High Frequency Ultrasound RF Time Series Analysis for Tissue Characterization

Transcription

1 High Frequency Ultrasound RF Time Series Analysis for Tissue Characterization by Mohsen Najafi Yazdi A thesis submitted to the Graduate Program in Electrical and Computer Engineering in conformity with the requirements for the degree of Master of Applied Science Queen s University Kingston, Ontario, Canada March 2012 Copyright c Mohsen Najafi Yazdi, 2012

2 Abstract Ultrasound-based tissue characterization has been an active field of cancer detection in the past decades. The main concept behind various techniques is that the returning ultrasound echoes carry tissue-dependent information that can be used to distinguish tissue types. Recently, a new paradigm for tissue typing has been proposed which uses ultrasound Radio Frequency (RF) echoes, recorded continuously from a fixed location of the tissue, to extract tissue-dependent information. This is hereafter referred to as RF time series. The source of tissue typing information in RF time series is not a well known concept in the literature. However, there are two main hypotheses that describe the informativeness of variations in RF time series. Such information could be partly due to heat induction as a result of consistent eradiation of tissue with ultrasound beams which results in a virtual displacement in RF echoes, and partly due to the acoustic radiation force of ultrasound beams resulting in micro-vibration inside tissue. In this thesis, we further investigate RF time series signals, collected at high frequencies, by analyzing the properties of the RF displacements. It will be shown that such displacements exhibit oscillatory behavior, emphasizing on the possible micro-vibrations inside tissue, as well as linear incremental trend, indicating the effect of heat absorbtion of tissue. i

3 The main focus of this thesis is to study the oscillatory behavior of RF displacements in order to extract tissue-dependent features based on which tissue classification is performed. Using various linear and nonlinear tools, we study the properties of such displacements in both frequency and time domain. Nonlinear analysis, based on the theory of dynamical systems, is used to study the dynamical and geometrical properties of RF displacements in the time domain. Using Support Vector Machine (SVM), different tissue typing experiments are performed to investigate the capability of the proposed features in tissue classification. It will be shown that the combination of such features can distinguish between different tissue types almost perfectly. In addition, a feature reduction algorithm, based on principle component analysis (PCA), is performed to reduce the number of features required for a successful tissue classification. ii

4 Acknowledgments Writing a thesis, by its own, is a procedure of putting one s ideas and contributions, on a scientific topic, into a structural, detailed but still easily followable piece of document. Although a thesis is named after the one who writes it, I believe that the ownership also belongs to those who made the path smooth ( differentiable!!! ) for this achievement. We all know that the concept path, in this context, is a multi-dimensional object. The variables in such a space varies from technical assistants to financial aids to supportive friendships and to many other aspects. I want to take this time to thank those who made different parts of this system working by meeting me in their offices, talking to me on phone, hanging out at Starbucks, and saying Hey, what s up buddy? How is your thesis going? at 2:30am in the lab. I really appreciate each and every single of these examples as they led me into an enjoyable state. A state where I am writing these few paragraphs, while having a smile on my face. A smile of accomplishment! iii

5 Contents Abstract Acknowledgments List of Tables List of Figures Glossary of Terms i iii vi viii xi Chapter 1: Introduction Motivation Objective Contributions Organization of Thesis Chapter 2: Background Overview Magnetic Resonance Imaging Ultrasound Imaging Ultrasound-based Tissue Typing Spectral Analysis Texture Analysis Elastography RF Time Series Analysis Summary Chapter 3: Methodology Overview Displacement Estimation Spline-based Time-delay Estimation Frequency Analysis and Feature Extraction iv

6 3.3.1 Fourier Analysis Spectral Fast Orthogonal Search Nonlinear Analysis and Feature Extraction Phase Space Reconstruction Lyapunov Exponent Correlation Dimension Recurrence Probability Density Entropy Testing for Nonlinearity Classification Support Vector Machines Feature Reduction Summary Chapter 4: Experiments and Results Overview Data Acquisition Data Acquired Using Single-element Transducer (Vevo770) Data Acquired Using Linear-Array Transducer (Vevo2100) Displacement Estimation Displacement Frequency Spectrums Feature Extraction Frequency Features Spectral FOS Features Phase Space Reconstruction Maximal Lyapounov Exponents Correlation Dimensions Recurrence Probability Density Entropies Testing for Nonlinearity Tissue Classification Features Summary Classification Experiments Non-Fourier Features Classification Results All Features Classification Results Feature Reduction Summary Chapter 5: Summary and Conclusion Summary of Contributions Future Work Bibliography 101 v

7 List of Tables 3.1 Different kinds of motions according to the maximal LE of a system Comparison of MLE of RF displacements of LE dataset, fps = 1000 Hz, and their corresponding surrogate datasets List of non-fourier features with their values (mean ± std) for different tissue types of dataset SE List of non-fourier features with their values (mean ± std) for different tissue types of dataset LA at frame rate f s = 100 Hz List of non-fourier features with their values (mean ± std) for different tissue types of dataset LA at frame rate f s = 1000 Hz One-way tissue classification accuracies (mean ± std) for different features performed on SE01 dataset (non-fourier features) One-way tissue classification accuracies (mean ± std) for different features performed on LA01 dataset; f s = 100 Hz (non-fourier features) One-way tissue classification accuracies (mean ± std) for different features performed on LA01 dataset; f s = 1000 Hz (non-fourier features) Two-way tissue classification accuracies (mean ± std) for different features performed on SE dataset (non-fourier features) vi

8 4.9 Two-way tissue classification accuracies (mean ± std) for different features performed on LA dataset; f s = 100 Hz (non-fourier features) Two-way tissue classification accuracies (mean ± std) for different features performed on LA dataset; f s = 1000 Hz (non-fourier features) One-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on SE01 dataset One-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on LA01 dataset; f s = 100 Hz One-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on LA01 dataset; f s = 1000 Hz Two-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on SE dataset Two-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on LA dataset; f s = 100 Hz Two-way tissue classification accuracies (mean ± std) for Fourier and all 17 features performed on LA dataset; f s = 1000 Hz Two-way tissue classification accuracies (mean ± std) for PCA features performed on SE dataset Two-way tissue classification accuracies (mean ± std) for PCA features performed on LA dataset; f s = 100 Hz Two-way tissue classification accuracies (mean ± std) for PCA features performed on LA dataset; f s = 1000 Hz vii

9 List of Figures 4.1 Single-element RVM high-resolution transducers for in vivo small animal imagining used in Vevo770 (image taken from VisualSonics website) The experimental setup for RF time series data collection using Vevo Chicken breast data collection using Vevo2100 operating in RF-mode Estimated RF displacement of a scanline of porcine kidney (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo Estimated RF displacement of a scanline of quartz (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 25 Hz Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 100 Hz Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 1000 Hz viii

10 4.9 Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and porcine kidney (black dot-line) for SE01 dataset RF displacement frequency spectrum of bovine liver for SE01 (blue) and SE02 (green) datasets RF displacement frequency spectrum of chicken breast for SE01 (blue) and SE02 (green) datasets RF displacement frequency spectrum of porcine kidney for SE01 (blue) and SE02 (green) datasets RF displacement frequency spectrum of quartz for SE01 (blue) and SE02 (green) datasets Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and steak (black dot-line) for LA01 dataset collected at fps = 100 Hz Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and steak (black dot-line) for LA01 dataset collected at fps = 1000 Hz Dividing the displacement spectrum RF displacements of SE01 (top) and LE01 (bottom) datasets Spectral FOS displacement estimation using 37 candidate frequencies Estimated probability density function of s 14 for SE dataset Estimated probability density function of s 14 for LA dataset Averaged false nearest neighbors of chicken breast for different embedding dimensions, m ix

11 4.21 Projection of reconstructed phase space of RF displacement estimated for steak in 3D averaged Γ( n) for all scanlines of SE dataset Estimated probability density function of s 15 for SE dataset Estimated probability density function of s 15 for LA dataset averaged ln (C(ɛ)) vs. ln(ɛ) for all scanlines of Vevo700 data set Estimated probability density function of s 16 for SE dataset Estimated probability density function of s 16 for LA dataset Estimated probability density function of s 17 for SE dataset Estimated probability density function of s 17 for LA dataset PCA accuracy versus the number of features x

12 Glossary of Terms ARF Acoustic Radiation Force. ARMA Autoregressive Moving Average. CD Correlation Dimension. DFT Discrete Fourier Transform. DRE Digital Rectal Examination. FNN False Nearest Neighbors. FOS Fast Orthogonal Search. MLE Maximal Lyapunov Exponent. MRI Magnetic Resonance Imaging. MSE Mean Square Error. PCA Principal Component Analysis. PCa Prostate Cancer. xi

13 PSA Prostate Specific Antigen. RBF Radial Basis Function. RF Radio Frequency. ROI Region of Interest. RPDE Recurrence Probability Density Entropy. SNR Signal to Noise Ratio. SVD Singular Value Decomposition. SVM Support Vector Machine. TRUS Transrectal Ultrasound. xii

14 1 Chapter 1 Introduction 1.1 Motivation According to the statistics published by the American Cancer Society, 565,650 Americans are estimated to annually die of cancer corresponding to over 1,500 deaths per day. Of these statistics, Prostate cancer (PCa) in men (25%) and breast cancer in women (26%) are the most common types of cancer followed by lung and bronchus cancer (14%). The leading cancer-related cause of death, in both men and women, is lung cancer, estimated to take 160,000 lives, followed by PCa, estimated to take 28,000 lives in the United States in 2008 [32]. PCa, in particular, if diagnosed early is managable in long-term. However, there are many challenges in the process of screening, diagnosis, and staging of PCa. Measuring the blood levels of prostate specific antigen (PSA) 1 is the first widely accepted step for PCa screening [8, 17]. Men with high levels of PSA are considered at higher risk for PCa and usually undergo Digital Rectal Examination (DRE) and Transrectal 1 PSA is a glycoprotein produced by epithelium of prostate gland

15 1.1. MOTIVATION 2 Ultrasound (TRUS) examinations. F. Lee et al. have shown that transrectal ultrasound is more sensitive than digital rectal examination for PCa detection [43]. In an examination of 784 men over 60, to compare the usefulness of TRUS versus DRE, it appeared that the overall detection rate of TRUS was twice higher than that of DRE (2.6% vs. 1.3%). Both of these methods have low sensitivity and specificity for diagnosis of PCa. As such, the gold standard diagnosis of PCa is histopathological analysis of biopsy samples of the prostate under TRUS guidance. This could result in missing cancerous regions and/or the need of repeating the biopsy procedure for accurate diagnosis. The importance of computer-aided diagnosis approaches to enhance this procedure is unquestionable. In the next chapter of this thesis, we will provide an overview of image processing techniques of Magnetic Resonance Imaging (MRI) and ultrasound images for PCa detection. Due to the advantages of ultrasound imaging over MRI, namely being real-time, safe, and relatively cheap, we narrow down our discussion to ultrasound-based tissue typing by introducing various methods such as spectral and texture analysis of B-mode ultrasound images, elastography, and Radio Frequency (RF) time series analysis. The latter technique, which is the basis of analysis of the work presented in this thesis, is a relatively new paradigm proposed by Moradi et al. [58] which is at its early stages of clinical development. In this method, a specific location in the tissue is continuously irradiated with ultrasound incident pulses and the time series of the backscattered signals are analyzed to extract tissue-dependent information. This approach has been used for tissue characterization and cancer detection in both clinical frequencies (6.6 MHz to 12 MHz) [62, 59] and high frequencies (40 MHz to 55

16 1.2. OBJECTIVE 3 MHz) [61]. At high frequencies, due to the short wavelength and the high axial resolution of RF data, the information in the RF time series were shown to be highly correlated with the microstructure of the tissue [61]. 1.2 Objective Most of the previous RF time series studies were focused on its application to tissue classification and did not elaborate on the mechanism underlying the tissue-typing capabilities of RF time series. The source of tissue-dependent information in RF time series could be partly due to heat induction as a result of consistent irradiation of tissue with ultrasound signals, and partly due to micro-vibration of the tissue induced by acoustic radiation force (ARF) of ultrasound beams [58]. An analytical model based on mechanical and thermo-physical properties of tissue was previously proposed to relate changes in tissue microstructure to variations in RF time series [16]. In this study, 3 seconds of RF time series data at the clinical frequency of 6.6 MHz were used to demonstrate that there is a detectable change in the speed of sound due to the ultrasound-induced raise in tissue temperature. The induced change in sound speed creates displacements in the backscattered signals that can be used for tissue typing. In this thesis we focus our investigation on the the mechanism of interaction of RF time series with tissue, at high frequencies, through displacement estimation. It will be shown that high frequency ultrasound RF displacements exhibit two main characteristics: 1) oscillatory behavior 2) linearly incremental trend which are associated with the effects of ARF and heat induction of ultrasound beams, respectively. In addition, we use a variety of linear and nonlinear analysis tools to study the

17 1.3. CONTRIBUTIONS 4 properties of displacements in RF signals in order to extract tissue-dependent features for tissue typing. 1.3 Contributions The specific contributions of this thesis are: Estimating the displacements of high frequency ultrasound RF time series which exhibit 1) oscillatory behavior of RF echoes at different frames, emphasizing on the potential of micro-vibration inside the tissue as a source of tissue-dependent information of RF time series, and 2) linearly incremental trend of such signals which indicates the heat induction of ultrasound beams as another source of tissue typing information. Using various linear and nonlinear time series analysis methods to study the behavior of such displacements in both frequency and time domains and proposing different features extracted from RF displacements for tissue typing. Experimentally demonstrating, through successful tissue classification, that linear and nonlinear features of RF displacements, proposed in this thesis, are highly tissue dependent. Applying a feature reduction technique to minimize the number of features required for a successful tissue classification. 1.4 Organization of Thesis The rest of the thesis is organized as follows:

18 1.4. ORGANIZATION OF THESIS 5 Chapter 2 provides an overview of computer-aided diagnosis of PCa. Stating the importance of ultrasound in PCa diagnosis, a short introduction to ultrasound imaging is presented. Subsequently, different methods in ultrasound-based tissue typing such as spectral analysis, texture analysis, elastography, and RF time series analysis are reviewed. Chapter 3 sketches the methodology used in this thesis to study the properties of high frequency RF time series. Displacement estimation, as the first step in our analysis, is discussed in details. Subsequently, Fourier analysis and spectral Fast Orthogonal Search (FOS) are used to study the properties of such displacements in the frequency domain and extract tissue-dependent features for tissue typing. In addition, nonlinear time series analysis, based on the theory of dynamical systems, is as an approach to investigate the behavior of RF displacements in the time domain. Three nonlinear features, namely maximal Lyapunov exponent (MLE), correlation dimension, and recurrence probability density entropy (RPDE), are extracted from RF displacement for tissue classification. Also, Support Vector Machine (SVM), as the classifier used in this thesis, is described. Lastly, a feature reduction algorithm, based on principal component analysis (PCA), is presented to minimize the number of features needed for accurate tissue typing. Chapter 4 illustrates the results of high frequency RF time series analysis. These results experimentally show the underlying sources of information for tissue typing in RF displacements. Classification results of animal tissue types are also presented which indicate that the proposed linear and nonlinear features are capable of distinguishing between different tissue types almost perfectly. Chapter 5 closes the thesis with a summary and discussion of the results and

19 1.4. ORGANIZATION OF THESIS 6 possible future work.

