Sound Field Simulation in Breast Cancer Research

Submitted by Dinah Brandner, BSc

Submitted at Institute for Measure- ment Technology, JKU and Radiology Department, Stanford University

Supervisor Dr. Bernhard Zagar and Sound Field Simulation Dr. Katherine Ferrara in Breast Cancer July 2020 Research

Master Thesis to obtain the academic degree of Diplom-Ingenieurin in the Master’s Program Mechatronik

JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696

Abstract

According to the American Cancer Society, it is estimated that one in eight women will develop invasive breast cancer in their lifetime. Key for a full recovery is early detection, which is why regular screening is recommended, as many early breast carcinoma show no symptoms like pain or discomfort, and are too small to be detected by self-observation. The gold standard for breast screening still is the mammography, being the most accurate in diagnosis, besides magnetic resonance imaging (MRI) and ultrasonic imaging. However, since screening by mammography or MRI is associated with risks and comparatively high costs, efforts are being made to increase the diagnostic accuracy of the noninvasive ultrasound imaging. This can be done by estimation of the acoustic properties of the different breast tissue types out of ultrasound image data, as being able to determine these properties additionally to the structural (morphologic) information, is expected to improve the diagnostic accuracy. But, in order to verify that the method used for estimation leads to unbiased results of low variance, methods for producing phantoms of known locally variable acoustic properties need to be developed. Part of this thesis is therefore devoted to the improvements of methods for the production of phantoms with certain acoustic properties, using tissue-mimicking materials. Since it is known from literature that the majority of malignant processes are highly acoustically attenuating, formulars based on Al2O3 are characterized with respect to their absorption behavior, as this material, dependend on the chosen concentration, allows to select the attenuation parameters of a to be produced phantom. Also artificial intelligence can help in the improvement of ultrasonic diagnostic, when employed to assist the radiologist in making a diagnosis by drawing attention to even small morphologies that may seem suspicious. Using huge annotated databases and self-learning algorithms, computers can recognize trained patterns with great statistical certainty. These annotated databases are not yet available for ultrasound images or not in sufficient quality. Therefore, ultrasound simulations of annotated volume data derived from other imaging

3 modalities are useful for the generation of a breast cancer ultrasound database. Two open- source ultrasound simulation software, Field II, and k-Wave, were tested for suitability of providing valid simulation results that match as closely as possible in any aspect ultrasound images obtained from clinical equipment. As it turned out k-Wave was designed to fulﬁll all necessary requirements, and was therefore investigated in great detail, and several simulation results are presented.

4 Zusammenfassung

Nach Angaben der American Cancer Society wird geschätzt, dass jede achte Frau im Laufe ihres Lebens invasiven Brustkrebs entwickelt. Der Schlüssel für eine vollständige Genesung ist die Früherkennung, weshalb ein regelmäßiges Screening empfohlen wird, da viele Brustkrebs- Frühstadien keine Symptome wie Schmerzen oder Beschwerden aufweisen und zu klein sind, um durch Selbstbeobachtung erkannt zu werden. Der Goldstandard für das Screening ist nach wie vor die Mammographie, die neben der Ma- gnetresonanztomographie (MRT) und der Ultraschallbildgebung die genaueste Diagnose er- laubt. Da jedoch das Screening mittels Mammographie oder MRT mit Risiken und vergleichs- weise hohe Kosten verbunden ist, wird versucht die Diagnosegenauigkeit der nicht-invasiven Ultraschallbildgebung zu erhöhen, um diese künftig vermehrt einzusetzen. Dies kann durch Abschätzung der akustischen Eigenschaften der verschiedenen Brustgewe- bearten anhand der Daten eines Ultraschallbildes erfolgen, denn es besteht die Möglichkeit die diagnostische Genauigkeit zu verbessern, wenn man neben den strukturellen (morpholo- gischen) Informationen auch noch die zugehörigen akustischen Eigenschaften bestimmt. Um jedoch zu veriﬁzieren, dass die zur Schätzung verwendete Methode zu erwartungstreuen Er- gebnissen geringer Varianz führt, müssen Methoden zur Anfertigung von Phantomen bekann- ter, örtlich variabler akustischen Eigenschaften entwickelt werden. Ein Teil dieser Arbeit ist daher der Entwicklung von Verfahren zur Produktion von Phantomen mit vorherbestimmba- ren akustischen Eigenschaften gewidmet. Da aus der Literatur bekannt ist, dass der Großteil der bösartigen Prozesse akustisch stark dämpfende Eigenschaften besitzt, werden auf Al2O3 basierende Rezepturen hinsichtlich deren Absorptionsverhalten charakterisiert. Auch künstliche Intelligenz kann bei der Verbesserung der Ultraschalldiagnostik helfen, wenn sie eingesetzt wird, um die Radiologin oder den Radiologen bei der Erstellung einer Diagno- se insofern zu unterstützen, als ihre bzw. seine Aufmerksamkeit auf selbst kleine, verdäch- tig erscheinende Morphologien zu lenken. Aus großen annotierten Datensätzen und unter Anwendung selbst lernender Algorithmen können Computer trainierte Muster mit großer

5 statistischer Sicherheit erkennen. Diese annotierten Datensätze stehen für Ultraschallbilder noch nicht oder nicht in ausreichender Qualität zur Verfügung. Daher sind Ultraschall Si- mulationen von aus anderen Abbildungsmodalitäten gewonnenen annotierten Volumendaten hilfreich, um eine Brustkrebs Ultraschall-Datenbank erstellen zu können. Aus diesem Grund wurden zwei Open-Source-Ultraschall-Simulationsprogramme, Field II und k-Wave, auf ihre Eignung getestet, Simulationsergebnisse zu liefern, die in jeder Hinsicht mit Ultraschallbildern übereinstimmen, die von klinischen Geräten gewonnen werden könnten. Wie sich herausstell- te, ist k-Wave sehr gut geeignet alle notwendigen Anforderungen zu erfüllen, und wurde daher sehr detailliert untersucht. Einige Simulationsergebnisse werden in dieser Arbeit präsentiert.

6 Preface

This thesis was written in partial fulfillment of the requirement for a master of science (MSc) - degree in Mechatronics at the Johannes Kepler University Linz. Part of the work was done during a research stay with the Stanford University, where I was invited to join the Radiology Department, and in particular the Ferrara Lab led by Dr. Katherine W. Ferrara. The Lab is working in cancer research in general and among other things in ultrasound imaging and applying artificial intelligence in diagnostics. When I first visited Dr. Ferrara in August 2019, the idea was for me to work on the generation of a 3D ultrasound image database to be used in deep learning. As deep learning requires a huge training database of sufficient quality that currently is not available for the ultrasonic imaging modality but is readily available for the CT and MRI imaging modalities, ultrasound simulations mapping MRI contrast to ultrasound images turned out to be the way to go. This chain of reasoning brought me to the topic of my master thesis, the sound field simulation in breast cancer research. As I am not the ordinary mechatronic student, being fascinated by all aspects of technology, I was really happy I could spend some time delving into medical topics. Throughout this journey many people helped and supported me for which I am deeply grateful and whom I would like to thank. First I would like to thank my family and friends, who are always by my side, believing in me and supporting me in difficult times. Special thanks to my parents, who always support and encourage me to find my path in life. A big thank you also to my brother and sister, who always encourage me to reach for the stars, who are there for me in times of disappointment and with whom I can celebrate small and big wins. Special thanks also to my best friend Nici, I would not be where I am right now without her. Also I would like to thank Professor Katherine Ferrara and all members of her team, for welcoming me with open arms, supporting and helping me, to make my stay there, although

7 cut short due to the Covid-19 pandemic, a time I will never forget. Special thanks also to the LCM and in particular the COMET K-2 Program of The Austrian Research Promotion Agency (FFG) for funding my research at Stanford. And of course I would like to thank my advisor at the JKU, Dr. Bernhard Zagar, who introduced me to Dr. Katherine Ferrara, and who supported me through the whole process of my research stay and preparation of my master thesis, which was deﬁnitely not easy for me. It was an exciting journey and I am glad to have experienced it.

Per aspera ad astra. Lucius Annaeus Seneca -Wave simulated ultrasound pressure ﬁeld. k

Contents

List of symbols 11

1 Introduction 17

2 Medical imaging modalities 23

2.1 Introduction...... 23 2.2 Mammography...... 23 2.2.1 Fundamentals of x-rays...... 24 2.2.2 Operating principle of mammograms...... 26 2.2.3 Limitation and risks...... 27 2.3 Magnetic resonance imaging...... 28 2.3.1 Fundamentals of MRI...... 29 2.3.2 Operating principle of MRI...... 30 2.3.3 Limitation and risks...... 33 2.4 Ultrasonic imaging...... 34 2.4.1 Fundamentals of ultrasound...... 34 2.4.1.1 Deﬁnition of acoustic waves...... 38 2.4.1.2 Interactions of acoustic waves with human tissue...... 41 2.4.2 Operating principles of ultrasonic imaging...... 49 2.4.3 Limitation and risks...... 54

3 Tissue-mimicking phantoms 55

I 3.1 Introduction...... 55 3.2 Tissue-mimicking materials (TMM)...... 56 3.2.1 Phantom making...... 56 3.3 Measurement methods...... 57 3.3.1 Measurement setup...... 58 3.3.2 Data processing...... 59 3.4 Measurement results...... 63 3.5 Discussion and conclusion...... 65

4 Sound ﬁeld simulations 67

4.1 Introduction...... 67 4.2 The Field II simulation software...... 68 4.2.1 The method of spatial impulse response...... 69 4.3 The k-Wave simulation software...... 72 4.3.1 The simulation strategy of k-Wave...... 73 4.4 Simulation Results...... 78 4.4.1 First simulation results...... 78 4.4.2 Simulation results of tissue-mimicking phantoms...... 83 4.4.3 Limitations of the k-Wave software...... 86 4.4.4 Guideline to numerically stable ultrasound simulations...... 93

5 Discussion 95

A Program listings 106

A.1 File for the calculation of the attenuation coefficient of tissue-mimicking phantoms...... 106 A.2 k-Wave simulation files...... 111 A.2.1 Simulation file...... 111 A.2.2 The function kspaceFirstOrder3D_video_v3 ...... 116 A.2.3 File to generate the video...... 119

II B Decimal to IEEE-754 conversion example 121

III

List of symbols

Latin symbols

Symbol SI-Unit Description ~a m s−2 Acceleration A m2 Area

A0 Pa Initial wave amplitude

Aph Pa s Magnitude spectrum of phantom data

Aref Pa s Magnitude spectrum of reference data

Aw Pa Wave amplitude

Awx Pa Wave amplitude in x-direction 2 Ax m Area normal to the x-axis B~ T Magnetic ﬂux density vector

B~0 T Magnetic ﬂux density vector

Bz T Magnetic ﬂux density vector z-component c m s−1 Speed of sound −1 c0 m s Speed of sound in the quiescent state −1 c1 m s Speed of sound of medium 1 −1 c2 m s Speed of sound of medium 2 −1 cb m s Speed of sound of the background medium −1 cr1 m s Speed of sound of scattering region 1 −1 cr2 m s Speed of sound of scattering region 2 −1 cr3 m s Speed of sound of scattering region 3 −1 cw m s Speed of sound of water d m Thickness of handmade phantom

dx m Side length of the volume element

dy m Side length of the volume element

11 Symbol SI-Unit Description dz m Side length of the volume element

~ex – Unit vector in x-direction

~ey – Unit vector in y-direction

~ez – Unit vector in z-direction

Ephoton J Photon energy F~ N Force f Hz Frequency fexcite Hz Center frequency of the transducer

Fx N Force in x-component h Js Planck’s constant (6.63×10−34) I A Current −2 Ii W m Sound intensity incident −2 Ir W m Sound intensity reﬂected −2 Iref W m Reference sound intensity −2 It W m Sound intensity transmitted −2 Iv W m Variable sound intensity I~ W m−2 Sound intensity j – Imaginary unit k rad m−1 Wave number, angular spatial frequency −1 kx rad m Wave number for the x-component −1 ky rad m Wave number for the y-component −1 kz rad m Wave number for the z-component K Pa Bulk modulus L dB Level (rel. to a reference value) −2 Li dB Sound intensity level (rel. to 1 pW m ) −5 Lp dB Sound pressure level (rel. to 2 × 10 Pa) m kg Mass N – Number of coil windings

Nx – Number of grid points in x-direction

Ny – Number of grid points in y-direction

Nz – Number of grid points in z-direction ∅b m Diameter of background medium ∅r1 m Diameter of scattering region 1 ∅r2 m Diameter of scattering region 2

12 Symbol SI-Unit Description

∅r3 m Diameter of scattering region 3 p Pa Sound pressure px Pa Sound pressure in x-direction pi Pa Sound pressure of incident wave pr Pa Sound pressure of reﬂected wave pt Pa Sound pressure of transmitted wave P W Sound power

Pref W Reference sound power

Pv W Variable sound power

Ra – Pressure amplitude reﬂection coeﬃcient

Ri – Intensity reﬂection coeﬃcient −1 −2 SF N kg =ˆ m s Input force per unit of mass ∆t s Time step t s Time tend s Duration of longst path through computational domain T s Temporal period

Ta – Pressure amplitude transmission coeﬃcient

Ti – Intensity transmission coeﬃcient ◦ Tw C Water temperature ~u m s−1 Particle velocity ν Hz Frequency of the radiation V m−3 Volume ~x m Position vector xb m Background position in x-direction xr1 m Position of scattering region 1 in x-component xr2 m Position of scattering region 2 in x-component xr3 m Position of scattering region 3 in x-component ~y m Position vector yb m Background position in y-direction yr1 m Position of scattering region 1 in y-component yr2 m Position of scattering region 2 in y-component yr3 m Position of scattering region 3 in y-component ~z m Position vector Z N s m−3 Acoustic impedance

13 Symbol SI-Unit Description −3 Z0 N s m Acoustic impedance of medium 1 −3 Z1 N s m Acoustic impedance of medium 2

Greek symbols

Symbol SI-Unit Description α dB cm−1 Attenuation coeﬃcient −γ −1 α0 dB MHz cm Power law prefactor −γ −1 αb dB MHz cm Power law prefactor of the background −γ −1 αr1 dB MHz cm Power law prefactor of scattering region 1 −γ −1 αr2 dB MHz cm Power law prefactor of scattering region 2 −γ −1 αr3 dB MHz cm Power law prefactor of scattering region 3 −γ −1 αw dB MHz cm Power law prefactor of water

αF – Order of diﬀerentiation

θi rad Angle of incidence

θr rad Angle of reﬂection

θt rad Angle of transmission Θ rad Angle of the wavefront λ m Acoustic wavelength −1 µa m Amplitude attenuation factor ρ kg m−3 Mass density −3 ρ0 kg m Mass density in the quiescent state −3 ρb kg m Mass density of the background −3 ρr1 kg m Mass density of scattering region 1 −3 ρr2 kg m Mass density of scattering region 2 −3 ρr3 kg m Mass density of scattering region 3 −3 ρw kg m Mass density of water γ – Power law exponent −1 −1 γc rad s T Gyromagnetic ratio ω rad s−1 Angular temporal frequency −1 ω0 rad s Larmor-frequency

14 Operators

Operator SI-Unit Description ∂/∂t s−1 Partial derivative by time ∂2/∂t2 s−2 Second partial derivative by time < {x} − Real part of the complex number x grad(f(x, y, z)) m−1 Gradient (directional derivative) of scalar field f D s−1 Differentiation operator F s Fourier-transform operator L – Linear integro-differential operator ∇ m−1 Nabla operator ∆ m−2 Laplace operator

Chapter 1

Introduction

According to the American Cancer Society [1], breast cancer is the second most common form of cancer in american women after skin cancer and the second major cause of cancer-related death after lung cancer. In 2020, it is projected that 279,100 new breast cancer cases occur and 42,690 deaths, only in the United States. It is estimated that one in eight women will develop invasive breast cancer in their lifetime and one in a hundred patients diagnosed with breast cancer are male. The reason for women being more affected is that most forms of breast cancer start in the parts of the breast responsible for the milk production and transport. Changes in the breast are very common and most of them are benign, which means non- cancerous. These changes are not life threatening, however, every change increases the chance of getting breast cancer later on. And as there are no symptoms linked directly to breast cancer and some forms of breast cancer show no symptoms at all, it is hard to tell bengin and malignant, meaning cancerous, breast conditions apart, and further diagnostic steps are necessary to take. As the 5-year survivability dependents on the lesion size, early detection is the key to prevent the development of life-threatening breast cancer and to increase the chance of a full recovery. Therefore to recognize changes immediately, regular screening is recommended, as many early breast carcinoma show no symptoms like pain or discomfort, and are too small to be detected by self-observation [3]. Therefore finding a screening method that is cost effective, fast, risk free and of highest accuracy is of significant interest. The accuracy of a screening method is defined by two parameters, the sensitivity, the true positive rate (TPR), which states the probability of being tested positive out of all diseased

17 patients, and the speciﬁcity, the true negative rate (TNR), which states the probability of being tested negative out of all not diseased patients. These rates are deﬁned as [4]:

TP TN sensitivity = TPR = specificity = TNR = , (1.1) TP + FN TN + FP where TP represents the patients truly positive for cancer, TN the truly negative, FP the falsely positive and FN the falsely negative. As a false positive diagnosis, which means being tested positive for cancer, although not being sick, is mentally challenging, a high specificity is important. But critically essential is a high sensitivity, as a false negative diagnosis is fatal. That is why, especially for women with dense breasts, as they have a 4- to 6-fold increased risk of developing breast cancer, it is important to find a screening method with both, high sensitivity and high specificity.

Figure 1.1: Representation of the composition of a human breast (left) [5] and a mammogram with a palpable mass in the upper breast (right) [6].

