BEHAVIORAL STUDY OF ULTRASOUND WAVE PROPAGATION IN BIOLOGICAL TISSUES

Abstract

In contemporary times, the ultrasound imaging has become an established non-invasive clinical tool to image human anatomy for critical diagnosis. It is the second most widely used imaging modality after X-ray. The basic principle behind its application is the transmission of an acoustic field in the human body and the reception of reflections from tissue layers and body structures, or from volume scattering within the tissue. The biological tissue has a complex structure which shows an unpredictable behavior for ultrasonic waves. The acoustical properties of various tissues are different and phenomenon like scattering (generated because of various anatomical structures) and attenuation (which is also frequency dependent) affects the ultrasound propagation. It has also been observed that the propagation of ultrasound in biological tissues is nonlinear in nature. Harmonic images can be generated artificially by injecting microbubbles (ultrasound contrast agents (UCA)) in the blood which distort the acoustic energy resulting in generation of higher harmonics, known as contrast harmonic imaging (CHI). The nonlinearity also leads to the distortion of the transmitted beam while spreading energy in higher harmonics, and images are acquired preferably by receiving those of second harmonic, known as tissue harmonic imaging (THI). The harmonic image often demonstrates improved contrast resolution due to higher harmonic frequencies and therefore detects smaller objects. Initially harmonic imaging has been evolved from UCA only and later tissue nonlinearity has been revealed. By using THI, the significant improvements are visible in abdominal, pelvic, and cardiac sonography. High frequencies are, however, attenuated more in biological tissues as the beam propagates, leading to reduced depth of penetration inside the objects under scan. A new imaging technique named "super-harmonic imaging" (SHI) has been proposed recently. It takes advantage of the higher harmonics (third to fifth) arises either from nonlinear propagation or from ultrasound-contrast-agents. It provides further enhancement in resolution with acceptable penetration depth and signal-to-noise ratio (SNR). The dedicated phased array transducer and propagation model that enables the full exploitation of SHI are the topics of ongoing research. After dealing with general introduction regarding ultrasound imaging, important properties of sound, sound propagation and technical literature, the first stage of the work presented here deals with modeling and analysis of linear wave propagation for preliminary understanding of xv ultrasound wave propagation in biological tissues. We have followed the Tulphome- Stepanishen approach according to which the acoustic field at any point in the medium can be predicted by convolving the scattered field integrated over the transducer surface and spatial response of the transducer. The basic mechanisms like beam generation, propagation and focusing for single element transducer and array transducers, has been studied using FIELD program. Moreover, the beam optimization has been incorporated to improve the image quality parameters for array transducer. It has included various transducer parameters like element size, element spacing and apodization. Thus, for particular application an optimized combination of these parameters improves the image quality. The main factors focused during beam optimization are beam-width, number and magnitude of side-lobes relative to that of main-lobe. High intensity ultrasonic pulses are used in medical ultrasonics, which exhibit finite amplitude effects while propagating through human tissues but these effects can't be predicted by linear approximation methods. The signal source having finite dimensions generates signals of finite amplitude and these signals experience nonlinear characteristic while propagating through human tissue medium along with diffraction and attenuation. Nonlinearity is a property of a medium by which the shape and amplitude of a signal at a location are no longer proportional to the input excitation and it arises from nonlinear relationship between pressure and density variations. The numerical models are essential tools in order to understand nonlinear wave propagation and its behaviour. Thus, the second study starts with the derivation of nonlinear wave equation. This is further modulated to Burger's equation which has only attenuation and nonlinearity terms. Burgers' equation is used to model the nonlinear propagation of single sinusoidal wave with finite amplitude. In the first case of lossless medium, the analytical solution of Burgers' equation is achieved by using Fubini method in pre-shock region while weak shock theory is applied in post-shock region. Similarly in case of lossy medium, the analytical solution of Burgers' equation is achieved by using linear diffusion equation via Hopf-Cole transformation in pre-shock region and Fay's equation in post-shock region. Using the same analytical solution, conventional, tissue harmonic imaging and superharmonic imaging are evaluated numerically and their axial propagation is observed. The analytical solution for Burger's equation is also compared with time domain solution of modified KZK equation in which the diffraction term is removed. This transformed form then has been solved stepwise using with Poisson solution and Crank Nicolson Finite Difference method (CNFD). xvi The simulations are carried for circular transducer with overall system parameters for soft human tissues as selecting suitable transducer frequency, acoustic pressure, coefficient of nonlinearity, acoustic speed, density and attenuation coefficient for lossy medium. The propagation is observed right from surface of transducer to far away from transducer by using both the solutions in lossless and lossy media. It has been confirmed that the nonlinear propagation distorts the waveform and generates numerous harmonics. The comparison has been done close to transducer and for varying values of shock parameter. It has been found that both methods are in excellent agreement at shock formation distance. After the shock formation distance, the shock persists but amplitude reduces due to multivalued waveform in lossless medium. While in lossy medium perfect shock has not been formed by both methods and at deeper regions, the waveform regains its shape after heavy attenuation. With the same methodologies, THI and SHI analysis has been done along axial axis. The SNR and penetration depth both have improved for superharmonic field. The depth of penetration which is most concerned issue in harmonic imaging is also improved in superharmonic propagation as compared to second harmonic components as the amplitude of superharmonic is found to be higher in the possible imaging depth. On the basis of this initial