3D printed acoustically transparent pressure sensor for breast cancer detection

sensor for breast cancer detection D.C. (Dan) Sîrboiu

BSC ASSIGNMENT

Committee:

prof. dr. ir. G.J.M. Krijnen M.K. Welleweerd, MSc dr. ir. T.H.J. Vaneker July 2020

032RaM2020 Robotics and Mechatronics

EEMCS University of Twente P.O. Box 217 7500 AE Enschede

The Netherlands

This bachelor thesis investigates if acoustically transparent pressure sensors can be produced

using 3D printing. The sensor is designed in SOLIDWORKS and then 3D printed using Ulti-

maker S5. The material used is Layfomm-40. The sensor needs to be acoustically transparent

in order for ultrasound to pass through. The behaviour of the bottom surface of the sensor is

modeled using Macaulay’s method for deflection of beams. A Finite Element Analysis in Ansys

is made in order to validate the mathematical model. Based on the simulation results it is not

feasible to model the bottom surface of the sensor using Macaulay’s method. Moreover, testing

for acoustic transparency is done on the structure. This shows that the material has potential

when it comes to acoustic transparency.

Acknowledgements

I would like to thank my supervisors Gijs Krijnen and Marcel Welleweerd for all their guidance, support, feedback and help. Additionally, Remco Sanders and Sander Smits who helped me with ordering filament and technical issues. Also, Martijn Schouten, Gerjan Wolterink, and Dimitrios Kosmas who provided help with the 3D printer and the simulation program. Addi- tionally, I would like to thank Jolanda Boelema-Kaufmann who provided help in many aspects.

Lastly, I am grateful to my family who supported me constantly throughout this thesis.

List of symbols

This page contains an overview of the symbols used in this work, their associated meanings, and their units.

E Young’s modulus (Pa) I mass moment of inertia (m

⁴

) R

A

reaction force at point A (N) R

_B

reaction force at point B (N) M

A

reaction moment at point A (N m) M

B

reaction moment at point B (N m)

M

_x

reaction moment at cross-section x (N m) l original length of the beam (m)

y deflection of the beam (m)

x length of the beam at cross-section (m) w uniform distributed load (N m

⁻¹

) σ sum of all forces/moments

y

⁰

first time derivative of the deflection (m s

⁻¹

) y

⁰⁰

second time derivative of the deflection (m/s

²

) y

max

maximum deflection of the beam (m)

b width of the beam (m)

F force (N)

Contents

Acknowledgements ii

List of Symbols iii

1 Introduction 1

1.1 Context . . . . 1

1.2 State of the art . . . . 1

1.3 Research goal . . . . 3

1.4 Approach . . . . 3

2 Literature study 4 2.1 Ultrasound and its applications . . . . 4

2.2 Sound propagation . . . . 5

2.3 Sound intensity and attenuation . . . . 8

2.4 Breast cancer diagnosis procedures . . . . 8

2.5 3D printing technology . . . . 13

2.6 Sensing . . . . 14

3 Sensor design 20 3.1 Design and manufacturing . . . . 20

3.2 Post processing and acoustic transparency testing . . . . 24

3.3 Sensing mechanism . . . . 25

4 Model and simulations 26 4.1 Mathematical model . . . . 26

4.2 Finite Element Analysis . . . . 30

4.3 Model discussion and conclusion . . . . 31

4.4 Conclusions . . . . 33

5 Fabrication and measurements 34 5.1 Fabrication . . . . 34

5.2 Measurement results . . . . 36

5.3 Conclusions . . . . 39

6 Overall discussion and conclusion 40 6.1 Recommendations for future work . . . . 40

A Matlab code 42

A.1 Code used for mathematical model . . . . 42

A.2 Code used for Ansys simulations . . . . 44

A.3 Code for the first comparison . . . . 45

A.4 Code used for the second comparison . . . . 46

A.5 Code used for the third comparison . . . . 47

A.6 Code used for the fourth comparison . . . . 47

B Additional figures from Ansys simulation 49

Bibliography 52

1 Introduction

1.1 Context

Breast cancer is the most common type of cancer among women. Each year 2.1 million women are diagnosed with breast cancer [1]. For a woman, the chances of developing cancer range between 12% and 14% [2]. Detection of a tumor in the early stage is crucial in order to in- crease the chances of survival and breast cancer outcomes [1]. Currently, two early strategies are used for breast cancer: early diagnosis and screening. Early diagnosis can be achieved by the patient through manual palpation. When the woman finds a hard mass in the breast she can visit a general practitioner who directs her to a hospital for further investigation [2]. The screening strategy consists of a mammography or a clinical breast exam [1]. In order to get a good picture during mammography, the breast is positioned between two plates which are pressed against each other in order to reduce movement and flatten the breast. Then an image of the internal part of the breast is captured using low-energy X-rays. When abnormalities are detected, the patient can be sent to an ultrasound screening in order to distinguish between solid masses and cysts. The advantages of ultrasound is that it is cheap and generally avail- able [3]. However, it has its shortcomings too, such as tumors smaller than 5 mm cannot be detected using ultrasound. Another drawback is that the outcomes are operator dependent, which means that depending on the radiologist, it is improbable that every woman will benefit from the same level of expertise [2]. Another method used for detecting cancer is Magnetic Res- onance Imaging(MRI). The advantages of MRI is that it has higher resolution and is less reliant on the operator. Using contrast agents, MRI can detect lesions which would have never been detected using ultrasound and mammography [3].

1.2 State of the art

At the moment, there are a few sensors on the market, based on different sensing technolo- gies. One device that is used is called SureTouch. This device uses a flexible capacitive sensing array, which consists of 192 high resolution sensors. The sensor is four times more effective than manual palpation and can detect lesions as small as 5 mm. In the picture below, Sure- Touch device can be seen performing an analysis on a phantom breast which has a hard volume within [4].

Figure 1.1: SureTouch device detecting a hard lesion within the breast. On the screen, the pressure distribution measured by the sensor array is shown [4]

Another capacitive sensor is proposed by Fernandes and Jiang [5]. They suggest a novel capaci- tive sensing scheme along with a structured dielectric layer constructed using soft-lithography.

The sensor uses a novel normal and shear force capacitive sensing structure with a patterned dielectric layer. A change in capacitance is detected when one or more changes of the following are recorded: a change in the dielectric thickness, a change in the overlap area between the top and bottom electrode or a change in the permittivity of the dielectric layer. The schematic of the sensor is shown in figure 1.2.

Figure 1.2: Schematic representation of the capacitive sensor [5]

Another approach is proposed by Pandya et al. [6]. They propose a sensor based on piezore- sistive microcantilevers (PMCs). By performing indentations on micro-scale tissue specimens, detection and characterization of benign and diseased breast tissue can be achieved. During a biopsy procedure an indentation is made within the breast tissue sample. The PMC is inserted within this sample as can be seen in figure 1.3.