There are several medical imaging modalities, which will be described in more detail in Chapter 2. The gold standard for breast cancer screening currently is the mammography, which uses low dose x-rays (ionizing radiation) to examine the human breast and allows the identiﬁcation of changes in a breasts tissue years before a palpable mass is developed. But due to the utilized ionizing radiation, which can lead to biochemical changes in living

18 cells, the chance of possible harm caused by this form of radiation and the development of cancer later on increases. The sensitivity and specificity for digital mammography are 97% and 64.5%, with a decrease in sensitivity in women with dense breasts [7]. Therefore the mammography remains the most cost-effective approach, as the benefits outweigh the risks, if done occasionally [1]. The second most common medical imaging modality for breast screening is the magnetic resonance imaging (MRI), which uses a strong magnetic field and radio waves to form an image of the different tissues of the breast. In contrary to the mammography, this imaging modality does not use ionizing radiation, therefore is considered risk free, but for obtaining high- resolution images, to detect the onset of small lesions, the injection of intravenous contrast agents is required and this in turn entails risks. But because of its high sensitivity of 99%, MRI has been increasingly used in high-risk patients. The specificity ranges from 37-97%, also the application is very expensive and time consuming, therefore this imaging modality was not yet considered for the application on average-risk patients and as a regular screening method [8]. The only screening method considered riskfree is ultrasonic (US) imaging, where pulses of ultrasound waves are sent through the body and the received echos from the different tissues of the breast are used to reconstruct an image of the internal structures. Due to its cost-effective and rapid application, this imaging modality would be a great approach for a regular and riskfree examination, replacing the mammography as the gold standard for breast screening. Therefore it is necessary to find a way to increase the sensitivity and specificity, which are with 82% and 84% lower than those of mammography and MRI [9]. Ultrasound is mechanical oscillation of an elastic medium, which propagates through gases, fluids and solids. So–called transducers generate pulses of ultrasound, these pulses travel through the body in form of waves, the ultrasound waves. Due to the ultrasound waves the tissues particles oscillate around their mean position and therefore supporting the ultrasound wave to travel through. Now every tissue type has its individual acoustic properties, which means the ultrasound wave behaves differently in the different tissue types. A breast, as shown in Fig. 1.1, is composed of all sorts of tissue, there is fatty, fibrous, glandular, supportive and connective tissue, there can be cysts, benign or malignant masses or lumps, and in women also the milk ducts. All of these tissue types have individual acoustic properties, like speed of sound, indicating how fast the ultrasound wave travels through this tissue, the mass density or the attenuation coefficient, indicating how much the ultrasound waves amplitude decreases while traveling trough. As ultrasound is mechanical oscillation these mechanical (acoustic) parameters might be estimated by appropriately processing US images. Therefore, in order

19 to provide the clinician with as much information as possible to aid her diagnostic decision making the idea is to also add (overlay) these parameters to the already pictured structures in an ultrasound scan, making it possible to identify cancerous tissues more easily. As for example they are higher attenuating compared to average breast tissue, and as a result increasing the sensitivity and speciﬁcity in the diagnosis.

Figure 1.2: Representation of a B-mode image of a tissue-mimicking phantom with two inclusions of diﬀerent acoustic properties (left) reconstructed by compounding 8 views using the Octo-scanner from the Ferrara Lab/Stanford University (right) [10].

The overall goal would be to use a B-mode image (Fig. 1.2) and estimate the different acoustic properties for all the structures depicted in the image. But to verify that the method to estimate the acoustic properties is providing an unbiased estimate with a variance close to the Cramér-Rao Lower Bound (CRLB) [11], one needs to apply the estimator to known properties and compare results. Therefore phantoms are prepared, as depicted in Fig. 1.2 (left), with inclusions of known acoustic properties. But to be able to prepare such phantoms, it is necessary to find a way to produce inclusions with certain acoustic properties. Tissue- mimicking materials are essential, therefore Chapter 3 is dedicated to finding a connection between a tissue-mimicking material and the acoustic properties of a phantom. By verifying that the method, used to estimate the acoustic properties of different tissues from B-mode data, leads to low variance, unbiased results, the sensitivity and specificity in breast screening using ultrasonic imaging can be increased. Another important role, for high accuracy in the diagnosis, plays the expertise of the doctor or radiologist. The fact that computers can recognize patterns even in huge amounts of data, could help in this matter. Only thing is, computers can recognize patterns easier, but humans also comprehend what these patterns mean, humans have a cognitive understanding [2]. So what we need is artificial intelligence, the computer has to learn to recognize and also

20 comprehend things, building its own neural network, this method is called deep learning [12]. A neural network can be trained to do certain things, like recognizing handwritten numbers, but to acquire this ability, a huge database with annotated, high-resolution images is necessary. Just to give an idea on how big a database needs to be: the MNIST-Database [13], used in the recognition of handwritten numbers, contains 60 k images for training and 10 k images for testing. So for medical scans, which are way more complex, massive amounts of training data are required for high accuracy in the diagnosis, which is currently not available for ultrasonic imaging in suﬃcient (annotated) quality. There are databases available of MRI images, like the data set from the I-SPY 1 TRIAL [14], for the use of deep leaning, but the conversion of MRI images to ultrasound images, would require the segmentation of every image into its diﬀerent tissue structures and afterwards the simulation of these segmented images [15]. So to generate a database of ultrasound images for the use in deep learning, and also for testing methods to estimate the acoustic properties, simulations are essential. Therefore Chapter 4 is dedicated to the creation of ultrasound simulations.

Chapter 2

Medical imaging modalities

2.1 Introduction

There are several medical imaging modalities, adapted to their respective ﬁeld of application. The three main imaging modalities used in breast screening are mammography, magnetic resonance imaging (MRI) and ultrasonic imaging. This chapter is intended to give an overview of the individual imaging modalities, including their capabilities, advantages and disadvan- tages. Since this work is aiming at the simulation of ultrasound images, the ultrasonic imaging modality will be described in more detail.

2.2 Mammography

Mammography, whos foundations were laid by the german surgeon Albert Salomon in 1913 [16], is a type of medical imaging, which uses low dose x-rays to examine the human breast. It still is the gold standard in breast cancer early detection, as even the smallest changes in the breast tissues, which can be a first indication of a pathological event in the breast, can be detected even years before a palpable lump is noticed. There are two forms of mammograms, the screening mammogram, for patients who do not have any breast symptoms or problems and the diagnostic mammogram, for patients with symptoms, like nipple discharge or unusual lumps or whos screening mammogram showed signs of changes [1]. With mammograms, as shown in Fig. 2.1, one can find changes in the breast, like calcifications, which are depostions of calcium in the breast, cysts, which are fluid-filled bladders or

23 masses, areas in the breast, which are more dense or appear in an abnormal shape, making them look diﬀerent than the rest of the breast tissue, but with mammograms one can not tell whether an abnormal area is cancerous. So they are not used to specify tissues, but once an abnormal area is found, further tests can be initiated. Often ultrasound is used for further evaluation of abnormal areas [1].

Figure 2.1: On the left there are schematic representations of mammograms: calcifications appear as small bright spots randomly distributed in the breast, whereas a cyst, which is a fluid- filled bladder in the glandular lobe and therefore less dense, appears less bright. Cancer on the other hand can’t be identified exactly, only the abnormalty of an area can lead to an assumption, that more testing is needed. On the right a schematic representation of the operating principle can be seen, where the two plates compress the breast and the x-ray beam penetrates through the breast from above [17].

2.2.1 Fundamentals of x-rays x-rays were discovered by the German scientist Wilhelm Conrad Röntgen in 1895. The ’x’ in this context stands for the unknown, as Röntgen at ﬁrst did not know where this radiation was coming from [18]. Now it is known that x-rays same as light are a form of electromagnetic radiation, which is the emission of energy as an electromagnetic wave, but other than visible

24 light (in the frequency range from 405 to 790 THz), x-ray photons have higher energy and frequencies ranging from 30 PHz to 30 EHz1. High-energetic electromagnetic waves are considered ionizing radiations, which means they can interact with tissue and cause damage by ionization but they can also pass through most objects, since the interaction with matter probability diminishes for higher energy photons so most make it through the object and do not interact and ionize. Most importantly they can pass through the human body conveying information on the mass attenuation coeﬃcient along their path. Mammograms same as computed tomography (CT) and projection radiography take advantage of this characteristic. As the german physicist Albert Einstein in 1905 proposed, electromagnetic radiation is gran- ular, consisting of photons. Each photon has an amount of energy associated with it:

h · c EPhoton = = h · ν, (2.1) λ where h is Planck’s constant (6.63×10−34 Js), ν is the frequency of the radiation, λ is the wavelength of the radiation, and c the speed of light in vacuum (299.79×106 m/s). The typical non–SI unit of photon energy is electronvolt (eV), 1 eV equals 1.602176634 × 10−19 J, so when talking about a soft x-ray radiation, the photon energy is around 15 – 30 keV, whereas hard x-ray radiation (with energies beyond 30 keV), as used in projection radiography and computed tomography, lies in the order of around 70 – 90 keV [19]. As x-rays propagate through the body, the high photon energy might cause (with very low probability, though) the ejection of electrons from the atoms along the path, as the photon’s energy is greater than the electron’s binding energy. Ionization occurs, which leads to biochemical changes in living cells, therefore the advantage of x-rays in medical imaging faces/opposes significant health risks. However, there are treatments that take advantage of ionization, they are called radiation therapy, where high doses of radiation are used to kill cancer cells and shrink tumor [20]. The amount of energy absorbed per unit mass by an irradiated object, in this case the body tissue, over a period of exposure is called absorbed dose. It depends on the intensity of the irradiation and on the absorption capacity of the irradiated substance for the given type of radiation and energy. The SI unit of absorbed dose is the Gray (1 Gy = 1 J/kg) [21]. The Gy is a rather large unit and does not measure the health effect of different types of

1Frequencies even higher than those are attributed to γ-rays or even cosmic rays.

25 radiation2, but rather the thermal effect on the tissue. The derived unit of ionizing radiation measuring the destructive effect on human tissue is the Sv, which measures the equivalent dose (proportional to the destructive effect). This equivalent dose for the absorbed dose of 1 Gy can be up to 20 Sv in case of α-radiation (see also Sec. 2.2.3).

2.2.2 Operating principle of mammograms x-rays measure the x-ray attenuation along the path from the x-ray source (the x-ray tube) to the x-ray detector. As the human body consists of diﬀerent kinds of tissue, with their individual radiological density, it is possible to conclude from the absorption measured to the tissue type. The denser the tissue, the more the x-rays are attenuated. The x-ray beam transmits through the body and loses energy due to scattering and absorption, weighted by the mass attenuation coeﬃcient of the tissue and also the thickness of the associated object (organ, bone, vessel)3. That is why bones appear brighter than the lungs on an x-ray scan. As shown in Fig. 2.2, we distinguish between 5 x-ray densities, from air as the most radiolucent, and therefore appearing as a dark spot, to metal as the most radiopaque [4].

Figure 2.2: The 5 x-ray densities, from air as most radiolucent to metal as most radiopaque. x-rays are mostly known for their ability to allow to identify bone fractures, this method is called projection radiography. It is the representation of a 3D volume of the body onto a 2D imaging surface. Simply put what happens is, that the semitransparent body gets illuminated by x-rays, the resulting shadow cast, when scanning straight through the body, represents the projection radiograph. This imaging modality lacks in depth resolution, meaning the resulting image is degraded due to superimpositions of shadows from over- and underlying tissues. CT completely eliminates these artifacts, as it scans through the body from diﬀerent

2Besides x-rays there is also α-radiation (fast He-nuclei produced by the α-decay of heavy elements), β- radiation (fast electrons emitted from radioactive nuclei), fast neutrons emitted from nuclei, etc. 3Please note that still all x-ray images are displayed and thus viewed in negative as it was the case when photographic ﬁlm negatives were still used.

26 angles and by solving an inverse problem the 3D geometry can be reconstructed [4]. As shown in Fig. 2.1, for a mammogram a machine is used with two plates, which compress the breast and ﬂatten the tissue, therefore it is possible to get a clear image with a lower dose of x-rays, as the distance between source and detector is smaller. Also as the tissues are spread apart, overlapping shadows, which could lead to the obscuring of abnormalties, like in the projection radiography, are reduced. Three-dimensional (3D) mammography, also known as digital breast tomosynthesis, can further improve the resolution and the accuracy of mammograms. It works in a similar way as the computed tomography, the breast again is compressed by two plates and the x-ray tube is guided in an arc across the breast while the detector moves virtually in opposite direction. In the process, 9-25 very low-dose images are taken. From them a 3D data set is calculated, leading to 40-150 slices per breast, depending on the breast volume. With the resulting tomograms, the breast is displayed largely without superimposition [22].

2.2.3 Limitation and risks

A limiting aspect for mammograms is the density of the breast. A dense breast is more fibrous and glandular than a fatty breast. It is not dangerous, however, dense tissue appears white on the mammogram (Fig. 2.3), same as breast masses or tumor, because of its more solid texture. Therefore the mammographic sensitivity decreases with density and the possibility of a false-negative diagnosis, due to missed underlying cancerous masses, rises. As dense breasts are common, especially in younger women, and the risk of developing breast cancer is higher in dense breasts, it is important to ensure that a regular screening does not do any harm to the tissue. During a mammogram, breasts are exposed to a low amount of ionizing radiation, which can lead to biochemical changes in living cells, meaning the possibility of developing cancer due to the radiation rises with the dose the breasts are exposed to. To determine the ionizing radiation exposure dose and the resultant effects on the human body, called equivalent dose, serves the SI derived unit called Sievert (Sv), named after the swedish medical physician and physicist Rolf Maximilian Sievert (1 Sv = 1 J/kg) [23]. The equivalent dose is obtained by multiplying the absorbed dose in unit Gray (Gy) by the radiation weighting factor, which in a simplified way describes the relative biological effectiveness of the radiation in question. It depends on the type of radiation and energy. For example, the radiation weighting factor for beta, gamma and Röntgen (x-ray) radiation is 1, so the equivalent dose in Sv is equal to the absorbed dose in Gy. For other types of radiation, factors of up to 20, as for instance

27 Figure 2.3: Comparing mammograms of a fatty (left) and a dense breast (right). A dense breast appears brighter than a fatty breast, because of its more solid texture and therefore decreasing the sensitivity, as the chance increases that cancerous masses or lumps are being obscured [1]. for alpha radiation, might be encountered [21]. Now to put the exposure of radiation4 from a mammogram into perspective: a normal mammogram done on both breasts exposes the patient to a partial irradiation5 of 0.4 mSv, a person in the US on average gets exposed to approximately 3 mSv per year, due to natural surrounding radiation, which means the radiation the breasts are exposed to equivals to the dose a person is exposed to within 7 weeks on a natural basis. Therefore the beneﬁts outweigh the potential risks if done occationally, but if done on a regular basis like twice a year, which is recommended for early detection, the radiation exposure adds up and the risks increase [1].

2.3 Magnetic resonance imaging

In 1973, Lauterbur and Mansﬁeld [24, 25] independently described the use of nuclear magnetic resonance to form an image. For their work, they shared the Nobel Prize for Medicine in 2003. This imaging modality was named NMR imaging, standing for nuclear magnetic resonance imaging. Due to the widespread concern over the word nuclear the acronym was soon changed to MRI. This imaging modality is a powerful tool because of its ﬂexibility and sensitivity to a broad range of tissue properties [26].

4Key facts: Being exposed to an equivalent dose of 6 Sv leads to death within 14 days. In the Chernobyl disaster, workers were exposed to a maximum equivalent dose of 13 Sv. Instant death occurs at 80 Sv [21]. 5As opposed to a full body irradiation.

28 Magnetic resonance imaging produces cross-sectional images of the structure inside the body, with high spatial resolution as well as high contrast, this 2D data can be used to calculate 3D datasets. But instead of x-rays, MRI uses a magnetic ﬁeld combined with radio waves to excite hydrogenic protons, therefore the images reﬂect the proton density in the tissue instead of the tissues mass density, leading to better soft tissue imaging. In contrary to mammograms MRI is basically noninvasive, as it does not use ionizing radiation, but as with every MRI the injection of an intravenous contrast agent [27] is required to achieve high resolution, it still entails some risks. The only real noninvasive imaging modality is ultrasonic imaging, to which the next section will be dedicated.

2.3.1 Fundamentals of MRI

Atomic and subatomic particles possess a property known as spin angular momentum or simply spin. Electrons, neutrons, and protons all possess spin and are often imaged as tiny spinning balls. Although inaccurate this mechanical model is a quite good analogy, except that:

• The particle is not actually rotating.

• Spin is a fundamental property of nature and does not arise from a more basic mecha- nism.

• Spin interacts with electromagnetic fields (whereas classic angular momentum interacts with gravitational fields) by precessing in a magnetic field according to the so–called

Larmor–frequency [28]. This Larmor–frequency is given by (with B0, as the magnetic

ﬂux density and γc, as the gyromagnetic ratio)

ω0 = γc · B0 . (2.2)

For a free spinning proton this angular frequency amounts to ω0 = 42.6 MHz per T.

• The magnitude of spin is quantized, which means that it can only take on a limited set of discrete values.

An MRI system is imaging the spatial distribution of protons in the imaged tissue by recording the strength of their respective precessing response to an exciting magnetic ﬁeld ﬂipping their spins.

29 Typically the magnetic moments (the orientation of the spin-axes) of protons are randomly oriented so that their overall signal coherently superposed cancels completely. In an external magnetic field they reorient themselfs either parallel or antiparallel to this field with a small unsymmetry to be seen. Out of 106 protons about a single proton more (at 1 T field strength) is aligned in field direction than aligned opposed to it, that gives a net effect ultimately resulting in a detectable RF signal if one considers the large overall number of protons per unit volume precessing in a magnetic field. This way the volume density of protons — the average water content — is imaged by an MRI system. If one were to flip the direction of this magnetic field periodically one would cause all spinning tops (the protons) periodically to try to attain a new spin orientation at those instances in time. The coherent superposition of their resultant RF field would thus perfectly sum up and result in a maximally strong radio signal at the location of the field coil. Over time, however, and due to various physical processes energy would dissipate – slightly differently rapid for individual atoms and different tissues — and thus this coherent signal will decay exponentially towards zero, requiring the periodic excitation via a current pulse applied through the field coil to maintain the MRI RF signal.

2.3.2 Operating principle of MRI

To describe the operating principle of MRI, Fig. 2.4 serves as a schematic representation of an MRI scanner. It consists of a large scale superconducting magnet with a typical B0 ﬁeld strength of 1.5 T to 3 T (for clinical use), or up to 11 T currently used in research. Gradient coils allow for a local variation in ﬁeld strength and the RF coils pick-up the resultant Larmor–frequency HF signal.

Gradient-coils

One of the most important concepts for obtaining an image using MRI is the use of magnetic field gradients. The gradient became a fundamental part of MRI in 1973, when Mansfield and Lauterbur proposed the idea of using a gradient that spatially encodes the positions of the nuclei of hydrogen within a sample by causing a variation in Larmor–frequency as a function of their position [26]. Typical MRI–systems allow for the acquisition of volume data resolved in all three directions (the x–direction typically is oriented parallel to the floor and orthogonal to the bore-axis, the y–direction is oriented vertical and the z–direction is oriented along the bore). To accomplish this spatial resolution the main magnetic field must deliberately be modulated by so–called

30 Figure 2.4: Cross-sectional view of an MRI scanner (from [29]).

gradient coils oriented parallel to these directions. All magnetic gradients are produced by so–called Helmholtz–coils [30, 31] operated in the Maxwell–mode, where the currents in the coil pair have opposing directions, as shown in Fig. 2.5. The resultant magnetic field can be seen in Fig. 2.6. There one can observe that in the left half a positive additional field component increases the field of the main coil whereas in the right half a negative gradient field diminishes the main field. For a properly chosen geometry of the coils [31] the field gradient in the central part is close to a constant. The x– and y– gradient coils usually have a Golay–type (saddle–type) [26] coil configuration (see Fig. 2.7), the z– gradient coil is usually a standard Maxwell configuration. It is important to note that, although the coils are varying the magnetic field along their respective direction, their B– field is always oriented parallel to the field of the main magnet and modulating it along the z–coordinate.

These gradient systems are capable of producing Bz–gradients from 20 to 100 mT/m and normally are resistive electro magnets driven by high performance amplifiers that allow for a well defined and rapid temporal variation of the currents that are on the order of several hundred A. Thus an effective heat removal needs to be ensured which is accomplished through

31 Figure 2.5: Maxwell–coil configuration. Note the inversely flowing currents resulting in just a gradient of the B-field in a paraxial volume (from [32]).

