Maria G. Vavva

Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone

The classical linear theory of elasticity has been largely used for the ultrasonic characterization of bone. However, linear elasticity cannot adequately describe the mechanical behavior of materials with microstructure in which the stress state has to be defined in a non-local manner. In this study, the simplest form of gradient theory (Mindlin Form-II) is used to theoretically determine the velocity dispersion curves of guided modes propagating in isotropic bone-mimicking plates. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g, introduced in the strain energy, and (b) the micro-inertia term h, in the kinetic energy. The plate was assumed free of stresses and of double stresses. Two cases were studied for the characteristic length: h=10(-4) m and h=10(-5) m. For each case, three subcases for g were assumed, namely, g>h, g<h, and g=h. The values of g and h were of the order of the osteons size. The velocity dispersion curves of guided waves were numerically obtained and compared with the Lamb modes. The results indicate that when g was not equal to h (i.e., g not equal h), microstructure affects mode dispersion by inducing both material and geometrical dispersion. In conclusion, gradient elasticity can provide supplementary information to better understand guided waves in bones.

Ultrasonic monitoring of bone fracture healing

Quantitative ultrasound has attracted significant interest in the evaluation of bone fracture healing. Animal and clinical studies have demonstrated that the propagation velocity across fractured bones can be used as an indicator of healing. Researchers have recently employed computational methods for modeling wave propagation in bones, aiming to gain insight into the underlying mechanisms of wave propagation and to further enhance the monitoring capabilities of ultrasound. In this paper, we review the relevant literature and present the current status of knowledge.

The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone.

Guided wave propagation has recently drawn significant interest in the ultrasonic characterization of bone. In this work, we present a two-dimensional computational study of ultrasound propagation in healing bones aiming at monitoring the fracture healing process. In particular, we address the effect of fluid loading boundary conditions on the characteristics of guided wave propagation, using both time and time-frequency (t-f) signal analysis techniques, for three study cases. In the first case, the bone was assumed immersed in blood which occupied the semi-infinite spaces of the upper and lower surfaces of the plate. In the second case, the bone model was assumed to have the upper surface loaded by a 2mm thick layer of blood and the lower surface loaded by a semi-infinite fluid with properties close to those of bone marrow. The third case, involves a three-layer model in which the upper surface of the plate was again loaded by a layer of blood, whereas the lower surface was loaded by a 2mm layer of a fluid which simulated bone marrow. The callus tissue was modeled as an inhomogeneous material and fracture healing was simulated as a three-stage process. The results clearly indicate that the application of realistic boundary conditions has a significant effect on the dispersion of guided waves when compared to simplified models in which the bones surfaces are assumed free.

Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones.

Quantitative ultrasound has recently drawn significant interest in the monitoring of the bone healing process. Several research groups have studied ultrasound propagation in healing bones numerically, assuming callus to be a homogeneous and isotropic medium, thus neglecting the multiple scattering phenomena that occur due to the porous nature of callus. In this study, we model ultrasound wave propagation in healing long bones using an iterative effective medium approximation (IEMA), which has been shown to be significantly accurate for highly concentrated elastic mixtures. First, the effectiveness of IEMA in bone characterization is examined: (a) by comparing the theoretical phase velocities with experimental measurements in cancellous bone mimicking phantoms, and (b) by simulating wave propagation in complex healing bone geometries by using IEMA. The original material properties of cortical bone and callus were derived using serial scanning acoustic microscopy (SAM) images from previous animal studies. Guided wave analysis is performed for different healing stages and the results clearly indicate that IEMA predictions could provide supplementary information for bone assessment during the healing process. This methodology could potentially be applied in numerical studies dealing with wave propagation in composite media such as healing or osteoporotic bones in order to reduce the simulation time and simplify the study of complicated geometries with a significant porous nature.

A numerical study on the propagation of Rayleigh and guided waves in cortical bone according to Mindlin’s Form II gradient elastic theory

Cortical bone is a multiscale heterogeneous natural material characterized by microstructural effects. Thus guided waves propagating in cortical bone undergo dispersion due to both material microstructure and bone geometry. However, above 0.8 MHz, ultrasound propagates rather as a dispersive surface Rayleigh wave than a dispersive guided wave because at those frequencies, the corresponding wavelengths are smaller than the thickness of cortical bone. Classical elasticity, although it has been largely used for wave propagation modeling in bones, is not able to support dispersion in bulk and Rayleigh waves. This is possible with the use of Mindlins Form-II gradient elastic theory, which introduces in its equation of motion intrinsic parameters that correlate microstructure with the macrostructure. In this work, the boundary element method in conjunction with the reassigned smoothed pseudo Wigner-Ville transform are employed for the numerical determination of time-frequency diagrams corresponding to the dispersion curves of Rayleigh and guided waves propagating in a cortical bone. A composite material model for the determination of the internal length scale parameters imposed by Mindlins elastic theory is exploited. The obtained results demonstrate the dispersive nature of Rayleigh wave propagating along the complex structure of bone as well as how microstructure affects guided waves.

