# Investigating Pulse-Echo Sound Speed Estimation in Breast Ultrasound with
Deep Learning

Walter A. Simson<EMAIL_ADDRESS>Magdalini Paschali  Vasiliki
Sideri-Lampretsa  Nassir Navab  Jeremy J. Dahl  Chair for Computer Aided
Medical Procedures and Augmented Reality, Technical University of Munich,
Munich, Germany  Department of Psychiatry and Behavioral Sciences, Stanford
University School of Medicine, Stanford, CA, USA  Institute for Artificial
Intelligence and Informatics in Medicine, Technical University of Munich,
Munich, Germany  Department of Radiology, Stanford University School of
Medicine, Stanford, CA, USA  Chair for Computer Aided Medical Procedures,
Whiting School of Engineering, Johns Hopkins University, Baltimore, MD, USA

###### Abstract

Ultrasound is an adjunct tool to mammography that can quickly and safely aid
physicians with diagnosing breast abnormalities. Clinical ultrasound often
assumes a constant sound speed to form B-mode images for diagnosis. However,
the various types of breast tissue, such as glandular, fat, and lesions,
differ in sound speed. These differences can degrade the image reconstruction
process. Alternatively, sound speed can be a powerful tool for identifying
disease. To this end, we propose a deep-learning approach for sound speed
estimation from in-phase and quadrature ultrasound signals. First, we develop
a large-scale simulated ultrasound dataset that generates quasi-realistic
breast tissue by modeling breast gland, skin, and lesions with varying
echogenicity and sound speed. We developed a fully convolutional neural
network architecture trained on a simulated dataset to produce an estimated
sound speed map from inputting three complex-value in-phase and quadrature
ultrasound images formed from plane-wave transmissions at separate angles.
Furthermore, thermal noise augmentation is used during model optimization to
enhance generalizability to real ultrasound data. We evaluate the model on
simulated, phantom, and in-vivo breast ultrasound data, demonstrating its
ability to accurately estimate sound speeds consistent with previously
reported values in the literature. Our simulated dataset and model will be
publicly available to provide a step towards accurate and generalizable sound
speed estimation for pulse-echo ultrasound imaging.

###### keywords:

Ultrasound, Sound Speed, Simulation, Deep Learning, Breast Imaging

##  1 Introduction

Breast cancer is the most common cancer in women, with 2.3 million cases in
2020 (2021) (WHO). Ultrasound is commonly used as an adjunct to mammography to
examine patients with indeterminate lesions. Sonography quickly and safely
aids clinicians in differentiating cysts from solid lesions but lacks high
specificity in discerning benign and malignant findings (Smithuis et al.,
2010).

Research has shown that different types of breast tissue, for instance, breast
gland and malignant lesions, can vary substantially in their sound speed by up
to 100 m/s (Bamber, 1983). However, in current diagnostic ultrasound imaging
systems, a constant sound speed (e.g., 1540 m/s) is assumed for all tissues to
create B-mode images (Feldman et al., 2009). Since breast tissue is
heterogeneous in its sound speed, this assumption can degrade image quality
and reduce the efficacy of ultrasound imaging in diagnosing breast cancer.
Thus, developing a technique that can accurately estimate the sound speed of
breast tissues can substantially aid imaging by integrating sound speed
estimates in the image reconstruction pipeline to increase image quality
through aberration correction. Methods of accurate and local sound speed
estimation have the potential to provide additional quantitative measures to
physicians to aid in diagnosis (Bamber, 1983; Sanabria et al., 2018).

Sound speed estimation in pulse-echo ultrasound imaging remains a challenging
problem. Unlike total tomographic sound speed reconstruction, uses a complete
angular sampling from $[-\pi,\pi]$ and a known distance between transmitter
and receiver (Kak and Slaney, 2001), pulse-echo sound speed estimation
utilizes a limited angular sampling and lacks accurate position or distance
information, thereby complicating the sound speed estimation. An accurate
pulse-echo sound speed estimation would greatly increase the applicability and
utility of such methods in clinical practice.

##  2 Related Work and Contributions

Pulse-echo models of sound speed estimation can be performed with a physical
model to solve for a sound speed distribution or with a machine learning model
and data-driven approach. In both cases, radio frequency (RF) or in-phase and
quadrature (IQ) data can be taken as input, and the method returns a predicted
spatial sound speed distribution in the medium of interest.

(Anderson and Trahey, 1998), proposed a method by which a global average sound
speed between the transmitting interface and a focal point is derived from
features extracted from the channel data of a focused transmit. The method was
experimentally validated on homogeneous phantoms with wire and speckle-
generating targets.

Building on the work of Anderson and Trahey, Jakovljevic et al. (2018)
proposed a model that estimated local sound speed along a wave propagation
path from a sequential series of global average sound speed measurements at
discretized depths. The model was shown to work when the medium was composed
of layers with different sound speeds.

Ali and Dahl proposed the IMPACT method that was better able to estimate local
sound speed in volumes with lateral inhomogeneities by tomographically
maximizing the coherence factor along with an innovative phase aberration term
given reconstructions with all sound speeds within a range from 1400 to 1700
m/s (Ali et al., 2021). Sanabria et al. (2018) aimed at solving the inverse
problem directly with a novel anisotropically-weighted total-variation
regularization method and displayed high-resolution sound speed
reconstructions for accurate time of flight measurements.

More recently, Stähli et al. (2020) extended the CUTE method (Jaeger and
Frenz, 2015) by solving for sound speed maps with a system of spatially
distributed phase shift measurements. Specifically, phase shift measurements
were taken between pairs of transmit and receive angles (Tx and Rx) set around
a common mid-angle. The method further took into account the erroneous
position of the echos in the reconstruction. This approach created accurate
sound speed maps in a series of phantoms, displayed marked improvements over
previously published base-lines, and exhibited realistic sound speed estimates
in an in-vivo reconstruction of a liver model.

Though all of the above-mentioned methods have made great contributions
towards real-time sound speed estimation, the problem of sound speed
estimation remains a challenging and important task. Recently, there has been
growing interest in methods for sound speed estimation built upon the advances
in computer vision with the advent of performant neural networks as universal
estimators.

