1. Ultrasound Imaging: Lecture 2 Absorption Reflection Scatter Speed of sound Signal modeling Signal Processing Statistics Interactions of ultrasound with tissue Image formation Jan 14, 2009 Steering Focusing Apodization Design rules Beams and Arrays

2. Anatomy of an ultrasound beam Near field or Fresnel zone Far field or Fraunhofer zone Near-to-far field transition, L L Lateral distance (mm) Depth (mm)

3. Anatomy of an ultrasound beam Lateral Resolution (FWHM) FWHM Lateral distance (mm) Depth (mm)

4. Anatomy of an ultrasound beam Depth of Field (DOF) DOF Lateral distance (mm) Depth (mm)

5. Array Geometries Schematic of a linear phased array Definition of azimuth, elevation Scanning angle shown, , in negative scan direction.

6. Some Basic Geometry Delay determination: –simple path length difference –reference point: phase center –apply Law of Cosines –approximate for ASIC implementation In some cases, split delay into 2 parts: –beam steering –dynamic focusing

7. Far field beam steering For beam steering: –far field calculation particularly easy –often implemented as a fixed delay

8. Beamformation: Focusing Basic focusing type beamformation Symmetrical delays about phase center.

9. Beamformation: Beam steering Beam steering with linear phased arrays. Asymmetrical delays, long delay lines

10. Anatomy of an ultrasound beam Electronic Focusing

11. Grating Lobes Linear array: –32 element array –3 MHz –‘pitch’ l = 0.4 mm – = 0.51 mm – L= N l = 13 mm How to avoid: –design for horizon-to- horizon safety How many elements? What Spacing? Main Lobe Grating Lobe

12. Array design Linear array: –32 element array –3 MHz –pitch l = 0.4 mm – = 0.51 mm – L array = N l = 13 mm How to avoid: –design for horizon-to- horizon safety How many elements? What Spacing?

13. Apodization Same array: –32 element array –3 MHz –pitch l = 0.4 mm – = 0.51 mm – L array = N l = 13 mm With & w/o Hanning wting. Sidelobes way down. Mainlobe wider No effect on grating lobes.

14. Summary of Beam Processing Beam shape is improved by several processing steps: –Transmit apodization –Multiple transmit focal locations –Dynamic focusing –Dynamic receive apodization –Post-beamsum processing Upper frame: fixed transmit focus Lower frame: the above steps.

15. I INTERACTIONS OF ULTRASOUND WITH TISSUE Some essentials of linear propagation Recall the equation of motion (1) Assume a plane progressive wave in the +x direction that satisfies the wave equation ie (2)

16. Substituting 2 into 1 we have Acoustic impedance (3)

17. Where = Characteristic Acoustic Impedance Define a type of Ohm’s Law for acoustics Electrical: Acoustical: Extending this analogy to Intensity we have

18. Propagation at an interface between 2 media

19. Define Reflection/Transmission Coef You will show: Example: Fat – Bone interface (4) (5)

20. THE DECIBEL (dB) SCALE Where A = measured amplitude A ref = reference amplitude In the amplitude domain 6 dB is a factor of 2 -6 dB is a factor of.5 (i.e. 6dB down) 20 dB is a factor of 10 -20 dB is a factor of.1 (i.e. 20dB down) (6)

21. 0 -10 -20 -30 -40 -50 Reflection Coefficients Air/solid or liquid Brass/soft tissue or water Bone/soft tissue or water Perspex/soft tissue or water Tendon/fat Lens/vitreous or aqueous humour Fat/non-fatty soft tissues Water/muscle Fat/water Muscle/blood Muscle/liver Kidney/liver, spleen/blood Liver/spleen, blood/brain Water/soft tissues R = 1.0 R =.1 R =.01 Reflection Coef. dB

22. 3) ULTRASOUND IMAGING AND SIGNAL PROCESSING Thus far we have been concerned with the ultrasound transducer and beamformer. Let’s now start considering the signal processing aspects of ultrasound imaging. Begin by considering the sources of information in an ultrasound image a)Large interfaces, let a = structure dimension - specular reflection - - reflection coefficient where -strong angle dependance -refraction effects density speed of sound

23. b)Small interfaces - - Rayleigh scattering Compressibility Density and (7) Morse and Ingard Theoretical Acoustics p. 427

24. SCATTER FROM A RIGID SPHERE * *

25. SCATTER FROM A RIGID SPHERE (Mie Scatter) *

26. ATTENUATION = absorption component + reflectivity component The units of are cm -1 for this equation. However attenuation is usually expressed in dB/cm. A simple conversion is given by

27. Attenuation in Various Tissues

28. Speed of Sound in Various Tissues 0% 5% 10% 15% -5% -10% Assumed speed of sound = 1540 m/s

29. SUMMARY ULTRASONIC PROPERTIES Table 1 MaterialSpeed of SoundImpedanceAttenuationFrequency ms -1 Kg m -2 s -1 X 10 6 At 1 MHz (dB cm -1) Dependency water1490 @ 23ºC1.49 0.002 2 muscle1585 @ 37ºC1.70 1.3-3.31.2 fat1420 @ 37ºC1.380.63 1.5-2 liver1560 @ 37ºC1.650.701.2 breast1500 + 80 @ 37ºC------0.751.5 blood1570 @ 37ºC1.700.181.2 skull bone4080 @ 37ºC7.60 20.001.6 air 331 @ STP 0.0004 12.00 2 PZT4300 @ STP 33.00--------