Figure 1.3: Schematic representation of the PMC within breast tissue sample [6]

The detection is done by measuring the change in the spring constant of the PMC when a force of known value is applied using AFM cantilever. It is assumed that the AFM tip behaves as a simple harmonic oscillator.

A more recent study was done by Wang, in 2017 [7]. Another type of sensor that is currently

in use for early-stage breast cancer detection is a piezoelectric biosensor. This sensor consists

of a piezoelectric microcantilever(PEM) with antibodies that bind to HER2 gene. HER2 stands for human epidermal growth factor receptor 2 and represents a gene that can play a role in the development of breast cancer [8]. The PEM is used to monitor the presence of the HER2 gene in the human blood.

1.3 Research goal

The goal of this work is to investigate if acoustically transparent pressure sensors can be created using 3D printing technology. It is desired to achieve a less invasive and more accurate sensing mechanism compared to the methods that are currently in use. The focus will be on the sensing principle which will be based on a combination of ultrasound imaging and a sort of mechanical imaging.

1.4 Approach

The approach is to use a mathematical model of the sensor and investigate how the bottom surface of the sensor deforms under different loading conditions. Measurements will be carried out to test if the obtained samples are acoustically transparent. Moreover, the stiffness of the structures will be tested to see if they are suitable for the application. The sensors that are going to be created will be made of one material material. A couple of printing parameters will be changed, one at a time, and their impact on the stiffness of the structure will be studied.

Measurement results will be used to check whether or not the selected material can be used

for manufacturing a sensor. Moreover, measurement results are going to be compared with

simulation results for model validation.

2 Literature study

2.1 Ultrasound and its applications

Ultrasound is the sound at frequencies above 20kHz, which is the upper limit of the human hearing range. Ultrasound technology is used in various fields such as science, industry and healthcare. Ultrasound applications can be categorized according to the intensity of the sound wave that is required in a process in low-intensity and high-intensity. On one hand, low- intensity, high-frequency applications, in general, focus on the transmission of energy through a medium in order to get information about that medium. This is done without altering the composition of the medium in which the propagation takes place. The power that is used for this kind of applications is usually in the milliwatt range while the frequencies that are used are high, mainly in the range of megahertz or higher. On the other hand, high-intensity, low- frequency applications, i. e. power ultrasonics, aim to produce a permanent change in the medium, or its contents, through which the sound wave travels. This change can affect the chemical, physical or biological properties of the medium. The power required for power ul- trasonics ranges from tens of watts up to thousands of watts, according to the intensity level that is specific to the application. The operating frequencies for high-intensity applications usually range from 20 kHz up to 100kHz. Figure 2.1 shows an overview of the frequency range that is used for ultrasound applications together with some application examples.

Figure 2.1: Frequency range of a number of ultrasound applications and acoustic phenomena [9]

High-intensity ultrasound applications are based on the effects of nonlinear phenomena which arise from the high-intensity waves. The presence of nonlinear phenomena results in different mechanisms that may be activated by ultrasonic energy and they include heat, mechanical rupture, diffusion, chemical effects, friction, and interface instability and they produce a permanent change in the medium. The effects that are produced include wave distortion, cavitation, acoustic saturation, and radiation pressure. One effect that can be found in applications that involve liquid is cavitation. When liquids are sonicated at high intensities, sound waves that propagate into the liquid are formed out of high-pressure (compression) and low-pressure (rarefaction) cycles and their rates depend on the frequency of the sound wave. During the low-pressure cycle, the sound waves create small gas bubbles (cavities).

The bubbles are formed out of dissolved gasses found in the liquid or the vapor of the liquid.

These bubbles then grow until they reach a volume which can not absorb energy anymore and

then they violently collapse producing high local temperatures and stresses, with very short

lifetimes [10]. This effect is used in sonochemistry and ultrasound cleaning. More applications

in which high-intensity ultrasound is used are welding of both metals and polymers, drying,

degassing, cleaning, emulsification, dispersion, noninvasive surgery, machining of brittle ma-

terials, and formation and processing of nanomaterials [11] [12]. The noninvasive surgery application is also known as high-intensity ultrasound (HIFU) treatment. An example of this is using high-intensity focused ultrasound to destroy the kidney stones found within a kidney while the healthy tissue that is found in the path is not affected by the sound waves [13].

Among the low-intensity applications are nondestructive characterization and testing of materials, such as crack detection in pipes or ceramics. Other applications are medical imag- ing, livestock and meat grading as well as sensing in industrial processes. There is one more application that usually requires significant power but is classified as a low-intensity appli- cation. That is SONAR, which stands for Sound Navigation And Ranging. Main uses of sonar include navigation and fish localization [11].

2.2 Sound propagation

Sound waves are waves made out of alternating cycles of compression and expansion (rar- efaction) and they can be classified as longitudinal waves, which propagate in all materials, or transverse waves, which propagate only in solids. This classification is done by looking at particle vibrations. If the particles vibrate in a direction parallel to the direction of propagation then the wave is considered to be longitudinal. Conversely, if the particles vibrate in a direc- tion perpendicular to the propagation direction, the wave is considered to be transverse. The parameters of an ultrasound wave are wavelength, frequency, pressure, velocity, power, and intensity. The last two parameters are measures of the “strength” of the sound wave. As the ul- trasound wave travels through a medium, it gets attenuated. This attenuation is due to various factors such as absorption and deflection( which includes the processes of reflection, refrac- tion, and scattering) of the acoustic energy of the beam but also due to beam divergence [14].

The subject of sound attenuation will be discussed in more detail later on in section 2.4. During

medical ultrasound imaging, the sound travels through the tissue and reflects when it encoun-

ters another medium with different acoustic properties. The reflected sound waves are called

echos and they are used to construct the medical image [15]. The magnitude of the echo de-

pends on 2 factors: the orientation of the reflecting surface with respect to the sound beam and

the difference in acoustic impedance between tissues(mediums). The magnitude of the ultra-

sound pulse has to be high enough such that the echo will not die out before reaching back to

the transducer. If that is the case, no image will be generated by the ultrasound probe. From

an acoustical point of view, soft tissues such as fat and muscle behave like liquids and the only

tissue through which transverse sound waves can propagate are compact bones. Propagation

of the wave is done at a certain sound velocity, which is a vector quantity and is characterized

by a direction and a magnitude. If the direction of propagation of the wave is not specified, the

term that is used is speed. Acoustic speed is influenced by how rigid or how compressible the

material is. Propagation speed is high if the material is rigid(solid), as metal or bones, or low

if the material is less rigid(fluid), as water or air. For example, propagation speed in water and

soft tissue is greater than in air and lower than in bone or metals. This can be seen in Table 2.1,

where a classification of ultrasound speeds according to tissue type and density is done. Also,

it is important to mention that sound speed varies also with temperature [14].