Two-dimensional simulations of wave propagation in healing long bones based on scanning acoustic microscopy images

Ultrasonic evaluation of bone fracture healing has been recognized as an important assessment method which reflects material, mechanical and structural properties. The objective of the present study is to examine the monitoring role of quantitative ultrasound by conducting axial transmission measurements of the propagation velocity of the first arriving signal (FAS) for two dimensional numerical models and various excitation frequencies (0.1-1 MHz). The models were derived from scanning acoustic microscopy (SAM). Although this is a preliminary study, the observed interaction between the FAS and the change in the material properties was quantified and the results give rise to the investigation of guided wave propagation, as well.

A mathematical model for bone healing predictions under the ultrasound effect

The bone healing process involves a sequence of cellular actions and interactions, regulated by biochemical and mechanical signals. Experimental studies have shown that ultrasound accelerates bone solidification and enhances the underlying healing mechanisms. We present a mathematical model for deriving predictions of bone healing under the presence of ultrasound. The model consists of i) partial differential equations which describe the spatiotemporal evolution cells, growth factors, tissues and ultrasound acoustic pressure and ii) velocity equations of endothelial tip cells which determine the development of the blood vessel network. The results showed that ultrasound accelerates bone healing primarily by enhancing blood vessel growth. Thus the proposed model could be useful for the ultrasonic evaluation of bone fracture healing.

A study on Rayleigh wave dispersion in bone according to Mindlin's Form II gradient elasticity

The classical elasticity cannot effectively describe bones mechanical behavior since only homogeneous media and local stresses are assumed. Additionally, it cannot predict the dispersive nature of the Rayleigh wave which has been reported in experimental studies and was also demonstrated in a previous computational study by adopting Mindlins Form II gradient elasticity. In this work Mindlins theory is employed to analytically determine the dispersion of Rayleigh waves in a strain gradient elastic half-space. An isotropic semi-infinite space is considered with properties equal to those of bone and dynamic behavior suffering from microstructural effects. Microstructural effects are considered by incorporating four intrinsic parameters in the stress analysis. The results are presented in the form of group and phase velocity dispersion curves and compared with existing computational results and semi-analytical curves calculated for a simpler case of Rayleigh waves in dipolar gradient elastic half-spaces. Comparisons are also performed with the velocity of the first-order antisymmetric mode propagating in a dipolar plate so as to observe the Rayleigh asymptotic behavior. It is shown that Mindlins Form II gradient elasticity can effectively describe the dispersive nature of Rayleigh waves. This study could be regarded as a step toward the ultrasonic characterization of bone.

A mechano-regulatory model for bone healing predictions under the influence of ultrasound.

The bone healing process involves a sequence of cellular action and interaction, regulated by biochemical and mechanical signals. Experimental studies have shown that ultrasound accelerates bone solidification and enhances the underlying healing mechanisms. An integrated computational model is presented for deriving predictions of bone healing under the presence of ultrasound.

A meshless Local Boundary Integral Equation (LBIE) method for cell proliferation predictions in bone healing

Bone healing involves a series of complicated cellular and molecular mechanisms that result in bone formation. Several mechanobiological models have been developed to simulate these cellular mechanisms via diffusive processes. In most cases solution to diffusion equations is accomplished using the Finite Element Method (FEM) which however requires global remeshing in problems with moving or new born surfaces or material phases. This limitation is addressed in meshless methods in which no background cells are needed for the numerical solution of the integrals. In this study a new meshless Local Boundary Integral Equation (LBIE) method is employed for deriving predictions of cell proliferation during bone healing. First a benchmark problem is presented to assess the accuracy of the method. Then the LBIE method is utilized for the solution of cell diffusion problem in a two-dimensional (2D) model of fractured model. Our findings indicate that the proposed here LBIE method can successfully predict cell distributions during fracture healing.