Figure 2.6: Gradient–ﬁeld of a Maxwell–coil with N = 100 windings, and current I = 1 A plotted along the z-axis.

32 water cooling. It needs to be mentioned that eddy currents might be induced by the rapid variation of coil currents, an eﬀect which is circumvented by shield coils. High performance MRI systems allow to temporally vary the gradients by up to 200 T/(m s) thus the driver ampliﬁers need to cope with the high induced voltages resulting.

RF-coils

These coils have a twofold functionality, ﬁrst they are responsible for perturbing the nuclear spin of precessing protons by emitting a high power HF excitation signal and secondly they are recording the resultant MRI signal emanating from the freely precessing nuclear spins at their respective (locally varying) Bz–ﬁeld. In case a better spatial resolution is sought these two functionalities might be separated and the pick-up coils are then placed close to the interesting body part. Their signal is digitized and then appropriately processed to reconstruct the slice plane addressed by the z–gradient coil current. This processing will not be discussed in this work.

Figure 2.7: Golay–type gradient coil principle (left), and optimized conductor geometry (right), (from [32]).

2.3.3 Limitation and risks

The magnetic ﬁeld in general is not harmful to the patient, but due to the irradiated high- frequency waves, metal implants such as screws can be heated to a critical level, same as the

33 irradiated tissue. In addition, the strong attraction during the MRI can turn metal objects into lethal projectiles. Main risk is entailed by the injection of intravenous gadolinium-based contrast agents, which is necessary to receive high-resolution images, to detect the onset of small lesions and therefore increasing the sensitivity as well as the speciﬁcity. Limiting aspect for MRI being the gold standard for breast cancer screening is the high range of speciﬁcity (37-97%), the involved high costs, the limited availability of MRI scanners and also the high expenditure of time.

2.4 Ultrasonic imaging

Ultrasonic diagnostics, also known as sonography, has become established, beside mammography and magnetic resonance imaging (MRI), as the third imaging modality in breast cancer screening. Compared to mammography and MRI, where the density of tissue, the radio opac- ity, and the density of hydrogenic protons, respectively, is imaged, and where we are interested in the transmission of the beam of radiation and its variable attenuation by different kinds of tissue, in ultrasonic imaging we are interested in the parts of the beam of sound that gets reflected or scattered back by variations in the acoustic impedance of different tissue types. For better understanding the medical ultrasound imaging modality, the physics of ultrasound and its interaction with human tissue needs to be detailed first.

2.4.1 Fundamentals of ultrasound

Sound is mechanical oscillations (vibrations) of an elastic medium, which propagates through gases, ﬂuids or solids. Pulses of sound are being generated by transducers, these pulses are sent through the medium in form of waves6. Imagine the medium, in this case human tissue, being jelly, when compressing and releasing it, it starts to wobble due to its elastic properties, causing the volume to alternately compress and expand, same happens with human tissue. The expansion of the volume past its point of equilibrium causes the neighboring volume of tissue to compress, resulting in the generation of a wave, as the process continues in time and space. Thus, compared to electromagnetic waves, sound waves are tied to a propagation medium. Where the mediums’ particles oscillate around their mean position, due to the sound wave, and therefore supporting the wave to propagate through [33]. In gases and ﬂuids the wave propagates in form of a longitudinal wave (Fig. 2.8, left), in solids also transverse 6The science behind sound and its properties is called acoustics.

34 waves can appear, due to the signiﬁcant shear forces between particles. These kind of waves are being attenuated while propagating through the medium, by reason that they dissipate energy into the medium, which leads for example to heating of the carrier liquid. Therefore the amplitude of the oscillation decreases with longer propagation time and path length.

Figure 2.8: Representation of a longitudinal wave (left) and a transverse wave (right) propagating in positive x-direction. λ being the wavelength [34].

In acoustics there are different forms of sound associated with different frequency ranges, for humans only frequencies between 20 Hz and 20 kHz are perceptible, whereas the ultrasonic frequency range starts with 20 kHz (Fig. 2.9) and ends at approximatley 1 GHz. In this section the physical properties of sound are discussed in more details. Starting off with the basic parameters, which are leading to the wave equation, a partial differential equation characterizing the way the wave propagates through matter.

Figure 2.9: Frequency range from Infrasound to Ultrasound, with examples of application [35].

Sound ﬁeld parameters

When an ultrasound wave propagates through a medium, the mediums’ particles oscillate

35 around their mean position, leading to a ﬂuctuations of the state variables, like pressure, p, mass density, ρ, and particle velocity, ~u. These changes can be described by ﬁrst-order equations, which are based on the conservation of momentum, conservation of mass and the state equation:

∂~u 1 Equation of motion : = − ∇ p, (2.3) ∂t ρ0 ∂ρ Continuity equation : = −ρ ∇ · ~u, (2.4) ∂t 0 2 Pressure-density relation : p = c0ρ. (2.5)

Derivation of Equation 2.3

The ﬁrst equation (Eqn. 2.3) is based on the conservation of momentum. Repulsive forces are caused by changes of the pressure within the propagation medium. According to Newtons ~ d~u law, force is mass times acceleration, F = m · ~a = m · dt , which indicates that a force acting for a certain time, results in mass undergoing a change of speed, leading to the integral form:

Z Z F~ · dt = m · d~u. (2.6)

Mass is given by the considered volume and its density, m = ρ0 · V . Analyzing changes of infinitely small volume elements, leads to dm = ρ0 · dV = ρ0 · dx · dy · dz. Density is constant, which means changes of the volume lead to changes of the mass. Another way to define force is pressure times a specific area A, F~ = p · A~, where changes of the area lead to changes of the force, dF~ = p · dA~. Considering for the moment only the component of force in x-direction:

dFx = p · dAx = p · dy · dz, (2.7) as shown in Fig. 2.10. Pressure is a scalar quantity, therefore acts equally in all directions, but the pressure propagation is limited by the speed of sound and is thus not inﬁnitely fast, therefore a spatial variation of p must be possible over an inﬁnitely small volume element dx · dy · dz. Making it possible that the entering and exiting forces acting on the volume

36 element diﬀer, resulting in a net force acting to accelerate the contained mass dm, and which also leads to a change of the pressure over the volume element’s size:

dFx = (px − px+dx) · Ax = (px − px+dx) · dy · dz, (2.8)

where px is the pressure at position x and px+dx the pressure at position x + dx on the right hand side of the volume element, as shown in Fig. 2.10.

Figure 2.10: Representation of the change in force in x-direction, due to change in pressure throughout the volume element [36, 37].

Introducing the gradient, which indicates how much a scalar is locally varying in magnitude, in this case how much the pressure changes over a variation in position (distance dx):

∂p dm dFx = − · dV = − grad p · dV = − grad p · . (2.9) ∂x ρ0

The negative sign results from the fact that the gradient points in the direction of the increase of a scalar quantity and not to the decrease. When a force acts on the volume element from the left side, which leads to an acceleration in positive x direction, the force from left has to be larger than the force on the right side of the volume element, which indicates: px > px+dx, in this case the positive x-direction is deﬁned as to point to the right. The time derivative of velocity is acceleration, and acceleration is caused by a force acting on a mass, and can be calculated from the particle velocity. With this the equation for the

37 x-direction, can be formed into:

dFx ∂ux 1 = = − grad px. (2.10) dm ∂t ρ0

Considering now all directions and the gradient operator, and introducing the nabla operator ~ ∂ ∂ ∂ ∇ = [ ∂x ex + ∂y ey + ∂z ez] operating on a scalar field, resulting in a vector field, leads to Eqn. 2.3. The second equation (Eqn. 2.4) represents the condition of local mass conservation. These two basic fluid dynamic equations, combined with the state equation of a compressible medium, which conveys the connection between pressure, p, and density, ρ, completely describe the dynamic of the assumed lossless medium. Leading to the known linear second-order partial differential wave equation for the sound pressure, assuming small fluctuations of sound pressure, mass density and particle velocity:

2 1 ∂ p 2 2 2 = ∇ p = ∆p, (2.11) c0 ∂t where ∆ is the Laplace operator. It is also possible to write the wave equation for the particle velocity [37, 38].

2.4.1.1 Deﬁnition of acoustic waves

A wave is a spatially propagating periodic change (oscillation) in the equilibrium state of a system (medium) with respect to at least one location- and time-dependent physical quantity. Compared to electromagnetic waves, which can travel through vacuum, mechanical waves like the ultrasound waves are bound to a medium. In media, the propagation of a local disturbance is mediated by the coupling of neighbouring oscillators (oscillating physical quantities, like sound pressure, mass density and particle velocity). A wave transports energy, but not matter, i.e. the neighbouring oscillators transport the disturbance through space without moving themselves on average over time. The sound pressure propagates in form of a wave with phase velocity c, the speed of sound. Often harmonic (sinusoidal) time dependencies are used to describe sound and vibration problems, therefore the basic sound pressure, p, of a (plane) wave with angular temporal frequency ω in rad/s, and angular spatial frequency k in rad/m, propagating in positive

38 x-direction, is considered here. The later discussed numerical simulation software for sound pressure fields, k-Wave (see Sec. 4.3), applies, via the Fourier transform, the superposition of plane waves of different temporal and spatial frequencies propagating in all possible directions in 3D space to solve for the total sound pressure field in the simulated volume meeting the respective initial value and boundary conditions. Thus the solution of the above partial differential equations is solved in the spatial frequency domain or k-space. A three-dimensional wave (3D space and time) can be written as (<{· · · } is the real-part operator)

n j(kxx+kyy+kzz−ωt)o p(x, y, z, t) = Aw · < e , (2.12)

Aw being the wave amplitude. To render things simple and allow for a graphical representation a one-dimensional wave is considered

n j(kx−ωt)o p(x, t) = Aw · < e = Aw cos(k · x − ω · t), (2.13) where k = ω/c is the so-called wave number, and ω is the angular frequency, ω = 2πf = 2π/T , with temporal period T . Equally, the wave number k relates to the spatial period λ, also known as wavelength:

ω 2π k = = . (2.14) c λ

Taking these equations into account, the relation between wavelength, λ, speed of sound, c, and frequency, f, can be described by:

c λ = . (2.15) f

Figure 2.11 represents a one-dimensional wave shown vs. time and space. At any ﬁxed position in space one can observe a sinusoidal variation of the pressure vs. time (see magenta line), and at any instance in time a wave propagates vs. spatial position (see cyan line).

39 Figure 2.11: One-dimensional wave in space displayed for frequency, f = ω/2π = 106 Hz, speed of sound, c = 1500 m/s, and wavenumber k = 4189 (waves in) rad/m. The magenta line indicates the time signal for a wave for a position held ﬁxed in space (here 0.5 mm), the cyan line indicates the spatial signal derived from a wave for a ﬁxed time (here 0.5 µs) vs. space.

Sound waves are longitudinal pressure waves, which means the oscillating particles combine to form high pressure regions called "compression" and separate to form low pressure regions called "rarefaction", therefore supporting the wave to propagate through. The particle velocity ~u, being the alternating velocity with which the particles are oscillating around their rest position, is not to be mistaken for the speed of sound c, being the phase velocity of the sound waves. The speed of sound in a ﬂuid is given by:

s K c = (2.16) ρ0

40 where K is the bulk modulus in Pa which measures how resistant to compression a substance is. Compressibility, the reciprocal of K, indicates whether it is easy to reduce a volume of a medium when applying pressure, the higher the compressibility the easier the volumes size changes with pressure, therefore the speed of sound decreases with an increase of compressibility. Reason for the increase of speed of sound with an increase in mass density, is that more dense media contain more particles, which can interact in form of compression and rarefaction faster, therefore the sound waves propagate through faster. Speed of sound also depends on temperature, in ﬂuids it decreases with increasing temperature, the only exception being water. The speed of sound in clear water increases with Tw −1 ◦ until it reaches the maximum of c = 1557 m s at Tw = 74 C, then it starts to decrease again [39, 40, 41]. An increase in pressure always leads to an increase in speed of sound for all ﬂuids [42].

2.4.1.2 Interactions of acoustic waves with human tissue

The foundation of ultrasound imaging is the different ways the sound waves interact with the bodies’ tissues, meaning the sound waves behave differently in different tissue types. As a sound wave propagates through the body it can get absorbed, reflected, scattered or refracted. The sound wave gets attenuated, meaning its intensity and amplitude decreases with the penetration depth. The sound intensity I~ is defined as:

I~ = p · ~u, (2.17) depending on the sound pressure p and particle velocity ~u. With it, the power, P , ﬂowing through each unit of area A can be calculated:

Z P = I~ · dA.~ (2.18) A

The sound power of a sound source is obtained by integrating the intensity contributions over all area elements dA~ of any closed surface around the source. The unit of the sound power is W, therefore the sound intensity has the SI-unit Wm−2.

41 Reﬂection and Refraction

When a sound wave impinges on the interface between two media, part of the wave gets reflected and the rest transmits through (assuming lossless media), as shown in Fig. 2.12. As a result the sound power decreases, as some sound power is reflected. The degree of reflection at an interface of different tissues is dependent on the acoustic impedances of the interfacing tissues. The acoustic impedance of a lossless medium (resulting in a real-valued quantity) is defined as:

p Z = = ρ · c , (2.19) u depending on the speed of sound and the mass density. It is also possible to describe the acoustic impedance with the sound pressure and the particle velocity. If the acoustic impedance of two media are similar, sound will travel through their interface easier, compared to two media of different acoustic impedance. Recall, when a billiard ball hits another billiard ball, of same mechanical properties, the hit ball continues to roll forward, whereas the hitting ball stops completely. But when a billiard ball hits a wall, it will be reflected and roll backwards and the wall would not move at all, due to different mechanical properties (conservation of momentum Eqn. 2.3). This is the reason why with every ultrasonic examination coupling gel is used, which has approximately the same acoustic impedance as soft body tissue, therefore less sound energy is lost on the way from transducer into the tissue.

The pressure amplitude reﬂection coeﬃcient, Ra, of the pressure signal of a sound wave at the transition from one acoustic medium with impedance Z0 to another medium with impedance

Z1, is given by (indices i, incident, r, reﬂected, t, transmitted):

Z1 cos(θi) − Z0 cos(θt) pr Ra = = . (2.20) Z1 cos(θi) + Z0 cos(θt) pi

Assuming the wavelength of sound is small compared to the lateral extension of the interface between two media, the law of reﬂection states that, the incident angle is equal to the reﬂection angle: θi = θr. Taking Snell’s law, also known as the law of refraction, into account, the transmission angle θt can be calculated [33]:

42 sin(θi) c1 = . (2.21) sin(θt) c2

As one can see from Eqn. 2.21, θt ≈ θi = θr, if the speed of sound in both media are the same, which is approximately true for most tissues in the human body, shown in Table 2.1, except for bones and air (in the lungs).

Figure 2.12: Transmission and reflection of a sound wave impinging on an interface between two media of different acoustic impedance, the angle of the incident and the reflected wave, for most breast tissue types, are equal according to the law of reflection [33].

The pressure amplitude transmission coeﬃcient, Ta, in this case is given by:

2Z1 cos(θi) pt Ta = 1 + Ra = = . (2.22) Z1 cos(θi) + Z0 cos(θt) pi

The intensity reﬂection and transmission coeﬃcient, Ri and Ti, for plane waves, derived from the intensity of a plane wave according to Eqn. 2.19 and Eqn. 2.17,(I = p2/Z), are given by:

43 2 2 Z1 cos(θi) − Z0 cos(θt) Ir Ri = Ra = = , (2.23) Z1 cos(θi) + Z0 cos(θt) Ii and

2 2 Z0 4Z0Z1 cos (θi) It Ti = Ta = 2 = . (2.24) Z1 (Z1 cos(θi) + Z0 cos(θt)) Ii

Table 2.1 gives some of the physical properties of different biological tissues important for ultrasonic imaging. It shows that for soft tissues the acoustic properties are roughly the same, which leads to a reflection coefficient in the range of −0.1 to 0.1, and therefore limiting medical ultrasound in the examination of soft tissue structures [33].

Tissue ρ in kg m−3 c in m s−1 Z in kg m−2 s−1 Bone 1912 4080 7.8 × 106 Muscle 1080 1580 1.7 × 106 Liver 1060 1550 1.64 × 106 Blood 1057 1575 1.62 × 106 Kidney 1038 1560 1.62 × 106 Brain 994 1560 1.55 × 106 Water 1000 1480 1.48 × 106 Fat 952 1459 1.38 × 106 Air 1.2 330 403

Table 2.1: Physical properties of diﬀerent biological tissues, where ρ, is the mass density, c, the speed of sound, and Z, the acoustic impedance [34].

An alternative derivation to describe the law of refraction is given by Fermat’s principle [28] and is a link between propagation of waves and their description through rays or beams. There the sound wave will be approximated in form of a beam being oriented orthogonally to the local wavefront. Fermat’s principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. According to Feynman [43] this idea is the consequence of the fact that all waves travel all possible ways concurrently but only those whose neighboring paths have very similar propagation times (and thus will have very similar phases) will constructively interfere. All other possible paths will result in destructive interference and will cancel or almost cancel. This leads as a consequence to the calculus of variations as

44 a ﬁeld of mathematical analysis. Thus when such a beam impinges on a plane interface between medium 1 and medium 2, it gets refracted according to the law of refraction that can now be derived using this idea and principle of nature.

Scattering

Low sound frequencies lead to correspondingly high wavelengths, whereby most structures and objects placed in the sound field, if smaller than the wavelength, are acoustically ’invisible’ and only scattering and diffraction happens. From Fig. 2.12 it can be derived that if the incidence angle is greater than zero, the reflection bounces off in another direction with the same angle, and the transducer, so it might seem, does not receive a reflected signal. However, due to the scattering of the ultrasound wave the structure, which wants to be depicted, is still visible. Scattered waves radiate in all directions in form of spherical waves (cf. Fig. 4.16), they are caused by small changes in density, compressibility and absorption. As there are numerous small tissue structures in a volume, the backscattered signal represents the sum of all constructive and destructive interferences of the individual spherical waves. That is the reason why an ultrasonic image of for instance a homogeneous liver tissue appears grainy and not homogeneous at all, because all cells, fibers and other small changes in the tissue result in a backscattered signal. Speckles are the term for this kind of pattern [33]. Absorption

While the sound wave’s power decreases, as part of the sound wave propagating through the medium gets reflected, scattered or refracted, and only part of the sound wave transmits through the medium. There is a process where the sound power decreases and dissipates in form of heat, this process is called absorption. It results from three different sound absorption mechanisms, absorption due to inner friction, due to thermal conductance and molecular absorption, but these mechanisms will not be further described in this work. It is noted, however, that these three mechanisms combined result in a frequency dependency of the attenuation coefficient to be discussed later.

Decibels and Levels, Neper

Often logarithmic rather than linear measures are used for sound pressure, sound power and intensity, as their range of magnitude in practice is rather large. The most common known measure is decibel, dB, which is the representation of the logarithm of a ratio between a quantity and its reference. Resulting in an expression called level [44].

