close

Вход

Забыли?

вход по аккаунту

?

Microwave imaging of breast tissues

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI films
the text directly from the original or copy submitted. Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of
computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality illustrations
and photographs, print bleedthrough, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and continuing
from left to right in equal sections with small overlaps.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6” x 9” black and white
photographic prints are available for any photographs or illustrations appearing
in this copy for an additional charge. Contact UMI directly to order.
ProQuest Information and Learning
300 North Zeeb Road, Ann Arbor. Ml 48106-1346 USA
800-521-0600
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NO RTH W ESTERN UNIVERSITY
Microwave Imaging of Breast Tissues
A DISSERTATION
SUBM ITTED T O T H E GRADUATE SCHOOL
IN PARTIAL FULFILLM ENT OF T H E REQUIREM ENTS
for the degree
D O CTO R O F PHILOSOPHY
Field of Electrical and Com puter Engineering
By
Milica Popovic
EVANSTON, ILLINOIS
December 2001
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3033551
Copyright 2001 by
Popovic, Milica
All rights reserved.
__ ___
(EQ
UMI
UMI Microform 3033551
Copyright 2002 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
© Copyright by Milica Popovic 2001
All Rights Reserved
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A BSTR A C T
Microwave Imaging of Breast Tissues
M ilica P op ovid
The work presented in this thesis is m otivated by tke need to calibrate a new
pulsed-microwave breast tumor detection system for patient-specific skin param e­
ters. A two-dimensional time-domain inverse-scattering algorithm is presented for
determ ining the skin thickness and the relative perm ittivity er- akin and electric con­
ductivity (Takin in the microwave range. The algorithm traces a search trajectory in
the (£r- Skin» 0 5 fcm) param eter space. The minimal param eter estim ation error along
this trajectory yields a set of approximate param eter values. It is shown th a t the
inverse-scattering technique depends on the shape and the duration of the illum inat­
ing electrom agnetic wave pulse chosen for the electrical param eter reconstruction.
The tim e-dom ain nature of the inverse algorithm allows for limiting the region of
inversion using causality. Thus, when the param eters of the skin are estim ated, the
skin thickness can be determined by comparing the m easurement with a sim ulated
all-skin response. After this step, the skin param eters are known and the same in­
version scheme can be used to determine the electrical param eters of the underlying
breast fatty tissue. Finally, the time-domain inverse-scattering algorithm is tested
for robustness in the presence of broadband Gaussian noise.
Ul
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A cknow ledgm ents
I owe my deepest gratitude to my advisor, Prof. Allen Taflove, for his unpar­
alleled guidance and support through my graduate research.
I would like to thank Prof. Alan Sahakian for his valuable advice and excep­
tional encouragement.
I am grateful to Prof. Todd Kuiken of R ehabilitation Institute of Chicago for
the opportunity to do research in his field.
I am boundlessly thankful to my parents, O lja and Branko Popovic for their
love and ever-present stim ulation for learning pursuits. Along with my late grand­
parents, their lives taught me of rewards brought by persistent work and intellectual
curiosity. I thank my M other for her early and continuing enthusiasm in teaching me
and supporting me in learning foreign languages. I thank my Father for nourishing
my interest for science and for being the most patient and instructive tennis coach.
My sisters, Zorana Popovic and Sofija Baronijan, are supportive siblings one
can only wish for. They taught me how to enjoy life by balancing work and play.
To them , I am further indebted for allowing me to enjoy early-age learning and
discoveries over again through my perfect nieces and nephew: Nina, Jelena, Fiona
and Leon.
Finally, I wish to thank my friends for th eir m oral support and for making
my graduate years a m emorable experience on the social front. I am grateful to my
tennis partners for many enjoyable games.
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
To m y parents, Olja and Branko Popovic
Ithaca
When you set out on your journey to Ithaca,
pray th at th e road is long,
full of adventure, hill of knowledge.
The Lestrygonians and the Cyclops,
the angry Poseidon - do not fear them:
You will never find such as these on your path,
if your thoughts remain lofty, if a fine
emotion touches your spirit and your body.
The Lestrygonians and the Cyclops,
the fierce Poseidon you will never encounter,
if you do not carry them within your soul,
if your soul does not set them up before you.
Pray th at the road is long.
That the summer mornings are many, when,
with such pleasure, with such joy
you will enter ports seen for the first time;
stop a t Phoenician markets,
and purchase fine merchandise,
mother-of-pearl and coral, amber and ebony,
and sensual perfumes of all kinds,
as many sensual perfumes as you can;
visit many Egyptian cities,
to learn and learn from scholars.
Always keep Ithaca in your mind.
To arrive there is your ultimate goal.
But do not hurry the voyage at all.
It is better to let it last for many years;
and to anchor a t the island when you are old,
rich with all you have gained on the way,
not expecting th at Ithaca will offer you riches.
Ithaca has given you the beautiful voyage.
W ithout her you would have never set out on the road.
She has nothing more to give you.
And if you find her poor, Ithaca has not deceived you.
Wise as you have become, with so much experience,
you must already have understood what Ithacas mean.
- Constantine P. Cavafy (1911)
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T able o f C o n ten ts
A b str a ct
iii
T ab le o f C on ten ts
vi
L ist o f F igu res
v iii
L ist o f T ables
xii
1
In tro d u ctio n
1
2
B ackgroun d an d L itera tu re R ev iew
3
2.1 Malignant Tum or C ateg o ries.........................................................................
4
2.2 Dielectric Properties of Breast Tissues in the Microwave Range . . . .
6
2.3 Other Electrical or Microwave Techniques for Breast T um or Detection 7
2.4 Confocal Microscopy in the Microwave R a n g e ........................................
8
2.5 Reported Sim ulations and First E x p e rim e n ts............................................ 9
2.6 Skin Properties and Thickness in the Area of the H um an B reast . . . 13
2.7 Numerical Inverse-Scattering M ethods ......................................................... 16
3
F ea sib ility S tu d ies for th e C on focal M icrow ave T ech n iq u e
18
3.1 Studies W ith Fixed-Focus Elliptical R e f l e c t o r ............................................ 18
3.1.1 G e o m e tr y .................................................................................................18
3.1.2 Calculated Power Density W ithin the Inhomogeneous Breast
T i s s u e ....................................................................................................... 20
3.1.3 Elliptical D etector Response as a Function of T um or Size . . . 24
3.1.4 Angiogenesis Model (Tumor w ith V ascu larizatio n )........................26
3.1.5 Forw ard-Scatter S t u d y ..........................................................................29
3.1.6 Reflector Response as a Function of Dielectric Perm ittivity
C o n tr a s t....................................................................................................32
3.2 Synthetic Aperture:
17-Element A ntenna A rra y ................................................................................ 34
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
S in g le-P a ra m eter R e co n str u c tio n o f D ielectric P r o p e r tie s o f N ea rS u rface B rea st T issu e s
43
4.1 Rationale and Motivation ................................................................................43
4.2 Basic T ec h n iq u e ................................................................................................... 46
4.3 Results in the Absence of N o i s e ......................................................................48
5
T w o-P ara m eter R e co n str u c tio n o f D ie le ctr ic P r o p e r tie s o f N earSurface B rea st T issu e s
50
5.1 M e th o d s ................................................................................................................ 50
5.1.1 Basic T echnique...................................................................................... 50
5.1.2 Excitation Waveforms and Observation W in d o w s ....................... 52
5.2 Results in the Absence of N o i s e ..................................................................... 55
5.2.1 £r-akin Reconstruction with TVial Values of cra k in ...........................55
5.2.2 (Takin Reconstruction with Trial Values of er- a k in ...........................59
5.3 Results in the Presence of Zero-Mean Gaussian N o i s e ..............................63
5.3.1 G eneration of Simulated N o ise ........................................................... 63
5.3.2 120 —p s Differentiated Gaussian Pulse and 5 —p s Rise-Time
R am p Excitations: cr,*in Reconstruction w ith Trial Values of
Sr-akin in the Presence of N o i s e .........................................................65
5.3.3 10 — ps Differentiated Gaussian Pulse E xcitation: crakin Re­
construction with TVial Values of er- akin in the Presence of
N o ise .......................................................................................................... 70
5.4 D isc u ssio n .............................................................................................................75
5.4.1 Im pact of the E xcitation Signal Shape and D uration Upon the
Robustness of the Inverse-Scattering A l g o r i t h m ........................... 75
5.4.2 Determ ining Skin T h ic k n e s s .............................................................. 76
6
C o n clu sion s a n d F u tu re W ork
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
L ist o f F igu res
2.1
Schematic representation of female hum an breast anatom y [1].
...
5
2.2
Results of a confocal microwave imaging algorithm for breast cancer
detection reported in [ 2 ] ...................................................................................10
2.3
Magnified cross-section of the human skin
2.4
Schematic representation of regions on compressed mediolateral (ML)
and craniocaudad (CC) views where measurements for skin thickness
.................................................. 14
15
3.1
Geometry of the elliptical reflector adjacent to the heterogeneous
breast tissue used in the 2-D FDTD m odel.................................................... 19
3.2
Gray-scale visualization of the FDTD-com puted normalized electric
field power density a t 6 G H z .........................................................................20
3.3
Normalized power density as a function of depth within the breast
along the central elliptical sensor axis for an excitation of (a) 3 G H z ,
(b) 6 G H z and (c) 9 G H z ................................................................................ 22
3.4
Normalized power density as a function of lateral distance from the
in-breast focus located 38 m m from the air-breast interface a t 3, 6
and 9 G H z............................................................................................................. 23
3.5
Time waveforms for the FDTD-calculated backscattered signal with
and without tum or located a t the in-breast focus......................................... 25
3.6
Signal-to-clutter (S / C ) ratio as a function of tum or size for center
frequencies of 3, 6 and 9 G H z of the monopole source................................ 25
3.7
Tumor and the surrounding tissue param eters a t 6 G H z - model for
tum or with vascularization................................................................................. 27
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.8
FD TD -com puted backscattered responses for a 5 —ram diam eter tu­
mor w ith and without vascularization a t 6 G H z...........................................28
3.9
Signal-to-clutter (5 /C ) ratio as a function of the tum or diam eter d at
6 G H z : comparison between tum ors w ith and without vascularization. 28
3.10 Geometry used for the forw ard-scatter s tu d y ................................................29
3.11 Forward-scatter study: 5 —m m tum or response for different positions
with respect to the axis of the elliptical reflectors..................................... 30
3.12 Differentiated Gaussian pulse used for studies in 3.1.6 and 3.1.7. . . .
32
3.13 Relative signal-to-clutter (5 /C ) ratio as a function of dielectric per­
m ittivity contrast between breast and tum or tissue......................................33
3.14 2-D FD T D model of the 17-position coherent-addition anten n a array
34
3.15 Algorithm used for processing d a ta obtained for the 2-D F D T D model
of the 17-position coherent-addition antenna array. ..................................36
3.16 FDTD-com puted time-domain waveforms resulting from tim e-shifting
and sum m ing backscattered responses for the 17-position antenna ar­
ray system shown in Figure 3.14....................................................................... 37
3.17 Geometry showing the antenna array adjacent to the skin and the
path used for estimating propagation tim e delays.........................................38
3.18 Geom etry used for determining tim e delay for an off-axis antenna
element a distance away from the array axis..................................................39
3.19 Sample 2-D FD TD calculated gray-scale image of the tu m o r of Fig­
ure 3.14....................................................................................................................42
4.1
Effect of skin thickness on propagation tim e delays for each antenna
element of Figure 3.17.......................................................................................... 44
4.2
Propagation tim e delays for each antenna element calculated for three
different values of relative perm ittivity of skin.............................................. 45
4.3
Propagation tim e delays for each antenna element calculated for three
different values of relative perm ittivity of breast tissue............................... 45
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.4
Time-domain inverse-scattering iterative algorithm for recovery o f di­
electric param eters er and a from a measured signal....................................46
4.5 Backscattered signals for different values of £r-akin and askin.......................47
4.6
Inverse-scattering FD TD computation: Convergence of £r-akin to its
correct value............................................................................................................49
4.7
Inverse-scattering FD TD com putation Convergence of o Skm to its cor­
rect value................................................................................................................. 49
5.1
Trajectory of the gradient method leading to approxim ate values
of^aWn and £r—skin..................................................................................................51
5.2
120 —ps differentiated Gaussian pulse (full w idth between 1/e points ). 52
5.3 10 —ps differentiated Gaussian pulse with 5 —ps rise tim e........................ 53
5.4 5 —ps rise tim e ram p signal.................................................................................53
5.5
Estim ate of £r-akin for trial values of <
j jfcin for the 120 —ps differen­
tiated Gaussian pulse excitation case................................................................56
5.6
Estim ate of £r-akin for trial values of <JSkin for the 5 —ps rise-time
ramp excitation case............................................................................................. 57
5.7
Estim ate of £r-akin for trial values of aakin for the 10—p s differentiated
Gaussian pulse excitation case............................................................................58
5.8
Estim ate of o akin for trial values of er_afctn for the 120 —ps differen­
tiated G aussian pulse excitation........................................................................ 60
5.9
Estim ate of a akin for trial values of £r- akin for the 5 — ps rise-tim e
ram p excitation...................................................................................................... 61
5.10 Estim ate of <rakin for trial values of £r- Skin for the 10—p s differentiated
Gaussian pulse excitation.................................................................................... 62
5.11 Sample noisy backscattered waveforms for the 10 —p s differentiated
Gaussian pulse excitation.................................................................................... 64
5.12 Sample trajectories in (er-jfcin> a skin) space for the 120 —ps differen­
tiated Gaussian pulse for signal-to-noise S / N = 20,30,35 and 40 d B .
