# American Institute of Mathematical Sciences

September  2016, 5(3): 449-461. doi: 10.3934/eect.2016013

## The Westervelt equation with a causal propagation operator coupled to the bioheat equation.

 1 Physics and Engineering Physics Department, Tulane University, New Orleans, LA 70118, United States, United States

Received  October 2015 Revised  January 2016 Published  August 2016

The Westervelt wave equation is frequently used to describe non-linear propagation of finite amplitude sound. If one assumes that the medium can be treated as a thermoviscous fluid, a loss mechanism can be incorporated. In this as in previous work the authors replaced the typical loss mechanism incorporated in the Westervelt equation with a causal Time Domain Propagation Factor (TDPF) which incorporates the full dispersive effects (both frequency dependent phase velocity and attenuation) in the numerical solution while remaining in the time-domain. In the present work we investigate heat deposition due to finite amplitude propagation through a dispersive medium (e.g., human tissue). To this end, the Westervelt equation with and without the TDPF is coupled to the Pennes bioheat equation and the coupled equations are solved using the method of finite differences to determine the resulting heat deposition. We show that non-linear effects are large and that proper treatment of dispersion results in significant changes as compared to modeling the medium as a thermoviscous fluid.
Citation: Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
##### References:
 [1] J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159. [2] R. N. Bracewell, The Fourier Transform and its Applications, $2^{nd}$ edition, McGraw-Hill, New York, 1986. [3] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04823-8. [4] I. M. Hallaj and R. O. Cleveland, FDTD simulation of finite-amplitude pressure and temperature fields for biomedical ultrasound, Journal of the Acoustic Society of America, 105 (1999), L7-L12. doi: 10.1121/1.426776. [5] Y. Jing and R. O. Cleveland, Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media, Journal of the Acoustic Society of America, 122 (2007), 1352-1364. doi: 10.1121/1.2767420. [6] P. M. Jordan and C. I. Christov, A simple finite difference scheme for modeling the finite-time blow-up of acoustic acceleration waves, Journal of Sound and Vibration, 281 (2005), 1207-1216. doi: 10.1016/j.jsv.2004.03.067. [7] R. D. L. Krönig, On the theory of dispersion of X-rays, Journal of the Optical Society of America, 12 (1926), 547-557. [8] G. V. Norton and R. D. Purrington, The Westervelt equation with viscous attenuation versus a causal propagation operator: A numerical comparison, Journal of Sound and Vibration, 327 (2009), 163-172. doi: 10.1016/j.jsv.2009.05.031. [9] H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forarm, Journal of Applied Physiology, 1 (1948), 93-122. [10] A. D. Pierce, Acoustics. An Introduction to its Physical Principles and Applications, Acoustical Society of America, NewYork, 1989. [11] R. D. Purrington and G. V. Norton, A numerical comparison of the Westervelt equation with viscous attenuation and a causal propagation operator, Mathematics and Computers in Simulation, 82 (2012), 1287-1297. doi: 10.1016/j.matcom.2010.05.017. [12] J. B. Schneider and O. M. Ramahi, The complementary operator method applied to acoustic finite-difference time-domain simulations, Journal of the Acoustic Society of America, 104 (1998), 686-693. doi: 10.1121/1.423343. [13] M. Solovchuk, T. W. H. Sheu and M. Thiriet, Simulation of nonlinear Westervelt equation for the investigation of acoustic streaming and nonlinear propagation effects, Journal of the Acoustic Society of America, 134 (2013), 3931-3942. doi: 10.1121/1.4821201. [14] T. L. Szabo, Time domain nonlinear wave equations for lossy media, Proceedings of the 13th International Symposium on Nonlinear Acoustics, Bergen, June 1993, 89-94. [15] T. L. Szabo, Time domain wave equations for lossy media obeying a frequency power law, Journal of the Acoustic Society of America, 96 (1994), 491-500. doi: 10.1121/1.410434. [16] T. L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law, Journal of the Acoustic Society of America, 97 (1995), 14-24. doi: 10.1121/1.412332. [17] T. L. Szabo and J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media, Journal of the Acoustic Society of America, 107 (2000), 2437-2446. doi: 10.1121/1.428630. [18] T. L. Szabo, Diagnostic Ultrasound Imaging, Elsevier Academic Press, San Diego, 2004. [19] T. Watson, Ultrasound Therapy: The Basics, International Society for Electro-Physical Agents in Physical Therapy, 1995.

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##### References:
 [1] J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114 (1994), 185-200. doi: 10.1006/jcph.1994.1159. [2] R. N. Bracewell, The Fourier Transform and its Applications, $2^{nd}$ edition, McGraw-Hill, New York, 1986. [3] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04823-8. [4] I. M. Hallaj and R. O. Cleveland, FDTD simulation of finite-amplitude pressure and temperature fields for biomedical ultrasound, Journal of the Acoustic Society of America, 105 (1999), L7-L12. doi: 10.1121/1.426776. [5] Y. Jing and R. O. Cleveland, Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media, Journal of the Acoustic Society of America, 122 (2007), 1352-1364. doi: 10.1121/1.2767420. [6] P. M. Jordan and C. I. Christov, A simple finite difference scheme for modeling the finite-time blow-up of acoustic acceleration waves, Journal of Sound and Vibration, 281 (2005), 1207-1216. doi: 10.1016/j.jsv.2004.03.067. [7] R. D. L. Krönig, On the theory of dispersion of X-rays, Journal of the Optical Society of America, 12 (1926), 547-557. [8] G. V. Norton and R. D. Purrington, The Westervelt equation with viscous attenuation versus a causal propagation operator: A numerical comparison, Journal of Sound and Vibration, 327 (2009), 163-172. doi: 10.1016/j.jsv.2009.05.031. [9] H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forarm, Journal of Applied Physiology, 1 (1948), 93-122. [10] A. D. Pierce, Acoustics. An Introduction to its Physical Principles and Applications, Acoustical Society of America, NewYork, 1989. [11] R. D. Purrington and G. V. Norton, A numerical comparison of the Westervelt equation with viscous attenuation and a causal propagation operator, Mathematics and Computers in Simulation, 82 (2012), 1287-1297. doi: 10.1016/j.matcom.2010.05.017. [12] J. B. Schneider and O. M. Ramahi, The complementary operator method applied to acoustic finite-difference time-domain simulations, Journal of the Acoustic Society of America, 104 (1998), 686-693. doi: 10.1121/1.423343. [13] M. Solovchuk, T. W. H. Sheu and M. Thiriet, Simulation of nonlinear Westervelt equation for the investigation of acoustic streaming and nonlinear propagation effects, Journal of the Acoustic Society of America, 134 (2013), 3931-3942. doi: 10.1121/1.4821201. [14] T. L. Szabo, Time domain nonlinear wave equations for lossy media, Proceedings of the 13th International Symposium on Nonlinear Acoustics, Bergen, June 1993, 89-94. [15] T. L. Szabo, Time domain wave equations for lossy media obeying a frequency power law, Journal of the Acoustic Society of America, 96 (1994), 491-500. doi: 10.1121/1.410434. [16] T. L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law, Journal of the Acoustic Society of America, 97 (1995), 14-24. doi: 10.1121/1.412332. [17] T. L. Szabo and J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media, Journal of the Acoustic Society of America, 107 (2000), 2437-2446. doi: 10.1121/1.428630. [18] T. L. Szabo, Diagnostic Ultrasound Imaging, Elsevier Academic Press, San Diego, 2004. [19] T. Watson, Ultrasound Therapy: The Basics, International Society for Electro-Physical Agents in Physical Therapy, 1995.
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