акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

6

Nonlinear Acoustical Imaging

6.1Application of Chaos Theory to Acoustical Imaging

6.1.1Nonlinear Problem Encountered in Diffraction Tomography

The concept of the application of chaos theory to acoustical imaging was first introduced by Gan [1] in the series of International Symposiums on Acoustical Imaging in 1990. The purpose is to solve the nonlinear problems encountered in the expression for the scattered field in diffraction tomography. This is unavoidable because diffraction tomography is most commonly applied to medical imaging such as breast imaging, and as the breast tissue is inhomogeneous there will be multiple scatterings of the sound wave in the medium. Usually the Born and Rytov approximations are used to solve the Lippmann–Schwinger integral, as can be referred to in the chapter on imaging modalities (see Chapter 4).

There is a mathematical theorem stating that every nonlinear differential equation has a chaos solution subject to certain conditions. Since the Lippmann–Schwinger integral is a nonlinear differential equation it will have chaos solutions. We next make use of the concept that the geometrical interpretation of chaos are fractal imaging. Also, as chaos is very sensitive to the initial conditions, the fractal images obtained should show fine changes in the condition of the object.

We shall give a background on chaos theory and fractal images before proceeding to illustrate how they can be implemented in the solution for the Lippmann–Schwinger integral and in forming the image. A medical application will also be considered, especially breast imaging.

6.1.2Definition and History of Chaos

The word ‘chaos’ is commonly interpreted as a state of confusion or disorder and is equivalent to randomness. This definition can be applied to most cases, but is not applicable when considering the mathematical theory of chaos. A meteorologist, Edward Lorenz, first discovered the theory of chaos in the 1960s. During that time, he was attempting to predict the weather through the use of a mathematical model. Using a computer, he managed to generate a theoretical sequence of weather predictions [2].

In a separate attempt to observe the sequence of pattern again, he thought of saving time by running the program from the middle of the sequence. However, to his surprise, the results

Acoustical Imaging: Techniques and Applications for Engineers, First Edition. Woon Siong Gan. © 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

Figure 6.1 3D reconstruction of cylinders (Rytov approximation) (Chow [3])

deviated from the previous sequence. He realized that the error was created because of a slight truncation in the decimal places of the values used. Stumbling upon this discovery, Lorenz began to establish the theory of chaos by relating this to the effect of a butterfly in the prediction of weather [2]. The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, the atmosphere diverges from what it would have done. Therefore, in one month’s time, a tornado that would have devastated the Indonesian coast does not happen. Or, perhaps the event that was not going to happen, happens [4]. This observation is commonly known in chaos theory to be the sensitive dependence on initial conditions.

Lorenz concluded that it is impossible to predict the weather since any slight factor can affect it so drastically. He went on to simplify the model he used for weather prediction and devised a set of three equations that are simple but yet have a sensitive dependence on initial conditions. Thinking that the equations would also generate a sequence of random behaviour, Lorenz was shocked to discover that an infinite, double spiral was generated when the equations were plotted (see Figure 6.1 – Lorenz Attractor). The output actually fluctuated within the spirals and it was later discovered that these equations described the water wheel system [5] exactly.

6.1.3Definition of Fractal

Benoit Mandelbrot created the word ‘fractal’ from the Latin adjective fractus. This word carries the meaning of being irregular and fragmented. It was created to describe the statistical average or a rough geometric figure that has the property of self-similarity. The self-similarity of a single item describes the repetitive nature of itself when it is viewed under different length scales.

log ρ

1

Slope df – d

3/4

9/16 27/64

81/256

one will notice that the density of the object actually decreases from unity to 3/4. By repeating

1.ρ (L) decreases monotonically with L, without a limit, such that any lowest amount of density is possible.

2.The decrement follows a simple power law. The power law follows a generic form y = Axα and has two parameters, A (amplitude) and α (exponent). In our case, the amplitude is chosen as unity so that the power law is simply given as ρ (L) = Lα . The value of the exponent α is given by the slope of Figure 6.6, that is,

By comparing equations (6.2) with (6.4), we realize that the Sierpinski Gasket has a fractal dimension loglog 32 = 1.58 . . ., a dimension that is intermediate between that of a line (1D) and that of an area (2D).

6.1.6.2Random Fractals

Random fractals [10] are those that occur in nature. Examples are clouds, coastlines, lightning and mountains [11]. In terms of human anatomy, random fractals also occur in areas like the regional distribution of pulmonary blood flow and mammographic parenchymal pattern as a risk for breast cancer. They are not necessarily geometrical in shape but they form a statistical average of some properties (e.g. density) that decreases linearly with length scale when plotted on a double logarithm paper.

To illustrate this concept the unbiased random walk problem that occurs in statistical mechanics will be considered. At time t = 0, an ant is dropped onto an arbitrary vertex of an infinite one-dimensional lattice with constant unity or xt=0 = 0. The ant carries an unbiased two-sided coin and a clock. The ant flips the coin at each ‘tick’ of the clock and heads towards the east (xt=1 = +1) if the coin indicates a head, and vice versa (xt=1 = −1). As time progresses, the average of the square of the displacement of the ant increases monotonically because of the law of nature. The explicit form of this increase concerning the mean square displacement is given by

6.1.6.3 Other Definitions

By comparing equation (6.5) with (6.6), the displacement of the randomly walking ant can be defined in two ways:

√

The characteristic lengths L2 and L4 display the same asymptotic dependence on time. The leading exponent in the equation is called the scaling exponent while the nonleading exponent is called the corrections-to-scaling exponent. By considering any length L4 (provided k is even), a general equation is obtained as follows:

The subscripts on the amplitudes indicate their dependence on k. This equation displays a robust feature of random systems. Despite different definitions of characteristic length, the same scaling exponent describes the asymptotic behaviour.

