3D MULTI-FREQUENCY CONDUCTIVITY IMAGING VIA …etd.lib.metu.edu.tr/upload/12615977/index.pdf · The...
Transcript of 3D MULTI-FREQUENCY CONDUCTIVITY IMAGING VIA …etd.lib.metu.edu.tr/upload/12615977/index.pdf · The...
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
ii
iii
Approval of the thesis
3D MULTI-FREQUENCY CONDUCTIVITY IMAGING
VIA CONTACTLESS MEASUREMENTS
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
Electrical and Electronics Engineering Dept., METU ______________
Prof. Dr. Osman Eroğul
Biomedical Engineering Center, GMMA ______________
Prof. Dr. Kemal Leblebicioğlu
Electrical and Electronics Engineering Dept., METU ______________
Assoc. Prof. Dr. Yeşim Serinağaoğlu Doğrusöz
Electrical and Electronics Engineering Dept., METU ______________
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
3D MULTI-FREQUENCY CONDUCTIVITY IMAGING
VIA CONTACTLESS MEASUREMENTS
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
xiii
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
4.3.2.1 Experimental Setup and Operation principle ............................................................... 53
4.3.2.2 Sensitivity to Conductivity Variations ........................................................................... 54
4.3.3 Data Acquisition with Array Coil Sensor ......................................................................... 56
4.3.3.1 Experimental Setup and Operation principle ............................................................... 57
4.3.3.1.1 Characteristics of the Coils Composing the Sensor Array ................................. 60
xiv
4.3.3.1.2 The Controller Card .......................................................................................... 67
4.3.3.2 Sensitivity to Conductivity Variations ........................................................................... 72
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=0.2). The nonlinearity error is 11.2% of the full scale. ............ 30 Figure 3-7: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS
simulations, respectively (β=0.3). The nonlinearity error is 16.6% of the full scale. ............ 31 Figure 3-8: 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:
(a) field profile, (b) reconstructed conductivity distribution. ................................................ 93 Figure 5-8: 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 3rd
coil of the sensor array:
(a) field profile, (b) reconstructed conductivity distribution. ................................................ 94 Figure 5-9: 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 4th
coil of the sensor array:
(a) field profile, (b) reconstructed conductivity distribution. ................................................ 95 Figure 5-10: 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:
(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
coil: (a) field profile, (b) reconstructed conductivity
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
coil: (a) field profile, (b) reconstructed conductivity
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.
14
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.
22
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.
Figure 3-6: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS
simulations, respectively (β=0.2). The nonlinearity error is 11.2% 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
Figure 3-7: The analytical and numerical sensitivities determined from Eq. (3-10) and ANSYS
simulations, respectively (β=0.3). The nonlinearity error is 16.6% of the full scale.
Figure 3-8: 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.
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)
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.).
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.
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.).
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.
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.).
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
perturbation diverges from the wires,
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
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.
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
4.3.2.1 Experimental Setup and Operation principle
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.
4.3.2.2 Sensitivity to Conductivity Variations
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.
4.3.3.1 Experimental Setup and Operation principle
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).
Coil #1 Coil #2 Coil #3 Coil #4
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)
Coil #1 Coil #2 Coil #3 Coil #4
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)
Figure 4-13: Impedance of Coil #2 as a function of frequency
(a) transmitter, (b) upper receiver, (c) lower receiver.
64
(c)
Figure 4-13: (Continued)
(a)
Figure 4-14: Impedance of Coil #3 as a function of frequency
(a) transmitter, (b) upper receiver, (c) lower receiver.
65
(b)
(c)
Figure 4-14: (Continued)
66
(a)
(b)
Figure 4-15: Impedance of Coil #4 as a function of frequency
(a) transmitter, (b) upper receiver, (c) lower receiver.