20 7 Chapter 2 Background 2.1 Overview As mentioned earlier, PCa is the most commonly diagnosed cancer and the second leading cancer-related cause of death among North American men [32]. Hence, diagnosis of PCa in the early stages of the diseases is very curial. Measuring the PSA level and the results of DRE tests are used to identify those individuals who need to undergo a biopsy procedure and histopathologic analysis of biopsy samples for confirmation of PCa. Normally, patients with PSA levels of over 4 ng/ml in the blood serum are considered as being at a higher risk of having PCa. Nevertheless, it has been shown that individuals with lower levels of PSA may have the same risk level [17, 84]. PCa detection through DRE palpation is beneficial only for relatively large lesions [39]. Transrectal ultrasound, as an alternative step in monitoring PCa, is able to detect cancer with only 40% accuracy [86]. The multifocal nature of PCa, and the fact that TRUS is not sensitive enough to identify cancer and biopsy procedures are not tailored to patients, might result in missing cancerous regions. This could yield

21 2.2. MAGNETIC RESONANCE IMAGING 8 false negative results or the need of repeating the biopsy procedure [78]. Due to the above mentioned shortcomings, researchers have tried to improve the detection rate of PCa by using computer-aided techniques. In the remainder of this chapter, we will review some of the current computer-aided methodologies used for detection of PCa. After a discussion on Magnetic Resonance Imaging (MRI) techniques, we will focus on ultrasound-based tissue typing, the basis of this thesis. 2.2 Magnetic Resonance Imaging Due to the high resolution of MR images, prostate cancer detection using this modality has the potential to be more accurate than ultrasound imaging [72]. Reconstruction of high-resolution 3D MRI models of the prostate are also beneficial for planning of radiotherapeutic treatments [77]. Noworolski et al. have shown that the cancerous regions of the prostate in peripheral zones can be detected using dynamic contrast enhance MRI [66]. MRI has also been used for guidance during biopsy [13, 15, 28]. Hata et al. have used T2-weighted MR images to guide biopsy needles to suspected targets [28]. The error in needle placement in their study, mainly due to the deflection of the needle, was reported to be 1.9 mm. In another study, MRI has been used as a preoperative modality for prostate biopsy [35]. In this study, Kaplana et al. have registered MRI to ultrasound images in real time during biopsy to improve the targeting accuracy. Although MRI provides superior images compared to ultrasound, there are still some questions on the effectiveness of this modality in the diagnosis process. For instance, Nakashima et al. have shown that the accuracy, sensitivity, and positive

22 2.3. ULTRASOUND IMAGING 9 predictive value of PCa detection using MRI for tumor foci greater than 1.0 cm in diameter are 79.8%, 85.3%, and 92.6%, respectively [63]. However, the same quantities drop to 24.2%, 26.2%, and 75.9% for tumor foci smaller than 1.0 cm. This suggests that there are some limitations on prostate cancer detection using MRI. In addition to the inability of MRI for prostate cancer detection in many cases, the time consuming procedure of MR imaging is a bottle-neck. Specially, constructing 3D models of the prostate using MR images could take an hour [52]. Moreover, MR imaging is relatively expensive compared to ultrasound imaging, which is currently the standard-of-care for guiding prostate biopsies. 2.3 Ultrasound Imaging Ultrasound imaging has been a widely-used modality for studying biological tissues over the past two decades[64, 70, 74, 76]. What makes the use of ultrasound imaging so common is its real time nature, high temporal resolution, relatively low cost, and safety. Using a set of piezoelectric crystals, an ultrasound machine transmits (receives) ultrasound beams to (from) the tissue. The time-lag between the transmission and reception of ultrasound beams is used to measure the distance of a scatterer inside the tissue from the surface of the transducer. Ultrasound beams are constructed using a beam forming process in which different time delays and weighting amplitudes are associated with each crystal in the transducer. This process makes it possible to have different shapes of ultrasound beams in order to produce better images. Beam formers control the quality of ultrasound images by changing different parameters of data acquisition such as the lateral and

23 2.4. ULTRASOUND-BASED TISSUE TYPING 10 axial resolutions 1, depth of field 2, frame rate 3, and image contrast. After the reception of reflected ultrasound beams, the analog Radio Frequency (RF) signals 4 are sampled and digitized to generate digital RF signals. B-mode images, commonly used in clinics, are produced by log compression of the envelop of the RF signals. A-mode images, that are mainly used in high-frequency ultrasound tissue typing, are produced when only one crystal is activated for ultrasound imaging. In this case, one will receive RF data for a scanline from the tissue. All these types of ultrasound data have been used widely in the literature for tissue typing. In the next section, we will review the most common ultrasound-based tissue typing techniques. 2.4 Ultrasound-based Tissue Typing Tissue characterization using acoustic properties of ultrasound RF echo signals such as attenuation, backscattering coefficients, and texture features, has been an active field of research during the past three decades [50]. Although ultrasound images have low signal to noise ratio (SNR), this modality is real time, safe, relatively inexpensive and has high temporal resolution. In the remainder of this section, we will describe different ultrasound-based methodologies that have been used in the literature for tissue typing. These include spectral analysis, texture analysis, elastography, and RF time series analysis. As mentioned in the previous chapter, the latter technique is the basis for the rest of the thesis. 1 These parameters specify how far apart two objects should be to be distinguishable by ultrasound beams. 2 The distance between the nearest and farthest point of the image. 3 Frequency of producing unique consecutive images. 4 These signals are normally referred to as radio frequency ultrasound signals since the frequency of medical ultrasound signals are in the range of radio frequency signals.

24 2.4. ULTRASOUND-BASED TISSUE TYPING Spectral Analysis Spectral analysis of conventional ultrasound images have been applied for tissue characterization in the past two decades [21]. Lizzi et al., as pioneers of this technique, have proposed an analytic model which relates the spectra of received ultrasound signals to the physical properties of tissue [49, 50]. The model can be applied to deterministic tissue structures such as large blood vessels and surface layers of the kidney, as well as to stochastic structures such as tumors. The beam patterns included in this model are those generated by focused transducers typically used in high-resolution clinical ultrasound devices. Generally, the advantages of spectral analysis are as follows: 1) Distinction of system parameters and tissue properties, 2) Description of wave propagation and tissue scatters to derive spectral features, and 3) Using averaged spectra to study stochastic properties of tissue structures. The first step in spectral analysis is to remove induced sources of noise in RF signals. This procedure is done by applying a Hamming window, which reduces the nearest sidelobe of the main frequency of RF signals. In the next step, calibration, the power spectrum of regions of interest (ROIs) are divided by the power spectrum of RF echoes from a planar object, e.g. a piece of glass, to compensate for the artifacts introduced by the transmitter and the receiver with the assumption that they have linear effects on the consequent signals. It has been shown that different tissue structures exhibit distinguishable patterns in the calibrated spectrum, and can be classified to be deterministic, stochastic, or a combination of both, accordingly. In deterministic tissue structures, which have two well-defined boundaries, periodic spectral peaks occur. The reason is that in such tissue structures, the RF signal has a form of a 1 e(t) + a 2 e(t 2w ) where e(t) is the echo from a single surface, c is the c

25 2.4. ULTRASOUND-BASED TISSUE TYPING 12 speed of sound in tissue, w is the thickness of tissue (the distance between two boundaries), and a 1 and a 2 are the reflection coefficients of the first and second surfaces, respectively 5. Assuming that the reflection coefficients are the same, the power spectrum of the RF magnitude will be proportional to cos ) (π ff0 2 where f 0 = c. As a 2w consequence, the successive peaks in the spectrum will occur at frequencies septated by f 0 = c 2w. In the case of stochastic tissue structures, however, numerous small scatterers, with randomly distributed positions, exist in the ROIs that lead to a monotonic power spectrum. Nevertheless, it has been shown that stochastic tissue exhibits quasi-linear spectrum[49]. Hence, regression analysis seem to be a good approach to extract features from them. Mixed tissue segments include both deterministic and stochastic components. Fellingham et al. have shown that for normal liver, as an example of mixed tissue structures, stochastic elements occur in an almost periodic matrix with a periodicity of near 1 mm[22]. In this case, the spectrums have small peaks that can be used to determine the periodic spacing interval within the tissue. Several methods such as autocorrelation functions and cepstrum analysis (Fourier transform of the log spectrum) were used to estimate this spacing interval [1, 49, 94, 95]. In summary, spatial periodicity can be used to detect deterministic structures whereas the slope, m, and the intercept, I, of a regression line fitted to the spectrum is useful to determine stochastic structures. Attenuation of ultrasound signals in tissue, which is almost linearly-dependent to the frequency[14], can have a huge impact on the features thereof. The measured spectrum of ROIs (in db) is the ideal spectrum minus 2αχf where α is the attenuation 5 Note that 2 is added as it is a two-way travel.

26 2.4. ULTRASOUND-BASED TISSUE TYPING 13 coefficient (measured in db/cm MHz), χ is the depth of ROI inside tissue, and f is the ultrasound frequency 6 [48]. According to this fact it follows that: I = I (2.1) m = m 2αχ (2.2) M = M 2αχf c (2.3) where f c is the central frequency of spectrum, I, m, and M are the intercept, slope, and midfit of the non-ideal spectrum, respectively. Feleppa et al. have used spectral features for PCa detection [20]. In the literature, mean central frequency is proposed as an alternative spectral feature for tissue characterization [65]. Spectral analysis has its importance in the sense that it relates spectral features to micro-structural properties of the tissue. However, practically, noise and system-dependent artifacts introduce difficulties for tissue classification using this method [49, 65] Texture Analysis The interaction of ultrasound beams with tissues such as reflection, attenuation, and scattering are dependent on tissue density, elasticity, and micro-structure [7]. Texture analysis is associated with finding tissue-dependent features using first and secondorder statistics of ultrasound B-mode images [7, 68]. In other words, in has been shown that the statistical information of B-mode intensity is tissue dependent as ultrasound beams propagating in tissue are affected by density, size, and distribution of scatterers. 6 Note that 2 accounts for two-way travel.

27 2.4. ULTRASOUND-BASED TISSUE TYPING 14 The first-order statistical features are the mean, µ, standard deviation, σ, and signal to noise ratio, SNR = µ σ of the gray-level intensities of B-mode images. First, statistical texture features are altered by changing the parameters of the ultrasound scanners. In addition, tissues with the same pathology can exhibit different acoustic properties [60, 82]. To overcome such problems, second-order statistical texture features have been proposed in the literature which are related to the spatial properties of the image. Co-occurrence and 2D autocorrelation matrices of B-mode images are calculated to be second-order statistical features. Autocorrelation is computed in axial and lateral directions. The speckle size is then estimated from the full width at half maximum of this autocorrelation function [7]. The co-occurrence matrix illustrates a histogram representation of gray-level intensities of B-mode image. To estimate such a matrix, one has to compute the second-order joint conditional probability density function p d,θ (i, j). Each p d,θ (i, j) is the probability of going from a gray level i to a gray level j in a direction θ at a given distance d. The co-occurrence N N matrix P d,θ (i, j), where N is the number of gray levels in the image, is the representation of these features. The non-normalized co-occurrence matrix of a B-mode image is computed as follows: C d,θ (i, j) = Ψ {(m, n), (k, l) I I f(m, n) = j, f(k, l) = i, m k = d cos(θ), n l = d sin(θ)} (2.4) where Ψ is the number of elements in the above set, and f(m, n) is the gray level of a pixel (m, n).

