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Radiation force

RADIATION FORCE, SHEAR WAVES, AND MEDICAL ULTRASOUND L. A. Ostrovsky Zel Technologies, Boulder, Colorado, USA, and Institute of Applied Physics, Nizhny Novgorod, Russia FNP, July 2007. Radiation force. Lord Rayleigh, 1902. Leon Brillouin, 1925. Paul Langevin, 1920s. Robert Wood.

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Radiation force

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  1. RADIATION FORCE, SHEAR WAVES, AND MEDICAL ULTRASOUNDL. A. OstrovskyZel Technologies, Boulder, Colorado, USA, andInstitute of Applied Physics, Nizhny Novgorod, RussiaFNP, July 2007

  2. Radiation force Lord Rayleigh, 1902 Leon Brillouin, 1925 Paul Langevin, 1920s RobertWood Alfred Lee Loomis Vilhelm Bjerknes 1906 1926-27

  3. RADIATION FORCE (RF), RADIATION STRESS, RADIATION PRESSURE - All are average forces generated by sound (ultrasound), acting on a body, boundary, or distributed in space. Momentum flux in a plane wave: In the absence of average mass flux: Rayleigh radiation pressure: - Nonlinearity parameter In an acoustic beam where Langevin radiation pressure:

  4. Non-dissipative, bulk radiation force Elastic nonlinearity leads to demodulation/rectification effect in modulated ultrasound that can be described in terms of nonlinear, non-dissipative radiation force. Nonlinear acoustic wave equation first derived by Westervelt (JASA, 1963) for parametric arrays [also suggested by Zverev and Kalachev in Russia in 1959]dependson theRayleigh force and takes into account physical nonlinearity in the equation of state and “geometrical” nonlinearity:

  5. In non-viscous case, = -  - For a harmonic wave, the forcing in (25) is constant in time: = FS. For a damping beam:

  6. Pumping and probing transducers Low frequency detector Shear Wave Elasticity Imaging (SWEI) (Sarvazyan et al, 1998)

  7. Shear Displacements(Sarvazyan, Rudenko et al, 1998) Simulated (dissip. force) MRI Ultrasound Measured (left) and calculated (right) space-time distribution of shear wave remotely induced in tissue by an ultrasonic pulse

  8. Ultrasound-induced displacements in tissue samples (Sutin, Sarvazyan) Doppler measurement data Time reversal (TRA) Blue –radiated signal Red –recorded TRA focused signal

  9. THEORY: Inhomogeneous Medium ELASTIC MEDIA: GENERAL NONLINEAR STRESS (Ostrovsky, Il’inskii, Rudenko, Sarvazyan, Sutin, 2007) Here ui is the displacement vector and σikis the stress tensor. Then Linear part:

  10. ELASTIC MEDIA: GENERAL NONLINEAR STRESS (Ostrovsky et al, JASA, 2007) AVERAGESTRESS COMPONENTS: Narrow-angle ultrasonic beam: G1− G2 = G3 =  + 3μ + A + 2B = Q In fluids and waterlike media,

  11. Radiation Force: Shear force component: Narrow-angle beam: KZK equation: (in terms of Mach number) 22 From here (similar to the known expression but with nonlinear Ma):

  12. WAVES In a smoothly inhomogeneous medium: Nonlinear wave equation for the displacement vector, u where are the velocities of linear longitudinal and transverse waves, respectively, and the linear term S is related to spatial parameter variations: Medium parameters may slowly depend on coordinate x that is directed along the primary beam axis. Here, S is of the 2nd order and further neglected.

  13. AVERAGING Let us represent u as a sum of two vectors, potential, U1 so that xU1 = 0, and solenoidal, U2, for which ( · U2 ) = 0. As a result, Potential: Solenoidal:

  14. FOR THE NARROW ACOUSTIC BEAM: Hence, for or

  15. AS A RESULT, in a harmonic beam: ADDING LINEAR LOSSES  = f /17.3 1/cm (f in MHz) Non-dissipative radiation force Dissipative radiation force Q = - ( + 3μ + A + 2B) For tissues, Q- c2

  16. CYLINDRICAL (PARAXIAL) BEAM Beam radius at a half-intensity level near focus: R = 0.3 cm Acoustic pressure in the focus: 2 MPa Length of medium acoustic parameter variation: 0.5 cm Shear wave velocity 3 m/s  = 15 Pas, so that  = 0.015 m2/s

  17. EXAMPLES  = 0.015 m2/s 20 ms 40 ms  = 0.0015 m2/s

  18. 3-D PLOT

  19. Spatial distribution of force

  20. LONGITUDINALDISTRIBUTION F = 10 cm D0 = 3 cm Inhomogeneous/Non-dissipative Homogeneous/Dissipative

  21. Application to lesion visualization (E. Ebbini) Tissue Displacement (0.9 mm away from the focal place) Effect increase in lesion can be explained by non-dissipative radiation force.

  22. NONLINEAR PRIMARY BEAM

  23. “GEOMETRICAL” STAGE (no diffraction) Implicit form:  Or x : shock formation

  24. Hence, At small amplitude (b <<1) : Applicability: until diffraction becomes significant (outside the focal length at the 1st harmonic): Fx/ Fx(r = F) M0=10-4 ,  = 15°, F = 10 cm, f = 1 MHz

  25. Focal Area (r < RF): Linear, diffracting non-sinusoidal wave (Ostrovsky&Sutin, 1975) Kirchhoff approximation (from S  RF2): RF x < 0 2 S At the focus (r = 0):

  26. Thus, focal force is Wave profiles: Force growth in the focal area: r = RF x = 0, z  0 (focal plane) r = 0 (focus)  = 0.7 (From Sutin, 1978)

  27. At the beam axis: RF IN SHOCK WAVES Sawtooth stage (b(RF)>1) Shock amplitude: Ostrovsky&Sutin, 1975; Sutin,1978

  28. At geometrical stage (See also Pishchalnikov et al, 2002) Near (before) the focus:

  29. NON-DISSIPATIVE, NONLINEAR RF (geometrical stage) At small b: In spite of a higher power of M0, this force can prevail over the dissipative one

  30. CONCLUSIONS • Nonlinear distortions in a focused ultrasonic beam can significantly enhance the resulting shear radiation force • Diffraction near the focus makes the force even stronger • The effects are different when the shocks form before the focal area • Nonlinear distortions can be of importance in biomedical experiments:. Mo=10-4 ,  = 15°, F = 10 cm, f = 1 MHz

  31. CONCLUSIONS Acoustic radiation force (RF) is a rather general notion referring to the average action of oscillating acoustic field in the medium. In water-like media such as biological tissues, shear motions generated by RF are much stronger than potential motions. This effect is used in medical diagnostics. To generate shear motions, at least one of the following factors must be accounted for: dissipation, inhomogeneity, and nonlinearity of a primary beam. The latter two are the new effects we have considered.

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