67
(c)
Figure 4-15: (Continued)
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
4.3.3.2 Sensitivity to Conductivity Variations
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
Figure 4-20: (Continued)
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 75 kHz, NL = % 1.8
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 100 kHz, NL = % 7.4
measured
fitted
74
(a) Slope = 1.01 mV / mho, Nonlinearity (NL) = %1.9
(b) Slope = 2.3 mV / mho, Nonlinearity (NL) = %2.3
Figure 4-21: The Sensitivity of Coil #2 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
-1
0
1
2
3
4
5
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 50 kHz, NL = % 1.9
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 75 kHz, NL = % 2.3
measured
fitted
75
(c) Slope = 4.13 mV / mho, Nonlinearity (NL) = %3.1
Figure 4-21: (Continued)
(a) Slope = 0.968 mV / mho, Nonlinearity (NL) = %5.5
Figure 4-22: The Sensitivity of Coil #3 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
-5
0
5
10
15
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 100 kHz, NL = % 3.1
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 50 kHz, NL = % 5.5
measured
fitted
76
(b) Slope = 1.95 mV / mho, Nonlinearity (NL) = %2.7
(c) Slope = 3.67 mV / mho, Nonlinearity (NL) = %4.5
Figure 4-22: (Continued)
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 75 kHz, NL = % 2.7
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 100 kHz, NL = % 4.5
measured
fitted
77
(a) Slope = 0.84 mV / mho, Nonlinearity (NL) = %4.6
(b) Slope = 1.77 mV/mho, Nonlinearity (NL) = %3.6
Figure 4-23: The Sensitivity of Coil #3 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
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 = % 4.6
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 75 kHz, NL = % 3.6
measured
fitted
78
(c) Slope = 3.1 mV / mho, Nonlinearity (NL) = %3.4
Figure 4-23: (Continued)
Table 4-6: The slopes of the sensitivities at different operating frequencies (mV/mho).
Coil #1 Coil #2 Coil #3 Coil #4
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
Conductivities between 1 / 220 - 1 / 10000 Mho
mili
Volts
Conductivities at f= 100 kHz, NL = % 3.4
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
Frequency Response of the System: Coil #2
measured
expected
80
(c) NRMS=6.05%
(d) NRMS=6.56%
Figure 4-24: (Continued)
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 #3
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
Frequency Response of the System: Coil #4
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)
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.
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
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
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
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
1
2
3
4
5
6
C1 C2 C3 C4
C1 C2 C3 C4
93
(a)
(b)
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: (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
0.3
0.35
0.4
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
1
2
3
4
5
6
C1 C2 C3 C4
C1 C2 C3 C4
94
(a)
(b)
Figure 5-8: 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 3rd
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
0.05
0.1
0.15
0.2
0.25
0.3
0.35
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
1
2
3
4
5
C1 C2 C3 C4
C1 C2 C3 C4
95
(a)
(b)
Figure 5-9: 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 4th
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
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
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
1
2
3
4
5
6
C1 C2 C3 C4
C1 C2 C3 C4
96
(a)
(b)
Figure 5-10: 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: (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
0.3
0.35
0.4
0.45
0.5
0.55
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
1
2
3
4
5
6
C1 C2 C3 C4
C1 C2 C3 C4
97
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)
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.
Field Measurements
step number (The Sensors Axis)
step
num
ber
(The
sca
nnin
g ax
is)
5 10 15 20
5
10
15
20
25
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
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
1
2
3
4
5
6
C1 C2 C3 C4
C1 C2 C3 C4
99
(a)
(b)
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.
Field Measurements
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
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
1
2
3
4
5
C1 C2 C3 C4
C1 C2 C3 C4
100
(a)
(b)
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.
Field Measurements
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0
0.02
0.04
0.06
0.08
0.1
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
1
2
3
4
5
6
7
C1 C2 C3 C4
C1 C2 C3 C4
101
(a)
(b)
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.