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
5.13 Sample variations of error vs. estimated crskin for the 120 —ps differ­
entiated Gaussian pulse excitation................................................................... 67
5.14 Sample trajectories in ( e y crakin) space for the 5 —ps rise-time
ram p excitation for signal-to-noise ratios of S / N = 20,30,35 and 40 d B . 68
5.15 Sample variations of error vs. estimated <Tskin for the 5 —ps rise-time
ram p excitation.....................................................................................................69
5.16 Sample trajectories in the (er-*fc»n> 0«*m) space for the 10 —ps differ­
entiated Gaussian pulse excitation with S / N = 20 d B
71
5.17 Sample trajectories in the (£>-**«», crskin) space for the 10 —ps differ­
entiated Gaussian pulse excitation with S / N = 30 d B
72
5.18 Sample trajectories in the (er-akm, <*akin) space for the 10 —ps differ­
entiated Gaussian pulse excitation with S / N = 35 d B
73
5.19 Sample trajectories in the (eT-akin, 0akin) space for the 10 —ps differ­
entiated Gaussian pulse excitation with S / N = 40 d B
74
5.20 Waveforms illustrating a strategy for estim ating the skin thickness
making use of the previously determined values of er-akin and cr3kin. . 77
xi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
L ist o f T ables
3.1
Dielectric param eters for breast tissue extended from the 3 — G H z
d ata of [3], [4] by Debye approximation........................................................... 20
xii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C hapter 1
In trod u ction
The investigations presented here are m otivated by the development of a new ultrawideband confocal microwave technology to detect and image early-stage breast
cancers [5], [6], [7], [2], [8], [9], [10], [11], [12], [13], [14], [15] . The new technology
exploits the dielectric property contrast between normal breast tissues and malig­
nant tumors a t microwave frequencies. Here, microwave im aging is performed by a
planar antenna array contacting only one side of the breast. T he antenna array ele­
ments collect impulsive backscattered signals, which are digitally delayed to achieve
coherent sum m ation a t the site of a potential tumor. T his scheme depends upon
knowledge of the average dielectric properties of the local breast tissues. Patientspecific calibration of the microwave imager requires knowledge of these properties.
In the present research, a time-domain inverse-scattering technique is studied
to measure the skin thickness and dielectric param eters in the area of the hum an
breast. This stu d y is motivated by several well-documented findings. Skin thickness
varies from patient to patient, and also with location on the body of an individual
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2
patient. A number of factors can cause thickening of m am m ary skin. Although
this study is primarily m otivated by assisting patient-specific calibration of the mi­
crowave breast cancer imaging system, determ ining breast skin thickness can also
help to diagnose possible pathologies in the underlying tissue and in the patient in
general.
Calculations are first presented to illustrate the im portance of knowledge of
the correct skin thickness for determining the tim e delays needed for the imageform ation signal-processing algorithm.
Then, results are presented to show the
development of a two-dimensional tim e-dom ain inverse-scattering algorithm for si­
m ultaneous estim ation of electrical perm ittivity
e r -s k in
and conductivity
c rs k i n
of the
skin layer. This algorithm locates a search trajectory in the (er-*fcm> <?skin) param ­
eter space. The minimum param eter estim ation error along this trajectory yields
the set of approximate param eter values.
Results are then presented which show th a t the search trajectory and the
convergence error depend on the shape and the duration of the impinging pulse.
T he time-domain nature of the inverse algorithm allows for limiting the spatial
region of inversion by causality. Thus, when the electrical param eters of the skin
are estim ated, the skin thickness can be determ ined by comparing the measurement
w ith a simulated all-skin response. After this step, the same inversion scheme can
be used to determine the electrical param eters of the underlying fatty breast tissue.
Finally, the robustness of the algorithm is tested in the presence of Gaussian noise
for various signal-to-noise ratios.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C hapter 2
B ackground and L iteratu re
R eview
The use of an active, pulsed confocal microwave system employing backscattering
to detect and image breast cancer has not been investigated until recently [5], [6],
[7], [2], [8], [9], [10], [11], [12], [13], [14], [15]. The new technique has no relation
to previously unsuccessful microwave tumor-detection approaches involving passive
thermography or active tomography.
Although X-ray mammography is recognized as a stan d ard tool for detect­
ing and characterizing breast tumors, it has im portant lim itations. First, screening
mammography suffers horn a high false-positive rate [16], and a false negative rate
ranging from 4% to 34%, depending on the definition of a false negative and the
length of follow-up after a “normal” mammogram. Second, the sensitivity of mam­
mography decreases for radiographically dense breasts [17], [18], [19]. Furthermore,
while the blurring caused by body motion and X-ray scatter is minimized by breast
3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
compression, this procedure is uncomfortable to the patient and could cause burst­
ing of pockets of the potentially malignant tissues. Finally, X-ray mammography
exposes the patient to ionizing radiation.
A lthough this exposure is conducted
under strict federal regulation [The Mammography Q uality Standards A c t o f 1992
/MQSA/][8], there is a 1/1,000,000 risk of inducing breast carcinoma with the levels
of radiation exposure currently used [8].
The new early-stage breast cancer detection technology [9], [10], [11], [12],
[6], [7], [5], [8] is founded upon (1) the dielectric properties of breast tissues at
microwave frequencies and (2) the principles of confocal microscopy. This foundation
is sum m arized in this chapter.
2.1
Malignant Tumor Categories
Figure 2.1 shows the m ain regions in female hum an breast anatom y [1]. Breast
cancer usually begins in the lobules,the milk-producing glands of the breast, or in
the ducts th a t bring the milk to the nipple. The lobules and ducts are surrounded
by fatty and connective tissue, nerves, blood vessels, and lym phatic vessels, which
carry lym phatic fluid from the lymph nodes. A bout 80% of breast cancers start in
the ducts; the rest start in the lobules. If breast cancer breaks out of a lobule or
a duct, it can invade the surrounding fatty tissue and spread to other parts of the
body via the lym ph or blood vessels [20].
There are number of risk factors for breast cancer. These include gender,
age, family history, hormone history, toxic exposures, weight, physical activity and
lifestyle in general [20].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.1: Schematic representation of female hum an breast anatomy [1].
T he likelihood of breast cancer spreading to other tissues or parts of the body
depends on the type of cancer involved [20]. Principle cancer types include:
• Ductal carcinoma in situ (DCIS). This is the m ost common type of in
situ breast cancer, meaning breast cancer which remains “in place” and has not
yet spread from its point of origin to other tissues and organs. DCIS is confined
to the ducts, and is therefore a t an early and curable stage. In screening centers,
20 —30% of newly diagnosed breast cancers are DCIS. DCIS occurs often on several
points along the duct, appearing as a cluster of calcifications or white flecks, on
a mammogram. It is treated with excision or mastectomy, possibly followed by
radiation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
• Invasive (infiltrating) ductal carcinoma (IDC) . This is the m ost common
type (70 —80%) of invasive breast cancer. It begins in a duct, breaks through the
duct wall, and invades fatty tissue in the breast. From there, it can metastasize to
other parts of the body. It is usually detected as a mass on a mammogram.
• Medullary, tubular and mucinous carcinoma .
These are less common
types of ductal carcinoma, together accounting for less than 10% of breast cancers.
Medullary and tubular carcinoma are both invasive but may have b etter prognoses
than do invasive ductal or invasive lobular carcinomas.
• Inflammatory breast cancer. This unusual and agressive form of breast
cancer involves cancer cells blocking the lymphatic channels. Therefore, the fluid
cannot drain from the lymph vessels of the skin.
• Paget’s disease o f the breast. This rare ductal carcinoma arises in the ducts
near the nipple. Diagnosis of this form of breast cancer calls for a mammogram, a
clinical breast exam, and a biopsy of the nipple skin.
2.2
Dielectric Properties of Breast Tissues in the
Microwave Range
Extensive dielectric measurements up to 3 G H z of both normal and m alignant,
human breast tissues have been reported in the literature [4], [21], [22], [23], [24].
At 6 G H z , normal breast tissue dielectric properties are sim ilar to those of fat and
vary in an approximate ±10% range about £r- b r e a s t = 9 for the perm ittivity and
a brea st
= 0.4 S /m for the conductivity. Malignant tum or dielectric properties are
similar to those of muscle. They have nominal values of £r-tumor = 50 and <7tumor = 7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7
S /m . This dielectric contrast causes m alignant tum ors to have significantly larger
microwave backscattering cross sections relative to normal tissues of comparable
size and geometry. For breast skin tissue, the values found in the literature are
approximately
e r -s k in
= 36 and
<rSk in
= 4 5 /m in the microwave frequency range
[22], [23], [24], [25].
For evaluation of the 1/e depth of microwave penetration, the d a ta of [4], [3]
can be extended to higher frequencies by either a Debye model [26] or an empirical
model [27]. In general, this result agrees w ith published penetration depths for lowwater-content fat tissue [21]. The path losses do not exceed 4 d B /c m up to 10 G H z.
2.3
Other Electrical or Microwave Techniques for
Breast Tumor Detection
Breast tissue is characterized by complex spatial shapes and m aterial inhomo­
geneities. These properties present a substantial challenge to applying electrical
or microwave techniques to tumor detection (see the review of Larsen and Jacobi
[28]). Two low-resolution microwave backscatter m ethods [29], [30] involve illu­
m inating the breast with large unfocused beams. These techniques are deficient
because the returns from the tumor can be masked by clutter from adjacent regions
of the breast. One reported low-frequency m ethod [31], built and tested on human
subjects, uses electrical impedance d a ta m easured between electrodes placed upon
the breast. This approach suffers from low resolution in locating the tum or. Here,
small errors in the sensed d ata can be amplified by the required matrix-inversion
process, which results in degraded accuracy.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
2.4
Confocal Microscopy in the Microwave Range
Microwave attenuation in norm al breast tissues does not exceed 4 d B /c m for fre­
quencies less th an 10 G H z. This attenuation is sufficiently low to perm it a quasioptical tum or-detection m ethod based upon the operating principle of the confo­
cal microscope [32]. A confocal microscope selectively images sm all particles in a
translucent medium. This approach provides spatial selectivity of b o th the trans­
m itted and received signals and thus reduces the problem of background clutter.
To im plem ent this concept a t microwave frequencies, a planar array of n resistively loaded, ultrawideband antenna elements is placed a t the breast surface in
direct contact with the skin. Each element is excited one at a tim e by a subnanosec­
ond RF tone burst having a signal bandw idth exceeding 10 G H z. T he backscattered
response from the breast tissues w ithin a selected nanosecond-regime tim e window is
measured a t the transm itting antenna element, digitized, and stored in a computer
file. This process is repeated for each element of the array. W hen all n backscattered
waveforms are available, a postprocessing com puter algorithm sim ilar to th a t used
in seismic geophysical surveying [33] is used to time-shift and sum the waveforms
according to the assumption th a t a backscattering center is located a t a particular
trial point w ithin the breast. If indeed such a backscattering center exists a t the
trial point, the n waveforms add coherently to generate a significant composite re­
sponse. If no backscattering center exists a t the trial point, the n waveforms add
incoherently to generate a weak com posite response. The position of the trial point
is scanned system atically throughout the breast by adjusting the assumed distri­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9
butions of the time windows and time-shifts of the stored backscattered waveforms
across the antenna array. The outcome is a 3-D distribution of composite-response
waveforms, which can be further postprocessed to provide the location and shape
of the microwave backscattering centers within the breast.
The above described technique of time-shifting and coherently summing in­
dividual backscattered responses achieves the effect of an electronically scannable
confocal antenna. The use of an adjustable tim e window for each backscattered
response provides range-gating, yielding additional spatial selectivity in the depth
direction. Because this signal processing does not involve m atrix inversion, the
m ethod is relatively insensitive to measurement errors.
2.5
Reported Simulations and First Experiments
Initial investigations pertaining to the pulsed microwave breast cancer detection
system are now reviewed.
Two-Dimensional (2-D) FDTD Simulations. These FD TD simulations sug­
gest th a t the pulsed, confocal, tumor-detection process is robust [6]. Specifically, it
was found th at the signal-to-clutter (S/C) ratio of a 5 —m m tum or embedded at
the synthetic focus is little degraded from the initial results even if:
• the average dielectric contrast between the tumor and the normal-tissue
background is decreased by 50% ;
• the random heterogeneity within the breast tissue is elevated to ±40%
around its mean value ;
• a simulated vein (2 m m in diameter) is interposed midway between the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
sensor array and the tumor (in this case, time-gating perm its distinguishing the
vein response from th at of the tum or) ;
• a Debye frequency dispersion of er and u is assumed for the media included
in the sim ulation geometry.
A com putationally efficient and robust image-reconstruction algorithm for
breast cancer detection was recently presented [2] using the ultrawideband confocal
microwave imaging system of [6]. The algorithm was tested on a two-dimensional
MRI-based FD T D model of the cancerous breast. These simulations demonstrate
the feasibility of detecting and imaging small malignant lesions within the breast
(Figure 2.2).
Figure 2.2: Results of a confocal microwave imaging algorithm for breast cancer
detection reported in [2]: two-dimensional microwave breast image reconstructed
from the processed backscatter waveforms com puted for a two-dimensional dielectric
properties breast model derived from an MRI scan. In this model, a 2mm-diameter
malignant tum or has been inserted a t a depth of 3.1 cm.
