1. 1 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

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

3. 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: - Nonlinearity parameter In the absence of average mass flux: Rayleigh radiation pressure: In an acoustic beam where Langevin radiation pressure:

4. 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] depends on the Rayleigh force and takes into account physical nonlinearity in the equation of state and “geometrical” nonlinearity:

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

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

7. 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. 8 Ultrasound-induced displacements in tissue samples ( Sutin, Sarvazyan ) Doppler measurement data Time reversal (TRA) Blue –radiated signal Red –recorded TRA focused signal

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

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

11. 11 Shear force component: Narrow-angle beam: KZK equation: (in terms of Mach number) Radiation Force: From here (similar to the known expression but with nonlinear M a ): 2

12. 12 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: WAVES 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. 13 AVERAGING Let us represent u as a sum of two vectors, potential, U 1 so that x U 1 = 0, and solenoidal, U 2, for which ( · U 2 ) = 0. As a result, Potential: Solenoidal:

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

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

16. 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 m 2 /s

17. 17 EXAMPLES = 0.015 m 2 /s 40 ms20 ms = 0.0015 m 2 /s

18. 18 3-D PLOT

19. 19 Spatial distribution of force

20. 20 Inhomogeneous/Non-dissipative LONGITUDINAL DISTRIBUTION F = 10 cm D 0 = 3 cm Homogeneous/Dissipative

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

22. 22 NONLINEAR PRIMARY BEAM

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

24. 24 Applicability: until diffraction becomes significant (outside the focal length at the 1 st harmonic): Hence, At small amplitude (b <<1) : F x / F x (r = F) M 0 =10 -4,  = 15 °, F = 10 cm, f = 1 MHz

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

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

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

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

29. 29 NON-DISSIPATIVE, NONLINEAR RF (geometrical stage) At small b: In spite of a higher power of M 0, this force can prevail over the dissipative one

30. 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:. M o =10 -4,  = 15°, F = 10 cm, f = 1 MHz

31. 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.