Field Measurements
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
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
1
2
3
4
5
C1 C2 C3 C4
C1 C2 C3 C4
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)
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
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
4
4.5
C1 C2 C3 C4
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
y (
S/m
)
Scanning axis [mm]
FWHM of the Array Coil system
FWHM = 19 mm
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
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).
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
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 20 40 60 80 100 120 1400
1
2
3
4
5
6FWHM of the Array Coil System
Scanning axis [mm]
Co
nd
ucti
vit
y (
S/m
)
FWHM = 21 mm
C1 C2 C3 C4
107
(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
0.45
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
1
2
3
4
5
6
C1 C2 C3 C4
108
(c)
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
coil: (a) field profile, (b) reconstructed conductivity
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
7FWHM of the Array Coil System
Scanning axis [mm]
Co
nd
ucti
vit
y (
S/m
)
FWHM = 19 mm
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
109
(b)
(c)
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
coil: (a) field profile, (b) reconstructed conductivity
distribution, (c) FWHM calculation by using the signal spread along axial direction (4th
coil).
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
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 20 40 60 80 100 120 1400
1
2
3
4
5FWHM of the Array Coil System
Scanning axis [mm]
Co
nd
ucti
vit
y (
S/m
)
FWHM = 20 mm
C1 C2 C3 C4
110
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)
Figure 6-2: (Continued)
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
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 50 kHz.
Field Measurements
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Reconstructed Image, f=50kHz
mm (The Sensors Axis)
mm
(T
he s
cannin
g a
xis
)
20 40 60 80 100 120
20
40
60
80
100
120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
C1 C2 C3 C4
C1 C2 C3 C4
117
(a)
(b)
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:
(a) field profile, (b) reconstructed conductivity distribution, at 75 kHz.
Field Measurements
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Reconstructed Image, f=75kHz
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
4
4.5
5
5.5
C1 C2 C3 C4
C1 C2 C3 C4
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
step number (The Sensors Axis)
ste
p n
um
ber
(The s
cannin
g a
xis
)
5 10 15 20
5
10
15
20
25
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
mm (The Sensors Axis)
mm
(T
he s
cannin
g a
xis
)
Reconstructed Image, f=100kHz
20 40 60 80 100 120
20
40
60
80
100
120
1
2
3
4
5
6
7
C1 C2 C3 C4
C1 C2 C3 C4
119
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.
120
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
4
6
8
10
12
14
16
0
1
2
3
4
5
0 2 4 6 8 10 12 14 161
1.5
2
2.5
3
3.5
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
5 10 15 20
5
10
15
20
25
0
1
2
3
4
5
0 5 10 15 20 251
1.5
2
2.5
3
3.5
4
4.5
5 3D Image (XZ View)
0
1
2
3
4
5
C1 C
2 C
3 C
4
C1 C
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
5
10
15
20
25
0
1
2
3
4
5
6
7
0 5 10 15 20 25 301
1.5
2
2.5
3
3.5
4
4.5
5 3D Image
0
1
2
3
4
5
6
7
C1 C
2 C
3 C
4
128
(a)
(b)
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
5 10 15 20
5
10
15
20
25
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20 251
2
3
4
5
6
7
8
9
10 3D Image (XZ View)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
C1 C
2 C
3 C
4
129
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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Voltage vs Distance (Sensor #1)
Distance (mm)
Norm
aliz
ed V
oltage (
Volts)
130
(b)
(c)
Figure 7-7: (Continued)
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Voltage vs Distance (Sensor #2)
Distance (mm)
Norm
aliz
ed V
oltage (
Volts)
1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Voltage vs Distance (Sensor #3)
Distance (mm)
Norm
aliz
ed V
oltage (
Volts)
131
(d)
Figure 7-7: (Continued)
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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Voltage vs Distance (Sensor #4)
Distance (mm)
Norm
aliz
ed V
oltage (
Volts)
132
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.
142
143
REFERENCES
[1] WIKIPEDIA, The Free Encyclopedia, Medical Imaging, www.wikipedia.org. January
2013.