45 The decibel is a unit to express a ratio of a power variable Pv with respect to a reference power Pref.

Pv L = 10 · log10 in dB (2.25) Pref

A decibel is a rather large ratio so that 1 dB represents a ratio R of

L/10 1/10 R1 dB = 10 = 10 ≈ 1.26 (2.26) and 10 dB represents a ratio of 10.

The sound pressure level Lp is deﬁned as

! p2 p Lp = 10 log10 2 = 20 log10 dB, (2.27) pref pref where the reference sound pressure commonly used for ambient sound is pref = 20 µPa. The sound power level LP is deﬁned as:

P LP = 10 log10 dB, (2.28) Pref where the commonly used reference sound power is Pref = 1 pW and P is the sound power of the source analyzed. The sound intensity level Li is deﬁned as:

Iv Li = 10 log10 dB, (2.29) Iref

−2 where the reference sound intensity commonly used is Iref = 1 pWm .

Another logarithmic measure is called Neper (Np)7, which will be used in further sections when talking about the attenuation of acoustic waves. It is important to note that the Np is defined in terms of ratios of so–called field quantities, as opposed to dB which is defined in terms of power quantities. Thus the conversion from dB to Np is as follows:

7The unit’s name is derived from the name of John Napier, the inventor of logarithms.

46 20 dB = ln (10) Np; 1 Np ≈ 8.686 dB . (2.30)

Eﬀects of the attenuation on the sound wave

An ultrasound pulse is strongly altered in biological tissue due to attenuation and dispersion. While attenuation is the decrease of amplitude of the acoustic wave, as a result of absorption, scattering, reflection and refraction, dispersion is a property of the medium that causes waves of different wavelengths to travel at different phase velocities. It results in a phenomenon of a wave separating spatially into its frequency components as it propagates, caused by the phase velocity in the material being frequency-dependent (the higher the frequency of the wave the faster the components in normally dispersive media) [45]. Considering a plane wave moving in the +x-direction, the amplitude decay can be defined as:

−µax Awx = A0 · e , (2.31)

−1 where µa is the amplitude attenuation factor with unit m . Leading straight to the attenuation coeﬃcent:

α = 20 · (log10e) · µa ≈ 8.686 µa, (2.32) with unit dB m−1. So when calculating the amplitude loss, one ﬁrst has to convert α into

µa, before using the amplitude loss equation. When attenuation is only due to absorption, α is called the absorption coeﬃcient, which is frequency dependent. An increase in frequency, leads to a decrease in pentration depth. This dependency is given by:

γ α = α0 · f , (2.33) where γ > 1 in biological tissue, but only insigniﬁcantly larger than one and therefore the rough approximation of γ = 1 is often used. Table 2.2 shows this frequency dependency for diﬀerent kinds of tissue.

47 −1 −1 Tissue α0 = α/f in dB MHz cm Distilled water 0.0035 Breast: fatty tissue 0.72 Breast: glandular tissue 1.18 Breast: connective tissue 4.7 Cyst 0.152 Tumor 1.28 Skin 1 - 3 Bone 20.0 Lung 41.0

Table 2.2: Frequency dependency of various biological tissues [4, 46].

To demonstrate, how strongly an ultrasound wave gets attenuated while passing through tissue, a small example is given: Suppose a 4 MHz acoustic pulse propagates through 3 cm of fat, then encounters the interface to the kidney. Table 2.2 shows that the frequency dependency for fat is 0.63 dB MHz−1 cm−1, therefore the absorption coeﬃcient at 4 MHz is:

α = 0.63 dB MHz−1cm−1 × 4 MHz = 2.52 dB cm−1 . (2.34)

As can be noted, a loss of 2.52 dB is encountered when traveling through only 1 cm of fat, and this even though no reﬂection loss is considered. Moving on to the amplitude attenuation factor, by transforming Eqn. 2.32:

−1 2.52 dB cm −1 µa = = 0.2897 cm , (2.35) 8.686 dB which leads to an amplitude ratio of:

Aw −1 x = e−0.2897 cm × 6.0 cm = 0.1758, (2.36) A0 where the used distance of 6.0 cm indicates, that the acoustic pulse has to go back and forth in medical imaging.

48 To also take the reﬂectivity into account, one needs to calculate the intensity reﬂectivity

Ri, which is the square of the reﬂection coeﬃcient Ra for a fat to kidney tissue interface, 6 −2 −1 using the acoustic impedances as shown in Table 2.1, Z0 = 1.38 × 10 kg m s and Z1 = 1.62 × 106 kg m−2s−1, neglecting the incidence and transmission angles, as the speed of sound in biological tissue is assumed to be constant at approximately 1540 m/s:

2 Z1 − Z0 Ri = = 0.0064 . (2.37) Z1 + Z0 √ The amplitude reﬂection coeﬃcient Ra, which is Ra = Ri = 0.08, combined with the amplitude ratio of the absorption attenuation, leads to the amplitude loss of:

Awx 20 log10 = 20 log10(0.1758 × 0.08) = −37.04 dB . (2.38) A0

This result shows, that the amplitude of the received echo from a 4 MHz acoustic pulse propagating through 6 cm of fat being reflected back when impinging on the interface to the kidney, is already seventy times smaller than the original amplitude of the acoustic pulse. Increasing the frequency to obtain higher spatial resolution, leads to even more loss of amplitude. Therefore for the use in medical ultrasound imaging a high performance amplifier is used to circumvent this rapid over time decrease in amplitude. This amplifier is called time-gain- compensation (TGC) amplifier and it must be able to exponentially increase its gain with a rate (in this numerical example) of 2.52 dB per 6.5 µs, which is the time for the sound to travel 1 cm. This is necessary to keep the signal level at the receiving transducer constant independent of the imaged depth, which leads to an evenly bright ultrasonic image.

2.4.2 Operating principles of ultrasonic imaging

Medical ultrasound systems typically operate with frequencies ranging from 1 to 10 MHz, except for some experimental systems which use frequencies up to 70 MHz. As already mentioned, the penetration depth dramatically decreases with an increase in frequency. Higher operating frequencies, however, lead to better spatial resolution in imaging systems. There- fore one has to ﬁnd the right balance between these two parameters. In this work ultrasound is used to image breast lesions that for their superﬁcial location allow for frequencies providing

49 a very high spatial resolution. Typical ultrasound imaging systems use pulsed sources that emit into the tissue only a few cycles (1 cycle is defined by the wavelength in space or, alternatively, by the period T in time) of a sinusoidal signal. These bursts are generated by transducers that are excited either by an almost Dirac-delta-like high-voltage electrical impulse of a few ns duration, or they are excited by a few cycles of a sinusoidal electrical signal. Piezoelectric transducers [37] exhibit the reciprocal piezoelectric effect converting electrical excitation signals into mechanical signals (typically compression waves) and vice versa mechanical excitations impinging onto their surface into electrical output signals. Almost all physical ways to contactless interrogating a volume for internal structures or sources as seen from a vantage point outside the volume involve the utilization of some kind of wave phenomenon. Be it mechanical waves as in ultrasonic pressure waves in case of (medical) ultrasound, that allow to image the spatially varying acoustic impedance along the propagation paths. Or be it x-rays to interrogate internal structures via the attenuation of penetrating electromagnetic waves or all the information gathered by the eyes using scattered off light in the visible spectral range or sound in the audible spectral range registered through the auditory perception of sources within the volume. In ultrasound that means first, one needs to excite an internal structure by pressure waves and then propagate some information conveyed by the amplitude, the phase, or the (Doppler–) frequency of the interrogation signal back to the transducer located at the volume’s surface. While propagating through the various tissues along the propagation path, the ultrasound wave can get absorbed, refracted, reflected, or scattered by structures within the tissue. Organ boundaries or complex tissues lead to reflection or backscattering. The transducer, which is both transmitter and receiver of ultrasound energy, detects the reflected or backscattered signals and converts these acoustic signals back into electrical signals. Basically each received echo has two properties which need to be determined to a very high degree of precision by the receiver. First there is the round trip propagation delay that is the time from the generation of the sound wave until it returns back to the transducer and some measure of the strength of the echo signal. From the round trip delay the traveled distance of the wave packet can be inferred by multiplying the time by one half of the average speed of sound (taken along the propagation path).

50 Generating so-called B-mode images is done by determining the complex envelope8 of the received signal, doing some depth dependent amplitude correction to accommodate / correct for US attenuation due to dissipation of US-power to heat or scattering within the tissue and imaging the signal’s strength as graylevel image over depth and a lateral dimension dependent on the ultrasonic beam’s orientation relative to the body. There are different kinds of ultrasonic systems available, with interchangeable transducers (see e.g. Fig. 2.13) and different modes to operate with, as the imaging requirements vary in different parts of the human anatomy [33].

Figure 2.13: Diﬀerent ultrasound transducers for acquiring B-mode images. The phased array beam proﬁles are indicated in the lower left image which also shows three focal zones in both receive and transmit (from [33, page 34]).

8From system theory it is known that in amplitude modulation systems the sought after information is conveyed by the envelope and not by the carrier, which is only necessary to transport energy to the proper location for interrogation.

51 Phased array transducers and coherent signal processing

For a single US transducer, dependent on the shape of its face plate, an US beam will form that basically is its (angular) diffraction pattern and will exhibit a rather complex geometric shape. An example of which is shown in Fig. 2.14. Here a 10 mm, 1 MHz center frequency single element transducer was characterized by using a 50 µm diameter Tungsten wire reflector moved in 2D in front of the transducer as a target positioned in a water tank, to record its two way response. The transducer was rotated to get the 3D sound field [47].

Figure 2.14: Measurement of the sound ﬁeld of a piston-type 1 MHz US transducer with 10 mm diameter. The imaged iso-surfaces are for −3 dB, −6 dB, −12 dB, −18, dB, and −24 dB relative to peak.

As the angular beam width is inversely proportional to the size of the transducer (in both dimensions) large transducers would allow for a narrow and thus spatially better resolving

52 beam. If several tansducer elements are used, which are individually controllable, instead of a single larger one a technique known as digital beamforming [33] can be applied. Where the transducer elements are controlled in phase, so that the superposition of the individual wave fronts, according to the Huygens’ principle, leads to a steering in a specﬁc direction, as shown in Fig. 2.15.

Figure 2.15: Typical time delayed excitation of the transducer elements, TE, to generate a wavefront propagating under a subtended angle Θ used in phased-array US transducers. The backscattered waves are delay-and-sum processed to focus on receive [48], too.

The Huygens’ principle (sometimes named Huygens’–Fresnel Principle) [28] states that every point on a wavefront is itself the source of spherical wavelets. These wavelets while propagating superpose (correct in both amplitude and phase) and will thus show constructive and destructive interference. The resultant wavefront will develop into a (propagating) shape – the US pulse – that only allows for constructive interference at all times. So it needs to be propagating in a way to connect all farthest traveled waves (since those can’t possibly destructively interfere, because they must be in phase) [44]. The transmission energy density will be ampliﬁed in the desired direction, whereas the undesired directions are canceled out (nulled) by destructive interference. The steering and focusing occurs due to a method, called delay-and-sum processing, where some transducer elements ﬁre later than others, controlled by a computer (Fig. 2.15), leading

53 to a wavefront which either moves in a certain direction (steering), or superpose, such that at a certain point the wavefronts overlap (focusing), as shown in Fig. 2.16.

Figure 2.16: Electronic focusing and steering of an ultrasound beam (from [33, page 35]).

2.4.3 Limitation and risks

Ultrasonic imaging is still lacking in resolution, which results in a lower diagnostic accuracy as for the mammography or the MRI. However, since screening by mammography or MRI is associated with risks and comparatively high costs, eﬀorts are being made to increase the diagnostic accuracy of the noninvasive ultrasound imaging, so as recommended regular screenings are possible in a risk-free manner.

54 Chapter 3

Tissue-mimicking phantoms

3.1 Introduction

Every type of tissue in a human breast has its own acoustic properties, which means that the ultrasound wave behaves differently while traveling through the different tissues. As ultrasound is mechanical oscillation, these mechanical (acoustical) parameters can possibly be recorded through proper processing. Therefore the idea is to add numbers (estimated parameter values) to the already imaged structures, making it possible for the human observer to identify cancerous tissues more easily, as they are mostly higher attenuating than the surrounding breast tissue. So being able to record and identify the different properties like the attenuation coefficient of a specific lesion, can lead to higher sensitivity and specificity in the diagnosis. Therefore a method is needed, to estimate the acoustic parameters of certain areas of the breast using B-mode data. To verify that this method leads to correct results, first it is important to know what is inside of the breast, as this can only be judged by highly trained physicians, phantoms are being used for this matter. With tissue-mimicking materials it is possible to produce phantoms, which mimick human breasts in all acoustic aspects. Figure 1.2 (left) on page 20 gives an example of such an agar-based phantom, it contains two inclusions of different acoustic properties, inclusion 1 has an attenuation coefficient of 1.2 dB/(MHz cm), inclusion 2 an attenuation coefficient of 2.3 dB/(MHz cm). To be able to produce phantoms like these, first it is necessary to find recipes and governing equations to produce phantoms of certain acoustic properties. There are materials, which change the acoustic properties, glycerol for instance changes the speed of sound of a phantom by varying the concentraction of glycerol used, the change in aluminium oxide concentration on the other hand changes the attenuation coefficient [49].

55 This chapter describes the production and characterization of homemade phantoms and whether there is a way to ﬁnd a connection between the concentration of aluminium oxide and the resulting attenuation coeﬃcient.

3.2 Tissue-mimicking materials (TMM)

The basic components for an agar-based phantom, are water, agar for the solidification and silicon carbide for the scattering effect, to obtain a realistic looking ultrasound image, which also exhibits this grainy pattern, the so–called speckles. The water should be degassed, which means that dissolved gases are removed, that could otherwise form microbubbles that would lead to the distortion of the measurements of the phantoms’ acoustic properties. To describe the production and characterization of homemade phantoms, the ingredients of the International Electrotechnical Commission (IEC) TMM [50], as listed in Table 3.1, are varied and afterwards their effect on the acoustic properties of the produced phantoms determined.

Component Weight composition Weight composition (%) original (%) adapted Distilled, degrassed, deionised water 82.97 93.9 Glycerol 11.21 0 Benzalkonium chloride 0.46 0 Agar 3.0 3.4 Silicon carbide (17 µm) 0.53 0.6 Aluminium oxide (3.0 µm) 0.95 1.1 Aluminium oxide (0.3 µm) 0.88 1.0

Table 3.1: Weight composition of the IEC TMM (by %) according to [50] (left) and adjusted for the use in this thesis (right), therefore the proportions of glycerol and benzalkonium chloride were divide among the rest.

3.2.1 Phantom making

For safety reasons, as aluminium oxide is a fine powder, which stays in the lungs once inhaled, it is adviced to use a face mask, goggles and gloves at every step of the way! The first step is to pour 80 ml water into a flask. The flask should have a volumetric capacity of 125 ml and a side-arm tubulation for vaccuming. Initial testing has shown, that it is necessary to put the aluminium oxide into the water first, so it can become a homogeneous suspension by stirring, using a magnetic stirrer (SH-2). Before stirring, carefully put in the stirr bar, which needs to be removed before heating the mixture in the microwave. Now

56 instead of 3.4 g agar, for this study only 1.5 g agar was used for each phantom, not varying with the concentration as the rest of the ingredients, as the first trials, using 3.4 g agar with the small amount of water (80 ml), led to almost immediate solidification. After putting agar into the suspension and waving the flask for distribution, the mixture needs to be heated for the agar to liquefy, therefore the flask needs to be put into the microwave. It takes approximately one minute, at highest power setting, but caution is advised as it overflows pretty fast once it reaches the boiling point. Next step is to add the silicon carbide and again the stirr bar. The magnetic stirrer has an integrated heating plate, which needs to operate continuously to prevent solidifying. As due to the stirring microbubbles reappear, which could lead to the distortion of the measurements of the phantoms acoustic properties, it is necessary to vacuum again, therefore the vaccum tube is placed onto the side-arm of the flask and a plug is placed on top, to prevent air from entering. Again caution is advised, while sucking out the bubbles, as an overflow can lead to blockage of the vacuum tube. During vaccuming it is time to turn off the heating. Once all the bubbles are gone and the temperature has dropped to approximately 46◦C, the mixture is ready to be poured into a 3D printed mould (2 cm × 5 cm × 6 cm). If the mixture is poured at a higher temperature, it takes more time to solidify and distortions of the result could occur due to the loss of an even particle dispersion, due to segregation. With a volume of this size, the phantom takes approximately half an hour to solidify.

3.3 Measurement methods

There are different methods to measure the unknown acoustic properties like the attenuation coefficient or the speed of sound of a tissue-mimicking material. There are all sorts of scientific papers describing the measurement by either the reflection [49], or the transmission signal [51]. In this work a setup is chosen to acquire the transmission signal. It is now possible to measure for instance the attenuation coefficient in two different ways, first by exciting a transducer with a one cylce pulse of 5 MHz1, as a short pulse in the time domain leads to a broadband excitation in the frequency domain. The attenuation can be estimated by performing fast Fourier transforms on the received RF signals, with the sample in the way and as reference without the sample in the way and calculating the log-difference between the spectra leading to the attenuation coefficient [49, 51]. Second method would be to excite a transducer with a 20 cycle burst of 4, 5, and 6 MHz. A long burst leads to a narrow bandwidth in the frequency domain, thus no spectral overlap is

1A frequency close to the operating frequency of the Octo-scanner is chosen.

57 encountered. The quotient (equivalent to the differences in the log-scale power spectral densities) of the peaks between the spectra at different frequencies then leads to the attenuation coefficient slope [51]. In the further course of this work the method with one cycle pulse excitation is used2 to obtain the attenuation coefficient of phantoms with different concentration of aluminium oxide 0.3 µm and 3 µm.

3.3.1 Measurement setup

Figure 3.1 shows the schematic setup for the measurement of the attenuation coefficient of different phantoms. A pulse generator (Olympus-5072PR) is used to excite the ultrasonic transducer (ROHE-5604) with an electrical pulse, the transducer then converts this pulse into mechanical energy, resulting in an ultrasonic wave. This wave progagates through the water and gets detected by a very sensitive needle hydrophone (Onda-HNA-0400), which converts the received signal back into an electrical voltage, which afterwards gets amplified (Onda- AH-1100) before being displayed on an oscilloscope. The broadband, piston-type transducer has a center frequeny of 5 MHz and a piston diameter of 6 mm, it is placed in a watertank filled with degassed water of 23 ± 0.5◦C. Also the hydrophone is placed in the watertank and is aligned coaxially to the transducer.

Figure 3.1: Schematic setup for the measurement of the attenuation coeﬃcient of diﬀerent phantoms [51]. In order to aqcuire the hydrophone signal which is time-delayed by the propagation delay over 5 cm of water (= 32.5 µs) a trigger signal from the pulse generator and an appropriately set trigger delay at the oscilloscope is necessary.

According to Chapter 2, the sound ﬁeld of a transducer is composed of a main lobe, which contains the highest signal strength and many side lobes of less power, see e.g. Fig. 2.14, 2Since the excitation then is as close as possible to the excitation used by the Octo-scanner hardware.

58 separated by nulls, angles at which no signal strength gets emitted. For the measurement it is important to adjust the hydrophone so it covers the main lobe, therefore a positioning system is used with which it is possible in combination with the oscilloscope to ﬁnd the maximum of the signal, by ﬁnely moving the hydrophone. The distance between the transducer and the hydrophone should be 5 cm, so the distance to the 2 cm thick phantom is 1.5 cm on either side.

Figure 3.2: Close-up of the Measurement Setup [51]. Watertank ﬁlled with degassed water, the piston-type transducer (a), which is connected to the pulse generator, and the needle hydrophone (c) (mounted on the positioning system), which is connected with the oscillosope via a power ampliﬁer, are 5 cm appart, in between the handmade phantom (b) (size: 2 cm × 5 cm × 6 cm).

3.3.2 Data processing

To obtain the unknown acoustic properties of the tissue-mimicking phantoms, the ultrasonic wave was detected once with the phantom in the way and, as a reference, once without. As shown in Eqn. 3.1, the attenuation coeﬃcient is calculated from the log-diﬀerence between the two spectra,

20 Aph(f) α(f) = − log10 , (3.1) d Aref (f) where d is the thickness of the phantom and Aph(f) and Aref (f) are the magnitude spectra of

59 the phantom data and the reference data. The method used here to measure the attenuation coefficient of the homemade phantoms, is called spectral log-difference technique [52]. This technique can also be applied for the estimation of the attenuation coefficient using B-mode data, which is described in more details in our papers [10, 53]. The data was acquired from the oscilloscope. Before measuring some adjustments were made to the oscilloscope, first the average was set to 8, which means that the oscilloscope calculates the ensemble average of eight acquired waveforms before displaying, to reduce noise artefacts. Second to find the signal the trigger from the pulse generator was connected to channel two of the oscilloscope, the trigger level was set to 2 V. The chosen record length was 1 M points, leading to a sampling rate of 1 GHz. In Fig. 3.3 the two overlayed signals are shown, which were obtained by the oscilloscope, once with the phantom in the way and as a reference, once without the phantom in the way. The code, that was used to process the data, can be found in the Appendix A.1.

Figure 3.3: Representation of the two signals (sound pressure vs. time) obtained by the oscilloscope, in blue the reference signal, meaning the measurement without a phantom in the way and in red the signal measured with the phantom in the way. Already observable is the damping of the sound energy due to the higher attenuation of the phantom (here: 0.321 dB/(MHz cm)) compared to that of water (0.1 dB/(MHz cm)).

60 As the attenuation coefficient is calculated by the log-difference of the spectra, it is not necessary to have that many sampling points, therefore the signals where first downsampled by 50, afterwards the signals were filtered using a bandpass filter (Fig. 3.4), which only allows frequencies between 1.5 MHz and 7.5 MHz to pass through.

Figure 3.4: Representation of the bandpass ﬁlter which was used on the signals, allowing only the parts of the signals of the frequency band between 1.5 and 7.5 MHz to pass through.

Performing a fast Fourier transform on both signals, leads to the two spectra, as shown in Fig. 3.5. To calculate the attenutation coefficient, the log-difference between those two spectra needs to be determined, therefore for more precise results only the parts between 3.5 and 6.5 MHz where used, leading to an attenuation coefficient slope vs. frequency, as depicted in Fig. 3.6. −1 −1 The attenuation coefficient, α0, in dB MHz cm is then obtained by calculating the gradient of the attentuation coefficient slope of Fig. 3.6, as an example, the calculation of the attenuation coefficient of a phantom with an aluminium oxide 0.3 µm concentration of 2.19% per volume is given:

3.744 − 3.098 −1 −1 α0 = = 0.321 dB MHz cm . (3.2) 6.006 − 3.994

61 Figure 3.5: Represenation of the spectrum with the phantom in the way (blue) and as a reference, the spectrum without the phantom in the way (red).

Figure 3.6: Representation of the attenuation coefficient slope of a tissue-mimicking phantom with an aluminium oxide 0.3 µm concentration of 2.19% per volume. The attenuation coefficient −1 −1 is represented by the slope, which leads for this specific example to: α0 = 0.321 dB MHz cm .

62 By measuring several tissue-mimicking phantoms with different concentrations of aluminium oxide 0.3 µm and 3 µm, it is possible to find a relation between the attenuation coefficient and the concentration of aluminium oxide. The next section will give an overview of the measurement results.

3.4 Measurement results

The attenuation coefficient for different aluminium oxide tissue-mimicking phantoms were determined, as a change of the concentration of aluminum oxide leads to a change of the attenuation coefficient. For this study, 3 different approches were chosen, producing homogeneous tissue-mimicking phantoms using only aluminium oxide 0.3 µm, only aluminium oxide 3 µm and both aluminium oxide 0.3 µm and 3 µm combined. For every measurement 3 phantoms were produced to be able to define a range in which the attenuation coefficient for these particular phantoms will be located. The resulting ranges are depicted by error bars, as represented in Fig. 3.7, which shows the results obtained from the measurement of phantoms using different concentration of aluminium oxide 0.3 µm.

Figure 3.7: Attenuation coeﬃcient vs. concentration of aluminum oxide 0.3 µm for homogeneous tissue-mimicking phantoms.

63 Due to better homogeneity of the tissue-mimicking phantoms produced using aluminium oxide 0.3 µm, the measurements of the attenuation coefficient show less variance (Fig. 3.7), as compared to the measurements of the phantoms using aluminium oxide 3 µm. Imagine a volume filled with billiard balls and the same volume filled with marbles, and you randomly choose one path through the volume. Due to the smaller size of the marbles they fill the volume more homogeneously than the billiard balls, therefore the possibility increases that by choosing only one path you sometimes hit more and sometimes less balls, with an increase in ball size. Same for the phantoms, with the increase in grain size, the possibility increases that the measurement is more dependent on a particular positioning and thus the sound path, resulting in an increased variance. More accurate measurement results would be obtained by measuring the phantoms more than once, and at different positions and averaging over all results.

Figure 3.8: Attenuation coeﬃcient vs. concentration of aluminum oxide 3 µm for homogeneous tissue-mimicking phantoms. Greater grain size leads to more variance, due to the fact, that for this measurement no spatial averaging was performed, therefore it is possible that with every repeated measurement more or less grains are placed in the sound path, with smaller grains the likelihood is higher that with every measurement approximately the same number of grains is in the path, as they are more evenly distributed.

64 Figure 3.9: Attenuation coeﬃcient vs. concentration of aluminum oxide 0.3 µm + 3 µm for homogeneous tissue-mimicking phantoms. Same as for the phantoms using only aluminum oxide 3 µm, the variance increases with the amount of aluminum oxide 3 µm.

3.5 Discussion and conclusion

With this study, it was possible to define how a change in the aluminium oxide concentration changes the attenuation coefficient for specific phantoms made. As the results show, the attenuation coefficient is almost linear dependent on the volume concentration of aluminium oxide. Due to lack of homogeneity, which is a result of the bigger grain size, we have observed more variance using the 3 µm particles. Ideal for lowest variance of attenuation properties would be to just use aluminium oxide 0.3 µm, but a decrease in grain size leads to an increase in volume of the powder to get the same weight, which above a certain point (above 6 g) is hard to handle safely, keeping in mind that aluminium oxide once inhaled into the lungs, stays there. Concluding, most breast tissues’ attenuation coefficient, as shown in Table 2.2, range between 0.1 and 2 dB/(MHz cm). By reason that using aluminium oxide 0.3 µm leads to most accurate measurement results, but phantoms with higher attenuation coefficients than 1 dB/(MHz cm)

65 are hard to produce without risking to emit particles into breathing air. The best method would be the combination of both aluminium oxide 0.3 and 3 µm, as depicted in Fig. 3.9. Using the formula of subsection 3.2.1, it is possible to produce phantoms mimicking certain types of breast tissue.

66 Chapter 4

Sound ﬁeld simulations

4.1 Introduction

For high accuracy in the diagnosis of malignant processes, the expertise of a doctor or radiologist plays an important role. Artificial intelligence, as already mentioned, might be employed to point the clinician to conspicious structures that might be overlooked due to their small- ness. As computers can be programmed to recognize trained patterns even in huge amounts of data, they are able to learn complex concepts by assembling them from simpler concepts provided they are given sufficient training sets [54]. This method of artificial intelligence is called deep learning. A computer creates its own neural network consisting of many layers, which can subsequently be trained to recognize and comprehend certain things, like handwritten numbers [13]. In medical diagnostics, to acquire this ability, a huge database with annotated, high-resolution images is necessary. As they are not yet available in the necessary quality, ultrasound simulations are required to generate ultrasound images derived from other imaging modalities available in sufficient number and annotated quality. In this work two ultrasound simulation software, Field II [55, 56], and k-Wave [38, 57], were tested for suitability with respect to simulating the Stanford University’s Octo-scanner system, as shown in Fig. 1.2 (right), a concept for tomographic ultrasound imaging. As the powerful but somewhat simpler and thus faster performing Field II software uses the spatial impulse response technique, it only allows to simulate linear systems and media, respectively, meaning it is able to simulate spatially varying scattering strength and scatterer density, but is not able to simulate spatially varying medium parameters like speed of sound, mass density, and attenuation coefficient. A feature that is needed for simulating realistic

67 breast cancer lesions. Thus after initial tests, it was not further considered, and will therefore only be discussed brieﬂy in this work. The much more powerful k-Wave simulation environment, on the other hand, is able to simulate all possible parameters of the propagating medium, allowing them to be spatially varying on a voxel basis. Meaning, the to be simulated volume is discretized into many small volume elements (dV = dx · dy · dz), the so–called voxels, and for every voxel the user can set individual acoustic properties. k-Wave is also able to simulate non-linear wave propagation, at the cost of a dramatic increase in necessary processing power. Even though the processing times are orders of magnitude higher than those for the Field II software, we switched to this software for its greater capabilites. k-Wave will therefore be discussed in much greater detail, thereby covering its internal processing, the to be expected execution times, possible numerical stability problems, and obtained results.

4.2 The Field II simulation software

The overall goal of simulations is to provide realistically simulated ultrasound images, perhaps even derived – by some to be determined rules (e.g. like suggested by the authors of [58]) – from 3D input data stemming from 3D mammography or, preferably, MRI data. All the properties of the simulations should match as close as possible the in vivo data, like shape and intensity of ultrasound speckles, shadowing behind highly attenuating structures, correct frequency dependency of spatially resolved attenuation coeﬃceint, correct carrying over of ultrasound power from B-mode to second harmonics due to non-linearities in wave propagation (cf. 2nd harmonic imaging, see Fig. 4.8), and so forth. Thus the goal is to generate, by simulation, ultrasound images that in as many as possible aspects mimic the real images, taken directly from a patient. Initially it was assumed that the open source software developed by Jøgren Arendt Jensen [59] was able to simulate all these necessary features expected from an ultrasonic scan of breast tissue, like tissue type dependent attenuation, speed of sound, mass density, compressibility, etc. But already in the evaluation phase and after contacting the auther via email, it turned out that this is not true, since the simulation method chosen for Field II – the method of spatial impulse response [60] – is heavily relying on the superposition principle, that clearly, only allows for linear media properties precluding any spatially varying parameter and any frequency dependent attenuation.

68 4.2.1 The method of spatial impulse response

The Field II system uses the spatial impulse response technique, which is a development by Tupholme and Stepanishen [61, 62, 63]. The line of argument presented here follows closely the description given in [56], where the author Jensen argues that Field II relies on linear systems theory to determine the ultrasound field for both the pulsed and the continuous wave regime for any distribution of – typically randomly placed – scatterers. So first the spatial impulse response, which is the wave (= a 3D disturbance in the medium evolving over time) that is accordingly scaled by the local stattering strength and shifted via a spatial convolution operation to the locations within the simulated volume of all randomly placed scatterers. By convolving the spatial impulse response with a specific excitation function, the ultrasound field can be determined. The name spatial impulse response results form the fact that the impulse response as seen at the position of the transducer will vary depending on the position of the scatterer relative to the transducer.

Parameter Value Unit transducer center frequency 3.5 · 106 Hz sampling frequency 108 Hz speed of sound 1,540 m s−1 width of TE 200 µm kerf 50 µm focus (x,y,z-coordinates) [0, 0, 70] mm number of TE 192 1 number of active TE 64 1 apodization hanning number of placed scatterers 416,000 number of simulated scan lines 50 x-dimension −20 - +20 mm y-dimension 0 - 10 mm z-dimension (depth) 35 - 90 mm execution time (standard PC) 9,300 s

Table 4.1: Parameters of the simulation in Field II (for recommended values see e.g. [56]) whose results are shown in the following ﬁgures. TE stands for transducer element.

Waves propagating between two points in space, when the propagation medium is linear and at rest, obey the fundamental property of reciprocity. Which indicates that the waveform observed at the receiver end does not change by interchanging the source and receiver locations, due to waves traveling in a symmetric manner, therefore ensuring reciprocity [65]. As

69 a result of the acoustic reciprocity the received response can be determined by the wave from a small oscillating sphere located at the postition of the to be placed scatterers. Thus, as already mentioned, by convolving the transducer excitation function with the spatial impulse response of the emitting aperture and of the receiving aperture and then considering the electro-mechanical transfer function of the transducer, to obtain the received voltage signal trace, the total received response in pulse-echo mode can be determined. Since linear systems theory states that the property of superposition is valid, any excitation can be used. Now by Fourier transforming the spatial impulse response for a particularly chosen frequency even results for a harmonic continuous excitation can be determined easily. Using this approach can lead to all the diﬀerent commonly found ultrasound ﬁelds for linear propagation, although restricted to the case where all parameters of the propagating medium are homogeneous and isotropic. Realizing this limitation, only a few runs of simulation on a dataset derived from MRI volume data [14] were performed, before redirecting the focus to the much more powerful simulation tool k-Wave. A more elaborate description on Field II can be found in [59].

Figure 4.1: MRI data from the I-SPY 1 trial [14] of a tumor in the breast cropped to display only the lesion (in order to speed-up Field II simulation time).

70 Simulation Parameters

Since Field II is calculating the spatial impulse response of scatterers, due to the sending and receiving aperture, it is not required to discretize the simulated volume into voxels, which for instance is a requirement of all ﬁnite element based simulation software like k-Wave. Field II requires to set a list of parameters, as are stated in Table 4.1. The values chosen here lead to the simulation result as depicted in Fig. 4.4. As a simulation target volume data readily available from MRI-scans was initially used (here from the I-SPY 1 database, patient ID 1011, (Directory 2-T2-FSE-Sagittal-35538) [14]). The scatterer strengths were derived from the MRI contrast as is depicted in Fig. 4.1. The scatterers were placed randomly with their spatially averaged density being homogeneous with a numerical value of 20 mm−3 that is about 1 scatterer per λ3-sized volume element. As an MRI image depicts the density of hydrogenic protons in a medium and the ultrasound images the diﬀerent acoustic impedances, there is no physical meaning behind this ultrasound simulation, but it was usefully for testing the software. The execution time of this simulation on a standard PC (8 cores) was about 155 minutes.

Figure 4.2: Same MRI dataset as displayed in Fig. 4.1 to make visible the signal dynamic, the MRI contrast, going from zero to about 350 in signal strength.

71 Figure 4.3: Randomly placed scatterers ﬁll the simulated volume as input to the Field II simulation program. The MRI contrast is taken as scattering strength (here displayed colorcoded). Remarks: there is no physical meaning behind that chosen mapping from MRI to US contrast. The density of the scatterers was deliberately decimated by a factor of 30 for display to prevent the MATLAB’s rendering engine from completely ﬁlling in the volume.

4.3 The k-Wave simulation software

"k-Wave is an open source, third party, MATLAB toolbox designed for the time-domain simulation of propagating acoustic waves in 1D, 2D, or 3D. The toolbox has a wide range of functionality, but at its heart is an advanced numerical model that can account for both linear and nonlinear wave propagation, an arbitrary distribution of heterogeneous material parameters, and power law acoustic absorption", quoted from [38]. The toolbox can be downloaded for free from the k-Wave website [57]. Furthermore it has to be noted that this MATLAB toolbox allows to utilize, if available, the very powerful CUDA1 environment that uses the very many processor cores of current NVIDIA graphics processing units (GPU) resulting in a speed increase of up to a few hundred as compared to CPU processing. 1Compute Uniﬁed Device Architecture [66].

72 Figure 4.4: Result of the Field II simulation. In this particular case displayed as B-mode image. The dataset is derived from the MRI image displayed in Fig. 4.1. Please note the depth scale being inverted relative to the ﬁgures before. The lesion is barely visible centered at 65 mm in depth.

4.3.1 The simulation strategy of k-Wave k-Wave numerically solves the governing first-order partial differential equations 2.3 to 2.5 (on page 36) since doing so allows to easily introduce mass and force sources which in the coupled form in the second-order partial differential equation 2.11 are not so easily introduced. Furthermore to absorb acoustic waves that reach the margins of the computational domain, the action of a so-called perfectly matched layer (PML) is introduced. The complexity involved in solving these equations depends on the properties of the propagation medium as selected by the user.

73 It is necessary to choose a simulation environment allowing to set parameters like the mass density, ρ(x, y, z), the speed of sound, c(x, y, z), and the absorption parameters α(x, y, z) = γ α0 ·ω spatially resolved. Although it is assumed that the wave propagation at diagnostically used ultrasound pressures is mainly linear still there is some nonlinearity experienced and thus higher order harmonics are typically generated (at very small amplitudes, though). The simulation environment should also be able to handle second order harmonic generation. k-Wave was designed to fulﬁll all these requirements and was thus chosen, especially as the resulting B-mode image simulation data can be used to estimate the spatially varying parameters, mass density, speed of sound, and attenuation coeﬃcient. An important feature, as being able to determine these parameters additionally to the structural (morphologic) information, is expected to improve the diagnostic accuracy. Taking all these aspects into account, the governing equations, which k-Wave uses to numerically solve the posed simulation problems, change to:

∂~u 1 = − ∇ p , (4.1) ∂t ρ0 ∂ρ = − (2ρ + ρ ) ∇ · ~u − ~u · ∇ρ , (4.2) ∂t 0 0 2 ! 2 ~ B ρ p = c0 ρ + d · ∇ ρ0 + · − Lρ . (4.3) 2A ρ0

Included are also the acoustic absorption and non-homogeneities in the material parameters. d~ is the acoustic particle displacement resulting from a convective term, ~u is the particle velocity, p is the sound pressure and ρ the mass density. The operator L in the pressure- density relation is a linear integro-differential operator that accounts for acoustic absorption and dispersion that follows a frequency power law [38]. For efficiency reasons, the terms ∇ ρ0 cancel each other, and are therefore in the discrete equations solved in k-Wave not included. As k-Wave does not consider all nonlinearity effects, which could occur in an elastic medium, there are only two terms included to account for a possible nonlinearity. First, the finite- amplitude effect (as opposed to assume that the excursion form the ambient pressure is close to zero!), which contributes to a change in sound speed, is characterized by the term B/A, the nonlinearity parameter [67]. Second, a convective nonlinearity, which states that the particle velocity also contributes to the wave velocity, is included by the additional term 2ρ in Eqn. 4.2, written as spatial, not temporal gradient, for efficiency reasons in numerical encoding [57, 68].

74 Combining all three coupled equations leads to a generalised form of the Westervelt equation [69, 70, 71]. For any useful simulation environment there needs to be the possibility to excite the system by adding source terms at proper locations. In k-Wave this is accomplished by adding a force source term to Eqn. 4.1:

∂~u 1 = − ∇ p + SF , (4.4) ∂t ρ0

−1 where SF is the input force per unit of mass (= acceleration) and is thus entered in N kg or equivalently in m s−2. There exists also the possibility to add a mass source term but this is not considered here.

The k-Wave pseudospectral method

Since in k-Wave the number of voxels is enormous, in particular if one considers to simulate 3D-structures, any method to speed up numerical calculations is essential. Solving differential equations in several intermediate steps the calculation of gradients (spatial difference quotients) and time derivatives (temporal difference quotients) are necessary, typically for each and every voxel considered. Doing so on a local basis, going from voxel to voxel is detrimental. Thus one is seeking globally acting differential operators. As is well known from linear system’s theory the Fourier transform is providing a theoretical basis via the differentiation property, as can be stated like [64]:

F {DαF [x(t)]} (ω) = jαF ωαF F{x(t)} (ω) . (4.5)

F, is the Fourier-transform operator, D, is the differentiation operator, αF , is the order of differentiation, that can also be a non-integer, x(t), is the time (or spatial) signal to be differentiated, and ω, is the temporal or spatial angular frequency. Thus employing or utilizing this properly combined with the numerically very efficient fast Fourier transform algorithm allows to calculate approximations of time and spatial derivatives for all voxels concurrently. Another interesting property of the Fourier transform is the translation / time shifting property, which states that:

75 F{x(t − τ)} (ω) = e−jωτ F{x(t)} (ω) , (4.6) thus in the frequency domain the complete sound wave ﬁeld can be propagated in a single time step by just applying a phase term ωτ; ∀ ω before applying the inverse Fourier transform to convert back to the time/spatial domain. k-Wave is utilizing extensively both properties and it is stated in the manual that for the full-blown diﬀerential equations up to 14 Fourier transforms are processed over the 3D voxel space for each step in time. More information and details can be found in [38].

Perfectly matched layer

The k-space grid needs to be given boundary conditions in order to prevent the simulation to have waves leaving the simulated volume at one margin to be wrapped around and enter at the opposite margin again. This property is due to the periodicity of the Fourier transform employed. This wrap-around is prevented in k-Wave by enclosing the simulated volume by a so-called perfectly matching layer (PML) that is governed by a set of equations guaranteeing sufficient absorption of the outgoing wave and, equally important, preventing reflection of waves back into the simulated volume. The PML is automatically generated and set in the software but the user needs to be aware of the fact that the PML reduces the usable volume significantly.

Simulation Parameters

The software provides three simulation functions, namely kspaceFirstOrder1D, kspace- FirstOrder2D, and kspaceFirstOrder3D, implementing the first-order k-space model for elastic media as described by the equations presented above. By calling these functions, the simulation of the wave propagation in one, two or three dimensions will be started. There are four input structures, which need to be set in advance, the kgrid, the medium, the source and the sensor. Table 4.2 lists all of the to be set parameters. The first input, the kgrid, defines the computational grid. As the governing equations need to be solved for discrete positions in space, the continuous medium, by calling the function makeGrid, will be divided up into an evenly distributed mesh of cuboid grid points. To guarantee the sampling theorem in space, the grid must be of sufficient density, meaning per wavelength λ at least 3 grid points must be provided. To give an example, if the used center frequency of the transducer (fexcite) is 5 MHz, leading to a wavelength of 294 µm (for a minimum speed of sound of 1470 m/s), the grid spacing (dx, dy, dz) must be smaller than

76 100 µm. Also for efficiency reasons, as the internal calculations are based on the Fourier transform, the grid dimensions (Nx, Ny, Nz) should be a powers of two2. Last thing, which needs to be kept in mind is that the simulation duration must be sufficient to accomodate longest path at lowest speed of sound, this can be set by the parameter tend. All these settings contribute to the excessive computational time, making it practically indis- pensable to use supercomputers for the simulations, as a simulation of a B-mode sector scan image with 192 lines-of-sight of a phantom (80 mm diameter), like the one in Fig. 1.2 (left), with a grid dimension of 1024×1024×128, would take up to 2 years on a CPU of an average computer. As second input structure, the medium has to be defined, meaning for each grid point the acoustic parameters of the to be simulated tissue need to be set, like the isentropic sound speed, the mass density, the nonlinearity parameter B/A, the power law absorption coeffcient or the prefactor, and the power law absorption exponent. For clarifying these parameters, remember from Chapter 2, when an ultrasound wave propagates through a medium, it generally loses some acoustic energy to random thermal motion, resulting in acoustic absorption. This can be modelled as a sound-absorbing fluid in which the absorption follows a frequency power law of the form:

γ α = α0 ω , (4.7) where, α, is the absorption coefficient in coherent units of Np/m, in medical terms in dB/cm, −γ −1 α0, is the power law prefactor in Np (rad/s) m , in medical usage in dB/(MHz cm), with the power law exponent, γ, typically set to γ = 1.0. In k-Wave only values between 0 and 3, excluding 1.0, are allowed to be set, therefore after initial testing for all simulations the power law exponent (alpha_power) was set to 1.01, which is closest possible to the preferred 1.0. The third input structure needs to include the positions and the properties of the acoustic sources. In general there are three different source types available to be set in k-Wave, an initial pressure distribution, a time varying pressure source or a time varying particle velocity source. But it is also possible, to define and use a diagnostic ultrasound transducer as a source (and/or concurrently as a sensor). A setting that was used for the simulations in this work. Table 4.2 lists all of the to be defined parameters of the transducer, like the excitation frequency and strength, number of transducer elements and their size, and also

2Alternatively, also a product of powers of small primes can be used since internally the so–called Fastest Fourier transform in the world (cf. www.ﬀtw.org) is used.

77 whether the transducer uses linear- or phased-array, steering to a certain angle. The final input structure, is the sensor, which also needs to be defined in position and properties. The sensor is used to detect the acoustic field at each time step (the smaller the time step, the longer the computational time, but the higher the chance of numerical stability) of the simulation, but it is also possible to set the physical quantity, which is requested to be recorded, like the time varying acoustic intensity or the maximum pressure. Again it is possible to use a diagnostic ultrasound transducer also as the sensor, and also to define a single transducer which operates as emitter and receiver all at once. If one were to use k-Wave, reading or studying the manual closely is essential, anyway, so here the reader is referred to the k-Wave manual [38].

4.4 Simulation Results

In this section, various simulation results will be presented. The selected parameters are listed in Table 4.2. The transducer, which is used both as source and sensor, is always to be placed at the top center at a depth of 0 mm. An example of the transducer used for this simulations can be seen in Fig. 4.5, only part of the transducer elements are active at once. For the complete listing of associated Matlab m-ﬁles see Appendix A.2 on page 111 pp.

Figure 4.5: Representation of the structure of a transducer used in the k-Wave simulations [38].

4.4.1 First simulation results

The ﬁrst simulation results, were obtained by using an example, which is provided additionally to the k-Wave toolbox (’example_us_bmode_linear_transducer’). The idea was to get familiar with the functionalities provided by the software.

78 Parameter SI unit Subsec. 4.4.1 Subsec. 4.4.2 Subsec. 4.4.3

Fig. 4.6 Fig. 4.10 Fig. 4.13 k-grid grid spacing µm 156.25 99.292 50 x-size mm 40 96.71 30.4 Nx-size — 256 974 608 Ny-size — 128 462 770 Nz-size — 128 88 78 ∆t ns 30.44 19.3425 9.2025 k-transducer fexcite MHz 2.4 3.00 5.00 # of cycles — 4 3 3 strength kPa 24 250 500 # of elements — 128 128 128 elem.-width µm 156.25 297.87 250 elem.-kerf µm 0.0 0.0 50 focus azim. mm 20 70 25 focus elev. mm 19 70 25 ◦ steering ^ ±(... ) 0 0 - 13 0 - 13 k-medium α-power — 1.5 1.01 1.01 αw dB/(MHz cm) – 0.1 – αb dB/(MHz cm) 0.75 0.3 varying αr1 dB/(MHz cm) 0.75 2.3 – αr2 dB/(MHz cm) 0.75 1.2 – αr3 dB/(MHz cm) 0.75 – – cw m/s – 1540 – cb m/s 1540 1550 ± 37 1550 ± 37 cr1 m/s 1565 1400 - 1600 – cr2 m/s 1565 1450 - 1630 – cr3 m/s 1565 – – 3 ρw kg/m – 1000 – 3 ρb kg/m 1000 1020 ± 24 1020 ± 24 3 ρr1 kg/m 1000 933 - 1066 – 3 ρr2 kg/m 1000 967 - 1087 – 3 ρr3 kg/m 1000 – – ∅b mm – 90 – ∅r1 mm 8.6 12 – ∅r2 mm 5 11 – ∅r3 mm 4.5 – – xb mm – 90 – xr1 mm 27.5 78 – xr2 mm 30.5 70 – xr3 mm 15.5 – – yb mm – 0 – yr1 mm 20.5 −10 – yr2 mm 37 +9 – yr3 mm 30.5 – –

Table 4.2: Listing of phantom simulation parameters, for simulation results represented in subsections 4.4.1, 4.4.2 and 4.4.3. Subscripts denote: (... )w, water, (... )b, background, (... )r1, highly scattering region 1, (... )r2, highly scattering region 2, (... )r3, highly scattering region 3. 79 Figure 4.6 shows the simulated phantom (with a k-grid size of 256 × 256 × 128 voxels), that is composed of three highly scattering regions placed within a low scattering background, the biggest scattering region is not filled with any scatterers, mimicking a fluid-filled bladder, a cyst.

Figure 4.6: Simulated phantom consisting of a low scattering background medium surrounding three highly scattering regions, the left of which is ﬁlled with water, mimicking a cyst.

All scattering regions are assigned a random speed of sound normally distributed, mean 1565 m/s with a standard deviation of 75 m/s, clipped at 1400 m/s and 1600 m/s, a density of 3 1043 kg/m and an α0-value of 0.75 dB/(MHz cm). Only the cyst, which is ﬁlled with water 3 (diameter 11 mm), has a speed of sound of 1450 m/s, a density of 1000 kg/m and an α0- value of 0.99 dB/(MHz cm). As perfectly matched layer (PML) 20 grid points were chosen for the x-direction and 10 grid points each for the y- and z-direction. For this simulation a transducer center frequency of 2.4 MHz was used, where 32 out of 128 transducer elements focused at x = 20 mm in azimuth and at x = 19 mm in elevation using digital beamforming for each of the 192 simulated lines-of-sight.

80 Figure 4.7: Representation of operating principle of a transducer used in the k-Wave simulations [38]. Only parts of the transducer elements are active at once while focusing.

An example for the transducer output time signal for a single line-of-sight is given in Fig. 4.9. This form of presentation of an ultrasonic signal is called A-mode (amplitude mode). As the name implies, the echo strength (amplitude) vs. penetration depth is displayed. First the raw beamformed signal (barely visible) is shown at the top graph, which is then time gain compensated, considering the fact that the sound energy decreases with the increase in penetration depth and of course the attenuation. The TGC makes up for that assumed average attenuation (second plot). After removing the second harmonic generated, by filtering all of the lines-of-sight with a band-pass filter centered at the fundamental frequency (3rd graph) and generating the complex envelope (4th graph) of the signals via the Hilbert- transform, which is transformed into a dB scale (log-compression), the B-mode (brightness mode) image (Fig. 4.8) can be generated by placing scan-line information, columnwise as the phased-array scanning is progressing through the scanned section resulting in a virtual slice plane through the imaged or simulated volume. Figure 4.8 shows the B-mode images that resulted from the simulation of the phantom depicted in Fig. 4.6. One can clearly observe the TGC-amplifier’s overcompensation of the lacking attenuation of the water-filled cyst. But the speckles generated seem not to match those seen in medical ultrasound B-mode images. Presumably the selected numerical aperture is too large. Also depicted is the B-mode image of the second harmonic generated by nonlinearities of the propagation medium for the phantom depicted in Fig. 4.6. The first simulation results already looked quite promising, but as the next results will show, visibly correct B-mode images, do not automatically indicate numerical stability and physical correctness.

81 Figure 4.8: Fundamental and second harmonic B-mode (log-compressed) image generated by simulation of the phantom depicted in Fig. 4.6.

82 Figure 4.9: Several intermediate signals of the post-processing in k-Wave are depicted for the center line-of-sight of Fig. 4.6. Top: beamformed raw signal (32 out of 128 transducer elements firing for each line-of-sight). 2nd plot: action of the TGC-amplifier. 3rd plot: band limiting by filtering out only fundamental frequency ± rel. bandwidth (there is also a second harmonic generated, which is not shown here). 4th plot: Magnitude of the complex envelope (generated within k-Wave presumably via Hilbert-transform). Plot 5 shows the log-compressed signal used in B-mode images.

4.4.2 Simulation results of tissue-mimicking phantoms

The goal is to generate a database of 3D ultrasound images for breast cancer research. Stan- ford University’s Ferrara Lab is currently working on an Octo-scanner, as depicted in Fig. 1.2 (right), a concept for tomographic ultrasound imaging. Due to a lack of breast cancer patients for producing enough high quality, annotated images for the to be built database, simulations are essential of breast mimicking phantoms. Figure 4.10 gives an example of such a phantom. The phantom was produced as described in Chapter 3, it is agar-based, which means the phantom itself has an attenuation coefficient of approximately 0.3 dB/(MHz cm), the two inclusions were made using different concentrations of aluminium oxide 0.3 and 3 µm, resulting in 1.2 and 2.3 dB/(MHz cm). For the simulations the same settings were chosen. Here, as the phantom to be scanned is placed in a watertank filled with degassed water, part of the surrounding water is also simulated (the light blue region in Fig. 4.10).

83 Figure 4.10: Geometry of the simulated arrangement. A single Octo-scan-transducer (128 elements) is placed at the top center operated with phased-array excitation to scan 131 lines- of-sight within a sector of ±13◦. The light blue color depicts water (no scatterers, attenuation: αw = 0.1 dB/(MHz cm)). The phantom center is placed at a depth of 90 mm and has a diameter of 90 mm. It has scatterers placed randomly resulting in an attenuation coefficient of α0 = 0.3 dB/(MHz cm). There are two strongly scattering regions, the left has a diameter of 12 mm and an attenuation coefficient of α0 = 2.3 dB/(MHz cm), the right has a diameter of 11 mm and an attenuation coefficient of α0 = 1.2 dB/(MHz cm).

84 Figure 4.11: Resultant B-mode images of simulating the phantom depicted in Fig. 4.10 with background (α0 = 0.3 dB/(MHz cm)), and highly scattering regions of α0 = 1.2, and 2.3 dB/(MHz cm), respectively. (top) shows the B-mode image at the fundamental frequency, (bottom) shows the second harmonic B-mode image.

85 Compared to the ﬁrst simulation results, this time a transducer using a phased-array was chosen instead of a linear-array, resulting in the B-mode images as depicted in Fig. 4.11. The results again seem promising, the speckles this time match those seen in medical ultrasound B-mode images and also the shadowing due to the attenuation of the inclusions indicate physical correctness. But when looking closer at the data, as depicted in Fig. 4.12 for a single line-of-sight, one can observe that the amplitude of the signal is way too high, which indicates that something must be wrong. Therefore the next step was to use a simple simulation setting, trying to ﬁgure out the problem, that leads to the observed numerical instabilities.

Figure 4.12: Representation of the center line-of-sight of the B-mode image (Fig. 4.11), showing that for an induced source strength of 250 kPa, the resulting sound pressure is on the order of several MPa, although as was described in Subsection 2.4.1 the sound pressure usually decreases tremendously with the increase in penetration depth.

4.4.3 Limitations of the k-Wave software

To clarify what might go wrong in the simulations as to result in numerical instability, a series of simulations were planned with a geometry as the one depicted in Fig. 4.13. Placing 5 very small almost impulse like scatterers at depths 5, 10, 15, 20, and 25 mm in the medium,

86 which otherwise has a homogeneous speed of sound of 1540 m/s and a homogeneous density of 1000 kg/m3, only varying the homogeneous attenuation coeﬃcient from run to run. The chosen grid spacing was 50 µm, resulting in a time step of 9.2025 ns, which is automatically selected by k-Wave.

Figure 4.13: Representation of the simulated medium. 5 scatterers were placed at depths 5, 10, 15, 20, and 25 mm. For demonstration in this picture the scatterers are displayed larger than they actually are. In reality each scatterer is represented by 3 grid points only.

By increasing the attenuation coefficient step-by-step and looking at the center line-of-sight, first thing that also came to attention was the not sufficiently diminishing time signal, which means, the echo received back from the scatterers is too high. Furthermore it seemed that by increasing the medium’s attenuation coefficient beyond a certain limit (here α0 = 2.8 dB/(MHz cm)) no useful results could be obtained any more as the simulation seems to result in numerical instability, as depicted in Fig. 4.14. The results presented in Fig. 4.14, led to believe that the very reason we switched from Field II to k-Wave, being the ability to accomodate any type of attenuation, now turns out to give us problems. But as the simulations using a smaller attenuation coefficient lead to numerically stable results, the next guess was, that the GPU single-precision floating point number format, which was used to speed up the simulation, was not precise enough.

87 Figure 4.14: Representation of the center line-of-sight from Fig. 4.13. An increase of the attenuation coeﬃcient α0 seems to result in numerical instability at a certain point in simulation time. For this setting with a grid spacing of 50 µm and a time step of 9.2025 ns, the limit to numerical instability is approximately α0 = 2.8 dB/(MHz cm), above that the simulation kind of explodes within a few time steps as represented in the bottom image, leading to a sound pressure above 10 to the power of 26 Pa.

88 So one explanation might be that with a higher attenuation coefficient the sound pressure also decreases faster, which can lead to concurrent sound pressure levels many magnitudes apart (like 1 MPa excitation pressure and downrange only a few Pa) causing problems. Since k-Wave employs global processing via an FT, these numbers need to be encoded in the same floating-point format denormalized for the summation operation in the FFT, which clearly leads to problems as shown in the numerical example below, demonstrating that a GPU with single-precision arithmetic is not able to sum correctly numbers like 1000000 and 0.01 (GPU single-precision: 1000000 + 0.01 = 1000000.00, see AppendixB), possibly resulting in numerical instability. With this knowledge, the next step was to perform a study using different settings for the simulation, varying the grid spacing and the time step and switching between GPU single- precision and CPU single- and double-precision, with the same medium as before (Fig. 4.13).

Figure 4.15: Represenation of the maximum attenuation coeﬃcient possible to be selected for diﬀerent time step sizes and grid spacings using a GPU with single-precision. What is noticeable is that the decrease in time step (increment) tends to result in numerically stable simulations, same as the increase in grid spacing. The excitation frequency used was 5 MHz, with a source strength of 500 kPa.

Figure 4.15 shows, that contrary to what one would actually expect, that due to a better compliance of the sampling theorem, a decrease in grid spacing seems to result in numerical instability more easily. A decrease in time step on the other hand, which means the sound

89 field gets recorded more often, guarantees numerical stability more easily. This leads to believe that the key for numerically stable simulations is a decrease in time step, lower than automatically selected by k-Wave. Due to the benefit of a lower computational time, k-Wave seems to select a time step right on the edge of numerical stability. Now looking at the line-of-sight, which corresponds to the signal the transducer receives as echo from the scatterers, we can only guess what is happening inside of the medium. To make sure the simulated sound wave propagates through the medium in a physically correct way, a video was produced, by putting together frames of plane plots (x-y-plane), which were saved to disk space during the simulation process, demonstrating the sound wave propagating through the medium. Fig. 4.16, shows a few frames of the video sequence, where coloring indicates pressure strength (from dark blue as the lowest pressure to bright yellow indicating highest pressure), therefore to cover the huge dynamic range, same as for the B-mode images, a log-compression has to be applied on the data before displaying. For this video, whos few frames are displayed in Figures 4.16 and 4.17, a setup was used, that was known, from the study before, resulted in a numerically stable simulation, with an attenuation coefficient α0 = 2.9 dB/(MHz cm), a grid spacing of 50 µm and a time step of 5 ns.

Figure 4.16: Representation of 3 frames from the video produced to make sure that the sound wave propagates in a physically correct manner through the medium. The frames were taken at the time steps t = n·∆t for n = 1500, 1650, and 1950. In the ﬁrst frame (left) the sound wave just hits the ﬁrst scatterer at a depth of 5 mm, in the next frame (middle) what can be seen is that part of the sound wave that scatters back in all directions, as was described in Subsection 2.4.1 will happen when a sound wave hits a scatterer. So the main part of the sound wave continues traveling forth, the other part propagates in form of a concentric circle (2D), or rather a sphere (3D).

These frames (Fig. 4.16) show that the sound wave seems to propagate in a physically correct manner. As when impinging on a scatterer (a small) part of the wave scatters in form of an evolving sphere in every direction, as was described in Subsection 2.4.1, when deﬁning a scatterer. The next Fig. 4.17, also shows that the simulation is so precise as to even

90 demonstrate scattering of backscattered waves. The spherical waves from the scatterers build up mutually until the main excitation sound wave leaves the computational domain (see Fig. 4.18), which indicates a physically correct propagation of the sound wave through the medium.

Figure 4.17: Continuation of the sound wave propagating through the medium of Fig. 4.16. The part of the sound wave continuing to propagate loses sound energy everytime it hits a scatterer (here at depth 10 mm), therefore the spherical sound waves appear less bright with time (brightness corresponds to pressure strength), as can be seen in Fig. 4.18. The frames were taken at time steps t = n · ∆t for n = 2450, and 2550. The very minute patterns, which can be seen on the lower half of the frames are a result of the periodicity the Fourier transform imposes on the solution, resulting in a residual mirroring of the sound waves. It can be minimized by increasing the perfectly matched layer size. But the mirrored wave contains only pressures of a few Pa and can therefore be neglected. Testing, whether the reason for the numerical instability is the single-precision of the GPU, a comparison was made using a simulation, which was known would end up in numerical instability using the GPU single-precision. The results showed that using a CPU with double- precision, will end up in numerical instability later in simulation time than when using the CPU single-precision, but still it would. So this is not the key for numerical stable simulations, which is good news, as the computational time for a simulation using a GPU is only a fraction of that of a simulation using a CPU. When a simulation results in numerical instability, as depicted in Fig. 4.14, the sound pressure kind of explodes within a few time steps, in the video this is demonstrated by a fully yellow frame. When zooming into the data from this yellow frame, a checkerboard pattern seems to appear (Fig. 4.19). As in this work the goal is to ﬁnd a way to prevent this from happening, the

91 reason which leads to the checkerboard pattern was not further investigated. One explanation might be that due to the fact that the diﬀerentiation action using FT methods scales the spectral components by their frequencies see Eqn. 4.5, thus the highest frequencies get scaled most. So it does not wonder if those seem to dominate the ﬁnal pattern seen. But a guideline on how to obtain numerically stable simulations, will be presented in the next subsection.

Figure 4.18: The frame was taken at time step t = n · ∆t for n = 3200. As can be seen at a depth of 10 mm, the spherical sound waves build up mutually, resulting in even more spherical sound waves. The focal point for this simulation was at 25 mm, visible, due to the change of the wavefront from crescent-shape to a point with the increase in penetration depth.

Figure 4.19: Representation of the zoomed in view of the pressure ﬁeld, which seems to result in a checkerboard pattern, when the simulation ends up in numerical instability.

92 4.4.4 Guideline to numerically stable ultrasound simulations

There are 5 parameters, which need to be taken into consideration for numerically stable simulations.

• First the time step size, which normally is automatically selected by k-Wave, but rather on the edge of numerical stability to give as fast a computational time as possibe (persumably).

• Therefore, depending on the grid spacing chosen, which is the second parameter, that needs to be considered, smaller time step sizes, like 50 – 80 % of what is selected automatically by k-Wave, need to be chosen to keep the simulation stable and to minimize unwanted numerical errors. The grid spacing must be chosen small enough, to garantee the sampling theorem in space (smaller than 80 µm for a transducer center frequency of 7 MHz for water), but the decrease in grid spacing leads to less numerical stability. For the simulation of breast tissue, it must be guaranteed that simulations with an attenuation coeﬃcient of up to 5 dB/(MHz cm) are possible. See Fig. 4.15, to select the best time step size according to the chosen grid spacing, and vice versa.

• Thirdly the alpha power, (α, as was mentioned before), the alpha power, being the power law exponent γ, is for breast tissue typically 1.0. k-Wave only allows values between 0 and 3, with the only exception of exactly γ = 1.0. Therefore the chosen alpha power for the simulations presented above was 1.01, as this value is closest possible to 1.0, but by increasing the alpha power to 1.05, simulations, which resulted in numerical instability for 1.01, now ended up being numerically stable. If observing a video one could see that stripped patterns seem to appear with an increase in alpha power, which would need further examination, therefore it is recommended to stick to an alpha power of γ = 1.01.

• Fourthly the precision used for the simulation. A comparison showed that double- precision tends to result in numerical instability later in simulation time than for single- precision, but still, switching to double-precision is not the key for numerically stable simulations. Also it may not be possible to use a GPU providing double-precision, therefore it is recommended to stick to the single-precision GPU-based simulations that are 2 orders of magnitude faster than on a CPU.

• Last issue that needs to be pointed out is the memory constraint. As the computation of each time step happens at the same time for the whole computational domain (medium),

93 the calculation task can not be split into subproblems of lesser size, but needs to be held completely in memory, which leads to an immense memory requirement. For the simulations presented here a GPU NVIDIA GeForce RTX 2080 was used, with 11 GB fast video memory, limiting the to be simulated computational domain to more or less a maximum of 1024 × 512 × 128 grid points. The NVIDIA TITAN RTX (GPU) provides the largest memory of 24 GB so far, by using the possibility of connecting two of these GPUs together a maximum of 48 GB memory is thus available and therefore the largest possible problem to be solved using a double GPU is a computational domain of size 2048 × 1024 × 128 grid points. A CPU can easily provide enough memory, but a simulation like this would require an excessive computational time of several years.

So, as already shown in Fig. 4.15, the key for numerically stable simulations using the k- Wave software, is using small time steps, with these it is possible to simulate breast tissue considering all acoustic properties in a numerically stable way. But for physical correctness it is not sufficient enough looking at the sound wave traveling through the medium, as was demonstrated in the sequence in Fig. 4.16 and 4.17, due to the fact that as was presented in Subsection 4.4.2, visibly correct simulations do not automatically indicate numerical stability nor physical correctness. Therefore it is necessary to look at the B-mode data more closely, which is why several papers are dedicated to finding a method to estimate the attenuation coefficient of B-mode data. This method then applied on the simulated B-mode data, would be an even further proof of physical correctness, but this investigation would go beyond the scope of this work. The Stanford group with me being part of, has submitted two papers [10, 53] on that topic, but those results are not included here for time constraints.

94 Chapter 5

Discussion

As the statistics show, breast cancer is a serious matter. Fortunately the percentage of breast cancer related deaths decrease recently, by technological impovement in screening and treatment. This thesis was intended to present possibilities of improving the accuracy in the diagnosis using ultrasonic imaging, as this is the only medical imaging modality, which entails no risks. As already mentioned a few times troughout this work, ultrasound is mechanical oscillation of an elastic medium. An ultrasound wave, which is generated by transducers, behaves differently in different types of breast tissue, due to varying acoustic properties (like speed of sound, mass density, attenuation coefficient, etc.). Improvements of the diagnostic accuracy are expected, when in addition to presenting the radiologist an ultrasound image, she or he could also choose to overlay (colorcoded) estimates of these acoustic properties. Once a method is found to estimate these properties from US-scan data, it needs to be verified that this method is providing an unbiased estimate of the properties with a variance close to the Cramér-Rao Lower Bound (CRLB). The estimator has to be applied to known properties and the results need to be compared with measured values. Therefore part of the master thesis was dedicated to the investigation of different tissue-mimicking materials, trying to find some kind of formular, for the production of tissue-mimicking phantoms with a set of desired acoustic properties. There are different materials that have specific impacts on the acoustic properties when added to the phantom recipe. Like the change in concentration of glycerol changes the speed of sound of the produced phantom. As most cancer masses are highly attenuating, the goal in this work

95 was to find a formular to produce phantoms with specific attenuation coefficients. Aluminium oxide of different grain sizes was used, that is known to influence the attenuation coefficient dependent on its concentration in the produced phantom. Many agar-based phantoms were actually produced using different concentrations of aluminium oxide 0.3 µm and 3 µm, these were measured using a piston-type transducer as source and a needle hydrophone as sensor in a watertank filled with degassed water, once with the phantom in the way, once without. The attenuation coefficient then was determined by calculating the log-difference between the spectra of the measured and Fourier transformed signals. As the Figures 3.7, 3.8, and 3.9 show, it was possible to determine the functional relationship between the concentration of aluminium oxide and the resulting attenuation coefficient. Basically the connection is linear as function of mass concentration, with more variance resulting from phantoms produced using aluminium oxide 3 µm. This is attributed to the greater deviation from macroscopic homogeneity due to bigger grain size. Finally a recipe was derived to produce phantoms mimicking a wide range of breast tissue types of sufficient quality. The produced phantoms can then be used to verify that the methods to estimate the acoustic properties lead to unbiased results. Another way for testing the estimators is by using ultrasound simulations, that are deemed essential for the generation of a database of annotated, high resolution ultrasound images. Therefore the second part of the thesis is dedicated to the production of ultrasound (B- mode) images. Two open-source software, Field II and k-Wave, were tested for suitability of simulating breast tissues considering all their acoustic properties. Field II soon turned out to not accomodate the requirements, like spatially varying speed of sound, attenuation coefficient or mass density, as its simulation principle relys on the spatial impulse response and a linear propagation medium and was therefore not further considered. k-Wave on the other hand, designed to solve the full-blown nonlinear differential equations governing sound propagation, is able accomodate all these requirements. In using k-Wave its limitations soon became apparent by exhibiting numerical stability is- sues. Therefore a study was performed using different simulation settings, trying to figure out what the trigger for these numerical instabilites was. As it turned out k-Wave automatically selects a time step size, meaning in how small a time increment the sound field is reevalu- ated. Unfortunately for some settings parameters right on the edge of numerical stability are automatically selected presumably to result in as fast a computation as possible. Subsection 4.4.4 is dedicated to give a guideline on how to achieve numerically stable simulations. To guarantee that the simulations, besides numerical stability, lead to also physically correct results, videos were produced by putting together frames of plane plots, which were saved to

96 disk space during the simulation process, demonstrating the sound wave propagating through the medium. It turned out that the sound wave propagates exactly how it is supposed to through the medium. But as the simulations from Subsection 4.4.2 indicate, visibly correct simulations do not always guarantee numerical stability nor physical correctness. To also check for numerically and physically correct results necessary for the subsequent testing of acoustic parameter estimators (see also my continuing on work [53], not included here due to the limited time frame for a master thesis) the estimation of the acoustic attenuation coeﬃcient was checked for correctness by comparing the simulation parameters to estimation results of the same parameter. Concluding k-Wave turns out to provide a viable simulation environment, providing physically correct results for the generation of a 3D ultrasound image database to be used in deep learning in breast cancer research. The only drawback is the excessive simulation time requiring enormous processing power only provided by cluster computers and/or a GPU-accelerated standard computer.

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104 Appendices

105 Appendix A

Program listings

In this Appendix the MATLAB files written and used for the calculation of the attenuation coefficient of the produced tissue-mimicking phantoms as described in Chapter 3 are presented. And also the MATLAB files used to produce the simulations using the k-Wave toolbox and the generation of the video, demonstrating the sound wave propagating through the simulated medium, whos few frames are displayed in Figures 4.16, 4.17 and 4.18. Fur- thermore the attached DVD contains all the documentation in LATEX, all images generated, the movies generated and all software developed during the course of this work.

A.1 File for the calculation of the attenuation coeﬃcient of tissue-mimicking phantoms

This MATLAB file was used, to calculate the attenuation coefficients for the different phantoms made. First the data was loaded, after performing the Fourier transform of the signal with the phantom in the way and on the signal without the phantom in the way, the log- difference could be calculated between the two spectra, which led to the attenuation of the phantom.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Measurement of the attenuation coefficient of tissue-mimicking phantoms %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% datafiles1{1} = ’Method: 1 cycle’; datafiles1{2} = ’Frequency: 5 MHz’;

106 datafiles1{3} = ’Concentration: Aluminium Oxide 0.3e-6 m: ...’; % adjust datafiles1{4} = ’data1: Signal with Phantom’; datafiles1{5} = ’data2: Reference Signal’; datafiles1{6} = ’Output: Attenuation coefficient slope’;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Loading data data1 = csvread(’Cycle1Data1.csv’,16,0); %loading the reference data data2 = csvread(’Cycle1Data2.csv’,16,0); %loading data with phantom %Plot data figure; plot(data1(:,1),data1(:,2)); % plot reference signal hold on; plot(data2(:,1),data2(:,2)); % plot of signal with phantom legend(’Reference Signal’,’Phantom Signal’); xlabel(’Time’); ylabel(’Sound pressure’); %Crop data [m i] = max(data1(:,2)); CD1 = [data1(:,1) data1(:,2) data2(:,2)]; CD1 = CD1(i-5000:i+5000,:); figure; plot(CD1(:,1),CD1(:,2)); hold on; plot(CD1(:,1),CD1(:,3)); legend(’Reference Signal’,’Phantom Signal’); xlabel(’Time’); ylabel(’Sound pressure’);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%downsample and cropping again according to the specific frequency used %therefore calculate the time period T, which indicates the time for 1 cylce downsample = 50; %downsample, for the spectrum we don’t need that much ,→ sampling points fg=5e6; T = 1/fg; indexlength = 2*round(T*10^9)+200;

107 index = zeros(2,indexlength); if(round(indexlength/downsample)-indexlength/downsample<0) inddown = round( ,→ indexlength/downsample)+1; else inddown = round(indexlength/downsample); end; t = zeros(indexlength,2); thelp = zeros(indexlength,1); data = zeros(indexlength,2); datahelp = zeros(indexlength,1); datanew = zeros(inddown,2); data_filtered = zeros(inddown,2); tnew = zeros(inddown,2); nfft = 1024; %transform length, fft ignores the remaining signal values past ,→ the nth entry, should be power of 2 Y1 = zeros(nfft,2); f1 = zeros(nfft,2); P1 = zeros(nfft,2); Y1_filtered = zeros(nfft,2); P1_filtered = zeros(nfft,2); for l = 1:1:2 [m i] = max(CD1(:,l+1)); index(l,1:indexlength) = i-indexlength/2:i+indexlength/2-1; %crop signal, so ,→ we have just the pulse, so the spectrum is less noise t(:,l) = CD1(index(l,1:indexlength),1)*1e6; %first column indicates time, ,→ times 1e6 to get MHz data(:,l) = CD1(index(l,1:indexlength),l+1); %second column is the amplitude ,→ data datahelp(1:length(data)) = data(:,l); datahelp(1:inddown) = datahelp(1:downsample:length(data(:,l))); datanew(1:inddown,l) = datahelp(1:inddown); datanew(:,l) = datanew(:,l) - mean(datanew(:,l)); %always subtract the mean ,→ so there is no DC signale in the spectrum thelp(1:length(data)) = t(:,l); thelp(1:inddown) = thelp(1:downsample:length(t(:,l))); tnew(1:inddown,l) = thelp(1:inddown); figure; plot(tnew(:,l),datanew(:,l))

108 xlabel(’time/us’) Ts = CD1(2,1)-CD1(1,1); fs = 1/Ts/downsample; % in MHz low_freq = 1.5e6; high_freq = 7.5e6; transition_freq = 0.4e6; BP_freq=[low_freq low_freq+transition_freq high_freq-transition_freq ,→ high_freq]; [data_filtered(:,l) fre_res w]= band_pass(datanew(:,l),fs,BP_freq); % ,→ applying bandpass filter figure; subplot(211) plot(w/pi*fs/2/1e6,20*log10(abs(fre_res))); ylabel(’Magnitude/dB’); subplot(212) plot(w/pi*fs/2/1e6,unwrap(angle(fre_res))); xlabel(’Frequency/MHz’) ylabel(’Phase’) Y1(:,l) = fftshift(fft(datanew(:,l),nfft)); %Fast Fourier Transform f1(:,l) = ((-(nfft-1)/2:(nfft-1)/2))*fs/nfft/1e6; %to get frequency scale P1(:,l) = 10*log10(abs(Y1(:,l))); Y1_filtered(:,l) = fftshift(fft(data_filtered(:,l),nfft)); %Fast Fourier ,→ Transform P1_filtered(:,l) = 10*log10(abs(Y1_filtered(:,l)))+eps; %to get amplitude ,→ spectrum (Attention: log10(10)=1, log(10)=2.3026) %eps: Floating-point relative accuracy, returns the distance %from 1.0 to the next larger double-precision number, that is, 2e-52 figure; plot(f1(nfft/2:end,2),P1_filtered(nfft/2:end,2)); hold on; plot(f1(nfft/2:end,1),P1_filtered(nfft/2:end,1)); %overlay the two amplitude ,→ spectra for measurement legend(’Phantom’,’Referenz’); xlabel(’Frequency in MHz’); ylabel(’Amplitude in dB’); title(’Spectra filtered’);

109 i = find(f1(:,1)> 3.5); indexsp = i(1):i(1)+169; figure; plot(f1(indexsp,2),P1_filtered(indexsp,2)); hold on; plot(f1(indexsp,1),P1_filtered(indexsp,1)); %overlay the two spectra legend(’Phantom filtered’,’Reference filtered’); xlabel(’Frequency in MHz’); ylabel(’Amplitude in dB’); title(’Cropped Spectra’); end;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d = 2; %phantom thickness 2 cm alpha = -20/d*log10(abs(Y1(indexsp,2))./abs(Y1(indexsp,1))); %attenuation ,→ coefficient in dB/cm alpha_filtered = -20/d*log10(abs(Y1_filtered(indexsp,2))./abs(Y1_filtered( ,→ indexsp,1))); %attenuation coefficient in dB/cm figure; plot(f1(indexsp,1),alpha_filtered,’-*’) xlabel(’Frequency in MHz’); ylabel(’Difference in Amplitude in dB/cm’); title(’Attenuation coefficient slope Al2O3 0.3um: 130%’); xlim([3.5 7]); hold on; %linear line through alpha p = polyfit(f1(indexsp,1),alpha,1); f = polyval(p,f1(indexsp,1)); p_filtered = polyfit(f1(indexsp,1),alpha_filtered,1); f_filtered = polyval(p_filtered,f1(indexsp,1)); plot(f1(indexsp,2),f_filtered,’-.’) legend(’1 pulse’,’1 pulse linear approximation’,’Location’,’Southeast’);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

110 Function band_pass.m

This MATLAB function ﬁlters the signals only allowing frequencies between 1.5 MHz and 7.5 MHz to pass.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [filtered_p,h,w] = band_pass(p,Fs,F) % function filtered_p = band_pass(p,Fs,F) % filter specification A=[0 1 0]; % band type: 0=’stop’, 1=’pass’ dev=[1e-3 1e-3 1e-3]; % ripple/attenuation spec [M,Wn,beta,typ]= kaiserord(F,A,dev,Fs); % window parameters b=fir1(M,Wn,typ,kaiser(M+1,beta),’noscale’); % filter design [h,w] = freqz(b); % filtering sz=size(p); filt_delay=round((length(b)-1)/2); filtered_p=filter(b,1,[p; zeros([filt_delay sz(2:end)])],[],1); % correcting the delay filtered_p=filtered_p((filt_delay+1):end,:,:,:); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A.2 k-Wave simulation ﬁles

These ﬁles are intended to provide one of the environments used to produce the simulations presented in Chapter 4. Including the function to generate the mentioned video, demonstrating the sound wave propagating through a deﬁned medium.

