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Ultrasound bent-ray tomography with a modified total-variation regularization scheme

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Ultrasound bent-ray tomography with a modified total-variation regularization scheme
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Intrator, Miranda Huang ( author )
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Breast -- Cancer -- Tomography ( lcsh )
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The sound-speed distribution of the breast can be used for characterizing breast tumors, because they typically have a higher sound speed than normal breast tissue. This is understood to be the result of remodeling of the extracellular matrix surrounding tumors. Breast sound-speed distribution can be reconstructed using ultrasound bent-ray tomography. We have recently demonstrated that ultrasound bent-ray tomography, using arrival times of both transmission and reflection data, significantly improves image quality. To further improve the robustness of tomographic reconstructions, we develop an ultrasound bent-ray tomography method using a modified total-variation regularization scheme and implement it using transmission data. Regularization is often used in solving inverse problems by introducing constraints on inversion results such as smoothness. Tikhonov regularization is a widely used regularization scheme that tends to smooth tomographic images, but oversmoothing can obscure critical diagnostic detail such as tumor margins. Total-variation regularization is another common regularization scheme that helps preserve tumor margins, but at the cost of increased image noise. Our new ultrasound bent-ray tomography with the modified total-variation regularization scheme employs a Tikhonov-Total-Variation hybrid regularization method, reducing image noise while preserving margins. We validate our new method using ultrasound transmission data from numerical phantoms, and compare the results with those obtained using Tikhonov regularization.
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Thesis (M.S.)--University of Colorado Denver.
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Department of Bioengineering
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by Miranda Huanga Intrator.

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Full Text
ULTRASOUND BENT-RAY TOMOGRAPHY
WITH A MODIFIED TOTAL-VARIATION REGULARIZATION SCHEME
by
MIRANDA HUANG INTRATOR
B.A., University of California at Santa Cruz, 2007
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Bioengineering
2015


This thesis for the Master of Science degree by
Miranda Huang Intrator
has been approved for the
Bioengineering Program
by
Kendall Hunter, Chair
Daewon Park, Advisor
Lianjie Huang, Advisor
Date: July 10, 2015
1


Intrator, Miranda Huang (M.S., Bioengineering)
Ultrasound bent-ray tomography with a modified total-variation regularization scheme
Thesis directed by Assistant Professor Daewon Park and Dr. Lianjie Huang
ABSTRACT
The sound-speed distribution of the breast can be used for characterizing breast tumors, because
they typically have a higher sound speed than normal breast tissue. This is understood to be the re-
sult of remodeling of the extracellular matrix surrounding tumors. Breast sound-speed distribution
can be reconstructed using ultrasound bent-ray tomography. We have recently demonstrated that
ultrasound bent-ray tomography, using arrival times of both transmission and reflection data, signif-
icantly improves image quality. To further improve the robustness of tomographic reconstructions,
we develop an ultrasound bent-ray tomography method using a modified total-variation regular-
ization scheme and implement it using transmission data. Regularization is often used in solving
inverse problems by introducing constraints on inversion results such as smoothness. Tikhonov
regularization is a widely used regularization scheme that tends to smooth tomographic images, but
oversmoothing can obscure critical diagnostic detail such as tumor margins. Total-variation regular-
ization is another common regularization scheme that helps preserve tumor margins, but at the cost
of increased image noise. Our new ultrasound bent-ray tomography with the modified total-variation
regularization scheme employs a Tikhonov-Total-Variation hybrid regularization method, reducing
image noise while preserving margins. We validate our new method using ultrasound transmission
data from numerical phantoms, and compare the results with those obtained using Tikhonov regu-
larization.
The form and content of this abstract are approved. I recommend its publication.
Approved: Daewon Park and Lianjie Huang
11


Declaration of original work
By Miranda Huang Intrator
This page is to assert that this thesis for the Master of Science degree was independently composed
and authored by myself, using solely the referenced sources and support from my advisors, fellow
students, the Department of Bioengineering at the University of Colorado Denver Anschutz Med-
ical Campus and the Geophysics Group of the Earth and Environmental Sciences Division at Los
Alamos National Laboratory. The thesis describes the contribution of a new ultrasound bent-ray to-
mography (USRT) method using a modified total-variation (MTV) regularization scheme, which is
the hrst time to incorporate an MTV regularization scheme into USRT. A computationally efficient
and accurate reconstruction algorithm is developed and implemented for reconstruction of numeri-
cal phantoms, comparing results with those obtained using a Tikhonov regularization scheme. This
research will improve USRTs capability for breast cancer characterization in the clinic.
This statement is approved
by
Miranda Huang Intrator
and
Daewon Park, Advisor
and
Lianjie Huang, Advisor
m


ACKNOWLEDGEMENTS
I would like to thank everyone who helped me, directly and indirectly, in achieving this milestone.
First and foremost I would like to thank my advisors, Dr. Daewon Park, Dr. Lianjie Fluang, and Dr.
Kendall Flunter. The time I spent in Dr. Parks Translational Biomaterials Reserach Laboratory was
invaluable in learning critical benchtop skills -1 thank Dr. Park for his patience and his trust in me,
and for allowing me to pursue research as a Graduate Research Assistant under the supervision of
Dr. Lianjie Fluang at Los Alamos National Laboratory. I thank Dr. Lianjie Fluang for his enthusiasm
and kind support during my research in his group. I am also deeply indebted to the postdocs there,
who provided invaluable assistance with all matters large and small Drs. Youzuo Lin, Ting Chen,
Junseob Shin, Kai Gao and Nghia Nguyen. Last but certainly not least, my family has served
to encourage, galvanize and console me in all matters since the very beginning. I will always
be deeply grateful to, and inspired by, them. This research was supported by Dr. Daewon Park
at the University of Colorado at Denver Anschutz Medical Campus, by Dr. Lianjie Fluang at
Los Alamos National Laboratory, and by the Breast Cancer Research Program of the U.S. DoD
Congressionally Directed Medical Research Programs through Contract MIPR0LDATM0144 to
Los Alamos National Laboratory.
IV


TABLE OF CONTENTS
Chapter
1 Breast cancer facts and figures 1
1.1 Screening with mammography........................................................ 1
1.1.1 Does mammography prevent deaths? ......................................... 2
1.1.2 Breast density: newly identified risk factor ............................. 3
1.2 Ultrasound tomography for breast cancer screening................................. 4
2 Ultrasound acoustics 7
2.1 Modeling sound propagation in biological tissue................................... 8
2.1.1 The scalar-wave equation.................................................. 8
2.1.2 The ray approximation..................................................... 9
2.1.3 HuygensPrinciple......................................................... 9
2.1.4 Fermats Principle....................................................... 10
2.1.5 Snells Law.............................................................. 10
2.2 B-mode ultrasound imaging of biological tissues ................................. 12
2.3 Opportunities for improvement of B-mode ultrasound imaging....................... 13
3 USRT for breast characterization 15
3.1 The forward and inverse problem.................................................. 16
3.1.1 Inverse and ill-posed problems........................................... 16
3.1.2 The forward problem...................................................... 17
3.2 Regularized inversion............................................................ 19
3.2.1 USRT with Tikhonov regularization........................................ 19
3.2.2 Tikhonov regularization.................................................. 21
3.2.3 USRT with total-variation regularization................................. 21
3.2.4 Total-variation regularization........................................... 22
v


4 USRT with MTV regularization
24
4.1 Implementation of USRT with MTV regularization............................... 24
4.2 Numerical phantoms............................................................. 26
5 Sound speed reconstruction results 28
5.1 Reconstruction of two different phantoms....................................... 28
5.2 Reconstruction of phantoms with varying target and background sound speeds . . 32
6 Conclusions and future work 37
References 37
vi


1 Breast cancer facts and figures
After more than three decades of screening efforts, breast cancer remains the most common type
of cancer and one of the leading causes of cancer-related death in women worldwide (Ferlay et al.,
2013). An estimated 231,840 new cases of invasive breast cancer are expected to be diagnosed
among US women alone during 2015; about 2,350 new cases are expected in men. The standard of
care in the US is annual screening by mammography for all women over the age of 40 (American
Cancer Society, 2015), however a rapidly increasing number of studies are questioning this standard
(Biller-Andorno and Jiini, 2014; Bleyer and Welch, 2012; Esserman et al., 2009, 2014; Independent
UK Panel on Breast Cancer Screening, 2012; Ong and Mandl, 2015; Ripping et al., 2015; The
Canadian Task Force on Preventive Health Care, 2011).
1.1 Screening with mammography
Mammography uses X-ray to image the breast it compresses the breast between two plates to
image tissue radiodensity. There are a number of concerns with mammography X-rays are ionizing
radiation, which is itself a risk factor for cancer. Mammography also often fails to differentiate
between benign and malignant masses ultrasound outperforms mammography for tumor detection
in dense breasts (Fasching et al., 2006; Kolb et al., 2002). Mammography has difficulty visualizing
lesions in dense breasts, requiring follow-up imaging that is routinely accomplished by ultrasound
imaging. Furthermore, patients with dense breasts are considered higher-risk than their non-dense
breast counterparts (Boyd et al., 2007; Elias et al., 2014; McCormack and dos Santos Silva, 2006).
Mammography can also result in false-negative results, missing cancers ultrasound can often
detect these occult lesions (Gordon and Goldenberg, 1995; Uchida et al., 2008). Mammography
compresses the breast which is uncomfortable or painful for many patients, which may interfere
with patient compliance (Sapir et al., 2003). Compression of the breast may interfere with image
interpretation and diagnosis as well.
1


1.1.1 Does mammography prevent deaths?
The most pressing concern today, however, is that population-wide mammography screening has
been associated with a rise in the incidence of breast cancer but not with a decline in presentation of
advanced-stage breast cancers or overall breast cancer mortality (Bleyer and Welch, 2012; Esserman
et al., 2009). This is most likely due to: (1) the large number of false-positive findings and breast
cancer overdiagnosis associated with mammography (Esserman et al., 2009; Pace and Keating,
2014), and (2) the fact that mammography is not conducted in a manner that detects the most
aggressive and lethal cancers (Drukker et al., 2014; Esserman et al., 2009).
A false-positive is when the mammographic finding raises suspicion of breast cancer, and leads
to additional imaging or biopsy, but ultimately does not lead to a cancer diagnosis. Overdiagnosis is
defined as the detection of a tumor through screening that would not have become clinically evident
in the absence of screening.
Early detection and screening have been shown to be most successful when pre-malignant le-
sions can be detected and eliminated, as with the removal of adenomatous polyps during colon-
cancer colonoscopy screening. Colon cancer has thus enjoyed a significant decrease in invasive
cases. In contrast, ductal carcinoma in situ (DCIS), which is considered a precancerous lesion, was
rare prior to screening but now represents 25-30% of all breast cancer diagnoses (Esserman et al.,
2009).
A 2014 study assessing the risks and benefits of mammography found that per 10,000 women
screened with mammography in a ten-year period, there are orders of magnitude difference between
the number of deaths averted by mammography screening (tens), the number of overdiagnoses
(hundreds), and the number of false-positives (thousands) and unnecessary biopsies (also thou-
sands) (Pace and Keating, 2014). Another study in the same year concluded that screen-detected
(by mammography) cancers were significantly more often low-risk tumors (68%), of which 54%
were considered to be ultra-low risk when compared to interval (biologically aggressive) cancers
(Drukker et al., 2014). To clarify the distinction between indolent (non-lethal) and lethal aggressive
cancers requiring treatment, some leading researchers are even calling for a change in terminology,
terming non-lethal breast lesions as IDLE (indolent lesions of epithelial origin) instead of calling
2


them cancer (Essennan el al., 2009, 2014).
Breast cancer also represents an enormous personal and societal financial burden. Out-of-pocket
expenses for a breast cancer patient average about $1500 per month (Arozullah et al., 2004), and a
recent study estimated the total costs of false-positives and overdiagnosis in the US is approximately
$4 billion annually (Ong and Mandl, 2015). The total cost of mammography screening is estimated
at nearly $8 billion annually, with the two largest drivers of cost being (1) screening frequency, and
(2) percentage of women screened (ODonoghue et al., 2014).
Thus, the data now points to a new picture of mammography, which has effectively increased
the detection of indolent cancers such as ductal carcinoma in situ (DCIS) and often misses the most
aggressive (interval) cancers. This screening paradigm has increased breast cancer-related morbidity
because of overdiagnosis of non-life-threatening cancers, while failing to reduce mortality due to
undetected aggressive breast cancers. There is a clear and pressing need to develop improved breast
cancer screening methods.
1.1.2 Breast density: newly identified risk factor
Mammographic breast density (MBD) is a measure of radiodense fibroglandular tissue in the breast.
This biomarker is very strongly associated with increased cancer risk (Boyd et al., 2007, 2010; Elias
et al., 2014; McCormack and dos Santos Silva, 2006), stronger, in fact, than most other established
breast cancer risk factors, except for age and some genetic factors. Furthermore, MBD has been
indicated as a promising biomarker for assessing risk of more aggressive interval cancers (Bertrand
et al., 2013; Boyd et al 2007; Buist et al 2004; Kerlikowske et ah, 2007; Mandelson et ah, 2000;
McCormack and dos Santos Silva, 2006; Nothacker et ah, 2009; Porter et ah, 2007). Women with
dense breasts may indeed benefit from shorter screening intervals. Breast density is also one of
the leading factors in false-negative mammographic findings (Ma et ah, 1992; Mandelson et ah,
2000; Nothacker et ah, 2009; Porter et ah, 2007; Roubidoux et ah, 2004). Epidemiological studies
confirm that patients can categorized by risk according to breast density, and that breast density
may represent an intermediate phenotype due to genetic factors (Boyd et ah, 2010; Martin and
Boyd, 2008; Nothacker et ah, 2009; Provenzano et ah, 2008; Vachon et ah, 2007). Breast density
3


is a heritable trait (Boyd et al., 2002, 2010), and has been linked to a number of other risk factors,
such as high tumor grade and large tumor size at presentation (Roubidoux et al., 2004). The fact
that mammographys usefulness is controversial, in conjunction with the fact that its performance
is suboptimal for high-risk populations where screening is needed most, motivates us to develop a
method of breast cancer screening that is safe for more frequent use with higher risk populations.
1.2 Ultrasound tomography for breast cancer screening
Supplemental breast ultrasound for women with dense breasts may enable the detection of small,
otherwise occult, breast lesions, but this may result in an increased biopsy rate (Nothacker et al.,
2009; Ucliida et al., 2008), and additional breast ultrasound screening for dense breasts brings the
interval cancer rate down to the interval cancer rate for non-dense breasts (Corsetti et al., 2011). An-
other more recent meta-analysis correlating imaging features with breast tumor malignance (HER2
overexpression) indicated that suspected malignance as assessed by breast ultrasound may indeed
indicate malignance (Elias et al., 2014). However, mammographic specificity declines with increas-
ing breast density (Kolb et al., 2002), and therefore other modalities must be developed to serve a
newly identified high-risk population via better sensitivity to this critical biomarker.
Ultrasound is widely available, non-invasive, non-ionizing, low-cost and is a real-rime imag-
ing modality. It is well-known that supplementing mammography with breast ultrasound has better
detection rates than mammography alone in particular, for high-risk patients with dense breasts
(Kolb et al., 2002; Ma et al., 1992; Mainiero et al., 2013). Indeed, the sensitivity of mammography
alone decreases from 100% in fatty breasts to 45% in extremely dense breasts (Berg et al., 2004).
Whole-breast bilateral (both breasts) sonography has been shown to detect small non-palpable inva-
sive breast cancers not visualized by mammography, particularly in dense breasts (Buchberger et al.,
2000; Gordon and Goldenberg, 1995; Kaplan, 2001; Kolb et al., 1998,2002), and high-risk patients
were two to three more likely to have cancers seen only sonographically (Berg, 2003; Buchberger
et al., 2000; Gordon and Goldenberg, 1995; Kaplan, 2001; Kolb et al., 2002). In patients diag-
nosed with invasive breast cancer, survival is a function of tumor size (Michaelson et al., 2002), and
ultrasound has proven better at measuring tumor size in dense breasts (Schreer, 2009). Automated
4


whole breast ultrasound plus mammography showed significantly better cancer detection than mam-
mography alone in high-risk patients (Kelly et al., 2010), and has also been successfully used for
computer-aided diagnosis in a pilot study of 147 patients, with promising sensitivity (84.5%) and
specificity (85.5%) values (Moon et al., 2011).
While breast ultrasound has long been an adjunct imaging method to mammography for breast
cancer, current techniques lack the sensitivity and specificity to serve as a standalone modality
(Berg et al., 2008). B-mode ultrasound is qualitative instead of quantitative, has limited resolu-
tion and contrast, and has speckle noise (which may obscure small tissue structures) and artifacts,
all preventing it from wholly replacing mammography (Duric et al., 2005). Furthermore, hand-
held ultrasound transducers make image quality highly operator-dependent (Yaffe, 2008), further
exacerbating image interpretation (Mendelson et al., 2001).
To address these issues, the imaging modality of ultrasound tomography has been an area of
recently accelerating development. In ultrasound tomography (UST), the breast is immersed in
water and automatically scanned in slices to build a 3D image of the breast. The breast is not
compressed as it is with mammography, and the dedicated breast scanning instrument results in less
operator-dependence or operator-independent to obtain high-quality diagnostic images. In the early
1970s, Kossoff et al. (1973) reported the composite (mixed fatty and glandular tissue) breast to have
an average sound speed of 1510 m/s in pre-menopausal and 1468 m/s in post-menopausal women.
Li et al. (2009a) acquired patient data using a dedicated breast UST scanner with a ring-shaped
array of transducers in pendant-mode (in which the patient lies face-down on a UST table with
breast suspended in a warm water bath the breast is then considered pendant). Their sound-speed
tomograms reconstructed using a bent-ray UST algorithm could reliably differentiate between fatty
(14229 m/s) and glandular (148721 m/s) breast tissue, as well as between benign (151327 m/s)
and malignant (154817 m/s) breast lesions, suggesting there is great promise for the use of this
modality as a breast cancer characterization method (Li et al., 2009a). There may be complications
imaging close to the chest wall using the ring-shaped array. It is important to be able to image this
region because this is where metastasized tumors may appear in the lymph nodes (Nebeker and
Nelson, 2012). For this reason, a UST scanner with a parallel transducer array was designed and
manufactured by Lianjie Huangs group at Los Alamos National Laboratory that may better image
this lymph node region and different breast sizes. This scanner is now used for a clinical study at
5


