0005=2×10-3

meaning a 0.2% error in the value of

the conductivity. Since this seems a sufficient accuracy, there is no need to trade off the resolution

against accuracy.

Suppose now that the imaging system has a noise level of 1 in 103 3101 CC VV , and a

resolution of 1 in 20 05.0ba rr . Figure 3-9 shows that, assuming α =10, the sensitivity must be

about 0.005. The accuracy is therefore 10-3

/0.005=0.2. This means that 20% error in the value of

the conductivity. To reduce the error, say 5%, one may refer to Interaction 1 in Section 3.2.1.2.

This relationship indicates that improving the accuracy by a factor of 4 balances degrading the

resolution by a factor of 1.59. Therefore the resolution should be increased to 1.59×0.05=0.08 to

maintain the required accuracy.

The required accuracy can also be maintained without limiting the resolution. For that case one

may refer to Interaction 3 in Section 3.2.1.2. This relationship indicates that improving the

accuracy by a factor of 4 balances improving noise by a factor of 4. Therefore the noise level of

the system should be improved to 25×10-5

(1 in 4×10-3

).

3.2.2 The Sensitivity of the Imaging System with Impedance Analysis of the

Sensor

In the previous analysis, we modeled the object as a conductive loop and analyzed the sensitivity

of the system by defining the sensitivity as “a fractional change in voltage for a fractional change

in contrast”. In this subsection, another definition of the sensitivity is used to analysis the sensor

performance.

38

In this approach, the well-known Geselowitz relationship which simply expresses “the local

contribution of a volume element to the total impedance change” is employed. The Geselowitz

relationship is generally used for EIT sensitivity calculations based on the reciprocity theorem

which is used to calculate the sensitivity of the system. In this approach, the sensitivity is defined

as the ratio of the impedance change due to the conductivity variations as s

ZS . By using

reciprocity theorem, the impedance of the receiver coil can be expressed as [77],

dVEEII

Z 12

21

s (3-15)

where 1E

and 2E

are the electric fields created in the conductive object and in the receiver coil

due to the transmitter coil currents and the currents flowing in the conductive object, respectively.

Figure 3-12: Coaxial Coil System [77]. Here h1 and h2 are the vertical distance between the

perturbation and the transmitter and receiver coils, respectively, r is the horizontal distance

between the center of the coils and the perturbation. (with the courtesy of Prof. Dr. N.G.

GENCER)

For the system shown in Figure 3-12, the E

fields in Eq. 3-15 can be expressed as,

23

2

2

2

222

23

2

1

2

11112

44 hr

rkISN

hr

rkISNEE

(3-16)

Receiver

coil

Transmitter

coil

39

Substituting the E

fields into Eq. 3-15 we obtain the sensitivity expression,

dVkSSNN

hr

r

hr

rZS 2

21212

32

2

223

2

1

22

444

1

s

(3-17)

where k2=w

2μ0ϵ0

As it can be seen from Eq. (3-17), the sensitivity is a function of the horizontal and vertical

distance between the sensor and the perturbation (object), operating frequency of the system, and

number of turns, and the area of the coils. The relations are depicted in Figure 3-13, Figure 3-14

and Figure 3-15 below.

As a conclusion, the following results may be obtained considering Figure 3-13, Figure 3-14 and

Figure 3-15:

The relation between the sensitivity and the variable parameters are given in Eq. 3-17,

The sensitivity is maximum just below the coil wires, and it decreases as the

perturbation diverges from the wires,

The sensitivity decreases as the vertical distance between the sensor and the perturbation

increases,

The sensitivity increases as the operating frequency increases,

The sensitivity is also linearly related to the area of the coils.

40

Figure 3-13: The Sensitivity (Eq. (3-17)) variations with respect to r, for h=1 cm and f=50 kHz.

Figure 3-14: The Sensitivity (Eq. (3-17)) variations with respect to h, for r=1 cm and f=50 kHz.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.2

0.4

0.6

0.8

1

1.2x 10

-17 S vs r for the height of 1 cm

the horizontal distance between the perturbation and the center of the sensor in m

the s

ensitiv

ity

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

-16 S vs h for the distance of 1 cm

the vertical distance between the perturbation and the sensor in m

the s

ensitiv

ity

41

Figure 3-15: The Sensitivity (Eq. (3-17)) variations with respect to f, for r=1 cm and h=1 cm.

3.2.3 The Sensitivity of The Imaging System with Signal-to-Noise Ratio

(SNR) Analysis

One of the quality measures of an imaging system is the signal-to-noise ratio (SNR) of the system.

The performance of an imaging system increases as the SNR increases. In this study, the SNR

analysis of a single coil sensor is performed. The details of this analysis are explained in the

following subsection.

3.2.3.1 SNR Analysis of the Single coil Sensor

Here, we define the signal as the voltage induced in the receiver coil due to the conductive object,

and the noise as the thermal noise of the conductive object measured at the receiver coil (Figure

3-16).

Thus, the SNR may be expressed as,

2

2

n

rVSNR

s (3-18)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

0

0.2

0.4

0.6

0.8

1

1.2x 10

-15 S vs f for r = 1 cm and h1 = 1 cm and h

2 = 2 cm

frequency in Hz

the s

ensitiv

ity

42

Figure 3-16: Sensor-Perturbation geometry for SNR calculations [77]. Here h is the vertical

distance between the perturbation and the receiver coil, r is the horizontal distance between the

center of the receiver coil and the perturbation and a is the radius of the coil. (With the courtesy of

Prof. Dr. N.G. GENCER)

The noise voltage is nothing but the thermal noise of the receiver coil. This thermal noise arises

from the resistance of the receiver coil. In order to find the noise voltage level, the resistance of

the receiver coil should be determined first. The noise can be expressed by using the well-known

thermal noise formula which is,

fKTRbn 42s (3-19)

The voltage induced in the receiver coil can be expressed as,

2

3

42

16

1

R

raVIkV bgr s (3-20)

Substituting these equations into Eq. 3-18, we end up with the final SNR equation,

1/2 2

1/2 2 2 3/2

( )

4 (4 ) ( )

g bkI V a r

SNRKT f r h

s

(3-21)

As it can be seen from the Eq. 3-21, the SNR is a function of the horizontal and vertical distance

between the sensor and the perturbation (object), operating frequency of the system and the

conductivity and the volume of the perturbation. The relations are depicted in the figures below.

Receiver coil

Volume Element

(perturbation)

43

Figure 3-17 : The SNR (Eq. (3-21)) variations with respect to r, for h=1 cm and f=50 kHz.

Figure 3-18: The SNR (Eq. (3-21)) variations with respect to h, for r=1 cm and f=50 kHz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05SNR vs r for the height of 1 cm

horizontal distance between the perturbation and the sensor in m

SN

R

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.02

0.04

0.06

0.08

0.1

0.12

0.14SNR vs h for the height of 1 cm

vertical distance between the perturbation and the sensor in m

SN

R

44

Figure 3-19: The SNR (Eq. (3-21)) variations with respect to f, for h=1 cm and r=1 cm.

As a conclusion, the following results may be obtained from the Figure 3-17, Figure 3-18 and

Figure 3-19:

The relation between the SNR and the variable parameters are given in Eq. 3-21,

The SNR reaches its maximum value just below the coil wires and it decreases as the


The SNR is greater for the perturbation within the coil area than for the perturbation

outside the coil area,

The SNR decreases as the vertical distance between the sensor and the perturbation

increases,

The SNR increases as the operating frequency increases.

The SNR is also linearly related to the current of the coil, the area of the coil, the square root of

the perturbation conductivity and the square root of the perturbation volume.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16SNR vs f for h=1 cm, r=1 cm

frequency in Hz

SN

R

45

CHAPTER 4

HARDWARE STUDIES

4.1 Principle of Data Acquisition

The basic block diagram of the Low-Frequency Electrical Conductivity Imaging Data Acquisition

System is shown in Figure 4-1. The operation principle of the system is the following: the power

amplifier fed by the function generator excites the transmitter coil which produces a primary field

distribution in the body. In turn, so-called primary voltage is induced on the receiver coils. The

receiver coils are identical and differentially connected, thus, in the absence of the conductive

body, the sensor output VRec12 is approximately 0 V.

When the object is brought nearby the sensor, the secondary field created due to the conductive

objects results a phase shift at the receiver coil output. The degree of the phase shift is related with

the conductivity of the object. Thus, by measuring this phase shift, it is possible to obtain the

conductivity information of the object under inspection.

The phase shift is picked-up by employing a Lock-in amplifier which relies on Phase Sensitive

Detection (PSD) principle. The PSD method requires two inputs, one is for the reference signal

and the other one is for the input or measured signal. The phase difference between two signals

(the transmitter and receiver coil signals in our case) is converted to a corresponding DC voltage

as shown in Figure 4-2. In this study, the resultant voltage is measured using a digital multimeter.

46

Figure 4-1: The block diagram of the Low-Frequency Electrical Conductivity Imaging Data

Acquisition System.

Figure 4-2: The block diagram of the circuit which performs Phase Sensitive Detection.

The details of the blocks of the data acquisition system will be explained in the following

subsections.

Monitor

Resistance

Phase

Shifter Buffer

Buffer

Multiplier

Receiver

Coils

Low Pass

Filter

Amplifier

47

4.2 Sensor Design

The sensor coils used in the array have been designed and constructed in the light of the

simulation studies. To obtain optimum impedance, each coil is constructed as a Brook’s coil. The

sensor coils are capable of operating within multi-frequency. Besides, they give response to the

conductivity variations near the average tissue conductivity.

To determine the relation between the sensitivity and the impedance parameters of the coils, we

need to recall the Eq. 3-17 which is obtained by substituting Eq. 3-16 into Eq. 3-15,

dVkSSNN

hr

r

hr

rZS 2

21212

32

2

223

2

1

22

444

1

s

(3-17)

where k2=w

2μ0ϵ0.

We also have another definition of the sensitivity, which we defined as “a fractional change in

voltage for a fractional change in contrast” in Eq. (3-10):

32

3

1cbca

cbcc

c

c

cc

cVV

V

VS

(3-10)

Here cac and cbc are the coupling coefficients. Eq. (3-10) and Eq. (3-17) can be used to

determine the radius of the coil. The following approach is used for this purpose:

The sensitivity is calculated by employing Eq. (3-10) for a realistic conductivity-

perturbation model, for instance a timorous tissue in a woman breast.

The area, Si, of the coil can be expressed as, Si= π ai2, where ai is the radius of the i

th

sensor. Substituting the sensitivity calculated in Eq. (3-17) into Eq. (3-10) and

rearranging the terms in Eq. (3-10) it is possible to end up with a the radius versus

distance equality.

The radius of the sensor, while the sensitivity is 30x10-2

, the frequency is 100 kHz, the

heights of the receiver and transmitter coils are 0.5 cm and 1 cm, respectively, and the

width of the square shaped perturbation is 2cm, is obtained as 0.9 cm, and it is shown in

the Figure 4-3 below.

48

Figure 4-3: The sensitivity versus radius of the coils. As the distance between the sensor and the

object increases the sensitivity decreases. Thus, the optimum radius of the coil would be the point

where the distance from the sensor to the object and the radius of the coils composing the sensor

are the same. This corresponds to 9mm in our design.

