Biomedical Imaging I

1. Biomedical Imaging I Class 9 – Ultrasound Imaging Doppler Ultrasonography; Image Reconstruction 11/09/05

2. θ R observer velocity source velocity Doppler Effect • Change in ultrasound frequency caused by motion of source (which can be a scatterer) and/or receiver relative to the background medium • Effect of receiver motion is different from that of source motion (Why?). Combining both effects gives: • If v = v’,

3. How about if the medium is moving past a stationary source and detector? Ultrasound transmitter Artery Ultrasound receiver Limb Clinical Application of Doppler Effect see: C. Holcombe et al., “Blood flow in breast cancer and fibroadenoma estimated by colour Doppler ultrasonography,” British J. Surgery82, 787-788 (1995). • If v = v’, • But why would it ever be the case that source and detector both are moving in the same direction with the same speed?

4. Net Doppler Shift Source Receiver c λ f = wave frequency [s-1] λ = wavelength [m] c = wave (or phase, or propagation) velocity [m-s-1] c = λf

5. Net Doppler Shift Case 1: source in motion relative to medium and receiver v v = source speed [cm-s-1] T = 1/f = wave period [s] vT = v/f = distance source travels between emission of successive wavefronts (crests) [m] λ´ = λ–v/f = c/f – v/f = (c – v)/f f’ = c/λ´ = [c/(c – v)]f

6. Net Doppler Shift Case 2: receiver in motion relative to medium and source v v = receiver speed [m-s-1] c’ = c + v = wave propagation speed in receiver’s frame of reference [m-s-1] λ´= λ c´ = c + v f’ = c´/λ = [(c + v)/c]f

7. Net Doppler Shift Case 3: source and detector in motion relative to each other and medium vs vr f’net = f’sf’r/f = [c/(c – vs)][(c + vr)/c]f = [(c + vr)/(c – vs)]f fd = f’net – f = [(c + vr)/(c – vs) - 1]f

8. Nonlinear Features of Ultrasound Wave Propagation • Pressure p can be expressed as a function of density η: • In combination with the fact that , we get • Note that if B  0, then c is a function of p. Wave crests (regions of compression) propagate faster than wave troughs (regions of rarefaction)! • Observable significance of this dependence is...? condensation

9. Nonlinear Features of Ultrasound: Shock Waves

10. Ultrasound Computed Tomography • Recommended supplemental reading: • A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,”Ultrasonic Imaging4, 336-350 (1982). • J. F. Greenleaf, “Computerized tomography with ultrasound,”Proceedings of the IEEE71, 330-337 (1983). • H. Schomberg, W. Beil, G. C. McKinnon, R. Proksa, and O. Tschendel, “Ultrasound computerized tomography,”Acta Electronica26, 121-128 (1984). • J. Ylitalo, J. Koivukangas, and J. Oksman, “Ultrasonic reflection mode computed tomography through a skullbone,”IEEE Transactions on Biomedical Engineering37, 1059-1066 (1990). • Kak and Slaney, Chapters 6 (Tomographic Imaging with Diffracting Sources) and 8 (Reflection Tomography)

11. Ultrasound Computed Tomography • Elementary Forms of Ultrasound CT Types • Ultrasonic Refractive Index Tomography • Projection: • Ultrasonic Attenuation Tomography • Projection: • Three methods for estimating attenuation line integral: • Energy-Ratio Method • Division of Transforms Followed by Averaging Method • Frequency-Shift Method • All of the foregoing are predicated on an assumption of negligible refraction/diffraction/scattering of ultrasound beams in the medium

12. Ultrasound Computed Tomography Kak and Slaney, pp. 153, 154

13. Ultrasound Computed Tomography Ultrasound Refractive Index CT Image Photograph Kak and Slaney, p. 154

14. Ultrasound Computed Tomography Ultrasonic Attenuation CT Images Division of Transforms Method f-shift Method E-ratio Method Kak and Slaney, pp. 156-158

15. Energy-Ratio Method • x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse, yw(t) = detected pulse for transmission through water • FT  X(f), Y(f), Yw(f) • Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|. • E(fk) = energy (or power), at frequency fk, in H(f). • Consider any two specific frequencies, f1 and f2, for which E(f1) and E(f2) can be reliably and accurately determined. Then in principle:

16. Division of Transforms Followed by Averaging Method • x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse, yw(t) = detected pulse for transmission through water • FT  X(f), Y(f), Yw(f), Yw(f) • Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|. • HA(f) = -ln|H(f)| = -ln|Y(f)/Yw(f)|. • In principle:

17. Frequency-Shift Method • x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse • FT  X(f), Y(f), Yw(f) • f0 = frequency at which Yw(f) is maximal • fr = frequency at which Y(f) is maximal • σ2 = width of |Yw(f)| • In principle:

18. Ultrasound Computed Tomography ...compared with x-ray CT mammograms of the same patient. Ultrasound CT mammography... Kak and Slaney, pp. 159-160

19. Diffraction Tomography with Ultrasound • What we can (attempt to) do when the “negligible refraction/diffraction/scattering” criterion mentioned earlier is violated • Based upon treating ultrasound propagation through medium as a wave phenomenon, not as a particle (i.e., ray) phenomenon • For homogeneous media, the Fourier Diffraction Theorem is analogous to the central-slice theorem of x-ray CT • Heterogeneous media are treated as (we hope) small perturbations of a homogeneous medium, to which an assumption such as the Born approximation or Rytov approximation can be applied

20. Fourier Diffraction Theorem I Arc radius is  ultrasound frequency, or wavenumber Kak and Slaney, pp. 219

21. Fourier Diffraction Theorem II Arc radius dependenceon wavenumber Tomographic measurements fill up Fourier space Kak and Slaney, pp. 228, 229