Biomedical Imaging I

1. Biomedical Imaging I Class 8 – Ultrasound Imaging Physics of Matter-Energy Interactions 11/02/05

2. Different Forms of Energy • Electromagnetic • Acoustic

3. Different Forms of Energy • Electromagnetic • Photons (quantum description), electromagnetic waves (classical description) • Connection: classical EM fields and waves are constituted of large numbers of photons • Does not require a material medium through which to propagate • Mechanisms of propagation through material media are different from that of propagation through free space • In some materials: [see W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, NJ, 1995), §1.1] • electric and magnetic field vectors rotate • Phase shifts develop between electric and magnetic field vectors • Acoustic

4. Different Forms of Energy • Electromagnetic • Acoustic • Requires a material medium through which to propagate • Consists of oscillatory motions of the atoms/molecules of which a material is constituted. • Oscillating particles have kinetic energy  square of amplitudes of their motions • Through action of intermolecular forces, particles transfer their energy to adjacent particles  energy wave traveling through material.

5. Transfer/Transformation of Energy • Light becomes sound — photoacoustic phenomena • Sound becomes light — sonoluminescence • Absorbed electromagnetic (EM) and acoustic energy both become heat • Nevertheless, EM and acoustic energy are FUNDAMENTALLY DISTINCT PHENOMENA!

6. Acoustic Wave Energy Ranges • Just as there are infrared, visible, and ultraviolet ranges in the EM spectrum, so there are infrasound (“infra” = “below,”“beneath”), audible (i.e., sound) and ultrasound (“ultra” = “beyond,”“above”) ranges of acoustic wave frequencies • Note that the ratio of the highest to the lowest audible frequencies is 103, or almost 10 octaves. On the other hand, the ratio of the highest to the lowest frequencies of visible light is a bit less than 2 (i.e., less than one octave). Infrasound Ultrasound Audible 20 Hz 20 kHz

7. “Mass–and–Spring” Representation of Matter U – particle displacement (10-10 m (H2O)) ux = dx/dt – particle velocity (10-2 m-s-1 (H2O)) U’ – distance energy propagates during one particle oscillation cycle (~10-5 m (H2O)) c – phase velocity, energy propagation velocity (1.5×103m-s-1 (H2O)) time Two 2Umax

8. Longitudinal Waves of Molecular Motion every Nth particle, N = U’/U, shown greatly enlarged Now T/2 ago T ago 3T/2 ago 2T ago … molecule oscillation time ~ (2×10-10 m)/(10-2 m-s-1) = 2×10-8 s  T distance wave propagates ~ (2×10 -8 s)·(1.5× 10-3 m-s-1) = 3×10-5 m

9. Definition of Stress ts1s2 force per unit area, acting in direction s1, applied to face that is  the s2 axis s1 = s2: compressional stress acting in s1 direction; by definition, is positive if directed outward from the volume element. i.e., tzz(z0+Δz) i.e., tzz(z0) s1  s2: shear stress in s1 direction; positive if || the positive s1 direction. pressure on any face = -1 × corresponding compressional stress force = stress × surface area; e.g., fzz = tzzΔxΔy

10. V y U x z W Definition of Strain strain: material’s reaction to applied stress(es); how much deformation it  W(z0 + Δz) undergoes; simply displacing the cube some distance from its initial location is not strain! = W(z0) εzz = compressional strain in z direction:  U(z0 + Δz) x0 + Δx x0 εxz = shear strain in x direction:

11. Moduli of Elasticity pressure volume Young’s modulus (E) tzz/εzz Bulk modulus (B)  (compressibility)-1 = [-(1/σ)(σ/p)]-1 = -σ/(σ/p) Lamé constants: μ, ν — Related to classical moduli via:

12. Moduli of Elasticity Lamé constants: μ, ν — Related to classical moduli via: — Inverting preceding formulae gives:

13. Shear Modulus vs. State of Matter Lamé constants: μ, ν — μ = shear modulus — When displacements U, V, and W all are small: — tzz (ν + 2μ)εzz = (ν + 2μ)(W/z) — tyz μεyz = μ(V/z) — txz μεxz = μ(U/z) — In fluids (liquid or gas), μ 0: E  0, B  ν — i.e., fluids (including most biological soft tissues) can not support shear waves!

14. Inability of Fluid Media To Support Shear Waves Direction of wave propagation Directions of particle displacement Why do soft tissues behave more like fluids than like solids, with respect to (lack of) shear wave propagation?