The sensitivity of the coils with respect to the perturbation position is also investigated by

employing Eq. (3-17). To do this so, two neighboring coils and a perturbation are placed together

as seen in Figure 4-4. The sensitivity of the coils with respect to the perturbation position is

plotted in Figure 4-5. The following results are derived from the figures:

The sensitivity is maximum just below the coil wires, and it decreases as the


The sensitivity decreases as the vertical distance between the sensor and the perturbation

increases,

The sensitivity increases as the operating frequency increases,

The sensitivity is also linearly related to the area of the coils.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

X: 0.9

Y: 0.9

the radius of the sensor vs d

distance (cm)

the r

adiu

s (

cm

)

49

Figure 4-4: Representation of the geometry for the two neighboring sensors and a perturbation.

Figure 4-5: Sensitivity versus distance of two neighboring coils shown in Figure 4-4. The

perturbation is placed at the 5th

cm of the x-axis. The coils are placed 1 cm above the perturbation.

It is determined that as the distance between the coils increases the total sensitivity decreases,

while the perturbation stays at the same position. As a conclusion, the sensitivity is determined to

be maximum when the coils are almost touched to each other, while the perturbation is placed at

the intersection point of the coils (x=5cm) which yields an intersection of two maxima of pink

curve and green curve.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Sensitivities Of Two Neighboring Coils

Norm

aliz

ed S

ensitiv

itie

s

x axis (cm)

Total Sensitivity

Coil #2Coil #1

Sensitivity of the Coil #1

Sensitivity of the Coil #2

50

4.3 Data Acquisition Systems

Throughout this thesis, four data acquisition systems have been designed and implemented. The

sensitivities of the systems to the conductivity variations are investigated by using resistance

phantoms. In the following sub-sections, these implementations will be explained.

4.3.1 Data Acquisition System with CM-2251 Data Acquisition Card

4.3.1.1 Experimental Setup and Operation principle

The block diagram of the system is shown in Figure 4-6. In this implementation, a multi-frequency

electrical conductivity imaging system which employs a differential coil sensor is realized. The

operation principle of the system is explained in Section 4.1. In the configuration depicted in

Figure 4-6, the only difference is the use of CM-2251 Data Acquisition Card in comparison to the

configuration shown in Figure 4-1. In this setup, the Lock-in amplifier and the Digital Multimeter

are replaced with the CM-2251 Data Acquisition Card. With this replacement, a) the start-up

adjustments such as nulling of the output signal at the beginning of experiments (i.e., calibration)

becomes unnecessary, b) the mobility of the system increases, c) some additional noise sources are

eliminated, d) and the cost of the setup decreases.

The CM-2251 is a multifunction analog I/O board with up to 32 inputs sampled at up to 1 MHz

with 16 Bit precision. Having an onboard DSP processor, the CM-2251 is capable of performing

real-time tasks. Since the board is PCI Plug-and-Play and auto-calibrating, the measurements are

repeatable and reliable.

Being acquired with the CM-2251 Data Acquisition Card, the measured signal and the reference

signal are digitized and directly sent to the PC. The system is capable of operating at a multi-

frequency range between 20-60 kHz.

51

Figure 4-6: Multi-frequency data acquisition system with CM-2251 Data Acquisition Card.

4.3.1.2 Sensitivity to Conductivity Variations

The sensitivity of the system which is the response of the system to conductivity variations at a

selected operating frequency is tested by using resistance phantoms. The resistance range of the

phantoms is between 0.15-10 kΩ corresponding to conductivity range of 0.125-8.37 S/m, which is

around the tissue conductivity range. The sensitivity of the system is determined at the operating

frequencies of 20 kHz, 30 kHz, 50 kHz and 60 kHz. The sensitivity of the overall system, which is

defined as the slope of the measured voltages as a function of conductivity curve, at these

operating frequencies is given in Table 4-1 below. The measured and theoretical sensitivities are

plotted in Figure 4-7.

Cm C Cm

Vref.

Vmeas.

Iref.

VA12

Power

Amplifier

Signal

Generator

CM-2251

Data Acq.

Card

52

Figure 4-7: Theoretical and measured sensitivities of the system at operating frequencies. The

sensitivity plots are normalized. The figure reveals that the sensitivity is proportional to the square

of the frequency, as expected.

Table 4-1: Comparison of the Sensitivities at Different Operating frequencies.

Frequency

( kHz )

Measured

Sensitivity

( mV / S )

Theoretical

Sensitivity

( mV / S )

20 4.29 4.29

30 8.81 9.66

50 27.49 26.83

60 36.65 38.64

4.3.2 Data Acquisition System with Single Coil Sensor

The theory behind the single coil sensor is explained in Section 2.1.2. Knowing that the

impedance change will be maximum in a single coil sensor, it is concluded that the single coil

sensor is the most efficient sensor in the sense of sensitivity.

In the following subsections the experimental setup of the data acquisition system using single coil

sensor and the sensitivity of the system to the conductivity variations will be explained.

20 25 30 35 40 45 50 55 600

5

10

15

20

25

30

35

40

frequency (kHz)

Norm

aliz

ed s

lopes (

mV

/S)

Frequency Response of the System

Theoretical

Measured

53


The block diagram of the system is shown in Figure 4-8. In this implementation, a multi-frequency

electrical conductivity imaging system using a single coil sensor is realized. The operation

principle of the system is as follows: the power amplifier, fed by the function generator, excites

the transmitter coil, which is also a receiver coil. This induces eddy currents on the conductive

object to be imaged. In turn, these currents create secondary flux, the intensity of which is related

to the conductivity of the object. This secondary flux results a change in the inductance or in the

impedance of the sensor. This impedance change is measured via Lock-in Amplifier.

The power amplifier used in the system is a constant voltage source. Thus, since the source

voltage is constant in the sense of amplitude and phase, a change in the inductance results a

change in the phase of the current with respect to the voltage. It is observed that, the change in the

amplitude of the current due to the change in the inductance can be negligible at the operating

frequencies.

As a conclusion, the degree of the phase shift is related with the conductivity of the object. And by

measuring this phase shift, it is possible to obtain the conductivity information of the object under

study.

The phase shift is picked by employing a Lock-in amplifier using Phase Sensitive Detection (PSD)

principle. The PSD method requires two inputs, one is for the reference and the other is for the

measurement. The phase difference between two input signals (the transmitter and receiver coil

signals in our case) is converted to a DC voltage as shown in Figure 4-2. In this study, the

resultant voltage, i.e. the output of the Lock-in Amplifier, is measured using a 6,5 digit digital

multimeter.

54

Figure 4-8: Multi-frequency data acquisition system with single coil sensor.


Before discussing the results it may be handy to introduce the relation between the phase and the

inductance of the sensor.

Considering Figure 4-8, the impedance from the output of the power amplifier can be modelled as

the circuit composing of,

The monitoring resistance Rm of 1Ω,

The capacitance C of 11nF (@100kHz),

The self-inductance Lc of 190,8μH and the self-resistance Rc of 1.46Ω of the coil,

which are connected in series. The impedance value is expressed as

CLjRR

LjRCj

RZ

ccm

ccm

1

1

(4-1)

The current flowing through the coil is

Sensor

N = 80 turns

rcoil = 1.46

L coil = 190.8 H

dwire = 0.75 mm

C = 11 nF @ 100 kHz

C = 4 nF@ 120 kHz

Rm = 1 Ω

Agilent 34410A

Digital Multimeter

Referance Ch. Input Ch.

EG&G Model 5209

Lock-in Amp. Ouput

GPIB Port

PC

Agilent 33220A

Signal Generator

Phonic XP3000

Power Amplifier

55

2

2

2

2 1

1

1

1

1

CLRR

CL

j

CLRR

RRV

CLjRR

VZ

VI

ccm

c

ccm

cm

ccm

(4-2)

And the phase of the current is

cm

c

VIRR

CL

1

arctan (4-3)

Here ϕv is the reference phase obtained at the output of the power amplifier and ϕi is the phase of

the coil current that contains the conductivity information of the conductive object, monitored on

the monitoring resistance. As seen from Eq. (4-3), the relation between the phase and the

inductance is quite simple. Since the conductivity of the object affects the inductance of the

sensor, it is possible to determine the conductivity of the object by employing the phase sensitive

detection method utilizing the phases of the coil current and voltage. The inductance of the coil

can be found by using the Eq. (4-3) and then the conductivity information can be extracted from

the inductance. Although it is not determined in the scope of this thesis, the relation between the

inductance of the coil and the conductivity of the conductive object can be found from starting Eq.

(3-1), and then the inverse problem can be formulated.

The sensitivity of the system is tested by using resistance phantoms. The resistance values of the

phantoms vary between 0.26-10 kΩ corresponding to conductivity range of 0.125-5.7 S/m, which

is around the average tissue conductivity range. The sensitivity of the system is determined at the

operating frequency of 100 kHz. The sensitivity of the system is given in Figure 4-9. In order to

compare, some tissue conductivities are given in Table 4-2. It is evident that the distance between

the probe and the object is very crucial. In addition, by considering Figure 4-9 and Table 4-2,

theoretically it can be concluded that it is possible to detect an inner-bleeding in the head by

employing such a system. The results reveal that the system has a potential to be used especially in

first aid applications, for example in ambulances, to detect/determine an inner-bleeding.

56

Figure 4-9: PSD output in volts as a function of conductivity.

Table 4-2: Electrical conductivities of several tissues.

Tissue Conductivity (S/m)

Skull 0.02

Gray Matter 0.1

CSF 2

Blood 0.7

Fat 0.04

Muscle 0.35

4.3.3 Data Acquisition with Array Coil Sensor

The coils that constitute the sensor are constructed using the theory explained in CHAPTER 3

above. The array is composed of 1×4 differential coils, which means that there are 1×4×3=12 coils

in the array: 8 of which are receiver coils and 4 of which are transmitter coils. The receiver coils

57

have 400 turns with a wire diameter of 0.20 mm and the transmitter coils have 100 turns with a

wire diameter of 0.45 mm.

The sensor is completely handmade and the key point for such a construction is that the coils

should be identical. For instance at the operating frequency of 100 kHz, the difference in voltage

(i.e. non-ideality voltage) between the two receiver coils due to the non-identical construction of

the coil is at the range of tens of volts. More realistic example is the following: the sensor used in

the work of Ulker et. al. [61] had a voltage difference larger than 50 Volt at the operating

frequency of 14 kHz. Since the signals due to conductivity variations are very small in comparison

to voltages arising due to the non-idealities, the voltage difference between the receiver coils

should be dropped below 1 Volt so as the signal due to conductive object can be determined. The

coils that are constructed in the scope of this study to constitute the array of coil sensors are

identical in the sense that the maximum voltage difference between the receiver coils making the

differential coil is below 1 Volt, even at the operating frequency of 100 kHz. The non-ideality

voltage increases as the frequency increases.


In this part of the thesis, the operation principle of the data acquisition system is explained. A

photograph of the sensor and the coil geometry is shown in Figure 4-10. The block diagram of the

data acquisition system is given in Figure 4-11.

58

Transmitter Coil:

#of turns = 100

ϕ Coil = 0.45 mm

Brook’s coil properties:c = b = 7mma = 3c/2 = 10.5mm

Sensor Properties:

Receiver Coils(Differentially connected) #of turns = 400 ϕ Coil = 0.20 mm

Figure 4-10: The 1x4 array coil sensor. Each coil is constructed as a Brook’s coil, which makes

the impedance thus the sensitivity of the coils maximum.(The figure on the left is taken from

http://info.ee.surrey.ac.uk/Workshop/advice/coils/air_coils.html, January 2013)

59

Excitation Lock-in Amplifier DC OutputSignal Input Reference Signal(50 kHz) Signal (From the Rm )

DijitalMultimeter

GPIB

R1 R2 R3 R4

V1

A1’

A1

V2

A2’

A2

V3

A3’

A3

V4

A4’

A4

“Transmitter Coil” Relays

R1’ R2’ R3’ R4’

Control UnitC & Drivers)

“Receiver Coil”Relays

C1=34 nF C2=34 nF C3=35 nF C4=33 nF

RmReference Signal

(to the Lock-in’s Reference input)

Power Amplifier

Figure 4-11: The block diagram representation of the data acquisition system which comprises a 1x4 array coil sensor, relays, a controller, necessary

instruments and a PC.