[2] Boas D, Brooks D, Miller E, DiMarzio C, Kilmer M, Gaudette R, Zhang Q. Ieee Signal
Processing Magazine 2001;18:57.
[3] Ophir J, Garra B, Kallel F, Konofagou E, Krouskop T, Righetti R, Varghese T.
Ultrasound in Medicine and Biology 2000;26:S23.
[4] Webster J. Electrical Impedance Tomography: IOP Publishing, 1990.
[5] Webb AG. Introduction to Biomedical Engineering: IEEE, 2002.
[6] Niederhauser J, Jaeger M, Lemor R, Weber P, Frenz M. Ieee Transactions on Medical
Imaging 2005;24:436.
[7] RANGANATH M, DHAWAN A, MULLANI N. Ieee Transactions on Medical Imaging
1988;7:273.
[8] CERULLO L. Ieee Journal of Quantum Electronics 1984;20:1397.
[9] PURVIS W, TOZER R, ANDERSON D, FREESTON I. Iee Proceedings-a-Science
Measurement and Technology 1993;140:135.
[10] Gencer NG. Electrical Impedance Tomography Using Induced Currents. vol. PhD:
METU, 1993.
[11] GENCER N, IDER Y, KUZUOGLU M. Clinical Physics and Physiological
Measurement 1992;13:95.
[12] FREESTON I, TOZER R. Physiological Measurement 1995;16:A257.
[13] Gencer N, Ider Y, Williamson S. Ieee Transactions on Biomedical Engineering
1996;43:758.
[14] TARJAN P, MCFEE R. Ieee Transactions on Biomedical Engineering 1968;BM15:266.
[15] Gencer NG, Tek MN. Ieee Transactions on Medical Imaging 1999;18:617.
[16] Korjenevsky A, Cherepenin V, Sapetsky S. Physiological Measurement 2000;21:89.
[17] Scharfetter H, Lackner HK, Rosell J. Physiological Measurement 2001;22:131.
[18] Karbeyaz BU, Gencer NG. Ieee Transactions on Medical Imaging 2003;22:627.
[19] Barber DC, Brown BH. Journal of Physics 1984;17:11.
144
[20] Morucci J, Rigaud B. Critical Reviews in Biomedical Engineering 1996;24:655.
[21] ISAACSON D. Ieee Transactions on Medical Imaging 1986;5:91.
[22] Bronzino J. The Biomedical Engineering Handbook: IEEE Press, 1995.
[23] IDER Y, GENCER N, ATALAR E, TOSUN H. Ieee Transactions on Medical Imaging
1990;9:49.
[24] BARBER D, BROWN B. Clinical Physics and Physiological Measurement 1992;13:3.
[25] Zlochiver S, Radai MM, Abboud S, Rosenfeld M, Dong XZ, Liu RG, You FS, Xiang
HY, Shi XT. Physiological Measurement 2004;25:239.
[26] Ireland RH, Tozer JC, Barker AT, Barber DC. Physiological Measurement 2004;25:775.
[27] Eyuboglu BM, Brown BH, Barber DC, Seager A. Localization of cardiac related
impedance changes in the thorax. vol. 8: Clin. Phys. Physiol. Meas., 1987. p.A167.
[28] Eyuboglu BM, Brown BH. Methods of cardiac gating applied potential tomography. vol.
9: Clin. Phys. Physiol. Meas., 1988. p.71.
[29] Eyuboglu BM, Brown BH, Barber DC. Problems of cardiac output determination from eit
scans. vol. 9: Clin. Phys. Physiol. Meas., 1988. p.71.
[30] Piperno G, Frei E, Moshitzky M. Breast cancer screening by impedance measurements.
vol. 2: Frontiers Med. Biol. Eng., 1990. p.111.
[31] Cherepenin V, Karpov A, Korjenevsky A, Kornienko V, Mazaletskaya A, Mazourov D,
Meister D. Physiological Measurement 2001;22:9.
[32] Kerner T, Paulsen K, Hartov A, Soho S, Poplack S. Ieee Transactions on Medical
Imaging 2002;21:638.
[33] Nirmal KS, Hartov A, Kogel C, Poplack SP, Paulsen KD. Multi-frequency electrical
impedance tomography of the breast: New clinical results. vol. 25: Physiol. Meas., 2004. p.301.
[34] Cherepenin V, Karpov A, Korjenevsky A, Kornienko V, Kultiasov Y, Mazaletskaya A,
Mazourov D. Physiological Measurement 2002;23:33.
[35] Merwa R, Hollaus K, Oszkar B, Scharfetter H. Physiological Measurement 2004;25:347.
[36] Holder D. Physiological Measurement 2002;23.
[37] Kaufman AA, Keller GV. Induction Logging: Elsevier Science Publishers, 1989.
[38] Libby H. Introduction to Electromagnetic Nondestructive Test Methods: John Willey &
Sons Inc., 1971.
[39] Blitz J. Electrical and Magnetic Methods of Nondestructive Testing: IOP Publishing Ltd.,
1991.
[40] HENDERSON R, WEBSTER J. Ieee Transactions on Biomedical Engineering
1978;25:250.
145
[41] Alzeibak S, Saunders NH. Physics in Medicine and Biology 1993;38:151.
[42] Netz J, Forner E, Haagemann S. Physiological Measurement 1993;14:463.
[43] Korzhenevskii A, Cherepenin V. Radiotekhnika I Elektronika 1997;42:506.
[44] Korjenevsky A, Sapetsky S. Methods of measurements and image reconstruction in
magnetic induction tomography. Proc. 2nd EPSRC Engineering Network Meeting on Biomedical
Applications of EIT, 2000.
[45] Griffiths H. Measurement Science & Technology 2001;12:1126.
[46] Rosell J, Casanas R, Scharfetter H. Physiological Measurement 2001;22:121.
[47] Scharfetter H, Riu P, Populo M, Rosell J. Physiological Measurement 2002;23:195.
[48] Hollaus K, Magele C, Merwa R, Scharfetter H. Physiological Measurement 2004;25:159.
[49] Merwa R, Hollaus K, Brunner P, Scharfetter H. Physiological Measurement
2005;26:S241.
[50] Watson S, Morris A, Williams RJ, Griffiths H, Gough W. Physiological Measurement
2004;25:271.
[51] Morris A, Griffiths H, Gough W. Physiological Measurement 2001;22:113.
[52] Watson S, Igney CH, Dossel O, Williams RJ, Griffiths H. Physiological Measurement
2005;26:S319.
[53] Igney CH, Watson S, Williams RJ, Griffiths H, Dossel O. Physiological Measurement
2005;26:S263.
[54] Gencer NG, Tek MN. Imaging tissue conductivity via contactless measurements: A
feasible study. vol. 6: TUBITAK Electrical J., 1998. p.183.
[55] Gencer NG, Tek MN. Physics in Medicine and Biology 1999;44:927.
[56] Ahmad T, Gencer NG, Ieee. Development of a data acquisition system for electrical
conductivity images of biological tissues via contactless measurements. 23rd Annual International
Conference of the IEEE-Engineering-in-Medicine-and-Biology-Society, vol. 23. Istanbul, Turkey,
2001. p.3380.
[57] Colak IE. An Improved Data Acquisition System for Contactless Conductivity Imaging.
vol. MSc: METU, 2005.
[58] OZKAN KO. Multi-frequency Electrical Conductivity Imaging via contactless
measurements. Electrical and Electronics Engineering, vol. MSc: Middle East Technical
University, 2006.
[59] Ahmad T. Experimental Studies on Development of a New Imaging System For
Contactless Subsurface Conductivity Imaging of Biological Tissues. vol. MSc: METU, 2001.