Tissue/Material Sound speed (m s

⁻¹

) Density (g/cm

³

)

Air 343 0.001225

Water 1480 1

Blood 1575 1.055

Fat 1450 0.95

Liver 1590 1.06

Kidney 1570 1.05

Brain 1550 1.03

Heart 1570 1.045

Muscle(along the fiber) 1575 1.065

Muscle(across the fiber) 1590 1.065

Skin 1730 1.15

Bone axial(longitudinal wave) 4080 1.9

Bone axial(shear waves) 2800 1.9

Table 2.1: Speed of sound in different types of tissue and their corresponding densities [16]

2.2.1 Sound waves at interfaces

So far we have discussed the propagation of a sound wave through a homogeneous medium.

Now we will extend our discussion to sound propagation from one medium to a different medium. When a sound wave arrives at an interface, it behaves exactly like a light wave, mean- ing that when a wave encounters an interface part of the energy of the beam is reflected and the remaining energy is further transmitted through the second medium. Also, as for light, the law of reflection states that the angle of reflection θ

r

is the same as the angle of incidence θ

i

.This can be seen below in Figure 2.2.

Figure 2.2: Reflection and transmission of a sound wave when it meets an interface [14]

The direction of the reflected wave, the echo, is dependent on the orientation of the interface to the sound wave. Its amplitude depends on a couple of factors: orientation of the interface with respect to the incident sound wave and the relative difference in acoustic impedance between the tissues on either side of the interface. The deviation of the transmitted wave from the incident beam is called refraction. The angle of transmittance θ

t

is defined by Snell’s law to be:

sin(θ

t

)/ sin(θ

i

) = V

1

/V

2

(2.1)

where θ

t

is the transmittance angle, θ

i

is the incident angle, V

₁

is the velocity of the sound for the incident medium, and V

2

is the velocity of the sound for the transmitting medium. After refraction of the wave, the wave will propagate through the second medium until it dies out, is attenuated, or until meets another interface where the previous behavior repeats itself, part of the sound wave is reflected and part of it is transmitted further. When a second interface is reached, echoes are generated from those positions. Those positions can be away from the initial direction of the incident beam which leads to artifacts in the ultrasound image. In gen- eral, in order to increase the magnitude of the echos, the angle of incidence can be increased.

However, this does not apply to pulse-echo imaging, where the same transducer that produces the ultrasound pulse is used also for receiving the echos. In the case of pulse-echo imaging, the only echos that can be detected are the ones traveling in the direction of the transducer. So in this case, the magnitude of the echos is maximum when the sound beam is perpendicular to the surface (angle of incidence is zero) and these echos are used for the construction of the ultrasound image. The amount of energy that is reflected at an interface is given by the ampli- tude reflection coefficient r which is defined as the ratio of the amplitude of the echo to that of the incident wave and given by the formula:

r = (Z

2

− Z

1

)/(Z

2

+ Z

1

) (2.2)

where Z

1

is the acoustic impedance of tissue 1 and Z

2

is the acoustic impedance of tissue 2.

Acoustic impedance is defined as a measure of the resistance that a material offers to the pas- sage of ultrasound waves and it is expressed in Rayls (kg/m

²

s). The acoustic impedance of a medium is defined as the product between the density of that medium and the speed of sound in that medium [14]. A table with the acoustic impedances of various tissues is presented below.

Tissue Acoustic impedance (*10

⁸

kg/m

²

s)

Air 0.000420

Water 1.48

Blood 1.66

Fat 1.38

Liver 1.69

Kidney 1.65

Brain 1.60

Heart 1.64

Muscle(along the fiber) 1.68

Muscle(across the fiber) 1.69

Skin 1.99

Bone axial(longitudinal wave) 7.75

Bone axial(shear waves) 5.32

Table 2.2: Acoustic impedances of various tissue types [16]

It is observed that the mismatch between the acoustic impedance of air and skin is very high.

This leads to almost a total reflection of the sound wave when the skin is encountered which is undesired because in this situation no sound will penetrate the skin and hence, no

information about the tissues under the skin can be collected. In order to avoid total reflection

an ultrasound coupling gel is used. The gel ensures an air-free sound path and will match the

acoustic impedance of the skin; that is, the difference between the acoustic impedance of the

skin and the gel is small. This results in the sound wave being transmitted through the body in

order to collect the desired information about the region of interest. As the sound travels

through the body, it encounters other surfaces(tissues) which generate reflections of the

sound wave. Depending on how big surface irregularities are, surface reflection can be either specular or diffuse. The surface is seen as smooth and the reflected wave is considered to be specular or mirror like when the irregularities are small compared to the wavelength of the ultrasound beam. Conversely, if the irregularities are large compared to the wavelength, the surface is seen as rough and the echo, being diffuse, is scattered in all directions. The

parenchyma of tissues is made out of cell and tissue elements, that are small when compared to the wavelength of the sound wave, which will lead to the scattering of the sound wave [14].

2.3 Sound intensity and attenuation

In order for a sound wave to travel through the tissue, it needs energy in order to overcome the internal friction intrinsic to any material. As the sound wave propagates, it transfers a portion of its energy to the tissue. This decrease in the acoustic energy per unit distance traveled by a sound wave is known as attenuation. There are three factors which influence the attenuation of the sound wave: divergence, deflection, and absorption.

Divergence of a sound wave spreads the beam over a large area thereby reducing the intensity along the beam axis, which will result in a decreased magnitude of the echos that are used to construct the medical image. Deflection includes the processes of reflection, refraction, and scattering which also reduce the sound intensity.

In the body, the greatest cause of attenuation is absorption. During the propagation of the sound beam, acoustic energy is transferred from the sound wave to the tissue. This energy is ultimately degraded into heat. One factor that influences absorption is the frequency of the ultrasound beam. The relation between absorption and frequency is linear for most tissues within the body. Both absorption and attenuation values are expressed in units of decibels per centimeter per megahertz. The decibel is given by the equation below [14]:

dB = 10 ∗ log

₁₀

µ I

I

0

¶

(2.3) where I is the intensity and I

0

is the incident or reference intensity. There is another way of expressing attenuation; that is, in terms of the half-value layer. The half-value layer represents the distance required for a sound wave to travel for its intensity to be reduced in half. Table 2.3 gives an overview of the attenuation coefficients and half-value layers for various materials [14].

Material Attenuation (dB cm

⁻¹

MHz

⁻¹

) Half-Value Layer (cm)

Water 0.0022 1360.0

Blood 0.15 20.0

Soft tissue 0.75 4.0

Air 7.50 0.4

Bone 15.00 0.2

Table 2.3: Attenuation and Half-Value Layer for various materials [14].