60

The controller comprises a microcontroller, two 2×4 multiplexer and 8 transistors (Figure 4-18

and Figure 4-19). The relays are controlled by the transistors driven by the multiplexer. Finally the

multiplexer is controlled by the microcontroller. The operation sequence is the following (see

Figure 4-11):

i. The GUI is communicated with the microcontroller via the serial port of the PC.

ii. The microcontroller selects Coil #1 with the help of the multiplexer. The

multiplexer controls Relay #1, designated as R1, by driving it with a transistor.

iii. R1 drives the transmitter of the first sensor while R1’ drives the receivers.

iv. After then, the field measurements are performed by taking 100 data at one point

by using the multimeter. The final value of the field measurement at one point is

determined by taking the average of the 100 measurements at that point.

v. Then the same procedure is repeated for Coil #2, #3 and #4.

In the remaining of this subsection, the impedance characteristics of the coils and the PCB Cards

which are designed and constructed to conduct the data acquisition procedure are explained.

4.3.3.1.1 Characteristics of the Coils Composing the Sensor Array

Mechanical properties of the coils which constitute the sensor array used in the multi-frequency

system are given in Table 4-3. The impedances of the coils were obtained by employing Agilent

4294A Impedance Analyzer and are plotted as a function of frequency in Figure 4-12 to Figure

4-15. The transmitter coil is fed by the XP3000 2800 Watts power amplifier. Resonance

frequencies of the coils composing the sensor array are given in Table 4-4. Appropriate capacitors

are employed in series with the transmitter coils to cancel out the inductance of the transmitter coil

at each operating frequencies. The capacitor values for each operating frequency are given in

Table 4-5. The impedance versus frequency plots of the coils of the sensor array are given in

Figure 4-12 for Coil #1, Figure 4-13 for Coil #2, Figure 4-14 for Coil #3, and Figure 4-15 for Coil

#4.

Table 4-3: Mechanical Properties of the Coils Composing the Sensor array.

Coil #1 Coil #2 Coil #3 Coil #4

Transmitter Receiver Transmitter Receiver Transmitter Receiver Transmitter Receiver

# of Turns 100 400 100 400 100 400 100 400

ϕ (mm) 0.45 0.20 0.45 0.20 0.45 0.20 0.45 0.20

61

Table 4-4: Resonance Frequencies of the Coils Composing the Sensor array (in MHz).


Transmitter 4.05 4 3.85 3.85

Receiver Above (A) 0.375 0.360 0.365 0.355

Receiver Below (A’) 0.375 0.360 0.370 0.360

Table 4-5: Capacitors employed in series with the transmitter coils to cancel out the inductance of

the transmitter coil at operating frequencies.

Operating frequency

(kHz)

Capacitance (nF)


25 199,74 209,77 195,74 199,73

50 40,11 42,18 39,12 40,19

75 16,11 17,11 16,11 17,11

100 8,96 8,96 8,96 8,96

(a)

Figure 4-12: Impedance of Coil #1 as a function of frequency

(a) transmitter, (b) upper receiver, (c) lower receiver.

62

(b)

(c)

Figure 4-12:(Continued)

.

63

(a)

(b)



64

(c)

Figure 4-13: (Continued)

(a)



65

(b)

(c)


66

(a)

(b)



67

(c)


4.3.3.1.2 The Controller Card

As explained in the above subsections, the array consists of 1x4 differential coils. Thus the system

needs 4 relays for transmitters and 4 relays for receivers. The relays are being employed here to

select the active coil at a time, in other words one differential coil is expected to be “ON” at the

desired time while the other three are “OFF”. This can be achieved with the help of relays

controlled by the Controller and the PC. The relays are designed and constructed as 2 separate

PCB’s consisting of 8 relays and necessary electronics. The PCB layouts and photographs of the

transmitter and receiver coil selection (or driver) circuits are shown in the figures below (Figure

4-16-Figure 4-19).

68

(a)

(b)

Figure 4-16: Relay Card for the Receiver coils: a) PCB layout of the card,

b) Photograph of the card

69

(a)

(b)

Figure 4-17: Relay Card for the Transmitter coils: a) PCB layout of the card,

b) Photograph of the card

70

(a)

(b)

Figure 4-18: The analog multiplexer and the relay driver card. The card is composed of a

multiplexer, transistors, resistors and capacitances: a) PCB layout of the card,

b) Photograph of the card (The transmitter and receiver coils are driven with two independent

cards.)

71

(a)

(b)

Figure 4-19: The controller and controller-to-PC communication card. The controller-to-PC

communication is performed via the serial port (RS-232 protocol): a) PCB layout of the card,

b) Photograph of the card.

Analog multiplexer circuit and relay driver circuit is implemented on the same PCB, as shown in

Figure 4-18 above. The card is composed of ADG408 multiplexer from Analog Devices, BD256

PNP transistors, resistors and capacitances. Since one card can drive 8 Relays, 2 driver cards have

been constructed (Figure 4-18/b).

Figure 4-19 shows the controller part of the system. The card is composed of a microcontroller

and necessary electronics for the microcontroller to work, and a circuit block for the

communication with PC. The communication is performed through the RS232 port. Despite its

complexity, the controller card is very flexible and functional.

72


The sensitivity of the system, i.e. the response of the system to conductivity variations at an

operating frequency, is tested by using resistance phantoms. The resistance range is from 0.15 kΩ

to 10 kΩ, corresponding to conductivity range from 0.125 to 8.37 S/m, which covers the typical

conductivity range of biological tissues. The sensitivity of the system is determined at the

operating frequencies of 25 kHz, 50 kHz, 75 kHz and 100 kHz. The sensitivity of the overall

system can be derived from the slope of the measured voltages as a function of conductivity curve

and tabulated for all operating frequencies in Table 4-6. The measured and theoretical sensitivities

are also plotted in Figure 4-24. The results show that the response of the system to conductivity

variations obeys the theory stating that 1) the sensitivity increases as the conductivity increases

and 2) the sensitivity increases as the operating frequency increases.

(a) Slope= 0.8434 mV / mho, Nonlinearity (NL) = %3.3

Figure 4-20: The Sensitivity of Coil #1 at operating frequency of

(a) 50 kHz, (b) 75 kHz, and (c) 100 kHz.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Conductivities between 1 / 220 - 1 / 10000 Mho

mili

Volts

Conductivities at f= 50 kHz, NL = % 3.3

measured

fitted

73

(b) Slope=1.7889 mV / mho, Nonlinearity (NL) = %1.8

(c) Slope = 3.3576 mV / mho, Nonlinearity (NL) = %7.4


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

0

1

2

3

4

5

6

7


mili

Volts


measured

fitted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-4

-2

0

2

4

6

8

10

12

14


mili

Volts


measured

fitted

74

(a) Slope = 1.01 mV / mho, Nonlinearity (NL) = %1.9

(b) Slope = 2.3 mV / mho, Nonlinearity (NL) = %2.3



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

0

1

2

3

4

5


mili

Volts


measured

fitted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-2

0

2

4

6

8

10

12


mili

Volts


measured

fitted

75






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-5

0

5

10

15


mili

Volts


measured

fitted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


mili

Volts


measured

fitted

76

(b) Slope = 1.95 mV / mho, Nonlinearity (NL) = %2.7



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

0

1

2

3

4

5

6

7

8

9


mili

Volts


measured

fitted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-2

0

2

4

6

8

10

12

14

16


mili

Volts


measured

fitted

77


(b) Slope = 1.77 mV/mho, Nonlinearity (NL) = %3.6



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4


mili

Volts


measured

fitted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-1

0

1

2

3

4

5

6

7

8

9


mili

Volts


measured

fitted

78



Table 4-6: The slopes of the sensitivities at different operating frequencies (mV/mho).


f=50kHz 0.843 1.01 0.97 0.84

f=75kHz 1.79 2.3 1.95 1.77

f=100kHz 3.36 4.13 3.67 3.1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-4

-2

0

2

4

6

8

10

12


mili

Volts


measured

fitted

79

(a) NRMS=2.56% (Coil #1)

(b) NRMS=1.79%

Figure 4-24: Theoretical and measured sensitivities of the coils at different operating frequencies.

The sensitivity plots are normalized. The figures reveal that the sensitivity is proportional to the

square of the frequency, as it is expected. (a) Coil #1, (b) Coil #2, (c) Coil #3 and (d) Coil #4.

50 55 60 65 70 75 80 85 90 95 1001

1.5

2

2.5

3

3.5

4

Frequency ( kHz )

Norm

aliz

ed S

lopes

Frequency Response of the System: Coil #1

measured

expected

50 55 60 65 70 75 80 85 90 95 1001

1.5

2

2.5

3

3.5

4

4.5

Frequency ( kHz )

Norm

aliz

ed S

lopes


measured

expected

80

(c) NRMS=6.05%

(d) NRMS=6.56%


50 55 60 65 70 75 80 85 90 95 1001

1.5

2

2.5

3

3.5

4

Frequency ( kHz )

Norm

aliz

ed S

lopes


measured

expected

50 55 60 65 70 75 80 85 90 95 1001

1.5

2

2.5

3

3.5

4

Frequency ( kHz )

Norm

aliz

ed S

lopes


measured

expected

81

CHAPTER 5

SINGLE FREQUENCY STUDIES

5.1 Introduction

In this section the images reconstructed from the multi-frequency array coil sensor system will be

presented. The data are collected by performing 2D scanning over the region to be imaged. The

details of the data collection and acquisition process are explained in CHAPTER 4. The scanning

is performed in two schemes, namely, 1D movement and 2D movement. Since the sensor used in

the system is an array, it intrinsically makes a 1D scanning along the sensor (array) alignment

direction. Thus, even for a 1D movement of the sensor, the system achieves a 2D scanning. That is

why we prefer to use the term movement instead of scanning.

5.1.1 Inverse Problem Solution and Comparison of the Solution Methods:

The inverse problem is solved by employing several iterative methods, namely the Steepest

Descent Method, Newton-Rapson Method or Conjugate Gradient Method. In computational

mathematics, an iterative method attempts to solve a problem (for example an equation or system

of equations) by finding successive approximations to the solution starting from an initial guess.

This approach is in contrast to the direct methods which attempt to solve the problem by a finite

sequence of operations, and, in the absence of rounding errors, would deliver an exact solution

(like solving a linear system of equations Ax = b by Gaussian elimination). Iterative methods are

usually the only choice for nonlinear equations. However, the iterative methods are often useful

even for linear problems involving a large number of variables (sometimes of the order of

millions), where the direct methods would be prohibitively expensive (and in some cases

impossible) even with the best available computing power.

82

In this thesis study, three iterative methods are employed and compared to determine the best

approach for the inverse problem solutions. The data used for this comparison was obtained with

the differential coil system implemented in the Master of Science studies of the author [58]. The

inhomogeneous conductive body is obtained by immersing an agar block of conductivity of 6 S/m

in a saline solution (0.2 S/m).

Knowing that we have a linear system of equations Ax=b, following algorithms are adopted for

assessment:

i) The Steepest Descent Method:

)(1 kkkk xfxx (5-1)

where k is the iteration number, 2

)( bAxxf is the function to be minimized,

kkk xfxf minarg is the step size.

Here,

vSxSSxf T

k

T

k )(

and

k

T

k

k

T

k

k

TT

k

k

TT

k

kxfHxf

xfxf

xfSSxf

vxSSSvxS

where, H is an nxn, positive definite, symmetric Hessian matrix.

ii) The Modified Newton Method:

kkkkk xfxx

1

1 H (5-2)

where kk xf2H is the Hessian matrix. 2

)( bAxxf is the function to be

minimized and kkk xfxf minarg is the step length.

iii)The Conjugate Gradient Method

83

kkkk dxx 1 (5-3)

where, k is the iteration number, kkkk dgd 11 is the direction of search,

kkk xfxf minarg is the step length. )( 11 kk xfg is the gradient,

2)( bAxxf is the function to be minimized and

k

T

k

kk

T

k

kgg

ggg

11 is Polak-

Ribiere formula,

The results are shown in the below figures. It is observed that, although it does not offer the

smallest error, the Steepest Descent Method converges with only 2 iterations.

Table 5-1: Comparison of the Inverse Problem Solution algorithms

Method Error Number of

iterations

Elapsed time

(sec.)

Steepest-Descent 0.00428 3 0.4

Newton-Raphson 0.043 3000 270

Conjugate-Gradient 8.8748e-011 69 6.7

With this observation, the Steepest Descent Method was chosen for image reconstruction in the

rest of this study. Please note that, one cannot conclude on the best approach by using a single

data. Different inhomogeneity locations and computational resources should also be taken into

account.

84

(a)

(b)

Figure 5-1: The inverse problem solution with the Steepest Descent Method (a) Conductivity

distribution (b) Error versus number of iterations.

Steepest Descent Method

50 100 150 200 250 300

50

100

150

200

250

300

1

2

3

4

5

6

7

0 5 10 15 20 25 30 35 40 45 504.27

4.28

4.29

4.3

4.31

4.32

4.33

4.34x 10

-3 Steepest Descent Method (Error function)

(S/m)

85

(a)

(b)

Figure 5-2: The inverse problem solution with the Newton Rapson (a) Conductivity distribution

(b) Error function.

Newton-Raphson Method

50 100 150 200 250 300

50

100

150

200

250

300

0

1

2

3

4

5

6

0 500 1000 1500 2000 2500 30000.0425

0.043

0.0435

0.044

0.0445

0.045

0.0455

0.046

0.0465Newton-Raphson Method (Error Function)

(S/m)

86

(a)

(b)

Figure 5-3: The inverse problem solution with the Conjugate-Gradient (a) Conductivity

distribution (b) Error function.

Conjugate-Gradient Method

50 100 150 200 250 300

50

100

150

200

250

300

1

2

3

4

5

6

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-4 Conjugate-Gradient Method(Error function)

(S/m)

87

5.1.2 Characteristics of the Imaging system

To reveal the characteristics of the imaging system, the coefficient matrix is analyzed using the

Singular Value Decomposition (SVD) technique. In this subsection, the SVD technique will be

reviewed first. The characteristics of the imaging system will then be presented using the SVD

tools.

If A is an m×n matrix with m > n, then A can be written using the so-called singular value

decomposition of the form

TVU (5-4)

where U is an m × m orthonormal matrix and V is an n x n orthonormal matrix. ∑ is an m × n

diagonal matrix such that

rdiag ss ,...,1 (5-5)

where r = min(m; n) and σ1 ≥…≥ σr ≥ 0. σk is called the kth

singular value of A. The first r columns

of V and U are the right- and left-singular vectors and r denotes the rank of A [67]. The condition

number is the measure of the singularity of a matrix and is defined as

min

max

s

s (5-6)

where σmax and σmin are the maximum and minimum singular values, respectively. The condition

number of the sensitivity matrix used in this work is, κ=569,7.

5.2 Multi-Frequency Array-Coil System

It is well known that conductivity of biological tissues may vary with frequency. That is why

multi-frequency data acquisition is a great challenge as well as a tool which enables the imaging

system with the increased number of opportunities. It can occur that in contactless conductivity

imaging of tumors or detection of other inhomogeneities, peculiarities which cannot be

distinguished or detected in one frequency may be detected in another. Besides, as shown

theoretically in CHAPTER 2, the signal obtained from the object under investigation is related

with the square of the frequency.

88

By using the multi-frequency coil-array system, both the single-frequency and the multi-frequency

experiments were performed. The operating frequencies of the system were selected as 50 kHz for

single frequency experiments and as 50kHz, 75kHz and 100kHz for multi-frequency experiments.

The imaging area was scanned involving 1D and 2D movements of the scanner.

The signal-to-noise ratio (SNR) and the spatial resolution of the system were determined at each

operating frequency, to investigate the performance of the system.

In the following subsections the inverse problem solution procedure for sensor array is explained

first. Then, the results of the single-frequency experiments will be presented. After that, the results

of the multi-frequency experiments will be covered.

5.2.1 Inverse Problem Solution for Sensor Array:

In the 1x4-coil sensor array system, the measurement sets are obtained with a 4N samples in a

single direction for a movement in the form of a strip or with (4×M)×N samples for a 2D

movement. In another words, the area under investigation is scanned by 1×N steps or M×N steps.

Thus the number of samples is 4×N or (4×M)×N and the sensitivity matrix is of dimension

(4×N)×(4×N) or (4×M×N)×(4×M×N).

As an example, assume that the sensor is to be translated through a distance of 20 cm with a step

length of 5 mm, meaning that we need 40 steps and let us say the movement is in the x-direction.

Since the system constitutes 4 coils, we acquire 4 samples along the y-direction for each sample

point along the x-direction. In addition, please note that the length of the 4-coil sensor array is

13.5 cm meaning that the scanning length along the y direction is 13.5 cm. As a conclusion, we

can scan a 20 cm × 13.5 cm area with 40 × 4 steps. The step size along the x- direction is 5mm and

that along the y-direction is 35mm. The sensitivity matrix in this case will be of dimension

160×160.

After constituting the sensitivity matrix, the inverse problem solution procedure is similar to the

one explained in Section 5.1.1.

89

5.2.2 Image Reconstruction at Single Frequency

Experiments using agar phantoms are performed to understand the performance of the system for

biological subjects. Cylindrical shaped agar phantoms with conductivity of 5 S/m are placed in a

saline solution of conductivity 0.2 S/m. The experiments are carried at an operating frequency of

50 kHz.

The imaging area is scanned with 1D and 2D movement of the sensor. Since the coils composing

the array are aligned along the y-direction, 1D scanning is defined as the movement of the sensor

along the x-direction and 2D scanning is defined as the movement of the sensor along both the x-

direction and y-direction.

5.2.2.1 1D Scanning (Movement)

The imaging area is scanned along a single direction, namely x-direction. Thus the scanning is

performed on a grid of 26×4 data points. Here 26 is the number of the steps along the scanning

(movement) direction, and 4 is the number of the coils that constitute the array, aligning along the

y-direction. Here, step size is 5 mm, and the array length is 13 cm. As a conclusion, the scanning

area of 13 cm × 13 cm is scanned on a grid of 26×4 data point. To reconstruct the conductivity

distributions of the subjects, the inverse problem is solved by employing the Steepest Descent

method (the effect of ϕ term is ignored in inverse problem solution). The inverse problem

solution procedure is explained in Section 5.2.1. The results of these experiments are given in

Figure 5-1 to Figure 5-7.

90

(a)

(b)

Figure 5-4: Field Profile and reconstructed image of a cylindrical agar phantom with a radius of

7.5 mm and a height of 20 mm placed below the intersection of two neighboring coils: (a) field

profile, (b) reconstructed conductivity distribution.

Field measurements

The Sensors

Scannin

g S

tep N

um

ber

0.5 1 1.5 2 2.5 3 3.5 4 4.5

5

10

15

20

25

0.05

0.1

0.15

0.2

0.25

Reconstructed Image-Steepest Descent Method with Uniform Search

mm (The Sensors Axis)

mm

(T

he s

cannin

g a

xis

)

20 40 60 80 100 120

20

40

60

80

100

120

0.5

1

1.5

2

2.5

3

3.5

C1 C2 C3 C4

91

(a)


7.5 mm and a height of 20 mm placed below the center of the 2nd

coil of the sensor array: (a) field


Field measurements

The Sensors

Scannin

g S

tep N

um

ber

0.5 1 1.5 2 2.5 3 3.5 4 4.5

5

10

15

20

25

0.05

0.1

0.15

0.2

0.25

0.3



mm

(T

he s

cannin

g a

xis

)

20 40 60 80 100 120

20

40

60

80

100

120

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

C1 C2 C3 C4

92

(a)

(b)

Figure 5-6: Field Profile and reconstructed image of two cylindrical agar phantoms with a radius

of 7.5 mm and a height of 20 mm, placed below the center of the 1st coil of the sensor array: (a)

field profile, (b) reconstructed conductivity distribution.

Field measurements

The Sensors

Scannin

g S

tep N

um

ber

0.5 1 1.5 2 2.5 3 3.5 4 4.5

5

10

15

20

25

0.1

0.2

0.3

0.4

0.5

0.6

0.7



mm

(T

he s

cannin

g a

xis

)

20 40 60 80 100 120

20

40

60

80

100

120

0

1

2

3

4

5

6

C1 C2 C3 C4

C1 C2 C3 C4

93

(a)

(b)





Field measurements

The Sensors

Scannin

g S

tep N

um

ber

0.5 1 1.5 2 2.5 3 3.5 4 4.5

5

10

15

20

25

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4



mm

(T

he s

cannin

g a

xis

)

20 40 60 80 100 120

20

40

60

80

100

120

1

2

3

4

5

6

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94

(a)

(b)


of 7.5 mm and a height of 20 mm, placed below the center of the 3rd



Field measurements

The Sensors

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ber

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(a)

(b)


of 7.5 mm and a height of 20 mm, placed below the center of the 4th



Field measurements

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(a)

(b)





Field measurements

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5.2.2.2 2D Scanning (Movement)

In this work, the area to be imaged is also scanned along both the x- and y- directions as part of 2D

Scan studies.

Similar to the experiments carried on for a single direction of scanning (movement), cylindrical

shaped agar phantoms with conductivity of 5 S/m are placed in a saline solution of conductivity

0.2 S/m.

Here the scanning is performed on a grid of 26×24 data points. Here 26 is the number of the steps

along the x-direction, and 24 is the result of the multiplication of 4 which is the number of the

coils and 6 which is the number of the steps along the y-direction. Thus with only 6 steps, the scan

along the y-axis is achieved with a resolution of 5mm.

Finally, the scanning area of 13 cm × 13 cm is scanned on a grid of 26×24 data point. To

reconstruct the conductivity distributions of the subjects, the inverse problem is solved by

employing the Steepest Descent method (the effect of ϕ term is ignored in inverse problem

solution). The results of these experiments are given in Figure 4-8 to Figure 4-11.

98

(a)

(b)


7.5 mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15 mm, a length of

30 mm and a height of 20 mm placed below the sensor array: (a) field profile, (b) reconstructed

conductivity distribution.

Field Measurements

step number (The Sensors Axis)

step

num

ber

(The

sca

nnin

g ax

is)

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(a)

(b)



30 mm and a height of 20 mm placed below the sensor array: (a) field profile, (b) reconstructed

conductivity distribution.

Field Measurements


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ber

(The s

cannin

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xis

)

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100

(a)

(b)


of 7.5 mm and a height of 20 mm, placed below the 3rd

coil of the sensor array: (a) field profile,

(b) reconstructed conductivity distribution.

Field Measurements


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(The s

cannin

g a

xis

)

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25

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101

(a)

(b)


of 7.5 mm and a height of 20 mm, placed below the intersection of two neighbor coils: (a) field


Field Measurements


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p n

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(The s

cannin

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102

5.2.2.3 System Performance

5.2.2.3.1 Signal to Noise Ratio

Signal-to-noise ratio (SNR) is an important quantity in determining the performance of an imaging

system. SNR in the experimental data is defined as the ratio of the signal variance to the noise

variance, and is often expressed in decibels:

2

2

10log10N

SSNRs

s (5-7)

where 2

Ss and 2

Ns are the variances of the signal and noise, respectively. In this thesis, these

variances are calculated by obtaining 100 samples of data at a point in an area of 13 cm × 13 cm.

The data is obtained on a 26×4 grid with and without conductive body. When the conductive body

is placed, the measurements also contain a noise component. Thus, in order to obtain the true

signal data, average value of the noise component is subtracted from the average value of signal at

each data point. The SNR of each coil in the array is obtained individually for the x-direction (the

movement direction). The SNR values of the each coil are tabulated in Table 5-2 below. Finally,

the SNR of the system is determined as the minimum value among them as 22.25 dB.

Table 5-2: SNR values of the coils comprising the sensor array.

Coil Number SNR (dB)

1 38.3458

2 26.2796

3 36.2701

4 22.2457

5.2.2.3.2 Spatial Resolution

Spatial resolution is a measure of how well the system can distinguish two closely located point or

line objects. One common method to find the spatial resolution is finding the Full Width Half

Maximum (FWHM) of the Point Spread Function (PSF). To determine the FWHM of the system,

a cylindrical agar phantom of conductivity 5 S/m (blood conductivity) is prepared and placed in a

saline solution of conductivity 0.2 S/m (average tissue conductivity). The radius and height of the

phantom is 7.5 mm and 20 mm, respectively. 13 cm x 13 cm area is scanned and data are acquired

103

on a 26×4 grid (i.e. step/pixel size along the scanning (movement) direction is 5 mm). The field

profile and the reconstructed conductivity image are shown in (a) and (b) parts from Figure 5-15 to

Figure 5-18. FWHM of the PSD in the x-direction is found for each coil in the array. as 19 mm for

Coil #1, as 21 mm for Coil #2, as 19 mm for Coil #3 and as 20 mm for Coil #4. The results are

shown in (e)’s from Figure 5-15 to Figure 5-18. To determine the spatial resolution of the system

along the other scanning (alignment) direction, namely, y-direction, another experiment is

performed with the same experimental conditions being established. Finally, the spatial resolution

of the system turns out to be 21 mm for this scheme.

104

(a)

(b)

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105

(c)

Figure 5-15: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm

radius placed below the center of the 1st coil: (a) field profile, (b) reconstructed conductivity

distribution, (c) FWHM calculation by using the signal spread along axial direction (1st coil).

(a)

0 20 40 60 80 100 120 1400

1

2

3

4

5

Co

nd

ucti

vit

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)

Scanning axis [mm]

FWHM of the Array Coil system

FWHM = 19 mm

Field measurements

The Sensors

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106

(b)

(c)

Figure 5-16: Field Profile and reconstructed image of a cylindrical agar phantom of 7.5 mm radius

placed below the center of the 2nd

coil: (a) field profile, (b) reconstructed conductivity distribution,

(c) FWHM calculation by using the signal spread along axial direction (2nd

coil).



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6FWHM of the Array Coil System

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C1 C2 C3 C4

107

(a)

(b)

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108

(c)


radius placed below the center of the 3rd


distribution, (c) FWHM calculation by using the signal spread along axial direction (3rd

coil).

(a)

0 20 40 60 80 100 120 1400

1

2

3

4

5

6


Scanning axis [mm]

Co

nd

ucti

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y (

S/m

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Field measurements

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109

(b)

(c)


radius placed below the center of the 4th


distribution, (c) FWHM calculation by using the signal spread along axial direction (4th

coil).



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C1 C2 C3 C4

111

CHAPTER 6

MULTI-FREQUENCY STUDIES

6.1 Introduction

As discussed in CHAPTER 1, magnetic fields created by the induced currents can be used to

determine the conductivity of an object and to reconstruct low resolution tomographic images. The

operating frequencies of tomographic systems are above 100 kHz (on the order of MHz), however,

for the operating frequencies below 100 kHz there are some certain advantages: 1) the

displacement currents in the conducting body can be negligible, 2) the propagating effects can be

ignored, and 3) the effects of the stray capacitances can be neglected [15]. In addition to this, since

the change of conductivity as a function of frequency differs in different tissues, multi-frequency

studies should be investigated. Thus, tissues that cannot be distinguished at a particular frequency

can be resolved at another frequency. This may enable the detection of tissues at different health

states.

In this Chapter multi-frequency images obtained from the multi-frequency array coil sensor

system will be presented. The data are collected by performing 2D scanning over the region to be

imaged. The details of the data collection and acquisition process are explained in CHAPTER 4.

The scanning is performed with 2D movement.

6.2 Image Reconstruction at Multi-Frequency

To determine the performance of the system at different operating frequencies, cylindrical shaped

agar phantoms with conductivity of 5 S/m are prepared and placed in a saline solution of

conductivity 0.2 S/m. Experiments are carried on at operating frequencies of 50 kHz, 75 kHz and

100 kHz.

112

In this work, a 2D movement is also performed and the imaging area is scanned along the x- and y-

directions.

In the scope of multi-frequency imaging, 2D scanning is performed on a grid of 26×24 data points.

Here 26 is the number of the steps along the x-direction, and 24 is the result of the multiplication

of 4 which is the number of the coils and 6 which is the number of the steps along the y-direction.

The size of the scanning area is 13cm x 13cm. To reconstruct the conductivity distributions of the

subjects, the inverse problem is solved by employing the Steepest Descent method (the effect of

ϕ term is ignored in inverse problem solution).

The results of these experiments are given in Figure 4-16 to Figure 4-18.

6.3 System Performance

The theory behind the SNR determination is given in Section 5.2.2.3.1. The SNR of the system is

determined at each operating frequency. The results are tabulated in Table 6-1 and plotted in

Figure 6-1.

Table 6-1: SNR of the multi-frequency system at different operating frequencies.

Frequency (kHz) SNR (dB)

50 22.2457

75 33.3685

100 49.9861

113

Figure 6-1: SNR of the system as a function of frequency

Another performance criterion of an imaging system is the spatial resolution. To determine the

spatial resolution of the system at different frequencies, the FWHM of the system is determined at

all operating frequencies. The details are explained in Section 5.2.2.3.2. Here, 13 cm × 13 cm area

is scanned and data is acquired on a 26×24 grid. The image spread of the cylindrical object along

the x-direction is plotted at 50 kHz in Figure 6-2/a, at 75 kHz in Figure 6-2/b and at 100 kHz in

Figure 6-2/c. The FWHM of the PSD in the x-direction is found as 31mm at 50kHz, as 25mm at

75 kHz and as 16mm at 100 kHz. The FWHM values are tabulated in Table 6-2 at each operating

frequency.

Table 6-2: FWHM of the multi-frequency system at different operating frequencies.

Frequency (kHz) FWHM (mm)

50 31

75 25

100 16

50 55 60 65 70 75 80 85 90 95 10020

25

30

35

40

45

50SNR vs Frequency

Frequency (kHz)

SN

R (

dB

)

114

a)

b)

Figure 6-2: FWHM calculation by using the signal spread along axial direction (4th

coil):

a) FWHM=31mm at 50 kHz, (b) FWHM=25mm at 75 kHz, (c) FWHM=16mm at 100 kHz

0 5 10 15 20 25 300

0.5

1

1.5FWHM at 50 kHz

0 5 10 15 20 25 300

1

2

3

4

5

6FWHM at 75 kHz

FWHM=31 mm

FWHM=25 mm

115

c)


0 5 10 15 20 25 300

1

2

3

4

5

6

7

8FWHM at 100 kHz

FWHM=16 mm

116

(a)

(b)

Figure 6-3 Field Profile and reconstructed image of a cylindrical agar phantom with a radius of


45 mm and a height of 20 mm placed below the sensor array:

(a) field profile, (b) reconstructed conductivity distribution, at 50 kHz.

Field Measurements


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(The s

cannin

g a

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)

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Reconstructed Image, f=50kHz


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117

(a)

(b)

Figure 6-4 Field Profile and reconstructed image of a cylindrical agar phantom with a radius of


45 mm and a height of 20 mm placed below the sensor array:

(a) field profile, (b) reconstructed conductivity distribution, at 75 kHz.

Field Measurements


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(The s

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118

(a)

(b)

Figure 6-5 Field Profile measurements and reconstructed image of a cylindrical agar phantom

with a radius of 7.5 mm and a height of 20 mm, and a bar shaped agar phantom with a width of 15

mm, a length of 45 mm and a height of 20 mm placed below the sensor array: (a) field profile, (b)

reconstructed conductivity distribution, at 100 kHz.

Field Measurements


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p n

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(The s

cannin

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)

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6.4 Summary and Comments

In Chapter, field profiles and reconstructed conductivity distributions of agar phantoms obtained,

with the multi-frequency electrical conductivity imaging via contactless measurement system, at

50 kHz, 75 kHz and 100 kHz are presented. The steepest descent algorithm was employed to

reconstruct the conductivity distributions of the objects. The system is quite sensitive to

conductivity variations in the vicinity of the average tissue conductivity of 0.2 S/m at each

operation frequency. Also, the conductivity response of the system around the body conductivity

(0-6 S/m) is linear at each operation frequency. Theoretically, the measured signal linearly

increases with the square of the frequency.

The frequency response of the system was investigated at 50 kHz, 75 kHz and 100 kHz and the

linearity of the system was obtained as 0.84 mV/Mho, 1.77 mV/Mho and 3.1 mV/Mho,

respectively (see Table 4-6). The results verify the theory. The SNR of the system was

investigated at each operation frequency and it was observed that, as expected, the SNR of the

system increases as the frequency increases (see Table 6-1).

The FWHM of the point spread function was used to determine the spatial resolution of the

system. It was seen that, the spatial resolution of the system increases as the frequency increases

(see Table 6-2).

The conductivities of the biological tissues are strictly dependent on the operation frequency. This

study shows the feasibility of the contactless, multi-frequency conductivity imaging of the

biological tissues.

121

CHAPTER 7

3D IMAGE RECONSTRUCTION

7.1 Introduction

Three-dimensional (3D) imaging is one of the most important abilities that an imaging system

must possess. In traditional 2D imaging, 3D images are represented as 2D. Thus, the reconstructed

image has a width and a height but it has no depth. This prevents to determine the exact location

and size of the inhomogeneity. With 3D imaging it is also possible to distinguish the foreground

objects from background objects. 3D image imaging has not been studied in the low frequency

subsurface imaging with a real data yet.

In this Chapter, 3D images obtained from the multi-frequency array coil sensor system will be

presented. The data are collected by performing 2D scanning over the region to be imaged. The

details of the data collection and acquisition process are explained in CHAPTER 4. The scanning

is performed with 2D movement. Constructing the 3D sensitivity matrix and then solving the 3D

image reconstruction problem will also be explained.