A.2.1 Simulation ﬁle

By running this file, the simulation will be executed. First the four input structures were defined, being the kgrid, the medium, the source and the sensor. A diagnostic ultrasound transducer was used concurrently as source and sensor. At the end of the file the function

111 kspaceFirstOrder3D_video_v3 is called, a function provided by k-Wave and altered to save the plane plots need for the generation of the video.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clearvars; % simulation settings DATA_CAST = ’single’; % set to ’single’ or ’gpuArray-single’ to speed up ,→ computations RUN_SIMULATION = true; % set to false to reload previous results instead of ,→ running simulation % set the size of the perfectly matched layer (PML) pml_x_size = 20; % [grid points] pml_y_size = 20; % [grid points] pml_z_size = 15; % [grid points] % set total number of grid points not including the PML sc = 1; Nx = 648/sc - 2*pml_x_size; % [grid points] Ny = 810/sc - 2*pml_y_size; % [grid points] Nz = 108/sc - 2*pml_z_size; % [grid points] % set desired grid size in the x-direction not including the PML x = 30.4e-3; % [m] % calculate the spacing between the grid points dx = x / Nx; % [m] dy = dx; % [m] dz = dx; % [m] % create the k-space grid kgrid = kWaveGrid(Nx, dx, Ny, dy, Nz, dz);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEFINE THE MEDIUM PARAMETERS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% define the properties of the propagation medium c0 = 1630; % [m/s] % note: this sound speed is % called c_ref in the k-wave manual

112 rho0 = 1000; % [kg/m^3] medium.alpha_power = 1.01; % create the time array t_end = (Nx * dx) * 2.2 / c0; % [s] kgrid.makeTime(3000, [], t_end);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEFINE THE INPUT SIGNAL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% define properties of the input signal source_strength = 5e5; % [Pa] according to Kathy tone_burst_freq = 5e6 / sc; % [Hz] tone_burst_cycles = 3; % create the input signal using toneBurst input_signal = toneBurst(1/kgrid.dt, tone_burst_freq, tone_burst_cycles); % scale the source magnitude by the source_strength divided by the % impedance (the source is assigned to the particle velocity) input_signal = (source_strength ./ (c0 * rho0)) .* input_signal;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEFINE THE ULTRASOUND TRANSDUCER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% define the physical properties of the phased array transducer transducer.number_elements = 128 / sc; % total number of transducer elements transducer.element_width = 5; % width of each element [grid points] transducer.element_length = 70 / sc; % length of each element [grid points] transducer.element_spacing = 1; % spacing (kerf width) between the elements ,→ [grid points] % calculate the width of the transducer in grid points transducer_width = transducer.number_elements * transducer.element_width ... + (transducer.number_elements - 1) * transducer.element_spacing; % use this to position the transducer in the middle of the computational ,→ grid transducer.position = round([1, Ny/2 - transducer_width/2, Nz/2 - transducer

113 ,→ .element_length/2]); % properties used to derive the beamforming delays transducer.sound_speed = c0; % sound speed [m/s] transducer.focus_distance = 35e-3; % focus distance [m] transducer.elevation_focus_distance = 35e-3; % focus distance in the ,→ elevation plane [m] transducer.steering_angle = 0; % steering angle [degrees] transducer.steering_angle_max = 13; % maximum steering angle [degrees] % apodization transducer.transmit_apodization = ’Hanning’; transducer.receive_apodization = ’Rectangular’; % define the transducer elements that are currently active transducer.active_elements = ones(transducer.number_elements, 1); % append input signal used to drive the transducer transducer.input_signal = input_signal; % create the transducer using the defined settings transducer = kWaveTransducer(kgrid, transducer); % print out transducer properties transducer.properties;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEFINE THE MEDIUM PROPERTIES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cw = 1540; % m/s rhow = 1000; % kg/m^3 % aw = 2.8; % in dB/(MHz^y cm) should give 5 dB/cm (@ 5 MHz) and 3 cm are ,→ simulated aw = 2.9; % in dB/(MHz^y cm) should give 5 dB/cm (@ 5 MHz) and 3 cm are ,→ simulated sound_speed_map = cw * ones(Nx,Ny,Nz); % so, since no variance is assumed density_map = rhow * ones(Nx,Ny,Nz); % here, there should be no scattering alpha_map = aw * ones(Nx,Ny,Nz); % at all in water!! % place discrete scatters at multiples of 5 mm @ grid spacing of 0.05 mm in ,→ x % of size (2, 3, Nz) along center line of grid

114 xscat=[100,200,300,400,500]; % 0.5 mm, 1.0 mm, 1.5 mm, 2.0 mm, 2.5 mm yscat=[Ny/2-1,Ny/2,Ny/2+1]; % 3 voxels wide in y-direction zscat=1:Nz; % all planes alog z-axis sound_speed_map(xscat,yscat,zscat) = 1350; density_map (xscat,yscat,zscat) = 1350/1.5; % copy stuff over into medium! medium.sound_speed = sound_speed_map; medium.density = density_map; medium.alpha_coeff = alpha_map; % now clear intermediate variables to save memory clear region region3d sound_speed_map density_map alpha_map save single_prec_sim29.mat % save workspace so that a very same simulation ,→ can be executed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % RUN THE SIMULATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% range of steering angles to test steering_angles = 0; % preallocate the storage number_scan_lines = length(steering_angles); scan_lines = zeros(number_scan_lines, kgrid.Nt); % set the input settings input_args = {... ’PMLInside’, false, ’PMLSize’, [pml_x_size, pml_y_size, pml_z_size], ... ’DataCast’, DATA_CAST, ’DataRecast’, false, ’PlotSim’, false,... ’LogScale’, false, ’PlotFreq’, 5, ... ’RecordMovie’, true, ’MovieName’, ’20200511_29_movie’}; % run the simulation if set to true, otherwise, load previous results if RUN_SIMULATION % loop through the range of angles to test for angle_index = 1:number_scan_lines % update the command line status disp(’’); disp([’Computing scan line ’ num2str(angle_index) ’ of ’ num2str(

115 ,→ number_scan_lines)]); % update the current steering angle transducer.steering_angle = steering_angles(angle_index); sensor_data = kspaceFirstOrder3D_video_v3(kgrid, medium, transducer, ,→ transducer, input_args{:}); % extract the scan line from the sensor data save_sensor = transducer.scan_line(sensor_data); end else disp(’end’); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A.2.2 The function kspaceFirstOrder3D_video_v3

MATLAB can access this function if it is saved in the folder, where the orignal function kspaceFirstOrder3D is saved. Here only the parts which were altered are presented.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % PREPARE VISUALISATIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% pre-compute suitable axes scaling factor if plot_layout || plot_sim [x_sc, scale, prefix] = scaleSI(max([kgrid.x_vec; kgrid.y_vec; kgrid.z_vec]) ,→ ); %#ok end % run subscript to plot the simulation layout if ’PlotLayout’ is set to true if plot_layout % kspaceFirstOrder_plotLayout; %commented out end % initialise the figure used for animation if ’PlotSim’ is set to ’true’ if plot_sim % kspaceFirstOrder_initialiseFigureWindow; %commented out

116 end % initialise movie parameters if ’RecordMovie’ is set to ’true’ if record_movie % kspaceFirstOrder_initialiseMovieParameters; %commented out end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % LOOP THROUGH TIME STEPS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% plot data if required if plot_sim && (rem(t_index, plot_freq) == 0 || t_index == 1 || t_index == ,→ index_end) % update progress bar % waitbar(t_index / kgrid.Nt, pbar); %commented out % drawnow; %commented out % ensure p is cast as a CPU variable and remove the PML from the % plot if required if strcmp(data_cast, ’gpuArray’) p_plot = double(gather(p(x1:x2, y1:y2, z1:z2))); else p_plot = double(p(x1:x2, y1:y2, z1:z2)); end % update plot scale if set to automatic or log if plot_scale_auto || plot_scale_log kspaceFirstOrder_adjustPlotScale; end % add display mask onto plot if strcmp(display_mask, ’default’) p_plot(sensor.mask(x1:x2, y1:y2, z1:z2) ~= 0) = plot_scale(2); elseif ~strcmp(display_mask, ’off’) p_plot(display_mask(x1:x2, y1:y2, z1:z2) ~= 0) = plot_scale(2); end % update plot planeplot(scale * kgrid.x_vec(x1:x2), scale * kgrid.y_vec(y1:y2), scale * ,→ kgrid.z_vec(z1:z2), p_plot, ’’,...

117 plot_scale, prefix, COLOR_MAP, t_index, kgrid.dt); % save movie frames if required if record_movie % set background color to white % set(gcf, ’Color’, [1, 1, 1]); %commented out % save the movie frame % writeVideo(video_obj, getframe(gcf)); %commented out end % update variable used for timing variable to exclude the first % time step if plotting is enabled if t_index == 1 loop_start_time = clock; end end end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % CLEAN UP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function planeplot(x_vec, y_vec, z_vec, data, data_title, plot_scale, prefix ,→ , color_map, t_index, delta_t) % Subfunction to produce a plot of a three-dimensional matrix through the % three central planes. % subplot(2, 2, 1); %commented out % imagesc(y_vec, x_vec, squeeze(data(:, :, round(end/2))), plot_scale); % ,→ commented out % title([data_title ’x-y plane’]); %commented out % axis image; %commented out % subplot(2, 2, 2); %commented out % imagesc(z_vec, x_vec, squeeze(data(:, round(end/2), :)), plot_scale); % ,→ commented out % title(’x-z plane’); %commented out % axis image; %commented out % xlabel([’(All axes in ’ prefix ’m)’]); %commented out % subplot(2, 2, 3); %commented out % imagesc(z_vec, y_vec, squeeze(data(round(end/2), :, :)), plot_scale); %

118 ,→ commented out % title(’y-z plane’); %commented out % axis image; %commented out % colormap(color_map); %commented out % drawnow; %commented out %added to save data for video plot_data = data(:, :, round(end/2)); prime =’’’’; save_string=[’save (’,prime,’data_plane_’,int2str(t_index),prime,’,’,prime, ,→ ’plot_data’,prime,’);’] eval(save_string);

A.2.3 File to generate the video

This ﬁle needs to be saved and then executed right where the plane plot data were saved to disk during the simulation process. By calling this ﬁle an mpeg-4 video will be generated.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% prime=’’’’; fig = figure(’Position’,[40, 40, 1150, 900]); % set width of plot to have x- ,→ and y-scale approx. equal y_axis = 0.03*(0:1023); x_axis = 0.03*(-256:255); % make y-axis symmetric about zero imax = 7290; % the number of time steps, differs when using different ,→ simulation settings v = VideoWriter(’video.mp4’,’MPEG-4’); % set a filename v.Quality = 50; % set quality of video open(v); for i=5:10:imax, % do not use all the data to avoid out of memory error! load_string=[’load (’,prime,’data_plane_’,int2str(i),’.mat’,prime,’);’]; eval(load_string); mm = max(max(plot_data)); subplot(1,2,1); imagesc(x_axis,y_axis,(abs(hilbert(plot_data)))); title([’complex envelope, time in ns: ’,int2str(round(i*5.625))]) subplot(1,2,2);

119 imagesc(x_axis,y_axis,log10(abs(plot_data)+0.01)); % add 0.01 to prevent ,→ logs of zeros title([’log10(abs(wave)), time in ns: ’,int2str(round(i*5.625)),’; p_{max}= ,→ ’, int2str(round(mm)), ’ Pa’]) caxis([-2,5]); drawnow; frame = getframe(fig); writeVideo(v,frame); end; close(v); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

120 Appendix B

Decimal to IEEE-754 conversion example

The IEEE-754 (a technical standard for floating-point arithmetic) single-precision floating- point format is a computer number format, usually occupying 32 bits (binary32) in computer memory. The IEEE-754 standard specifies a binary32 as having, 1 bit representing the sign, 8 bits representing the exponent (in offset binary, bias = 127) and 24 (23 there of explicitly stored) bits representing the mantissa [73].

Figure B.1: Binary representation of a 32-bit ﬂoating-point number. The hidden 1. is never saved but still used in all calculations, therefore numbers (mantissa) always have to be normalized, i.e. they must always be set to 1.xxx.

This appendix is used to give an example of how a GPU with single-precision adds up two numbers (here the decimal numbers 1000000 and 0.01). Showing the limitation in precision. First the two decimal numbers need to be converted (using the chinese remainder theorem [74]) into the binary ﬁxed-point number, as represented in Fig. B.1, starting oﬀ with the decimal number 1000000: One can see, that 19 times the decimal number 1000000 can be divided by 2 until it results in a number starting with 1, which as indicated before is hidden.

121 1000000 : 2 = 500000 ... 500000 : 2 = 250000 15.2587890625 : 2 = 7.62939453125 250000 : 2 = 125000 7.62939453125 : 2 = 3.814697265625 ... 3.814697265625 : 2 = 1.9073486328125 hidden 1.0 + 0.9073486328125

1.9073486328125 · 219 = 1000000

The exponent 19 needs to be combined with the bias, being 127 in the IEEE-754 format (19+127=146), and again converted into a binary ﬁxed-point number:

146 : 2 = 73 0 remainder (Least-Signiﬁcant-Bit) 73 : 2 = 36 1 remainder 36 : 2 = 18 0 remainder 18 : 2 = 9 0 remainder .. .. 2 : 2 = 1 0 remainder 1 : 2 = 0 1 remainder 0 : 2 = 0 0 remainder (Most-Signiﬁcant-Bit)

The remaining decimal places (1.9073486328125 - 1.0 (hidden) = 0.9073486328125) must now be converted into a binary form, by multiplying by 21, substracting 1 if greater than 1.0, until the result of the multiplication is 0:

0.9073486328125 · 2 = 1.814697265625 1 remainder (Most-Signiﬁcant-Bit) 0.814697265625 · 2 = 1.62939453125 1 remainder 0.62939453125 · 2 = 1.2587890625 1 remainder 0.2587890625 · 2 = 0.517578125 0 remainder 0.517578125 · 2 = 1.03515625 1 remainder . . 0.5 · 2 = 1.0 1 remainder 0 · 2 = 0 0 remainder

The sign is positiv, therefore the sign bit is set to 0, with this the decimal number 1000000 can now be represented by: The hidden bit is left out.

1Actually again dividing, but this time by negative powers of 2.

122 Now for the decimal number 0.01:

0.01 · 2 = 0.02 0.02 · 2 = 0.04 0.04 · 2 = 0.08 0.08 · 2 = 0.16 0.16 · 2 = 0.32 0.32 · 2 = 0.64 0.64 · 2 = 1.28 hidden 1.0 + 0.28

The decimal number 0.01 was multiplied 7 times by 2 until resulting in a number greater than 1, the hidden 1.0.

1.28 · 2−7 = 0.01

The exponent −7 (negative sign due to the decimal number being smaller than 1.0) needs to be combined with the bias, 127 leading to 127-7=120), and again converted into binary ﬁxed-point number:

120 : 2 = 60 0 remainder (Least-Signiﬁcant-Bit) 60 : 2 = 30 0 remainder 30 : 2 = 15 0 remainder 15 : 2 = 7 1 remainder 7 : 2 = 3 1 remainder 3 : 2 = 1 1 remainder 1 : 2 = 0 1 remainder 0 : 2 = 0 0 remainder (Most-Signiﬁcant-Bit)

The remaining decimal places (1.28 - 1.0 (hidden) = 0.28) must now be converted into a binary form, by multiplying by 2, substracting 1 if greater than 1.0, until the result of the multiplication is 0: The multiplications by 2 results in a periodic pattern, which indicates that after 23 bits the rest needs to be rounded up. The sign bit is again 0, with this the decimal number 0.01 can be represented by:

123 0.28 · 2 = 0.56 0 remainder (Most-Signiﬁcant-Bit) 0.56 · 2 = 1.12 1 remainder 0.12 · 2 = 0.24 0 remainder 0.24 · 2 = 0.48 0 remainder 0.48 · 2 = 0.96 0 remainder .. .. 0.32 · 2 = 0.64 0 remainder 0.64 · 2 = 1.28 1 remainder 0.28 · 2 = 0.56 0 remainder (Least-Signiﬁcant-Bit)

Now the real problem arises, as the summation of two numbers in IEEE 754 format requires for the two numbers to have the same exponent. Denormalisation is necessary. If one were to increase the exponent of the smaller number by 1, the mantissa including the hidden bit would shift by 1:

But the exponent for the decimal numbers 1000000 and 0.01 differs by 146 - 120 = 26, which means the mantissa would need to be shifted 26 bits, but only 23 are significant. Therefore the mantissa ends up being all zeros, leading to a summation result of 1000000.0 instead of 1000000.01. This example demonstrates, that for high values of source strength of the transducer and high attenuation coefficients, that can lead to concurrent sound pressure levels many magnitudes apart (like 1 MPa excitation pressure and downrange only a few Pa), the precision might be not sufficient resulting in possible numerical instabilities.

124

SWORN DECLARATION

I hereby declare under oath that the submitted Master’s Thesis has been written solely by me without any third-party assistance, information other than provided sources or aids have not been used and those used have been fully documented. Sources for literal, paraphrased and cited quotes have been accurately credited.

The submitted document here present is identical to the electronically submitted text document.

Linz, 07/14/2020