2.4 Breast cancer diagnosis procedures

Breast cancer is the most common cancer in women worldwide [17]. Because of this, proce-

dures of detecting breast cancer were developed over the years. Nowadays there are various

imaging modalities that are used to examine the breast tissue. Medical imaging is the process

and technique of making a visual map of the internal structure of the body. Medical imaging is

used by radiologists to diagnose and treat diseases in the body by using images. In the following

subsections multiple methods will be discussed such that a clear idea on imaging procedures

can be formed.

2.4.1 Manual palpation

Manual palpation is based on qualitative assessment of the low-frequency stiffness of tissue.

The human finger acts as a pressure sensor that detects changes in mechanical properties of the tissue, such as stiffness, when the breast is pressed by the human hand. This method is dependent on perception as well as lump size and breast characteristics. For dense breasts, hard inclusions are more difficult to detect. Another parameter that can raise difficulties in detection is the depth of the inclusion. If the lesion is located well below the skin, close to the rib cage, changes in pressure patterns can hardly be observed which can lead to a wrong diagnostic. Manual palpation is executed by a trained physician called radiologist, but because this method is dependent on perception and the skill of the radiologist, more advanced and reliable methods are needed in order to get a more accurate diagnostic. In the figure 2.3 one can see the Young’s moduli of different tissue types [18]. The measurements were carried out with constant quasi-static loading at a frequency of 0.1 Hz, but at different levels of static preloading:

5% and 20%.

Figure 2.3: Young’s moduli of different types of breast tissue [19]

2.4.2 Mammography

Mammography is probably one of the most frequently used screening methods world-wide.

During the procedure, a dedicated mammography unit is used. This unit consists of parallel

plates that are used to compress the breast such that it evens out the thickness of the breast tis-

sue. This is needed because during the mammography X-rays are used. It is more beneficial to

have a decreased breast thickness in order to reduce the amount of scattered radiation, which

degrades image quality, as well as the amount of the required radiation dose. Because X-rays

can pass easier through some tissues compared to others, cancerous lumps and calcifications

can be easily differentiated from normal breast tissue by appearing brighter on the mammo-

gram. Mammography can detect ductal carcinoma in situ, which is ductal cancer, before it can

be detected through palpation. Studies show that mammography screening reduces mortal-

ity by 25%. This method, as all the others, has some drawbacks. It can miss cancer in dense

breasted women and the images close to the rib cage present poor quality. Also, this procedure

is dependent on the training of the radiologist who interprets the mammogram, which can lead

to wrong diagnostic [4].

2.4.3 Magnetic Resonance Imaging(MRI)

Magnetic Resonance Imaging(MRI) uses strong magnets in order to polarize and excite the nu- cleus of the hydrogen atom, which is a proton and is present in all human tissue. The image is built by making use of the magnetic properties of the nuclei. The nucleus of the hydrogen atom is capable of absorbing and emitting magnetic energy created by the magnets. The signal emitted by the nuclei is then detected and spatially encoded and an image is formed [4]. Image contrast is dependent on the mobility of the hydrogen atoms. Moreover, the magnetic environ- ment of the atoms in fat and water contributes to the measured signal. The measured signal determines the brightness of tissues in image [20]. MRI, being the best evaluation modality, is proposed as a screening method for patients who present low mammogram sensitivity: mainly young women or dense breasted women [4]. MRI also presents some drawbacks. One small is that it cannot detect calcifications, which are an indicator of cancer [21]. A major drawback is that is expensive as it requires expensive apparatus, a well trained radiologist that can interpret and operate the device and high running costs [4].

2.4.4 Ultrasound(US)

Unlike MRI and mammography, ultrasound does not use radiation to create the image, but sound waves. Ultrasound is usually used in combination with mammography in order to assess uncertain breast masses. The radiologist moves the ultrasound probe over the region of inter- est and interprets the image. Hence this method is very user dependent. Ultrasound presents high sensitivity, but because it is subjective it can lead to high rates of false positives [22]. An ultrasound device uses focused sound waves in order to create the image by measuring the reflected sound waves. The magnitude of the reflected waves depends on the difference in acoustic impedance Z of the tissues found within the breast. The acoustic impedance Z is a function of speed of sound in tissue and density of tissue. In the table below it can be seen what the sound velocities are depending on tissue type. Based on the reflected waves, an image of the underlying tissue is created. Sound waves are created by either a piezoelectric transducer or nowadays by capacitive transducers [23] that is mounted at the tip of the probe, and are af- terwards transmitted into the body. Depending on the acoustic impedance of different tissues, a fraction of the waves are reflected back and recorded by the transducer. In figure 2.4 sound velocities for different types of breast tissue are presented.

Figure 2.4: Sound velocities for different types of breast tissue [4]

2.4.5 Biopsy

In order to execute a biopsy, the location of the cancerous tissue must be known. For this

purpose, an imaging technique must be used in order to locate the cancerous tissue and usually

the techniques that are used are either ultrasound or MRI. During biopsy, the radiologist makes

a small incision in the breast through which a small needle is inserted. After the insertion of

the needle in the right position some samples of the suspicious tissue are collected. Afterwards these samples are analyzed in the lab and a diagnosis is given.

MRI guided biopsy

During an MRI guided biopsy, MRI images are used to confirm that the needle was inserted in the right position, in the cancerous tissue. Because MRI is not a real time imaging modality, the radiologist needs a few tries until the needle is inserted precisely in the region of interest. After every needle insertion an MRI image is taken to see if the needle was inserted in the region of interest. This is repeated several times until the right location is reached. Therefore the procedure takes a considerable amount of time. The procedure can be seen in the figure below:

Figure 2.5: Left: Set-up during an MRI guided biopsy. Right: Offline visual confirmation of an MRI guided biopsy [4]

US guided biopsy

During an ultrasound guided biopsy, real time imaging of the breast is used in order to confirm that the needle was inserted in the region of interest. This method, in contrast to MRI, is a real time procedure so the time required to perform the biopsy reduces significantly. This is also the tremendous advantage of this procedure. The radiologist places the ultrasound probe on the breast and identifies the region of interest. Then, a small incision is made in the breast through which the insertion of the core needle occurs. While the needle is inserted it is tracked down by the radiologist on the real time image and steered such that it ends up in the correct position for biopsy. The downsides of this procedure are the visual limitations of ultrasound which are small lesions and especially in dense breasted women these can barely be localized.

In the figure below the procedure can be seen:

2.4.6 Elasticity imaging

Elasticity imaging (EI) is the method that allows visualisation through quantitative evaluation of the mechanical properties of tissue like elasticity and viscosity. These parameters are very sensitive to structural changes in tissue due to pathological changes. The change in Young’s modulus of tissue during the development of tumors could reach thousands of percent [20].

The principle of EI is to produce stress by static or dynamic loading of the breast and measure the strain caused by the loading using MRI or ultrasound. Therefore EI is also referred to as

“strain imaging” and from this the local elastic moduli are reconstructed based on the theory of elasticity. The principle of strain imaging is depicted below in Figure 2.7.