28 2.4. ULTRASOUND-BASED TISSUE TYPING 15 Because of edge effects, the number of paired occurrences in a given B-mode image varies with d. Hence, it is necessary to normalize the matrix C d,θ (i, j) by dividing it by the total number of paired occurrences as follows: P d,θ (i, j) = i C d,θ (i, j) j C d,θ(i, j) (2.5) This matrix is used to derive contrast, correlation, smoothness, and homogeneity of the image [81]. A combination of texture-based and clinical features, such as location and shape of the hypoechoic region, can be a promising method for PCa detection[7]. It has been reported that cancerous regions can be detected using these features with a specificity of around 90% and a sensitivity of around 92%. However, these results are limited to lesions that are hypo-echoic in ultrasound images. Multi-channel filtering is another texture-based technique used in the literature for tissue characterization [57]. In this method, TRUS images are processed using multiple resolution decomposition techniques such as a proper filter bank, to obtain appropriate texture features based on which tissue classification is performed. In this approach, texture features are extracted from both spatial and frequency domain of gray level intensities of B-mode images. It is usually more desirable to have filters with small bandwidths in frequency domain as they allow us to extract more accurate features. On the other hand, accurate localization of texture features in spatial domain requires filters with wide bandwidths. To address this problem, Gabor function is used in the process of texture segmentation which includes a proper filter bank design that should be tuned for different spatial-frequencies and orientations[91]. The Gabor function in the spatial domain is a Gaussian modulated sinusoid with

29 2.4. ULTRASOUND-BASED TISSUE TYPING 16 a real impulse response of: h(x, y) = { 1 exp 1 ( )} x 2 + y2 cos(2πω 2πσ x σ y 2 σx 2 σy 2 0 x) (2.6) where σ x and σ y are standard deviations in the x and y directions, respectively; and ω 0 is the modulating frequency. After applying such filter to the B-mode images, texture features are extracted to be used for tissue classification. One major disadvantage of texture analysis is that the information related to the scatterers is lost in the B-mode image due to speckle in the images. To tackle this problem, statistical distributions are used in the literature to find the probability density function (pdf) of backscattered signals. It has been shown that the Nakagami distribution is suitable to describe random behavior of ultrasound signals [90] Elastography Elastography is an approach that directly measures the mechanical properties of the tissue. This technique has been used in a wide range of medical applications such as breast and PCa diagnosis [24, 29, 69]. The main idea behind this method is that the cancerous regions of the tissue are stiffer than normal ones due to increased density. Hence, by calculating the stiffness of tissue in different ROIs, one is able to distinguish between healthy and unhealthy parts of tissue. However, this technique has some drawbacks for areas deep in tissue and for those cases that have small stiffness differences [3]. In this method, an internal mechanical excitation such as the motion of cardiac structures, or an external force such as acoustic radiation force (ARF), is applied to the tissue which results in shear and longitudinal displacements of the tissue under

30 2.4. ULTRASOUND-BASED TISSUE TYPING 17 study. Time-delays are estimated from displacement, and the strain profile of the tissue is computed by comparing the peaks in ultrasound RF echo signals before and after applying the force [31, 67, 73]. The comparison is computed by crosscorrelation analysis of corresponding pairs of RF A-lines. Pesavento et al. have proposed an algorithm that, in real time, accurately estimates strain with the help of phase differences in corresponding pairs of RF signals [73]. Salcudean et al. have reported a vibro-elastography system that is specifically designed to study mechanical properties of the prostate tissues [80]. Their system uses a transrectal ultrasonography transducer (TRUS) to apply low-frequency vibrations to tissue. Assuming that the tissue response to external forces is linear, they have applied mechanical forces, with frequencies up to 20 Hz, to estimate the frequency responses of the tissue. In another study, Baghani el al., have shown that when the entire top surface of a finite block of a soft-tissue-mimicking material is vibrated by compressional force, longitudinal motions of tissue are ultrasound-trackable as the speed of wave propagation in these cases is much less than the speed of sound (around 1 to 10 m/s) [6]. Elastography has been known as a valuable tool for tissue typing and significant improvements have been reported in the literature for computational and engineering methods used in this technique [19, 45, 54, 71, 80]. However, its drawbacks include: 1) restricted value of signal to noise ratio, 2) difficulties when multi-focal imaging is used, 3) inability to detect very small cancerous regions, and 4) high computational cost that restricts its real-time applications.

31 2.4. ULTRASOUND-BASED TISSUE TYPING RF Time Series Analysis Recently, a new paradigm for tissue classification, namely RF time series analysis, has been proposed by Moradi et al. [58]. They have shown that if a specific location of the tissue undergoes continuous ultrasound emissions, the time series generated by recording the RF echoes from that location in time is tissue dependent [59]. They have utilized RF time series of both high frequency (40-55 MHz) and clinical frequency ( MHz) ultrasound signals to distinguish between different animal tissue types [59, 61]. In another study, they have used this technique for PCa detection from the RF echoes of the tissue [62]. It has been shown that the spectral features extracted from the discrete Fourier transform of such time series are highly informative for tissue typing. Besides the electronic noise that might be induced by the ultrasound machine, the possible two main reasons of tissue-dependency of RF time series are as follows: 1) induced heat generated in tissue as a result of ultrasound absorbtion and 2) microvibrations in the micro-structure of the tissue caused by acoustic radiation force of ultrasound beams. Daoud et al. [16] have shown that variations in RF time series at clinical frequencies (6.6 MHz), which are tissue dependant, are mostly due to virtual axial displacement observed in RF data. The source of such displacement is the change in the speed of sound as the tissue absorbs ultrasound energy; these results are in agreement with an analytic model based on the mechanical and thermo-physical properties of tissue [51]. In that model, Maass-Moreno et al. have assumed c(t, z) to be the ultrasound velocity profile in the tissue as a function of temperature, T, and axial depth z where

32 2.4. ULTRASOUND-BASED TISSUE TYPING 19 z = 0 corresponds to the surface of the transducer. Hence, the time it takes for ultrasound waves to travel to a specific point inside the tissue at depth z and return, namely t(z), is computed as: z t(z) = 2 0 dα c(t, α). (2.7) As the tissue is more radiated, it will absorb more energy which yields an increase in the speed of sound inside the tissue. Consequently, a negative time delay, δ, will occur in the successive RF echoes which shows itself as some displacement in RF signals toward the transducer. One can compute such delay as: z ( ) 1 δ(t, z) = 2 0 c(t, α) 1 dα. (2.8) c 0 (T 0, α) By measuring the phase shift in the RF signals as well as quantifying the distortion of them, Daoud et al. were able to perform tissue classification with high accuracy. This study emphasizes the importance of the displacement of RF time series as a source of tissue-dependent information. In this thesis, we focus our attention on high frequency ultrasound RF time series analysis by computing displacements of RF echoes. We will show, in Chapter 4, that such displacements have two main characteristics: 1) oscillatory behavior that emphasizes the possibility of micro-vibration of tissue induced by ARF of ultrasound beams, and 2) linearly incremental trend that emphasizes the changes in the speed of sound inside the tissue as a result of heat absorbtion, similar to the effect observed by Daoud et al. [16].

33 2.5. SUMMARY Summary In this chapter, ultrasound-based computer-aided diagnosis tools for tissue typing were reviewed. Different techniques such as spectral and texture analysis as well as elastography and RF time series were discussed. In spectral analysis, the spectrum of received ultrasound signals is related to the physical properties of the tissue according to an analytic model. This model is capable of detecting different types of tissue structure, namely deterministic and stochastic. However, the effect of ultrasound attenuation in tissue should be taken into account for accurate results. In texture analysis, statistical information of gray-level B-mode images are used for tissue characterization. It has been shown that co-occurrence and 2D autocorrelation matrices of B-mode images do indeed contain tissue-dependent information as they measure the density, size, and distribution of scatters. In elastography, the mechanical properties of tissue, such as stiffness are directly measured by continuous palpation of tissue and tracking the deformation of ultrasound echoes as a result. In this method, an internal or external mechanical excitation is applied to the tissue which results in shear and longitudinal displacements of the tissue under study. Finally, RF time series was discussed as a newly developed technique where variations of RF echoes of a specific location inside the tissue over the course of data acquisition are studied for tissue typing. It has been discussed that in low frequency imaging, the heat that is generated inside the tissue, as a result of energy absorbtion, yields to a change in the speed of sound which consequently leads to a virtual displacement in RF echoes.

34 2.5. SUMMARY 21 In the next two chapters, we will focus our attention on high frequency ultrasound RF displacement estimation and linear and nonlinear tools to study the properties of such signals. We will depict that such displacements are oscillatory and have a chaotic behavior. In addition, we will illustrate that these signals also exhibit a linearly incremental trend which is a sign of changes in the speed of sound. Also, successful tissue classification will be performed using linear and nonlinear features extracted from these displacements.

35 22 Chapter 3 Methodology 3.1 Overview As mentioned in the previous chapter, RF time series analysis is a promising approach for tissue characterization. However, the source of tissue-dependence of RF time series is still an unknown concept in the literature. Nevertheless, such dependence could be partly due to heat induction as a result of consistent radiation of tissue with ultrasound beams, and partly due to the acoustic radiation force of ultrasound beams. Daoud et al. have shown that tissue dependency of variations in RF time series at clinical frequencies (6.6 MHz) is mostly due to virtual axial displacement that has been observed in RF data over the time of imaging [16]. The source of such displacement is the change in the speed of sound as the tissue absorbs ultrasound energy which is in agreement with an analytic model, based on mechanical and thermo-physical properties of tissue [51]. The main focus of this thesis is to investigate tissue dependency of RF time series

36 3.1. OVERVIEW 23 at high frequencies 1. We will start our analysis by evaluating the displacement of RF echoes at different frames with respect to frame one 2. In the next chapter, we will show that such displacements have oscillatory behavior as well as linearly incremental trend for high frequency RF time series. Oscillations are indications of possible microvibrations while incremental trends suggest that heat induction results in a detectable change in the speed of sound inside the tissue. In particular, we will illustrate that linear and nonlinear properties of such oscillations are highly tissue dependent which can be used for successful tissue typing. This chapter is mainly dedicated to describing linear and nonlinear analysis methods we have used to study the properties of high frequency RF time series displacements in order to extract tissue-dependent features. The results of such analysis along with the classification results are provided in the next chapter, where we present our experimental outcomes. The rest of this chapter is organized as follows: In Section 3.2 we describe a well-known algorithm to estimate RF displacements, based on which our analysis is performed to extract a variety of features. The main discussion of this chapter starts in Section 3.3 where we provide different linear and nonlinear tools to study the properties of RF displacements. Fourier analysis and spectral Fast Orthogonal Search (FOS) are used to study the properties of RF displacements in frequency-domain. In Section 3.4, we switch our focus to nonlinear time series analysis by introducing 1 The wavelength of ultrasound beams are much shorter in high frequency imaging which manifests the micro-structure of tissue better. 2 We are interested to study the dynamical changes in the RF echoes over the course of imaging. Since RF time series displacements are able to track such changes, we will first try to estimate these displacements before extracting tissue-dependent features.

37 3.1. OVERVIEW 24 the concept of Phase Space Reconstruction 3. Having a suitable RF phase space, three different nonlinear features are extracted from RF displacements. Lyapunov exponents (LE) and correlation dimensions (CD) are discussed to study the dynamical and geometrical properties of the underlying unknown system which gave birth to RF displacements. Recurrence probability density entropy (RPDE) is also evaluated to study the likelihood of RF displacements to be periodic signals. We will close this section by describing the concept of surrogate data, used to test the nonlinearity in the dynamics of RF displacements 4. Section 3.5 is devoted to briefly describe the classifier used in this thesis, namely Support Vector Machine (SVM). Section 3.6 is devoted to describe a PCA-based algorithm to minimize the number of features, needed for a successful tissue typing, in order to reduce the computational time. Section 3.7 closes this chapter by providing a summary of our discussion and pinpointing the important parts of our analysis. Before we start our discussion, it is important to mention that the main goal of this chapter is to introduce a variety of linear and nonlinear tools to summarize tissue-dependent information of RF displacements into a set of features. Hence, we may not go into the details of the theory behind our analysis if not necessary for our practical purposes. 3 The nonlinear analysis tools used in this thesis are based on the theory of dynamical systems. In such analysis, it is important to first transform our 1D observation to a suitable higher dimensional space. This process is called phase space reconstruction. Later in this chapter, we will talk in more details of how and why phase space reconstruction should be performed. 4 This test shows whether or not the results of nonlinear analysis are valid for the data under study.

38 3.2. DISPLACEMENT ESTIMATION Displacement Estimation The first step in our analysis is to estimate RF displacements. Such displacements are evaluated using cross correlation of RF lines at different frames of the time series with respect to the first frame. Basically, we try to find the time-delay (phase-shift) in the RF echo signals at a specific frame with respect to frame one. Time-delay displacement estimation of ultrasound echoes has been used in many applications such as blood flow estimation, tissue velocity and elasticity estimation, and radiation force imaging[37, 41, 73, 99]. Such estimators measure the displacement of backscattered signals, which appear as time-shift or phase-shift between sequences of echo signals, with respect to a reference signal 5. Generally, these techniques identify the maximum of a pattern matching function. The shape of a reference echo signal within a specific window is assumed to be the pattern and a matching algorithm is used to find the best match in the delayed echo signals. Finite sampling intervals induce some inaccuracy in such estimators which can be overcome by using interpolation methods such as parabolic fitting, spline fitting, cosine fitting or grid slope. Viola et al. have proposed a continuous time-shift estimator in which the reference signal is interpolated by a polynomial whereas the delayed echoes are kept discrete [93]. Pinton et al. have introduced another algorithm which uses the continuous representation of both reference and delayed echo signals to derive the pattern matching function [75]. In this thesis, we use spline-based time-delay estimators, proposed by Pinton et 5 We use RF signal at frame one as the reference signal as we are interested to study the changes in RF echo during the time of imaging. The main goal is to extract tissue-dependent information out of these changes for tissue typing.