6.1.7Fractal Approximation

Recently, scientists at the Mount Sinai School of Medicine, New York City, have successfully shown that fractal patterns inside cells can reveal breast cancer [12]. Moreover, other works have confirmed that scattering of ultrasound is chaotic in nature, and a fractal growth model can be used to adequately represent the scattered field within the breast [13].

Hence, a new approximation method – Fractal Approximation (FA) – is proposed by Leeman and Costa [14], based on the assumption that the scattered field, us (r ), can be approximated by a modified version of the incident field, u0 (r ). Modifications are done by introducing the fractal growth probability distribution function, P(r, t ). This distribution function is based on a fractal growth model, commonly known as the Diffusion Limited Aggregation (DLA) [15].

6.1.8Diffusion Limited Aggregation

Today, scientists are constantly discovering that many diverse natural phenomena produce similar fractal (that is, self-similar under different scaling factors) shapes. The patterns created by water as it seeps into the soil or flows through coffee grinds are described by fractals known as percolation clusters. Electrical discharges and the growth of some crystals also generate comparable fractal patterns.

In 1981, scientists Leonard M. Sander [16] and Thomas A. Witten devised a model for fractal growth by what they called the DLA. Their model used a random and irreversible growth process to create a particular type of fractal. DLA alone has about 50 realizations in physical systems, and much of the current interest on fractals in nature focuses on DLA [16].

Many clusters in nature grow by a process called aggregation, where one particle after another comes into contact with a cluster and remains fixed in place. If these particles diffuse toward the growing cluster along random walks, the resulting process is called DLA [17].

The main assumption made in this research is that the scattering paths of ultrasound in the breast actually follow the fractal-like structure of these DLAs. Therefore, it is possible to model the internal scattered field, us (r ), using the fractal growth model.

To grow or generate a DLA cluster, a seed particle is first placed at the origin. Then, one at a time, random walkers are released from some distant locations (around the circumference of a circle surrounding the site of the origin).

When one of these particles makes contact with the seed at the origin, it sticks and forms an aggregate and the next particle is released. If the particle touches the boundary of the circle before reaching the origin, it is considered void and removed. The totally random motion of these particles creates self-similar clusters like those shown in Figure 6.7. Note that the dimension of the clusters increases, as represented by s.

6.1.9Growth Site Probability Distribution

As described above, the generation of a DLA cluster is a matter of probability, where each step of a random walker is actually described and governed by a probability distribution known as the Growth Site Probability Distribution (GSPD) [18].

There are a few types of GSPD available and the one employed in this research is similar to the diffusion process of a random walker that is slowed down by entanglements like large

Figure 6.7 A two-dimension DLA cluster at six different stages of growth (Yu and Chan [19])

holes, bottlenecks and dangling ends [18]. This is chosen since the scattering of ultrasound is also affected by the surrounding medium and the internal structure of the object, which is an inhomogeneous medium.

When this distribution is being averaged over all the starting points of the random walkers (which are the coordinates of each sample point in the projection of the forward-scattered field), the probability distribution becomes a stretched version of the Gaussian distribution [18].

This distribution, P (r, t ) , is given by the following expression:

where P (0, t ) is the average probability of finding a random walker at the starting point,r2 (t ) 1/2 is the root mean square distance of the walker from the starting point, and dw is the fractal dimension of a random walk (also known as the diffusion exponent).

There is a strong evidence that the above equation is valid for a large class of random fractals [13]. By rearranging the above expression, the average GSPD, P (r, t ) , is given by

u r

P (r, t ) P (0, t ) · exp − (6.12)

r2 (t ) 1/2

where u = dw .

dw−1

When the time span is long, the average probability at the starting point, P (0, t ) , is proportional to the inverse of the number of distinctly visited sites, S(t ). On fractals, S(t ) scales as r2 (t ) d f /2 and thus

where d f is the fractal dimension of the DLA cluster and dw is the diffusion exponent.

As mentioned earlier, the motion of the random walker is assumed to be slowed down by entanglements, and thus the root mean square distance of the walker is given by a more general

At this point, there are two unknown parameters that need to be defined: d f , the fractal dimension of the DLA cluster and dw, the diffusion exponent.

6.1.10Approximating the Scattered Field Using GSPD

After obtaining the general equation for GSPD, the modelling of the scattered field within the breast can now be done.

Recall that the general expression for the incident field (assumed to be a plane wave) is given by

By multiplying the incident field with the GSPD, the scattered field within the object is given by

In this work, A is taken to be unity while R is the absolute distance between the coordinates of a scatterer within the breast (r = x , y T ) and a sampling point (starting point of random walk). The value of t for every scatterer is taken to be 1 while P (R, t ) or the average GSPD is obtained by taking the average of each probability value obtained for all R (all scatterers with respect to a sampling point).