30. 2.2Modeling the signal from a point scatterer Imagine that we have a transducer radiating into a medium and we wish to know the received signal due to a single point scatterer located at position By modifying the impulse response equation (Lecture 1 Equ. 25 ) we can write:

31. transmit + receive electromechanical IR’s scatterer IR transmit IR receive IR pulse (t) easily measured

32. Now consider a complex distribution of scatterers Isochronous volume rxrx riri (1) (2) (3) (4) At any point in the isochronous volume there exists a transmit – receive path length divided by c for a time, t, such that z l1l1 l2l2

33. If we look at the four field points shown on the previous page we would see the following impulse responses (1) (2) (3) (4)

34. The total signal for a given ray position r x is given by (9) scatterer strength

35. The resultant signal is the coherent sum of signals resulting from the group of randomly positioned scatterers that make up the isochronous volume as a function of time. A useful model of the signal is: EnvelopeModulated carrier Phase Grayscale information for B-scan Image How do we calculate a(t) and (t)? Velocity information for Doppler (10)

36. 3.3Hilbert Transform The Hilbert transform is an unusual form of filtration in which the spectral magnitude of a signal is left unchanged but its phase is altered by for negative frequencies and for positive frequencies Definition (11)

37. In the frequency domain Consider the Hilbert transform of Cos (12)

38. The application of two successive Hilbert transforms results in the inversion of the signal – we have 2 successive rotations in the negative frequency range and 2 rotations in the positive frequency range. Thus the total shift in each direction is.

40. The Hilbert transform is interesting but what good is it? ANALYTIC SIGNAL THEORY Consider a real function. Associate with this function another function called the analytic signal defined by: where = Hilbert Transform The real part of the analytic signal is the function itself whereas the imaginary part is the Hilbert transform of the function. Note that the real and imaginary components of the analytic signal are often called the “in phase”, I, and “quadrature”, Q, components. (13)

41. Just as complex phasors simplify many problems in AC circuit analysis the analytic signal simplifies many signal processing problems. The Fourier transform of the analytic signal has an interesting property. (14)

42. Equation 14 gives us an easy way to calculate the analytic signal of a function: 1)Fourier transform function 2)Truncate negative frequencies to zero 3)Multiply positive frequencies by 2 4)Inverse Fourier Transform Recall that our resultant ultrasound signal can be expressed as: Its analytic signal is then (15)

43. which on the complex plane looks like: Where and the phase is given by (16) (17) a(t) envelope

44. Demodulation: estimate using 1)Analytic signal method using FFT (slow) 2)Analytic signal using baseband quadrature approach 3)Sampled quadrature Baseband Quadrature Demodulation Low Pass Low Pass Baseband Inphase Signal Baseband Quadrature Signal

45. Use shift and convolution theorems to calculate spectra (slowly varying)

46. Similarly Baseband Analytic Signal No carrier Phase preserved

47. Thus and Sampled Quadrature Begin with the signal of the ultrasound waveform

48. Sample with period * ** Recall that the quadrature signal is the Hilbert Transform of the inphase component of the analytic signal i.e. for a cos wave it is a negative sine wave. Thus we see that...

49. If the inphase and quadrature signals are slowly varying we can get the quadrature signal simply by sampling the inphase signal 90º or ¼ period later Samplingt = nT for I samples t =nT+T/4 for Q sample let (18)

50. 1 Overall Imager Block Diagram 2 345 6

51. Imaging System Signals 2 345 6 1

52. Coarse and Fine Beamforming Delays

53. SIGNAL STATISTICS Recall that the ultrasound signal is the sum of harmonic components with random phase and amplitude. It can be shown that the probability density function for such a situation is Gaussian with zero mean i.e. (19) The quadrature signal will also be Gaussian with the same standard deviation (20)

54. Since p(y) and p(z) are independent random variables the joint probability density function is given by (21) The probability of a joint event (corresponding to a particular amplitude of the envelope) is the probability that:

55. total area = The probability that a lies between a and a + da is

56. So that the probability density function for the radio frequency signal is given by Rayleigh Prob. Density function few white pixels many gray pixels few black pixels

57. The speckle in an ultrasound image is described by this probability density function. Let’s define the signal as and the noise as the deviation from this value Thus Recall

58. Thus: SNR = 1.91 and is invariant (25) Note that the SNR in ultrasound imaging is independent of signal level. This is in contrast to x-ray imaging where the noise is proportional to the square root of the number of photons.

59. Speckle Noise in an Ultrasound Image

60. Let’s make several independent measurements of s o and s i These measurements will form distributions

61. The parameter used to define image quality includes both the observed contrast and the noise due to speckle in the following fashion: Define Contrast: Define Normalized speckle noise as: and finally, define our quality factor as the contrast to speckle noise ratio (CSR) (26)

62. Suggested Ultrasound Book References: General Biomedical Ultrasound (and physical/mathematical foundations): “Foundations of Biomedical Ultrasound”, RSC Cobbold, Oxford Press 2007. General Biomedical Ultrasound (bit more applied): “Diagnostic Ultrasound Imaging: inside out” TL Szabo Academic Press 2004. Ultrasound Blood flow detection/imaging: “Estimation of blood velocities with ultrasound” JA Jensen Cambridge university press 1996 Basic acoustics: “Theoretical Acoustics” PM Morse and KU Ingard, Princeton University Press (many editions). Bubble behaviour: “The Acoustic bubble” TG Leighton Academic Press 1997. Nonlinear Acoustics: “Nonlinear Acoustics” Hamilton and Blackstock, Academic Press 1998.