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

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 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 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.

Quantitative Assessments of Mechanical Responses upon Radial Extracorporeal Shock Wave Therapy

“…Numerical simulations are capable of obtaining comprehensive information of the pressure field, which is difficult for experimental measurements, especially inside biological tissues with complex geometries. [12b,30] Finite element method has been used to simulate the generation and propagation of the ballistic type rESW . However, the finite element method has not been widely used in studying rESWT.…”

An experimentally-calibrated damage mechanics model for stone fracture in shock wave lithotripsy

“…Within this interpretation the appearance of a secondary fracture area can be interpreted as arising from reflection of incoming shocks at the interface formed by the first fracture. Elastic wave propagation in inhomogeneous media using methods closely related to the approach in this work, but without consideration of a damage model, have also been reported [10]. Work directly addressing fatigue in SWL [20, 37] has mostly concentrated on analytical estimates in order to provide estimates of the number of cycles needed for appearance of the first fracture, with results that can vary by orders of magnitude [20] in the predicted number of shocks, mainly attributed to the need for incorporation of experimental data to better characterize crack growth.…”

Computational and In Vitro Studies of Blast-Induced Blood-Brain Barrier Disruption

“…To handle this range of materials we use the stiffened gas equation of state (SGEOS), also known as the Tammann EOS. This equation of state is very useful to model a wide range of fluids even in the presence of strong shock waves and was successfully used in [31,32] to model shock wave propagation in tissue and bone. The Tammann EOS is given by…”

“…It is worth mentioning that for sufficiently weak shocks the Tammann EOS can be further simplified to the Tait EOS, which neglects the energy coupling. In [31] this was shown to be adequate for modeling shocks in fluids and solids in the context of shock wave therapy. In this work, we will employ the Tammann EOS since it provides a more comprehensive approach and conserves the energy coupling that could be useful to relate to thermodynamic quantities.…”

“…Table 3.1: Parameters for the Tammann EOS to model the different materials. The parameters for air and water were taken from [31]. Since the polystyrene is a solid, γ was chosen very close to 1, and p ∞ was adjusted to yield the right speed of sound in polystyrene.…”