7.2 3D Inverse Problem Solution

In the previous studies, the depth of the imaging medium is neglected and the inverse problem is

solved for 2D case, thus the image is reconstructed as a 2D image. However, the medium has a

third dimension; the depth (Figure 7-1). In order to reconstruct the images in 3D, the sensitivity

matrix needs to be constructed properly. For this purpose, the medium is divided into 5 layers

along the depth direction, defined as z-direction, where the thickness of each layer is 5mm (Figure

7-2). Then the sensitivity matrix is constructed for each layer by employing the method explained

in Section 5.1.1 and Section 5.2.1. The sensitivity matrix will be of dimension of 5MN×5MN.

122

Accordingly, in Eq. 2-15, the conductivity vector (σ) is of dimension of 5MN and the data vector

(v) is of dimension of (5MN)×(5MN). After constituting the sensitivity matrix, the inverse

problem solution procedure is similar to the one explained in Section 5.1.1.

(a)

(b)

Figure 7-1: 3D visualization of the medium and the inhomogeneity to be imaged.

(a) XY-view, (b) XZ-view

123

Figure 7-2: 3D inverse problem solution: the medium is divided into voxels.

7.3 3D Image Reconstruction

The inverse problem solution procedure is explained in above section. In order to determine 3D

images, four individual experiments have been performed. The experiments were carried on at

operating frequency of 50 kHz.

In the first experiment, the data obtained from the previous work (MSc. Thesis) of the author has

been used. The data was obtained from a cylindrical shaped agar object with a radius 7.5mm of

and a depth of 27 mm. The medium is divided into 5 equal layers each of which has a length of

5.4mm. The reconstructed 3D image is shown in Figure 7-3.

In the second experiment, cylindrical shaped agar phantom and an agar bar with conductivity of 5

S/m are prepared and placed together in a saline solution of conductivity 0.2 S/m. The radius of

the cylindrical phantom is 7.5mm; the width and the length of the agar bar are 15mm and 30mm,

respectively. The height of the objects is 20mm. The data were collected by employing the multi-

frequency array-coil system the details of which is explained in Section 4.3.3. Here, the medium is

again divided into 5 layers along the depth direction where the thickness of each layer is 4mm.

The reconstructed 3D image is shown in Figure 7-4.

124

In the third experiment, data were obtained from two cylindrical shaped agar phantoms with a

radius of 7.5mm and a height of 20 mm. The distance between the agars from center to center is 2

cm. The medium is divided into 5 equal layers each of which has a length of 4mm. The

reconstructed 2D and 3D images are shown in Figure 7-5.

In the fourth experiment, the data were obtained from a cylindrical shaped agar phantom by using

the multi-frequency array-coil system the details of which is explained in Section 4.3.3. The radius

of the cylindrical object is 7.5mm and the height of the object is 20mm. Similarly the medium is

divided into 10 equal layers each of which has a length of 2mm. The reconstructed 3D image is

shown in Figure 7-6.

The results reveal that if the sensors are sensitive enough, it is possible to reconstruct 3D images

of biological tissues.

125

(a)

(b)

Figure 7-3: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a

height of 27 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction

2D Image

2 4 6 8 10 12 14 16

2

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4

4.5

5 3D Image (YZ View)

-2

-1

0

1

2

3

4

5

126

(a)

(b)

Figure 7-4: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a bar

shaped agar phantom with a width of 15 mm and a length of 30 mm. The height of the objects is

20 mm. (a) 2D Image Reconstruction, (b) 3D Image Reconstruction (XZ-crosssection)

2D Image

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2 C

3 C

4

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2 C

3 C

4

127

(a)

(b)

Figure 7-5: Reconstructed images of two cylindrical agar phantoms with a radius of 7.5 mm and a

height of 20 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction (XZ-crosssection)

2D Image

5 10 15 20

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Figure 7-6: Reconstructed images of a cylindrical agar phantom with a radius of 7.5 mm and a

height of 20 mm (a) 2D Image Reconstruction, (b) 3D Image Reconstruction

2D Image

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7.4 3D Imaging Performance

In order to determine the 3D imaging performance of the system, the response of the system (each

sensor) against distance from conductive object is investigated. A cylindrical object with a

conductivity of 5 S/m, a radius of 7.5 mm and a height of 20 mm is employed in the experiments.

The experiments were carried on at operating frequency of 50 kHz. At the beginning, the object is

placed 1 mm below the sensor. After then the object is moved by 1 mm apart from the sensor and

measurements were taken. The movement procedure was done until the measurements made no

sense. The response of each sensor to the distance is given in the figures below.

(a)

Figure 7-7: Response of the system (each sensor) against distance from conductive object. (a)

Sensor #1, (b) Sensor #2, (c) Sensor #3, (d) Sensor #4

1 2 3 4 5 6 7 8 9 100

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(d)


The results show that the measured voltage level is inversely proportional to the distance, i.e. as

the distance between the sensor and the object increases the measured voltage decreases. Distance

versus measured voltage is previously investigated in Section 3.2.2. The relationship is plotted in

Figure 3-14 by using Equation (3-17). As it is seen from the figures the theoretical and

experimental results are consistent with each other.

7.5 Summary and Comments

It is determined from the theoretical and experimental studies that the distance is an important

restriction for 3D sensor performance. This restriction, of course, has a negative effect on the 3D

imaging performance of the system. The 3D performance of the system is directly related to the

sensitivity of the sensor with respect to the distance and can be increased by increasing the number

of turns and the area of the coils and the operating frequency. However there are tradeoffs, for

instance the penetration depth is inversely related to the frequency or the resolution is inversely

related to the area of the coil sensor. 3D imaging in the magnetic induction-magnetic

measurements modality is a hot research topic and should further be investigated.

1 2 3 4 5 6 7 8 9 100

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133

CHAPTER 8

CONCLUSION AND DISCUSSION

8.1 Summary

In a magnetic induction imaging system it is not straightforward to make quantitative statements

about the relationships between the resolution, accuracy, conductivity contrast, and noise.

However, knowing these relationships is essential in designing effective imaging systems. In this

study, a theoretical work was conducted to reveal the relationships between these parameters. For

this purpose, a simple detection system was analyzed that uses spatially uniform (sinusoidally

varying) magnetic fields for magnetic-induction. A circular coil was used for magnetic field

measurement. A thin cylinder with a concentric inhomogeneity was used as a conductive body. An

analytical expression was developed that relates coil and body parameters to the measurements. A

set of six rules were found that reveal the relationships between resolution, accuracy, conductivity

contrast, and noise. The results were interpreted by numerical examples.

As hardware studies, prototype multi-frequency electrical conductivity imaging systems were

developed to image electrical conductivity of biological tissues via contactless measurements.

Different types of sensors, namely, single coil sensor, PCB sensor, differential coil sensor and

sensor array were designed and implemented for the systems. The sensors were quite sensitive to

conductivity variations in the vicinity of the average tissue conductivity (0.2 S/m).

An HP-VEE based user interface program was prepared to perform the scanning experiments

using a PC. A LabVIEW based user interface program was prepared to perform experiments by

using the Acquitek CM-2251 data acquisition card. GPIB and serial communications were

appealed during the scanning and the data collection process.

In the data analysis step, a MATLAB based graphical user interface (GUI) program was prepared.

134

A novel test method employing resistive ring phantoms was employed to investigate the

sensitivity of the system.

The SNR of the array coil system was investigated for each of the coils in the sensor array. The

SNR of the overall system was considered as the minimum SNR value among them which turns

out to be 22.25 dB. The sensitivity to conductivity variations of the coils constituting the sensor

array was determined individually. The overall sensitivity of the system to conductivity variations

was considered as the smallest sensitivity value among them which is 0.82 V/S.

The spatial resolution of the prototype system was found as 31 mm at 50 kHz, 25 mm at 75 kHz

and 16 mm at 100 kHz with cylindrical agar phantoms.

The average data acquisition time (scanning and recording the output for 100 samples) is

calculated as 72.5 msec/mm2.

To understand the imaging performance, different agar phantoms are scanned. The field profiles

and the reconstructed conductivity distributions of the objects are obtained. The reconstructed

images show the exact location, actual size and geometry of the objects.

8.2 Discussion

This thesis aims to investigate the potential of a sub-surface conductivity imaging methodology

for biomedical applications. For this purpose, relationships between sensitivity and the

conductivity contrast, the spatial resolution and the noise of the system were determined. In

addition to this, three novel prototype data acquisition systems were developed. Imaging

experiments were carried with the multi-frequency data acquisition system employing array coil

sensor. The performance of the system was also investigated.

In this section, firstly the interrelationships are concluded. After that, imaging within the tissue

conductivity range is discussed. Finally, the results obtained and deduced from this study are

interpreted.

8.2.1 Interrelationships between the image quality measures and affecting

factors for magnetic induction imaging.

Knowing the relationships between resolution, accuracy, conductivity contrast, and noise is

important in designing effective imaging systems. In this thesis, relationships between the

135

sensitivity of the sensor and the parameters affecting the quality of the measurements in

subsurface conductivity imaging (or magnetic induction tomography) were determined. For this

purpose, a simple detection system that uses the magnetic-induction and magnetic-measurement

technique was analyzed. A mathematical model of the sensor and conductive medium, comprising

a disk like conductive object and a concentric inhomogeneity, was developed. The

interrelationships between the sensitivity of the sensor and the parameters affecting the quality of

the measurements were determined. These relations can be used to design an imaging system or to

investigate/improve the performance of an existing system.

In determining the interrelationships, several assumptions are used to simplify the formulation

while preserving the general characteristics of the detection system. Firstly, it was assumed that a

circular receiver coil is above a thin cylindrical body with a concentric inhomogeneity. The

cylindrical bodies were further modelled as a resistance and inductance connected in series in the

electrical analogue. Choice of simple geometries allowed us to develop analytical expressions for

the pick-up voltage which relates several configuration parameters to the measurement. Although

this configuration does not represent the general case, the results obtained with this simple model

were sufficient to understand the general behavior of the system. A concentric inhomogeneity was

chosen since it is the most difficult detection problem using a circular receiver coil. Study on the

effects of inhomogeneity location remains as a future study.

In this study, the effects of an arbitrary transmitter coil configuration were not studied. Instead, a

spatially uniform (sinuzoidally varying) magnetic field was assumed that can easily be realized by

a distant transmitter coil configuration. In practice, discrepancy from a uniform field can be

expected and should be further studied.

The validity of the analytical model (Equation (3-10)) is tested with the numerical model

implemented using the Finite Element Method (ANSYS Version 11). It is deduced that for the

small values of spatial resolution (for β ≤ 0.1) the nonlinearity error is less than 1.4 %.

In this study, the effects of the two factors on the image quality, i.e., the number of measurements

and the reconstruction algorithm, are not also considered. These factors should also be considered

before a practical realization of an imaging system.

8.2.2 Image Reconstruction within the Biological Tissue Range

One of the aims of this thesis is to investigate the potential of a sub-surface conductivity imaging

methodology for biomedical applications. However, image reconstruction of objects with a

136

conductivity of biological tissue range (0.01 – 1 S/m) could not been achieved in this study. As it

was discussed in CHAPTER 3, the sensitivity is related with the area of the coil, the square of the

operating frequency, the excitation current, spatial resolution, conductivity resolution (or in other

words conductivity of the object) and the number of turns of the coils. While working within the

biological tissue conductivity range, the sensor (the system) should be sensitive enough to

measure the data produced by the tissue or inhomogeneity.