Three-Dimensional (3-D) FD TD Simulations. In work reported in [7], threedimensional FD T D modeling work dem onstrated the following key points.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11
• An ultrawideband m iniaturized bow-tie an ten n a element located a t the
surface of the breast is feasible [34]. Optimized resistive loading [35], [36] perm its
efficient radiation of pulses having banawidths of several G H z . The initial electro­
magnetic wave reflection from the ends of the antenna is 110 d B below the excita­
tion, perm itting backscattered returns to be sensed im m ediately after radiation of
the illum inating pulse.
• For a 5 — m m diam eter spherical tum or em bedded in normal breast tissue
3.8 cm beneath the skin surface directly below the feedpoint of a single bow-tie
element, the co-polarized backscattered pulse observed a t the feedpoint is 94 d B
below the excitation.
• For a pair of perpendicular bow-tie antenna elem ents forming a M altese
cross, the typical cross-polarized backscattered responses of small off-axis spherical
tum ors and on-axis cylindrical tum ors are 10 — 15 d B below the respective co­
polarized responses.
These results point to the feasibility of a microwave sensor consisting of an
array of resistively loaded Maltese-cross antenna elements.
The tum or’s cross-
polarized backscatter collected by a single bow-tie elem ent for a 0.5 — cm tum or
embedded 3.8 cm deep within the breast is 105 — 110 d B below the peak applied
power. However, the backscatter collected by a single elem ent is augmented by the
processing gain of the n-position synthetic-aperture array, which yields an improve­
ment in signal-to-noise (S/N) ratio of lQlogi0n dB. T hus, assuming 10< n <100,
the processing gain would range between 10 and 20 d B . T his implies th a t th e sys­
tem noise floor should be 100 — 110 d B below the peak applied power to achieve
a composite S /N of 10 dB. Results to date suggest th a t this noise floor can be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
achieved using existing microwave instruments.
Reference [5] reports the use of the confocal microwave imaging technique
for detection of breast cancer; however, the geometric configuration and algorithms
are different from those used in [6]. Here, the proposed system encircles the breast
with an array of antennas. The patient lies in a prone position with the breasts
immersed in a liquid having electrical properties similar to those of breast tissue
or skin, but of low loss. The antenna array is placed in the liquid and positioned
around and offset from the breast. For d a ta acquisition, each antenna (one a t a
time) transm its an ultrawideband pulse, and the backscattered returns are recorded
a t the same antenna. The antennas are spaced to reduce their m utual coupling
and the whole array can be rotated to a new position for obtaining additional data.
Furthermore, translating the array vertically allows for scans of different sections
through the breast.
Although the breast was modeled in [5] as a finite cylinder of randomly inhomogeneous fatty tissue surrounded by an outer skin layer, this is a reasonable
approximation for feasibility studies of tum or detection in 2-D cross sections. For
the antenna model of [5], a simple resistively loaded dipole based on [37] was selected.
Although such antennas have poor efficiency and directivity, they can be physically
small and possess wide bandwidth. T his leads to reduced com putational model­
ing cost relative to the bowtie antenna. The previously m entioned time-shifting
and summing algorithm applied to backscattered signals is again used for tum or
image formation; however, after the first calibration, a skin-subtraction algorithm
is implemented. Initial computational studies for this system immersed in a lowloss skin-like liquid show adequate sensitivity to tumors. Improvements, however,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
13
are needed for the skin-subtraction algorithms and design of suitable antennas for
practical systems.
Reference [8] reported experimental studies where the breast is subjected to
continuous wave (CW) illum ination by a 16-element transceiving monopole antenna
array in the 300 — 1000 M H z range. Preliminary results show th at this imaging
system can detect subtle tissue abnormalities. Further, these results show th a t the
average relative perm ittivity of the breast may correlate w ith radiological breast
density, and may be considerably higher than previously published values based on
ex vivo specimens.
2.6
Skin Properties and Thickness in the Area of
the Human Breast
Although microwave wavelengths below 20 G H z are too long to differentiate the
fine inhomogeneities w ithin the human skin, it is nevertheless useful in this work
to understand the internal structure of the human skin [38]. The magnified image
shown in Figure 2.3 depicts a cross-section of a basal layer from the forearm of a
three-week old infant. This image is obtained by osmium fixation and lead stain­
ing, with magnification of 5,000 x . Human skin consists of a multi-layered cellular
epidermis, designated as feature E in Figure 2.3, derived m ainly from the surface
endoderm and an underlying dermis of mesodermal origin. These two distinct p arts
of the skin meet at th e epidermal-dermal junction ( feature j in Figure 2.3), a re­
gion where im portant morphogenetic and other influences operate throughout fetal
and post-natal life. T he dermis contains a variety of elements indicated by lettered
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
features in Figure 2.3 as follows: vessels (v); nerves (n); and individual cells (c) set
in a strong meshwork of collagen fibers (f).
T he literature offers several studies
Figure 2.3: Human skin consists of a m ulti-layered cellular epidermis (E ), and an
underlying dermis. These two distinct parts m eet a t the epidermal-dermal junction
(j). The dermis contains various elements, vessels (v), nerves (n), and individual
cells (c) set in a strong collagen fiber meshwork (f). The photograph shows a basal
layer of epidermis and derm is from forearm skin of a three-weeks old infant a t
magnification of 5,000 x [38].
on skin thickness in the area of the human breast. Skin thickening was evidenced
after breast reconstruction [40], [41] and tissue expansion after masectomy [42], [39].
Increased skin thickness can often be a consequence of therapeutic radiation of the
breast [43], [39]. In most cases, the reported skin thickness was evaluated from the
film-screen mammograms. A comprehensive norm al range of breast skin thickness
and the causes of thickening shown on film-screen mammography is given in [39]. As
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15
MEDIOLATERAL
(ML) VIEW
CRANIOCAUDAD
(CC) VIEW
Superior
i Mm (U tfa u if
0.7-.
In ferior
mJHm M M m m
& 7 -2 .7 m
Figure 2.4: Schematic representation of regions on compressed m ediolateral (ML)
and craniocaudad (CC) views where measurements for skin thickness were obtained
on film-screen mammograms [39].
shown in Figure 2.4, breast skin thickness ranges from 0.7 m m - 2.7 m m depending
upon the location (Figure 2.4). Specifically, the inferior skin has m axim um thickness
(m ean value = 1.7 mm); the m edial and superior skin thicknesses b o th average 1.5
m m ; and the lateral skin has m inimum thickness (mean value = 1.3 m m ).
The reported causes of m am m ary skin thickening are commonly categorized
by th eir local or generalized nature. The localized causes can be carcinom a, inflam­
m ation, traum a, fat necrosis, post-biopsy and dermatological conditions. The gener­
alized causes found in the literature are breast cancer, Hodgkin’s disease, reticulum
cell sarcom a, m etastatic disease, radiation therapy, inflammation, surgery, primary
skin disorders, anasarca and any cause of lym phatic obstruction. Therefore, careful
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
m onitoring of breast skin thickness can also be used as a pre-diagnostic tool of the
underlying tissue pathologies or health concerns of th e patient in general.
2.7
Numerical Inverse-Scattering Methods
Numerical inverse-scattering studies found in the literature are based on either
frequency- or time-domain approaches. In general, a feedback process in conjunction
with an optimization process is used for determining the set of unknown parame­
ters th a t minimizes the difference between the calculated and the observed fields.
In the frequency-domain algorithm s, the interaction of the entire medium with the
incident field m ust be considered simultaneously [44], [45], [46], [47]. Therefore, for
the frequency-domain-based approaches, solving for a general, lossy inhomogeneous
medium results in a prohibitively large number of unknown parameters. In contrast,
time-domain approaches exploit causality to limit th e region of inversion. As the
work of this thesis focuses on time-domain inverse-scattering methods, this section
summarizes the literature relevant to th a t approach only.
Reference [48] reported a one-dimensional FD T D formulation of an inverse
scattering scheme for remote sensing of inhomogeneous lossy layered media. Here, a
layer-stripping procedure was used to simultaneously recover the conductivity and
perm ittivity profiles. This work assumed an incident plane wave pulse and zero
noise present on the received pulse.
Reference [49] reported on a two-dimensional F D T D inverse-scattering scheme
for remote sensing of conducting and dielectric targets. Here, a forward scattering
FD TD element was used in a numerical feedback loop w ith a nonlinear optim ization
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
routine. W ith this technique, as the illum inating wavefront sweeps across the target,
causality is exploited to reconstruct the actual target surface contour in a sequential
and cumulative m anner. To test the complexity of a recoverable structure, a dielec­
tric target with reentrant features resembling the letter “J ” was reconstructed from
a single-point observation, having plane wave illum ination as a source. Numerical
experiments tested th e performance of the algorithm and dem onstrated a promising
degree of robustness.
Inverse-scattering problems commonly appear in the area of geoscience and
remote sensing [50], [51], [52]. G round-penetrating radar (G PR ) is used to obtain
information on subsurface features from d a ta collected over th e surface. Most G PRs
use broad beam width antennas. Therefore, the energy reflected from a subsurface
target is recorded over a large lateral aperture. M igration algorithm s can be used for
target reconstruction by refocusing the recorded scattering events to their true spa­
tial locations through backpropagation. Reference [50] presents FD TD reverse-time
migration algorithms for ground-penetrating radar (GPR) d a ta processing. O ther
signal processing techniques in conjunction with FDTD algorithm s are reported in
[51] for detection of buried dielectric cavities. However, the propagation losses lim it
deep penetration into the ground to low-frequency signals.
Recently, a new class of inverse-scattering techniques employing global inver­
sion algorithms has emerged. Two of these algorithm s involve the neural network
m ethod [51] and the use of genetic algorithm s [52]. Global inversion algorithms based
on stochastic strategies offer many advantages over local inversion algorithms includ­
ing strong search ability, simplicity, robustness, and insensitivity to ill-posedness.
However, they suffer from a high number of function evaluations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h ap ter 3
F ea sib ility S tu d ies for th e
C on focal M icrow ave T echniqu e
3.1
Studies With Fixed'Focus Elliptical Reflector
This section describes num erical studies modeling the characteristics of a fixed-focus
elliptical microwave sensor. This system focuses a microwave signal a t a site in the
breast and efficiently collects the backscatter. Since this sensor can be synthesized
by an electronically scanned impulse antenna array, the results obtained here are
applicable to the impulse antenna array discussed later.
3.1.1
G eom etry
Figure 3.1 illustrates the 2-D geometry of the elliptical microwave sensor adjacent
to the randomly heterogeneous breast tissue model considered in this study. The
reflector aperture is 80 m m in diam eter with an in-breast focus 38 m m deep w ithin
the breast. The elliptical reflector resting on the breast tissue is filled with dielectric
m aterial having the same average dielectric constant as the underlying breast tissue.
18
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
This minimizes reflections a t the breast surface. A monopole source is located a t the
focal point within the reflector. The breast tissue region consists of 5m m x 5m m
blocks differing randomly in electrical properties (e, a) by ±10% about the mean
value. The resulting tissue heterogeneity is represented in Figure 3.1 as square
zones of different gray scale.
Figure 3.1: Geometry of the elliptical reflector adjacent to the heterogeneous breast
tissue used in the 2-D FD TD model.
Measured d ata of [4], [3] up to 3 G H z for norm al and m alignant breast tis­
sues are extended up to 9 G H z by means of a Debye approximation. The mean
extrapolated values incorporated in the simulation are shown in Table 3.1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
Table 3.1: Dielectric param eters for breast tissue extended from the 3 — G H z d a ta
of [3], [4] by Debye approxim ation.
Frequency (G /fz)
3
6
9
3.1.2
^■r—b r e a s t
9.96
9.84
9.65
& b reast
(S /III)
0.21
0.38
0.63
C alculated Pow er D en sity W ith in th e Inhom ogeneous
B reast T issu e
For power-deposition studies, a continuous sinusoidal waveform excitation of the
monopole source is assumed.
Figure 3.2: Gray-scale visualization of the FDTD-com puted normalized electric field
power density at 6 G H z . Darker color indicates high field values, while paler shades
represent low-field regions.
Figure 3.2 shows a gray-scale visualization of the FDTD-computed norm alized
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
electric field power density a t 6 G H z . The source is seen clearly a t the in-reflector
focus. Furthermore, the power density within the the breast tissue is visibly con­
centrated around the in-breast focus.
Figures 3.3(a), 3.3(b) and 3.3(c), graph th e normalized power density within
the breast tissue as a function of distance from the tissue surface along the central
reflector axis a t 3, 6 and 9 G H z , respectively. A dditional studies have shown th a t the
attenuation in the zone between the breast surface and the in-breast focus increases
significantly above 9 G H z despite the increased gain of the ellipsoidal reflector.
Figure 3.4 shows the normalized power density vs. lateral distance a t the
depth of the in-breast focus. This indicates the expected sharpening of th e incident
beam with frequency. The slight departure from even sym m etry results from the
assumed random heterogeneity of the tissue. F urther sharpening would be expected
above 9 G H z . However, the cost of increased sp atial resolution at higher frequencies
is increased attenuation of the signal. In the next section, we present further results
to confirm th a t operation of the elliptical reflector w ithin the frequency window
centered a t 6 G H z optimally balances resolution and power losses.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
slope:
-0.19 dB/mm
in-breast
focus location
slope:
-0.12 dB/mm
Distance from the air-breast interface along the central reflector axis (mm)
slope:
-0.17 dB/mm
Figure 3.3: Normalized power density as a function of depth within the breast along
the central elliptical sensor axis for an excitation of (a) 3 G H z, (b) 6 G H z and (c)
9 GHz .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
3 GHz
i ------ 6 GHz
\ --------- 9 GHz
-3
-5
-2 0
-15
-1 0
-5
0
5
10
Lateral distance from in-breast focus (mm)
20
Figure 3.4: Normalized power density as a function of lateral distance from the
in-breast focus located 38 m m from the air-breast interface a t 3, 6 and 9 G H z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
3.1.3
E llip tical D etector R esponse as a F u n ction o f Tumor
S ize
Next we study the backscatter response from a circular tu m o r located at the in­
breast focus of the detector. This desired signal must be detected in the presence
of clutter due to backscatter from the surrounding random ly inhomogeneous nor­
mal tissues. The tum or electrical properties are assumed to be eT-tumor = 50 and
&tumor = 7 [3], [4] on the basis of measured d ata at 6 G H z . T he tum or diameter, d,
is varied from 1 m m to 10 mm. T he source excitation is assum ed to be a 120 —ps
Gaussian pulse (full width between 1/e points) m odulating a sinusoidal carrier at
either 3, 6 or 9 G H z . In order to keep the same range resolution, the pulse width is
fixed for all center frequencies studied. The time waveform of the magnetic field is
observed at the source location for the backscattered signal w ithout and with tumor
present. We “zoom in” the portion of the graph where these two signals begin to
diverge, as shown in Figure 3.5.