15. Non–Rigorous Derivation of Linearized Equation of Motion 1) Estimate force, in z direction, on each face by multiplying corresponding stress and area: fzz = tzzΔxΔy, fzx = tzxΔyΔz, … 2) Sum differences between forces on each pair of parallel faces (because only non-zero force difference produces deformation) 3) Estimate net acceleration, in z direction, on cube as second temporal derivative of displacement W: acc.  2W/t 2 4) Mass of cube is its density, η, times its volume, ΔxΔyΔz 5) Equating result of step 2 with product of cube’s mass (step 4) and acceleration (step 3) gives:

16. Compressional and Shear Waves Special case #1 — tzx = tzy = 0 (i.e., no shear stress, medium is fluid): Special case #2 — tzz = tzy = 0 (i.e., shear wave, so medium is a solid):

17. Sinusoidal Solutions to Wave Equations Definition: Acoustic impedance  pressure/(particle velocity) Z± = p±/uz± = ±ηc (uz = W/t = iωW) [Compare electric circuit analogue, impedance = voltage/current]

18. Notice how similar these values are to each other and to that for water, metal gas acrylic and how different they are from these. soft tissues hard tissue Sinusoidal Solutions to Wave Equations

19. Frequency-domain Form of Wave Equation Formally identical to homogeneous potential equation (Zwillinger, p. 418) or homogeneous Helmholtz equation (Kak & Slaney, p. 210) One-dimensional: More generally: 1, 2, or 3-dimensional Laplacian operator For solutions that are not identically zero, addition of a source term is required: Source location Same dimensionality as 2

20. peak average pulse average spatial peak temporal average spatial average Intensity, Radiation Force • Intensity • Average Intensity: I = (½)ηcω2W02 = (½) p0u0 • Assumes sinusoidal, plane-wave irradiation • Temporal Average • Spatial Average • Combined Spatial and Temporal Averaging • Spatial Average Temporal Average, I(SATA) • Spatial Peak Temporal Average, I(SPTA) • Spatial Peak Pulse Average, I(SPPA)

21. Intensity, Radiation Force • Intensity • Average Intensity: I = (½)ηcω2W02 = (½) p0u0 • Assumes sinusoidal, plane-wave irradiation • Temporal Average • Spatial Average • Combined Spatial and Temporal Averaging • Spatial Average Temporal Average, I(SATA) • Spatial Peak Temporal Average, I(SPTA) • Spatial Peak Pulse Average, I(SPPA) • Radiation Force (fr) per Unit Area • Ultrasound exerts pressure at interfaces within the medium • fr = D(I/c), for plane-wave irradiation • D = 2 at strongly reflecting interface, D = 1 at strongly absorbing interface

22. pi pr Z1, u1 pt Z2, u2 Reflection and Refraction • Behavior or ultrasound at an interface between materials of different Z is analogous to behavior of light at interface between materials of different refractive index. • Fraction of pressure reflected = Reflection Coefficient, R; fraction of pressure transmitted = Transmission Coefficient, T • Intensity reflection and transmission coefficients are derived from the preceding equations and p = Zu and I = p02/(2Z):

23. Orders of magnitude larger than the value for water, because of absorption and scattering by other constituents of tissue. Attenuation • For plane wave propagating through homogeneous medium, • β = pressure attenuation coefficient: • Attenuation is caused by absorption and scattering • Two mechanisms for absorption: viscosity (drag) and relaxation • Scattering is really the same phenomenon as reflection, at interfaces of volumes whose dimensions are small relative to the ultrasound wavelength

24. Clinical Potential of Attenuation Measurements Note, overall attenuation coefficient β, not only absorption or only (back)scattering Infarcted myocardium Healthy myocardium That is, ultrasound attenuation and backscatter measurements can be used (among many other things) to assess extent of tissue death in myocardial infarction

25. Absorption Mechanisms • Viscosity-induced absorption • Some of the energy in the propagating beam is transferred, in the transverse direction, to adjacent portions of the material • Viscosity coefficient (ζ): • Attenuation coefficient: • Relaxation • Critical parameter is ratio of particle relaxation time  to wave period U´/c. • Negligible energy loss occurs if the ratio is either <<1 or >>1; substantial loss can occur when it is ~1. • Attenuation coefficient:

26. Tissue Absorption • Experimentally determined values of β: • Curves do not resemble homogeneous single-component result from previous slide; why not? • Tissue is a mixture of many components, each with its own fR: • Overall net attenuation curve is the sum of those for all of the components

27. Scattering (Far–Field) Note, this is microscopic, or single-particle cross section

28. (Infarcted Myocardium) (Healthy Tissue) Tissue Scattering • If all particle dimensions are << ultrasound wavelength, then σs ω4 or f4. • If particle is cylindrical with radius << ultrasound wavelength, then σs ω3 or f3. time-integrated measurements measurement gated to heart cycle (Healthy Myocardium)