39 3.2. DISPLACEMENT ESTIMATION 26 al., to evaluate the RF displacements 6. In this method, the sampled echo signals are interpolated by spline polynomials and the coefficients of the polynomials are used to estimate the time-delay of each frame RF signal with respect to frame one Spline-based Time-delay Estimation Assume that v(x, n) is the RF echo value measured from a location in tissue at depth x and frame n. According to the sampling frequency of the ultrasound machine, we only record v at a resolution of x. Hence, we can represent the waveform of v with a cubic spline 7 as follows: v(k x, n) = r kn (x) = a kn x 3 + b kn x 2 + c kn x + d kn (3.1) where k is an integer. For our analysis, r k1 (x) is considered to be the reference signal and all displacements are estimated with respect to this waveform 8. The normalized cross correlation function φ(α, n) is defined as follows: φ(α, n) = X/2 X/2 r k1(x)r kn (α + x)dx X/2 X/2 r2 k1 (x)dx X/2 X/2 r2 kn (α + x)dx (3.2) where X is the length of the window on which correlation is computed. This function determines the correlation between r kn and r k1 at different lags of r kn. The α max which maximizes φ(α, n) indicates the time-delay where v(x, 1) and v(x α, n) are 6 We will use the cubic spline polynomial for data fitting and normalized cross correlation to estimate the displacement of RF echo signals at different frames. 7 A cubic spline is a piecewise cubic polynomial such that the function and its first and second derivatives are continuous at the interpolation points. 8 This is actually the RF echo signal at frame one which is chosen to be the reference signal.

40 3.2. DISPLACEMENT ESTIMATION 27 most similar. Hence, the displacement signal ψ(n) is computed as follows: { } ψ(n) = arg max (φ(α, n)) α (3.3) where ψ(1) = 0 and v(x ψ(n), n) v(x, 1). It is also important to study the value of φ(α max, n). The closer this value is to unity the more similar RF signals are. Hence, for future reference, we also introduce the following function 9 : ζ(n) = max (φ(α, n)) (3.4) α In the next chapter, we will show that ψ(n) is oscillatory for high-frequency ultrasound RF time series collected from animal tissues. In addition, we will show that for long RF time series, ψ(n) will have an almost-linear trend which is due to the change in the speed of sound, i.e. the virtual displacement. Also, it will be sketched that ζ(n) drops significantly for long RF time series which is an indication of distortion in RF echoes due to heat expansion induced in the tissue as a result of ultrasound imaging. However, our main focus will be on studying the oscillatory nature of RF displacements and trying to summarize such changes into a set of tissue-dependent features based on which our classification is performed. The rest of this chapter is devoted to describe the linear and nonlinear methods to analyze these displacements and introduce a variety of features that will vary for 9 Note that ψ(n) indicates the best delayed version of v(x, n) that matches v(x, 1) whereas ζ(n) shows how close this version is to v(x, 1). Hence, if the value of ζ(n) is very low, i.e. near zero, ψ(n) does not really have a meaningful interpretation.

41 3.3. FREQUENCY ANALYSIS AND FEATURE EXTRACTION 28 different tissue types Frequency Analysis and Feature Extraction The main goal of this section is to introduce two different linear methods, namely Fourier analysis and spectral FOS, to extract frequency-based features out of RF displacements for tissue typing Fourier Analysis Oscillatory behavior of RF displacements encourages us to use Fourier analysis to reveal tissue-dependent information of RF displacements. Having the displacement signal ψ(n) for n = 1, 2,..., N, where N is the number of RF frames acquired, one can easily compute the discrete Fourier transform (DFT) of zero-mean displacement ˆψ(n) as follows: Ψ(f k ) = 1 N N 1 n=0 ˆψ(n + 1)e j2πf kn (3.5) where f k = kfs N for k = 0, 1,..., N 1 are different harmonics of the fundamental frequency f 1 = fs N, and f s is the sampling frequency (frame per second), and j 2 = 1. To extract consistent features, we normalize the amplitude of Ψ(f k ) as follows: ˆΨ(f k ) = Ψ(f k) (3.6) Ψ max 10 It is important to mention that the set of all features is significantly different for distinct tissue types. Hence, we do not expect that every single feature, by its own, has the ability to distinguish tissue types with high accuracy. However, at the end of Chapter 4, where we present the classification results, we will show that some features like Lyapunov exponents and RPDEs do well in tissue classification.

42 3.3. FREQUENCY ANALYSIS AND FEATURE EXTRACTION 29 where. is the absolute value operator 11 and Ψ max = max ( Ψ(f k ) ). f k The features we will use for tissue classification, namely s i, are the average of the ˆΨ(f k ) over I windows of K frequencies as follows: s i = 1 K i K 1 k=(i 1) K ˆΨ(f k ) (3.7) where s i is the ith feature and i = 1, 2,..., I. The number of features, I, and the length of the windows, K, will be determined in the next chapter according to the spectrum of the displacements. It will be shown that the main information of RF displacements is in the low-frequency components. Hence, we will choose I and K such that we capture this tissue-dependent information Spectral Fast Orthogonal Search Fast Orthogonal Search (FOS) has initially been proposed by M. Korenberg as technique for nonlinear system identification [40]. It has also been used for estimation of frequency spectrum of digital signals even in the presence of massive noise [2, 12]. It is well known that the resolution of discrete Fourier transform is limited by the number of samples under study which makes it impossible for detection of subharmonic frequencies 13. Subharmonic detection has been used to estimate the outer scale in wave-front slope data. It has also been used to model the speed of induction 11 In this thesis we used L 2 norm to evaluate the absolute values. 12 It will be shown that I = 13 and K = 3 is a good choice for this purpose. Although this might not be the best choice, we will depict that the set of these 13 features can distinguish different tissue types almost perfectly. On the other hand, finding the best choice for I and K requires to solve an optimization problem which does not seem necessary in this case since what we care the most is the ability of our features in tissue typing. 13 Subharmonic frequencies are those that are between integer-multiples of the fundamental frequency f = N f s where f s is the sample frequency.

43 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 30 motors as a function of current [30, 55] FOS rebuilds a target signal, y(n), with a weighed sum of arbitrary candidate functions, p i (n) as follows: M y(n) = {a m p m (n)} + e(n) (3.8) m=0 where a m is the weight of p m (n) in the model, M + 1 is the number of candidate functions used in the model to rebuild y(n), and e(n) is the error between the target signal and its FOS model 14. Using pairs of sine and cosine functions, as candidates, FOS has been used for spectral analysis[56]. In such analysis, it is assume that p 2m = cos(ω m n) and p 2m+1 = sin(ω m n) for m = 1, 2,..., M. Hence, one can recast Equ 3.8 into the following: M y(n) = a 0 + {a m cos(ω k n) + b m sin(ω m n)} + e(n) (3.9) m=1 where 2M + 1 is the number of terms used in the model 15. In the next chapter, we will use sine and cosine functions with different frequencies as candidates to rebuild RF displacements. The feature extracted from the FOS model for tissue typing will be s 14 = M i=m fmam M m=1 Am, where A m = a 2 m + b 2 m. 3.4 Nonlinear Analysis and Feature Extraction In linear methods, it is assumed that the governing equations of the underlying system which gave birth to our observations are linear. Such systems have a very limited 14 Note that usually it is considered that p 0 (n) = The number of terms in the model is highly application dependent; it depends on the complicity of the signal under study and the desired accuracy of the model.

44 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 31 variety of different dynamical behaviors. The solution of a linear differential equation, for instance, can either exponentially decay (grow) or exhibit a (damped) periodic behavior [27]. Consequently, in linear analysis, all irregular behaviors of a system are associated with some random external input to the system or noise of measurement. However, chaos theory states that noise is not the only possible source of irregularity in the output of a dynamical system [34]. In other words, there might be useful information in irregular behaviors of a dataset that helps the investigator to understand different aspects of the unknown underlying system that results in such observations. Nonlinear time series analysis, as a modern approach of studying our observations, links chaos theory to the process of extracting information from the system under study. In other words, instead of studying the properties of observed signals, we are interested to extract as much information as possible from the underlying dynamical systems which produce the signals [34]. Although linear features, described in the previous section, will be able to distinguish different tissue types almost perfectly, it is of interest to study the dynamical and geometrical properties of the underlying system which produces RF time series. In the next few sections, we describe the process of such analysis; but still the main focus will be to extract nonlinear features using these tools. The first step will be to transfer our one dimensional RF displacements into a higher dimensional space which mimics the dynamical and geometrical properties of the underlying unknown system producing the RF displacements. Having the suitable RF displacements phase spaces, we can easily extract nonlinear features of the system by evaluating Lyapunov exponent, correlation dimension, and RPDE of the RF displacements. We will show in the next chapter that Lyapunov exponents

45 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 32 and RPDEs are promising features for tissue typing whereas correlation dimensions are not Phase Space Reconstruction The nonlinear time series analysis tools used in this thesis are based on the theory of dynamical systems. In such analysis, the behavior of a system is usually described by a set of state variables in a vector field called phase space. For a deterministic system, if one knows the exact value of these variables i.e. the exact position of the system in the phase space, all future states of the system are determined. Hence, the dynamics of the system can be studied by analyzing the corresponding phase space points. Most often, the behavior of phase space is described by a set of first-order difference equations as follows 16 : X(n + 1) = f (X(n)) (3.10) where X(n) = [x 1 (n), x 2 (n)..., x d (n)] T is the state of the dth order system at time step n, x i (n) is the ith state variable, and f(.) is a smooth vector function i.e. f : R d R d which maps each point in the phase space onto its next position in time. We will focus our attention on those systems whose dimension is finite i.e. X(n) R d where d is the number of degrees of freedom of the system. It is of interest to mention that the dynamics of those systems described by partial differential equations, such as Navier-Stokes and Lorenz equations, live in infinite dimensional phase spaces. Even in those cases, usually it is sufficient to study only a finite dimensional part of the 16 Since RF displacements have a finite number of samples in time with a finite sampling rate, we assume that they are the outputs of a discrete system. However, for continuous signals the governing equations are usually described by a set of first-order differential equations.

46 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 33 phase space associated with active state variables of the system [87]. In order to apply nonlinear analysis to our observations 17, the first step is to convert the one dimensional RF displacements into a higher dimensional space whose dynamical and geometrical properties is equivalent to the unknown system. Hence, we assume that the unknown dynamical system, whose output is RF displacement, is described by a set of equations like in Moreover, it is considered that RF displacements are measured through a smooth nonlinear observation function, Q : R d R as follows: ψ(n) = Q (X(n)) (3.11) Having such assumptions, one is interested to study how x i (n)s are interacting with one another, i.e. the properties of function f(.) in equation 3.10, as well as how they affect ψ(n), i.e. the properties of the observation function Q(.). However, having only one dimensional sequence of samples, namely ψ(n), one has no direct access to the values of X(n) to be able to fully understand the interaction of different modes of the system. To address this problem, a variety of algorithms have been proposed in the literature to build a phase space for the scalar observation time series at hand, whose dynamical and geometrical properties are identical to the unknown underlying system[36, 38, 47]; such algorithms are usually referred to as phase space reconstruction methods. Using scalar time series, for example RF displacements, the ultimate goal in these methods is to define a new set of state variables, Y (n) = 17 In this case, the observation is the RF displacements evaluated for different ROIs of distinct tissue types.

47 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 34 [y 1 (n), y 2 (n)..., y m (n)] T, whose dynamical interactions are equivalent to X(n) in equation Several techniques have been proposed for phase space reconstruction such as derivatives coordinates[44], principal components [25], and delay reconstruction [36]. The latter algorithm, which will be used in this thesis, has received attention from researchers due to two main reasons: 1) It is the most straightforward method for phase space reconstruction. 2) The noise level is constant through the entire reconstructed space. Delay Reconstruction Having the RF displacement time series, namely ψ(n), we are interested to build a vector field, Ψ(n) R m, that can accurately mimic the dynamical and geometrical properties of the unknown system which produced ψ(n) 19. The delay reconstruction technique suggests that the following vector field, with suitable parameter to be chosen later, can accomplish this task [36]: Ψ(n) = [ψ(n), ψ(n + N 0 ),..., ψ(n + (m 1)N 0 )] T (3.12) where m is the embedding dimension of the reconstructed phase space, N 0 is the time lag between successive delay components, and ψ(n) is the RF displacement at frame n. 18 Note that m and d might not be identical in most cases. However, y i (n)s will exhibit the same dynamical behavior as that of x i (n)s. So, the reader should appreciate the fact that equivalence in dynamics and geometry does not necessarily imply identical number of state variables. 19 Here, we use the capital letter Ψ to emphasize the fact that the outcome of this process is a vector. We hope that this will not result in any confusion for the reader considering it as the Fourier transform of ψ.

48 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 35 Taken has proved the existence of such phase space 20, as long as the embedding dimension, m, is greater than twice the topological dimension, d T, of the original unknown system, i.e. m > 2d T [88] 21. In Taken s proof, it is assumed that the time series on hand has an infinite number of noise-free samples. Consequently, the dynamical and geometrical properties of delayed reconstructed phase space will be independent of the delay lag, N 0. However, in practice, the time series is neither infinitely long nor noise-free. Consequently, a small value of N 0 results in redundancy in the reconstructed phase space 22 whereas a large value of N 0 makes the delay components dynamically unrelated, a phenomenon called irrelevance by Casdagli et al. [11]. As a summary, there are two problems for delay reconstruction to be carefully addressed in order to obtain a suitable reconstructed phase space: 1) Finding a suitable delay time, N 0 2) Finding a proper embedding dimension, m. Delay Time Estimation There are a number of methods proposed in the literature to find a suitable delay time[4, 9, 10, 23, 46]; most of which try to measure the linear and/or nonlinear correlation of delay components in order to minimize redundancy and irrelevance 20 Such existence is studied in the sense that it will preserve the system invariants, e.g. Lyapunov exponents and correlation dimensions. 21 Note that the topological dimension is different than the degrees of freedom of a system. Hence, d T might be different than d in equation Usually the topological dimension is a fractional number, specially in case of chaotic behaviors. For example, the Lorenz system has three degrees of freedom (three state variables) whereas its correlation dimension is 2.05 ± Almost all the points in the reconstructed phase space will be on the main diagonal. In this case, the state variables are highly correlated. In other words, they cannot represent independent information about the system.