By this m ethod, we can decide on the time window in which we perform
the signal-to-clutter (5 /C ) calculation. W ithin the chosen window, the tumor and
clutter signals are squared. The ratio of their maxima, designated as the S / C ratio,
is calculated in decibels (dB) for each of the frequencies of interest, and for ten tumor
diameters from 1 m m to 10 mm. T his yields a S/C-vs-d curve. These curves are
shown for 3, 6, and 9 G H z in Figure 3.6. The results again suggest th a t operation
of the elliptical sensor in the range 3 — 9 G H z is advantageous. Above 9 GHz,
the microwave attenuation in the breast is too high, yielding reduced S / C ratios.
Below 3 G H z , the spatial resolution is too low, yielding reduced ability to localize
the tum or’s position.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
i
No tumor, 9 GHz
— — Tumor diameter 5mm, 9 GHz
I
i
i3
8
<0
1.6
1.8
2.0
2J2
2.4
2.6
2.8
3.0
3.2
time(ns)
Figure 3.5: T im e waveforms for the FDTD-calculated backscattered signal with
and without tu m o r located at the in-breast focus. The plots are for a 120 —ps
Gaussian pulse m odulating a sinusoidal carrier having center frequency of 9 GHz.
The assumed tu m o r diam eter is d = 5m m .
■— ■3G H z
•
• 6 GHz
* — * 9 GHz
-1
tumor diameter (mm)
Figure 3.6: Signal-to-clutter ( S / C ) ratio as a function of tu m o r size for center
frequencies of 3, 6 and 9 G H z of the monopole source.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
3.1.4
A ngiogenesis M odel (Tumor w ith V ascularization)
Electrical perm ittivity er and conductivity a in the tissue region around a breast
tum or is affected by the presence of increased vascularization. Literature values of
eT and a as a function of distance from the tumor [53] are used to form a simplified
model. The model assumes a 1 /r decrease of eT and a going radially outward from
the tum or into the surrounding breast tissue, where r is the distance from the tum or.
However, according to [53], the conductivity within a th in ring adjacent to the tum or
increases slightly before it begins to decrease. This is also taken into account within
the model. These approximations are illustrated in Figure 3.7 for the values of
param eters a t 6 GH z.
C lutter signals for a geometry with no tumor and for a geometry th at includes
a tum or are computed and the results inspected graphically. The source excitation
is a m odulated 120 —ps Gaussian pulse centered at 6 G H z . W ithin the tim e frame
of the tum or signal return, the S / C ratio is estim ated by comparing peak-to-peak
values. A typical graph for determining such an estim ate is given in Figure 3.8 for a
5 m m diam eter tumor. This analysis is repeated for tum ors 1 —10 m m in diam eter,
with results for the S / C ratio as a function of the tum or size shown in Figure 3.9.
The results presented in Figures 3.8 and 3.9 illustrate the effect of angiogenesis
and suggest th at the vascularization increases the level of the backscattered signal.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
60
w" 50
> 40
. tumor
1/r
20
0.0
10.0
12.5
2.5
5.0
7.5
r: Distance from the center of the tumor (mm)
15.0
(a)
E
o5
c
&
'>
‘*0
T5
C
o
O
10
9
a 10 % increase in conductivity
8
in the immediate tumor proximity
7
Rt / 4
6
~ 1/r
5 - tumor
4
Rt
3
2
1
0
0.0
2.5
5.0
7.5
10.0
12.5
r: Distance from the center of the tumor (mm)
15.0
(b)
Figure 3.7: Tumor and the surrounding tissue param eters at 6 G H z - model for
tum or with vascularization. The curves show param eter approxim ation for a tum or
5 m m in diameter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
40-07
no tumor
tumor (diameter 5mm)
tumor (5mm diameter) with vascularization
30-07
i
20-07 -
«
o
10-07
0e+00
3.0
time (ns)
4.0
3.5
Figure 3.8: FDTD-com puted backscattered responses for a 5 —m m diam eter tum or
with and w ithout vascularization a t 6 G H z .
25
20
no vascularization
tumor with vascularization
2m 15
®
1
g
2 10
a>
m
5
10
Tumor diameter (mm)
Figure 3.9: Signal-to-clutter ( S / C ) ratio as a function of the tum or diam eter d a t 6
GHz: comparison between tum ors with and without vascularization.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
3.1.5
Forw ard-Scatter S tu d y
This section describes the study of the elliptical reflector response if used to detect
the forward-scattered signal from the sim ulated tumor.
Figure 3.10: Geometry used for the forward-scatter study. Between the transm itting
and the receiving elliptical reflector lies a slab of heterogeneous breast tissue with
thickness equal to two focal lengths of the confocal reflectors.
This investigation has utility only for purpose of comparison. In practical im­
plementation, the detection of forward scattering would involve breast compression,
which we want to avoid by designing a system th a t relies solely on the backscattered
response. The geometry used for the forward-scatter study is shown in Figure 3.10.
A slab of heterogeneous breast tissue with random ±10% variations is placed be­
tween the transm itting and the receiving elliptical reflectors. The slab thickness is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
equal to two focal lengths of the elliptical reflectors. A m odulated 120—ps Gaussian
pulse centered at 6 G H z is sourced from the focus of the transm itter. After propa­
gating through the breast tissue slab, it is detected a t the focus of the receiver. To
obtain the signal for comparison with the backscattered signal, we place the tumor
in the middle of the slab, 38 m m deep from either of the reflector-breast interfaces.
ie-07
8e-08
no tumor
centered tumor (at in-bfbast focus)
6mm off-center
14mm off-center
6e-08
3JSi
o
4e-08
z
2e-08
2.5
2.7
2.9
3.1
3.3
3.5
time(ns)
Figure 3.11: Forward-scatter study: 5 —m m tum or response for different positions
with respect to the axis of the elliptical reflectors.
For the tum or located on the reflector axis, we expect to obtain the maximum
signal-to-clutter ratio. We note, however, th a t clutter and signal here are interpreted
differently from the backscatter case. For forward scatter, with no tum or present,
the received signal is a t the maximum level, as it travels without p artial reflection
from the tumor. Conversely, the minimum signal is detected a t the receiving antenna
when the tumor is located in the center of the slab and on the axis of the two elliptical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
reflectors. It is this decrease th a t carries the inform ation about the tum or’s presence.
Note th a t this decrease is less evident as we move the tum or off-center along the
middle of the slab, as illustrated in Figure 3.11, approaching the case of no tum or
present. For 5 —m m tum or, the S / C ratio is no b e tte r than th at for the same tum or
detected using the backscatter m ethod. This result suggests th a t we can proceed
with the backscatter m ethod, thus avoiding breast compression. Although theory
predicts greater forward scatter than backscatter for a plane wave impinging on a
scattering object, the focused wave generated using the elliptical reflector does not
obey this law.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
3.1.6
R eflector R esponse as a F unction o f D ielectric Per­
m ittiv ity C ontrast
In this section, we examine how the degree of contrast of the electric perm ittivities
of breast and tum or tissues affects the backscattered signal.
1.0
0.5
0.0
120
-0.5
- 1.0.
0.0
0.1
0.2
time(ns)
0.3
0.4
0.5
Figure 3.12: Differentiated Gaussian pulse used for studies in 3.1.6 and 3.1.7.
On a relative scale, we expect the S / C ratio to approximately follow the
theoretical curve for the plane-wave reflection coefficient:
Q=
where
q
1 + \fx
1 — y/x
(3.1)
is the reflection coefficient, and
x =
E r —t u m o r
E r —b r e a s t
is the ratio (contrast) of dielectric perm ittivities of tum or and breast tissue.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.2)
33
4
cotK. p«(1*k”V(1 -k ” )
3
m
-o
1
2
* ""
✓
contrast value reported
1 - in the literature - 5
p
5O
C
0
s
a
a>
2
•3
-4
1
2
3
4
5
6
7
8
9
10
11
12
13
X = ( Er-lunio/e r-tf*««|) ’
Figure 3.13: Relative signal-to-clutter {S/C) ratio as a function of dielectric per­
m ittivity contrast between breast and tum or tissue.
For this study, and the studies presented in the section of preliminary work,
we used a 120 — ps differentiated Gaussian pulse for th e source signal, since the
literature suggests it is advantageous for obtaining higher S / C ratios [6]. The time
waveform of this pulse is shown in Figure 3.12.
Figure 3.13 graphs, on a relative scale, the S / C ratio for a 4.8 mm-diameter
tum or located a t the in-breast focus, and the theoretical plane-wave reflection co­
efficient as a function of the dielectric perm ittivity contrast. The two curves are in
reasonable agreement, and, more importantly, dem onstrate th a t variations of the
perm ittivity contrast of up to ±70% of the value reported in the literature result
in a change of S / C no greater than ±2.5 dB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
3.2
Synthetic Aperture:
17-Element Antenna Array
As mentioned previously, the focusing feature of the elliptical reflector can be syn­
thesized by an electronically scanned impulse antenna array. The advantage of such
an array is its variable focus potentially controlled via software.
Figure 3.14: 2-D FD T D model of the 17-position coherent-addition antenna array,
showing a 1 —m m thick layer of skin, the heterogeneous breast tissue and a 5 m mdiam eter tum or located a t the synthetic focus 3 cm beneath the surface.
The geometry used for the synthetic aperture investigations is shown in Fig­
ure 3.14. As in the previously investigated cases, the modeled breast tissue region
consists of 5 —m m x 5 —m m blocks differing random ly in electrical properties (e, cr)
by ±10% about the m ean value. Additionally, a 1 —m m thick skin layer is included
in the geometry. The assumed parameters for the skin layer are £r_,*»„ = 36 and
crskin = 4 S / m . Seventeen elements of the antenna array, spaced by 5 m m , lie next
to the skin layer. A 5 — m m diam eter tum or is assum ed centered 3 cm beneath the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
skin-breast tissue interface.
2-D FD TD sim ulations are perform ed for four cases: 1) homogeneous breast
tissue with no tum or, 2) homogeneous breast tissue with tumor, 3) heterogeneous
breast tissue w ith no tum or, and 4) heterogeneous breast tissue w ith tum or. For
each case, a set o f 17 tim e responses is obtained (one for each antenna element).
The usage of each set of results in the data-processing algorithm is illustrated in
Figure 3.15. Here, the calculated responses of homogeneous breast tissues with and
w ithout tum or are used to estimate tim e delays between individual array element
responses. Then, the no-tumor response set is shifted and summed using the time
delays to yield th e exponential decay curve, which originates from th e tail of the
differentiated G aussian source function. Responses from each of the o th er two sets,
calculated for the heterogeneous tissue w ith and without tumor, are shifted and
summed using the same estim ated tim e delays to yield two signals: clu tter signal
and the tum or signal with clutter.
Finally, the calculated exponential decay is
subtracted from each of these signals, to result in the pure clutter signal and the
signal representing tum or response with clutter.
Overall, th e time-shifting and sum m ing of backscattered responses for the
17-position coherent-addition antenna array system of Figure 3.14 results in a set
of two FD TD -com puted time-domain waveforms: one with no tum or present and
one with the tum or embedded in the breast tissue. These curves are graphed in
Figure 3.16.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
tum or pi
no
determine
TIME DELAYS
shift and sum
using determined
TIME
DELAYS
result:
pure exponential decay
("tailing" from the
Differentiated Gaussian
source function)
shift and
shift and
TIME
DELAYS
TIME
DELAYS
result:
clutter signal
"riding" oa the
exponential decay
tumor+ciutier respoon
"riding" oa the
exponential decay
£ (-11
RESULT: CLUTTER signal
RESULT: TUMOR
+ cluttemsponse
Figure 3.15: Algorithm used for processing d a ta obtained for the 2-D FDTD model
of the 17-position coherent-addition antenna array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
/\
c
0.5
0.6
0.7
0.8
0.9
no tumor
— — tumor
u
1.0 1.1
time (ns)
1.2
1.3
1.4
1.5
Figure 3.16: FDTD-computed time-domain waveforms resulting from time-shifting
and summing backscattered responses for the 17-position antenna array system
shown in Figure 3.14.
O ur next goal is to use the same data used to obtain the curves in Figure 3.16
to form a 2-D gray-scale image. For a 10 x 17 m atrix of focal locations covering
the area formed by distance along the array and perpendicular to it, 5 cm deep
into the breast tissue, we determine S /C ratio for each point using the following
strategy. For each focal point, we determine the theoretical tim e delay for each
antenna element. The theoretical time delays for each focal point are calculated
using simplified geometrical optics principles and appropriate average propagation
velocities of the pulse within the skin V^,n and the breast tissue V^-ea*^
(3.3)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
V^reast =
V'^r—b r e a s t
(3.4)
where c is the speed of light in vacuum.
We next present analysis and calculation of the theoretical tim e delays.
skin
layer
breast tissue
tumor
antenna
array
elements
Figure 3.17: Geom etry showing the antenna array adjacent to the skin and the p a th
used for estim ating propagation tim e delays for a 5 —m m tum or embedded in th e
breast tissue 3 cm from the skin surface. Both breast and skin tissue are considered
to be homogeneous with £r- b r e a s t — 9 and £ r - s k i n = 36.
Figure 3.17 shows a simplified 2-D geometry of the 17-element microwave
antenna array system adjacent to th e breast skin, underlying breast tissue, and a
5 — m m spherical tum or in the breast tissue 3 cm below the skin surface on th e
central axis of the sensing antenna array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
skin
breast tissue
ds
Ls
tumor
db
Lb
Qb
antenna
Figure 3.18: Geom etry used for determining time delay for an off-axis antenna
element a distance away from the array axis. A 5 —m m diameter on-axis tum or is
assumed to be <4 deep into the breast tissue from the interface of tissue with skin
of thickness ds. The propagation p ath sketched in Figure 3.17 is here the sum of
distances marked as L s (the part of propagation path w ithin skin) and Lb (the p a rt of
propagation p ath w ithin the breast tissue). Refraction angles which the path forms
with respect to the axis within skin and breast tissue are 0, and Ob, respectively.
Vertical shift y which the path makes as it reaches the skin-tissue interface is also
used in the analysis. Relative perm ittivities of skin £>-»*»»» and breast £r-breaat are
related to their respective indices of refraction by n , = y/£r-akin and rib = y/£r-breoat-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
The geometry used for determining the tim e delay for an off-axis antenna
element located a distance a from the array axis is depicted in Figure 3.18. A
5 —m m diam eter on-axis tum or is assumed to be located a distance db in the breast
tissue beneath the skin - breast tissue interface. A skin of thickness ds is assumed.
The total length of the propagation path in Figure 3.18 is the sum of L s (the
p ath length within the skin) and Lb (the path w ithin the underlying breast tissue).
Refraction angles which the p ath forms with respect to the norm al axis within the
skin and breast tissues are 0S and 9b, respectively. T he vertical shift y which the
p ath makes as it reaches the skin - breast tissue interface is also used in the analysis.
The relative perm ittivities of skin, £r-skin, and breast, er-breast, are related to their
respective indices of refraction by n , = y/£r-skin and n* = y/e,—breastFor a particular point within the breast (in this analysis, the tum or location
shown in Figure 3.18 ), the distances needed for the tim e-delay calculation are L„, Lb
and y. These are found as follows. By Snell’s law,
n asinda = UbsinOb
(3.5)
From Figure 3.18, we have the following trigonometric relations:
sinOa =
La
(3.6)
and
sin6b =
Lb
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.7)
41
Equations (3.6) and (3.7) are combined with (3.5) to yield
„
y
72*
Ls
— 7lft
a~y
Lb
fo a \
(3.8)
From Figure 3.18 it can be seen th at
(a - y)2 + dft = L \
(3.9)
and
y 2 +<Ps = L 2,
(3.10)
Equations (3.8), (3.9), (3.10) constitute a system of three equations with
three unknowns: L „, Lb and y. Solving for y, we obtain the fourth-order polynomial
y4[nl - n2] + y3[2a(n2 - ng)] +
y2[nl(a2 + d2) - n 2(a2 + <Pb)\ + y [ - 2 a n ^ ] + a 2n ^ = 0
(3.11)
W hen solved for y, (3.11) results in a solution which, when substituted in Equations
(3.9) and (3.10), yield Lb and L s. Finally, the tim e delay t^ a y between th e antenna
emission and the reception of the backscattered response from a particular in-tissue
location is
=
2
=
2 x 1 -A £ = + ~ ^ = ]
\E r —skin
\J^r—breast
x
[ A +
vskin
*breast
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.12)
42
In the m anner of Figure 3.15, we now use these tim e delays to temporally
shift and sum the backscatter responses obtained for the individual elements of the
antenna array. Then, within a time-window of interest, we tim e-integrate the square
of this summed response. For each potential backscattering point tested, we assign
a gray-scale value to the integrated (energy) response, allowing the formation of an
image as the assumed backscattering point is scanned through the breast. A sample
of such an image is shown in Figure 3.19. As m entioned in the background section of
this work, improved shifting-and-summing algorithm s have recently been reported,
showing gray-scale images of tumors embedded in M Rl-scan-based geometries [2].
Figure 3.19: Sample 2-D FDTD calculated gray-scale of the tum or of Figure 3.14.
Bright pixels occur a t the test backscattering points where the temporally shifted
backscatter responses of the individual elements of th e antenna array add coherently
to yield a relatively large energy value upon tim e integration. Dark pixels denote
test backscatter points where the temporally shifted responses add incoherently and
yield relatively small energy values.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h ap ter 4
S in gle-P aram eter R eco n stru ctio n
o f D ielectric P ro p erties o f
N ear-Surface B rea st T issu es
4.1
Rationale and Motivation
The previous chapter described how th e signals received by each antenna element
of the synthetic-aperture array are shifted in tim e for coherent summation. We now
show th a t a wrong assumption of er-akin>£r-breast or skin thickness can lead to sig­
nificant errors in calculating propagation tim e delays and, consequently, incoherent
sum m ation of the individual received signals.
Figure 4.1 graphs calculated tim e delays for the extreme values of the reported
skin thickness range. We note that th e range of variation (i.e., uncertainty) in these
delays is com parable to the difference in tim e delays required for coherent addition
for two adjacent antenna elements for each thickness value considered.
A sim ilar variation is obtained if, for a fixed skin thickness of 2.7 m m , we
calculate tim e delays for several values of £r-skin, as shown in Figure 4.2. Due to
the relatively thin skin layer, the effect of the variation in er-akin on the variation in
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
Time delays for the on-axis location
3 cm within the breast tissue
1.0
O— O Skin thickness 0.5mm
□— Q Skin thickness 2.7mm
S. 0.9
p 0.8
jct
0.7
0.6
10
Antenna element # from the central axis
Figure 4.1: Effect of skin thickness on propagation tim e delays for each antenna
element of Figure 3.17, assum ing £r-akm = 36. Reported range of normal skin
thickness is 0.5 —2.7 m m .
the calculated time delays is not pronounced. However, when a similar analysis is
done varying er- b r e a s t of the thicker breast tissue layer by 50% around its nominal
value, the results graphed in Figure 4.3 suggest that th is variation could have a
significant im pact on determ ining the accurate time delays needed for the shiftingand-sum m ing signal-processing algorithm .
Overall, the results presented in Figures 4.1, 4.2 and
4.3 imply th at we
require a patient-specific calibration of the confocal microwave imager for nearsurface breast tissue dielectric param eters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
1.1
Time delays for the on-axis location
3 cm within the breast tissue
1.0
° ------
•
S. 0.9
0 *HWS «
•
= 36
*-------A Er-« d r = S 4
j= 0.8
0.7
0.6
skin thickness > 2.7mm
0
1
4
7
2
3
5
6
8
Antenna element # from the central axis
9
10
Figure 4.2: Propagation tim e delays for each antenna element calculated for three
different values of relative perm ittivity of skin er-s*m '• 18,36 and 54, assuming skin
thickness of 2.7 m m and relative perm ittivity of breast tissue £r-breast = 9-
1.3
Time delays for the on-axis location
3 cm within the breast tissue
= 4.5
=9
= 13.5
1.0
S. 0.9
0.8
isr —
0.7
0.6
0.5
sldn thickness = 15mm
0.4
10
Antenna element # from the central axis
Figure 4.3: Propagation tim e delays for each antenna element calculated for three
different values of relative perm ittivity of breast tissue £ r - b r e a s t ■ 4.5,9 and 13.5,
assum ing skin thickness of 2.7mm and £ r - s k i n = 36 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
4.2
Basic Technique
This work focuses on the development of a time-domain inverse-scattering algorithm
th at perm its a noninvasive measurement of the near-surface dielectric param eters of
a lossy layered half-space. The new algorithm extends the iterative technique of [48]
to the case of a 2-D half-space excited at its surface by an infinitely long monopole.
The principal logic of this scheme is shown in Figure 4.4.
Nonlinear
optimization routine:
intelligently perturbs
Error
criteria
Guess of
FDTD code computes
trial scattered pulse:
direct (forward)
analysis
For small error:
reconstructed a , e
Comparison of
trial pulse with
simulated
measurement
Input data: measured
(simulated) scattered pulse
Figure 4.4: Time-domain inverse-scattering iterative algorithm for recovery of di­
electric param eters ey and a from a measured (in this study, sim ulated measured)
signal. The scheme exploits causality to lim it the region of inversion.
Based on an initial guess for a set of dielectric parameters ( ey, a ), the FDTD
code computes a trial backscattered time waveform. Figure 4.5 depicts a simulated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
0.01
0.00
“
-
0.01
-
0.02
-0.03
0.0
Reference values:
S/m. e, lMn=36
Permittivity perturbation: e .mhj 72
Conductivity perturbation: 0 ^ = 8 S/m
0.1
0.2
0.3
time (ns)
0.4
0.5
Figure 4.5: Backscattered signals for different values of e T - a k i n and cra/cin.
noiseless measured signal (obtained for the reference values of
e r -s k in
and
),
and two trial signals - one calculated with a perturbed value ( “initial guess” ) of
£r-akin, and the other w ith a perturbed value of <x4*tn. The algorithm compares a
trial signal with the m easured signal and calculates the energy-normed error. Based
on this error, the dielectric parameters are perturbed by a gradient-based nonlinear
optim ization routine, and the FDTD code com putes a new trial backscattered signal
based on this new guess for (er , <r). This process is repeated w ith varying ey and a
perturbations as needed.
There are two ways to exit the iterative loop of Figure 4.4. For exit 1, the
energy-normed error decreases below a pre-determ ined threshold value. For exit
2, the error remains above the threshold but th e estim ated sT an d a parameters
oscillate within a narrow band of values a pre-determined num ber of times. Both of
these exit criteria are chosen according to physical meaningfulness an d to limitations
imposed by available com putational resources.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
4.3
Results in the Absence of Noise
Here, we illustrate the convergence properties of the tim e-dom ain inverse-scattering
algorithm of Figure 4.4 w ith only one unknown param eter, e r-akin or crakin. For
these investigations, the electrical param eters for th e underlying breast tissue are
assumed to be £ r -breoat = 9 and abreast = 0-4 5 /m , the nom inal values given in the
literature. T he 120-ps differentiated Gaussian pulse of Figure 3.12 is used as the
excitation waveform.
In Case 1 (Figure 4.6), we fix the value of
of 1 m m , while iterating for
£ r - ak i n
<Js k i n
= 4 and the skin thickness
■ Here, the FD T D code computes the backscat­
tered signal for a trial value of £ r - s J n n , compares it w ith the reference backscattered
signal obtained for the correct value of
error. Based on this error,
£ r -s k in
£ r - ak i n ,
and calculates the energy-normed
is perturbed, and th e FD TD code computes the
backscattered signal for this new guess for the perm ittivity. This is repeated until
one of th e criteria for exiting the iterative process is m et. Sample results for two
initial guesses of £r- akin are shown in Figure 4.6.
In Case 2 (Figure 4.7), we fix the value of er -**»n = 36 and the skin thickness
of 1 m m , while iterating for <xs*,n- Sample results are shown for two initial guesses
for crskin. In these and sim ilar simulations, we have found a robust convergence for
£r—akin an d <Jakin in the absence of noise.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
150
O— O Initial e, m =99: initial e ....-d e c re m e n ts
■ — ■ Initial e, ..= 1 5 0 : Initial e, ..-d e c re m e n ts
100
i
u>
20
25
Number of iterations
Figure 4.6: Inverse-scattering FDTD com putation: Convergence of £r-skm to its
correct value (36) for two different initial guesses, with assumed skin thickness of 1
m m and crafct„ = 4 S / m .
20
>
*----- * Initial 0 ^ = 2 0 S/m; Initial o.„,-decrem ent*3 S/m
o - — o initial 0 ^ = 1 9 S/m; initial o^.-decrem ent»1 S/m
15
15
20
25
40
Number of iterations
Figure 4.7: Inverse-scattering FDTD com putation Convergence of a skin to its correct
value (4 S /m )) for two different initial guesses, with assumed skin thickness of 1
m m and £r- akin = 36 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h ap ter 5
T w o-P aram eter R eco n stru ctio n o f
D ielectric P ro p erties o f
N ear-S u rface B reast T issu es
5.1
5.1.1
Methods
Basic Technique
This section describes an optimization scheme which allows simultaneous recovery
of the perm ittivity and conductivity of near-surface breast tissues.
We apply a
gradient-based vector m ethod to generate a trajectory in the (er , a ) space th a t
converges to the reference values of these param eters.
Figure 5.1 illustrates this technique applied to the breast skin. From the
initial guess, designated G U E S S ( 1,1), we make two further guesses in the direction
of the <Jakin and er-jjfc»n coordinates, respectively: G U E S S ( 1,2) and G U E S S ( 1,3).
C alculating the errors differences A e rro r(l,2 ) and A e rro r(l,3 ), yields an errorgradient vector to bring us to G U E S S {2,1).