2.4.7 Mechanical imaging

Mechanical imaging (MI) which is also known as “tactile imaging” or “stress imaging” repre-

sents another version of EI. While EI relies on applying stress and measuring the strain response

Figure 2.6: Left: Set-up during ultrasound guided biopsy. Right: Real time confirmation of ultrasound guided biopsy [4].

Figure 2.7: Strain imaging principle [24]

of the tissue, MI relies on applying static or dynamic displacements and measuring the surface stress responses of the tissue. When the breast that contains cancerous tissue is compressed, the hard inclusion can be identified by looking at the stress distribution at the surface of the skin. The pressure pattern response indicates spatial distribution of softer and tougher areas of the compressed region and provides information about the presence, location and dimension of the lesion. The resulting stress profile from the presence of a hard inclusion is highly depen- dent on the size, shape, depth and hardness of the inclusion. Essentially, MI closely imitates manual palpation, but in this case the stress distribution is measured by a pressure sensor and not by the mechanoreceptors in our fingers. Because MI can be used independent of other imaging techniques like US or MRI, it is viewed as the most cost-effective imaging modality. MI however has its drawbacks. When it comes to sensitivity and specificity MI can be compared to other imaging techniques like MRI or US in terms of efficiency. But when it comes to detecting the location and size of the hard inclusions, MI is able to detect inclusions which are located to maximum 35 mm under the skin. This is still better than manual palpation which has a thresh- old value of 27 mm but it is less than what MRI or US can detect.

There are two differences that differentiate MI from US and MRI [25]:

1. Principle of sensing:

MRI and US use the data of the strain induced by static or dynamic loading whereas MI

uses the stress patterns of the compressed region to reconstruct the internal mechanical

structure of the tissue by using an algorithm.

2. Operating conditions:

During an examination MI is performed much easier compared to both US and MRI.

Also, regarding the financial aspect MI is much cheaper compared to US and MRI which is the most expensive of the three. After measuring the stress patterns the elastogram is constructed. This is accomplished by using the inversion method which basically re- constructs the elastic parameters based on the measured stress pattern. For this recon- struction algorithms are used. They are based in general on the theory of elasticity. In the figure below the principle of MI is shown by using a pressure sensor array based on piezoresistive sensing [26]:

Figure 2.8: Left: Sensor used for detection. Right: Breast with tactile map plane and landmarks for orientation reference [26].

Mechanical imaging is capable of distinguishing between different types of hard inclusions.

Also, MI is capable of differentiating between fibroadenoma, fibrosis, carcinomas, cysts and other conditions [25].

2.5 3D printing technology

3D printing is the best known representative of additive manufacturing. Additive manufactur- ing is a type of non-conventional manufacturing method where physical models are produced with the help of Computer-Aided-Design (CAD) data building them layer by layer. The major advantages that 3D printing brings to the table, compared to other manufacturing technolo- gies, are minimum loss of material and the absence of fixing devices which result in a cheaper and faster production processes. Due to these advantages 3D printing is also known as rapid prototyping (RP). Literature shows that in the present there are over 20 RP processes that are divided into 3 main categories. Depending on the raw material that is used during the pro- cess they are divided in liquid-based-, powder-based-, and solid-based systems. Out of these categories, Stereolithography(SLA), Fused Deposition Modelling(FDM), and Laminated Object Manufacturing(LOM) are the ones which are used most often. In this paper, FDM is used for producing the sensor so it will be discussed in more detail below. Other production methods are beyond the scope of this assignment so they will not be discussed here. FDM presents sim- plified operations and this is why it is one of the widely spread methods. The raw materials that are used during FDM printing are thermoplastic materials in the form of solid filament so this process falls under solid-based methods. The filament is heated to a molten state by using electricity and is extruded through a nozzle that moves in X − Y direction. Nowadays there are also printers which build the physical model by moving the build plate in the X − Y direction or the nozzle in the X − Z direction but the advantages and disadvantages of these methods will not be discussed here as they are beyond the scope of this thesis. The extrusion head is controlled by computer software and deposits a thin layer of material on the build platform.

The temperature of the build platform is kept low in order for the liquid thermoplastic to be

able to cool down and solidify. After the first layer is printed, the build platform moves down a distance of one layer thickness and the second layer is deposited on top of the first one by the extrusion head [27]. This process can be seen in the figure below.

Figure 2.9: Schematic showing how FDM machine works [28].

The process is using 2 materials, one being the building material and the other support mate- rial(sacrificial material) which are being fed into the extrusion heads that deposits the melted filaments on the building platform. The downward displacement of the build plate defines the layer thickness. After the model is printed, depending on the application, the model goes through a post-processing phase which enhances either mechanical or other properties. Be- low, a figure of the RP stages is shown.

Figure 2.10: Rapid Prototyping stages [27]

2.6 Sensing

As humans, we use our sensory receptors of vision, taste, smell, touch and sound as a means

of experiencing and interacting with the environment around us. Utilizing one or a multitude

of these senses, we discover new and unstructured environments. For example, when we ma-

nipulate an egg, the size, shape, temperature, texture and color are transmitted to the brain

through the sensory receptors. If the applied force is too high the egg will break. Conversely,

if the applied force is too little the egg will slip. Thus, an exact force needs to be applied in or-

der to keep the egg intact. If this task was to be performed by a robotic manipulator, sensory inputs identical to those possessed by humans are crucial for providing the needed feedback information in order for the robot to explore and interact with objects. This paper focuses on pressure sensing which is information acquired through the sense of touch. For that reason, a more detailed description of the tactile sensing techniques will be given from now on.

2.6.1 Capacitive tactile sensors

A capacitive sensor is made out of two parallel plates and a dielectric material sandwiched be- tween them. For parallel plate capacitors, capacitance is expressed as C = A · ²

0

· ²

r

/d . Where C is the capacitance, A is the overlapping area of the two plates, ²

0

is the permittivity of free space,

²

r

is the relative permittivity of the dielectric material and d is the distance between the plates.

These sensors have high dynamic range, good spatial resolution and exhibit good frequency response. Because these sensors are very vulnerable to noise. This happens especially when they are put in mesh configurations because of cross-talk noise, field interactions and fringing capacitance. For this reason they require complex electronics to filter this noise out [29]. The figure below represents the working principle of a capacitive tactile sensor, where one can see how the voltage potential is created:

Figure 2.11: Schematic illustration of capacitive pressure sensors [30]

2.6.2 Piezoresistive tactile sensors

This type of sensors are usually made out of a sensitive pressure element that changes its re- sistance when a force is applied to it. The voltage-current characteristic of a simple resistive element can be expressed using Ohm’s law: V = I ·R, where V is the voltage, I is the current and R is the electric resistance of the element. Usually either the voltage or current is fixed and the change in resistance is measured which is a consequence of a change in either current or volt- age. In general, this resistive element is either a conductive rubber, an elastomer or conductive ink which is sensitive to pressure. Because the change in resistance can be quantified easily they need less electronics to be manufactured and integrated. They are also less susceptible to noise so they work better in mesh configurations compared to capacitive sensors. This is due to the fact that no cross talk or field interactions are present. However, the drawback of piezore- sistive sensors is that they suffer from hysteresis so they have a lower frequency response [29].

Figure 2.12 represents the working principle of a piezoresistive pressure sensor:

2.6.3 Piezoelectric tactile sensors

Another transduction method that is used for tactile sensing is piezoelectricity. The voltage

produced in response to applied mechanical stress is called piezoelectricity [30] and is derived

from oriented, permanent dipoles in the material. Different materials, like certain crystals and

Figure 2.12: Schematic illustration of a piezoresistive sensor [30]

some ceramics, produce a voltage potential when the crystal lattice is compressed or elon- gated. The crystals sensitivity is dependent on its structure, which allows it to differentiate between transverse, longitudinal and shear forces. The voltage, V, that is generated is directly proportional to the applied strain, force or pressure. Piezoelectric sensors exhibit an excel- lent high-frequency response, which makes them a perfect candidate for measuring vibrations.

However, they have some limitations. Due to the large internal resistance they cannot measure static forces and also they are limited when it comes to measuring dynamic forces. The internal impedance and the dielectric constant of the piezoelectric film determine the time constant at which the charge that is generated within the crystal decays. During sensor design, impedance matching between interface electronics and crystal must be achieved such that the response of the device is accurate [29]. Figure 2.13 below represents the working principle of a piezoelectric pressure sensor:

Figure 2.13: Schematic illustration of a piezoelectric sensor [30]

2.6.4 Inductive tactile sensors

Inductive sensors are based on a primary coil that induces a magnetic field which is detected by a secondary coil called sense coil. By modulating the collective inductance between the coils, this, in turn, modulates the phase and amplitude of the voltage sensed by the secondary coil.

For example, by modifying the length of an iron core in the case of a linear variable differential transformer. These sensors exhibit a very large dynamic range and often a robust construction.

However because they are bulky in size they have a low spatial resolution when arrayed. An-

other limitation that comes from their mechanical nature is low repeatability. This is because

the coils do not always come back to the same position between readings. Another drawback

comes from the complex electronics that has to be used in order to demodulate the alternat- ing voltage amplitude that comes from the alternating current that is induced in the primary coil [29]. Figure 2.14 shows the working principle of an inductive tactile sensor.

Figure 2.14: Schematic illustration of an inductive sensor [31]

2.6.5 Optoelectronic tactile sensors

Optoelectronic sensors use a lightsource, a transduction medium and a photodetector. The photodetector can be either a camera or a photodiode. In general, transduction takes place when modifications in the tactile medium modulate the transmission or reflectance intensity, or the spectrum of the light source. Modulation occurs when the applied force is varied. The advantages of these sensors are that they exhibit high spatial resolution and they are unaf- fected by lower frequency electromagnetic interference which is generated by electrical sys- tems. However, they also have drawbacks which are the lack of robustness and size. Also they demand a considerable amount of processing power but in exchange they have a wide range of frequency responses [29].

2.6.6 Strain gauges

Strain gauges are low cost sensors that are widely used in various applications. Strain gauges are sensors that measure mechanical strain, usually by a change in resistance [29]. Consider- ing their required lifetime, they are usually connected to the substrate by using special glues. In general they are very sensitive and very prone to changes in temperature and moisture. In order to overcome these drawbacks, strain gauges are used in Wheatstone bridge configurations [32].

If they are overloaded, they can break and the strain gauge cannot be recovered. 3D printed strain gauges in doped TPU have a strong hysteresis because of their mechanical nature and also they present a non-linear response. The advantage of strain gauges is that they have been widely used for a long time so the best applications for their use are well established [29]. In Figure 2.15 a diagram of a strain gauge is shown. The dimensions in the diagram are in mil- limeters, where 1 is the metallic cover, 2 is the metallic web, 3 represents force concentrators, 4 represents transversal walls, 5 represents longitudinal walls, 6 and 7 are the strain gauges and 8 represents the lateral opening.

2.6.7 Multi-component tactile sensors

Combining multiple different transducers in one sensor to overcome the drawbacks of each

device was also looked into by a few researchers [34], [35]. For instance, polyvinylidene fluoride,

PVDF, film is only able to measure dynamic loadings and has a limited ability to detect slip, but

it is not capable of measuring static loadings. Overcoming these limitations can be done by

Figure 2.15: Schematic illustration of a strain gauge sensor [33]

adding a capacitive or resistive element which results in a sensor that is able to detect both slip and static forces [34], [35]. When considering applications for which flexibility or large area coverage is required, fluid based tactile sensors are used most often. They combine different intrinsic methods to carry out the task [36], [37]. Figure 2.16 shows the working principle of a multi-component tactile sensor.

Figure 2.16: Schematic illustration of a multi-component sensor [37]

In the figure below and overview of the transduction techniques is presented, together with

their advantages and disadvantages:

Figure 2.17: Transduction techniques and their advantages and disadvantages [29]

3 Sensor design

The goal of the sensor is to detect cancerous tissue within the breast. The sensing method is based on a combination of ultrasound imaging and sort of mechanical imaging. Hence, its novelty. By using ultrasound the radiologist will be capable of seeing what is found within the breast. This is the reason why the sensor needs to be acoustically transparent. While the ra- diologist moves the ultrasound probe over the breast, he will apply a pressure with the probe.

Because the cancerous tissue is up to 5 times stiffer than the fatty tissue within the breast, the sensor is made in such a way that its mechanical impedance will be inbetween the mechanical impedance of the breast and mechanical impedance of the cancerous tissue. The explanation why that is will be given in section 3.3 where a detailed description of the sensing mechanism is given. When cancerous tissue is found underneath the sensor, it will deform showing the radiologist what is the location of the cancerous tissue. By using the ultrasound image, the ra- diologist can see what is the position of the cancer and depth within the breast and can decide how to proceed further

3.1 Design and manufacturing

In this section, the design and manufacturing of the sensor will be discussed. The sensor was designed using SOLIDWORKS. The first design of the sensor can be seen in figure 3.1, where the top surface of the sensor and the hinges are transparent for visualisation purposes.

Figure 3.1: First design of the sensor

For the second iteration of the design, the hinges were removed. This is can be seen in figure 3.2.

The hinges were removed because the goal of the thesis is to design a sensor but the clamping mechanism to the probe is not of paramount interest.

