Communications in Applied Mathematics and Computational Science

Computational models of material interfaces for the study of extracorporeal shock wave therapy

Kirsten Fagnan, Randall LeVeque, and Thomas Matula

Full-text: Open access

Abstract

Extracorporeal shock wave therapy (ESWT) is a noninvasive treatment for a variety of musculoskeletal ailments. A shock wave is generated in water and then focused using an acoustic lens or reflector so the energy of the wave is concentrated in a small treatment region where mechanical stimulation in principle enhances healing. In this work we have computationally investigated shock wave propagation in ESWT by solving a Lagrangian form of the isentropic Euler equations in the fluid and linear elasticity in the bone using high-resolution finite volume methods. We solve a full three-dimensional system of equations and use adaptive mesh refinement to concentrate grid cells near the propagating shock. We can model complex bone geometries, the reflection and mode conversion at interfaces, and the propagation of the resulting shear stresses generated within the bone. We discuss the validity of our simplified model and present results validating this approach.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 8, Number 1 (2013), 159-194.

Dates
Received: 6 December 2010
Accepted: 6 November 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732083

Digital Object Identifier
doi:10.2140/camcos.2013.8.159

Mathematical Reviews number (MathSciNet)
MR3153665

Zentralblatt MATH identifier
1278.92018

Subjects
Primary: 92-08: Computational methods 92C50: Medical applications (general) 65M08: Finite volume methods

Keywords
high-resolution finite volume methods computational biology shock wave therapy

Citation

Fagnan, Kirsten; LeVeque, Randall; Matula, Thomas. Computational models of material interfaces for the study of extracorporeal shock wave therapy. Commun. Appl. Math. Comput. Sci. 8 (2013), no. 1, 159--194. doi:10.2140/camcos.2013.8.159. https://projecteuclid.org/euclid.camcos/1513732083