the University of New Mexico Hospital (Huang et al., 2015).
6


2 Ultrasound acoustics
Sound can be rigorously defined as propagating differential pressure (that is, propagating zones
of compression and expansion, or rarefaction) with a given oscillation frequency, /. The sound
frequency spectrum is conventionally split into three ranges:
<20 Hz is referred to as infrasound,
20 Hz-20 kHz is referred to as the audible range, and
>20 kHz is referred to as ultrasound.
Human hearing is typically confined to the audible range of frequencies. Ultrasound is used in many
different areas of science, notably in medical imaging where it is called sonography. The process of
medical ultrasound imaging is, briefly, a piezoelectric transducer is used to convert electrical energy
to mechanical energy, creating a sound wave that propagates through the tissue being imaged. De-
pending on certain tissue parameters, some portion of the applied ultrasound is returned back to the
transducer, and this amount of energy is translated into a grayscale image where dark pixels repre-
sent less energy returned (anechoic or hypoechoic regions) and lighter pixels represent more energy
returned (hyperechoic regions). A single transducer generates a ID image of its line of sight into
the tissue. Transducer arrays are used to generate 2D image slices in medical ultrasound.
Typical medical imaging ultrasound transducers operate in the 2-15 MHz range (Laugier and
Hai'at, 2011). The size of objects that are detectable with these transducers depends on the ultra-
sound wavelength, which is related to frequency in the following way:
X = j = cT, (1)
where A is the wavelength in mm, c is the propagation mediums speed of sound (mm//xs), / is
frequency (MHz), and T is the period (ps). Most clinical ultrasound imaging systems assume
a constant sound-speed value c of 1.540 mm//xs. Thus, the typical wavelength range of clinical
ultrasound imaging ranges from 0.77 mm (at 2 MHz) down to 0.1 mm (at 15 MHz).
7


2.1 Modeling sound propagation in biological tissue
Biological tissues are viscoelastic materials, in which both bulk compression (corresponding to
longitudinal waves) and shear waves can propagate. In biological tissues, unlike in fluids, shearing
strain can be transmitted to adjacent layers of tissue because of the strong binding between particles.
However, because shear waves in soft tissue are highly attenuated at ultrasonic frequencies (con-
ventionally 20 kHz and up) they are typically ignored, leaving only compression waves to consider
(Laugier and Hai'at, 2011).
2.1.1 The scalar-wave equation
Physically, sound propagation produces a medium density variation as it travels through a tissue,
creating alternating zones of compression and rarefaction. This phenomenon is modeled using the
scalar-wave equation, representing these zones of varying pressure as a waveform. The scalar-wave
equation in two dimensions can be written as (Hill et al., 2004):
through tissue, the path it takes is governed by its interactions with the propagating medium, which
may have varying properties. The sensitivity of ultrasound to these variations results in an acoustic
pressure field, containing information about the propagating medium (in this case, biological tissue).
The main types of tissue interactions affecting this field are:
Refraction, resulting in a change in the sound-wave propagation direction when transmitting
through a medium interface, is based on tissue mechanical properties. Refraction data is also
sometimes referred to as transmission data, as it is transmitted through a medium as opposed
to reflected off it.
Reflection, resulting in a change in the propagation direction of the wave while propagating
at the same sound speed as that of the incident wave. Reflection data are used to created
pulse-echo (B-mode) clinical ultrasound images.
(2)
where V2 is the spatial Laplacian operator, and p is the acoustic pressure. When a sound wave passes
8


Attenuation, resulting in a reduction in amplitude and intensity of the wave, can be caused by
scattering or absorption of ultrasound energy. Wave frequency, dssue mechanical properties
and distance traveled all affect attenuation.
Reflection data and refraction (transmission) data are governed by different physics, and are thus
mathematically dealt with differently.
2.1.2 The ray approximation
Although ultrasound is typically conceptualized as a wave, its asymptotic high-frequency approx-
imation is the ray approximation. The ray approximation can be stated as: For sufficiently short
wavelengths, rays behave as straight lines when they pass through a medium in which the speed of
sound (or, speed of light, as originally stated for optics) is constant. At boundaries where the sound
speed changes, a ray obeys the laws of reflection and refraction.
The ray approximation of a sound wave is defined by choosing a line normal to a sound wave
wavefront pointing in the direction of wave propagation. However, we can add some complexity
to this simplified version of wave propagation by requiring only that the ray be locally straight -
over the rays journey through the tissue, it can change directions many times and we model this
as an effective zig-zag ray. This is referred to as a bent-ray, rather than a straight-ray, model.
Modeling sound waves as as bent rays allows us to divide the sound wave field up into discrete and
locally straight rays in many cells of constant sound speed, whose paths can be computationally
propagated through a system using ray tracing techniques. Three fundamental theories for this field
are Huygens Principle, Fermats Principle and Snells law.
2.1.3 Huygens Principle
Originating from a 1678 treatise on the behavior of light by Dutch physicist Christian Huygens
(1629-1695), Huygens Principle, or the Huygens-Fresnel Principle, describes the behavior of wave-
front propagation from a point source. It states that every point of a wavefront can be considered a
source of secondary wavelets that spread in all directions, with a speed equal to the speed of prop-
9


agation of the waves. That is, at any given instant, the wavefront of a propagating wave conforms
to an envelope of spherical wavelets emanating from every point on the wavefront in the instant
immediately prior. Mathematically, this is stated as:
U{r0)
U0eikr
ro
(3)
where U (ro) is the complex amplitude of a primary wave at some point a distance of ro from the
point source. The initial disturbance IJq produces a spherical wave of wavelength A and wavenumber
k. Note the magnitude of the wave amplitude decreases as distance traveled increases. Huygens
Principle allows us to conclude, also, that there is no wave diffusion (i.e. waves do not broaden as
they propagate).
2.1.4 Fermats Principle
Also known as the principle of least time, Fermats Principle describes the behavior of a ray between
two points, stating that the path a ray takes between two points is the path that takes the least time
for a wave to propagate or travel. From this, the laws of reflection and refraction can be derived
(see Fig. 1). The law of reflection states that a ray incident upon a specular reflective boundary is
reflected at an angle ((),) equal to the incident angle (0,) (both angles are measured relative to a ray
normal to the surface, dotted line in Fig. 1). Note this is not necessarily true for diffuse reflection,
or scattering. The law of refraction is also known as Snells Law.
2.1.5 Snells Law
Snells law (Fig. 1) describes the behavior of a ray crossing an interface, that is, it describes the
relationship between the angles of incidence (0j), refraction (a.k.a. transmission) (6t), and reflection
((),) of a ray. It states that the ratio of sines of the angles of incidence and refraction is equivalent
to the ratio of velocities (a, ct) in the two bordering (isotropic) media, or also equivalent to the
10


inverted ratio of the indices of refraction (rji, 772):
sin^j
sinth
Ci
ct
m
(4)
which can be rewritten as:
sin 9i sin 0r
a ^
sin$t
ct
(5)
These relationships allow us to distinguish between different media based on the speed of sound
traveling in that medium (c) and its density (p, units of kg m-3).
Figure 1: An incident (I) ultrasound wave passing through two different tissues having different mechanical
properties (and therefore different indices of refraction, ??i and 772). The different mechanical properties result in
the wave traveling at different velocities in the different tissues, producing refraction (also called transmission,
T; difference in velocity and direction of travel) at the material boundary. Reflection (R; difference in direction
of travel but not velocity) also occurs. The energy of the incident wave is divided between it's subsequent
reflecting wave and refracting waves, and some additional energy is lost due to attenuation. The relative
amounts of energy splitting off to the reflecting and refracting waves is a function of the ratio of the refractive
indices of the two media.
For biological tissue, the primary relevant properties that affect sound wave behavior are sound
speed and acoustic impedance. The formula for calculating the speed of sound in any given material
is given by the square root of the ratio of the bulk modulus (l>) to the density (p):
(6)
11


The inverse of the bulk modulus is the compressibility. When using the ray approximation (or in the
linear propagation regimes of tiny perturbations and/or small wave amplitudes), the speed of sound
(c) is a characteristic of the medium. That is, it is independent of the ultrasound wave amplitude
and can be determined from that tissues material properties alone. The compressibility is typically
determined experimentally, and controls that materials stiffness relative to different types of waves.
Density is related to the inertia of the material.
Specific acoustic impedance, Z, is a measure of a materials resistance to sound passing through
it. It is the product of the materials density and its sound speed, and is measured in rayls (kgcm_2s_1):
See Table 1 for typical acoustic impedance values of human tissue.
2.2 B-mode ultrasound imaging of biological tissues
Clinical ultrasound and ultrasound tomography use different approaches to deliver tissue informa-
tion. Table 2 summarizes the key differences between clinical and tomographic ultrasound imaging
- ultrasound tomographic image reconstruction is discussed in the next chapter. Brightness mode,
known as B-mode, is the predominating ultrasound imaging method used in the clinic for breast
imaging. B-mode images are obtained from the distribution of reflectivity within the tissue. This
reflectivity distribution is formed from measurements of ultrasound energy reflected back from the
normal incident angle and impedance differences in internal tissue structures. This reflectivity is
called the intensity reflection coefficient (IRC):
Z = pc.
(7)
(8)
See table 1 for reference tissue parameters.
12


2.3 Opportunities for improvement of B-mode ultrasound imaging
Ultrasound transducers are handheld, therefore the quality of diagnostic B-mode ultrasound images
is highly dependent on the skill of the ultrasound technician (Yaffe, 2008). B-mode ultrasound is
also non-quantitative, reconstructing acoustic impedance differences within the interrogated tissue
rather than directly imaging tissue parameters. B-mode imaging is a qualitative assessment of tis-
sue structures relative to their surroundings. Limited resolution and contrast prevent high-quality
diagnostic images, and image speckles can obscure small tissue structures. Ultrasound tomography,
discussed in the next chapter, may offers an alternative solution to these shortcomings.
13


Table 1: Sound speed, density, acoustic impedance, and attenuation coefficients (at 1 MHz) for selected
human tissues, listed by ascending sound speed value. For human tissue, the attenuation coefficient is a
strong function of frequency and can be expressed as a = 0.5/dB/MHz/cm. The attenuation coefficient has
been shown to vary linearly with density (Mast, 2000).
Tissue type Sound speed (c, m/s) Density (p, g/cm2) Acoustic impedance (Z, rayls) Attenuation coefficient (a, dB/cm)
Adipose (International Commission on Radiation Units and Measurements, 1998) 1450 1.12 162.400 0.29
Fatty (International Commission on Radiation Units and Measurements, 1998) 1465 0.99 144.303 0.40
Breast, subcutaneous fat (Duric et al., 2005) 1470 - - 0.89
Breast, internal fat (Duric et al., 2005) 1470 - - 0.92
Breast (International Commission on Radiation Units and Measurements, 1998) 1510 1.02 154.020 0.75
Breast, glandular parenchyma (Duric et al., 2005) 1515 - - 1.02
Breast, high attenuation tumor(Duric et al., 2005) 1549 - - 0.92
Kidney (International Commission on Radiation Units and Measurements, 1998) 1560 1.05 163.800 1.00
Breast, cyst (Duric et al., 2005) 1569 - - 0.06
Non-fatty (International Commission on Radiation Units and Measurements, 1998) 1575 1.06 166.163 0.60
Muscle, cardiac (International Commission on Radiation Units and Measurements, 1998) 1576 1.06 167.056 0.52
Muscle, skeletal (International Commission on Radiation Units and Measurements, 1998) 1580 1.05 165.900 0.74
Liver (International Commission on Radiation Units and Measurements, 1998) 1595 1.06 169.070 0.50
Skin (International Commission on Radiation Units and Measurements, 1998) 1615 1.05 169.575 0.35
14


3 USRT for breast characterization
The word tomography derives from the Greek tomo (slice) and graph (draw), and can be under-
stood as cross-sectional imaging from data collected by illuminating the object from many different
directions, that is, reconstructing an object from its projections (which are information derived from
transmitted energy). Ultrasound tomography began with the pioneering work of Greenleaf et al.
(Greenleaf et al., 1974, 1975) and Carson et al. (Carson et al., 1981). A number of prototypes were
subsequently built (Andre et al., 1997; Duric et al., 2007a, 2014; Huang et al., 2015; Marmarelis
et al., 2003; Ruiter et al., 2012; Waag et al., 1996; Wiskin et al., 1997). The working principle of
UST is similar to X-ray computed tomography (CT) as established by Johann Radon with his 1917
paper (Radon, 2005) and by Godfrey Hounsheld for his invention of the CT scanner in the 1970s
(for which he and Allan Cormack shared the 1979 Nobel Prize in Physiology or Medicine) (Raju,
1999).
UST propagates ultrasound instead of ionizing radiation through the patient, and ultrasound
tomography data are acquired for tomographic reconstruction of the distribution of sound speed (or
other tissue parameters) within the breast (Duric et al., 2007a,b, 2003; Huang et al., 2014, 2015;
Huthwaite et al., 2010; Intrator et al., 2015; Uabyed and Huang, 2014; Ui and Duric, 2008; Li et al.,
2009a,b; Lin and Huang, 2013, 2014; Lin et al., 2012; Littrup et al., 2002a,b; Nebeker and Nelson,
2012; Nguyen and Huang, 2014b; Quan and Huang, 2007; Simonetti and Huang, 2008; Simonetti
et al., 2008, 2007; Zhang and Huang, 2013, 2014; Zhang et al., 2012). Sound speed and tissue
density are linearly related for a range of biological tissues (Mast, 2000).
The purpose of ultrasound tomography is to gather tomographic data generated by probing a
medium with sound waves (pressure coupled with velocity), and to convert this to held informa-
tion (for example sound speed or attenuation coefficient). This conversion of tomographic data is
referred to as inversion, and methods for accomplishing it are often computationally expensive and
ill-conditioned. Inversion is a specific mathematical process, in which a held of object parameters is
reconstructed using a set of observahons that give indirect information about the parameters. In our
case, the parameter held we want to reconstruct is the held of sound speed values in a slice of the
breast. The observahons we use are ultrasound ray travel (or, arrival) times. The enhre ultrasound
15


waveform can also be used for ultrasound waveform tomography.
B-mode ultrasound and ultrasound tomography use different types of data for imaging B-
mode ultrasound uses 180 reflection data to image changes in acoustic impedance, whereas ultra-
sound tomography uses transmission and/or reflection data to reconstruct the tissue parameters such
as sound speed or attenuation coefficient. See Table 1 for typical sound speed values of human tis-
sue. Table 2 summarizes the key differences between B-mode and tomographic ultrasound imaging.
Table 2: Summarized technique comparison of B-mode breast ultrasound imaging and ultrasound sound
speed tomographic imaging modalities.
B-mode Ultrasound Ultrasound Ray Tomography
Transducer(s) transmits acoustic pressure (ultrasound pulse) into tissue.
Acoustic pressure from reflection, scatter- ing sound-tissue interactions are received by receiving transducers. Acoustic pressure from transmission (re- fraction, scattering) sound-tissue interac- tions are received by receiving transducers.
Receiving transducer(s) convert received acoustic pressure to voltage (waveform).
Delay-and-sum beamforming builds B- mode image of tissue reflectivity. First-arrival time picking algorithm applied, then regularized iterative inversion recon- structs a sound speed field.
Mismatches in tissue acoustic impedance are qualitatively visualized. Tissue sound speed is quantitatively recon- structed.
3.1 The forward and inverse problem
3.1.1 Inverse and ill-posed problems
Ultrasound tomography is a type of inverse problem (Hansen and OLeary, 1993; Tarantola, 2005)
- that is, sound propagation information through the field of interest is used to infer field properties.
The set of parameters to be determined, that describe the state of the field, are called the model, M.
If these parameters cannot be measured directly, some signals must be obtained to infer the state of
the unobservable (unmeasurable) set of model parameters. The set of measured data is called the
data, D. The method to infer M from D is called a (forward) modeling, H:
D = HM. (9)
16