Figure 3.2: Second design

For the third iteration, the shape of the strain gauges inside the sensor(the red components) was changed from circular to rectangular. This was done due to the cross contamination of the two materials that were used for printing the sensor. The third design can be seen in figure 3.3.

Figure 3.3: Third design

The fourth design of the sensor can be seen in figure 3.4. The difference between the third and fourth iteration is the thickness of the top and bottom surfaces, which are thinner for the forth iteration. This was needed because during post processing when the sensor was submerged in water, the PVA inside the top and bottom surfaces was not able to leave the structure and little blobs of not well dissolved PVA were forming inside the structure. This would result in unwanted surface irregularities. Moreover, when pressure would be applied on the sensor, the blobs of PVA would break. This destroys the surfaces and creates the possibility for air to leak inside the structure, which can distort the ultrasound image as it can be seen in the next chap- ter.

Figure 3.4: Fourth design

For the fifth iteration, it was decided to remove the strain gauges. This was done because of

the hard read-out procedure and interpretation of the results. Instead of four strain gauges, a

hollow box was created. This can be seen in figure 3.5. This design choice was made because it

was decided to replace strain gauge sensing with a sort of mechanical imaging sensing, which

gives the novelty of this assignment. The sensing mechanism will be discussed in more detail

in section 3.3.

Figure 3.5: Fifth design

Because the structure is hollow, support material was needed in order to print the top surface of the sensor. The support material used was PVA. This support material needs to be replaced from the sensor with water. For this purpose, a sixth iteration of the design was created and it can be seen in figure 3.6.

Figure 3.6: Sixth design

Due to the difficulty of removing the support, it was decided to place the holes of the sensor on the sides instead of the top surface. Furthermore, an increase in height of the structure was implemented. This was done in order to increase the space for deflection of the bottom surface as well as the rigidity of the sensor. The seventh iteration can be seen in figure 3.7.

Figure 3.7: Seventh design

Due to layer delamination of the corners that was occurring during post processing, an eighth iteration was made. It consists of rounding the edges of the sensor and it can be seen in figure 3.8.

Figure 3.8: Eighth design

Due to the printing position of the sensor, which can be seen later on in this section, the excess material around the holes was removed, resulting in the final design of the sensor, which is presented below in figure 3.9 using 3 different views and a cross section. The dimensions of the SOLIDWORKS model are: L=26 mm, W =15 mm, and H =7.8 mm, where: L is length, W is width, and H is height.

(a) Top view (b) Front view

(c) Isometric view (d) Cross section

Figure 3.9: Final design of the sensor presented in 3 different views and a cross section

The sensor is printed using FDM 3D printing. The printer that is used is an Ultimaker S5.

The material used for printing the sensor is called Layfomm-40. This is an experimental

material which is composed of a combination of Polyvinyl Alcohol(PVA) and Thermoplastic polyurethane(TPU). The sensor is printed in the upright position as shown in figure 3.10.

(a) Slicer view and printing direc- tion

(b) Cross section

Figure 3.10: View of the sensor in Cura slicer

The reason why the sensor is printed in the upright position and not as shown in figure 3.11, is due to the layer delamination of the structure during post processing.

(a) Slicer view and printing direc- tion

(b) Cross section

Figure 3.11: View of the sensor in Cura slicer

Another reason is due to the support material (PVA) inside the structure, which needs to dis- solve in water in order to be extracted. This results in longer times for post processing as well as a more tedious process which will be discussed in the next section. Moreover, printing hor- izontally results in breaking of the structure during post processing. This is happening due to the expansion of the enclosed support material. Printing the sensor in the upright position does not require any support material. This is due to the small gap (7.8 mm) between the walls which can be easily covered by the nozzle, so upright printing simplifies and reduces the time of post processing.

3.2 Post processing and acoustic transparency testing

After the print is finished, the sensor is filled with tap water by using a syringe. The holes were created for this purpose and also because the dissolved PVA inside the sensor needs to leave the structure. After the sensor is filled, it is submerged in warm tap water for 48 hours such that the PVA of the outer walls dissolves at the same time as the PVA of the inner walls. During these 48 hours the sensor is checked every 12 hours to see if the water inside is saturated with PVA.

If saturation occurred, the water is extracted using a syringe and it is replaced with fresh water.

Also, the water that the structure is submerged in is replaced every 12 hours to speed up the

process and prevent saturation. The reason why the sensor is printed this way is because layer

delamination occurs during the dissolving of the PVA if it is printed horizontally. After all the

PVA has dissolved, an expansion in the length of the sensor can be observed. This expansion

is around 100 % of the original length, so the length of the sensor has reached a new value of

L=52 mm. This matches the length of the ultrasound probe. The width of the sensor remains

unchanged (W =15 mm) and matches the width of the ultrasound probe as well. Also, the height of the sensor stays the same H =7.8 mm.

After all the PVA has dissolved, the structure becomes soft and spongy. Now, all the water is replaced by fresh tap water and the sensor is submerged again in water. Afterwards it is placed in a vacuum oven in order to take out all the air bubbles, that are left inside the structure. The next step is to seal the holes to ensure that no air is coming in anymore. For this purpose, the brim of the sensor and a soldering iron are used. The brim is cut in half and each half is placed over the two holes. The soldering iron is then used to melt the brim into the holes, resulting in sealing the holes.

Now that the structure is filled with water and no air is present inside, the sensor can be tested for ultrasound transparency. This is done by using a phantom that contains small balls inside.

First, the probe is placed directly on top of the phantom. Ultrasound gel is used between the phantom and the probe in order to remove any unwanted air bubbles that can affect the qual- ity of the ultrasound image. Then, an image is recorded. This image is the reference image.

Afterwards, the sensor is placed in-between the probe and the phantom. Ultrasound gel is ap- plied on both top and bottom of the sensor to remove the air bubbles and to achieve acoustic impedance matching. After this, the ultrasound image is recorded in order to be compared with the reference image. Next, pressure is applied on the sensor and an image is recorded again.

If the structure breaks and the sealed water leaks out and air leaks in, this can be seen on the ultrasound image.

3.3 Sensing mechanism

In this section, the sensing mechanism of the sensor will be elaborated. As stated previously, the sensor is filled with water. When pressure is applied on the sensor using the ultrasound probe and there is no cancerous tissue underneath the sensor the breast will deform slightly but the sensor will remain undeformed. Even though there is no cancerous tissue underneath the sensor, some artifacts can appear on the ultrasound image which result from other com- ponents of the breast. This is happening due to the different densities of these components which can reflect more or less of the ultrasound sent by the probe. When cancerous tissue is present underneath the sensor, both the breast and the sensor will deform slightly. Also, in the ultrasound image, an artifact will be present. This artifact will stand out from neighbouring parts due to the high density of the cancerous tissue.