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References

  • Chombo: software for adaptive solutions of partial differential equations, webpage, Applied Numerical Algorithms Group (ANAG), Lawrence Berkeley National Laboratory, 2009.
  • P. Augat, J. Merk, S. Wolf, and L. E. Claes, Mechanical stimulation by external application of cyclic tensile strains does not effectively enhance bone healing, Journal of Orthopaedic Trauma 15 (2001), 54–60.
  • M. Averkiou and R. Cleveland, Modeling of an electrohydraulic lithotripter with the KZK equation, Journal for the Acoustical Society of America 106 (1999), no. 1, 102–112.
  • D. S. Bale, R. J. Leveque, S. Mitran, and J. A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. Sci. Comput. 24 (2002), no. 3, 955–978.
  • M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82 (1989), 64–84.
  • M. J. Berger and I. Rigoutsos, An algorithm for point clustering and grid generation, IEEE Trans. Sys. Man & Cyber. 21 (1991), 1278–1286.
  • M. J. Berger and R. J. Leveque, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2298–2316.
  • M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53 (1984), no. 3, 484–512.
  • R. Biedermann, A. Martin, G. Handle, T. Auckenthaler, C. Bach, and M. Krismer, Extracorporeal shock waves in the treatment of nonunions, Journal of Trauma 54 (2003), no. 5, 936–42.
  • D. A. Calhoun, P. Colella, and R. J. LeVeque, chombo-claw software, web site.
  • D. R. Carter, G. S. Beaupre, N. J. Giori, and J. A. Helms, Mechanobiology of skeletal regeneration, Clinical Orthopaedics and Related Research 355 \normalfont(Suppl) (1998), S41–55.
  • T. Christopher, Modeling the Dornier HM3 lithotripter, Journal of the Acoustical Society of America 96 (1994), no. 5, 3088–3095.
  • L. E. Claes and C. A. Heigele, Magnitudes of local stress and strain along osseous surfaces predict the course and type of fracture-healing, The Journal of Biomechanics 32 (1999), 255–266.
  • L. E. Claes, H. J. Wilke, P. Augat, S. Rubenacker, and K. J. Margevicius, Effect of dynamization on gap healing of diaphyseal fractures under external fixation, Cinical Biomechanics 10 (1995), 227–234.
  • Clawpack Team, Clawpack software, web page, 2013.
  • R. O. Cleveland and O. Sapozhnikov, Modeling elastic wave propagation in kidney stones with application to shock wave lithotripsy, Journal of the Acoustical Society of America 118 (2005), no. 4, 2667–2676.
  • A. J. Coleman, J. Saunders, R. Preston, and D. Bacon, Pressure waveforms generated by a Dornier extra-corporeal shock-wave lithotripter, Ultrasound in Medicine and Biology 13 (1987), 651–657.
  • K. M. \vphantomFagnaFagnan, High-resolution finite volume methods for extracorporeal shock wave therapy, Ph.D. thesis, University of Washington, 2010.
  • K. Fagnan, R. J. LeVeque, T. J. Matula, and B. MacConaghy, High-resolution finite volume methods for extracorporeal shock wave therapy, Hyperbolic problems: theory, numerics, applications (S. Benzoni-Gavage and D. Serre, eds.), Springer, Berlin, 2008, pp. 503–510.
  • J. Freund, T. Colonius, and A. Evan, A cumulative shear mechanism for tissue damage initiation in shock-wave lithotripsy, Ultrasound in Medicine and Biology 33 (2007), 1495–1503.
  • Y. C. Fung, Biomechanics: mechanical properties of living tissues, Springer, 1993.
  • E. Garner, R. Lakes, T. Lee, C. Swan, and R. Brand, Viscoelastic dissipation in compact bone: implications for stress-induced fluid flow in bone, Journal of Biomechanical Engineering 122 (2000), 166–73.
  • A. E. Goodship and J. Kenwright, The influence of induced micromovement on the healing of experimental tibial fractures, Journal of Bone and Joint Surgery British Volume 67 (1985), 650–655.
  • M. Hamilton, Transient axial solution for the reflection of a spherical wave from a concave ellipsoidal mirror, Journal of the Acoustical Society of America 93 (1993), no. 3, 1256–1266.
  • M. V. Hillsley and J. A. Frangos, Review of “Bone tissue engineering: the role of interstitial fluid flow”, Biotechnology and Bioengineering 43 (1994), 573–581.
  • C. Huang and R. Ogawa, Mechanotransduction in bone repair and regeneration, The FASEB Journal 23 (2010), 3625–3632.
  • H. Isaksson, W. Wilson, C. C. van Donkelaar, R. Huiskes, and K. Ito, Comparison of biophysical stimuli for mechano-regulation of tissue differentiation during fracture-healing, The Journal of Biomechanics 39 (2006), 1507–1516.
  • M. J. Ivings, D. M. Causon, and E. F. Toro, On Riemann solvers for compressible liquids, Internat. J. Numer. Methods Fluids 28 (1998), no. 3, 395–418.
  • T. Keaveny, X. E. Guo, E. F. Wachtel, T. A. McMahon, and W. C. Hayes, Trabecular bone exhibits fully linear elastic behavior and yields at low strains, Journal of Biomechanics 27 (1994), 1127–1129, 1131–1136.