Feigin et al. (2019) proposed pulse-echo sound speed estimation with a deep
neural network based on VGG (Simonyan and Zisserman, 2014). The network was
trained to map the raw channel data from three plane wave transmits to a sound
speed distribution. Training data was generated by simulating ultrasonic
interrogations of media containing randomly positioned ellipses of varying
ultrasonic properties with the k-Wave software package (Treeby and Cox, 2010).
Each plane wave used a separate 64 element sub-aperture of a 128 element
transducer to interrogate the medium at a pre-defined angle. The trained
network was evaluated on a simulated test set and achieved a mean absolute
error of 12.5$\pm$16.1 m/s. Nonetheless, when applying the method to in-vivo
data the resulting estimated values were both outside the training data range
and the normal envelope of healthy tissue, indicating a large domain shift
between the simulation and in-vivo data.

A new investigation on the use of deep learning for ultrasound sound speed
reconstruction was presented by Jush et al. (2021a), in which the network
multi-input architecture of Feigin et al. (2019) is extended to map IQ data
with separate I and Q branches to sound speed distribution maps. The k-Wave
suite was again used to simulate random ellipses in media, although only one
plane wave transmission was simulated. The evaluation showed accurate sound
speed estimations, but the method was solely evaluated on simulated data
similar to the training set data and did not incorporate features commonly
observed in real-world ultrasound signals, such as thermal noise.

Two challenges for deep learning sound speed estimation models are data
collection and labeling. For supervised deep learning, there is not yet an
accurate way to manually label ultrasound signals with a local sound speed.
For this reason, full-wave simulations are used to create a paired dataset of
known sound speed distributions and their respective channel data. This
approach is nevertheless challenged by the requirement for simulations to be
carefully parameterized in order to accurately model transducer
characteristics and tissue property distributions. Previous works have
proposed a variety of methods to create in-silico phantoms for the generation
of ultrasound data. Some have generated randomly positioned ellipses of
varying ultrasonic properties to create in-silico phantoms that are randomly
parameterizable (Feigin et al., 2019; Jush et al., 2021b, 2020). These works
showcased phantoms that are fully parameterizable with geometric shapes but
did not incorporate anatomical priors of a target anatomy. Others have sampled
a virtual breast model containing complex anatomical geometries through a
novel combination of image-based landmark structures based on MRI data from
the NIH Visible Human Project and randomly distributed structures (Wang et
al., 2015). Karamalis et al. (2010) modified a fetus dataset presented by
Jensen and Munk (1997) for discretized Westervelt simulations. Salehi et al.
(2015) evaluated their hybrid ultrasound simulation approach using phantoms
based on segmented MRI volumes of patient data fur intra-operative
registration. While these works took care to model anatomical geometries in
their simulation phantoms, they were limited in scale by the availability of
MRIs as anatomical priors and could not be randomly generated. Therefore,
bridge anatomical realism and random parameterization is required to generate
large, heterogeneous, and quasi anatomically realistic datasets of in-silico
phantoms for full-wave ultrasound simulations.

###  2.1 Our Contributions

This study proposes a novel pipeline for sound speed estimation of medical
ultrasound for breast tissue. We create a large-scale synthetic ultrasound
dataset of breast ultrasound images, including breast gland, skin, and two
types of lesions. Afterward, a Deep Neural Network (DNN) is trained on IQ
demodulated synthetic aperture data to estimate sound speed. Finally, we show
the capabilities of the proposed method with an extensive evaluation of
simulated, phantom, and in-vivo data. Our contributions are:

  * 1.

A novel approach to generating randomized and quasi-realistic simulation input
data

  * 2.

A deep neural network (DNN) trained on beamformed IQ data generated from
angled plane wave transmission to estimate the distribution of sound speed in
a medium

  * 3.

First quantitative results of a DNN on phantom and in-vivo data consistent
with traditional sound speed measurements

  * 4.

Evaluation of temporal estimation consistency that displays invariance to
artifacts such as thermal noise on sequential measurements

Importantly, this work is meant to show the viability of using neural network
to infer sound speed given a small number of angled plane-wave acquisitions
which have been simulated from in-silico phantoms. An exhaustive evaluation
and comparison of network architectures and methods is left to future works.

##  3 Methods

The following section describes the methodology used in order to train a DNN
on quasi-realistic in-silico simulations for the purpose of generalizable
sound speed estimation with real transducer data. To this end, we will cover
the generation and parameterization of a quasi-realistic in-silico breast
phantom, the simulation process using the k-Wave suite, the proposed data
processing and augmentation steps for training a DNN, as well as the
architecture and structure of the DNN. The methods and notation are formally
described in this section, while the selected parameter values are presented
in Section 4.

###  3.1 In-Silico Phantom

Figure 1: (a) Simulated ultrasound B-mode image with background approximating
glandular breast tissue and an anechoic cyst (Smithuis et al., 2010). (b)
Sound speed of the simulated medium. (c) Sound speed target used for model
optimization with region-average sound speed values. Note that two sound speed
values are used for the background, and a single sound speed value is used for
the cyst.

The three dimensional ultrasound simulations developed in this work are
generated to model human breast tissue and are comprised of three basic
elements; a tissue property map, which defines the spatial distribution of
anatomies in the domain such as skin, lesion, cyst and breast gland, a scatter
distribution field, which defines the location and relative intensity of all
scatterers and a tissue property model, which defines voxel-wise tissue
properties of sound speed, density, non-linearity (B/A) and attenuation in the
simulation domain. The combination and random parameterization of these
elements generates a large heterogeneous dataset used for sound speed
estimation.

The cuboid in-silico phantom domain is defined on a Cartesian grid of points
$p_{i}\in{X\times Y\times Z},$ where
$X=\\{0,1x,2x,...,x_{d}\\},Y=\\{0,1y,2y,...,y_{d}\\}$, and
$Z=\\{0,1z,2z,...,z_{d}\\}$. $x$, $y$, and $z$ are the spatial resolution of
the grid and $x_{d}$, $y_{d}$ and $z_{d}$ are the respective grid dimensions.

####  3.1.1 Tissue Type Label Map

The in-silico simulations include phantoms containing the breast anatomy,
skin, breast gland, breast cysts, and breast lesions resembling fibroadenomas
and glandular tissue (Smithuis et al., 2010). The tissue type label map
assigns every voxel the class of its respective tissue type distributed in
space. Both cysts and lesions in the dataset are modeled by elliptical
inclusions but are differentiated by cysts being anechoic and lesions having
either positive or negative echogenicity. The cyst/lesion mask is defined as
an ellipse $E$ in space projected onto the aforementioned Cartesian grid and
randomly parameterized by the position of its center $(x_{c},y_{c})\in[X\times
Y]$, the lengths of its radii $r_{i=\\{1,2\\}}\in\mathbb{R}\quad\forall\quad
r_{i}<min(x_{d},y_{d}),$ and an orientation angle $\theta\in[0,\pi]$.

$\begin{split}E(x_{p},y_{p},r_{1},r_{2},x_{c},y_{c})=\frac{((x_{p}-x_{c})\cdot\cos(\theta)+(y_{p}-y_{c})\cdot\sin(\theta))^{2}}{r_{1}}+\\\
\frac{((x_{p}-x_{c})\cdot\sin(\theta)-(y_{p}-y_{c})\cdot\cos(\theta))^{2}}{r_{2}}.\end{split}$
(1)

The 2D ellipse is projected in the elevational plane to generate a 3D
inclusion.

The skin mask is defined as a simple linear mask at the top of the cuboid and
parameterized in depth for varying skin thickness within anatomical norms of
0.7 to 3 mm (Huang et al., 2008). All remaining unassigned voxels are assigned
to the breast gland class.

In total, six combinations of the above-mentioned tissue classes are defined;
namely, cyst with skin, lesion with skin, skin, breast gland, lesion, and
cyst. All classes are created over a breast gland background, and the breast
gland class represents a phantom without any other anatomical structures.

####  3.1.2 Scatterer Distribution

Next, a spatial distribution of scatterers and their relative intensity is
generated. In 3D, the discretized scatterer density

$\rho_{s}=\frac{n_{s}}{\lambda^{3}}\cdot x\cdot y\cdot z,\rho_{s}\in[0,1]$

where $n_{s}$ is the number of scatterers in an imaging resolution voxel
(IRV). The unit-less discretized scatterer density $\rho_{s}$ defines the
fraction of all voxels in a region that are labeled as scatterers such that
the number of scatterers per IRV is upheld. The size of the IRV in 3D is
approximated as $\lambda^{3}$ where $\lambda$ is the wavelength of the
transmit pulse.

The scatterer intensity distribution, i.e., how strongly the scatterer at a
given location deviates from average medium properties, is modeled by a
uniform distribution of intensity for every scatterer in the domain. For this,
every point in 3D space is assigned a sample value from the distribution
$U\sim\mathcal{U}_{[-0.5,0.5]}$ to create a spatial white noise distribution
for sound speed and density. This white noise distribution is jointly sampled
by a Bernoulli distribution $B\sim\mathcal{B}(1,\rho_{s})$ to determine the
location of the scatterers in the 3D grid. The Bernoulli distribution
represents the likelihood that a given point in 3D space is a scatterer and is
parameterized by the scatterer density $\rho_{s}$. By jointly sampling these
two distributions, a resulting final scatterer distribution is characterized
by randomly located discretized scatterers with random intensities.

####  3.1.3 Tissue Property Model

The tissue property model takes the previously defined spatial distribution of
tissue types as well as the spatial scatterer distribution in order to
generate realistic in-scilico phantoms of varying tissue classes. A different
method of mapping the tissue properties onto the spatial distribution is
employed for each tissue class. The model utilized for both the skin and
lesion classes is straightforward, while the breast gland model is slightly
more complex for added realism. Breast gland tissue class is modeled by using
a 2D Gaussian random field (GRF) to model the correlated variation of breast
gland tissue properties. A 2D Gaussian filter $g$ of the size $(x_{f},y_{f})$
is defined as
$g(u,v)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{u^{2}+v^{2}}{2\sigma^{2}}}$ where
$u\in[-\frac{x_{f}}{2},\frac{x_{f}}{2}]\text{ and
}v\in[\frac{-y_{f}}{2},\frac{y_{f}}{2}]$ and is convolved with a 2D random
field of size $F=[0,x_{d}+x_{f}]\times[0,y_{d}+y_{f}]\text{ where the field
elements }F_{i}\sim\mathcal{U}_{[0,1]}$ in order to generate a GRF that mimics
breast gland. The resulting GRF is then normalized ($\mu=0)$ and scaled to the
interval $[-0.5,0.5]$, and two sub-regions of foreground and background via a
randomly thresholded to create a random binary map of breast gland regions.
Values below the threshold are assigned to the background (low echogenicity),
and all above are assigned to the foreground (high echogenicity). The
foreground and background define the varying echogenicity regions in the
breast gland class of the resulting image. This value distribution is the
basis of the breast gland model and is later scaled with the mean sound speed
and augmented with the scaled scatterer map. For a given class region, the
tissue properties are assigned as a random uniformly sampled mean sound speed
value combined with a scaled scatterer field intensity to induce the
echogenicity to create the resulting in-silico phantom used for simulation.
The ranges of the sound speed and contrast for each given class can be seen in
Table 1 and were chosen following (Bamber, 1983). As shown empirically in
(Mast, 2000), the values of sound speed and density can be approximated to be
linear, and the density map is set to be proportional to the sound speed map
by a factor of $\alpha_{\rho}$. The attenuation and non-linearity values are
constant, as listed in Table 2.

Lastly, the in-silico phantom sound speed map in Figure 1 (a) is averaged by
region (two sound speeds for breast gland, one for cyst/lesion, and one for
skin) in order to form a target average sound speed map, as shown in Figure 1
(c), suitable for training our deep model. The averaged sound speed map is
used only as a training label and not for the k-Wave simulation. The original
in-silico phantoms are then utilized in k-Wave to generate simulated RF
channel signals from pulse-echo ultrasound.

Table 1: Mean sound speed range and scatter contrast per class used for our
breast ultrasound dataset simulation.

Tissue Class | Mean sound speed Range | Scatter Contrast
---|---|---
Cyst (Bamber, 1983) | [1500, 1620] | -
Lesion (Bamber, 1983) | [1488, 1512] |  $\pm$ 10-30 dB
Skin | [1540, 1670] | 10 dB
Breast Gland (Bamber, 1983) | [1480, 1528] | 12 dB

###  3.2 Data Processing

Neural networks perform best when the dataset they are trained on accurately
represents the dataset they will see at “inference time” or when they are
deployed. Therefore, a large heterogeneous dataset is desirable to train
robust neural networks. To increase the heterogeneity of the training data,
data augmentation is often applied. In computer vision, these augmentations
can include rotations, flips, and deformations of the training data.
Augmentation is applied at train time given an augmentation likelihood.
Furthermore, augmentation is randomly parameterized at every invocation. We
propose the use of ultrasound-specific augmentation techniques. We propose
augmenting convolved with the transducer’s impulse response with a random
relative bandwidth and the addition of thermal noise on the channel data via
thermal noise augmentation (TNA) Huang et al. (2021).

Thermal noise is an artifact resulting from electronic noise in ultrasound
devices but is missing from ultrasound simulations (Hyun et al., 2019). TNA is
performed by adding white thermal noise to channel data with an augmentation
likelihood $p_{TNA}$. TNA can be added on top of the clutter and aberration
noise generated by the forward process of the k-Wave simulation. Here, TNA is
parameterized by an upper and lower bound in noise amplitude relative to the
transmit signal’s Root Mean Square (RMS). This uniform parameterization
distribution is randomly sampled via the method proposed by Hyun et al.
(2019). Due to the constant TNA and attenuating tissue model, SNR reduces over
depth.

Next, a start delay is applied to the RF channel signal to align a defined t0
for every transmitted plane wave correctly. $t_{0}$ defines a standardized
position in space at which the propagation time $t=0$, i.e. the starting time
of wave propagation. This aligns multiple transmissions through a medium
during beamforming independently of steering and focus. The definition $t_{0}$
can vary between devices and simulation platforms. In this work, $t_{0}$ is
defined as the end of the pulse of the last firing element for a given
transmission. This definition removes the transmitted pulse from the training
data, which would otherwise introduce a large amplitude discrepancy and impair
network training. Then, the complex IQ signal is generated via the Hilbert
transform, and each plane wave is beamformed individually via dynamic receive
beamforming with an assumed sound speed $c_{0}$ to generate complex beamformed
IQ images similar to (Stähli et al., 2020). Lastly, the complex IQ components
from the same spatial location are mapped to the channel dimension of the
convolutional neural network.

###  3.3 Network Architecture

Figure 2: Overview of the proposed architecture. Our model is composed of an
encoder that individually processes three beamformed IQ images, whose features
are concatenated after their individual dense blocks, a bottleneck, and a
decoder that utilizes unpooling and produces the sound speed estimations.
Dense skip connections are used within each dense block and long-term skip
connections are placed between encoder and decoder to enhance gradient flow
and maintain feature quality.

We modify a deep fully convolutional neural network $F$ based on (Jégou et
al., 2017) and with modifications from (He et al., 2015) and (Ulyanov et al.,
2016) described below to take, as input, three beamformed IQ images of a
medium (one for each angled plane wave transmission) and output an estimated
sound speed map of the medium defined as

$F:\mathbb{C}^{N\times M}\mapsto\mathbb{R}^{N\times M},$

for an image size of $N\times M$ pixels. This network consists of three input
dense blocks (one for each angled plane wave), a bottleneck, and four decoder
dense blocks that output the model sound speed estimation. The overall
architecture can be seen in Figure 2. Our network utilizes dense network
blocks (Jégou et al., 2017) that incorporate dense skip connections to enhance
the gradient flow and maintain feature quality. In this work, we replace ReLu
activations with PReLu (He et al., 2015), which has been shown to improve
model fitting and reduce the risk of overfitting. Furthermore, batch
normalization layers are replaced with instance normalization (Ulyanov et al.,
2016), which has also been shown to enhance training dynamics in noise
sensitive applications. Our encoder and decoder are comprised of stacked dense
blocks connected via 2D max pooling and unpooling blocks, respectively. Every
dense block consists of three convolutional layers, the first two of which
have a kernel size of 5$\times$5 with stride 1 and the third one a kernel size
of 1$\times$1 and stride 1.

Furthermore, the encoder block input accepts three complex beamformed plane
waves discussed previously. The separate processing of each plane wave ensures
the extraction of robust features, such as phase coherence and feature
position. Later blocks can use these features to generate an accurate sound
speed map. After each plane wave is passed through an individual dense block,
the three plane wave features are concatenated along the channel dimension and
collapsed via a 1$\times$1 convolutional layer. Skip connections (Ronneberger
et al., 2015) are added between each dense block of the encoder and decoder to
prevent vanishing gradients and enhance network trainability.

Our model is trained to estimate a target sound speed map, given three
beamformed IQ images from angled plane waves as input. The Mean Square Error
(MSE) between the estimated and target sound speed map is used as the loss
function. We use weight decay with L2 regularization to avoid overfitting and
maintain weight sparsity (Goodfellow et al., 2016).

##  4 Experimental Setup

###  4.1 In-Silico Simulations

The in-silico tissue models described in Section 3.1 are parameterized with
the values for sound speed, non-linearity (B/A), density, and attenuation as
listed in Table 1. The k-Wave simulation parameters are summarized in Table 2.
The Gaussian filter for the background generation use the parameters
$x_{f}=y_{f}=400$ and $\sigma=600$. In this work, the density ratio
$\alpha_{\rho}$ is set to be $1.5\pm 10\%$, i.e. $[1.35,1.65]$ uniformly
sampled. The transducer modeled for the simulations is the Cephasonics
CPLA12875 (Cephasonics Ultrasound Solutions, Santa Clara, California, USA)
with a transmit frequency of 5 MHz, a sampling frequency of 40 MHz, and a
transmit duration of one tone-burst cycle. Geometrically, the transducer is
modeled to have 128 elements with a total aperture width of 37.5 mm, an
element height of 7 mm, an element width of 0.293 mm, and a kerf of 0 mm
between elements. A medium sound speed of 1540 m/s is used to calculate the
transmit delays for steering angles of -8, 0 and 8 degrees for each plane wave
transmission. Critically, in contrast to (Feigin et al., 2019), all three
transmissions are simulated from the same aperture of the center 64 elements,
as is more commonly used in plane wave imaging pulse sequences for coherent
compounding. On both transmit and receive, rectangular apodization is
employed. The medium dimensions in grid points are $N_{x}=548$, $N_{y}=648$,
and $N_{z}=126$, with a grid spacing of 58.594 $\mu$m in all directions such
that five grid points could fit laterally within one modeled piezo element.
The simulation domain’s total dimensions ($x_{d},y_{d},z_{d}$) are 32 mm
$\times$ 38 mm $\times$ 7.4 mm.

Table 2: Simulation Properties

Property | Value
---|---
Transmit Frequency | 5 Mhz $\pm$ 10%
Center Frequency | 5 Mhz
Density Ratio | 1.5% $\pm$ 10%
Alpha Coeff | 0.75 dB/MHz cm
Alpha Power | 1.5
B/A | 6
Bandwidth | 60%
Tone Burst Cycles | 1
Sampling Frequency | 87.6 Mhz
# Elements | 128 elements
Pitch | 293 $\mu$m
Kerf | 0 $\mu$m

A Perfectly Matched Layer (PML) of $7\times 17\times 9$ grid points is added
to the medium to prevent signal wraparound (Treeby and Cox, 2010). The modeled
transducer is centered on top of the phantom grid. In total, 5996 samples
consisting of three plane wave simulations are generated using the k-Wave
Toolbox (Treeby and Cox, 2010), and the C++ accelerated binary on an NVIDIA
Quadro RTX 6000 GPU with 64 CPU threads. The expected GPU run time per
simulation is 620 seconds, and 43 days 38 minutes and 40 seconds for the
entire dataset.

###  4.2 Data Processing Parameters

The simulated channel data is resampled from 87.6 MHz to 40 MHz. A Gaussian
band-pass filter centered at 5 MHz with a fractional bandwidth uniformly
sampled between 50% and 90% is applied to model the transducer’s impulse
response. TNA was performed with a magnitude range from -120 dB to -80 dB
relative to the ballistic pulse and an augmentation likelihood of
$p_{TNA}=20\%$. A $t_{0}$ was set to $2.75\ \mu\text{s}$ for the center
transmission and $5.0\mu\text{s}$ for the $\pm 8$ degree plane waves.

###  4.3 Network Training

Our deep model is trained with a batch size of 6, for 138 epochs and with a
learning rate of 0.001 with early stopping based on the loss of the simulated
validation set. The Adam optimizer (Kingma and Ba, 2014) is used with weight
decay activated with a decay rate of $e^{-4}$. The network is programmed in
Python using the PyTorch Library v1.7 (Paszke et al., 2019) and the Pytorch
Lightning framework v1.2.10 Falcon et al. (2019) and Weights and Biases
Biewald (2020) for experimental tracking. Our models are trained on an NVIDIA
Quadro RTX 6000 GPU. Our simulated dataset and code will become publicly
available upon acceptance.

###  4.4 Simulation Evaluation

Our model is evaluated on a simulated validation set of 514 samples equally
drawn from all classes. We report the Mean Absolute Error (MAE) between the
predicted and target sound speed for each class. Furthermore, to showcase the
importance of TNA, we compare the error distributions over all classes for two
otherwise identical models, trained with and without TNA. Lastly, we further
investigate the effect of thermal noise on the models by comparing the error
over depth for three levels of additive thermal noise and our baseline without
noise.

###  4.5 Phantom Evaluation and In-Vivo Demonstration

In order to evaluate the predictive efficacy of our model, phantom and in-vivo
studies are performed using a Cephasonics Griffin with 64 channels and a
CPLA12875 transducer.

The sound speed of a homogeneous CIRS Phantom Model 040GSE (CIRS Inc, Norfolk,
VA USA) is verified via speckle brightness (Nock et al., 1989) due to the
phantom age to ensure the phantom sound speed is consistent with the factory-
specified sound speed. The speckle brightness method of sound speed determined
the sound speed in the phantom to be 1558 m/s.

Next, a bovine steak is prepared, and its sound speed is measured to be 1566
m/s in a distilled water bath (24.6∘C, 1495.8 m/s (Marczak, 1997)) using the
method described in Kuo et al. (1990).

The steak is cut into two separate slices of 8 mm and 4 mm thickness and
stacked on the CIRS phantom to create a two-layered model. The regional mean
sound speed error is estimated for regions of interest (ROI) in the steak and
at proximal and distal locations in the CIRS phantom. The differentiation of
proximal and distal regions is performed to showcase the effect of depth-
dependent SNR on the model predictions.

Furthermore, to reduce selection bias and evaluate the temporal consistency of
our model, the regional sound speed estimates are averaged over 100
consecutive static frame measurements to quantify the influence of thermal
noise on the sound speed estimations and stability of the estimates.

In-vivo imaging is performed on the left breast of a healthy volunteer (Age:
28, BMI: 22.4) in three regions. The volunteer was selected and provided
written informed consent under a protocol approved by an ethical committee
from the Technical University of Munich. Channel data for each region are
acquired with the same configuration as for the phantom experiments.

##  5 Results

###  5.1 Validation Set Evaluation

Figure 3: Boxplot comparing sound speed relative estimation error
distributions per class for the simulated validation set. The central mark on
the box indicates the median value, and the top and bottom edges of the box
indicate ther interquartile range. The black whiskers indicate the extent of
the distribution without the outliers, which are denoted by circles on the
plot. Overall, the model trained with TNA achieves a lower relative error
standard deviation and fewer outliers for all classes.

Table 3 shows the classwise MAE on the validation set for a base network
trained without TNA and one trained with TNA. For both models, the classwise
MAE is small relative to the wide sound speed ranges on which the model is
trained and ranges from 8.50 m/s for the skin class with TNA to 16.40 m/s for
the base lesion class without TNA.

Table 3: Sound speed estimation MAE and standard deviation per class for
models trained with and without TNA. Estimations are shown in m/s.

Class | No TNA | TNA
---|---|---
Cyst & Skin | 16.1 $\pm$ 12.4 | 10.6 $\pm$ 5.10
Lesion & Skin | 15.5 $\pm$ 8.70 | 12.0 $\pm$ 5.70
Skin | 12.9 $\pm$ 9.10 | 8.50 $\pm$ 4.00
Breast Gland | 12.8 $\pm$ 8.84 | 7.90 $\pm$ 3.70
Lesion | 16.4 $\pm$ 8.70 | 12.7 $\pm$ 7.30
Cyst | 12.8 $\pm$ 6.30 | 10.6 $\pm$ 5.90
Overall | 14.3 $\pm$ 9.20 | 10.3 $\pm$ 5.60

Table 3 also highlights that TNA substantially improves estimation error
across the classes by 2.2 m/s for the cyst class to 5.5 m/s for the skin and
cyst class. The overall error and standard deviation are also greatly reduced
on the validation set with the addition of TNA.

Figure 3 shows the relative average error for each class for models trained
with and without TNA. Though the performance of both DNNs is good, TNA
contributes toward reducing the standard deviation (signified by box size) of
the relative error and number of outliers (signified by circles).

#### Effect of Thermal Noise over Depth

In Figure 4, we show the effect of additive thermal noise over transmission
depth on the predictions of networks trained with and without TNA. We select
three scales of additive noise, specifically $-80$ dB, $-100$ dB, and $-120$
dB relative to the transmit signal RMS, along with a baseline measurement
without noise. First, it can be seen that the network trained with TNA
(Bottom) is robust to thermal noise since the error remains low over the
entire transmission depth of the measurements. The network trained without TNA
(Top) is severely affected by thermal noise present in the channel signals for
all noise levels, with increasing error from -120 dB to -80 dB. The baseline
sound speed error shows that the network trained without TNA is able to
accurately estimate the sound speed over transmission depth on the validation
set when no noise is added. For noise levels -120 and -100 dB, the model
trained without TNA underestimates the sound speed in the medium. For the
noise level of -80 dB, the model underestimates to a depth of 1.6 mm and then
overestimates the sound speed in the medium. The network trained with TNA only
marginally underestimates the medium sound speed after 2 mm depth.

Figure 4: Relative sound speed estimation error over depth for the simulated
validation set for three levels of additive thermal noise and baselines
without added noise. Our model trained with TNA (Bottom) is substantially more
robust to the addition of thermal noise, while the one trained without TNA
(Top) is more sensitive to noise and its performance substantially decreases
along with the SNR over depth.

#### Qualitative Evaluation

Figure 5: Simulated B-modes with target sound speed and model estimation from
the six classes. Our simulations produce quasi-realistic B-mode images, and
our model successfully estimates sound speeds and contours of cysts, lesions,
skin, and background.

Qualitative results of simulated B-modes for all classes and their respective
sound speed estimates are shown in Figure 5. Our proposed simulation pipeline
creates B-mode images with an overall quasi-realistic breast-tissue
appearance. Consistent with the results shown above, our model is able to
successfully estimate sound speed distributions throughout the simulated
domain for all data classes. Contours of both anechoic and echogenic features
in the images are successfully recovered, and the sound speeds within the
regions are correctly estimated. In classes with cysts, reverberation
artifacts are often present on the boundaries of the simulated cysts.
Nonetheless, the model can generate accurate sound speed estimates
successfully.

###  5.2 CIRS Phantom Evaluation

In order to evaluate the generalization ability of our model, we evaluate its
performance on a layered phantom that was not represented in the training set.
One hundred ultrasound frames are acquired with a real transducer, and the
results for the layered phantoms with both a 4 mm and 8 mm bovine steak
phantom are shown in Table 4. As stated in Section 4.5, the measured sound
speed of the steak is 1566 m/s, and that of the CIRS phantom is 1558 m/s. The
evaluation of the model estimation is performed in three discrete ROIs that
extend across the entire image aperture and are depicted by red, yellow, and
green boxes in Figure 6 in order to evaluate the influence of thermal noise
and attenuation along with other real-world factors on the model’s estimation
at varying depths.

Table 4: Sound speed estimations and errors for the CIRS and steak phantom
predictions compared with the insertion and speckle brightness methods in m/s.
Estimations, errors and standard deviations are computed over 100 consecutive
frames.

|  Traditional Measurements (m/s) | 4mm Steak | 8mm Steak
---|---|---|---
| Estimation | Error | Estimation | Error
Steak (red) | 1566 | $1564.4\pm 3.60$ | $-1.60\pm 3.60$ | $1564.6\pm 2.70$ | $-1.40\pm 2.70$
CIRS Background (yellow) | 1558 | $1555.6\pm 4.43$ | $-2.40\pm 4.43$ | $1558.9\pm 2.49$ | $+0.90\pm 2.50$
CIRS Background (green) | 1558 | $1544.7\pm 6.90$ | $-13.9\pm 6.90$ | $1542.7\pm 4.80$ | $-15.3\pm 4.80$

Even though our model is solely trained on simulated breast ultrasound
signals, it is still able to infer the sound speed of these two-layered
phantoms in agreement with in-vitro sound speed measurement. The mean error
for the steak layers ranges from 1.60 m/s for the 4 mm steak to 1.40 m/s for
the 8 mm one. Furthermore, the sound speed for the top ROI of the CIRS phantom
is also successfully estimated with a mean error of 2.40 m/s for the 4 mm
steak and 0.90 m/s for the 8 mm one. As seen in Table 4, the standard
deviation values of the estimations among the 100 consecutive frames are also
low, ranging from 2.70 m/s to 6.90 m/s, showcasing the temporal consistency of
the model predictions for real-world data.

Figure 6: Sound speed estimations for CIRS and steak layered phantoms along
with B-mode images. The red ROI delineates the steak at a depth of 4 mm and 8
mm respectively. The yellow ROI delineates the top of the abutting CIRS layer
and is 8.6 mm thick (left) and 6.6 mm thick (right). A green ROI encloses the
bottom 2.9 mm of both phantoms. Model estimations are coherent and in
agreement with the measured sound speed of 1566 m/s for the steak and 1558 m/s
for the CIRS background.

Finally, we can see that the prediction for the bottom 2.9 mm of the CIRS
phantom (green ROI) has a larger error than the top, ranging from 13.90 m/s
for the 4 mm steak phantom to 15.30 m/s for the 8 mm layer steak phantom. The
total range of the 8 mm steak phantom is 1516.20–1653.70 m/s and
1518.90–1645.90 m/s for the 4 mm steak phantom. The prediction of the model
trained without TNA for the bottom region of the CIRS phantom is 1536.70 m/s
$\pm$ 4.72 m/s for the 4 mm steak phantom and 1510.70 m/s $\pm$ 8.11 m/s for
the 8 mm steak phantom. This improvement in accuracy and reduced standard
deviation shows model superiority when trained with TNA, which decreases the
error from 29.30 m/s to 13.90 for the 4 mm steak phantom and from 55.30 m/s to
15.30 m/s for the 8 mm steak phantom. These results are in agreement with
those on the validation set and are promising for the generalization of our
trained model beyond simulations to out-of-distribution heterogeneous tissues

###  5.3 In-vivo Demonstration

Figure 7 shows the predictions of our model for three breast regions in a
healthy volunteer. As with the phantom evaluation, we calculate the average
sound speed over 100 consecutive frames with a static probe for the in-vivo
measurements. With no specific ROI or ground truth, the estimated sound speed
of the entire field view is evaluated. The overall mean sound speed over 100
frames for R1, R2, and R3 are 1518.0$\pm$5.3, 1500.1$\pm$6.1, and
1499.0$\pm$3.4 m/s, respectively. These values are consistent with each other
and the literature on the sound speed of measured glandular breast tissue of
1505.0$\pm$47.3 m/s from the Foundation for Research on Information
Technologies in Society (IT’IS) (Hasgall et al., 2018) and 1510 m/s from
Nebeker and Nelson (2012). Also, the model predictions align with the values
of our simulated dataset, where the breast gland is modeled with sound speed
between 1480 m/s and 1528 m/s following (Bamber, 1983).

##  6 Discussion

Our proposed modeling of breast tissue creates quasi-realistic US B-mode
images from simulated signals. Networks trained on these simulated signals can
be deployed on real scanners and applied to in-vivo data with interpretable
results. The proposed 3D phantom allows the modeling of 3D wave simulations on
an in-silico phantom. In order to have clear and accurate 2D label maps for
each 3D phantom, some properties were projected in the elevational plane. This
projection allows for an accurate and known 2D sound speed distribution for a
3D phantom used for the non-linear wave simulations. This assumption of
elevational consistency is considered to be a fair approximation of many in-
vivo tissue distributions in the elevational plane of a linear ultrasound
transducer. Despite this approximation in the elevational plane, it is
essential to note that all simulations modeled the 3D wave propagation within
a 3D medium. Critically, the scatterer distribution field was modeled in 3D,
ensuring that despite the projection of 2D distributions, every slice in the
elevational plane was independent. The real-world transducer was also modeled
with a Gaussian impulse response which is representative of the technical
specification of the transducer.

A possible advantage of the complex spatial IQ representation is that our
model can process both the magnitude and phase information when predicting
spatial sound speed distributions, taking into account slight phase shifts,
without having to learn filters to extract the phase shift within the network.
Our fully-convolutional architecture takes a multi-scale context, i.e., the
large anatomical features and local phase shift features, into account when
generating sound speed estimates. High-frequency filters in shallow layers and
a large receptive field in deeper layers of the proposed architecture
(Goodfellow et al., 2016) enable multi-scale feature extraction. These multi-
scale features are collected via skip connections and combined with the
decoder weights to generate the final sound speed estimates.

Due to the constant sound speed assumption used for beamforming, the geometry
of some B-mode images in Figure 5 can be spatially distorted compared to the
true geometrical layout of their respective media. Geometric deformation is
especially prominent below the lesion regions for the Lesion & Skin and Lesion
classes in Figure 5 where the lower lesion boundary is not pictured in the
B-mode but is visible in the sound speed simulation medium. Notably, the
estimation of the sound speed maps does not simply correlate with the
echogenicity in the B-mode images. On the contrary, the estimated sound speed
maps correspond correctly to the spatial distribution of the target sound
speed maps and not the B-modes, indicating that signal echogenicity is not the
only feature used for sound speed estimation, and the network also considers
the relative spatial feature location. This could be attributed to the fact
that the scatterer sound speed standard deviation was randomly sampled at
simulation time to generate an echogenicity from a defined uniform
distribution. The echogenicity distribution was sampled independently of the
underlying mean sound speed values. Furthermore, the sound speed values of
anechoic cysts are also accurately estimated. Since anechoic cysts contain no
reflectors, there is no local spatial information in the pulse-echo signal to
indicate their sound speed. The correct estimation of the sound speed of
anechoic cysts shows that global context is used to estimate sound speed and
not only local echogenicity.

The real-world predictions of the layered CIRS and steak phantoms in Figure 6
show that both cases are consistent with a homogeneous background, which was
unseen in the training set. The sound speed difference between the steak and
CIRS layers is measured with the insertion method to be 8 m/s. Our model
estimates a sound speed with an accuracy of 8.8 m/s for the 4 mm steak phantom
and 5.7 m/s for the 8 mm phantom, close to the insertion and speckle
brightness methods. In the case of the homogeneous medium, the network did not
infer a sound speed distribution with the appearance of the breast tissue from
the training set, but rather correctly inferred a homogeneous sound speed,
indicting that the network had learned a collection of robust features that
generalize beyond the training data to real transducer data and out of
distribution property geometries. Normally, it would be expected for network
performance to deteriorate on out of distribution samples. Nevertheless, the
macro sound speed estimate is accurate even for out of distribution
homogeneous samples. Two regions of over-estimation (1620 m/s) can be seen in
the top 1-2 mm of both phantoms, especially the 8 mm steak phantom, resembling
the skin class from the training set. Despite this fact, when median absolute
distance outlier removal is applied frame-wise to the sound speed in the
region of interest, the sound speed estimate is $1562.3\pm 2.6$ over 100
frames, only modestly increasing the regional error by 2.3 m/s.

The in-vivo demonstration showed global sound speed estimates that were in
line with reference values from literature. Unlike the homogeneous phantom
models, the in-vivo estimates displayed the expected tissue variation in the
sound speed estimate which resemble the underlying breast tissue distribution.
Furthermore, all estimated values in the in-vivo estimation were within the
expected range for in-vivo breast tissue. It is important to note that the
same model, trained on the simulated dataset, evaluated on both phantom and
in-vivo data after being trained only on simulated data. For both of these
cases, the model could differentiate the sound speed regimes of $\sim 1550$
m/s and $\sim 1500$ m/s, respectively.

While physics-based models are often limited to correlating sound speed with
spatially local features, convolutional neural networks consider global
spatially distributed features when estimating sound speed. Specifically,
physics-based models often struggle to make accurate sound speed estimates
between 0 and 5 mm of a scan due to a multitude of complexities, such as lack
of coherent wave formation, contact interfaces, and limited angular
sensitivity can invalidate the underlying assumptions upon which the model is
based (Jakovljevic et al., 2018; Stähli et al., 2020). Because neural networks
model the training data and not a canonical model, it can be hypothesized that
spatial aberration relationships are features for sound speed prediction. The
network can interpret, for example, aberration in the middle of the image
relative to other measurement signals to indicate the aberrating medium sound
speed above.

We hypothesize that the slight sound speed underestimation in the bottom of
both the phantom and in-vivo scans results from lower SNR deeper in the
medium. Thermal noise is present in real-world transducers and TNA contributes
toward bridging the performance gap but does not entirely alleviate the
problem. Further, the thermal noise amplitude used in the TNA might not
directly match the amplitude in the US device, in part due to mismatched
attenuation values. Other noise sources in the signals from the lower regions
could lower the SNR and contribute to the performance loss.

Methods of increasing SNR such as higher angular sampling frequency by an
increased number of plane wave firings could potentially alleviate the problem
of low SNR at deeper imaging depths and lead to increased performance on less
superficial anatomies. An increase in the number of transmissions passed to
the network, of course, comes at a computational cost when generating
simulation data, which why it was not performed in this work.

Our phantom and in-vivo results display the proposed method’s robustness by
correctly predicting sound speed on out-of-distribution data and under the
influence of real-world factors. This robustness can be attributed to the
proposed modestly anatomically realistic simulations and the data pre-
processing pipeline with TNA that improve generalization to real-world
signals.

Furthermore, the robust evaluation of our method goes beyond the standard
protocol for deep model evaluation in medical imaging. This evaluation
pipeline includes testing our model on external data sources of real-world
phantom and real-world in-vivo data that were not included in the training
distribution and reporting the model predictions over 100 US sequential
frames. The real-world inference on in-vivo data showed that the proposed
method could generalize beyond the elliptical and linear contours of the
training data set and infer sound speed on biological sound speed
distributions. The low standard deviation of our errors shows the stability of
our predictions over 100 consecutive frames. This approach could set a new
precedent for evaluating the consistency of sound speed estimation for
physics-based and deep learning models.

Future work includes more realistic modeling of real transducers and in-vivo
artifacts. The dataset could be further extended to include irregularly shaped
lesions to model malignant tissue with irregular boundaries. Such modeling
will be crucial for developing robust and generalizable sound speed estimation
models with DNNs. Furthermore, the presented method utilizes three plane
waves, which reduces the SNR of the signal at both training and inference
time. It is expected that with the simulation of more plane waves to a
comparable number to Stähli et al. (2020), the performance could increase
further along with the computational cost. Finally, our dataset could be used
as a benchmark for sound speed estimation methods to increase their
comparability, similarly to challenges in beamforming such as PICMUS and CUBDL
(Bell et al., 2020; Liebgott et al., 2016).

##  7 Conclusion

In this paper, we proposed a novel pipeline for sound speed estimation of
breast ultrasound imaging. A large-scale quasi-realistic simulation dataset
was created, processed, and used to train our tailored fully convolutional
network architecture to produce sound speed estimations from beamformed IQ
plane wave ultrasound data. Our model, which will become publicly available
along with our simulated dataset, is a promising step toward precise and
generalizable sound speed estimation for ultrasound imaging and could be
further extended to other anatomies, such as thyroid or liver, or used as an
initialization for traditional ultrasound estimation techniques such as
(Stähli et al., 2020) to enhance convergence and alleviate the need for
strenuous regularization and hyper-parameter tuning. The proposed method is
simulator agnostic so long as a ground truth sound speed map can be generated,
and the simulator can accept input phantoms as proposed in this work. Our
method was evaluated on simulated, phantom, and in-vivo breast ultrasound
data. The estimated sound speeds were temporally consistent among frames and
were in agreement with traditionally measured sound speeds and clinical
literature. Future work could also utilize our predicted sound speeds to
improve beamforming quality by removing the assumptions of constant sound
speed and straight ray propagation.

Figure 7: In-vivo sound speed estimations along with B-mode images for three
breast regions R1-R3. Our model is able to estimate coherent maps for all
three breast regions, with breast gland sound speed values within the sound
speed range measured in (Hasgall et al., 2018; Nebeker and Nelson, 2012;
Bamber, 1983). Moreover, tissue contours around fat and connective tissue are
also correctly delineated by our model.

##  8 Declaration of Competing Interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  9 Acknowledgments

The work of Walter Simson was supported by grant ZF4190502CR8 of the Zentrale
Innovationsprogramm Mittelstand (ZIM). The work of Jeremy Dahl was supported
by grant R01-EB027100 from the National Institute of Biomedical Imaging and
Biotechnology.

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