49 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 36 caused by small and large values of N 0, respectively 23. One of the ways to find a suitable delay time is to evaluate the autocorrelation of the time series on hand, namely R ψψ (λ). The autocorrelation is reasonably capable of exhibiting the transition from redundancy to irrelevance. Usually, the delay time, for phase space reconstruction, is the first time lag, λ 0, where R ψψ (λ) drops to a specific fraction of its maximum value at R ψψ (0) [9]. The major advantage of using this technique to find the suitable delay time is its very low computation time. However, it has been shown for a number of cases, where this method failed to produce consistent results[4, 10, 23, 46]. It is argued that such inconsistency is due to none-well defined relationship between the spatial distribution of points in the reconstructed phase space and the temporal autocorrelation of a single time series. To overcome this problem, Fraser et al. have proposed an algorithm to measure such spatial distribution using mutual information [23]. Since mutual information, I(N 0 ), can measure a wider scope of dependency of successive samples, 24 it is expected that using such information, one can track the transition from redundancy to irrelevance much more accurately. The main idea behind mutual-information-based technique for finding the suitable delay time is its capability of measuring the likelihood of predicting ψ(n + N 0 ) based on the value of ψ(n). Hence, the lowest value of I(N 0 ) is associated with the delay time which has the least redundancy and therefor is the best choice for phase space reconstruction[23]. 23 Note that small and large values of N 0 in this context do not imply a specific interval of numbers. This comparison is made according to redundancy versus irrelevance of successive delay components which is completely application dependent. 24 Autocorrelation can only measure linear dependency whereas mutual information can also find hidden nonlinear correlations.

50 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 37 Although it might be concluded from the above discussion that mutual information has a superior outcome compared to autocorrelation-based technique, it appears that for our analysis, the latter method produces good enough results for further analysis toward a successful tissue classification 25. Embedding Dimension Estimation As mentioned earlier, one of the main problems one has to deal with for delay phase space reconstruction is to find the suitable embedding dimension that can mimic the dynamical and geometrical properties of the underlying unknown system. Kennel et. al. have proposed an algorithm to find the suitable dimension which is based on a geometrical construction [36]. To understand the main idea behind this method, it is noteworthy to mention that we assume the time series under study, i.e. RF displacement, is the projection 26 of the attractor of a multi-variable system onto the one-dimensional axis, namely the observation axis. In the delay phase space reconstruction procedure, one tries to unfold this projection back to a multi-variable vector field that is dynamically and geometrically equivalent to the original unknown system. Taken has proved that if m > 2d T then the trajectory is unfolded enough not to have any self-intersection, a necessary property of any dynamical system [88] Again, we would like to remind the reader that the main goal of this thesis is to extract features that are powerful enough to distinguish different tissue types with high accuracy. Hence, we avoid getting into complicated analysis if not necessary to reach our goal. Using autocorrelation we found that N 0 = 70 and N 0 = 10 are suitable delay times for phase space reconstruction of RF displacements whose sampling frequencies are f s = 1000 Hz and f s = 100 Hz, respectively. 26 This projection might be a linear or nonlinear process, performed through an observation function. 27 Note that self-intersection is very likely to occur if one projects a high dimensional attractor into a low dimensional one.

51 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 38 Kennel et. al. designed a statistic called False Neighbors to determine the embedding dimension in which the attractor is completely unfolded[36]. The main idea behind this test is as follows: Assume that m 0 is the suitable embedding dimension for delay reconstruction. Also, consider that one has reconstructed a phase space in an m-dimensional space where m < m 0. We can think of the latter space as a projection of the suitable m 0 -dimensional space onto fewer degrees of freedom. Hence, in the m-dimensional space, there should be some points that are in the vicinity of one another (they are neighbors) whereas in the suitable embedding space, they are far apart. Kennel et. al. call such points false neighbors. Hence, starting from an arbitrary dimension m 28, we would count the number of points in the entire space that are false neighbors as their distance in m+1 dimensional space is much larger. To put this discussion into a more mathematical structure, Kennel defined false neighbors as those points for which the following inequality holds: R 2 m+1(n, r) R 2 m(n, r) R 2 m(n, r) > R 0 (3.13) where R 0 is a threshold and R 2 m(n, r) = Ψ(n) Ψ r (n) is the distance between a point Ψ(n) in the phase space of dimension m and its rth nearest neighbor, namely Ψ r (n). It has been shown that it is sufficient to compute R 2 m(n, r) only for r = 1, i.e. to consider only false nearest neighbors[36]. Hence, each point in the phase space is labeled as a true or false neighbor for a variety of values of the embedding dimension m. The suitable embedding dimension 28 Usually we start with m = 2.

52 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 39 m 0 will then be the one for which the ratio Υ(m) = ϑ(m), where ϑ(m) is the total N number of false nearest neighbors and N is the total number of points in phase space, is zero (or almost zero), implying that the attractor is unfolded enough to remove any self-intersection 29. Having the suitable embedding dimension and delay time, we now assume that Ψ(n) can be properly reconstructed for a given RF time series displacement, ψ(n). In the next three sections, we describe how to quantify dynamical and geometrical properties of the vector field Ψ(n) to be used later in the next chapter for tissue classification. Lyapunov exponents will be calculated for RF displacements which indicate if they have any signature of chaos and to what extent (dynamical property). In addition, correlation dimensions will be evaluated which measure the degree of freedom of dynamics of RF displacements (geometrical property). Finally, RPDEs will be estimated to study the likelihood of RF displacements to exhibit periodic behaviors Lyapunov Exponent One of the most important features of a chaotic motion is its sensitivity to initial conditions that makes it impossible for long-term prediction. In a chaotic system, starting from two different initial conditions (no matter how close they are) the trajectories will diverge exponentially. The averaged exponent of such divergence is a characteristic of the underlying system which is called the Lyapunov exponent (LE) [5]. It is important to mention that one can define as many different LEs as there are phase 29 In the next chapter, it will be illustrated that m = 5 is a suitable embedding dimension for phase space reconstruction of RF displacements.

53 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 40 space dimension. In other words, LEs are extension of eigen values of a linear system for nonlinear ones. However, in this thesis, we focus our attention to the most important LE, the maximal Lyapunov exponent 30. Imagine that Ψ(n 1 ) and Ψ(n 2 ) are two very close points in the phase space with distance Ψ(n 1 ) Ψ(n 2 ) = δ 0 where δ 0 1. Moreover, assume that δ n is the distance between the two points Ψ(n 1 + n) and Ψ(n 2 + n). The rate of exponent divergence of the two trajectories initiated at Ψ(n 1 ) and Ψ(n 2 ) is defined as follows: λ = 1 n ln ( δ n δ 0 ) (3.14) If λ is a positive number, it is a sign of chaos; i.e. a small initial distance diverges exponentially as time evolves [97]. Negative values of λ can also occur for dissipative systems which illustrates a stable fixed point in the phase space. Finally, λ = 0 depicts a stable limit cycle (periodic solution). Table 3.1 summarizes different kinds of motions according to the maximal LE of a system. Table 3.1: Different kinds of motions according to the maximal LE of a system. Types of motions Maximal LE Stable fixed point λ < 0 Stable limit cycle λ = 0 Chaos λ > 0 Rosenstein et al. and Kantz have independently developed an algorithm to find the maximal LE of a given time series[33, 79]. They calculate this value by averaging local exponent divergence over the whole data set. What follows is the procedure suggested by them to estimate maximal LE of a time series: 30 The reason that we just focus on maximal LE is that long-term behavior of a chaotic motion is mostly dependent on this value.

54 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 41 For a point Ψ n0 = Ψ(n 0 ) in the reconstructed phase space of a time series, namely RF displacement, calculate Θ(Ψ n0 ) which is the set of all the points in the neighborhood of Ψ n0 with a distance smaller than ɛ. For each Ψ k Θ(Ψ n0 ), compute Ψ k+ n Ψ n0 + n Ψ k Ψ n0 and average over all members of Θ. The natural logarithm of this average is a good estimate of the exponent divergence rate of the time series in the phase space. Repeating this process for many values of n 0 and averaging over resulted divergence rates will remove any bias which might be introduced due to noise of measurement. Hence, the statistic to calculate is as follows: N m 1 1 Γ( n) = ln 1 N m 1 µ (Θ(Ψ n0 )) n 0 =1 Ψ k Θ(Ψ n0 ) Ψ k+ n Ψ n0 + n (3.15) Ψ k Ψ n0 where m is the dimension of the reconstructed phase space and µ (Θ(Ψ n0 )) is the number of members of Θ(Ψ n0 ). Basically, the function Γ( n) illustrates the overall exponent divergence of two trajectories in phase space, whose initial conditions are very close. If Γ( n) shows a linear behavior for a wide range of n, the slope of it is an estimate of maximal LE 31. We will use maximal LE as a nonlinear feature for tissue classification. In the next chapter, we will show that this feature, alone, can distinguish different tissue types relatively accurately. 31 In the next chapter, we will show that Γ( n) does indeed exhibit a linear behavior with a positive slope for different tissue types. Moreover, the slope of such increment in Γ( n) is statistically different for different tissue types which makes it a good feature for our ultimate goal, tissue classification.

55 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION Correlation Dimension Dynamical properties of the phase space also has a geometrical counter part. Chaotic motions, whose LEs are positive, will create strange objects in the phase space whose dimensions are not integer. For example, the butter-fly shape of the Lorenz attractor has a dimension between 2 and Fractal objects show self-similarity at different scales which can be quantified using dimension. A proper definition of dimension should be such that non-fractal objects such as a set of points, lines, surfaces, etc. still have their integer dimension. Among different ways of defining dimension, we will use correlation dimension proposed by Grassberger et al. [26]. The advantage of this definition of dimension is its applicability for reconstructed phase spaces, where the time series on hand has some measurement noise. First of all, let us define the correlation sum to be the ratio of all possible pairs of points in a vector filed, whose distances are smaller than a threshold ɛ, to the number of all points in the space as follows: C(ɛ, N) = 2 (N m 1)(N m 2) N m 1 i=1 N m 1 j=i+1 u (ɛ Ψ(i) Ψ(j) ) (3.16) where m is the dimension of the reconstructed phase space and u(.) is the Heaviside function. It is shown that C(ɛ, N) ɛ D when N where D is called the correlation dimension. Hence, if we consider d(n, ɛ) = ln(c(ɛ,n)), then the correlation dimension ln(ɛ) 32 It is important to mention that the dimension of the topological object, created by the trajectories in phase space, is different from that of the embedding space. Embedding space always has integer dimension whereas strange object living in it might have a fractional dimension.

56 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 43 D is defined as follows: [ ] D = lim lim d(n, ɛ) N ɛ 0 (3.17) It is obvious that in practice we cannot compute the two limits as the number of samples of the time series is finite. Hence, we have to plot C(ɛ, N) versus ɛ in a double logarithmic axes. If we see a linear increase, then the assumption of self-similarity is valid and the slope of the line is a good estimate of the correlation dimension D 33. In the next chapter, we will use the estimated correlation dimension of RF displacements as another nonlinear feature for tissue classification. It will be shown that correlation dimension is not that much different for different tissue types yielding to poor classification results Recurrence Probability Density Entropy Sometimes a signal is almost periodic or seems to be periodic in a small portion of time. For instance, consider the signal x(t) = a 1 cos (ω 1 t)+a 2 cos (ω 2 t) where the ratio r = ω 1 ω 2 is irrational. In this case, there is no real value T such that x(t) = x(t + T ) for all values of t. This kind of signal is called quasi-periodic or almost periodic. The requirement of periodicity is crucial for many linear time series analysis such as Fourier transform for accurate results. The concept of recurrence is an extension of periodicity which measures the repetitiveness of a signal, especially in the case of quasi-periodic or chaotic behaviors [18]. Due to oscillatory behavior of RF displacements and appreciating the fact that 33 It is of interest to mention that the estimated correlation dimension for a time series is very close to its topology dimension, d T, used in Taken s theorem. Hence, we expect that the suitable embedding dimension m > D, where D is the correlation dimension. In the next chapter we will show that the suitable embedding dimension for RF displacement phase space reconstruction is m = 5 where their correlation dimensions are slightly greater than 2, e.g. D =

57 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 44 they are not perfectly periodic, we are interested to evaluate this concept for such signals and use it as a feature for tissue typing 34. Assume that Ψ(n) R m is a point in the reconstructed phase space of RF displacement at time step n. The state-space recurrence of such a system is defined as follows: Ψ(n) Θ (Ψ(n + N(n)), ɛ) (3.18) where Θ (Ψ(n + N(n)), ɛ) is a closed ball of radius ɛ around the point Ψ(n) in the phase space. Moreover, Ψ(n) / Θ (Ψ(n + n 0 ), ɛ) for 0 < n 0 < N(n). N(n) is called the recurrence time of the trajectory at time step n which indicates how many time steps took for the trajectory to return to the vicinity of itself at time step n. As an special case, where the signal is periodic, N = N 0 is the period of the system and ɛ = 0 for all points. For each point Ψ(n 0 ), in the phase space, we consider a closed ball of radius ɛ around it. Subsequently, the trajectory is followed till it leaves this ball. The time at which the trajectory returns to the ball, n 1, is recorded. The recurrence time is then the time difference between these two points, i.e. N(n 0 ) = n 1 n 0. In fact, recurrence measures the time it takes for the trajectory to return to the vicinity of a point in the phase space where it had been in past. For perfectly periodic orbits, this number is the same for all points in phase space and is equal to the period of the signal. For quasi-periodic and chaotic orbits, however, this number slightly varies for different points in phase space. Repeating this procedure for all points in the phase space results in a histogram of recurrence times H( N). The histogram is normalized to obtain the recurrence 34 It will be shown in the next chapter that this feature can distinguish between different tissue types relatively accurately.

58 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 45 probability density function: p( N) = H( N) N H( N) (3.19) If the time series is periodic with period N 0 then p( N) = δ(n 0 ) where δ(.) is the Dirac function. On the other hand, if the time series is purely random then p( N) is almost uniform. Having this concept in mind, we will extract the entropy of this density function which is a measure of uncertainty in the values of p( N) as follows: E = N max N=1 p( N) ln (p( N)) (3.20) where N max is the maximum value of recurrence time found for the data under study. If the time series is periodic then E = 0 (since p( N) = δ(n)) whereas E = ln(n max ) when the signal is purely random. Hence, we will use the normalized entropy: Ê = N max N=1 p( N) ln (p( N)) ln(n max ) (3.21) as a measurement of repetitiveness of RF displacements. Note that if Ê is near zero, the signal is almost periodic, whereas if it is near one, the signal is almost random. In the next chapter, we will use this feature for tissue classification. It will be shown that RPDE can distinguish between different tissue types relatively accurately Testing for Nonlinearity All nonlinear analysis tools provided in this chapter look for nonlinear structures of a give time series and try to quantify them through dynamical and geometrical values. Recall that the reason encouraging us to use them was the fact that all irregular

59 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 46 behaviors in the output of a system are not necessarily due to some stochastic processes involved in measurement. However, before we can rely on the results obtained from nonlinear analysis described earlier, it is important to make sure that such irregularities are indeed associated with some nonlinear dynamics. In other words, we need to check whether or not a similar linear stochastic system would have the same dynamical and geometrical properties as the dataset under study [34]. Null Hypothesis For this purpose, we assume that the dataset under study, namely RF displacement, is the output of an autoregressive moving average (ARMA) process of the following form: x n = a 0 + M AR i=1 a i x n i + M MA i=1 b i η n i, (3.22) where η n are independent Gaussian random numbers with zero mean and unit variance 35. The objective is to check whether or not the null hypothesis is true for the dataset under study. For instance, assume that γ is a statistic value extracted from time series under study such as Lyapunov exponent or correlation dimension. We want to check if such statistic does indeed suggest that the data is not the output of an ARMA process; in other words, if the null hypothesis is wrong. To address this issue, we have to investigate the distribution of γ that we could get from comparable linear model. If the nonlinear quantity extracted from the original 35 It is noteworthy to mention that for any ARMA process with Gaussian inputs, autocorrelation function and thus power spectrum as well as higher moments and correlations are completely determined by the coefficients a i and b i.

60 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 47 time series cannot be consistently described by the linear one, then one can conclude that nonlinear structures do exist in the data and the null hypothesis is wrong. Since we do not usually have any information about the distribution of the values of γ for linear stochastic processes, we will estimate such distribution using the method of surrogate data to be described later. As a summary, we need to do the following steps to test the existence of nonlinear dynamics in our dataset: 1. Assuming that the dataset under study, namely RF displacement, is the output of an ARMA process described thereof (the null hypotheis). 2. Generating surrogate datasets such that they contain correlated random numbers while their power spectrums are identical to the power spectrum of the original data set. 3. Specifying a level of significance that we are looking after Computing the statistic γ 0 for the original dataset as well as γ i, i = 1, 2,..., S where S is the number of surrogate datasets, and investigating if γ 0 is drawn from the distribution defined by γ i. If it does, then the null hypothesis is accepted meaning that strange structures in the data set are not really due to nonlinear dynamics. Otherwise, the assumption of nonlinear nature of dataset 36 For instance, if we allow for a chance of 5% (α = 0.05) that we reject the null hypothesis although it is in fact true, then the test is said to be valid at a %95 significance level. In designing the statistical test, we have to be careful that the actual rejection probability on data from null hypothesis does not exceed the value specified by the level of significance we are aiming for.

61 3.4. NONLINEAR ANALYSIS AND FEATURE EXTRACTION 48 is valid 37. We still need to describe a solid way of generating surrogate datasets. Note that we need to construct them such that their power spectrums are identical to the power spectrum of the original datasets, namely RF displacements. Surrogate Datasets To achieve this goal[89, 83], we take the Fourier transform of the time series under study, ψ(n): Ψ(f k ) = 1 N N 1 n=0 ˆψ(n + 1)e j2πf kn (3.23) and multiple the complex components Ψ(f k ) by some random phases φ k drawn from a uniformly distributed density over [0, 2π), i.e. χ k = Ψ(f k )e jφ k. However, we should defined the constraint φ N k = φ k to obtain a real inverse transform. The inverse transform of χ k is a suitable surrogate data. Repeating this procedure, we can generate as many surrogate data sets as necessary to accomplish the null hypothesis test 38. Having the surrogate datasets and the original RF displacements we can easily test the null hypothesis. It will be demonstrated in the next chapter that the null hypothesis is rejected for RF displacements studied in this thesis; implying that nonlinear analysis described in this chapter does indeed capture nonlinear dynamics of 37 In this thesis, we use the maximal Lyapunov exponent as the statistics for which the null hypothesis is investigated for. In the next chapter, we will show that MLEs of RF displacements are farm away from the distribution of MLEs of surrogate datasets, according to the level of significance α = 0.05, which implies that RF displacements do indeed contain nonlinear dynamics. Hence, the nonlinear analysis used in this thesis to study the behavior of RF displacements is nontrivial. 38 For our analysis, we have generated 1000 surrogate datasets for each RF displacement. It is noteworthy to mention that the number of surrogate datasets is not obtained through an optimization process. It is chosen to be 1000 in order to have enough samples to estimate the probability density function.

62 3.5. CLASSIFICATION 49 RF displacements. 3.5 Classification So far, we have proposed different linear and nonlinear features for tissue classification. In this section, we will briefly describe the classifier used in this thesis, namely Support Vector Machines. For tissue classification, we assign a feature vector, S = (s 1,..., s I, s I+1, s I+2, s I+3, s I+4 ) T, to each ROI which corresponds to a point in the n = I + 4 dimensional feature space. The first I features are whose extracted directly from the Fourier spectrum of RF displacements whereas the rest of the features correspond to Spectral FOS, maximal LE, correlation dimension, and RPDE, respectively. In the next chapter, we will show that I = 13 is a good choice for our analysis. Hence, the total number of features used in this thesis for tissue classification is 17. There are a variety of classifiers in the literature which are suitable for different applications. Many of them such as Bayesian classifier, use the distribution function of the features to classify different types of inputs. However, due to multi-variable nature of our feature space and also heterogeneity and complexity of biological signals, we decided to perform tissue typing using a maximum margin classifier; in particular Support Vector Machines (SVM). The most important reason for this choice is the independency of SVM algorithm on the distribution of features. The rest of this section is dedicated to briefly describe how SVM algorithm is tuned for classification.

63 3.5. CLASSIFICATION Support Vector Machines Support vector machines (SVM) was originally introduced by Vapnik et al. [92] which belongs to the family of maximum margin classifiers. SVM has been used in different applications such as text categorization and protein function prediction due to its robust performance even working on sparse and noisy datasets. The goal in the process of training an SVM classifier, using a binary labeled training dataset, is to find a hyper-plane that 1) minimizes the error on the training set and 2) maximizes the distance between classes boundaries. Assume that S m = (s 1m,..., s nm ) T is the mth point in the n dimensional feature space, for m = 1, 2,..., M where M is the number of all training feature points. Also, assume that c m { 1, 1} determines which class the feature point S m belongs to, i.e. c m = 1 states that S m belongs to class one whereas c m = 1 indicates that S m belongs to class two. Hence, the objective is to find a hyper-plane that can distinguish the two classes with the constraints mentioned thereof. The simplest way to find such hyper-plane (W, b) 39 is to maximize the following quantity: γ(w, b) = min m {c m (W.S m b)} (3.24) In the above equation, the term (W.S m b) is the distance between S m and the decision boundary. When this value is multiplied by the label c m, it results in a positive value for all correctly classified points and a negative value for all incorrect 39 The hyper-plane in this case is simply the set of all x that satisfy W.x b = 0 where the dot denotes the inner product. In this case the decision boundaries will be W.x b = 1 and W.x b = 1 corresponding to class one and two, respectively. Hence, if W.S m b 1 one concludes that S m belongs to class one whereas if W.S m b 1, S m is considered to belong to class two.

64 3.5. CLASSIFICATION 51 ones. γ(w, b) is the minimum of this value over all the dataset which is called the margin. Assuming that γ(w, b) is positive for a dataset, it denotes the minimum distance between the two linear decision boundaries. Hence, the objective is to maximize this distance as much as possible to be able to identify the two classes more accurately. It has been shown that the following value of W maximizes γ(w, b)[85]: W = M α m c m S m (3.25) m=1 where α m s are positive real numbers that maximize: M α m m=1 M m=1 l=1 provided that α m > 0 and M m=1 α mc m = 0. M α m α l c m c l S m.s l (3.26) However, there are some cases that linear separation is not possible 40. In such cases, the feature points are first mapped to a higher dimensional space through a usually-nonlinear function, Φ(x). Subsequently, the nonlinear decision boundaries are found such that the classification accuracy is maximized in the new feature space 41. The function K(x i, x j ) = Φ(x i ).Φ(x j ) is called the kernel function which determines the type of SVM classifier. There are a variety of kernel functions that can be used to map the features into a new space, that makes it easier to separate, such as linear, sigmoid, polynomial, and Gaussian Radial Basis Function (RBF). 40 In such cases, γ(w, b) defined in equation 3.24 is negative. 41 In such cases, W = M where M m=1 α m M m=1 α mc m Φ(S m ) maximizes the distance between the decision boundaries; M m=1 l=1 α mα l c m c l Φ(S m ).Φ(S l )

65 3.6. FEATURE REDUCTION 52 The latter kernel, for example, is defined as follows: K(x i, x j ) = e β x i x j 2 (3.27) where. denotes the L 2 norm. This kernel is useful when the main difference between two classes is the distance of their points from the origin. In this case, a linear (or even polynomial) kernel cannot separate the classes properly. In this thesis, we have used MATLAB c svmtrain and svmclassify functions to train the classifier and test it for tissue typing Feature Reduction So far, we have discussed about various methods to analyze and extract features from the displacement of RF signals. Also, we have described how the SVM classifier is used for tissue typing. However, having a large number of features 43, training the SVM classifier might be very time consuming 44. Hence, we would like to reduce the number of features based on which the classifier is trained. In other words, we are interested to remove any redundant information in the original feature space. To address this problem, we will use a feature reduction scheme proposed in the literature [53] which is based on Principal Component Analysis (PCA). In this method, we apply principle component decomposition on the features space to find the directions in which the most variation occur. Consequently, we build up a new feature space, with much less dimensions, that can almost perfectly represent the 42 The kernel used to train the classifier was the linear kernel as our features were linearly separable. 43 In the next chapter, we will indicate that there are 13 fourier features which makes the total number of features Note that training the classifier is equivalent to solve an optimization problem. Hence, the larger the dimensions of the feature space is, the longer it takes to find the suitable hyper-plane.

66 3.6. FEATURE REDUCTION 53 information of the original feature space 45. Assume that Ω 1 and Ω 2 are the sets of all feature vectors for tissue type 1 and 2, respectively 46. We define the set Ω = Ω 1 Ω 2 where denotes the union of these two sets. Hence, Ω will be an n 2M matrix where n = I + 4 is the number of features and 2M is the total number of samples (feature points) corresponding to all ROIs of both tissue types. The goal of PCA is to identify the optimal linear combination of original bases that minimize redundancy, and accurately describe the data set with the lowest possible number of dimensions. This is obtained by defining the principal directions in the data set, along which maximum data variations are expected. To achieve this purpose, let C be an n 2M matrix which represents the projection of the original data set, Ω, along principal directions. Moreover, assume that C is related to Ω with the following equation: C = W Ω (3.28) where W is an 2M 2M matrix which transforms the original data set to the new one. For optimal representation of the original data set, the columns of matrix C should have minimum redundancy and linearly independent. This can be achieved by defining W in such a way that the covariance matrix of the transformed data, Ψ C = 1 n CCT, (3.29) 45 In the new space, each feature vector will be a linear combination of the original features. 46 Without loss of generality, we subtract the mean of features as it is required for PCA decomposition.

67 3.6. FEATURE REDUCTION 54 is diagonal; where C T is the transpose matrix of C. By substituting C = W Ω in equation 3.29, one can find the relation between the covariance matrix of the transformed data set, Ψ C, and the original one, Ψ Ω, as follows: Ψ C = W Ψ Ω W T (3.30) Furthermore, assume that the singular value decomposition (SVD) of the matrix Ω is as follows: Ω = USV T (3.31) where U and V are orthonormal 2M 2M and n n matrices, respectively; S is an 2M n diagonal matrix whose elements are called singular values of Ω 47. Having the SVD representation of Ω and using the orthonormality property of V, one can easily expand the covariance matrix, Ψ Ω, which results in: Ψ Ω = 1 n USST U T (3.32) Recasting equation 3.32 yields: U T Ψ Ω U = 1 n SST (3.33) As mentioned earlier, the goal of PCA is to find a transform matrix W such that Ψ C is diagonal. Appreciating the fact that SS T is a diagonal matrix and comparing equations 3.33 and 3.30, one can easily conclude that W = U T. Hence, the columns of U T are the principal components of the data set represented by Ω. It is of importance 47 Commonly, the elements of S are ranked in descending order.