  • D. Lacroix and P. J. Prendergast, A mechano-regulation model for tissue differentiation during fracture-healing: analysis of gap size and loading, The Journal of Biomechanics 35 (2002), 1163–1171.
  • J. O. Langseth and R. J. LeVeque, A wave propagation method for three-dimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000), no. 1, 126–166.
  • G. I. Lemoine, Numerical modeling of poroelastic-fluid systems using high-resolution finite volume methods, Ph.D. thesis, University of Washington, 2013.
  • G. I. Lemoine, M. Y. Ou, and R. J. LeVeque, High-resolution finite volume modeling of wave propagation in orthotropic poroelastic media, SIAM J. Sci. Comput. 35 (2013), no. 1, B176–B206.
  • R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys. 131 (1997), 327–353.
  • ––––, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, no. 31, Cambridge University Press, 2002.
  • ––––, Finite-volume methods for non-linear elasticity in heterogeneous media, Internat. J. Numer. Methods Fluids 40 (2002), no. 1-2, 93–104.
  • R. B. Martin, D. B. Burr, and N. A. Sharkey, Skeletal tissue mechanics, Springer, New York, 1998.
  • T. J. Matula, P. R. Hilmo, and M. R. Bailey, A suppressor to prevent direct wave-induced cavitation in shock wave therapy devices, Journal of the Acoustical Society of America 118 (2005), no. 1, 178–185.
  • E. F. Morgan, R. E. Gleason, L. N. M. Hayward, P. L. Leong, and K. T. Salisbury-Paolomares, Mechanotransduction and fracture repair, Journal of Bone and Joint Surgery American Volume 90 \normalfont(Suppl 1) (2008), 25–30.
  • G. Mouzopoulos, M. Stamatakos, D. Mouzopoulos, and M. Tzurbakis, Extracorporeal shock wave treatment for shoulder calcific tendonitis: a systematic review, Skeletal Radiology 36 (2008), no. 9, 803–811.
  • M. Nakahara, K. Nagayama, and Y. Mori, Shockwave dynamics of high pressure pulse in water and other biological materials based on Hugoniot data, Japanese Journal of Applied Physics 47 (2008), 3510.
  • J. A. Ogden, A. Toth-Kischkat, and R. Schultheiss, Principles of shock wave therapy, Clinical Orthopaedics and Related Research 387 (2001), 8–17.
  • S. H. Park, K. O'Connor, H. McKellop, and A. Sarmiento, The influence of active shear or compressive motion on fracture-healing, The Journal of Bone and Joint Surgery American Volume 80 (1998), 868–878.
  • P. J. Prendergast, R. Huiskes, and K. Soballe, Biophysical stimuli on cells during tissue differentiation at implant interfaces, The Journal of Biomechanics 30 (1997), 539–548.
  • A. G. Robling, F. M. Hinant, D. B. Burr, and C. H. Turner, Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts, Journal of Bone Mineral Research 17 (2002), 1545–1554.
  • ––––, Shorter, more frequent mechanical loading sessions enhance bone mass, Medicine and Science in Sports and Exercise 34 (2002), 196–202.
  • T. Saito, M. Marumoto, H. Yamashita, S. H. R. Hosseini, A. Nakagawa, T. Hirano, and K. Takayama, Experimental and numerical studies of underwater shock wave attenuation, Shock Waves 13 (2003), 139–148.
  • O. Sapozhnikov, M. Bailey, and R. O. Cleveland, The role of shear and longitudinal waves in the kidney stone comminution by a lithotripter shock pulse, Journal of the Acoustical Society of America 115 (2004), 2562.
  • O. Sapozhnikov, A. D. Maxwell, B. MacConaghy, and M. Bailey, A mechanistic analysis of stone fracture in lithotripsy, Journal of the Acoustical Society of America 121 (2007), no. 2, 1190–1202.
  • L. Saxon, A. Robling, I. Alam, and C. Turner, Mechanosensitivity of the rat skeleton decreases after a long period of loading, but is improved with time off, Bone 36 (2005), 454–464.
  • M. Tanguay, Computation of bubbly cavitating flow in shock wave lithotripsy, Ph.D. thesis, California Institute of Technology, 2004.
  • W. R. Taylor, E. Roland, H. Ploeg, D. Hertig, R. Klabunde, M. D. Warner, M. C. Hobatho, L. Rakotomanana, and S. E. Clift, Determination of orthotropic bone elastic constants using FEA and modal analysis, Journal of Biomechanics 35 (2002), 767–773.
  • C. H. Turner and F. M. Pavalko, Mechanotransduction and functional response of the skeleton to physical stress: the mechanisms and mechanics of bone adaptation, Journal of Orthopaedic Science 3 (1998), 346–355.
  • C.-J. Wang, K.-E. Huang, Y.-C. Sun, Y.-J. Yang, J.-Y. Ko, L.-H. Weng, and F.-S. Wang, VEGF modulates angiogenesis and osteogenesis in shockwave-promoted fracture healing in rabbits, Journal of Surgical Research 171 (2011), no. 1, 114–119.
  • C.-J. Wang, F.-S. Wang, J.-Y. Ko, H.-Y. Huang, C.-J. Chen, Y.-C. Sun, and Y.-J. Yang, Extracorporeal shockwave therapy shows regeneration in hip necrosis, Rheumatology 47 (2008), no. 4, 542–546.
  • S. Weinbaum, S. Cowin, and Y. Zeng, A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stress, Journal of Biomechanics 27 (1994), 339–360.