To increase the sensitivity of the sensor we can modify the excitation current, the operating

frequency, number of turns of the coils or the area of the coil. Since we are operating within the

quasi-static region, the operating frequency must be below 100 kHz. Beside this, the operating

frequency and the excitation current depend on the construction, i.e. radius, number of turns, wire

diameter etc., of the coil. Especially the impedance of the coil limits the excitation current and the

operating frequency. Considering the impedance-frequency plots (Figure 4-12 to Figure 4-15), it

can be seen that a coil acts as an inductance below the resonance frequency while it acts as a

capacitance above the resonance frequency. The reason for a coil to behave as a capacitance at

high frequency is the result of stray capacitances. There are capacitances between the wounding

wires (especially adjacent wires) called stray capacitance and as the frequency increases the

effects of these capacitances increases and at higher frequencies the coil even turns out to be a

capacitance.

In magnetic induction-magnetic measurement technique it is crucial that the coil acts as an

inductance within the operating frequency region. The reason is that the conductive object has an

effect on the inductance of the coil such that the more inductive the coil the more sensitive the coil

(Section 4.3.2). So the operating frequency should be below the resonance frequency.

Consequently following tradeoffs between the coil geometry and sensor sensitivity are

encountered:

- inductance and number of turns (and wire diameter),

- wire diameter and number of turns,

- number of turns and coil size

- coil size (or number of turns) and resolution,

It is well known that spacing between steps or pixel size influence the spatial resolution. So for an

array sensor the coil size must be as small as possible to increase the spatial resolution. However,

as mentioned above, there is a tradeoff between the coil size and the spatial resolution and as the

coil size decreases the sensitivity of the coil decreases. Furthermore, the sensitivity is related with

the area, thus the square of the radius, of the coil.

137

In the early stages of the design process we planned to realize a real-time system. To do this so we

planned to construct a matrix type sensor comprising 4×4 coils. Measurements were to be taken

from the coils without scanning. In an earlier work [58], the author had achieved to image a leech

as a biological tissue. However the experiments were conducted with a coil of diameter of 3 cm

and the imaging region (11.5 cm × 11.5 cm) was scanned by taking 16×16 points of measurements

with a step size of 7.2 mm (or pixel size of 7.2 mm × 7.2 mm). The technical specifications of the

sensor coil utilized at that system are the following: the radius of the coil is 3 cm, excitation

current is 400 mA-rms, number of turns of the transmitter coil is 80 and that of the receiver coil is

650. If the matrix type sensor were constructed with coils of the same specifications the system

would have a 4×4 measurement points with a step size of 6 cm and the imaging region would be

24 cm × 24 cm. Although the size of the imaging region is reasonable, the number of data points is

limited yielding low resolution images. Thus, instead of using 4×4 matrix array, we decided to

employ a 1×4 array with scanning. The technical specifications of the 1×4 array sensor system

are: radius of the coils is 1.5 cm (half of the radius used in [58]), number of turns of the transmitter

coil is 100 and that of the receiver coil is 400.

The sensitivity is related with the area of the coil, the square of the operating frequency, the

excitation current amplitude, the spatial resolution, the conductivity resolution, and the number of

turns of the coils. Consequently, the decrease in the coil radius may be balanced by increasing the

excitation current (IReference) k times where k can be calculated as,

mAmAIkI

nnS

nnSk

erenceNew

ceiverNewtterNewTransmiNewSensor

ceivererencesmittererenceTranorerenceSens

29604004.7

4.740010015.1

6508075.2

Re

2

2

Re

ReReReRe

Thus the excitation current should be increased 7.4 times to balance the decrease in the radius of

the coil. Since the excitation current was 400mA in the reference system, the array sensor coils

should be driven with an excitation current of 2960 mA at 100 kHz. However due to technical

limitations we were able to increase the excitation current up to 700 mA at 100 kHz.

138

Consequently, we could not achieve image reconstruction within the biological tissue conductivity

range, especially for σ < 1 S/m. Instead, we used agar phantoms with conductivity of 6 S/m, as in

the reference study. Increasing the operating current and operating frequency obviously increase

the sensitivity of the system. Especially working above 100 kHz seems to be a must in the future

of this modality. To operate above 100 kHz, the inverse problem must be solved without using

quasi-static assumptions.

8.2.3 Main Contributions of the Study

Main contributions of this study can be summarized as follows:

1) Following relationships were observed about sensitivity which we defined it as a

fractional change in voltage for a fractional change in contrast:

Interrelationships and rules:

1. resolution-accuracy:

degrading accuracy by K balances improving resolution by K1/3

.

2. resolution-noise:

improving noise by K balances improving resolution by K1/3

.

3. accuracy-noise:

improving noise by K balances improving accuracy by K

4. resolution-contrast:

extending contrast by K balances degrading resolution by K3

5. accuracy-contrast:

extending contrast by K balances degrading accuracy by K

6. noise-contrast:

extending contrast by K balances improving noise by K

These relationships can be used to investigate/improve the performance of a system and

in the design of new system.

2) Three data acquisition systems were designed and developed. These studies have made

the following contributions to this field:

i. A commercial data acquisition card was employed instead of measurement instruments,

namely lock-in amplifier and multi-meter, in the system. With this replacement:

Mobility of the system increases,

139

Cost of the system decreases,

The start-up adjustments such as nulling of the output signal at the beginning of

experiments becomes obsolete,

Some additional noise sources such as cabling are eliminated.

In addition:

Operating frequencies of the system were between 20 kHz and 60 kHz.

Sensitivity of the system was tested by using resistance phantoms corresponding

to the conductivity range of 0.125 - 8.37 S/m. The sensitivity of the system was

determined at operating frequencies of 20 kHz, 30 kHz, 50 kHz and 60 kHz as,

4.29 mV/S, 8.81 mV/S, 27.49 mV/S and 36.65 mV/S, respectively. The results

show that the response of the system to conductivity variations obeys the theory

stating that 1) the sensitivity increases as the conductivity increases and 2) the

sensitivity increases as the operating frequency increases.

The sensitivity tests reveal that the system is capable of distinguishing tissues

around the average tissue conductivity range of 0.2 S/m.

ii. A single coil was employed as a sensor for the first time in the literature:

It was shown that, by employing a single coil sensor, the system has a maximum

sensitivity to the conductivity variations.

In addition to this, by using single coil, the sensor made more robust than the

sensor employing differential coil. Beside this, because of the non-identical

construction of the coils in differential coil sensor there were some limitations,

such as nulling requirements of the system or amplification gain of the measured

signal, on the performance of the data acquisition system. These limitations

would not need to be taken into account in single coil sensor system.

The operating frequency of the system was 100kHz,

The sensitivity of the system was tested by using resistance phantoms

corresponding to the conductivity range of 0.125-5.7S/m. The sensitivity tests

reveal that the system is capable of distinguishing tissues around the average

tissue conductivity range of 0.2 S/m.

The results reveal that the system has a potential to be used especially in first aid

applications for instance in ambulances to detect an inner-bleeding.

iii. A multi-frequency data acquisition system with an array sensor consisting of four

differential coils have been developed for the first time in the subsurface conductivity

imaging:

140

Main challenge of using an array sensor is the improvement of the data

acquisition speed. Employing an array sensor, the scanning time decreases

almost four times comparing with a system employing single or differential coil

sensor. In other words, by employing a 1×4-array-sensor, the speed of the

system increases almost four times comparing with a system employing a single

or differential coil sensor.

Operating frequencies of the system were between 50 kHz and 100 kHz.

The sensitivity of the system was tested by using resistance phantoms

corresponding to the conductivity range of 0.125-8.37 S/m. The sensitivity of

the system was determined at operating frequencies of 50 kHz, 75 kHz and 100

kHz as, 0.84 mV/S, 1.77 mV/S and 3.1 mV/S, respectively. The results show

that the response of the system to conductivity variations obeys the theory

stating that 1) the sensitivity increases as the conductivity increases and 2) the

sensitivity increases as the operating frequency increases.

To understand imaging performance, multi-frequency images of agar phantoms

were reconstructed. For this purpose, different agar phantoms were scanned at

different operating frequencies. The field profiles and the reconstructed

conductivity distributions of the objects were obtained. The reconstructed

images show the location, actual size and geometry of the objects.

The SNR of the system was determined at operating frequencies of 50 kHz, 75

kHz and 100 kHz as, 22.25 dB, 33.37 dB and 49.98 dB, respectively. The spatial

resolution of the system was determined as 31 mm at 50 kHz, 25 mm at 75 kHz

and 16 mm at 100 kHz. It is deduced from these results that the system is

capable of distinguishing two cylindrical objects with a radius of 15 mm and a

conductivity of 6S/m, if the distance between the objects is greater than 31 mm

at 50 kHz, 25 mm at 75 kHz and 16 mm at 100 kHz.

It was determined that the results were consistent with the theory stating that the

performance of the system increases as the frequency increases.

3) 3D images of agar phantoms with translationally uniform conductivity distributions were

reconstructed for the first time in subsurface electrical conductivity imaging:

i.The experiments were performed at an operating frequency of 50 kHz.

ii.Agar phantoms had a translationally uniform conductivity distribution,

iii.Data acquisition process in 3D image reconstruction was similar to that in 2D image

reconstruction,

141

iv.Sensitivity matrix was constructed by taking into account the depth of the conductive

medium.

v.The reconstructed 3D images show the location, actual size and geometry of the objects.

vi.The system needs to be improved to detect translational non-uniform conductivity

distributions.

4) The results deduced from the experiments worth to be discussed here. One important

deduction is that the performance of the system increases as the operating frequency

increases, which is an expected result. This allows such a possibility that an

inhomogeneity (for biomedical applications a malignant tissue, for instance a tumor)

which cannot be detected at one frequency may be detected at another frequency.

Second deduction is that employing arrays, in other words increasing the number of

sensors, it is possible to increase data acquisition and thus image reconstruction time.

This improvement in time yields the possibility of the system to be used in first aid

applications such as detection of inner bleeding or the degree of burn in skin.

Final deduction is that the system has a maximum sensitivity to the conductivity

variations when both the transmitter and the receiver is the same coil. Single coil sensor

has considerable advantages in the sense of robustness. Thus, systems employing single

coil sensor may better be used in field applications such as probing an inner bleeding of a

casualty in a car accident case. The relation between the measured signal and

conductivity of the object to be imaged should be derived for a single coil sensor system.

143

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AND MAGNETIC INDUCTION TOMOGRAPHY SYSTEM COMPRISING SUCH A COIL

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RAG, editors, vol. AT2007/000359. WO, 2008.

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1988.

147

APPENDIX A

MAGNETIC FIELD MEASUREMENT USING

RESISTIVE RING EXPERIMENT

Secondary voltage induced in the receiver coils due to secondary flux generated by the resistive

ring is measured by using the circuit shown in Figure A-1. A 3-turn coil with diameter of 9 mm is

prepared and placed on the sensor. The terminals of the ring are shunted with resistors. The

primary flux created by the transmitter coil induces eddy currents in the 3-turn coil. However the

eddy currents cannot flow in the ring unless the terminals of the ring are shunted with a resistor.

Since the transmitter current's amplitude and frequency is constant, the amplitude of the current

flowing in the 3-turn ring is inversely proportional to the resistor shunting the ring. Consequently,

the secondary flux generated by the resistive ring is controlled with the shunting resistor. If the

resistor value is decreased, the current flowing in the ring will be increased and vice versa. The

operation principle of the circuit is explained in Section 4.1.