This procedure is repeated as we
follow the gradient trajectory in the (er, <
j ) space to the final point of correct values
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
o f S r - a k i n M i d O’afcin-
40
GRADIENT TRAJECTORY
leading to the set of
correct parameter values
35
GUESS(2f1)
£30
-£
w
GRADIENT
VECTOR
GUESS(1,3)
A error(1,3)
GUESS(1^2)
25
j
GUESS(1,1)
A error(1£)
20
4
5
6
7
8
10
(S/m)
Figure 5.1: T rajectory of the gradient m ethod leading to approxim ate values ofa,*,,,
a n d S r - a k in -
In the next section, we introduce the three excitation waveforms used in our
investigations. These studies will show th at th e search trajectory in the (sr , <
j ) space
depends strongly on the signal shape and duration of the excitation waveform.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
5.1.2
E xcitation Waveforms and O bservation W indows
In this section, the three E*-field excitation waveforms used for the dielectric pa­
ram eter reconstruction are visualized: (a) 120 — ps differentiated Gaussian pulse
(Figure 5.2); (b) 10 —ps differentiated Gaussian pulse w ith a 5 —ps rise-time (Fig­
ure 5.3); and (c) 5 —ps rise tim e ramp which has the same maximum as the peak
value of the 10 —ps differentiated Gaussian pulse of Figure 5.3 (Figure 5.4).
8e-03
4e-03
£
•%
"E
0e+00
>
U1H
5
I
-4e-03
£
100
300
200
Time (ps)
400
500
Figure 5.2: 120 —ps differentiated Gaussian pulse (full w idth between 1/e points ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
0.10
-5 pa
|
1
0.05
£
«
£
S
f
0.00
>
ur
5
-0.05
i
sJ
U
-
0.10
20
30
40
Time (ps)
Figure 5.3: 10 —ps differentiated Gaussian pulse with 5 —ps rise time.
0.10
0.08
0.06
The signal reaches the sam e
amplitude as the 5 -p s risetime
differentiated Gaussian pulse
used in this study
0.04
3
0.02
0.00
20
Time (ps)
Figure 5.4: 5 —ps rise tim e ramp signal. The maximum of the signal is deter­
mined by maximum of the 10 —ps differentiated Gaussian pulse of Figure 5.3 with
approximately the same rise time.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
It is im portant to note the following. The inverse-scattering algorithm pro­
cesses only a tim e window Twindow within which it com pares the trial and measured
signals. For breast-skin param eter recovery, it is critical th a t Twindow be sm all enough
to avoid wave reflection from the underlying skin-breast interface which corrupts
the received signal. However, Twindow must be large enough to allow the measured
backscattered pulse to contain meaningful information concerning the skin electrical
param eters. This, in turn, means th a t only a part of the measured backscattered
waveform m ight be usable for accurately generating a trajectory in the (£>-s*in,
(Jskin) space. For example, in the next section, we will see th at, for the 120 —ps
differentiated Gaussian pulse excitation case, only the leading edge of the measured
backscattered pulse is used in the reconstruction. Conversely, the entire measured
backscattered pulse is used for the 10 — ps differentiated Gaussian pulse excita­
tion case. Results will illustrate the significant consequences th a t this has on the
calculated trajectory in the (£r-skin, &skin) space.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
5.2
Results in the Absence of Noise
5.2.1
S r s k in
R econstruction w ith Trial Values o f
crs k in
This section reports results for the inverse-scattering algorithm iterating for er -«*m
using a range of trial values of <T4^ n varying ±50% around the reference value of
a akin = 4 S /m . In all simulations, the tim e step used in the FDTD code is A t =
0.33356 ps.
Figure 5.5(a) depicts the resulting trajectory in the (er-»«nj <7»fcm) space for
the 120 —ps differentiated Gaussian pulse excitation and Twindow = 400A t. This
trajectory passes through the reference value £r-akin = 36. Figure 5.5(a) also shows
the large error made in the er-skin recovery if Twindow = 600At, which is large enough
to include the unwanted reflection from the underlying skin - breast tissue interface.
Figure 5.5(b) graphs the error for Twindow for each value in the trial cr,*,n range for
its corresponding estim ated er-akin of Figure 5.5(a). Results are presented similarly
for the 5 —ps ram p excitation in Figure 5.6 and for TWindow = 100A t. Figures
5.5(b) and 5.6(b) show error function for er- akin recovery using trial values of askin
suffers from local m inim a, and thus cannot be used as a search criterion along the
trajectory in the (£>_,*,„, crSkin) space.
Figure 5.7(a) shows results for the 10 —ps differentiated Gaussian pulse ex­
citation where TWindow = 80At. Although the er-»kin estim ate remains th e same for
all trial a akin values, Figure 5.7(b) shows th a t the corresponding error function can
be used as a search criterion, since it exhibits a clear minimum for the reference
values of £r- akin and crsfcl„.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
120-ps Differentiated Gaussian Pulse
40
• — *400 timesteps
□ .... □ 600 timesteps
values: £,.^,*36.
39
,*4S/m
38
137
i
oT
■g 36
«3
I
35
34
d
- d -D- o - D - a - D - o - o - a - b - q
D -o -b -a -o -a -i
33
32
Trial
(S/m ): 2.0, 2.2.......6.0
(a)
120-ps Differentiated G aussian Pulse
1e-06
8 e -0 7
6 e -0 7
§J
U
4 e -07
2 e -07
0e+00
2.0
3.0
4.0
5.0
Trial o tkin (S/m): 2.0, 2.2.......6.0
6.0
(b )
Figure 5.5: E stim ate of er- akin for trial values of <x,*,n for the 120 —ps differentiated
Gaussian pulse excitation case, (a) Search trajectory in (er-«*m> &skin) space for
TWindow = 400At and TWindow — 600At, where A t = 0.33356 ps. (b) Error vs. trial
values of Gakin for Twindow ~~ 400A(.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
5 -p s Risetime Ramp Signal
41
40
39
38
1 37
cuw
Ia 36
1a> 35
111 34
33
-3|f-— ^Reference values:
Er
ikjn=36t olllin=4S/m
32
31
1
2
3
4
5
Trial a * * (S/m): 2.0, 2.2........6.0
6
7
(a)
5 -p s Risetime Ramp Signal
1e-07
8e-08
6e-08
4e-08
cv
20-08 - A r
0e+00 — ■
—
Trial a ^ (S/m): 2.0, 2.2........6.0
(b)
Figure 5.6: Estim ate of £r-skin for trial values of crskin for the 5 — ps rise-time ram p
excitation case, (a) Search trajectory in (ey-s/Wn, crskin) space for Twindow = 100At,
where A t = 0.33356 ps. (b) Error vs. trial values of crg/tm-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
10-ps Differentiated Gaussian Pulse
41
40
39
38
? 37
c*sT
"8 36
a
E 35
M
Cp
Ap
111 34
33
■^Reference values:
Er
5lun=36, o ^ M S /m
32
31
1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4
Trial
(S /m ): 2.0, 2.2.,..., 6.0
(a)
10-p s Differentiated Gaussian Pulse
2e-08
E
1e-08
Oe+OO
Trial
(S /m ): 2.0, 2.2,..., 6.0
(b)
Figure 5.7: E stim ate of £r-sfctn for trial values of crskin for the 10 —p s differentiated
Gaussian pulse excitation case, (a) Search trajectory in (er-«fctn, &skin) space for
TWindoui = SOA t, where A t = 0.33356 ps. (b) Error vs. trial values of cr3kin.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
5.2.2
crs k in
R econstruction w ith Trial V alues o f
£ r -s k in
This section reports results for the inverse-scattering algorithm iterating for (rakin
using a range of trial values of £r-skin varying ±50% around the reference value
of Er—stin = 3 6 S /m . In all simulations, the time step used in the FDTD code is
A t = 0.33356 ps.
Results for the excitation signal waveforms under investigation are shown in
Figures 5.8,
5.9, and 5.10. For the 120 —ps differentiated Gaussian pulse and
the 5 —p s rise-time ramp excitations, Tendon; includes only the leading edge of the
backscattered pulse waveform. This yields in Figures 5.8(a) and 5.9(a) linear search
trajectories in the (£r-*/km> &3 kin) space. For both of these excitations, Figures 5.8(b)
and 5.9(b) show a broad null of the error as a function of the estim ated trgkin value.
This null is approximately centered at the location of th e assum ed reference values,
implying th a t minimizing the error could be used as a search criterion along the
trajectory in the (£r- akin><*skin) space. However, the broad null suggests sensitivity
to param eter recovery in the presence of noise.
Figure 5.10(a) graphs the search trajectory in the (£>-**«»», &skin) space ob­
tained for th e 10—ps differentiated Gaussian pulse excitation. Here, Twindow includes
the entire backscattered pulse waveform, and the search trajecto ry shows a parabolic
behavior. We note a sharp null of the error at the location of the assumed reference
values of the skin parameters, which suggests robustness of param eter recovery in
the presence of noise.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
120-ps Differentiated Gaussian Pulse
26
22
■^Reference values: er_-un=36, allljn=4S/m
E
<0
Ie 14
T3
®
(0
10
E
2
6
-2
16
20
24
28
32
36
40
44
Trial er^Wn: 18, 20....... 54
48
52
56
(a)
120-ps Differentiated G aussian Pulse
1e-04
8e-05
1i11
4e-05
0e+00
Estimated
(S/m)
(b)
Figure 5.8: E stim ate of <Jak i n for trial values of e r - s k i n for the 120 —ps differenti­
ated Gaussian pulse excitation, (a) Search trajecto ry in (£r-skin, 0akin) space for
Twindow = 400Af, where A t = 0.33356 ps . (b) E rror vs. estim ated crskin values,
showing a broad null a t the location of the assum ed reference values of the skin
parameters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
5-ps Risetime Ramp Signal
+
16
20
♦-.... ♦ Reference values: er «tl=36, oM =4 S/m
24
28
Trial
32
36
40
44
= 18, 2 0 ........54
48
52
56
(a)
5-ps Risetime Ramp Signal
8 0 -0 4
Ar
6 0 -0 4
§
4 0 -0 4
2 0 -0 4 0e+00
-6
-2
14
Estimated a * * (S/m)
(b)
Figure 5.9: E stim ate of <ra*tn for trial values of e r - s k i n for the 5 —p s rise-tim e ramp
excitation, (a) Search trajectory in (er_a*<„, <rakin) space for Twindow = lOOAt, where
A t = 0.33356 p s . (b) E rror vs. estimated <74*,n values, showing a broad null at the
location of the assumed reference values of the skin parameters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
10-ps Differentiated Gaussian Pulse
50
Ap
•^Reference values: e, ^,=36: <1,^=4 S/m
40
a 20
20
24
28
32
44
Trial 6 ^ = 18,20...., 54
48
56
(a)
10- ps Differentiated Gaussian Pulse
0.06
0.04
2
LLl
0.02
0.00
Estimated
(S/m)
(b)
Figure 5.10: Estim ate of <rsjb,n for trial values of er-j*m for th e 10 —ps differenti­
ated Gaussian pulse excitation, (a) Search trajectory in (£>_«*,n» &skin) space for
values,
T 'w in d o w = 80At, where A t = 0.33356 ps . (b) E rror vs. estim ated
showing a sharp null a t the location of the assumed reference values of the skin
param eters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
5.3
Results in the Presence of Zero-Mean Gaus­
sian Noise
5.3.1
Generation o f Sim ulated N oise
We now test the robustness of the inverse-scattering scheme of the previous section
by adding zero-mean Gaussian noise to th e sim ulated measured backscattered sig­
nals. The signal-to-noise (S / N ) ratio is based on the peak value of th e noiseless
backscattered waveform S ignal m a x - A fraction of this value, defined by a desired
S / N decibel {dB) level, is used as the standard deviation N oises d of th e zero-mean
Gaussian noise. For example, for X d B level,
20
at
N m seSo =
S ig n a lM
mx/a0A
This defines the value for N o ises d to be fed to the
The
{dB)
= *
X
(51>
n \
(5.2)
noise-generating subroutine.
subroutine used in this study which generates Gaussian-distributed random
numbers was obtained from the public-domain National Institute of Standards and
Technology (NIST) website (http://gam s.nist.gov/serve.sgi/).
In this section, for each simulated backscattered waveform, we ad d noise to
obtain S / N levels of 20, 30, 35 and 40 d B . Sample noisy backscattered waveforms
for these S /N values are shown in Figure 5.11 for the case of the 10—ps differentiated
Gaussian pulse excitation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
10-ps Differentiated Gaussian Pulse with Zero-Mean Gaussian Noise
0.004
0.000
S/N = 20 dB
-0.004
-0.008
§
ca
10
20
30
40
50
60
70
80
90
100
0.004
0.000
§
§
£
-0.004
<5
-0.008
*
S
I
S/N = 30 dB
10
20
30
40
T
0.004
50
70
80
90
100
T
0.000
>.
X
73
O
S/N = 35 dB
-0.004
-0.008
20
30
40
20
30
40
50
60
70
80
90
100
50
60
70
80
90
100
o>
0.004
0.000
-0.004
-0.008
0
10
Time (ps)
Figure 5.11: Sample noisy backscattered waveforms for the 10 —ps differentiated
Gaussian pulse excitation. Zero-mean Gaussian noise is added to yield four values
of signal-to-noise ( S / N ) ratio: 20,30,35 and 40 dB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
5.3.2
120 — p s Differentiated G aussian P u lse and 5 — p s R iseT im e R am p Excitations: <Jskin R econstruction w ith
Trial V alues o f er- 8kin in th e Presence of N oise
Figure
5.12 graphs sample trajectories trajectories in the (er
crakin) space
generated by the inverse-scattering algorithm for the 120—p s differentiated Gaussian
pulse excitation case. Here, a single noise backscattered waveform is used for each
S / N ratio tested. We note, that even in the presence of noise, the trajectories in
the (£r-skini ^skin) space still retain their linearity.