The sensor is built in a way such that the acoustic impedance of it is matched with the breast

acoustic impedance. The mechanical impedance of the sensor has a value between the me-

chanical impedance of the fatty tissue of the breast and the mechanical impedance of the can-

cerous tissue. This is needed because if the mechanical impedance of the sensor is below the

mechanical impedance of the breast, when pressure will be applied on the sensor, the sen-

sor will deform before the breast deforms. This will result in a poor diagnosis. Conversely, if

the mechanical impedance of the sensor is above the mechanical impedance of the cancerous

lesion, when pressure is applied on the sensor, it will result in the deformation of the breast

but not the deformation of the sensor. This is happening because the sensor will push down

the lesion due to the increased stiffness. Therefore, the mechanical impedance of the sensor

needs to be between the mechanical impedance of the breast and cancerous tissue in order for

detection to occur properly.

4 Model and simulations

4.1 Mathematical model

The mathematical model that is used for modelling the sensor is the 1D model of a built-in beam. This model models the bottom surface of the sensor, which is getting in contact with the skin. It calculates what is the maximum deflection of the bottom surface when pressure is applied on the sensor using the ultrasound probe and when the sensor encounters cancerous tissue within the breast. The accuracy and the assumptions of the model will be discussed in more detail later on in this chapter. Now, the derivation of the deflection equation will be performed, using Macaulay’s method. In the end, the deflection equation will be presented together with an expression for the maximum deflection. Figure 4.1 shows a built in beam under 3 downwards point loads F

1

, P , F

2

and an upwards uniform distributed load(U.D.L.).

Point load P is the force due to the cancerous tissue and F

1

= F

2

= F are forces due to the fatty tissue of the breast. F

1

is located at a distance of l /4 from the left edge. P is located at a distance l /4 from F

1

and F

2

and F

2

at a distance of l /4 form the right edge.

F1 P F2

w

l/4 l/4 l/4 l/4

l

Figure 4.1: Schematic illustration of a built-in beam under upwards UDL, w, and 3 downwards point loads: F1, P, F2.

We start by drawing the free body diagram of the beam:

F1 P

l/4 l/4 l/4 F2

l/4

R

_A

R

_B

M

_B

M

_A

w

A

_X

A D C E B B

_X

Figure 4.2: Free body diagram of the beam

The governing equation used in Macaulay’s method is [33]:

M

_x

= E I y

⁰⁰

(x) (4.1)

where M

x

is the moment at a distance x from the left edge as shown in figure 4.3 , E is the Young’s modulus of the beam, I is the second area moment and y(x) is the deflection of the beam at point x.

In order to compute M

x

we use:

X M = 0 (4.2)

F1 P

l/4 l/4 l/4 F2

R _A M _A

w

M _X x

A D C E

Figure 4.3: Section of the beam at a distance x from the left edge

which results in the following expression:

M

_x

− R

A

x + M

A

− w x x 2 + F

1

µ x − l

4

¶ + P

µ x − l

2

¶ + F

2

µ x − 3l

4

¶

= 0 (4.3)

This gives forM

x

:

M

x

= R

A

x − M

A

+ w x

²

2 − F

1

µ x − l

4

¶

− P µ

x − l 2

¶

− F

2

µ x − 3l

4

¶

(4.4) Substituting equation 4.4 in equation 4.1 yields:

E I y

⁰⁰

(x) = R

A

x − M

A

+ w x

²

2 − F

1

µ x − l

4

¶

− P µ

x − l 2

¶

− F

2

µ x − 3l

4

¶

(4.5) Integrating equation 4.5 once with respect to x yields the slope equation:

E I y

⁰

(x) = E I θ(x) = R

A

x

²

2 − M

A

x + w x

³

6 − F

1

³ x −

^l₄

´

2

2 − P

³ x −

^l₂

´

2

2 − F

2

³ x −

^3l₄

´

2

2 +C

1

(4.6)

where F

1¡x−₄^l¢2

2

, P

^¡x−

l 2

¢2

2

, F

2¡x−^3l₄¢2

2

are the Macaulay’s terms which are integrated with respect to x −

^l₄

, x −

^l₂

and x −

^3l₄

and not x. Additionally, these terms should only be considered when x −

₄^l

, x −

₂^l

and x −

^3l₄

are positive. When they are negative, they should be neglected [33]. Integrating equation 4.6 once more yields the deflection equation:

E I y(x) = R

A

x

³

6 − M

A

x

²

2 + w x

⁴

24 − F

1

³ x −

^l₄

´

3

6 − P

³ x −

^l₂

´

3

6 − F

2

³

x −

^3l₄

´

3

6 +C

1

x +C

2

(4.7) The boundary conditions of the system are:

y(0) = y(l ) = y

⁰

(0) = y

⁰

(l ) = 0 (4.8)

Applying boundary conditions at end A gives the integration constants: at x = 0, y

⁰

(0) = 0 so C

1

= 0 and y(0) = 0 so C

2

= 0 which leads to the slope and deflection equations:

E I y

⁰

(x) = E I θ = R

A

x

²

2 − M

A

x + w x

³

6 − F

1

³ x −

₄^l

´

2

2 − P

³ x −

^l₂

´

2

2 − F

2

³ x −

^3l₄

´

2

2 (4.9)

E I y(x) = R

A

x

³

6 − M

A

x

²

2 + w x

⁴

24 − F

1

³ x −

^l₄

´

3

6 − P

³ x −

₂^l

´

3

6 − F

2

³ x −

^3l₄

´

3

6 (4.10)

Considering that the cancerous tissue is 3 times stiffer than the fatty tissue found within the breast we consider P = 3F . Applying the boundary conditions at end B yields the bending moment M

A

and the reaction force R

A

. At x = l , y

⁰

(l ) = 0 so:

0 = R

A

x

²

2 − M

A

x + w x

³

6 − F

³ x −

₄^l

´

2

2 − 3F

³ x −

₂^l

´

2

2 − F

³ x −

^3l₄

´

2

2 (4.11)

which gives:

M

_A

= R

A

l 2 + w l

²

6 − F 11l

16 (4.12)

At x = l , y(l ) = 0 so:

0 = R

A

x

³

6 − M

A

x

²

2 + w x

⁴

24 − F

1

³ x −

₄^l

´

3

6 − P

³ x −

₂^l

´

3

6 − F

2

³ x −

^3l₄

´

3

6 (4.13)

which gives:

R

A

= 5 2 F − w l

2 (4.14)

Substituting equation 4.14 into equation 4.12 yields the expression for M

A

:

M

A

= F 9l 16 − w l

²

12 (4.15)

For convenience, let us consider the free body diagram of the beam once more. In figure 4.4 A

_x

=B

x

=0, so they were left out.

F1 P

l/4 l/4 l/4 F2

l/4

R

_A

R

_B

M

_B

M

_A

w

A D C E B

Figure 4.4: Free body diagram of the beam without reactions A_xand B_x