6 resistances varied in the range of 260 Ω - 10000 Ω (Table A-1) are connected to the open ends

of the 3-turn coil (i.e. the coil is shunted with the resistors). In experiments, first, the ring is

shunted to a resistor and measurements are performed (100 samples are taken for each

measurement), then, another measurements are performed after making the ring open (eddy

currents cannot flow in the ring, thus, no secondary field is generated). The former data is

subtracted from the latter and the difference data is obtained, thus, the effect of voltage drift in the

receiver coils is cancelled out. No voltage drift is observed in the coils when the system become

stable (4 hours should be elapsed from the starting of the set-up). However, in this study always

the difference data is obtained and used. Mean value of the difference data is determined for each

resistance and plotted as a function of the inverse of the resistance values as shown in the Figures

in CHAPTER 5 and CHAPTER 6.

Table A-1: Resistor values (and corresponding 1/(resistor values)) used in the resistive ring

experiments.

Resistor Values (Ω) 265 553 1178 3303 5054 9829

1 / (Resistor Values) (x10-4

) 38 18 8 3 2 1

As an example mean value of the difference data obtained in the prototype system is plotted as a

function of the resistor values in Figure A-2/(a) and in Figure A-2/(b) the same data plotted for

1/(Resistor values).

148

Figure A-1: Magnetic field measurement using resistive ring experiment setup

(a)

Figure A-2: Receiver coil output voltage versus conductivity plot obtained by using

resistive ring. Mean value of the difference data is plotted as a function of: (a) resistor

values, (b) 1/(resistor values) (100 sample are taken for each measurement).

3-Turn Coil

(Conductive Ring)

Magnetic Sensor

R

Agilent

34410A

Digital

Multimete

r

Agilent 33220A

Function Generator

10-100 kHz Power

Amp.

Out

DAcC

Reference Buffer

Input Buffer

IREF 366 nF Rm = 2.2 Ω

149

(b)

Figure A-2: (Continued)

151

APPENDIX B

THERMAL NOISE1

Thermal noise comes from thermal agitation of electrons within a resistance, and it sets a lower

limit on the noise present in a circuit. The open-circuit rms noise voltage produced by a resistance

is :

RBTkVt 4 (B.1)

Where,

K = Boltzmann's constant (1.38x10-23

joules / 0K),

T = absolute temperature (0K),

B = noise bandwidth(Hz),

R = resistance (Ω).

At room temperature (290 0K), 4kT equals 1.6 x 10

-20 W/Hz.

B.1 Equivalent Noise Bandwidth

The noise bandwidth B is the voltage-gain-squared bandwidth of the system or circuit being

considered. The noise bandwidth is defined for a system with uniform gain throughout the

passband and zero gain outside the passband. Figure B-1 shows this ideal response for a low-pass

circuit and a band-pass circuit.

However, since practical circuits do not have these ideal characteristics, the area under the

equivalent noise bandwidth is made equal to the area under the actual curve. This is shown in

Figure B-2 for a low pass circuit. For any network transfer function, A(f) (expressed as a voltage

or current ratio), there is an equivalent noise bandwidth with constant magnitude of transmission

A0 and bandwidth of

0

2

2

0

1dffA

AB (B.2)

Table B.1 gives the ratio of the noise bandwidth to the 3-dB bandwidth for circuits with various

numbers of identical poles [78].

1 The theory about the thermal noise stated in this section is reprinted from "Noise Reduction

Techniques in Electronic Systems”, Ott, 1988 [78].

152

Table B-1: Ratio of the Noise Bandwidth B to the 3-dB Bandwidth f0

Number of Poles B / f0 High-frequency Rollof (dB per Octave)

1 1.57 6

2 1.22 12

3 1.15 18

4 1.13 24

5 1.11 30

Figure B-1: Ideal bandwidth of low-pass and band-pass circuit elements

Figure B-2: Actual response and equivalent noise bandwidth for low-pass circuit.

153

B.2 Noise In IC Operational Amplifiers

The noise characteristics of an op-amp can be modelled by using the equivalent input noise

voltage and current V n - I n. Typical op-amp circuit is shown in Figure B-3. The same circuit with

the equivalent noise voltage and current sources included is shown in Figure B-4/(a).

The equivalent circuit in Figure B-4/(a) can be used to calculate the total equivalent input noise

voltage, which is

2

12

22

2

11

2

2

2

1214 snsnnnsstn RIRIVVRRkTBV (B.3)

It should be noted that Vn1; V n2; I n1 and I n2 (given in the Datasheets of the components) are also

functions of the bandwidth B. The two noise voltage sources of Equation B.3 can be combined by

defining

2

2

2

1

2,

nnn VVV (B.4)

Figure B-3: Typical op-amp circuit with an absolute gain A = Rf/R1

Equation B.3 can then be written as

2

12

22

2

11

2,

214 snsnnsstn RIRIVRRkTBV (B.5)

Although the voltage sources have been combined, the two noise current sources are still required

in Equation B.5. If, however, Rs1 = Rs2, which is usually the case, then the two noise current

generators can be combined by defining

2

2

2

1

2,

nnn III (B.6)

For Rs1 = Rs2 = Rs, Equation B.5 reduces to

154

2

122,

214 snnsstn RIVRRkTBV (B.7)

The equivalent circuit for this case is shown in Figure B-4/(b). To obtain optimum noise

performance (maximum signal-to-noise-ratio) from an op-amp, the total equivalent input noise

voltage Vnt should be minimized [78] (pages 267-269).

B.3 Addition of Noise Voltages

When noise sources added together, the total power is equal to the sum of the individual powers.

Adding two noise voltage generators V1 and V1, together on a power basis, gives [78]

2

2

2

1

2 VVVtotal (B.8)

The total noise voltage can then be written as

2

2

2

1 VVVtotal (B.9)

155

(a)

(b)

Figure B-4: Typical op-amp circuit (Figure B-3) with the equivalent noise voltage and current

sources included; (a) Circuit of Figure B-3 with noise sources added; (b) Circuit of Figure B-4(a)

with noise sources combined at one terminal for the case Rs1 = Rs2 = Rs.

157

APPENDIX C

DETAILS ABOUT EXPERIMENTAL PROCEDURE

The experimental setup and the operation principle of the data acquisition systems developed

within this study were explained in CHAPTER 4. In this section details of the experimental

procedure will be explained.

All of the magnetic induction-magnetic measurement systems developed within this study may be

divided into 3 parts, namely, data processing part, data collection and acquisition part and

measurement part. The data processing part consists of a PC and necessary software for image

reconstruction. The data collection and acquisition part consists of necessary hardware and

instruments for data collection and acquisition. Finally, the measurement part consists of the

sensor, the most crucial part of the system, and sensor electronics.

Means of the system is explained in Section 4.3.3. A photograph of the sensor and the coil

geometry is shown in Figure 4-10. The block diagram of the data acquisition system is given in

Figure 4-11. The details of the data acquisition electronics are shown in Figure 4-16, Figure 4-17,

Figure 4-18 and Figure 4-19.

As a complementary to the imaging system, the details of the experimental procedure are

explained in this section. The experimental process is conducted as following:

i. Saline solution with a conductivity of 0.2 s/m is prepared and stored in a bottle.

ii. The object to be imaged (an agar phantom with a known conductivity) is prepared.

iii. The vessel is filled with a saline solution.

iv. The phantom is placed within the vessel at known coordinates.

v. The sensor at the scanner is moved to a beginning point (known location) above the imaging

area.

vi. After entering the step size, minimum error and number of iterations to the imaging software,

the program is started:

a. The scanning process is started by running the data collection software.

b. After the data collection process finishes, the image reconstruction software is

started and the image reconstruction is performed.

vii. When the scanning or data collection process finishes the image is reconstructed in seconds

and the reconstructed image is drown at the figure box on the user interface.

159

APPENDIX D

CHARACTERIZATIONS OF THE DATA

ACQUISITION SYSTEMS

Characterizations of the data acquisition systems realized during this thesis are necessary. In this

section, characterizations of the data acquisition systems in the sense of their specifications are

presented.

D.1 Characterization of the Data Acquisition System with CM-2251 Data

Acquisition Card

Technical specifications of the Data Acquisition system with CM-2251 Data Acquisition Card are

given in Table D-1 below.

Table D-1: Technical Specifications of the Data Acquisition System with CM-2251 Data

Acquisition Card

Sen

sor

Type Differential Coil Sensor

Properties

# of

turns

dwire

(mm)

Rcoil

(Ω)

Lcoil

(μH)

Resonance

Frequency (kHz)

Transmitter 80 0.75 1.96 245 2550

Receiver 650 0.2 306

Data Acquisition Electronics Commercial Data Acquisition Card (CM-2251)

Phonic XP3300 Power Amplifier

Operating Frequency (kHz) 20-60

Sensitivity (S/m) 0,125-8,37

D.2 Characterization of the Data Acquisition System with Single Coil Sensor

Technical specifications of the Data Acquisition system with Single Coil Sensor are given in

Table D-2 below.

160

Table D-2: Technical Specifications of the Data Acquisition System with Single Coil Sensor

Sen

sor

Type Single Coil Sensor

Properties

# of

turns

dwire

(mm)

Rcoil

(Ω)

Lcoil

(μH)

Resonance

Frequency (kHz)

80 0.75 1.46 191 1981

Data Acquisition Electronics Agilent 33220A Signal Generator

Phonic XP3000 Power Amplifier

EG-G Model 5209 Lock-in Amplifier

Agilent 34410A Digital Multimeter

Operating Frequency (kHz) 100


D.3 Characterization of the Data Acquisition System with Array Coil Sensor

Technical specifications of the Data Acquisition system with Array Coil Sensor are given in Table

D-3 below.

Table D-3: Technical Specifications of the Data Acquisition System with Array Coil Sensor

Sen

sor

Type 1x4 Array

Differential Coil Sensor

Properties

# of

turns

dwire

(mm)

Rcoil(Ω)

(@100kHz)

Lcoil(mH)

(@100kHz)

Resonance

Frequency (kHz)

Sensor

#1

Transmitter 100 0.45 9,17 0,187 4050

Receiver 400 0.2 144,6 5,03 375

Sensor

#2

Transmitter 100 0.45 9,15 0,192 4000

Receiver 400 0.2 159,4 4,97 360

Sensor

#3

Transmitter 100 0.45 10,09 0,180 3850

Receiver 400 0.2 150,7 4,78 370

Sensor

#4

Transmitter 100 0.45 9,26 0,188 3850

Receiver 400 0.2 151,2 4,79 360

Data Acquisition Electronics Agilent 33220A Signal Generator

SRS Power Amplifier

EG-G Model 5209 Lock-in Amplifier

Agilent 34410A Digital Multimeter

Operating Frequency (kHz) 25-100


161

APPENDIX E

AGAR PHANTOM PREPERATION

E.1 Equipment and Materials Needed

Agar powder

Distilled water

Table salt or NaCl

Heater

Beaker or Erlenmeyer flask

E.2 Preparation

i. Decide how much agar is needed according to the desired conductivity. If you add ½ gr

of NaCl and 1.5 gr agar powder to 100 ml distilled water, the conductivity of the solution

will be [18]:

S = 0.02 S/cm (E.1)

ii. Choose a beaker or erlenmeyer flask that is 2-4 times the volume of the solution.

iii. Place the flask above heater. Bring the solution to a boil while stirring. Agar powder will

be dissolved and solution change colour to light brown.

iv. Lower the heat and simmer until no air bubble remains.

v. Pour the agar solution into the phantom of desired shape and let the gel sit undisturbed

until cool (approximately 15 minutes). The agar will change colour from clear to slightly

milky [18, 58].

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