[60] Ulker B, Gencer NG. Ieee Engineering in Medicine and Biology Magazine 2002;21:152.
146
[61] ULKER B. Electrical Conductivity Imaging via Contactless Measuremens: An
Experimental Study. vol. MSc.: METU, December 2001.
[62] Schwan HP. Ann Biomed Eng 1988;16:245.
[63] Igney CHcoPHNLAAE, Pinter RcoPHNLAAE, Such OcoPHNLAAE. MAGNETIC
INDUCTION TOMOGRAPHY SYSTEM AND METHOD. In: Philips Intellectual P, Standards
Gmbh LH, Koninklijke Philips Electronics N. V GNLBAE, editors, vol. IB2006/054834. WO,
2007.
[64] Chen DS, Yan MS. METHOD AND DEVICE FOR CALIBRATING A MAGNETIC
INDUCTION TOMOGRAPHY SYSTEM. vol. 12919223. US, 2011.
[65] WATSON S. Magnetic induction tomography with two reference signals. 2010.
[66] Igney CHWA, Hamsch MWA, Mazurkewitz PRA, Luedeke K-MRA. METHOD AND
DEVICE FOR MAGNETIC INDUCTION TOMOGRAPHY. In: Koninklijke Philips Electronics
N.V GNLBAE, Philips Intellectual P, Standards Gmbh LH, editors, vol. IB2010/050422. WO,
2010.
[67] Eichardt RN, McEwan ALENSW. MAGNETIC INDUCTION TOMOGRAPHY
SYSTEMS WITH COIL CONFIGURATION. vol. 13258633. US, 2012.
[68] Chen DPCIClLTLRs, Yan MPCIClLTLRs, Jin HPCIClLTLRs. COIL ARRANGEMENT
AND MAGNETIC INDUCTION TOMOGRAPHY SYSTEM COMPRISING SUCH A COIL
ARRANGEMENT. In: Koninklijke Philips Electronics N.V GNLBAE, editor, vol.
IB2009/054766. WO, 2010.
[69] Scharfetter HL-K-GAG. CORRECTION OF PHASE ERROR IN MAGNETIC
INDUCTION TOMOGRAPHY. In: Technische UniversitÄT Graz RAG, Forschungsholding Tu
Graz Gmbh RAG, editors, vol. AT2009/000266. WO, 2010.
[70] Scharfetter HL-K-GAG. DEVICE AND METHOD FOR MAGNETIC INDUCTION
TOMOGRAPHY. In: Technische UniversitÄT Graz RAG, Forschungsholding Tu Graz Gmbh
RAG, editors, vol. AT2007/000359. WO, 2008.
[71] Scharfetter H. Device and method magnetic induction tomography. 2011.
[72] Ozkan KO, Gencer NG. Ieee Transactions on Medical Imaging 2009;28:564.
[73] GENCER N, KUZUOGLU M, IDER Y. Ieee Transactions on Medical Imaging
1994;13:338.
[74] Griffiths H, Stewart WR, Gough W. Electrical Bioimpedance Methods: Applications to
Medicine and Biotechnology 1999;873:335.
[75] Al-Zeibak S, Sunders NH. A feasibility study of in vivo electromagnetic imaging. 1993.
[76] Seagar AD, Barber DC, Brown BH. Theoretical limits to sensitivity and resolution in
impedance imaging. Clin. Phys. Physiol. Meas., vol. 8. Great Britain, 1987. p.13.
[77] GENCER NG. TUBITAK r-1001 Project Report, Project No: 106E170, February 2010.
[78] Ott HW. Noise Reduction Techniques in Electronic Systems: John Wiley and Sons Inc.,
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)
150
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.
156
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.
158
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
Sensitivity (S/m) 0,125-5,7
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
Sensitivity (S/m) 0,125-5,7
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].