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 & 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. doi: 10.1006/jcph.1994.1159. Google Scholar

[2]

R. N. Bracewell, The Fourier Transform and its Applications, $2^{nd}$, edition, (1986). Google Scholar

[3]

G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04823-8. Google Scholar

[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). doi: 10.1121/1.426776. Google Scholar

[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. doi: 10.1121/1.2767420. Google Scholar

[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. doi: 10.1016/j.jsv.2004.03.067. Google Scholar

[7]

R. D. L. Krönig, On the theory of dispersion of X-rays,, Journal of the Optical Society of America, 12 (1926), 547. Google Scholar

[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. doi: 10.1016/j.jsv.2009.05.031. Google Scholar

[9]

H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forarm,, Journal of Applied Physiology, 1 (1948), 93. Google Scholar

[10]

A. D. Pierce, Acoustics. An Introduction to its Physical Principles and Applications,, Acoustical Society of America, (1989). Google Scholar

[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. doi: 10.1016/j.matcom.2010.05.017. Google Scholar

[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. doi: 10.1121/1.423343. Google Scholar

[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. doi: 10.1121/1.4821201. Google Scholar

[14]

T. L. Szabo, Time domain nonlinear wave equations for lossy media,, Proceedings of the 13th International Symposium on Nonlinear Acoustics, (1993), 89. Google Scholar

[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. doi: 10.1121/1.410434. Google Scholar

[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. doi: 10.1121/1.412332. Google Scholar

[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. doi: 10.1121/1.428630. Google Scholar

[18]

T. L. Szabo, Diagnostic Ultrasound Imaging,, Elsevier Academic Press, (2004). Google Scholar

[19]

T. Watson, Ultrasound Therapy: The Basics,, International Society for Electro-Physical Agents in Physical Therapy, (1995). Google Scholar

show all references

References:
[1]

J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves,, Journal of Computational Physics, 114 (1994), 185. doi: 10.1006/jcph.1994.1159. Google Scholar

[2]

R. N. Bracewell, The Fourier Transform and its Applications, $2^{nd}$, edition, (1986). Google Scholar

[3]

G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04823-8. Google Scholar

[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). doi: 10.1121/1.426776. Google Scholar

[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. doi: 10.1121/1.2767420. Google Scholar

[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. doi: 10.1016/j.jsv.2004.03.067. Google Scholar

[7]

R. D. L. Krönig, On the theory of dispersion of X-rays,, Journal of the Optical Society of America, 12 (1926), 547. Google Scholar

[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. doi: 10.1016/j.jsv.2009.05.031. Google Scholar

[9]

H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forarm,, Journal of Applied Physiology, 1 (1948), 93. Google Scholar

[10]

A. D. Pierce, Acoustics. An Introduction to its Physical Principles and Applications,, Acoustical Society of America, (1989). Google Scholar

[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. doi: 10.1016/j.matcom.2010.05.017. Google Scholar

[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. doi: 10.1121/1.423343. Google Scholar

[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. doi: 10.1121/1.4821201. Google Scholar

[14]

T. L. Szabo, Time domain nonlinear wave equations for lossy media,, Proceedings of the 13th International Symposium on Nonlinear Acoustics, (1993), 89. Google Scholar

[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. doi: 10.1121/1.410434. Google Scholar

[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. doi: 10.1121/1.412332. Google Scholar

[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. doi: 10.1121/1.428630. Google Scholar

[18]

T. L. Szabo, Diagnostic Ultrasound Imaging,, Elsevier Academic Press, (2004). Google Scholar

[19]

T. Watson, Ultrasound Therapy: The Basics,, International Society for Electro-Physical Agents in Physical Therapy, (1995). Google Scholar

[1]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429

[2]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006

[3]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019

[4]

Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286

[5]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[6]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

[7]

Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495

[8]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[9]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

[10]

Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057

[11]

Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227

[12]

Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905

[13]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

[14]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558

[15]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637

[16]

Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34

[17]

P. M. Jordan. The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock's models. Evolution Equations & Control Theory, 2016, 5 (3) : 383-397. doi: 10.3934/eect.2016010

[18]

Pedro M. Jordan. Finite-amplitude acoustics under the classical theory of particle-laden flows. Evolution Equations & Control Theory, 2019, 8 (1) : 101-116. doi: 10.3934/eect.2019006

[19]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[20]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]