Note th a t m ultiple noisy backscattered waveforms can be generated by adding
different strings o f random numbers obtained from the NIST subroutine to the
calculated backscattered waveform. This procedure would lead to m ultiple sample
trajectories in th e (£r-skin, crskin) space. T his will be dem onstrated in the next
section, in the context of the 10 —ps differentiated Gaussian pulse excitation case.
Figures 5.13(a), 5.13(b), 5.13(c) and 5.13(d) show the error as a function
of the estim ated <Jakin for S / N ratios of 20,30,35 and 40 d B , respectively for the
120 —ps differentiated Gaussian pulse excitation. For S / N = 20 d B , the absence
of an error m inim um implies th at the inverse-scattering algorithm cannot rely on
the error as search criterion along the generated trajectory in (er-akin, &skin) space.
Even for higher S / N ratios, the error function shows broad minima, im plying a lack
of robustness in th e presence of noise.
Figures 5.14 and 5.15 show results analogous to these of Figures 5.12 and
5.13, but for the case of the 5 — ps rise-tim e ram p excitation in the presence of
noise. Again, the sample trajectories in the (er -«*tn, &skin) space are linear. Also
again, reducing th e S / N ratio to 20 d B elim inates the error m i n i m u m , while the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
error function exhibits only a broad minimum for higher S / N ratios.
120-ps Differentiated Gaussian Puise with Zero-Mean Gaussian Noise
t
1
i
1
i
1
i
■
i
1
i
1
r
• --- • S/N = 20 dB
o □ S/N = 30 dB
A - -A S/N = 35 dB
X- —X S/N = 40 dB
3|(fteference values: er_skin=36, a ^ ^ S / m
2
________ ■_________i________ i________ i________ >
16
20
24
T r ia l
i________ i________ i_________i________ i________
28
32
36
e r-skin = 18, 20...... 54
i_________■_________i—
40
44
Figure 5.12: Sample trajectories in (er- a**», <J3k i n ) space for th e 120 —p s differen­
tiated Gaussian pulse for signal-to-noise S / N = 20,30,35 and 40 dB . Twindow =
400At, where A t = 0.33356 ps.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
120-ps Differentiated Gaussian
120-ps Differentiated Gaussian
0.1466 r
S/N = 20 dB
0.1465
0.01674
0.1464
§J
U
§
0.01672
UJ
0.1463
0.01670
CJ-q-O
0.1462
o S/N = 30 dB
0.01666
0.1461
Estimated
Estimated cr^ (S/m)
(S/m)
(b)
(a)
120-ps Differentiated Gaussian
120-ps Differentiated Gaussian
0.00160
A - -
a
S/N = 35 dB
0.00155
x
x S/N = 40 dB
0.00150
0.00610
fc
0.00145
0.00140
0.00606
0.00135
0.00600
0
4
8
Estimated
(c)
12
(S/m)
0.00130
Estimated
(S/m)
(d)
Figure 5.13: Sample variations of error vs. estim ated o skin for the 120 —p s differen­
tiated Gaussian pulse excitation for S / N ratios of (a) 20 d B , (b) 30 dB , (c) 35 d B
and (d) 40 dB , for TWin<iow = 400At.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
5 -p s Risetime Ramp Signal with Zero-Mean Gaussian Noise
O
O S/N = 20 dB
S/N = 30 dB
▲ - -A S/N = 35 dB
X- - * S/N = 40 dB
■^Reference values: er_stdn=36, 0,^=4S/m
24
28
32
36
Trial
= 18, 20, .... 54
Figure 5.14: Sample trajectories in (er-skin, <Takin) space for the 5 —p s rise-time ramp
excitation for signal-to-noise ratios of S / N = 20,30,35 and 40 dB . T window = 400A£,
where A t = 0.33356 ps.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
5-ps Risetime Ramp Signal
5 -p s Risetime Ramp Signal
0.0700
0.0086
0.0695
■ S/N = 30 dB
S/N = 20 dB
0.0690
2
§
0.0685
UJ
UJ
0.0680
0.0075
0.0675
0.0670
0.0070
12
Estimated a ^ , (S/m)
Estimated
(a)
(S/m)
(b)
5-ps Risetime Ramp Signal
5 -p s Risetime Ramp Signal
0.0030
0.0014
*
-
x
-A S/N = 35 dB
K S/N = 40 dB
0.0028
0.0012
0.0026
I
I
UJ
0.0010
UJ
0.0024
0.0006
0.0022
0.0020
12
Estimated
(c)
(S/m)
0.0006
Estimated
(S/m)
(d)
Figure 5.15: Sample variations of error vs. estim ated cr3kin for the 5 —ps rise-time
ram p excitation for S / N ratio of (a) 20 dB, (b) 30 d B , (c) 35 dB and (d) 40 dB ,
for TWindow = 400A*.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
5.3.3
10 — p s Differentiated G aussian P u lse Excitation: (rskin
R econstruction w ith TWal Values o f e r - 8kin in the P res­
ence o f N oise
This section presents results for noisy backscattered signals resulting from the 10—ps
differentiated Gaussian pulse excitation. In contrast with the previous section, here
individual trajectories in the {er-skin, o’akin) space are generated for five independent
noise samples added to the backscattered signal for S / N ratios of 20, 30, 35 and
40 dB. Then, for each S / N level, the five d a ta sets are averaged before input to
the inverse-scattering algorithm to yield a final trajectory in the (£r-akin, Oakin)
space. The la tte r procedure simulates the “boxcar integration” often used as a
technique for m easuring signals contam inated with zero-mean noise wherein m ultiple
measurements averaged to enhanced the determ inistic signal. For all results shown,
TWindow = 80 A t, where A t = 0.33356 ps.
Figures 5.16(a), 5.17(a), 5.18(a), and 5.19(a) show the generated trajec­
tories in the (er _a*tn, oSkin) space for each noisy and the averaged backscattered
waveform for S / N = 20, 30, 35 and 40 d B , respectively. As expected, for higher
S / N ratios we observe a smaller deviation of the individual trajectories from the
trajectory obtained using the averaged backscattered signal.
More im portantly, however, Figures 5.16(b), 5.17(b), 5.18(b), and 5.19(b)
show th at the corresponding error as a function of the estim ated <xa*,n exhibits a
sharp minimum even for low values of S / N .
These results mean that using the
10 —ps differentiated Gaussian pulse excitation waveform and fully windowing its
backscattered response leads to a robust inverse-scattering scheme for recovering
£r—akin and <Tafctn, even in the presence of noise.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
S/N = 20 dB
10-ps Differentiated Gaussian Pulse with Zero-Mean Gaussian Noise
from averaged signals from
16
20
24
28
32
36
40
Trial
= 18, 20.......54
48
52
56
(a)
S/N = 20 dB
10-ps Differentiated Gaussian with Zero-Mean Gaussian Noise
3 o.OO ■‘ ‘ *
5
-5
0
111
i,
5
i . . . . i . . . . ............. .
... i .... i ...»
10
15
20
25
30
35
40
45
Estimated o ^ (S/m)
(b)
Figure 5.16: (a) Sample trajectories in the ( e y _ s p a c e for the 10 —ps
differentiated Gaussian pulse excitation with S / N = 20 dB . (b) E rro r vs. estimated
0akin values for the averaged-backscattered-waveform case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
S/N =30 dB
10-ps Differentiated Gaussian Pulse with Zero-Mean Gaussian Noise
60
50
J
c/5
40
1
o
3
41
E
e
Ul
30
\
Set 1
Set 2
Set 3
Set 4
S ets
J Estimate from averaged signals from sett 1-5
jfrrieterencs values: e, „,=36. eMs4S/m
20
10
0
-1 0
16
20
24
28
32
Trial er ^
36
40
44
= 18, 20.......54
48
52
56
(a)
S/N = 30 dB
10-ps Differentiated Gaussian with Zero-Mean Gaussian Noise
|
0.04
2
0.02
*
0.00
111
15
20
Estimated
25
30
(S/m)
(b)
Figure 5.17: (a) Sam ple trajectories in the (er- akin, &skin) space for the 10 — ps
differentiated Gaussian pulse excitation with S / N = 30 dB . (b) E rror vs. estimated
(Jtkin values for the averaged-backscattered-waveform case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
S/N = 35 dB
10-ps Differentiated Gaussian Pulse with Zero-Mean Gaussian Noise
60
Set2
Set 3
Set 4
S ets
03
40
averaged signals from st
jes: e, „=s36. a ^ 4 S / m
1-5
.§ 20
to
0
16
20
24
28
32
36
40
44
Trial e,_mn = 18, 20 ,.... 54
48
52
56
(a)
S/N = 35 dB
10-ps Differentiated Gaussian with Zero-Mean Gaussian Noise
|
111
0
5
10
15
20
Estimated
25
30
(S/m)
35
40
45
(b)
Figure 5.18: (a) Sample trajectories in the (er-stt»> &akin) space for the 10 —ps
differentiated Gaussian pulse excitation w ith S / N = 35 dB . (b) E rror vs. estim ated
crskin values for the averaged-backscattered-waveform case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
S/N = 40 dB
10-ps Differentiated Gaussian Pulse with Zero-Mean Gaussian Noise
60 --1---1---1--- ■
----1----1r
— Set 1
Set 2
- - Set 3
— Set 4
- - Set 5
CEstimate from averaged signals from cats 1-5
)j(neteisnce values: e, ,.,= 36. o^sSSAn
I 40
e
I
CO
J
2
0
co
Ui
16
20
24
28
32
36
40
44
Trial Er-edn = 18, 2 0 ,..., 54
48
52
56
(a)
S/N = 40 dB
10-ps Differentiated Gaussian with Zero-Mean Gaussian Noise
73
&
«5
i
«
0.06
0.04
0.02
£
0.00
UJ
10
Estimated
25
30
(S/m)
35
40
45
(b )
Figure 5.19: (a) Sample trajectories in the (er<*skin) space for the 10 — ps
differentiated Gaussian pulse excitation with S / N = 40 d B . (b) Error vs. estim ated
(Tskin values for the averaged-backscattered-waveform case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
5.4
Discussion
5.4.1
Im pact o f the E xcitation Signal Shape and D uration
U p on th e R obustness o f th e Inverse-Scattering A lgo­
rithm
The results of the previous sections show th a t the robustness of the time-domain
inverse-scattering algorithm for recovery of e r - a k it i and cr3kin depends on the duration
and shape of the excitation waveform. Specifically, of the excitations considered,
the 10 —ps differentiated Gaussian pulse provides superior robustness, even rel­
ative to the ram p signal having the sam e rise-time, if Tuindow includes the entire
backscattered waveform. Apparently, a bipolar excitation delivers more information
to the inverse-scattering algorithm about the signal-path propagation medium than
a comparably fast unipolar excitation. The sharp error minimum consistently ob­
tained along the generated trajectory in the (er-»*in> (Jskin) space for the 10 —ps
differentiated Gaussian pulse excitation, even in the presence of significant levels of
zero-mean Gaussian noise, cleariy illustrates this point. Signal-to-noise ratios as low
as 20 d B are adequate for reliable electrical param eter recovery using this excitation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
5.4.2
D eterm ining Skin Thickness
We now briefly describe how we can estimate the breast skin thickness once the
patient-specific values for £>_,*,»» and Gskin are determined. In contrast with deter­
mining e T- s k i n and cr5fctn, the excitation signal shape and duration is now less critical
for obtaining an estimate for skin thickness.
Due to the dielectric contrast between the skin and the underlying breast tis­
sue, the propagating pulse reflects off the skin - breast tissue interface. By comparing
the measured response for the finite skin-thickness case w ith a simulation which as­
sumes a homogeneous skin half-space, the observed tim e of the first reflection yields
information for the skin thickness.
To illustrate this strategy, Figure Fig. 5.20 graphs the results of four test
cases for an assumed 60 —ps differentiated Gaussian pulse excitation wherein mea­
sured backscattered waveforms for skin thicknesses of 0.6, 1.2, 1.8 and 2.4 m m
are simulated using 2-D FDTD models. These waveforms are subtracted from the
backscattered response assuming a skin half-space. Then, the estim ated time de­
lay of the peak value of the difference signal with respect to the peak of the skin
half-space response is used to estim ate the skin thickness. T he resulting breast skin
thickness values obtained in this m anner are 0.51,1.25,1.85 and 2.45 mm, respec­
tively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
All-skin response
Difference: (0.6mm-skin)-(all-skin) response
Difference: (1.2mm-skin)-(all-skin) response
Difference: (2.4mm-skin)-(all-skin) response
(0
c
c
®
c
0.00
CO
£a>
X
-
0.01
-
0.02
>k
£
(D
C
iQps
8.
E
5
74ps
-0.03
.2
98ps
o
®
c
O)
a
0
50
100
150
Time (ps)
200
250
300
Figure 5.20: Waveforms illustrating a strategy for estim ating th e skin thickness
making use of the previously determined values of £r -akin and <yskin ■ We compare
the simulated m easured response (magnetic field component at th e location of the
antenna) for four finite skin-thickness cases w ith a simulation which assumes a ho­
mogeneous skin half-space.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h ap ter 6
C on clu sion s and F u tu re W ork
This thesis has investigated a novel tim e-dom ain inverse-scattering technique
th a t has promise for perm itting calibration of the pulsed confocal microwave imager
for hum an breast cancers. T he new technique is based upon the use of an FD TD
forward-scattering program element embedded within a numerical feedback loop
containing a nonlinear optim ization routine. A gradient-based strategy is adopted
to perm it efficient searching of a m ultidimensional param eter space for the electrical
properties of the breast skin and the underlying breast tissue.