The inverse mapping is then:
M = H~lD.
(10)
Usually the forward problem is nonlinear, which is analytically and computationally difficult.
Therefore, forward problems are often linearized. A common method for linearization is via the
Taylor series expansion of the model: D = II Mi, I II \( A / Mo) + .... Keeping only first-order
terms gives:
where Di, = HMq, and H\ is a linearized mapping around the unperturbed/initial model Mo.
The forward problem is typically recast as discrete problems. Discretization is the process of
converting a signal (or, any function) into a numeric sequence. Suppose that a medium can be
sufficiently described by p parameters M = mi, m2,mp and that we have q measurements in
our data set I) (l\.(l2..... d,n we can then rewrite the forward problem as
where d and m are column vectors of data/parameter differences between an initial model and
measured data, with elements di dio for i = 1,..., q and rrij rrijo for j = 1, H is a q x p
matrix:
The finite difference is the discrete analog of the derivative. Finite difference methods are numerical
methods for approximating the solutions to differential equations using finite difference equations
to approximate derivatives. Finding a solution to this system of equations can be found using a least
squares approach.
3.1.2 The forward problem
The eikonal equation, following from Fermats Principle, is a nonlinear partial differential equation
describing wavefront (or ray) propagation in a sound speed model. Solving the eikonal equation
D-D0 = H\(M M0)
(11)
d = H m
(12)
(13)
17


via finite difference methods is widely recognized as one of the most efficient means of computing
wave traveltimes (Zelt and Barton, 1998). Most finite difference implementations of solving the
eikonal equation solve for first arrival traveltimes, although alternate algorithms exist for computing
arrival of, say, the most energetic signal arrivals.
Ultrasound bent-ray tomography (unlike X-ray tomography) assumes ultrasound propagation
paths may not be straight within the breast because of tissue inhomogeneities. To account for this
ray bending, we use a finite-difference scheme to solve the eikonal equation given by:
where t is the travel rime, v is the sound speed, and (sx, s,,) is the slowness vector. The imaging
region is defined over a rectangular grid of cells, with each cell having a constant sound speed (and
therefore constant slowness). This collection of cells with constant slowness is called the slowness
model. An initial slowness model with homogeneous sound speed can be used for inversion. To
calculate the true slowness using first-arrival times from a transmitting transducer element to each
grid cell, the raypath is traced back from a receiving element to the transmitting element following
the direction normal to the calculated traveltime field (Vidale, 1988; Zelt and Barton, 1998).
Because of the difference between the model and the true slowness distribution, there is a differ-
ence between the travel time calculated using the estimated model and the first-arrival time picked
from an ultrasound transmission signal. This time difference (I,) is linearly related to the differ-
ence between the model and true slowness perturbations along the ith raypath in the jth cell of the
slowness model:
N N
ti = '52lij/vj = '52kjsj, (15)
3 =1 3=1
where is the length of the ith raypath in the jth cell of the slowness model, v:l is the sound speed
in the jth cell, Sj is the slowness perturbation in the jth cell, and N is the total number of cells in the
model. Combining all possible transmission paths from eq. (15) leads to a set of linear equations
describing the forward modeling problem, which can be written in matrix form:
T = Hs, (16)
18


where T is an M-element column vector containing travel times (I ) of all M raypaths and s is an
iV-element column vector containing all the slowness values (sj) for each cell. The tomographic
matrix H is M x N (M rows and N columns), and its elements are the raypath segment lengths /,:/.
3.2 Regularized inversion
The least-squares inversion is an iterative method of solving inverse problems. When the problem
Ax = b is not well-posed (that is it is ill-posed, due to non-existence or non-uniqueness of x), then
the standard approach is linear (ordinary) LSQ, which seeks to minimize the sum of all squared
residuals (|| || is the Euclidian, or £2, norm, discussed below): ||Ax b\\2. In the case where the
system is underdetermined {A is singular, or non-invertible), LLSQ may provide us with some junk
answers.
For well-posed problems, the direct solution to LLSQ often works well: x = (ATA)~lATb. In
some cases however, a regularized version of the least squares solution may be preferable. Regular-
ization is a common tool in mathematics and statistics, used in particular to solve inverse problems
(among other applications). It refers to the process of introducing additional information in order
to solve an ill-posed problem, or to prevent over-fitting. This information is usually in the form
of a penalty for complexity in the solution, that is, a penalty is associated with certain coeffi-
cient values. Regularization adds the penalty associated with the coefficient values to the error of
the estimated x vector, and hence an accurate estimate with unlikely (unusually high or low) coeffi-
cients would be penalized while a somewhat less accurate but more conservative estimate with more
normally-distributed coefficients would not be penalized as much. Regularization is often used in
solving inverse problems by introducing constraints such as for smoothness.
3.2.1 USRT with Tikhonov regularization
Continuing from eq. (16), H is typically not square (and thus not invertible), and we therefore
cannot simply multiply both sides by H~l to solve for s. Instead, we multiply both sides by the
transpose matrix HT to get HTT = (HTH)s, resulting in s = (HTH)~lHTT. In addition to
19


being solved directly, linear systems can be solved iteratively. Since H is large, it may be desirable
to instead solve for the values in s iteratively, rather than directly, to converge on the true slowness
values using an iterative solver (such as the LSQR or LSMR algorithm), halting this process once
the slowness value updates become sufficiently small. Thus, the inverse of eq. (16) is typically
formulated as a least squares minimization problem:
£(s) = min{||f7s-D||^}, (17)
S
where D is a data vector of observed travel times, Hs is the corresponding forward modeling result,
||Hs D||I is the data misfit function, and || H2 represents the £2 norm. Solving eq. (17) yields
a vector s that minimizes the mean square difference between observed (D) and forward modeled
(Hs) travel times. Because of the error (difference) between these two quantities, direct solution is
not possible.
However, this inverse problem is still ill-posed because of limited data coverage. To address
this, a regularization technique can be used. The Tikhonov regularization is an .^-norm-based
regularization technique widely used for yielding a smoothed model. Incorporating a Tikhonov
regularization term into eq. (17) gives
E( s) = min{||i7s D||2 + A||Ls|||}, (18)
S
where A is a parameter controlling the tradeoff between contributions from the data misfit and
regularization terms. The L-curve technique (Hansen, 2000) can be used to optimize the value of
A. The misfit term is now modified by the Tikhonov regularization term A||Ls|||, where L is the
Laplacian operator V2 (specifically, the 2D discrete Laplacian operator for a 2D inverse problem).
Applying the Laplacian effectively minimizes large differences between adjacent cells, resulting in
a smoothed reconstructed image. However, use of the Tikhonov regularization alone can result in
inappropriately smoothed image reconstructions and a loss of tumor margin detail. To alleviate this
problem, we introduce a new regularization technique into the inverse problem.
20


3.2.2 Tikhonov regularization
The Tikhonov regularization (named for Andrey Tikhonov (1906-1993)) is a widely used regular-
ization scheme (Golub et al., 1999; Tikhonov, 1995) that tends to smooth tomographic images.
The Tikhonov regularization (also referred to as ridge regression in statistical applications) uses a
^2-norm. It is the most commonly used regularization scheme for ill-posed inverse problems. To
give preference to solutions with desirable properties, a regularization term in the minimization:
11 Ax b\ 12 + 11To;112, for some suitably chosen Tikhonov matrix T. This matrix can be chosen as
a multiple of the identity matrix (T = al), giving preferences to solutions with smaller norms. If
the underlying vector is believed to be mostly continuous, smoothness can be enforced with a low-
pass operator. Regularization improves the conditioning of the problem, enabling a direct explicit
numerical solution: x = (ArA + TTT)~1ATb.
The Tikhonov regularization reduces image noise but also decreases resolution. It uses the £2
norm (eq. (19)) that sums the squares of all components and takes the square root of that sum. This
is also referred to as the Euclidian norm. Consider some vector, for example x = (0.5,0.5). Here is
an example of computing the £2 norm of that vector compare this value (o (he £ \ norm computed
for the same vector below (eq. (25)):
||z||2 = = l/\/2 < 1 (19)
Note the £2 norm yields a smaller value than the £\ norm. This method prefers solutions where all
components of x are very small, and likewise tends to distribute error throughout the vector x.
3.2.3 USRT with total-variation regularization
The total-variation (TV) regularization has been used to solve inversion problems and preserve
tumor margins (Lin et al., 2012; Rudin et al., 1992).
Conventional TV regularization can be incorporated into eq. (17), giving
E(s) = min{||iTs D||| + Atv||s||tv}, (20)
S
21


where the TV norm for a 2D model is defined as
IM|tv = ^2 ^/|(Va;s)ij-|2 + |(Vys)jj|2, (21)
i,j=1
where (Vxs)%,j = &i+i,j sij and (V(/s),;.:/ = SjJ+i s,;.:/ are the spatial derivatives at grid point
(/', j) in Cartesian (a;, y) coordinates. The parameter A tv again regulates the tradeoff between the
data misfit and TV regularization terms.
To make the TV term differentiable at the origin, a small smoothing parameter e is typically
introduced, leading to the approximated TV regularization term:
n
IMItv = *22 + l(Vys)f + e. (22)
i,j=1
However, the solution of eq. (22) is highly dependent on the choice of e, and contains significant
image artifacts.
Note: Because, at first glance, the TV term may somewhat resemble an /:2-norm, consider the
following explanation using an example vector m, where m = (mI,m!/)andVm = ( V.rm, V(/m):
||m||i = |m| = ^Im^l2 + |my|2 (23)
||Vm||i = |Vm| = ^|Vxm|2 + |Vym|2 (24)
In the next section, development of a novel ultrasound bent-ray tomography method with a
modified total-variation regularization scheme is described, with the purpose of overcoming the
shortcomings of using TV regularization alone.
3.2.4 Total-variation regularization
USRT with the total-variation (TV) regularization can help preserve tumor margins, but at the cost
of increased image noise. Pioneered by Rudin et al. (1992), the TV regularization method is based
on the principle that signals with excessive (and possibly spurious) detail have a high total variation
22


value. That is, the integral of the absolute gradient of the signal is high. Thus, reducing the total
variation of the signal removes unwanted detail while preserving important details like edges. The
goal of TV is to find an approximation of an input matrix (e.g. an image) x that has smaller total
variation while still being close to x. This closeness is measured using least squares, and so the
TV problem amounts to minimizing the sum of square errors between x and its approximation, plus
a scalar (c) multiple of the total variation in x: 11 Ax b\ (2 + cV(x). To minimize this expression,
we differentiate with respect to x, deriving a corresponding Euler-Lagrange equation that can be
numerically integrated with the original signal as the initial condition.
The total-variation regularization is generally viewed as a good technique to denoise whilst pre-
serving edge information, and the Tikhonov regularization is generally considered better at smooth-
ing images. The TV regularization uses the £\ norm (eq. (25)), which sums the absolute value of all
components of x: \ \Ax b\ (2 +1\cV{x) \ \As an alternative to Euclidean geometry, where the norm
is the unique shortest distance between two points (in Cartesian coordinates), the £\ norm considers
the distance between two points as the sum of absolute differences in their coordinates, a concept
established by Hermann Minkowski in 19th C. Germany. Here is an example of computing the £\
norm of a vector compare this value to the £2 norm computed for the same vector above, eq. (19):
||aj||i = |0.5| + |0.5| = 1 (25)
Note the £\ norm yields a larger value than the £2 norm. Unlike Tikhonov-regularized solutions,
the TV method can allow a sparse x, that is, some values of x are exactly zero while others can be
relatively large. Both very large and very small values are tolerated better than with the £2 norm.
23


4 USRT with MTV regularization
Tumors tend to have higher sound speeds than normal breast tissue, which is understood to be the
result of remodeling of the extracellular matrix surrounding tumors (Macklin, 2010). This allows
them to be characterized using ultrasound bent-ray tomography (USRT). Sound speed is therefore a
critical diagnostic feature in ultrasound tomography-based diagnosis of breast lesions (Hopp et al.,
2012; Li and Duric, 2008; Li et al., 2009a). Margin information and tumor size is an important
diagnostic and prognostic feature for clinicians (Chen et al., 2004; Howlader et al., 2013), and
recovery of tumor size depends on accurate margin preservation in tomographic reconstructions.
Sound speed distribution of the breast, as well as tumor margins, can be reconstructed using USRT,
and it has been shown that USRT has the potential to distinguish breast tumors from normal breast
tissue (Li et al., 2009a). However, improvements in image quality must be made in order for this
technology to be clinically useful as a breast cancer detection and diagnostic tool.
4.1 Implementation of USRT with MTV regularization
USRT recently demonstrated the ability to significantly improve image quality, using arrival rimes
of both transmission and reflection data (Nguyen and Huang, 2014a). Ultrasound waveform to-
mographic reconstructions can be improved via a modified total-variation (MTV) regularization
scheme (Lin and Huang, 2013). However, ultrasound waveform inversion is computationally ex-
pensive, and it would be of interest to develop an alternative. Here, we aim to improve USRT image
reconstructions via a modified total-variation (MTV) regularization scheme, thereby developing a
more computationally efficient method of reconstructing breast sound speed (Intrator et al., 2015).
MTV is a hybrid regularization scheme, combining the smoothing capability of Tikhonov regu-
larization with the margin-preserving capability of total-variation (TV) regularization. This USRT
reconstruction scheme is offered as a proof of concept using transmission data, and can be extended
to include reflection data. Our new USRT with MTV regularization (USRT-MTV) is a Tikhonov-TV
hybrid, reducing image noise while preserving margins. We apply our new USRT-MTV method to
ultrasound transmission data from numerical phantoms, and compare the results with those obtained
24


using Tikhonov regularization. This is the first time to incorporate the MTV regularization scheme
into USRT.
The reconstruction algorithm is implemented by solving two decoupled minimization subprob-
lems. We minimize the misfit function using an alternating minimization algorithm (Bauschke et ah,
2006; Lin and Huang, 2013, 2015; Wang et ah, 2008), decomposing the inversion problem into
two subproblems: one is an U-norm-based Tikhonov regularization problem (Nguyen and Huang,
2014a), and the other is an G -norm-based TV regularization problem. We use an LSQR method to
solve the first subproblem, and apply the split Bregman method to the second subproblem (Gold-
stein and Osher, 2009; Osher et ah, 2005). The split Bregman method is preferred to other iterative
methods, as it avoids selection of the smoothing parameter in the TV term (significantly improving
algorithm robustness and computational efficiency), and as it also converges to the true rather than
approximated TV solution (Goldstein and Osher, 2009). We apply our new USRT-MTV method
to ultrasound transmission data from numerical phantoms, employing an automatic method to pick
arrival rimes of synthetic ultrasound transmission signals for tomographic reconstructions.
The transmission ultrasound tomography algorithm uses an LSQR iterative solver with Tikhonov
regularization to solve for As:
As = min ||HAs At||2 + A||LAs||2, (26)
As
where | |HAs At | |2 is the data misfit term describing the misfit between the data and the forward
modeled results, A| |As112 is the Tikhonov regularization term, L is the Laplacian operator V2,
and A is the damping parameter used to weight the relative contributions of the misfit term and
the regularization term. We obtain tomographic reconstructions of numerical phantoms using our
new USRT algorithm with MTV regularization (USRT-MTV), and compare these results with those
produced using USRT with Tikhonov regularization (USRT-Tikhonov).
To retain the smoothing benefits of Tikhonov regularization, while incorporating the margin-
preserving properties of TV regularization, we develop a modified TV (MTV) regularization method
for USRT. This method was previously developed by our group for use with ultrasound waveform
inversion (Lin and Huang, 2013, 2015); here it is adapted for use with USRT inversion.
In order to implement this nested regularization, we introduce a second parameter u, which
25


adds an additional step for solving the inverse problem. The minimization problem with MTV
regularization is formulated as follows:
E{s) = min{||f7s D||| + Ai||s u||| + A2||u||7v}, (27)
S,U
where Ai and A2 are both posirive regularization parameters. Equivalently, eq. (27) can be written
as
E(s) = min{ min{||i?s D||| + Ai||s u|||} + A211u11}- (28)
U S
It is now clearer that Ai controls the tradeoff between the data misfit term and the Tikhonov regu-
larization term (i.e. the amount of smoothing/noise reduction in the inversion), while A2 controls
the tradeoff between the Tikhonov-regularized data misfit term and the TV-regularized term (i.e. the
amount of margin preservation in the inversion). To solve eq. (28), the inversion is first performed
to minimize s with Tikhonov regularization. Next, this solution is piped to a second minimization
problem to minimize u using MTV regularization. This solution is used as the input for the next
iteration, minimizing s again. The iterative solver is halted once the updates and u'/ri (for
iterations k = 1, 2,...) are sufficiently small, as before. This alternating minimization method is
formulated in two minimization subproblems as follows:
s(fc) = min{77i(s)} = min j||77s T\\% + Ai||s u(fc1)|||}| , (29)
u(fc) = min{i72(u)} = min {||sfc_1 u||| + A2||u|Ny} . (30)
U U L J
4.2 Numerical phantoms
The mean sound speed in the breast has been found to range from approximately 1440 m/s (fatty
breast tissue) to approximately 1505 m/s (dense breast tissue) (Li and Duric, 2008). Dense breasts
present a great imaging challenge for mammography, the current gold standard for breast cancer
detection. Furthermore, dense breasts are correlated with increased breast cancer risk (Boyd et ah,
26


x (mm) x (mm)
-100 0 100 -100 0 100
1505 1520 1535 1550 1565 1505 1520 1535 1550 1565
Figure 2: Synthetic first-arrival times are generated for transmission tomography using two different numerical
phantoms. The phantom on the left contains a 30-mm diameter tumor-mimicking target with a sound speed
of 1560 m/s, in a background of 1500 m/s. The phantom on the right contains a tumor-mimicking target with
concentric regions of different sound speed, where the innermost region is 50 mm in diameter (1550 m/s), the
next larger is 100 mm in diameter (1530 m/s), and the next larger is 150 mm in diameter (1510 m/s). The
background sound speed is again 1500 m/s. Both phantoms are 223x223 mm.
2007). It is therefore imperative to develop better methods for imaging dense breasts. We use two
numerical phantoms, representing idealized cross-sections of a human breast with different types of
lesions, for this study. The first one is a simple phantom representing a tumor of homogeneous sound
speed, 30 mm in diameter (Fig. 2a). However, tumors are thought to have increased sound speed
because of remodeling of the extracellular matrix surrounding the tumor mass (Macklin, 2010). We
therefore also investigate a concentric numerical phantom (Fig. 2b) that may better recapitulate this
environment. Furthermore, this concentric numerical phantom with piecewise-constant sound speed
is better suited for comparing edge preservation. To gain a better understanding of how USRT might
detect breast tumors in dense breasts, we chose 1500 m/s as the background speed for our phantoms
to mimic a dense breast. Since breast tumors typically have a higher sound speed than surrounding
breast tissue (Li et al., 2009a), higher sound speeds are assigned to the circles representing tumors.
27


5 Sound speed reconstruction results
5.1 Reconstruction of two different phantoms
USRT-Tikhonov
USRT-MTV
1500 1520 1540 1560 1500 1520 1540 1560
Figure 3: Reconstructions of the different phantoms, comparing reconstructions achieved with USRT-Tikhonov
and USRT-MTV. It is evident that USRT-MTV provides a superior reconstruction of the sound speed field, more
accurately reconstructing the piecewise-constant nature of the phantoms.
In reconstructions of the phantom with the small target (Fig. 3), it is evident that use of the
parallel array leads to distorted coverage. The reconstruction of the phantom containing the small
28


Phantom
Figure 4: Horizontal sound speed profiles of reconstructions of phantom with small target, achieved using
the two different regularization schemes. The top figure, black line, is the actual phantom profile. The second
figure shows the profile of the reconstruction obtained using USRT-Tikhonov (blue line) and the third figure
shows the profile of the reconstruction obtained using USRT-MTV (magenta line). It is evident that USRT-MTV
yields the most accurate sound speed reconstruction.
target, using USRT-Tikhonov (Fig. 3, upper left), there is obvious reconstruction inaccuracy in the
center of the target, and some oversmoothing of the margins. The reconstruction of the phantom
containing the small target, using USRT-MTV (Fig. 3, upper right), the coverage distortion is much
less pronounced, and the margins are sharper.
The reconstruction of the phantom containing the concentric target, using USRT-Tikhonov (Fig.
3, lower left), again the parallel array leads to distorted coverage. The shortcomings of Tikhonov
regularization in not being able to preserve margin structure is much more evident than with the
reconstructions of the phantom with small target note the extreme distortion at the boundaries
where sound speed changes. The reconstruction of the phantom containing the concentric target,
using USRT-MTV (Fig. 3, lower right), the coverage distortion is much less pronounced, the mar-
gins at all boundaries are sharper, and the reconstruction of the overall piecewise-constant nature of
the phantom is improved.
In horizontal profile views of the reconstructions of the phantom containing the small target (Fig.
4), the significant reconstruction inaccuracies in the USRT-Tikhonov reconstruction (Fig. 4, middle
29


Phantom
Figure 5: Vertical sound speed profiles of reconstructions of phantom with small target, achieved using the two
different regularization schemes. The top figure, black line, is the actual phantom profile. The second figure
shows the profile of the reconstruction obtained using USRT-Tikhonov (blue line) and the third figure shows
the profile of the reconstruction obtained using USRT-MTV (magenta line). It is again evident that USRT-MTV
yields the most accurate sound speed reconstruction.
Figure 6: Horizontal sound speed profiles of reconstructions of phantom with concentric target, achieved using
the two different regularization schemes. The top figure, black line, is the actual phantom profile. The second
figure shows the profile of the reconstruction obtained using USRT-Tikhonov (blue line) and the third figure
shows the profile of the reconstruction obtained using USRT-MTV (magenta line). It is evident that USRT-MTV
yields the most accurate sound speed reconstruction.
30


y (mm)
Figure 7: Vertical sound speed profiles of reconstructions of phantom with concentric target, achieved using
the two different regularization schemes. The top figure, black line, is the actual phantom profile. The second
figure shows the profile of the reconstruction obtained using USRT-Tikhonov (blue line) and the third figure
shows the profile of the reconstruction obtained using USRT-MTV (magenta line). It is again evident that
USRT-MTV yields the most accurate sound speed reconstruction.
plot) are much more evident. The sound speed values in the center of the target are significantly
lower than the true value, and margin values are higher than the true value. Reconstruction of this
same phantom using USRT-MTV (Fig. 4, lower plot) demonstrates the more accurate sound speed
reconstruction, and improved margin reconstruction as well. Vertical profiles (Fig. 5) of the same
reconstructions further corroborate these results. The USRT-Tikhonov reconstruction profile (Fig.
5, middle plot) demonstrates the characteristic over-smoothing profile, overshooting the targets
sound speed at the margins and undershooting in the center. The USRT-MTV reconstruction (Fig.
5, lower plot), the profile looks quite similar to the horizontal profile, preserving sound speed much
more accurately than the USRT-Tikhonov counterpart.
In horizontal profile views of the reconstructions of the phantom containing the concentric target
(Fig. 6), the significant oversmoothing in the USRT-Tikhonov reconstruction (Fig. 6, middle plot) is
again evident, although reconstruction of the maximum (center region) sound speed is good relative
to the phantom with small target this may be due to the difference in sizes of the area of the small
target (30 mm diameter) versus the area of the center of the concentric target (50 mm diameter).
31


Reconstruction of this same phantom using USRT-MTV (Fig. 6, lower plot) demonstrates overall
improved sound speed reconstruction in all sound speed regions, as well as more faithful recon-
struction of margins where sound speed changes. The maximum sound speed is undershot. Vertical
profiles (Fig. 7) of the same reconstructions are more degraded. The USRT-Tikhonov reconstruction
profile (Fig. 7, middle plot) is nearly featureless, in particular at the transition from 1550 m/s (center
region) to 1530 m/s (middle region). The USRT-MTV reconstruction (Fig. 7, lower plot) contains
fewer margin features in this cross section, but is still better than USRT-Tikhonov at retaining some
of the piecewise constant nature of the phantom.
5.2 Reconstruction of phantoms with varying target and background sound speeds
We assess the performance of USRT-Tikhonov (Fig. 8) and USRT-MTV (Fig. 9) when reconstruct-
ing the phantoms having different tissue backgrounds and different tumor sound speeds. Phantoms
representing different background tissue sound speeds from relatively fatty breast tissue (1410
m/s), to relatively dense breast tissue (1500 m/s) are constructed for this study. Phantom tumor
sound speeds ranged from benign mass sound speeds (1480 and 1510 m/s), to malignant mass sound
speeds (1550, 1570 m/s). The sound speed overlap range in the middle represents sound speeds that
are potentially either benign and malignant masses.
For the reconstruction target with a benign sound speed of 1510 m/s in a background of 1480
m/s (Fig. 10), the USRT-Tikhonov reconstruction (middle plot; blue) significantly undershoot the
targets sound speed, and distorts the margins as well. USRT-MTV (lower plot; magenta) overshoots
the targets sound speed, but less significantly, and also avoids the margin distortion present in the
USRT-Tikhonov reconstruction.
For the reconstruction target in the overlap sound speed range between benign and malignant
(Fig. 11), USRT-Tikhonov again results in significantly undershooting the targets sound speed, and
distortion of tumor margins. USRT-MTV again overshoots sound speed, but, less so, and recon-
structs superior margin information.
Similar results are seen for reconstructions of the malignant target (Fig. 12), except the MTV-
regularized algorithm seems to handle the large difference between target and background less well
32


Benign mass
1480 m/s 1510 m/s
Target sound speed
Sound speed overlap
1530 m/s 1540 m/s

B
DB
Malignant mass
1550 m/s 1570 m/s
BB
BB
BB
BB
BB
BB
Figure 8: Comparison of the ability of USRT-Tikhonov to reconstruct targets of various sound speed (repre-
senting the sound speed range of benign to malignant masses) in backgrounds of various sound speed (rep-
resenting the sound speed range of fatty to glandular breast tissue). Note that the colorscale is constrained to
the value range 1500 m/s to 1570 m/s.
33


Benign mass
1480 m/s 1510 m/s
Target sound speed
Sound speed overlap
1530 m/s 1540 m/s
H
B
BB
Malignant mass
1550 m/s 1570 m/s
BB
BB
BB
BB
BB
BB
Figure 9: Comparison of the ability of USRT-MTV to reconstruct targets of various sound speed (representing
the sound speed range of benign to malignant masses) in backgrounds of various sound speed (representing
the sound speed range of fatty to glandular breast tissue). Note that the colorscale is constrained to the value
range 1500 m/s to 1570 m/s.
34


than the previous cases. Results for the entire comparison grid are similar, with USRT-MTV recon-
structing both sound speed and margins much more accurately than USRT-Tikhonov in all cases.
Of course, these reconstuction results could be different if different regularization parameters
are used. Optimal parameters are needed for USRT reconstructions.
Figure 10: Comparison of the ability of USRT-Tikhonov and USRT-MTV to reconstruct targets representing a
benign mass in a background of glandular (dense) breast tissue. The top two plots (black line) are horizontal
(x direction) and vertical (y direction) profiles of the actual phantom. The middle two plots (blue) and lower two
plots (magenta) are profiles of reconstructions achieved using USRT-Tikhonov and USRT-MTV, respectively. It
is evident that USRT-MTV yields the most accurate sound speed reconstruction in all cases.
35


_E 1'
x (mm)
1.
&
Figure 11: Comparison of the ability of USRT-Tikhonov and USRT-MTV to reconstruct targets representing a
mass of uncertain etiology in a background of glandular (dense) breast tissue. The top two plots (black line)
are horizontal (x direction) and vertical (y direction) profiles of the actual phantom. The middle two plots (blue)
and lower two plots (magenta) are profiles of reconstructions achieved using USRT-Tikhonov and USRT-MTV,
respectively. It is evident that USRT-MTV yields the most accurate sound speed reconstruction in all cases.
.co
E,
o
o
CD
Phantom
.CO
E
o
o
2
Phanton
USRT-MTV
USRT-TIkhorK
Figure 12: Comparison of the ability of USRT-Tikhonov and USRT-MTV to reconstruct targets representing a
malignant mass in a background of glandular (dense) breast tissue. The top two plots (black line) are horizontal
(x direction) and vertical (y direction) profiles of the actual phantom. The middle two plots (blue) and lower two
plots (magenta) are profiles of reconstructions achieved using USRT-Tikhonov and USRT-MTV, respectively. It
is evident that USRT-MTV yields the most accurate sound speed reconstruction in all cases.
36


6 Conclusions and future work
We have developed a new ultrasound bent-ray tomography (USRT) method using a modified total-
variation (MTV) regularization scheme; this is the first time to incorporate an MTV regularization
scheme into USRT. We decompose the inverse problem into two subproblems using an alternat-
ing minimization algorithm, resulting in a computationally efficient and accurate reconstruction
algorithm. Our use of the split Bregman algorithm avoids the non-differentiability of the second
subproblem, consequently improving the robustness of our method.
We have applied our new USRT-MTV method to ultrasound transmission data from numeri-
cal phantoms, and compared the results with those obtained using the Tikhonov regularization. Our
new USRT-MTV is a Tikhonov-TV hybrid, and reduced image noise while preserving margins. The
high-quality sound-speed tomography results of numerical phantom data demonstrate that USRT-
MTV significantly improves the robustness of USRT. Furthermore, image quality of USRT-MTV
reconstructions is less susceptible to distortion from incomplete data coverage from parallel trans-
ducer array geometry.
Future work includes applications of our new method to tumor-mimicking phantom data and
patient data acquired at the University of New Mexico hospital using UANLs custom-built breast
ultrasound tomography prototype (Huang et al., 2015). Some related research results have been
published in the following two proceedings papers (Fluang et al., 2015; Intrator et al., 2015):
M. Intrator, Y. Fin, T. Chen, J. Shin and U. Huang. Ultrasound bent-ray tomography with
a modified total-variation regularization scheme. In Proceedings of SPIE Medical Imaging,
2015.
U. Huang, J. Shin, T. Chen, Y. Fin, M. Intrator, K. Hanson, K. Epstein, D. Sandoval and M.
Williamson. Breast ultrasound tomography with two parallel transducer arrays: preliminary
clinical results. In Proceedings of SPIE Medical Imaging, 2015.
37


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Full Text

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ULTRASOUNDBENT-RAYTOMOGRAPHY WITHAMODIFIEDTOTAL-VARIATIONREGULARIZATIONSCHEME by MIRANDAHUANGINTRATOR B.A.,UniversityofCaliforniaatSantaCruz,2007 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience Bioengineering 2015

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ThisthesisfortheMasterofSciencedegreeby MirandaHuangIntrator hasbeenapprovedforthe BioengineeringProgram by KendallHunter,Chair DaewonPark,Advisor LianjieHuang,Advisor Date:July10,2015 i

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Intrator,MirandaHuangM.S.,Bioengineering Ultrasoundbent-raytomographywithamodiedtotal-variationregularizationscheme ThesisdirectedbyAssistantProfessorDaewonParkandDr.LianjieHuang ABSTRACT Thesound-speeddistributionofthebreastcanbeusedforcharacterizingbreasttumors,because theytypicallyhaveahighersoundspeedthannormalbreasttissue.Thisisunderstoodtobetheresultofremodelingoftheextracellularmatrixsurroundingtumors.Breastsound-speeddistribution canbereconstructedusingultrasoundbent-raytomography.Wehaverecentlydemonstratedthat ultrasoundbent-raytomography,usingarrivaltimesofbothtransmissionandreectiondata,significantlyimprovesimagequality.Tofurtherimprovetherobustnessoftomographicreconstructions, wedevelopanultrasoundbent-raytomographymethodusingamodiedtotal-variationregularizationschemeandimplementitusingtransmissiondata.Regularizationisoftenusedinsolving inverseproblemsbyintroducingconstraintsoninversionresultssuchassmoothness.Tikhonov regularizationisawidelyusedregularizationschemethattendstosmoothtomographicimages,but oversmoothingcanobscurecriticaldiagnosticdetailsuchastumormargins.Total-variationregularizationisanothercommonregularizationschemethathelpspreservetumormargins,butatthecost ofincreasedimagenoise.Ournewultrasoundbent-raytomographywiththemodiedtotal-variation regularizationschemeemploysaTikhonov-Total-Variationhybridregularizationmethod,reducing imagenoisewhilepreservingmargins.Wevalidateournewmethodusingultrasoundtransmission datafromnumericalphantoms,andcomparetheresultswiththoseobtainedusingTikhonovregularization. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:DaewonParkandLianjieHuang ii

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Declarationoforiginalwork ByMirandaHuangIntrator ThispageistoassertthatthisthesisfortheMasterofSciencedegreewasindependentlycomposed andauthoredbymyself,usingsolelythereferencedsourcesandsupportfrommyadvisors,fellow students,theDepartmentofBioengineeringattheUniversityofColoradoDenverAnschutzMedicalCampusandtheGeophysicsGroupoftheEarthandEnvironmentalSciencesDivisionatLos AlamosNationalLaboratory.Thethesisdescribesthecontributionofanewultrasoundbent-raytomographyUSRTmethodusingamodiedtotal-variationMTVregularizationscheme,whichis thersttimetoincorporateanMTVregularizationschemeintoUSRT.Acomputationallyefcient andaccuratereconstructionalgorithmisdevelopedandimplementedforreconstructionofnumericalphantoms,comparingresultswiththoseobtainedusingaTikhonovregularizationscheme.This researchwillimproveUSRT'scapabilityforbreastcancercharacterizationintheclinic. Thisstatementisapproved by MirandaHuangIntrator and DaewonPark,Advisor and LianjieHuang,Advisor iii

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ACKNOWLEDGEMENTS Iwouldliketothankeveryonewhohelpedme,directlyandindirectly,inachievingthismilestone. FirstandforemostIwouldliketothankmyadvisors,Dr.DaewonPark,Dr.LianjieHuang,andDr. KendallHunter.ThetimeIspentinDr.Park'sTranslationalBiomaterialsReserachLaboratorywas invaluableinlearningcriticalbenchtopskillsIthankDr.Parkforhispatienceandhistrustinme, andforallowingmetopursueresearchasaGraduateResearchAssistantunderthesupervisionof Dr.LianjieHuangatLosAlamosNationalLaboratory.IthankDr.LianjieHuangforhisenthusiasm andkindsupportduringmyresearchinhisgroup.Iamalsodeeplyindebtedtothepostdocsthere, whoprovidedinvaluableassistancewithallmatterslargeandsmallDrs.YouzuoLin,TingChen, JunseobShin,KaiGaoandNghiaNguyen.Lastbutcertainlynotleast,myfamilyhasserved toencourage,galvanizeandconsolemeinallmatterssincetheverybeginning.Iwillalways bedeeplygratefulto,andinspiredby,them.ThisresearchwassupportedbyDr.DaewonPark attheUniversityofColoradoatDenverAnschutzMedicalCampus,byDr.LianjieHuangat LosAlamosNationalLaboratory,andbytheBreastCancerResearchProgramoftheU.S.DoD CongressionallyDirectedMedicalResearchProgramsthroughContractMIPR0LDATM0144to LosAlamosNationalLaboratory. iv

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TABLEOFCONTENTS Chapter 1Breastcancerfactsandgures1 1.1Screeningwithmammography............................1 1.1.1Doesmammographypreventdeaths?....................2 1.1.2Breastdensity:newlyidentiedriskfactor.................3 1.2Ultrasoundtomographyforbreastcancerscreening.................4 2Ultrasoundacoustics7 2.1Modelingsoundpropagationinbiologicaltissue...................8 2.1.1Thescalar-waveequation...........................8 2.1.2Therayapproximation............................9 2.1.3Huygens'Principle..............................9 2.1.4Fermat'sPrinciple..............................10 2.1.5Snell'sLaw..................................10 2.2B-modeultrasoundimagingofbiologicaltissues..................12 2.3OpportunitiesforimprovementofB-modeultrasoundimaging...........13 3USRTforbreastcharacterization15 3.1Theforwardandinverseproblem...........................16 3.1.1Inverseandill-posedproblems........................16 3.1.2Theforwardproblem.............................17 3.2Regularizedinversion.................................19 3.2.1USRTwithTikhonovregularization.....................19 3.2.2Tikhonovregularization...........................21 3.2.3USRTwithtotal-variationregularization...................21 3.2.4Total-variationregularization.........................22 v

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4USRTwithMTVregularization24 4.1ImplementationofUSRTwithMTVregularization.................24 4.2Numericalphantoms.................................26 5Soundspeedreconstructionresults28 5.1Reconstructionoftwodifferentphantoms......................28 5.2Reconstructionofphantomswithvaryingtargetandbackgroundsoundspeeds...32 6Conclusionsandfuturework37 References 37 vi

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1Breastcancerfactsandgures Aftermorethanthreedecadesofscreeningefforts,breastcancerremainsthemostcommontype ofcancerandoneoftheleadingcausesofcancer-relateddeathinwomenworldwideFerlayetal., 2013.Anestimated231,840newcasesofinvasivebreastcancerareexpectedtobediagnosed amongUSwomenaloneduring2015;about2,350newcasesareexpectedinmen.Thestandardof careintheUSisannualscreeningbymammographyforallwomenovertheageof40American CancerSociety,2015,howeverarapidlyincreasingnumberofstudiesarequestioningthisstandard Biller-AndornoandJ uni,2014;BleyerandWelch,2012;Essermanetal.,2009,2014;Independent UKPanelonBreastCancerScreening,2012;OngandMandl,2015;Rippingetal.,2015;The CanadianTaskForceonPreventiveHealthCare,2011. 1.1Screeningwithmammography MammographyusesX-raytoimagethebreastitcompressesthebreastbetweentwoplatesto imagetissueradiodensity.ThereareanumberofconcernswithmammographyX-raysareionizing radiation,whichisitselfariskfactorforcancer.Mammographyalsooftenfailstodifferentiate betweenbenignandmalignantmassesultrasoundoutperformsmammographyfortumordetection indensebreastsFaschingetal.,2006;Kolbetal.,2002.Mammographyhasdifcultyvisualizing lesionsindensebreasts,requiringfollow-upimagingthatisroutinelyaccomplishedbyultrasound imaging.Furthermore,patientswithdensebreastsareconsideredhigher-riskthantheirnon-dense breastcounterpartsBoydetal.,2007;Eliasetal.,2014;McCormackanddosSantosSilva,2006. Mammographycanalsoresultinfalse-negativeresults,missingcancersultrasoundcanoften detecttheseoccultlesionsGordonandGoldenberg,1995;Uchidaetal.,2008.Mammography compressesthebreastwhichisuncomfortableorpainfulformanypatients,whichmayinterfere withpatientcomplianceSapiretal.,2003.Compressionofthebreastmayinterferewithimage interpretationanddiagnosisaswell. 1

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1.1.1Doesmammographypreventdeaths? Themostpressingconcerntoday,however,isthatpopulation-widemammographyscreeninghas beenassociatedwithariseintheincidenceofbreastcancerbutnotwithadeclineinpresentationof advanced-stagebreastcancersoroverallbreastcancermortalityBleyerandWelch,2012;Esserman etal.,2009.Thisismostlikelydueto:thelargenumberoffalse-positivendingsandbreast canceroverdiagnosisassociatedwithmammographyEssermanetal.,2009;PaceandKeating, 2014,andthefactthatmammographyisnotconductedinamannerthatdetectsthemost aggressiveandlethalcancersDrukkeretal.,2014;Essermanetal.,2009. Afalse-positiveiswhenthemammographicndingraisessuspicionofbreastcancer,andleads toadditionalimagingorbiopsy,butultimatelydoesnotleadtoacancerdiagnosis.Overdiagnosisis denedasthedetectionofatumorthroughscreeningthatwouldnothavebecomeclinicallyevident intheabsenceofscreening. Earlydetectionandscreeninghavebeenshowntobemostsuccessfulwhenpre-malignantlesionscanbedetectedandeliminated,aswiththeremovalofadenomatouspolypsduringcoloncancercolonoscopyscreening.Coloncancerhasthusenjoyedasignicantdecreaseininvasive cases.Incontrast,ductalcarcinomainsituDCIS,whichisconsideredaprecancerouslesion,was rarepriortoscreeningbutnowrepresents25-30%ofallbreastcancerdiagnosesEssermanetal., 2009. A2014studyassessingtherisksandbenetsofmammographyfoundthatper10,000women screenedwithmammographyinaten-yearperiod,thereareordersofmagnitudedifferencebetween thenumberofdeathsavertedbymammographyscreeningtens,thenumberofoverdiagnoses hundreds,andthenumberoffalse-positivesthousandsandunnecessarybiopsiesalsothousandsPaceandKeating,2014.Anotherstudyinthesameyearconcludedthatscreen-detected bymammographycancersweresignicantlymoreoftenlow-risktumors%,ofwhich54% wereconsideredtobeultra-lowriskwhencomparedtointervalbiologicallyaggressivecancers Drukkeretal.,2014.Toclarifythedistinctionbetweenindolentnon-lethalandlethalaggressive cancersrequiringtreatment,someleadingresearchersareevencallingforachangeinterminology, termingnon-lethalbreastlesionsasIDLEindolentlesionsofepithelialorigininsteadofcalling 2

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themcancerEssermanetal.,2009,2014. Breastcanceralsorepresentsanenormouspersonalandsocietalnancialburden.Out-of-pocket expensesforabreastcancerpatientaverageabout$1500permonthArozullahetal.,2004,anda recentstudyestimatedthetotalcostsoffalse-positivesandoverdiagnosisintheUSisapproximately $4billionannuallyOngandMandl,2015.Thetotalcostofmammographyscreeningisestimated atnearly$8billionannually,withthetwolargestdriversofcostbeingscreeningfrequency,and percentageofwomenscreenedO'Donoghueetal.,2014. Thus,thedatanowpointstoanewpictureofmammography,whichhaseffectivelyincreased thedetectionofindolentcancerssuchasductalcarcinomainsituDCISandoftenmissesthemost aggressiveintervalcancers.Thisscreeningparadigmhasincreasedbreastcancer-relatedmorbidity becauseofoverdiagnosisofnon-life-threateningcancers,whilefailingtoreducemortalitydueto undetectedaggressivebreastcancers.Thereisaclearandpressingneedtodevelopimprovedbreast cancerscreeningmethods. 1.1.2Breastdensity:newlyidentiedriskfactor MammographicbreastdensityMBDisameasureofradiodensebroglandulartissueinthebreast. ThisbiomarkerisverystronglyassociatedwithincreasedcancerriskBoydetal.,2007,2010;Elias etal.,2014;McCormackanddosSantosSilva,2006,stronger,infact,thanmostotherestablished breastcancerriskfactors,exceptforageandsomegeneticfactors.Furthermore,MBDhasbeen indicatedasapromisingbiomarkerforassessingriskofmoreaggressiveintervalcancersBertrand etal.,2013;Boydetal.,2007;Buistetal.,2004;Kerlikowskeetal.,2007;Mandelsonetal.,2000; McCormackanddosSantosSilva,2006;Nothackeretal.,2009;Porteretal.,2007.Womenwith densebreastsmayindeedbenetfromshorterscreeningintervals.Breastdensityisalsooneof theleadingfactorsinfalse-negativemammographicndingsMaetal.,1992;Mandelsonetal., 2000;Nothackeretal.,2009;Porteretal.,2007;Roubidouxetal.,2004.Epidemiologicalstudies conrmthatpatientscancategorizedbyriskaccordingtobreastdensity,andthatbreastdensity mayrepresentanintermediatephenotypeduetogeneticfactorsBoydetal.,2010;Martinand Boyd,2008;Nothackeretal.,2009;Provenzanoetal.,2008;Vachonetal.,2007.Breastdensity 3

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isaheritabletraitBoydetal.,2002,2010,andhasbeenlinkedtoanumberofotherriskfactors, suchashightumorgradeandlargetumorsizeatpresentationRoubidouxetal.,2004.Thefact thatmammography'susefulnessiscontroversial,inconjunctionwiththefactthatitsperformance issuboptimalforhigh-riskpopulationswherescreeningisneededmost,motivatesustodevelopa methodofbreastcancerscreeningthatissafeformorefrequentusewithhigherriskpopulations. 1.2Ultrasoundtomographyforbreastcancerscreening Supplementalbreastultrasoundforwomenwithdensebreastsmayenablethedetectionofsmall, otherwiseoccult,breastlesions,butthismayresultinanincreasedbiopsyrateNothackeretal., 2009;Uchidaetal.,2008,andadditionalbreastultrasoundscreeningfordensebreastsbringsthe intervalcancerratedowntotheintervalcancerratefornon-densebreastsCorsettietal.,2011.Anothermorerecentmeta-analysiscorrelatingimagingfeatureswithbreasttumormalignanceHER2 overexpressionindicatedthatsuspectedmalignanceasassessedbybreastultrasoundmayindeed indicatemalignanceEliasetal.,2014.However,mammographicspecicitydeclineswithincreasingbreastdensityKolbetal.,2002,andthereforeothermodalitiesmustbedevelopedtoservea newlyidentiedhigh-riskpopulationviabettersensitivitytothiscriticalbiomarker. Ultrasoundiswidelyavailable,non-invasive,non-ionizing,low-costandisareal-timeimagingmodality.Itiswell-knownthatsupplementingmammographywithbreastultrasoundhasbetter detectionratesthanmammographyaloneinparticular,forhigh-riskpatientswithdensebreasts Kolbetal.,2002;Maetal.,1992;Mainieroetal.,2013.Indeed,thesensitivityofmammography alonedecreasesfrom100%infattybreaststo45%inextremelydensebreastsBergetal.,2004. Whole-breastbilateralbothbreastssonographyhasbeenshowntodetectsmallnon-palpableinvasivebreastcancersnotvisualizedbymammography,particularlyindensebreastsBuchbergeretal., 2000;GordonandGoldenberg,1995;Kaplan,2001;Kolbetal.,1998,2002,andhigh-riskpatients weretwotothreemorelikelytohavecancersseenonlysonographicallyBerg,2003;Buchberger etal.,2000;GordonandGoldenberg,1995;Kaplan,2001;Kolbetal.,2002.Inpatientsdiagnosedwithinvasivebreastcancer,survivalisafunctionoftumorsizeMichaelsonetal.,2002,and ultrasoundhasprovenbetteratmeasuringtumorsizeindensebreastsSchreer,2009.Automated 4

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wholebreastultrasoundplusmammographyshowedsignicantlybettercancerdetectionthanmammographyaloneinhigh-riskpatientsKellyetal.,2010,andhasalsobeensuccessfullyusedfor computer-aideddiagnosisinapilotstudyof147patients,withpromisingsensitivity.5%and specicity.5%valuesMoonetal.,2011. Whilebreastultrasoundhaslongbeenanadjunctimagingmethodtomammographyforbreast cancer,currenttechniqueslackthesensitivityandspecicitytoserveasastandalonemodality Bergetal.,2008.B-modeultrasoundisqualitativeinsteadofquantitative,haslimitedresolutionandcontrast,andhasspecklenoisewhichmayobscuresmalltissuestructuresandartifacts, allpreventingitfromwhollyreplacingmammographyDuricetal.,2005.Furthermore,handheldultrasoundtransducersmakeimagequalityhighlyoperator-dependentYaffe,2008,further exacerbatingimageinterpretationMendelsonetal.,2001. Toaddresstheseissues,theimagingmodalityofultrasoundtomographyhasbeenanareaof recentlyacceleratingdevelopment.InultrasoundtomographyUST,thebreastisimmersedin waterandautomaticallyscannedin`slices'tobuilda3Dimageofthebreast.Thebreastisnot compressedasitiswithmammography,andthededicatedbreastscanninginstrumentresultsinless operator-dependenceoroperator-independenttoobtainhigh-qualitydiagnosticimages.Intheearly 1970s,Kossoffetal.1973reportedthecompositemixedfattyandglandulartissuebreasttohave anaveragesoundspeedof1510m/sinpre-menopausaland1468m/sinpost-menopausalwomen. Lietal.2009aacquiredpatientdatausingadedicatedbreastUSTscannerwitharing-shaped arrayoftransducersinpendant-modeinwhichthepatientliesface-downonaUSTtablewith breastsuspendedinawarmwaterbaththebreastisthenconsideredpendant.Theirsound-speed tomogramsreconstructedusingabent-rayUSTalgorithmcouldreliablydifferentiatebetweenfatty 9m/sandglandular 21m/sbreasttissue,aswellasbetweenbenign 27m/s andmalignant 17m/sbreastlesions,suggestingthereisgreatpromisefortheuseofthis modalityasabreastcancercharacterizationmethodLietal.,2009a.Theremaybecomplications imagingclosetothechestwallusingthering-shapedarray.Itisimportanttobeabletoimagethis regionbecausethisiswheremetastasizedtumorsmayappearinthelymphnodesNebekerand Nelson,2012.Forthisreason,aUSTscannerwithaparalleltransducerarraywasdesignedand manufacturedbyLianjieHuang'sgroupatLosAlamosNationalLaboratorythatmaybetterimage thislymphnoderegionanddifferentbreastsizes.Thisscannerisnowusedforaclinicalstudyat 5

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theUniversityofNewMexicoHospitalHuangetal.,2015. 6

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2Ultrasoundacoustics Soundcanberigorouslydenedaspropagatingdifferentialpressurethatis,propagatingzones ofcompressionandexpansion,orrarefactionwithagivenoscillationfrequency, f .Thesound frequencyspectrumisconventionallysplitintothreeranges: < 20Hzisreferredtoas infrasound 20HzkHzisreferredtoasthe audible range,and > 20kHzisreferredtoas ultrasound Humanhearingistypicallyconnedtotheaudiblerangeoffrequencies.Ultrasoundisusedinmany differentareasofscience,notablyinmedicalimagingwhereitiscalled sonography .Theprocessof medicalultrasoundimagingis,briey,apiezoelectrictransducerisusedtoconvertelectricalenergy tomechanicalenergy,creatingasoundwavethatpropagatesthroughthetissuebeingimaged.Dependingoncertaintissueparameters,someportionoftheappliedultrasoundisreturnedbacktothe transducer,andthisamountofenergyistranslatedintoagrayscaleimagewheredarkpixelsrepresentlessenergyreturnedanechoicorhypoechoicregionsandlighterpixelsrepresentmoreenergy returnedhyperechoicregions.Asingletransducergeneratesa1Dimageofit'slineofsightinto thetissue.Transducerarraysareusedtogenerate2Dimage`slices'inmedicalultrasound. Typicalmedicalimagingultrasoundtransducersoperateinthe2MHzrangeLaugierand Ha at,2011.Thesizeofobjectsthataredetectablewiththesetransducersdependsontheultrasoundwavelength,whichisrelatedtofrequencyinthefollowingway: = c f = cT; where isthewavelengthinmm, c isthepropagationmedium'sspeedofsoundmm/ s, f is frequencyMHz,and T istheperiod s.Mostclinicalultrasoundimagingsystemsassume aconstantsound-speedvalue c of1.540mm/ s.Thus,thetypicalwavelengthrangeofclinical ultrasoundimagingrangesfrom0.77mmat2MHzdownto0.1mmat15MHz. 7

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2.1Modelingsoundpropagationinbiologicaltissue Biologicaltissuesareviscoelasticmaterials,inwhichbothbulkcompressioncorrespondingto longitudinalwavesandshearwavescanpropagate.Inbiologicaltissues,unlikeinuids,shearing straincanbetransmittedtoadjacentlayersoftissuebecauseofthestrongbindingbetweenparticles. However,becauseshearwavesinsofttissuearehighlyattenuatedatultrasonicfrequenciesconventionally20kHzanduptheyaretypicallyignored,leavingonlycompressionwavestoconsider LaugierandHa at,2011. 2.1.1Thescalar-waveequation Physically,soundpropagationproducesamediumdensityvariationasittravelsthroughatissue, creatingalternatingzonesofcompressionandrarefaction.Thisphenomenonismodeledusingthe scalar-waveequation,representingthesezonesofvaryingpressureasawaveform.Thescalar-wave equationintwodimensionscanbewrittenasHilletal.,2004: 1 c 2 @ 2 @t 2 )-222(r 2 p =0 ; where r 2 isthespatialLaplacianoperator,and p istheacousticpressure.Whenasoundwavepasses throughtissue,thepathittakesisgovernedbyitsinteractionswiththepropagatingmedium,which mayhavevaryingproperties.Thesensitivityofultrasoundtothesevariationsresultsinanacoustic pressureeld,containinginformationaboutthepropagatingmediuminthiscase,biologicaltissue. Themaintypesoftissueinteractionsaffectingthiseldare: Refraction ,resultinginachangeinthesound-wavepropagationdirectionwhentransmitting throughamediuminterface,isbasedontissuemechanicalproperties.Refractiondataisalso sometimesreferredtoastransmissiondata,asitistransmittedthroughamediumasopposed toreectedoffit. Reection ,resultinginachangeinthepropagationdirectionofthewavewhilepropagating atthesamesoundspeedasthatoftheincidentwave.Reectiondataareusedtocreated pulse-echoB-modeclinicalultrasoundimages. 8

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Attenuation ,resultinginareductioninamplitudeandintensityofthewave,canbecausedby scatteringorabsorptionofultrasoundenergy.Wavefrequency,tissuemechanicalproperties anddistancetraveledallaffectattenuation. Reectiondataandrefractiontransmissiondataaregovernedbydifferentphysics,andarethus mathematicallydealtwithdifferently. 2.1.2Therayapproximation Althoughultrasoundistypicallyconceptualizedasawave,itsasymptotichigh-frequencyapproximationistherayapproximation.Therayapproximationcanbestatedas:Forsufcientlyshort wavelengths,raysbehaveasstraightlineswhentheypassthroughamediuminwhichthespeedof soundor,speedoflight,asoriginallystatedforopticsisconstant.Atboundarieswherethesound speedchanges,arayobeysthelawsofreectionandrefraction. Therayapproximationofasoundwaveisdenedbychoosingalinenormaltoasoundwave wavefrontpointinginthedirectionofwavepropagation.However,wecanaddsomecomplexity tothissimpliedversionofwavepropagationbyrequiringonlythattheraybelocallystraight overtheray'sjourneythroughthetissue,itcanchangedirectionsmanytimesandwemodelthis asaneffective`zig-zag'ray.Thisisreferredtoasabent-ray,ratherthanastraight-ray,model. Modelingsoundwavesasasbentraysallowsustodividethesoundwaveeldupintodiscreteand locallystraightraysinmanycellsofconstantsoundspeed,whosepathscanbecomputationally propagatedthroughasystemusingraytracingtechniques.Threefundamentaltheoriesforthiseld areHuygen'sPrinciple,Fermat'sPrincipleandSnell'slaw. 2.1.3Huygens'Principle Originatingfroma1678treatiseonthebehavioroflightbyDutchphysicistChristianHuygens -1695,Huygens'Principle,ortheHuygens-FresnelPrinciple,describesthebehaviorofwavefrontpropagationfromapointsource.Itstatesthateverypointofawavefrontcanbeconsidereda sourceofsecondarywaveletsthatspreadinalldirections,withaspeedequaltothespeedofprop9

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agationofthewaves.Thatis,atanygiveninstant,thewavefrontofapropagatingwaveconforms toanenvelopeofsphericalwaveletsemanatingfromeverypointonthewavefrontintheinstant immediatelyprior.Mathematically,thisisstatedas: U r 0 = U 0 e ikr 0 r 0 ; where U r 0 isthecomplexamplitudeofaprimarywaveatsomepointadistanceof r 0 fromthe pointsource.Theinitialdisturbance U 0 producesasphericalwaveofwavelength andwavenumber k .Notethemagnitudeofthewaveamplitudedecreasesasdistancetraveledincreases.Huygens' Principleallowsustoconclude,also,thatthereisnowavediffusioni.e.wavesdonotbroadenas theypropagate. 2.1.4Fermat'sPrinciple Alsoknownastheprincipleofleasttime,Fermat'sPrincipledescribesthebehaviorofaraybetween twopoints,statingthatthepatharaytakesbetweentwopointsisthepaththattakestheleasttime forawavetopropagateortravel.Fromthis,thelawsofreectionandrefractioncanbederived seeFig.1.Thelawofreectionstatesthatarayincidentuponaspecularreectiveboundaryis reectedatanangle r equaltotheincidentangle i bothanglesaremeasuredrelativetoaray normaltothesurface,dottedlineinFig.1.Notethisisnotnecessarilytruefordiffusereection, orscattering.ThelawofrefractionisalsoknownasSnell'sLaw. 2.1.5Snell'sLaw Snell'slawFig.1describesthebehaviorofaraycrossinganinterface,thatis,itdescribesthe relationshipbetweentheanglesofincidence i ,refractiona.k.a.transmission t ,andreection r ofaray.Itstatesthattheratioofsinesoftheanglesofincidenceandrefractionisequivalent totheratioofvelocities c i c t inthetwoborderingisotropicmedia,oralsoequivalenttothe 10

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invertedratiooftheindicesofrefraction 1 2 : sin i sin t = c i c t = 1 2 ; whichcanberewrittenas: sin i c i = sin r c i = sin t c t : Theserelationshipsallowustodistinguishbetweendifferentmediabasedonthespeedofsound travelinginthatmedium c anditsdensity ,unitsofkg m )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 Figure1: AnincidentIultrasoundwavepassingthroughtwodifferenttissueshavingdifferentmechanical propertiesandthereforedifferentindicesofrefraction, 1 and 2 .Thedifferentmechanicalpropertiesresultin thewavetravelingatdifferentvelocitiesinthedifferenttissues,producingrefractionalsocalledtransmission, T;differenceinvelocityanddirectionoftravelatthematerialboundary.ReectionR;differenceindirection oftravelbutnotvelocityalsooccurs.Theenergyoftheincidentwaveisdividedbetweenit'ssubsequent reectingwaveandrefractingwaves,andsomeadditionalenergyislostduetoattenuation.Therelative amountsofenergysplittingofftothereectingandrefractingwavesisafunctionofthe ratio oftherefractive indicesofthetwomedia. Forbiologicaltissue,theprimaryrelevantpropertiesthataffectsoundwavebehavioraresound speedandacousticimpedance.Theformulaforcalculatingthespeedofsoundinanygivenmaterial isgivenbythesquarerootoftheratioofthebulkmodulus B tothedensity : c = s B : 11

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Theinverseofthebulkmodulusisthecompressibility.Whenusingtherayapproximationorinthe linearpropagationregimesoftinyperturbationsand/orsmallwaveamplitudes,thespeedofsound c isacharacteristicofthemedium.Thatis,itisindependentoftheultrasoundwaveamplitude andcanbedeterminedfromthattissue'smaterialpropertiesalone.Thecompressibilityistypically determinedexperimentally,andcontrolsthatmaterial'sstiffnessrelativetodifferenttypesofwaves. Densityisrelatedtotheinertiaofthematerial. Specicacousticimpedance, Z ,isameasureofamaterial'sresistancetosoundpassingthrough it.Itistheproductofthematerial'sdensityanditssoundspeed,andismeasuredinraylskg cm )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 s )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 : Z = c: SeeTable1fortypicalacousticimpedancevaluesofhumantissue. 2.2B-modeultrasoundimagingofbiologicaltissues Clinicalultrasoundandultrasoundtomographyusedifferentapproachestodelivertissueinformation.Table2summarizesthekeydifferencesbetweenclinicalandtomographicultrasoundimaging ultrasoundtomographicimagereconstructionisdiscussedinthenextchapter.Brightnessmode, knownasB-mode,isthepredominatingultrasoundimagingmethodusedintheclinicforbreast imaging.B-modeimagesareobtainedfromthedistributionofreectivitywithinthetissue.This reectivitydistributionisformedfrommeasurementsofultrasoundenergyreectedbackfromthe normalincidentangleandimpedancedifferencesininternaltissuestructures.Thisreectivityis calledtheintensityreectioncoefcient IRC : IRC = Z 2 )]TJ/F19 10.9091 Tf 10.909 0 Td [(Z 1 Z 2 + Z 1 2 : Seetable1forreferencetissueparameters. 12

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2.3OpportunitiesforimprovementofB-modeultrasoundimaging Ultrasoundtransducersarehandheld,thereforethequalityofdiagnosticB-modeultrasoundimages ishighlydependentontheskilloftheultrasoundtechnicianYaffe,2008.B-modeultrasoundis alsonon-quantitative,reconstructingacousticimpedancedifferenceswithintheinterrogatedtissue ratherthandirectlyimagingtissueparameters.B-modeimagingisaqualitativeassessmentoftissuestructuresrelativetotheirsurroundings.Limitedresolutionandcontrastpreventhigh-quality diagnosticimages,andimagespecklescanobscuresmalltissuestructures.Ultrasoundtomography, discussedinthenextchapter,mayoffersanalternativesolutiontotheseshortcomings. 13

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Table1: Soundspeed,density,acousticimpedance,andattenuationcoefcientsat1MHzforselected humantissues,listedbyascendingsoundspeedvalue.Forhumantissue,theattenuationcoefcientisa strongfunctionoffrequencyandcanbeexpressedas =0.5/dB/MHz/cm.Theattenuationcoefcienthas beenshowntovarylinearlywithdensityMast,2000. Tissuetype Soundspeed c ,m/s Density ,g/cm 2 Acoustic impedance Z ,rayls Attenuation coefcient ,dB/cm AdiposeInternationalCommission onRadiationUnitsand Measurements,1998 1450 1.12 162.400 0.29 FattyInternationalCommissionon RadiationUnitsandMeasurements, 1998 1465 0.99 144.303 0.40 Breast,subcutaneousfatDuricetal., 2005 1470 0.89 Breast,internalfatDuricetal.,2005 1470 0.92 BreastInternationalCommissionon RadiationUnitsandMeasurements, 1998 1510 1.02 154.020 0.75 Breast,glandularparenchymaDuric etal.,2005 1515 1.02 Breast,highattenuationtumorDuric etal.,2005 1549 0.92 KidneyInternationalCommissionon RadiationUnitsandMeasurements, 1998 1560 1.05 163.800 1.00 Breast,cystDuricetal.,2005 1569 0.06 Non-fattyInternationalCommission onRadiationUnitsand Measurements,1998 1575 1.06 166.163 0.60 Muscle,cardiacInternational CommissiononRadiationUnitsand Measurements,1998 1576 1.06 167.056 0.52 Muscle,skeletalInternational CommissiononRadiationUnitsand Measurements,1998 1580 1.05 165.900 0.74 LiverInternationalCommissionon RadiationUnitsandMeasurements, 1998 1595 1.06 169.070 0.50 SkinInternationalCommissionon RadiationUnitsandMeasurements, 1998 1615 1.05 169.575 0.35 14

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3USRTforbreastcharacterization ThewordtomographyderivesfromtheGreek`tomo'sliceand`graph'draw,andcanbeunderstoodascross-sectionalimagingfromdatacollectedbyilluminatingtheobjectfrommanydifferent directions,thatis,reconstructinganobjectfromitsprojectionswhichareinformationderivedfrom transmittedenergy.UltrasoundtomographybeganwiththepioneeringworkofGreenleafetal. Greenleafetal.,1974,1975andCarsonetal.Carsonetal.,1981.Anumberofprototypeswere subsequentlybuiltAndreetal.,1997;Duricetal.,2007a,2014;Huangetal.,2015;Marmarelis etal.,2003;Ruiteretal.,2012;Waagetal.,1996;Wiskinetal.,1997.Theworkingprincipleof USTissimilartoX-raycomputedtomographyCTasestablishedbyJohannRadonwithhis1917 paperRadon,2005andbyGodfreyHounseldforhisinventionoftheCTscannerinthe1970s forwhichheandAllanCormacksharedthe1979NobelPrizeinPhysiologyorMedicineRaju, 1999. USTpropagatesultrasoundinsteadofionizingradiationthroughthepatient,andultrasound tomographydataareacquiredfortomographicreconstructionofthedistributionofsoundspeedor othertissueparameterswithinthebreastDuricetal.,2007a,b,2003;Huangetal.,2014,2015; Huthwaiteetal.,2010;Intratoretal.,2015;LabyedandHuang,2014;LiandDuric,2008;Lietal., 2009a,b;LinandHuang,2013,2014;Linetal.,2012;Littrupetal.,2002a,b;NebekerandNelson, 2012;NguyenandHuang,2014b;QuanandHuang,2007;SimonettiandHuang,2008;Simonetti etal.,2008,2007;ZhangandHuang,2013,2014;Zhangetal.,2012.Soundspeedandtissue densityarelinearlyrelatedforarangeofbiologicaltissuesMast,2000. Thepurposeofultrasoundtomographyistogathertomographicdatageneratedbyprobinga mediumwithsoundwavespressurecoupledwithvelocity,andtoconvertthistoeldinformationforexamplesoundspeedorattenuationcoefcient.Thisconversionoftomographicdatais referredtoasinversion,andmethodsforaccomplishingitareoftencomputationallyexpensiveand ill-conditioned.Inversionisaspecicmathematicalprocess,inwhichaeldofobjectparametersis reconstructedusingasetofobservationsthatgiveindirectinformationabouttheparameters.Inour case,theparametereldwewanttoreconstructistheeldofsoundspeedvaluesinasliceofthe breast.Theobservationsweuseareultrasoundraytravelor,arrivaltimes.Theentireultrasound 15

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waveformcanalsobeusedforultrasoundwaveformtomography. B-modeultrasoundandultrasoundtomographyusedifferenttypesofdataforimagingBmodeultrasounduses180 reectiondatatoimagechangesinacousticimpedance,whereasultrasoundtomographyusestransmissionand/orreectiondatatoreconstructthetissueparameterssuch assoundspeedorattenuationcoefcient.SeeTable1fortypicalsoundspeedvaluesofhumantissue.Table2summarizesthekeydifferencesbetweenB-modeandtomographicultrasoundimaging. Table2: SummarizedtechniquecomparisonofB-modebreastultrasoundimagingandultrasoundsound speedtomographicimagingmodalities. B-modeUltrasound UltrasoundRayTomography Transducerstransmitsacousticpressureultrasoundpulseintotissue. Acousticpressurefromreection,scatteringsound-tissueinteractionsarereceived byreceivingtransducers. Acousticpressurefromtransmissionrefraction,scatteringsound-tissueinteractionsarereceivedbyreceivingtransducers. Receivingtransducersconvertreceivedacousticpressuretovoltagewaveform. Delay-and-sumbeamformingbuildsBmodeimageoftissuereectivity. First-arrivaltimepickingalgorithmapplied, thenregularizediterativeinversionreconstructsasoundspeedeld. Mismatchesintissueacousticimpedance arequalitativelyvisualized. Tissuesoundspeedisquantitativelyreconstructed. 3.1Theforwardandinverseproblem 3.1.1Inverseandill-posedproblems UltrasoundtomographyisatypeofinverseproblemHansenandO'Leary,1993;Tarantola,2005 -thatis,soundpropagationinformationthroughtheeldofinterestisusedtoinfereldproperties. Thesetofparameterstobedetermined,thatdescribethestateoftheeld,arecalledthemodel, M Iftheseparameterscannotbemeasureddirectly,somesignalsmustbeobtainedtoinferthestateof theunobservableunmeasurablesetofmodelparameters.Thesetofmeasureddataiscalledthe data, D .Themethodtoinfer M from D iscalledaforwardmodeling, H : D = HM: 16

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Theinversemappingisthen: M = H )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 D: Usuallytheforwardproblemisnonlinear,whichisanalyticallyandcomputationallydifcult. Therefore,forwardproblemsareoftenlinearized.Acommonmethodforlinearizationisviathe Taylorseriesexpansionofthemodel: D = HM 0 + H 1 M )]TJ/F19 10.9091 Tf 11.139 0 Td [(M 0 + ::: .Keepingonlyrst-order termsgives: D )]TJ/F19 10.9091 Tf 10.909 0 Td [(D 0 = H 1 M )]TJ/F19 10.9091 Tf 10.909 0 Td [(M 0 ; where D 0 = HM 0 ,and H 1 isalinearizedmappingaroundtheunperturbed/initialmodel M 0 Theforwardproblemistypicallyrecastasdiscreteproblems.Discretizationistheprocessof convertingasignalor,anyfunctionintoanumericsequence.Supposethatamediumcanbe sufcientlydescribedby p parameters M = m 1 ;m 2 ;:::;m p andthatwehave q measurementsin ourdataset D = d 1 ;d 2 ;:::;d q ,wecanthenrewritetheforwardproblemas d = H m ; where d and m arecolumnvectorsofdata/parameterdifferencesbetweenaninitialmodeland measureddata,withelements d i )]TJ/F19 10.9091 Tf 11.01 0 Td [(d i 0 for i =1 ;:::;q and m j )]TJ/F19 10.9091 Tf 11.01 0 Td [(m j 0 for j =1 ;:::;p H isa q p matrix: H i;j = @d i @m j : Thenitedifferenceisthediscreteanalogofthederivative.Finitedifferencemethodsarenumerical methodsforapproximatingthesolutionstodifferentialequationsusingnitedifferenceequations toapproximatederivatives.Findingasolutiontothissystemofequationscanbefoundusingaleast squaresapproach. 3.1.2Theforwardproblem Theeikonalequation,followingfromFermat'sPrinciple,isanonlinearpartialdifferentialequation describingwavefrontorraypropagationinasoundspeedmodel.Solvingtheeikonalequation 17

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vianitedifferencemethodsiswidelyrecognizedasoneofthemostefcientmeansofcomputing wavetraveltimesZeltandBarton,1998.Mostnitedifferenceimplementationsofsolvingthe eikonalequationsolveforrstarrivaltraveltimes,althoughalternatealgorithmsexistforcomputing arrivalof,say,themostenergeticsignalarrivals. Ultrasoundbent-raytomographyunlikeX-raytomographyassumesultrasoundpropagation pathsmaynotbestraightwithinthebreastbecauseoftissueinhomogeneities.Toaccountforthis raybending,weuseanite-differenceschemetosolvetheeikonalequationgivenby: @t @x 2 + @t @y 2 = 1 v 2 = s 2 x + s 2 y ; where t isthetraveltime, v isthesoundspeed,and s x s y istheslownessvector.Theimaging regionisdenedoverarectangulargridofcells,witheachcellhavingaconstantsoundspeedand thereforeconstantslowness.Thiscollectionofcellswithconstantslownessiscalledtheslowness model.Aninitialslownessmodelwithhomogeneoussoundspeedcanbeusedforinversion.To calculatethetrueslownessusingrst-arrivaltimesfromatransmittingtransducerelementtoeach gridcell,theraypathistracedbackfromareceivingelementtothetransmittingelementfollowing thedirectionnormaltothecalculatedtraveltimeeldVidale,1988;ZeltandBarton,1998. Becauseofthedifferencebetweenthemodelandthetrueslownessdistribution,thereisadifferencebetweenthetraveltimecalculatedusingtheestimatedmodelandtherst-arrivaltimepicked fromanultrasoundtransmissionsignal.Thistimedifference t i islinearlyrelatedtothedifferencebetweenthemodelandtrueslownessperturbationsalongthe i th raypathinthe j th cellofthe slownessmodel: t i = N X j =1 l ij =v j = N X j =1 l ij s j ; where l ij isthelengthofthe i th raypathinthe j th celloftheslownessmodel, v j isthesoundspeed inthe j th cell, s j istheslownessperturbationinthe j th cell,and N isthetotalnumberofcellsinthe model.Combiningallpossibletransmissionpathsfromeq.15leadstoasetoflinearequations describingtheforwardmodelingproblem,whichcanbewritteninmatrixform: T = H s ; 18

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where T isan M -elementcolumnvectorcontainingtraveltimes t i ofall M raypathsand s isan N -elementcolumnvectorcontainingalltheslownessvalues s j foreachcell.Thetomographic matrix H is M N M rowsand N columns,anditselementsaretheraypathsegmentlengths l ij 3.2Regularizedinversion Theleast-squaresinversionisaniterativemethodofsolvinginverseproblems.Whentheproblem Ax = b isnotwell-posedthatisitisill-posed,duetonon-existenceornon-uniquenessof x ,then thestandardapproachislinearordinaryLSQ,whichseekstominimizethesumofallsquared residuals jjjj istheEuclidian,or ` 2 ,norm,discussedbelow: jj Ax )]TJ/F19 10.9091 Tf 11.159 0 Td [(b jj 2 .Inthecasewherethe systemisunderdetermined A issingular,ornon-invertible,LLSQmayprovideuswithsomejunk answers. Forwell-posedproblems,thedirectsolutiontoLLSQoftenworkswell: x = A T A )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 A T b .In somecaseshowever,aregularizedversionoftheleastsquaressolutionmaybepreferable.Regularizationisacommontoolinmathematicsandstatistics,usedinparticulartosolveinverseproblems amongotherapplications.Itreferstotheprocessofintroducingadditionalinformationinorder tosolveanill-posedproblem,ortopreventover-tting.Thisinformationisusuallyintheform ofapenaltyforcomplexityinthesolution,thatis,apenaltyisassociatedwithcertaincoefcientvalues.Regularizationaddsthepenaltyassociatedwiththecoefcientvaluestotheerrorof theestimated x vector,andhenceanaccurateestimatewithunlikelyunusuallyhighorlowcoefcientswouldbepenalizedwhileasomewhatlessaccuratebutmoreconservativeestimatewithmore normally-distributedcoefcientswouldnotbepenalizedasmuch.Regularizationisoftenusedin solvinginverseproblemsbyintroducingconstraintssuchasforsmoothness. 3.2.1USRTwithTikhonovregularization Continuingfromeq.16, H istypicallynotsquareandthusnotinvertible,andwetherefore cannotsimplymultiplybothsidesby H )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 tosolvefor s .Instead,wemultiplybothsidesbythe transposematrix H T toget H T T = H T H s ,resultingin s = H T H )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 H T T .Inadditionto 19

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beingsolveddirectly,linearsystemscanbesolvediteratively.Since H islarge,itmaybedesirable toinsteadsolveforthevaluesin s iteratively,ratherthandirectly,toconvergeonthetrueslowness valuesusinganiterativesolversuchastheLSQRorLSMRalgorithm,haltingthisprocessonce theslownessvalueupdatesbecomesufcientlysmall.Thus,theinverseofeq.16istypically formulatedasaleastsquaresminimizationproblem: E s =min s fk H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(D k 2 2 g ; where D isadatavectorofobservedtraveltimes, H s isthecorrespondingforwardmodelingresult, k H s )]TJ/F84 10.9091 Tf 11.259 0 Td [(D k 2 2 isthedatamistfunction,and kk 2 representsthe ` 2 norm.Solvingeq.17yields avector s thatminimizesthemeansquaredifferencebetweenobserved D andforwardmodeled H s traveltimes.Becauseoftheerrordifferencebetweenthesetwoquantities,directsolutionis notpossible. However,thisinverseproblemisstillill-posedbecauseoflimiteddatacoverage.Toaddress this,aregularizationtechniquecanbeused.TheTikhonovregularizationisan ` 2 -norm-based regularizationtechniquewidelyusedforyieldingasmoothedmodel.IncorporatingaTikhonov regularizationtermintoeq.17gives E s =min s fk H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(D k 2 2 + k Ls k 2 2 g ; where isaparametercontrollingthetradeoffbetweencontributionsfromthedatamistand regularizationterms.TheL-curvetechniqueHansen,2000canbeusedtooptimizethevalueof .ThemisttermisnowmodiedbytheTikhonovregularizationterm k Ls k 2 2 ,where L isthe Laplacianoperator r 2 specically,the2DdiscreteLaplacianoperatorfora2Dinverseproblem. ApplyingtheLaplacianeffectivelyminimizeslargedifferencesbetweenadjacentcells,resultingin asmoothedreconstructedimage.However,useoftheTikhonovregularizationalonecanresultin inappropriatelysmoothedimagereconstructionsandalossoftumormargindetail.Toalleviatethis problem,weintroduceanewregularizationtechniqueintotheinverseproblem. 20

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3.2.2Tikhonovregularization TheTikhonovregularizationnamedforAndreyTikhonov-1993isawidelyusedregularizationschemeGolubetal.,1999;Tikhonov,1995thattendstosmoothtomographicimages. TheTikhonovregularizationalsoreferredtoasridgeregressioninstatisticalapplicationsusesa ` 2 -norm.Itisthemostcommonlyusedregularizationschemeforill-posedinverseproblems.To givepreferencetosolutionswithdesirableproperties,aregularizationtermintheminimization: jj Ax )]TJ/F19 10.9091 Tf 11.124 0 Td [(b jj 2 + jj Tx jj 2 ,forsomesuitablychosenTikhonovmatrix T .Thismatrixcanbechosenas amultipleoftheidentitymatrix T = aI ,givingpreferencestosolutionswithsmallernorms.If theunderlyingvectorisbelievedtobemostlycontinuous,smoothnesscanbeenforcedwithalowpassoperator.Regularizationimprovestheconditioningoftheproblem,enablingadirectexplicit numericalsolution: x = A T A + T T T )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 A T b TheTikhonovregularizationreducesimagenoisebutalsodecreasesresolution.Itusesthe ` 2 normeq.19thatsumsthesquaresofallcomponentsandtakesthesquarerootofthatsum.This isalsoreferredtoastheEuclidiannorm.Considersomevector,forexample x = : 5 ; 0 : 5 .Hereis anexampleofcomputingthe ` 2 normofthatvectorcomparethisvaluetothe ` 1 normcomputed forthesamevectorbeloweq.25: jj x jj 2 = p 0 : 5 2 +0 : 5 2 =1 = p 2 < 1 Notethe ` 2 normyieldsasmallervaluethanthe ` 1 norm.Thismethodpreferssolutionswhereall componentsof x areverysmall,andlikewisetendstodistributeerrorthroughoutthevector x 3.2.3USRTwithtotal-variationregularization Thetotal-variationTVregularizationhasbeenusedtosolveinversionproblemsandpreserve tumormarginsLinetal.,2012;Rudinetal.,1992. ConventionalTVregularizationcanbeincorporatedintoeq.17,giving E s =min s fk H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(D k 2 2 + TV k s k TV g ; 21

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wheretheTVnormfora2Dmodelisdenedas k s k TV = n X i;j =1 q j r x s i;j j 2 + j r y s i;j j 2 ; where r x s i;j = s i +1 ;j )]TJ/F84 10.9091 Tf 10.99 0 Td [(s i;j and r y s i;j = s i;j +1 )]TJ/F84 10.9091 Tf 10.991 0 Td [(s i;j arethespatialderivativesatgridpoint i;j inCartesian x;y coordinates.Theparameter TV againregulatesthetradeoffbetweenthe datamistandTVregularizationterms. TomaketheTVtermdifferentiableattheorigin,asmallsmoothingparameter istypically introduced,leadingtotheapproximatedTVregularizationterm: k s k TV = n X i;j =1 q j r x s i;j j 2 + j r y s i;j j 2 + : However,thesolutionofeq.22ishighlydependentonthechoiceof ,andcontainssignicant imageartifacts. Note:Because,atrstglance,theTVtermmaysomewhatresemblean ` 2 -norm,considerthe followingexplanationusinganexamplevector m ,where m = m x ; m y and r m = r x m ; r y m : k m k 1 = j m j = q j m x j 2 + j m y j 2 kr m k 1 = jr m j = q jr x m j 2 + jr y m j 2 Inthenextsection,developmentofanovelultrasoundbent-raytomographymethodwitha modiedtotal-variationregularizationschemeisdescribed,withthepurposeofovercomingthe shortcomingsofusingTVregularizationalone. 3.2.4Total-variationregularization USRTwiththetotal-variationTVregularizationcanhelppreservetumormargins,butatthecost ofincreasedimagenoise.PioneeredbyRudinetal.1992,theTVregularizationmethodisbased ontheprinciplethatsignalswithexcessiveandpossiblyspuriousdetailhaveahightotalvariation 22

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value.Thatis,theintegraloftheabsolutegradientofthesignalishigh.Thus,reducingthetotal variationofthesignalremovesunwanteddetailwhilepreservingimportantdetailslikeedges.The goalofTVistondanapproximationofaninputmatrixe.g.animage x thathassmallertotal variationwhilestillbeingcloseto x .Thisclosenessismeasuredusingleastsquares,andsothe TVproblemamountstominimizingthesumofsquareerrorsbetween x anditsapproximation,plus ascalar c multipleofthetotalvariationin x : jj Ax )]TJ/F19 10.9091 Tf 11.086 0 Td [(b jj 2 + cV x .Tominimizethisexpression, wedifferentiatewithrespectto x ,derivingacorrespondingEuler-Lagrangeequationthatcanbe numericallyintegratedwiththeoriginalsignalastheinitialcondition. Thetotal-variationregularizationisgenerallyviewedasagoodtechniquetodenoisewhilstpreservingedgeinformation,andtheTikhonovregularizationisgenerallyconsideredbetteratsmoothingimages.TheTVregularizationusesthe ` 1 normeq.25,whichsumstheabsolutevalueofall componentsof x : jj Ax )]TJ/F19 10.9091 Tf 9.466 0 Td [(b jj 2 + jj cV x jj 1 .AsanalternativetoEuclideangeometry,wherethenorm istheuniqueshortestdistancebetweentwopointsinCartesiancoordinates,the ` 1 normconsiders thedistancebetweentwopointsasthesumofabsolutedifferencesintheircoordinates,aconcept establishedbyHermannMinkowskiin19thC.Germany.Hereisanexampleofcomputingthe ` 1 normofavectorcomparethisvaluetothe ` 2 normcomputedforthesamevectorabove,eq.19: jj x jj 1 = j 0 : 5 j + j 0 : 5 j =1 Notethe ` 1 normyieldsalargervaluethanthe ` 2 norm.UnlikeTikhonov-regularizedsolutions, theTVmethodcanallowasparse x ,thatis,somevaluesof x areexactlyzerowhileotherscanbe relativelylarge.Bothverylargeandverysmallvaluesaretoleratedbetterthanwiththe ` 2 norm. 23

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4USRTwithMTVregularization Tumorstendtohavehighersoundspeedsthannormalbreasttissue,whichisunderstoodtobethe resultofremodelingoftheextracellularmatrixsurroundingtumorsMacklin,2010.Thisallows themtobecharacterizedusingultrasoundbent-raytomographyUSRT.Soundspeedisthereforea criticaldiagnosticfeatureinultrasoundtomography-baseddiagnosisofbreastlesionsHoppetal., 2012;LiandDuric,2008;Lietal.,2009a.Margininformationandtumorsizeisanimportant diagnosticandprognosticfeatureforcliniciansChenetal.,2004;Howladeretal.,2013,and recoveryoftumorsizedependsonaccuratemarginpreservationintomographicreconstructions. Soundspeeddistributionofthebreast,aswellastumormargins,canbereconstructedusingUSRT, andithasbeenshownthatUSRThasthepotentialtodistinguishbreasttumorsfromnormalbreast tissueLietal.,2009a.However,improvementsinimagequalitymustbemadeinorderforthis technologytobeclinicallyusefulasabreastcancerdetectionanddiagnostictool. 4.1ImplementationofUSRTwithMTVregularization USRTrecentlydemonstratedtheabilitytosignicantlyimproveimagequality,usingarrivaltimes ofbothtransmissionandreectiondataNguyenandHuang,2014a.Ultrasoundwaveformtomographicreconstructionscanbeimprovedviaamodiedtotal-variationMTVregularization schemeLinandHuang,2013.However,ultrasoundwaveforminversioniscomputationallyexpensive,anditwouldbeofinteresttodevelopanalternative.Here,weaimtoimproveUSRTimage reconstructionsviaamodiedtotal-variationMTVregularizationscheme,therebydevelopinga morecomputationallyefcientmethodofreconstructingbreastsoundspeedIntratoretal.,2015. MTVisahybridregularizationscheme,combiningthesmoothingcapabilityofTikhonovregularizationwiththemargin-preservingcapabilityoftotal-variationTVregularization.ThisUSRT reconstructionschemeisofferedasaproofofconceptusingtransmissiondata,andcanbeextended toincludereectiondata.OurnewUSRTwithMTVregularizationUSRT-MTVisaTikhonov-TV hybrid,reducingimagenoisewhilepreservingmargins.WeapplyournewUSRT-MTVmethodto ultrasoundtransmissiondatafromnumericalphantoms,andcomparetheresultswiththoseobtained 24

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usingTikhonovregularization.ThisisthersttimetoincorporatetheMTVregularizationscheme intoUSRT. Thereconstructionalgorithmisimplementedbysolvingtwodecoupledminimizationsubproblems.WeminimizethemistfunctionusinganalternatingminimizationalgorithmBauschkeetal., 2006;LinandHuang,2013,2015;Wangetal.,2008,decomposingtheinversionprobleminto twosubproblems:oneisan ` 2 -norm-basedTikhonovregularizationproblemNguyenandHuang, 2014a,andtheotherisan ` 1 -norm-basedTVregularizationproblem.WeuseanLSQRmethodto solvetherstsubproblem,andapplythesplitBregmanmethodtothesecondsubproblemGoldsteinandOsher,2009;Osheretal.,2005.ThesplitBregmanmethodispreferredtootheriterative methods,asitavoidsselectionofthesmoothingparameterintheTVtermsignicantlyimproving algorithmrobustnessandcomputationalefciency,andasitalsoconvergestothetrueratherthan approximatedTVsolutionGoldsteinandOsher,2009.WeapplyournewUSRT-MTVmethod toultrasoundtransmissiondatafromnumericalphantoms,employinganautomaticmethodtopick arrivaltimesofsyntheticultrasoundtransmissionsignalsfortomographicreconstructions. ThetransmissionultrasoundtomographyalgorithmusesanLSQRiterativesolverwithTikhonov regularizationtosolvefor s : s =min s jj H s )]TJ/F15 10.9091 Tf 10.909 0 Td [( t jj 2 + jj L s jj 2 ; where jj H s )]TJ/F15 10.9091 Tf 10.672 0 Td [( t jj 2 isthedatamisttermdescribingthemistbetweenthedataandtheforward modeledresults, jj s jj 2 istheTikhonovregularizationterm, L istheLaplacianoperator r 2 and isthedampingparameterusedtoweighttherelativecontributionsofthemisttermand theregularizationterm.Weobtaintomographicreconstructionsofnumericalphantomsusingour newUSRTalgorithmwithMTVregularizationUSRT-MTV,andcomparetheseresultswiththose producedusingUSRTwithTikhonovregularizationUSRT-Tikhonov. ToretainthesmoothingbenetsofTikhonovregularization,whileincorporatingthemarginpreservingpropertiesofTVregularization,wedevelopamodiedTVMTVregularizationmethod forUSRT.Thismethodwaspreviouslydevelopedbyourgroupforusewithultrasoundwaveform inversionLinandHuang,2013,2015;hereitisadaptedforusewithUSRTinversion. Inordertoimplementthisnestedregularization,weintroduceasecondparameter u ,which 25

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addsanadditionalstepforsolvingtheinverseproblem.TheminimizationproblemwithMTV regularizationisformulatedasfollows: E s =min s ; u fk H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(D k 2 2 + 1 k s )]TJ/F84 10.9091 Tf 10.909 0 Td [(u k 2 2 + 2 k u k TV g ; where 1 and 2 arebothpositiveregularizationparameters.Equivalently,eq.27canbewritten as E s =min u f min s fk H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(D k 2 2 + 1 k s )]TJ/F84 10.9091 Tf 10.91 0 Td [(u k 2 2 g + 2 k u k TV g : Itisnowclearerthat 1 controlsthetradeoffbetweenthedatamisttermandtheTikhonovregularizationtermi.e.theamountofsmoothing/noisereductionintheinversion,while 2 controls thetradeoffbetweentheTikhonov-regularizeddatamisttermandtheTV-regularizedtermi.e.the amountofmarginpreservationintheinversion.Tosolveeq.28,theinversionisrstperformed tominimize s withTikhonovregularization.Next,thissolutionispipedtoasecondminimization problemtominimize u usingMTVregularization.Thissolutionisusedastheinputforthenext iteration,minimizing s again.Theiterativesolverishaltedoncetheupdates s k and u k for iterations k =1 ; 2 ;::: aresufcientlysmall,asbefore.Thisalternatingminimizationmethodis formulatedintwominimizationsubproblemsasfollows: s k =min s f E 1 s g =min s n k H s )]TJ/F84 10.9091 Tf 10.909 0 Td [(T k 2 2 + 1 k s )]TJ/F84 10.9091 Tf 10.909 0 Td [(u k )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 k 2 2 g o ; u k =min u f E 2 u g =min u n k s k )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 )]TJ/F84 10.9091 Tf 10.91 0 Td [(u k 2 2 + 2 k u k TV o : 4.2Numericalphantoms Themeansoundspeedinthebreasthasbeenfoundtorangefromapproximately1440m/sfatty breasttissuetoapproximately1505m/sdensebreasttissueLiandDuric,2008.Densebreasts presentagreatimagingchallengeformammography,thecurrentgoldstandardforbreastcancer detection.Furthermore,densebreastsarecorrelatedwithincreasedbreastcancerriskBoydetal., 26

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Figure2: Syntheticrst-arrivaltimesaregeneratedfortransmissiontomographyusingtwodifferentnumerical phantoms.Thephantomontheleftcontainsa30-mmdiametertumor-mimickingtargetwithasoundspeed of1560m/s,inabackgroundof1500m/s.Thephantomontherightcontainsatumor-mimickingtargetwith concentricregionsofdifferentsoundspeed,wheretheinnermostregionis50mmindiameterm/s,the nextlargeris100mmindiameterm/s,andthenextlargeris150mmindiameterm/s.The backgroundsoundspeedisagain1500m/s.Bothphantomsare223 223mm. 2007.Itisthereforeimperativetodevelopbettermethodsforimagingdensebreasts.Weusetwo numericalphantoms,representingidealizedcross-sectionsofahumanbreastwithdifferenttypesof lesions,forthisstudy.Therstoneisasimplephantomrepresentingatumorofhomogeneoussound speed,30mmindiameterFig.2a.However,tumorsarethoughttohaveincreasedsoundspeed becauseofremodelingoftheextracellularmatrixsurroundingthetumormassMacklin,2010.We thereforealsoinvestigateaconcentricnumericalphantomFig.2bthatmaybetterrecapitulatethis environment.Furthermore,thisconcentricnumericalphantomwithpiecewise-constantsoundspeed isbettersuitedforcomparingedgepreservation.TogainabetterunderstandingofhowUSRTmight detectbreasttumorsindensebreasts,wechose1500m/sasthebackgroundspeedforourphantoms tomimicadensebreast.Sincebreasttumorstypicallyhaveahighersoundspeedthansurrounding breasttissueLietal.,2009a,highersoundspeedsareassignedtothecirclesrepresentingtumors. 27

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5Soundspeedreconstructionresults 5.1Reconstructionoftwodifferentphantoms USRT-TikhonovUSRT-MTV Figure3: Reconstructionsofthedifferentphantoms,comparingreconstructionsachievedwithUSRT-Tikhonov andUSRT-MTV.ItisevidentthatUSRT-MTVprovidesasuperiorreconstructionofthesoundspeedeld,more accuratelyreconstructingthepiecewise-constantnatureofthephantoms. InreconstructionsofthephantomwiththesmalltargetFig.3,itisevidentthatuseofthe parallelarrayleadstodistortedcoverage.Thereconstructionofthephantomcontainingthesmall 28

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Figure4: Horizontalsoundspeedprolesofreconstructionsofphantomwithsmalltarget,achievedusing thetwodifferentregularizationschemes.Thetopgure,blackline,istheactualphantomprole.Thesecond gureshowstheproleofthereconstructionobtainedusingUSRT-Tikhonovbluelineandthethirdgure showstheproleofthereconstructionobtainedusingUSRT-MTVmagentaline.ItisevidentthatUSRT-MTV yieldsthemostaccuratesoundspeedreconstruction. target,usingUSRT-TikhonovFig.3,upperleft,thereisobviousreconstructioninaccuracyinthe centerofthetarget,andsomeoversmoothingofthemargins.Thereconstructionofthephantom containingthesmalltarget,usingUSRT-MTVFig.3,upperright,thecoveragedistortionismuch lesspronounced,andthemarginsaresharper. Thereconstructionofthephantomcontainingtheconcentrictarget,usingUSRT-TikhonovFig. 3,lowerleft,againtheparallelarrayleadstodistortedcoverage.TheshortcomingsofTikhonov regularizationinnotbeingabletopreservemarginstructureismuchmoreevidentthanwiththe reconstructionsofthephantomwithsmalltargetnotetheextremedistortionattheboundaries wheresoundspeedchanges.Thereconstructionofthephantomcontainingtheconcentrictarget, usingUSRT-MTVFig.3,lowerright,thecoveragedistortionismuchlesspronounced,themarginsatallboundariesaresharper,andthereconstructionoftheoverallpiecewise-constantnatureof thephantomisimproved. InhorizontalproleviewsofthereconstructionsofthephantomcontainingthesmalltargetFig. 4,thesignicantreconstructioninaccuraciesintheUSRT-TikhonovreconstructionFig.4,middle 29

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Figure5: Verticalsoundspeedprolesofreconstructionsofphantomwithsmalltarget,achievedusingthetwo differentregularizationschemes.Thetopgure,blackline,istheactualphantomprole.Thesecondgure showstheproleofthereconstructionobtainedusingUSRT-Tikhonovbluelineandthethirdgureshows theproleofthereconstructionobtainedusingUSRT-MTVmagentaline.ItisagainevidentthatUSRT-MTV yieldsthemostaccuratesoundspeedreconstruction. Figure6: Horizontalsoundspeedprolesofreconstructionsofphantomwithconcentrictarget,achievedusing thetwodifferentregularizationschemes.Thetopgure,blackline,istheactualphantomprole.Thesecond gureshowstheproleofthereconstructionobtainedusingUSRT-Tikhonovbluelineandthethirdgure showstheproleofthereconstructionobtainedusingUSRT-MTVmagentaline.ItisevidentthatUSRT-MTV yieldsthemostaccuratesoundspeedreconstruction. 30

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Figure7: Verticalsoundspeedprolesofreconstructionsofphantomwithconcentrictarget,achievedusing thetwodifferentregularizationschemes.Thetopgure,blackline,istheactualphantomprole.Thesecond gureshowstheproleofthereconstructionobtainedusingUSRT-Tikhonovbluelineandthethirdgure showstheproleofthereconstructionobtainedusingUSRT-MTVmagentaline.Itisagainevidentthat USRT-MTVyieldsthemostaccuratesoundspeedreconstruction. plotaremuchmoreevident.Thesoundspeedvaluesinthecenterofthetargetaresignicantly lowerthanthetruevalue,andmarginvaluesarehigherthanthetruevalue.Reconstructionofthis samephantomusingUSRT-MTVFig.4,lowerplotdemonstratesthemoreaccuratesoundspeed reconstruction,andimprovedmarginreconstructionaswell.VerticalprolesFig.5ofthesame reconstructionsfurthercorroboratetheseresults.TheUSRT-TikhonovreconstructionproleFig. 5,middleplotdemonstratesthecharacteristicover-smoothingprole,overshootingthetarget's soundspeedatthemarginsandundershootinginthecenter.TheUSRT-MTVreconstructionFig. 5,lowerplot,theprolelooksquitesimilartothehorizontalprole,preservingsoundspeedmuch moreaccuratelythantheUSRT-Tikhonovcounterpart. Inhorizontalproleviewsofthereconstructionsofthephantomcontainingtheconcentrictarget Fig.6,thesignicantoversmoothingintheUSRT-TikhonovreconstructionFig.6,middleplotis againevident,althoughreconstructionofthemaximumcenterregionsoundspeedisgoodrelative tothephantomwithsmalltargetthismaybeduetothedifferenceinsizesoftheareaofthesmall targetmmdiameterversustheareaofthecenteroftheconcentrictargetmmdiameter. 31

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ReconstructionofthissamephantomusingUSRT-MTVFig.6,lowerplotdemonstratesoverall improvedsoundspeedreconstructioninallsoundspeedregions,aswellasmorefaithfulreconstructionofmarginswheresoundspeedchanges.Themaximumsoundspeedisundershot.Vertical prolesFig.7ofthesamereconstructionsaremoredegraded.TheUSRT-Tikhonovreconstruction proleFig.7,middleplotisnearlyfeatureless,inparticularatthetransitionfrom1550m/scenter regionto1530m/smiddleregion.TheUSRT-MTVreconstructionFig.7,lowerplotcontains fewermarginfeaturesinthiscrosssection,butisstillbetterthanUSRT-Tikhonovatretainingsome ofthepiecewiseconstantnatureofthephantom. 5.2Reconstructionofphantomswithvaryingtargetandbackgroundsoundspeeds WeassesstheperformanceofUSRT-TikhonovFig.8andUSRT-MTVFig.9whenreconstructingthephantomshavingdifferenttissuebackgroundsanddifferenttumorsoundspeeds.Phantoms representingdifferentbackgroundtissuesoundspeedsfromrelativelyfattybreasttissue m/s,torelativelydensebreasttissuem/sareconstructedforthisstudy.Phantomtumor soundspeedsrangedfrombenignmasssoundspeedsand1510m/s,tomalignantmasssound speeds,1570m/s.Thesoundspeedoverlaprangeinthemiddlerepresentssoundspeedsthat arepotentiallyeitherbenignandmalignantmasses. Forthereconstructiontargetwithabenignsoundspeedof1510m/sinabackgroundof1480 m/sFig.10,theUSRT-Tikhonovreconstructionmiddleplot;bluesignicantlyundershootthe target'ssoundspeed,anddistortsthemarginsaswell.USRT-MTVlowerplot;magentaovershoots thetarget'ssoundspeed,butlesssignicantly,andalsoavoidsthemargindistortionpresentinthe USRT-Tikhonovreconstruction. Forthereconstructiontargetintheoverlapsoundspeedrangebetweenbenignandmalignant Fig.11,USRT-Tikhonovagainresultsinsignicantlyundershootingthetarget'ssoundspeed,and distortionoftumormargins.USRT-MTVagainovershootssoundspeed,but,lessso,andreconstructssuperiormargininformation. SimilarresultsareseenforreconstructionsofthemalignanttargetFig.12,excepttheMTVregularizedalgorithmseemstohandlethelargedifferencebetweentargetandbackgroundlesswell 32

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Targetsoundspeed BenignmassSoundspeedoverlapMalignantmass 1480m/s1510m/s 1530m/s1540m/s 1550m/s1570m/s Fattytissue 1410m/s 1420m/s Backgroundsoundspeed 1430m/s Glandulartissue 1460m/s 1480m/s 1500m/s Figure8: ComparisonoftheabilityofUSRT-Tikhonovtoreconstructtargetsofvarioussoundspeedrepresentingthesoundspeedrangeofbenigntomalignantmassesinbackgroundsofvarioussoundspeedrepresentingthesoundspeedrangeoffattytoglandularbreasttissue.Notethatthecolorscaleisconstrainedto thevaluerange1500m/sto1570m/s. 33

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Targetsoundspeed BenignmassSoundspeedoverlapMalignantmass 1480m/s1510m/s 1530m/s1540m/s 1550m/s1570m/s Fattytissue 1410m/s 1420m/s Backgroundsoundspeed 1430m/s Glandulartissue 1460m/s 1480m/s 1500m/s Figure9: ComparisonoftheabilityofUSRT-MTVtoreconstructtargetsofvarioussoundspeedrepresenting thesoundspeedrangeofbenigntomalignantmassesinbackgroundsofvarioussoundspeedrepresenting thesoundspeedrangeoffattytoglandularbreasttissue.Notethatthecolorscaleisconstrainedtothevalue range1500m/sto1570m/s. 34

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thanthepreviouscases.Resultsfortheentirecomparisongridaresimilar,withUSRT-MTVreconstructingbothsoundspeedandmarginsmuchmoreaccuratelythanUSRT-Tikhonovinallcases. Ofcourse,thesereconstuctionresultscouldbedifferentifdifferentregularizationparameters areused.OptimalparametersareneededforUSRTreconstructions. Figure10: ComparisonoftheabilityofUSRT-TikhonovandUSRT-MTVtoreconstructtargetsrepresentinga benignmassinabackgroundofglandulardensebreasttissue.Thetoptwoplotsblacklinearehorizontal xdirectionandverticalydirectionprolesoftheactualphantom.Themiddletwoplotsblueandlowertwo plotsmagentaareprolesofreconstructionsachievedusingUSRT-TikhonovandUSRT-MTV,respectively.It isevidentthatUSRT-MTVyieldsthemostaccuratesoundspeedreconstructioninallcases. 35

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Figure11: ComparisonoftheabilityofUSRT-TikhonovandUSRT-MTVtoreconstructtargetsrepresentinga massofuncertainetiologyinabackgroundofglandulardensebreasttissue.Thetoptwoplotsblackline arehorizontalxdirectionandverticalydirectionprolesoftheactualphantom.Themiddletwoplotsblue andlowertwoplotsmagentaareprolesofreconstructionsachievedusingUSRT-TikhonovandUSRT-MTV, respectively.ItisevidentthatUSRT-MTVyieldsthemostaccuratesoundspeedreconstructioninallcases. Figure12: ComparisonoftheabilityofUSRT-TikhonovandUSRT-MTVtoreconstructtargetsrepresentinga malignantmassinabackgroundofglandulardensebreasttissue.Thetoptwoplotsblacklinearehorizontal xdirectionandverticalydirectionprolesoftheactualphantom.Themiddletwoplotsblueandlowertwo plotsmagentaareprolesofreconstructionsachievedusingUSRT-TikhonovandUSRT-MTV,respectively.It isevidentthatUSRT-MTVyieldsthemostaccuratesoundspeedreconstructioninallcases. 36

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6Conclusionsandfuturework Wehavedevelopedanewultrasoundbent-raytomographyUSRTmethodusingamodiedtotalvariationMTVregularizationscheme;thisisthersttimetoincorporateanMTVregularization schemeintoUSRT.Wedecomposetheinverseproblemintotwosubproblemsusinganalternatingminimizationalgorithm,resultinginacomputationallyefcientandaccuratereconstruction algorithm.OuruseofthesplitBregmanalgorithmavoidsthenon-differentiabilityofthesecond subproblem,consequentlyimprovingtherobustnessofourmethod. WehaveappliedournewUSRT-MTVmethodtoultrasoundtransmissiondatafromnumericalphantoms,andcomparedtheresultswiththoseobtainedusingtheTikhonovregularization.Our newUSRT-MTVisaTikhonov-TVhybrid,andreducedimagenoisewhilepreservingmargins.The high-qualitysound-speedtomographyresultsofnumericalphantomdatademonstratethatUSRTMTVsignicantlyimprovestherobustnessofUSRT.Furthermore,imagequalityofUSRT-MTV reconstructionsislesssusceptibletodistortionfromincompletedatacoveragefromparalleltransducerarraygeometry. Futureworkincludesapplicationsofournewmethodtotumor-mimickingphantomdataand patientdataacquiredattheUniversityofNewMexicohospitalusingLANL'scustom-builtbreast ultrasoundtomographyprototypeHuangetal.,2015.Somerelatedresearchresultshavebeen publishedinthefollowingtwoproceedingspapersHuangetal.,2015;Intratoretal.,2015: M.Intrator ,Y.Lin,T.Chen,J.ShinandL.Huang.Ultrasoundbent-raytomographywith amodiedtotal-variationregularizationscheme.In ProceedingsofSPIEMedicalImaging 2015. L.Huang,J.Shin,T.Chen,Y.Lin, M.Intrator ,K.Hanson,K.Epstein,D.SandovalandM. Williamson.Breastultrasoundtomographywithtwoparalleltransducerarrays:preliminary clinicalresults.In ProceedingsofSPIEMedicalImaging ,2015. 37

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