68 3.7. SUMMARY 55 to note that the ith diagonal element of the matrix 1 n SST is the variance of the original dataset in the direction of the ith principle component, for i = 1, 2,..., min(2m, n). Another important property of SVD is that: χ k m n = U m k S k k V T n k, k = min(m, n) (3.34) is the closest approximation of Ω in the least square sense for a k-dimensional representation of Ω. The matrix χ k m n is commonly called the approximation of rank k for Ω. In equation 3.34, the matrices U m k and Vn k T are formed by the first k columns of U m m and V n n, respectively; similarly, the matrix S k k is the diagonal matrix made from the first k singular values of Ω. It will be shown in the next chapter that approximation of order 5 of Ω contains 95% of the information. Hence, we will train the SVM classifier using only 5 features which will significantly reduce the training time with ignorable loss in performance Summary In this chapter, we discussed a variety of linear and nonlinear methods to study the tissue-dependent information of RF time series. Starting with RF displacement estimation using cross correlation of RF signals at different frames with respect to frame one, we continued our analysis using frequency and time information of these signals. The main goal of this chapter was to provide a set of linear and nonlinear feature for tissue typing. 48 Note that each of these 5 new features is a combination of all 17 features described in this chapter.

69 3.7. SUMMARY 56 Fourier and spectral FOS analysis was used to summarizes tissue-dependent information of RF displacement into a set of frequency-based features. Moreover, nonlinear time series analysis, based on the theory of dynamical systems, was used to study the dynamical and geometrical properties of the unknown system which gave birth to RF displacements. The first step for such analysis was to reconstruct a suitable phase space of RF displacements that preserves the dynamical properties of the underlying system. Two main problems were mentioned for this construction: 1) finding a suitable delay time, N 0, to minimize redundancy and irrelevance of successive delay components, 2) finding a suitable embedding dimension such that the reconstructed phase space has no self-intersection. The first problem was addressed by using autocorrelation of RF displacements to find the suitable delay time while the second problem was discussed by using the concept of False Nearest Neighbors (FNN) as a test of finding self-intersection for a variety of embedding dimensions. Having a suitable reconstructed phase space of RF displacements, three different nonlinear features were discussed to be extracted from RF displacements. The first two features, namely Lyapunov exponent and correlation dimension, summarize the dynamical and geometrical properties of the underlying system, respectively. On the other hand, RPDE measures the likelihood of RF displacements to be perfectly periodic. All these three features, along with the frequency-based features, will be used in the next chapter for tissue classification. In addition, we briefly discussed a feature reduction algorithm, based on PCA decomposition of feature space, to minimize the number of features required for a successful tissue classification.

70 3.7. SUMMARY 57 In the next chapter, we will show that the set of features proposed in this chapter is a powerful distinguisher of different tissue types.

71 58 Chapter 4 Experiments and Results 4.1 Overview In this chapter, we provide results of the analysis of displacements of high-frequency RF time series from animal tissue. We use various linear and nonlinear methods, described in the previous chapter, to study the properties of the displacement signals and extract tissue-dependent features to distinguish between different tissue types. Starting with linear methods, i.e. Fourier analysis and spectral Fast Orthogonal Search (FOS), we investigate the frequency properties of such displacements. It will be shown that low frequency components of RF displacements are highly tissue dependent. As a result, we extract 13 frequency-based features described in the previous chapter for tissue classification. In addition, we use spectral FOS to extract another frequency-based feature to distinguish different tissue types. Subsequently, we switch to nonlinear analysis in order to study the dynamical and geometrical properties of the unknown system which produces such displacements. We begin the analysis by finding the suitable dimension in which the phase space of RF displacement signals should be reconstructed. We will show that m = 5 is a

72 4.1. OVERVIEW 59 suitable embedding dimension for further analysis. Having the proper reconstructed phase space of RF displacements, we compute maximal Lyapanouv exponents for the whole dataset and show that the behavior of RF time series displacements is chaotic. MLEs are also used, later in this chapter, for tissue typing. By calculating the correlation dimension, we illustrate that the attractor of such displacements, in the reconstructed phase space, lives on a manifold whose dimension is very close to 2. However, it will be shown that correlation dimension is not that much different for different tissue types yielding poor classification results. In addition, we estimate the Recurrence Probability Density Entropy (RPDE), as an extension to the concept of periodicity, and use it as a feature for classification. Finally, by generating surrogate data from RF displacements, we show that the assumption of nonlinearity of such signals are indeed valid. The rest of this chapter is organized as follows: Section 4.2 describes the process of data acquisition using high frequency ultrasound machines, namely Vevo770 and Vevo2100. In section 4.3, we provide the results of RF displacement estimation. It will be illustrated that such signals have oscillatory behavior in short time, emphasizing on the possibility of mico-vibrations inside the tissue. Moreover, it will be depicted that RF displacements have linearly incremental trend in long time which emphasizes on the heat absorbtion of tissue and changes in the speed of sound. Section 4.4 is dedicated to study the RF displacements in the frequency domain. It will be sketched that low frequency component of RF displacements are tissue dependent based on which frequency features will be extracted and used for tissue classification.

73 4.2. DATA ACQUISITION 60 The main analysis results will be provided in section 4.5. Linear and nonlinear features will be extracted and discussed in details. By the end of this section, we will have 17 features based on which tissue classification will be performed. In section 4.6, tissue typing is performed using SVM classifier with a comprehensive discussion on the results. It will be sketched that using the set of the linear and nonlinear features proposed in this thesis, one can distinguish between different tissue types, almost perfectly. Finally, section 4.7 uses a feature reduction algorithm, described in the previous chapter, to minimize the number of features needed for a successful tissue classification. Section 4.8 closes this chapter by a comprehensive summary and discussion of the results. 4.2 Data Acquisition For our analysis, we have used high frequency ultrasound RF time series acquired using VisualSonics Vevo770 and Vevo2100 scanners (VisualSonics, Toronto, Ontario). The Vevo770 High-Resolution in Vivo Micro-Imaging enables visualization of small animal anatomical organs with resolutions as high as 30 microns. It is used for high frequency ultrasound imagining of small animals such as cardiovascular studies, cancer biology,and blood flow analysis[42, 96, 98]. This machine uses single-element RMV series transducers (Fig. 4.1). The Vevo2100 extends the functionality and quality of images of by using high-resolution linear-array MS series transducers. Both platforms have a wide rang of imaging modes such as B-mode, M-mode, 3D-mode, and most importantly digital RF-mode. In the next two subsections, we provide a detailed

74 4.2. DATA ACQUISITION 61 Figure 4.1: Single-element RVM high-resolution transducers for in vivo small animal imagining used in Vevo770 (image taken from VisualSonics website). description of acquired RF datasets used in this thesis Data Acquired Using Single-element Transducer (Vevo770) Data was collected from three different tissue types: bovine liver, chicken breast, and porcine kidney using single-element RMV704 scanhead optating at a central frequency of 40 MHz, with an axial resolution of 40 µm, and a focal length of 6 mm. Two pieces of each tissue type were used for RF time series collection 1. While the tissue was placed in a water bath at room temperature, 1000 frames of 15 separate A-lines were acquired for each tissue specimen at a frame rate of f s = 1000 Hz; resulting in 30 scan lines for each tissue type. The length of each RF line is 2000 samples which corresponds to 4 mm of depth in the tissue extending between 3 and 7 mm. The experimental setup for data acquisition is shown in Fig To study the consistency of the results, another dataset with the same specifications was acquired from the same three tissue types on a different day. In addition, 1 Talk why you used two pieces

75 4.2. DATA ACQUISITION 62 Figure 4.2: The experimental setup for RF time series data collection using Vevo770. we acquired RF time series from 15 different A-lines of a piece of quartz to study the machine-dependent artifacts that might effect the RF time series. For future references, we name the fist and second datasets (day one and day two) hereafter as SE01 and SE02, respectively. Also, we name the combination of these two datasets as SE for future references Data Acquired Using Linear-Array Transducer (Vevo2100) RF time series were collected from three different tissue types: bovine liver, chicken breast, and steak using linear array MS400 scanhead operating at a central frequency of 40 MHz, with axial resolution of 20 µm, and focal length of 5 mm. Five pieces of each tissue type were used for RF time series collection; 1000 frames of 40 separate A-lines were acquired for each tissue specimen at three frame rates of f s = 25 Hz, f s = 100 Hz, and f s = 1000 Hz; resulting in 200 scanlines per tissue type per frame rate values. The length of each RF line is 5000 samples which corresponds to 5 mm

76 4.3. DISPLACEMENT ESTIMATION 63 Figure 4.3: Chicken breast data collection using Vevo2100 operating in RF-mode. of depth in the tissue extending between 2 and 7 mm. Fig 4.3 depicts data collection from chicken breast using Vevo2100 operating in RF-mode. Again, for the purpose of studying the consistency of results and also the machinedependent artifacts, we collected another dataset with the same specifications from the same three tissue types on a different day along with data from quartz. For future references, we name the fist and second datasets (day one and day two) hereafter as LA01 and LA02, respectively. Also, we name the combination of these two datasets as LA for future references. 4.3 Displacement Estimation As mentioned in the previous chapter, displacement estimation is an informative approach to better understand the source of tissue dependency of RF time series. For our analysis, we divided each scanline into 10 ROIs with the same number of samples;

77 4.3. DISPLACEMENT ESTIMATION 64 for each ROI, we estimated the RF displacement at different frames with respect to frame one. Fig. 4.4 shows the estimated RF displacement, ψ(n), and its corresponding correlation coefficient, ζ(n), of a scanline of porcine kidney from the dataset SE01. As seen in this figure, the displacement and its corresponding correlation coefficients are oscillatory. This result supports the hypothesis of micro-vibrations inside the tissue as a result of acoustic radiation forces of ultrasound beams to be one of the sources of tissue-dependent informativeness of RF time series. Fig. 4.5 shows the same information for a scanline of quartz. There are two main differences between these two displacements. First, one can easily see an exponential decay in the displacement of quartz whereas in displacement of porcine kidney, there is no such decay. In addition, it is clear from the graphs that the correlation coefficient of quartz displacement drops significantly (drops to around 0.6) compared to the porcine kidney displacement (drops to around 0.94). Hence, one can conclude that estimated RF displacement of tissues are not just a consequence of machine-dependent artifacts. The same process was performed for data collected from linear array transducer. Figs. 4.6 to 4.8 show the displacements and their corresponding correlation coefficients of a scanline of chicken breast from the dataset LA01 with the following parameters: fps = 25 Hz, fps = 100 Hz, and fps = 1000 Hz, respectively. The results emphasize the oscillatory nature of RF time series displacements in high-frequency ultrasound. It is clear from the graphs that these displacements are smoother (less noisy) with higher correlation coefficients compared to those collected from single-element transducers. This could be a result of more accurate imaging capabilities of the new machine with a much higher axial resolution. In addition, it is of great importance to mention the almost-linear increment

78 4.3. DISPLACEMENT ESTIMATION 65 Figure 4.4: Estimated RF displacement of a scanline of porcine kidney (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo770. in RF displacement for the data collected at fps = 100 Hz, corresponding to 10 seconds of data. Such increment is even more apparent from the RF displacement of the data collected at fps = 25 Hz which corresponds to 40 seconds of data. This increment is the consequence of heat absorbtion of tissue while being radiated by ultrasound beams. The induced heat leads to an increment in the speed of sound which results in a virtual displacement which yields to a trend in the RF displacement waveform. Nevertheless, we can see that in all three cases, RF displacements also exhibit an oscillatory behavior. Hence, one can assume that RF displacements are the consequence of a combination of two main sources: 1) micro-vibrations, which result in oscillatory behavior of RF displacements, and 2) heat absorbtion, which results in a virtual RF displacement that presents itself as an almost-linearly incremental trend.

79 4.3. DISPLACEMENT ESTIMATION 66 Figure 4.5: Estimated RF displacement of a scanline of quartz (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo770. Moreover, from the figures, it can be seen that displacement oscillations are not perfectly periodic. This encourages us to use nonlinear time series analysis techniques to better understand the behavior of such oscillations. Later in this chapter, we will show that such oscillations are indeed chaotic by evaluating Lyapanouv exponents of RF displacements. Furthermore, the drop in the correlation coefficient of RF displacements, Fig. 4.6, could suggest a heat expansion inside the tissue which results in distortion of RF echo signals at different frames with respect to frame one. The more the tissue is radiated with ultrasound beams, the more heat energy is transferred to the tissue which results in more expansion yielding more distortion in the RF signals which exhibits itself as a drop in correlation coefficient. Having shown that the RF displacements of animal tissues exhibit oscillatory

80 4.3. DISPLACEMENT ESTIMATION 67 Figure 4.6: Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 25 Hz. Figure 4.7: Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 100 Hz.

81 4.4. DISPLACEMENT FREQUENCY SPECTRUMS 68 Figure 4.8: Estimated RF displacement of chicken breast (top) and its corresponding correlation coefficient (bottom) as a function of time, collected from Vevo2100 at frame rate fps = 1000 Hz. behaviors in high-frequency ultrasound imaging, we would like to investigate whether or not such oscillations are tissue-dependent. In the next section, we illustrate that the frequency of RF displacement oscillations are indeed tissue-dependent. 4.4 Displacement Frequency Spectrums To show that the frequency spectrum of RF displacements are different for different tissue types, we have computed the DFT of the displacements of each ROI, normalized and averaged them over all scanlines for every tissue type. Fig. 4.9 shows the normalized-average displacement frequency spectrums of bovine liver (blue line), chicken breast (red dashed-line), porcine kidney (black dot-line), and quartz (green) for SE01 dataset. It is clear from the graph that RF displacement of different tissue types exhibit

82 4.4. DISPLACEMENT FREQUENCY SPECTRUMS 69 Figure 4.9: Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and porcine kidney (black dot-line) for SE01 dataset. distinct frequency spectrums. Bovine liver seems to have the fastest response whereas porcine kidney has the lowest frequency. Also, it is apparent from the figure that RF displacement estimated for quartz has one very consistent frequency component. It is of great importance to remind that these spectra are the average of the normalized DFTs of each ROI for different tissue types. Hence, if all ROIs of chicken, for example, had exactly the same dominating frequency, then the peak of the averaged spectrum would be 1. Hence, one can conclude that bovine liver and porcine kidney have more consistent frequency spectra compared to chicken breast. However, by extracting frequency-based features and performing tissue classification, we will show that relatively high accurate results can be obtained using frequency information of RF displacements for data collected by single-element high-frequency transducers.

83 4.4. DISPLACEMENT FREQUENCY SPECTRUMS 70 Figure 4.10: RF displacement frequency spectrum of bovine liver for SE01 (blue) and SE02 (green) datasets. Figure 4.11: RF displacement frequency spectrum of chicken breast for SE01 (blue) and SE02 (green) datasets.

84 4.4. DISPLACEMENT FREQUENCY SPECTRUMS 71 Figure 4.12: RF displacement frequency spectrum of porcine kidney for SE01 (blue) and SE02 (green) datasets. Figure 4.13: RF displacement frequency spectrum of quartz for SE01 (blue) and SE02 (green) datasets.

85 4.4. DISPLACEMENT FREQUENCY SPECTRUMS 72 Figure 4.14: Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and steak (black dot-line) for LA01 dataset collected at fps = 100 Hz. To investigate the consistency of frequency spectrums from experiment to experiment, we have calculated the DFT of all ROIs of dataset SE02 in the same way. Figs to 4.13 show the RF displacement spectra of SE01 and SE02 for bovine liver, chicken breast, porcine kidney, and quartz, respectively. It is clear from the graphs that the frequency responses are almost identical for both datasets. This result shows the repeatability of RF displacements analysis from experiment to experiment. The same process was performed for linear-array dataset collected at fps = 100 Hz and fps = 1000 Hz of LE01 2. Figs and 4.15 show the frequency spectrums of RF displacements bovine liver (blue line), chicken breast (red dashed-line), and steak (black dot-line) of LA01 dataset collected at fps = 100 Hz and fps = 1000 Hz, respectively. 2 It is important to mention that we do not investigate frequency spectrum of data collected at fps = 25 Hz as the sampling frequency is too low and the frequency spectrum is not reliable.

86 4.5. FEATURE EXTRACTION 73 Figure 4.15: Normalized-averaged displacement frequency spectrum of bovine liver (blue line), chicken breast (red dashed-line), and steak (black dot-line) for LA01 dataset collected at fps = 1000 Hz. Again, the results show that the dominating frequency of bovine liver is slightly higher than chicken breast. The frequency spectra of both single-element and lineararray datasets emphasize the fact that low frequency component of RF displacement are highly tissue dependent. In the next section, we provide the results of features extracted from these dataset to be used later in this chapter for tissue classification. We also discuss the outcomes of nonlinear analysis performed on RF displacements. 4.5 Feature Extraction This section is dedicated to show the results of linear and nonlinear feature extraction from RF displacements. We will start by Fourier and spectral FOS features and continue our discussion with the results of nonlinear time series analysis of RF displacements, introduced in Chapter 3.

87 4.5. FEATURE EXTRACTION 74 Figure 4.16: Dividing the displacement spectrum RF displacements of SE01 (top) and LE01 (bottom) datasets Frequency Features According to the oscillatory behavior of RF displacements and the frequency spectrum of these signals, 13 features were extracted from each ROI of every scanlines of all tissue types. Recall that s i = 1 i K 1 K k=(i 1) K ˆΨ(f k ) for i = 1, 2,..., I are the features extracted from Fourier analysis. Basically, these features are the average of normalized amplitude of displacement spectrums in each frequency region. According to the spectrums of displacements, it is apparent that most tissuedependent information of RF displacement exist in the low frequency component ( up to 40 Hz ). Hence, we chose I = 13 and K = 3 to capture such information to be used later for tissue classification 3. Fig shows these 13 regions on the displacement spectrum of SE01 (top) and LE01 (bottom) datasets. 3 In the process of choosing the suitable values of I and K, no optimization was performed. These values are chosen such that classification accuracy using all 13 features is high. Later in this chapter, we show that using Fourier features, one can distinguish different tissue types almost perfectly.

88 4.5. FEATURE EXTRACTION 75 Figure 4.17: Spectral FOS displacement estimation using 37 candidate frequencies Spectral FOS Features As mentioned in the previous chapter, we use Spectral FOS to rebuild RF displacements as it can find incommensurate frequencies and also remove noise from the displacements even in the case of a low SNR. According to the frequency spectrum estimated earlier, it turns out that the main information of the signals exist in the low frequencies (up to 40 Hz). Hence, for FOS analysis, we used 500 frequency candidates in the interval f [0, 50Hz] with a resolution of 0.1 Hz. Feeding the displacements to FOS, we could rebuilt the signals with an error of MSE = 0.01 (only 1%) using only 37 frequencies. Fig illustrates original RF displacement of a scanline of steak (blue line) and its FOS estimation (green dashedline) pulled out of LE01 with fps = 1000 Hz.

89 4.5. FEATURE EXTRACTION 76 Figure 4.18: Estimated probability density function of s 14 for SE dataset. The FOS model tries to rebuilt the displacement using the following series: where ω m = 2πfm f s ψ F OS (n) = a 0 + M a i cos (ω m n)) + b i sin (ω m n)) (4.1) m=1 is the chosen frequency candidate and a m s and b m s are the corresponding coefficients and 2M+1 is the total number of terms in the model. Recall that s 14 = M i=m fmam M m=1 Am is the feature extracted from the FOS model where A m = a 2 m + b 2 m. Fig 4.18 shows an estimation of probability density function of spectral FOS features for single-element datasets, SE, whereas Fig 4.19 shows the same information for linear-array datasets, LA Phase Space Reconstruction Although, linear analysis show tissue-dependency of RF displacements, it is of interest to study the behavior of displacement signals even further by using powerful methods of nonlinear time series analysis. One of the main reasons, as mentioned earlier, is

90 4.5. FEATURE EXTRACTION 77 Figure 4.19: Estimated probability density function of s 14 for LA dataset. the fact that such oscillations are not perfectly periodic, suggesting distortion due to nonlinear dynamics. The rest of this section is devoted to study such nonlinear dynamics and quantify them for RF displacements and extracting features for tissue classification. The first step is to reconstruct a phase space of the given RF displacements. Hence, we have to find a suitable embedding dimension, m, to make sure that the dynamical and geometrical properties of the reconstructed phase space is equivalent to the original unknown one i.e. to find the minimum embedding dimension in which the RF displacement attractors are unfolded enough to remove all self-intersections. Therefore, we calculated the ratio of false nears neighbors of RF displacements, Υ(m), for different embedding dimension to choose m 0 such that Υ(m 0 ) 0. Fig shows the averaged percentage of false nearest neighbors for different embedding dimensions (2 to 5 from top to bottom), for RF displacements of all ROIs of chicken breast of LA dataset. It is clear from the graph that, m = 5 has no false

91 4.5. FEATURE EXTRACTION 78 Figure 4.20: Averaged false nearest neighbors of chicken breast for different embedding dimensions, m. neighbors. Hence, the suitable embedding dimension for further analysis is chosen to be m = 5. Having calculated the FNN percentage for all ROIs of each tissue types, we concluded that m = 5 is a suitable dimension for all tissue types. This suggests that the dynamical responses of all tissue types to ultrasound beams, roughly, live in the same phase space (they all follow the same governing equations). It is important to mention that the unknown original phase spaces of RF displacements might not have 5 degrees of freedom. Nevertheless, analyzing the system in higher dimension is the trade-off we have to pay as we just observed a scalar time series of the system. To give the reader an insight about the geometry of reconstructed RF displacement phase space, we have plotted the projection of the attractor of the RF displacement of a scanline of steak from the dataset LA, in 3 dimensions (Fig. 4.21) 4. 4 Note that, the suitable embedding dimension is m = 5 and this is just a projection of the phase space in lower dimension. Hence, one cannot expect this trajectory to be self-intersected free. However, the reconstruction theory guarantees that this trajectory has no self-intersection in 5 dimensional space.

92 4.5. FEATURE EXTRACTION 79 Figure 4.21: Projection of reconstructed phase space of RF displacement estimated for steak in 3D. We can see that the trajectory makes a strange geometrical object which has complicated topological properties. If the RF displacements were perfectly periodic, one would see a closed orbit which were not necessarily a circle or an ellipsoid. However, it is apparent from the graph that the trajectory is wandering around in a bounded region in a strange fashion. This kind of motion in phase space strongly suggests chaotic behavior. By calculating the maximal Lyapunov exponent, we will demonstrate that the behavior of RF displacements is indeed chaotic. Also, we quantify the geometrical properties of such trajectory by calculating the correlation dimension Maximal Lyapounov Exponents As mentioned in the pervious chapter, we calculate MLE by evaluating the average of rate of divergence of nearby trajectories through out the whole reconstructed phase space. By changing the radius of neighborhood of each point, we evaluate the rate of divergence for each radius that results in a time series, Γ( n). If this function is

93 4.5. FEATURE EXTRACTION 80 Figure 4.22: averaged Γ( n) for all scanlines of SE dataset. linearly-dependent on n, then the assumption of chaotic behavior is valid and the slope of this function is the maximal Lyapounov exponent. Fig shows the average of zero-mean Ψ( n), over the reconstructed phase spaces of RF displacements of all scanlines of bovine liver (blue), chicken breast(dashed red), and porcine kidney (dash-doted black) for SE dataset. We can clearly see that, in all three cases, an almost-linear increment of Γ( n) exist which indicates that the dynamical behaviors of displacements are chaotic 5. MLEs are simply the slope of these linear functions which are calculated by regression. We can see that porcine kidney has the smallest, but still positive, MLE whereas chicken breast has the largest value. This result is in agreement with the result of frequency spectrum of RF displacements of porcine kidney, which have lower frequency components and therefore slower responses in the time domain, compared to chicken breast, for example; making their RF displacements seem more periodic (less chaotic). As mentioned earlier, we will use MLE, namely s 15 = dψ( n) d n as a feature for 5 This is not a surprise, as we expected this fact according to the strange attractor of RF displacements in phase space.

94 4.5. FEATURE EXTRACTION 81 Figure 4.23: Estimated probability density function of s 15 for SE dataset. Figure 4.24: Estimated probability density function of s 15 for LA dataset. tissue classification. Fig 4.23 shows an estimation of probability density function of this feature for single-element datasets, SE, whereas Fig 4.24 shows the same information for linear-array datasets, LA.

95 4.5. FEATURE EXTRACTION Correlation Dimensions In the previous subsection, we computed the maximal LE of the RF displacements for different tissue types and showed that they are positive; indicating that the dynamical behaviors of them are chaotic. Hence, we expect the geometrical objects of the attractors of such displacements to have dimensions greater than 2 as chaotic motions live on geometrical manifolds whose dimensions are 2 + ɛ where ɛ > 0[88]. Nevertheless, according to the relatively small values of the maximal LEs of the RF displacements, we expect the correlation dimensions, d, to be very close to 2 and do not differ that much for different tissue types. Recall that, correlation dimension is the slope of logarithm C(ɛ) versus logarithm of ɛ, where C(ɛ) indicates the average of number of points of the RF displacement trajectory in a neighborhood of radius ɛ of each point over the whole phase space. Fig shows the average of ln (C(ɛ)) vs. ln(ɛ) of bovine liver (blue), chicken breast(dashed red), and porcine kidney (dash-doted black) for SE dataset. One can easily see that ln (C(ɛ)) has an almost-linear dependency on ln(ɛ) for all tissue types which indicates self-similarity of the signals. However, the slope of this graph is almost the same (around 2) for all tissue types as expected 6. We will use correlation dimension as the 16th feature for tissue classification (s 16 = D). Nevertheless, we do not expect high accurate classification results as this feature does not vary that much between different tissue types. Fig 4.26 shows an estimation of probability density 6 It is of great importance to mention that these results are in agreement with the suitable embedding dimension, m = 5, which was achieved using FNN algorithm. The reasons comes from the fact that according to the reconstruction theory m > 2d T, where d T is the topology dimension of the original unknown system, satisfies no-self-intersection property. Hence, we need at least m > 2 (2 + ɛ) dimension, provided that m is an integer number, to make sure that the manifolds defined by RF displacements are unfolded enough not to have any self-intersection. Hence the embedding dimension should be at least 5.

96 4.5. FEATURE EXTRACTION 83 Figure 4.25: averaged ln (C(ɛ)) vs. ln(ɛ) for all scanlines of Vevo700 data set. Figure 4.26: Estimated probability density function of s 16 for SE dataset. function of this feature for single-element datasets whereas Fig 4.27 shows the same information for linear-array datasets Recurrence Probability Density Entropies As mentioned in Chapter 3, recurrence is an extension of periodicity which is useful to quantify the repetitiveness of chaotic and quasi-periodic signals. Hence, we can