A significant finding of this work is th a t the use of a short bipolar excitation
signal a t the surface of the breast skin is more robust than a unipolar ram p having a
com parable rise-time. The bipolar excitation, for example a differentiated Gaussian
pulse, provides a sharp null of the energy-normed error even when the backscat­
tered signal is contam inated w ith a significant level of additive zero-mean Gaussian
noise. This implies th at the proposed inverse-scattering technique could be useful
in practical measurement environments.
A logical near-term extension of this work is to construct an additional
software-element which would use the same technique to obtain the electrical proper78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
ties of the breast tissue underlying the skin after first determining the skin thickness,
perm ittivity and conductivity. A second extension of this technique would be to ex­
ploit its multidimensional search capabilities to determ ine the frequency dispersion
of the electrical properties of the skin and the underlying breast tissues.
In the longer term , extension of the m ethod to a full three-dimensional exci­
tation geometry is needed, since the confocal microwave imager will use an array of
dipoles or other three-dimensional antenna elements. Upon this extension to three
dimensions, a clear-cut experimental validation of the proposed inverse-scattering
technique will be possible.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
[1] W. K. Purves, G. H. Orians, and H. C. Heller, Life - The Science o f Biology.
Salt Lake City, Utah: W. H. Freeman and Co., 1995.
[2] X. Li and S. Hagness, “A conforcal microwave imaging algorithm for breast
cancer detection,” IE E E Microwave and Wireless Components Letters, vol. 11,
pp. 130-132, Mar. 2001.
[3] W. T. Joines, Y. Z. Dhenxing, and R. L. Jirtle, “The measured electrical prop­
erties of norm al and malignant hum an tissues from 50 to 900 m hz,” Medical
Physics, vol. 21, pp. 547-550, 1994.
[4] S. S. Chaudhary, R. K. Mishra, A. Swarup, and J. M. Thom as, “Dielectric
properties of normal and malignant hum an breast tissues at radiowave and mi­
crowave frequencies,” Indian Journal o f Biochemistry and Biophysics, vol. 21,
pp. 76-79, 1984.
[5] E. C. Fear and M. A. Stuchly, “Microwave detection of breast cancer,” IEEE
Transactions on Microwave Theory and Techniques, vol. 48, pp. 1854-63, Nov.
2000.
[6] S. C. Hagness, A. Taflove, and J. E. Bridges, “Two-dimensional F D T D analysis
of a pulsed microwave confocal system for breast cancer detection: fixed-focus
and antenna-array sensors,” IE E E Transactions on Biomedical Engineering,
vol. 45, pp. 1470-1479, Dec. 1998.
[7] S. C. Hagness, A. Taflove, and J. E. Bridges, “Three-dimensional FD T D analy­
sis of a pulsed microwave confocal system for breast cancer detection: design of
an antenna-array element,” IEEE Transactions on Antennas and Propagation,
vol. 47, pp. 783-791, May 1999.
[8] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clin­
ical prototype for active microwave im aging of the breast,” Microwave Theory
and Techniques, vol. 48, pp. 1841-53, Nov. 2000.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
[9] J. E. Bridges, “Non-invasive system for breast cancer detection.” U.S. P aten t
No. 5,704,355, Jan. 1998.
[10] J. E. Bridges, “Breast cancer detection, imaging and screening by electromag­
netic m illim eter waves.” U.S. Patent No. 5,807,257, Sept. 1998.
[11] J. E. Bridges, “Microwave m ethod and system to detect and locate cancer in
heterogeneous tissues.” U.S. Patent No. 5,829,437, Nov. 1998.
[12] J. E. Bridges, A. Taflove, S. C. Hagness, and A. Sahakian, “Microwave antenna
for cancer detection system.” U.S. P atent application subm itted Sept 9, 1997.
[13] M. Popovic, S. C. Hagness, A. Taflove, and J. E. Bridges, “2-D FDTD study of
a fixed-focus elliptical reflector system for breast cancer detection: frequency
window for optim um operation.” paper AP76-8 in Proc. 1998 IEEE A ntennas
and Propagation Society Int. Sym p. A tlanta, GA, Ju n e 21 - 26, 1998.
[14] S. C. Hagness, A. Taflove, and J. E. Bridges, “F D T D analysis of a pulsed
microwave confocal system for breast cancer detection.” in Proc. 19th Int.
Conf. IE E E Engineering in Medicine and Biology Society, Chicago, IL, Oct. 30
- Nov 2, 1997, pp. 2506 - 2508.
[15] S. C. Hagness, A. Taflove, and J. E. Bridges, “FD TD m odeling of a coherentaddition antenna array for early-stage detection of breast cancer.” paper AP457 in Proc. 1998 IE E E Antennas and Propagation Society Int. Symp. A tlanta,
GA, June 21 - 26, 1998.
[16] J. G. E. et a/., “Ten year risk of false positive screening mam m ogram s and clini­
cal breast exam inations,” New England Journal of Medicine, vol. 338, pp. 10891096, 1998.
[17] M. Laya, E. Larson, S. Taplicn, and E. W hite, “Effect of estrogen replacement
therapy on the specificity and sensitivity of screening mammography,” Journal
o f the National Cancer Institute, vol. 88, pp. 643-649, 1996.
[18] R. D. Rosenberg, W. C. Hunt, M. R. Williamson, F. D. Gilliland, P. W. W iest,
C. A. Kelsey, C. R. Key, and M. N. Linver, “Effects of age, breast density,
ethnicity and estrogen replacement therapy on screening mammographic sensi­
tivity and cancer stage at diagnosis: Review of 183 134 screening mammograms
in albuquerque,” Radiology, vol. 209, no. 2, pp. 511-518, 1998.
[19] V. P. J. et a1., “Imaging of the radiographically dense breasts,” Radiology,
vol. 188, pp. 297-301, 1993.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
[20] E. C. Robb-Nicholson, “Breast cancer update: P art i,” Harvard Women’s
Health Watch, vol. 8, pp. 3-5, Oct. 2000.
[21] C. C. Johnson and A. W. Guy, “Nonionizing electromagnetic wave effects in
biological m aterials and systems,” IE E E Proceedings, vol. 60, p. 694, 1972.
[22] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties of biological
tissues: I. literature survey,” Physics in Medicine and Biology, vol. 41, pp. 22312249, Nov. 1996.
[23] S. Gabriel, R. W . Lau, and C. Gabriel, “T he dielectric properties of biological
tissues: II. measurements on the frequency range 10 Hz to 20 GHz,” Physics
in Medicine and Biology, vol. 41, pp. 2251-2269, Nov. 1996.
[24] S. Gabriel, R. W . Lau, and C. Gabriel, “T he dielectric properties of biological
tissues: III. Param etric models for the dielectric spectrum of tissues,” Physics
in Medicine and Biology, vol. 41, pp. 2271-2293, Nov. 1996.
[25] J. P. G rant, R. N. Clarke, G. T. Symm, and N. M. Spyrou, “in vivo dielectric
properties of hum an skin from 50 mhz to 2.0 ghz,” Physics in Medicine and
Biology, vol. 33, no. 5, pp. 607-612, 1988.
[26] A. Taflove, Computational Electrodynamics: The Finite-Difference TimeDomain Method. Norwood, MA: Artech House, 1995.
[27] K. R. Foster and H. P. Schwan, “Dielectric properties of tissues and biologi­
cal materials: a critical review,” Critical Reviews in Biomedical Engineering,
vol. 17, pp. 25-104, 1989.
[28] L. E. Larsen and e. J. H. Jacobi, Medical Applications of Microwave Imaging.
New York: IEEE Press, 1986.
[29] A. W. Preece, H. Johnson, F. L. Green, and M. P. Robinson, “Dielectric imag­
ing for localization and detection of breast tum ors.” 1993 M T T International
Microwave Sym p. D ig e s t, Cat No. 93CH3277-1, pp. 1145-1146.
[30] R. E. Sepponen, “Medical diagnostic microwave scanning apparatus.” U.S.
P aten t 4,641,659, Feb 10, 1987.
[31] B. D. Sollish, E. H. Frie, E. Hammerman, B. Lang, and M. Moshitsky,
“Microprocessor-assisted screening technique,” Israel Journal o f Medical Sci­
ence, vol. 17, pp. 859-864, 1981.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
[32] J. W. Lichman, “Confocal microscopy,” Scientific American, vol. 171, pp. 4045, 1994.
[33] P. Kearey and M. Brooks, “Seismic reflection surveying,” 1984. Chap. 4 in Geo­
science Texts, VOL 4: A n Introduction to Geophysical Exploration , Boston,
MA: Blackwell Scientific.
[34] S. C. Hagness, A. Taflove, and J. E. Bridges, “W ideband ultralow reverberation
antenna for biological sensing,” Electronic Letters, vol. 33, no. 19, pp. 15941595, 1997.
[35] J. G. Maloney and G. S. Smith, “A study of transient radiation from the WuKing resistive monopole - FDTD analysis and experimental m easurem ents,”
IE E E Transactions on Antennas and Propagation, vol. 41, pp. 668-676, May
1993.
[36] J. G. Maloney and G. S. Smith, “O ptim ization of a conical antenna for pulse
radiation: An efficient design using resistive loading,” IEEE Transactions on
Antennas and Propagation, vol. 41, pp. 940-947, July 1993.
[37] T. T. Wu and R. W. P. King, “The cylindrical antenna with nonreflecting
resistive loading,” IE E E Transactions on Antennas and Propagation, pp. 369373, May 1965.
[38] A. S. Breathnach, A n Atlas of the Ultrastructure o f Human Skin. London: J.
& A. Churchill, 1971.
[39] J. T. L. Pope, M. E. Read, T. Medsker, A. J. Buschi, and A. N. Brenbridge,
“Breast skin thickness: normal range and causes of thickening shown on filmscreen mammography,” Journal of the Canadian Association o f Radiologists,
vol. 35, pp. 365-368, Dec. 1984.
[40] M. Wickman, “B reast reconstruction - past achievements, current statu s and
future goals,” Scandinavian Journal o f Plastic and Reconstructive and Hand
Surgery, vol. 29, pp. 81-100, 1995.
[41] M. Wickman, G. Jurell, and K. Sandelin, “Im mediate breast reconstruction:
short term experience in 75 consecutive cases,” Scandinavian Journal o f Plastic
and Reconstructive and Hand Surgery, vol. 29, pp. 153-159, 1995.
[42] M. Olenius and O. Johansson, “Variations in epidermal thickness in expanded
hum an breast skin,” Scandinavian Journal o f Plastic and Reconstructive and
Hand Surgery, vol. 29, pp. 15-20, 1995.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
[43] H. I. Libshitz, E. D. Montague, and J. D. D. Paulus, “Skin thickness in th e ther­
apeutically irradiated breast,” American Journal o f Roentgenology, vol. 130,
pp. 345-347, Feb. 1978.
[44] J. C. Bolomey, D. Lesselier, C.Pichot, and W. T abbara, “Spectral an d time
dom ain approaches to some inverse scattering problems,” IEEE Transactions
on Antennas and Propagation, vol. 29, no. 2, pp. 206-212, 1981.
[45] S. Coen, K. K. Mei, and D. J. Anelakos, “Inverse scattering technique ap­
plies to remote sensing of layered media,” IE E E Transactions on A ntennas and
Propagation, vol. 29, no. 2, pp. 298-306, 1981.
[46] W. Tabbara, “Reconstruction of perm ittivity profiles from a spectral analysis
of the reflection coefficient,” IE E E Transactions on Antennas and Propagation,
vol. 27, no. 2, pp. 241-244, 1979.
[47] M. Mostafavi and R. M ittra, “Remote probing of inhomogeneous m edia using
param eter optimization techniques,” Radio Science, vol. 7, no. 12, pp. 11051111, 1972.
[48] K. R. Umashankar, S. Chaudhuri, and A. Taflove, “Finite-difference timedom ain formulation of an inverse scattering scheme for remote sensing of inho­
mogeneous lossy layered m edia,” Journal o f Electromagnetic Waves and Appli­
cations, vol. 8, pp. 489-509, 1994.
[49] M. A. Strickel, A. Taflove, and K. R. Um ashankar, “Finite-difference timedom ain formulation of an inverse scattering scheme for remote sensing of con­
ducting and dielectric targets,” Journal o f Electromagnetic Waves and Appli­
cations, vol. 8, pp. 510-529, 1994.
[50] C. J. Leuschen and R. G. Plumb, “A matched-filter-based reverse-time mi­
gration algorithm for ground-penetrating radar d a ta ,” IE E E Transactions on
Geoscience and Remote Sensing, vol. 39, pp. 929-936, May 2001.
[51] J. F. Ma, W. H. Yu, and R. M ittra, “D etection of buried dielectric cavities
using the finite-difference time-domain m ethod in conjunction with signal pro­
cessing techniques,” IE E E Transactions on A ntennas and Propagation, vol. 48,
pp. 1289-1294, Sept. 2000.
[52] A. Qing, C. K. Lee, and L. Jen, “Electrom agnetic inverse scattering of twodimensional perfectly conducting objects by real-coded genetic algorithm ,”
IE E E Transactions on Geoscience and Rem ote Sensing, vol. 39, pp. 665-676,
Mar. 2001.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
[53] A. J. Surowiec, S. S. Stuchly, J. R. Barr, and A. Swafup, “Dielectric proper­
ties of breast carcinoma and the surrounding tissues,” IE E E Transactions on
Biomedical Engineering, vol. 35, pp. 257-263, 1988.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
3 327 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа