Linear Ultrasonic Wave Propagation in Biological Tissues

Transcription

1 Indian Journal of Biomechanics: Special Issue (NCBM 7-8 March 29) Linear Ultrasonic Wave Propagation in Biological Tissues Narendra D Londhe R. S. Anand 2, 2 Electrical Engineering Department, IIT Roorkee, Roorkee,, India Abstract Propagation of ultrasound in biological tissues is of nonlinear in nature. But linear approximation in far-field is promising solution to model and simulate the real time ultrasound wave propagation. The simulation of ultrasound imaging using linear acoustics has been most widely used for understanding focusing, image formation and flow estimation, and it has become a standard tool in ultrasound research. In this contest, in this paper the ultrasound field generated from linear array transducer and phased array transducer respectively and propagation through biological tissues is modeled and simulated using FIELD program. The frequency depended attenuation has also been considered in the present simulation. Keywords: Ultrasound imaging, biological tissues, array transducer, field.. Introduction While designing modern new scanners, the optimization of acoustic fields is the most relevant factor. In order to achieve this, the need is to focus on simulation and measurement of acoustic fields. The FIELD (Jensen, 996b) program developed by Jensen et al is based on Tulphome-Stepanishen (Tulphome, 969; Stepanishen, 97) approach of calculating the spatial impulse responses. It is very efficient and accurate approach to calculate all types of fields like emitted, received, pulsed, CW with all possible transducer geometries. It is based on the inhomogeneous wave equation derived for describing scattering and propagation in an inhomogeneous medium (Jensen, 99a). The model can be implemented with any kind of transducer geometry and excitation (Jensen et al, 992). Ultrasound propagating in human tissues experiences both absorption and scattering resulting in a frequency dependent attenuation (Jensen et al, 993). Apodizing of elements and weighting between elements are used for decreasing sidelobes and increasing the depth of field. The calculation of field at every point on the surface of aperture is expensive in time. So the aperture is divided into sections and field is calculated by summing the response from each section. The far field approximation is possible (Jensen et al, 993) only when the size of section is more than wavelength. The shape of sections also matters as the best fit need to be chosen, e.g. the rectangular sections is chosen for linear and phased array transducers while triangular sections are chosen for circular, annular array transducer and even for irregular shapes (Jensen 996, 996). The FIELD-II program written in Matlab, is the modified version of original FIELD program. The modified version facilitates time varying focusing and apodization which yields in simulation of imaging of tissues and blood flow estimation (Jensen, 996). Some standard simulation phantoms are used for designing and evaluating ultrasound transducers, beam formers and systems. The phantoms investigated provide indication of sidelobe levels, focusing abilities and imaging contrast ability to detect low contrast objects for real imaging (Jensen et al, 997; Jensen, 24). This approach has been extended later to study nonlinear ultrasound imaging using operator splitting method (Jensen et al, 22). 4

2 2. Basic Theory The basic simulation model is shown in Fig. consists of triangular aperture situated in infinite rigid baffle, the field point where emitted field is observed and coordinate system. The pressure field p generated by the aperture is then found by the Rayleigh integral, r, r v r 2 n 2 t c ρ p ( r, t ) = ds () 2π t S r r where v n is the velocity normal to the transducer surface, is the density of the medium, r 2 and r are the distances of aperture and observation field point respectively from centre of coordinate system and t is the time. This integral formulation assumes linear wave propagation in a homogeneous medium without attenuation. Further, the radiating aperture is assumed flat, so no re-radiation from scattering and reflection takes place. After exchanging the integration and the partial derivative and by introducing velocity potential function, all field quantities can be derived form it. The excitation pulse can be separated from the transducer geometry by introducing a time convolution with a delta function and by assuming the surface velocity is uniform over the aperture, the pressure for radius r and time, t is vn ( t) p ( r, t) ρ = h( r, t ) (2) t where * denotes convolution in time. The integral in this equation r r 2 δt t2 c h( r, t) = ds (3) 2π r r is called the spatial impulse response. S 2 2 Fig. Position of transducer, field point, and coordinate system. Equation (2) gives the emitted pulsed pressure for any kind of surface vibration v n (t). The continuous wave field can be found from the Fourier transform of above equation. The received response for a collection of scatterers can also be found from the spatial impulse response. Thus, the calculation of the spatial impulse response makes it possible to find all ultrasound fields of interest. 5