and primary study of superharmonic fields, it has been found that superharmonic imaging has an edge over second harmonic imaging. Another methodology based on time domain solution of KZK equation has been studied for simulating the behavior of ultrasound propagation for focused and unfocused transducers. In this method, the diffraction term is integrated over space using finite difference method and by trapezoidal rule in time while the attenuation term is integrated over time using finite difference method. Finally, nonlinearity is included using the time domain solution of Burger's equation. The total propagation distance is divided in small steps and above mentioned calculation steps are carried for each incremental distance. The waveforms at all ranges beyond half the focal length possess shocks. The peak positive pressure at 0.7 times of focal length is much larger than that predicted by linear theory for a focusing gain of 5, and it is approximately three times the peak negative pressure. The fields calculated in far off-axis and in deeper regions a>10 has not been found in good agreement with theoretical expectations. This may be happening due to parabolic nature ofKZK equation. In one another method called pseudospectral method which is valid for longer axial propagation distances has an edge over above mentioned KZK equation. The harmonic field xvii generation has been studied for linear and phased array transducers using pseudospectral method. Besides this, the beam optimization study is also conducted to encourage the instrumentation design. The parameters which influence image quality are instrumentation parameters and medium parameters. We have focused on instrumentation parameters like transducer frequency, applied pressure, apodization techniques and f-number (which depend on focal depth, number of elements and pitch of the array). Proper selection of these parameters and better understanding of the trade-offs of their effects on beam parameters leads to optimization of beam profiles. This leads to improved lateral and axial resolution and better penetration depth. The use of proper combination of transducer frequency and applied pressure improves both resolution and penetration depth. The selection of apodization technique also affects the sidelobe suppression and beamwidth which decides the lateral resolution. The variation of f-number using multiple foci may lead to reduced frame rate. The change in number of elements and pitch may also lead to sidelobe suppression but these are is physically not possible. So, the system needs to be designed in such a way that only selected elements can be excited. This will provide variation in aperture size and pitch also. In the next study, we have combined time and frequency domain methods by using time domain solution of Burger's equation to solve nonlinearity and frequency domain solution for diffraction and attenuation. We have followed the Christopher and Parker approach of angular spectrum formulation of diffraction phenomenon in combination with attenuation for solution in frequency domain. The nonlinearity is included using time domain solution of Burger's equation. This method has been found computationally less expensive than time domain solution of KZK equation. The simulation is performed for conventional short Gaussian pulse from phased array transducer. The lateral and axial views of field propagation confirm the harmonic generation and found in excellent agreement with analytical solution of Burger's equation. Another numerical representation for nonlinear wave propagation is Westervelt equation. In this work, we have studied newly developed nonlinear contrast source method based on the Neumann iterative method combined with a Green's function approach as an algorithm to compute nonlinear wave propagation. The method is validated for a one-dimensional nonlinear wave problem. We have evaluated the wavefield, including harmonic frequencies up to the fifth harmonic. The results are compared with a solution of the lossless Burgers' equation and found in perfect agreement. xviii Further, we have used amplitude modulated and frequency modulated excitations altogether to study their effects on THI and SHI. The results have been obtained using the above mentioned nonlinear models. The amplitude modulated long tone burst is investigated first as it propagates through the human tissue medium and numerical computations are performed to solve the nonlinear KZK equation in time domain using finite difference method. The effects of imaging system's parameter variation on the evaluation parameters like beamwidth, offaxis level, side-lobe level etc are calculated and compared for fundamental, second harmonic and superharmonic components. Again the transducer parameters variation has been done and its effect on beam pattern and trade-off between image quality parameters has been thoroughly investigated for optimized superharmonic beam. Further, the study of generation of superharmonic fields in soft tissues from focused circular piston by using time domain solution of KZK equation is done with frequency modulated excitations. The comparison of fundamental, second and superharmonic components are associated with amplitude and propagation distance on-axis and off-axis as well. The principles of ultrasound beam forming controls the quality of diagnostic imaging. Beam parameters associated with imaging quality are: (1) lateral and axial resolutions; (2) depth of field; (3) contrast and (4) frame rate. We have focused on trade-offs among the above four aspects of beam forming in fundamental, second harmonic and superharmonic fields. The numerical computations are performed with excitations like three cycle Gaussian pulse, amplitude modulated and frequency modulated pulses. The results are studied for all three excitations and were compared. Lastly, most widely used coded excitations in radar system such as linear and nonlinear frequency modulated waves have been used. The purpose of this application has been to use simulation means to analyze primary issues related to use of linear frequency modulated (LFM/chirp) and nonlinear frequency modulated (NLFM) waves for improving the SNR and penetration depth for both second harmonic and superharmonic imaging. For superharmonic component, linear summation of third to fifth harmonics is followed in this work. The simulations are based on pseudospectral model for nonlinear propagation and processing is done in MATLAB. The similar study is also done using the composite time-frequency domain algorithm for LFM and NLFM excitations. It has also been computationally checked and confirmed that coded excitations provides better SNR, resolution and penetration depth xix in second and superharmonic imaging as compared to the conventional excitations. Thus, this study has shown the new phase in ultrasound imaging i.e. coded superharmonic imaging (CSHI).