3 3. Wave Patterns for Simulated Transducers 3. Linear array transducer The linear array is the fundamental type of multi-element transducer and it scans the region of interest (ROI) by exciting the elements situated over the region. The field is focused on the region by introducing time delay in the excitation of the concerned individual elements, so initially concave beam is emitted. Here a 6 element linear array transducer is designed using FIELD-II program as shown in the Fig.2, height, width and kerf of individual element are taken as 5 mm, mm and.25 mm respectively. The transducer is situated at the center of the coordinate system. The electronic focusing is incorporated to achieve focal length of 8 mm from the center of transducer. For the above linear array transducer, an excitation signal of two cycles of sinusoidal pulses is given in Fig. 3. The impulse response pattern obtained for each element is shown in Fig. 3. Fig.2 Design of linear array transducer (Height=5mm, Width=mm, Kerf=.25mm) Excitation pulse Simulated impulse response of a single element a) excitation pulse normalized impulse response time, microseconds time, microseconds Fig.3 Excitation pulse Impulse response for each element. For this specified linear array, acoustic field generated is propagated through human liver tissues and is observed at a focal distance i.e. (,, 8) whose plot is shown in Fig. 4. Fig. 4 shows the field generated while Fig. 4 shows the pressure profile at focal distance. Similarly Fig.5 illustrates the lateral beam pattern generated by the assumed linear array transducer. The other simulation parameters chosen are central transducer frequency (f) =5 MHz, acoustic speed (c) =577 m/s and frequency dependant attenuation coefficient () =.4 db/ [MHz]. 6

4 Pressure Field in the Focal Plane Pressure vs. Time, at Lateral Location of Peak Pressure 6 axial distance, mm normalized pressure lateral position, mm time after transmit, microseconds Fig. 4 Emitted pressure field in focal plane Pressure waveform observed at (,, 8) Maximum Pressure Value at each Lateral Position (db re maximum) Lateral Beam Plot lateral position, mm Fig. 5 Lateral beam pattern at (,, 8) 3.2 Phased array transducer Phased array transducer incorporates electronic scanning, electronic focusing and electronic steering because of which it can scan region of interest (ROI) at different focusing depths and at variable angles. Here a 64(6x6) element rectangular phased array is designed using FIELD-II. Height, width and kerf of each element are taken as mm, mm and.2 mm respectively. The similar excitation pulse and spatial response shown in Fig. 3 are considered for the phased array transducer too. The generated field is propagated and then steered through the liver tissue medium through 9 degrees and observed at focal distance (,, 7). The elevation angle variation is from - to degrees so that it covers the large image sector. Simulation parameters are central transducer frequency (f) =5 MHz, acoustic speed (c) =577 m/s, frequency dependant attenuation coefficient () =.4 db/ [m Hz]. Fig. 6 shows the acoustic field generated in the focal plane while Fig. 6 shows pressure pulse observed at focal distance. The beam profile for the phased array under test is shown in Fig.7. Pressure Field in the Focal Plane Pressure vs. Time, at Lateral Location of Peak Pressure Axial Distance, mm normalized pressure Elevation Angle, degrees time after transmit, microseconds Fig. 6 Emitted pressure field waveform observed at (,, 7) Maximum Pressure Value at each elevation angle (db) Elevation Angle, degrees in focal plane, Pressure 7

5 Fig.7 Lateral beam pattern at (,, 7) 4. Conclusion Modern diagnostic ultrasound systems has attained a very high image resolution by using digital beam-forming which is done using electronic scanning, electronic focusing and electronic steering. This result in a beam which is perfectly focused at all depths with required elevation and/or azimuth angles. The focusing methodologies are mostly evaluated using linear acoustics. Linear wave propagation approach is quite appropriate for better understanding and optimization in instrumentation. In this paper, the emitted field and propagation in homogeneous tissue medium is modeled and simulated. The simulation is done using FIELD program which is the best tool to model linear acoustics in the present time. 5. Refrences. Jensen JA, Fox, PD. Simulation of non-linear fields, IEEE Ultrason Symposium Jensen JA, Gandhi D, O Brien WD. Ultrasound fields in an attenuating medium, IEEE Ultrason Symposium Jensen JA, Munk P. Computer phantoms for simulating ultrasound B-mode and CFM images, Acoustical Imaging Symposium, Boston, Massachusetts, USA, April 3-6, Jensen JA, Nikolov S. I., Fast simulation of ultrasound images, IEEE Ultrason. Symposium, Jensen JA, Svendsen NB. Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers, IEEE Trans Ultrason Ferroelectr Frequency Control 992; 39: Jensen JA. A model for the propagation and scattering of ultrasound in tissue, J Acoust Soc Am 99a; 89: Jensen JA. FIELD-A program for simulating ultrasound systems, Med Biol Eng Comp, th Nordic-Baltic Conference on Biomedical Imaging 996b; Jensen JA. Simulating arbitrary geometry ultrasound transducers using triangles, IEEE Ultrason Symposium Jensen JA. Simulation of advanced ultrasound systems using FIELD-II, IEEE Ultrason Symposium 24.. Jensen JA. Ultrasound fields from triangular apertures, J Acoust Soc Am 996; (4): Stepanishen PR. The time dependent force and radiation impedance on a piston in a rigid infinite planar baffle, J Acoust Soc Am 97; 49: Tulphome GE. Generation of acoustic pulses by baffled plane pistons, Mathematika 969; 6: