Effect of ultrasound on bone fracture healing: A computational mechanobioregulatory model.

Bone healing process is a complicated phenomenon regulated by biochemical and mechanical signals. Experimental studies have shown that ultrasound (US) accelerates bone ossification and has a multiple influence on cell differentiation and angiogenesis. In a recent work of the authors, a bioregulatory model for providing bone-healing predictions was addressed, taking into account for the first time the salutary effect of US on the involved angiogenesis. In the present work, a mechanobioregulatory model of bone solidification under the US presence incorporating also the mechanical environment on the regeneration process, which is known to affect cellular processes, is presented. An iterative procedure is adopted, where the finite element method is employed to compute the mechanical stimuli at the linear elastic phases of the poroelastic callus region and a coupled system of partial differential equations to simulate the enhancement by the US cell angiogenesis process and thus the oxygen concentration in the fractured area. Numerical simulations with and without the presence of US that illustrate the influence of progenitor cells' origin in the healing pattern and the healing rate and simultaneously demonstrate the salutary effect of US on bone repair are presented and discussed.

Effect of ultrasound on bone fracture healing: A computational bioregulatory model.

Bone healing is a complex biological procedure in which several cellular actions, directed by biochemical and mechanical signals, take place. Experimental studies have shown that ultrasound accelerates bone ossification and has a multiple influence on angiogenesis. In this study a mathematical model predicting bone healing under the presence of ultrasound is demonstrated. The primary objective is to account for the ultrasound effect on angiogenesis and more specifically on the transport of the Vascular Endothelial Growth Factor (VEGF). Partial differential equations describing the spatiotemporal evolution of cells, growth factors, tissues and ultrasound acoustic pressure and velocity equations determining the development of the blood vessel network constitute the present model. The effect of the ultrasound characteristics on angiogenesis and bone healing is investigated by applying different boundary conditions of acoustic pressure at the periosteal region of the bone model, which correspond to different intensity values. The results made clear that ultrasound enhances angiogenesis mechanisms during bone healing. The proposed model could be regarded as a step towards the monitoring of the effect of ultrasound on bone regeneration.

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

The classical elasticity cannot effectively describe bone's 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 Mindlin's Form II gradient elasticity. In this work Mindlin's 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 Mindlin's 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.

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.

Computational study of the influence of callus porosity on ultrasound propagation in healing bones.

In the process of fracture healing, several phases of recovery are observed as the mechanical stability, continuity and normal load carrying capacity are gradually restored. The ultrasonic monitoring and discrimination of different healing stages is a complex process due to the significant microstructure and porous nature of osseous and callus tissues. In this study, we investigate the influence of the callus pores' size and concentration on ultrasound propagation in a long bone at a late healing stage. Different excitation frequencies are applied in the range of 300 kHz-1 MHz. A 2D geometry is developed and axial transmission calculations are performed based on a Finite Element Method. The velocity of the first arriving signal (FAS) and the propagation of guided waves are used as the estimated parameters. It was shown that the FAS velocity can reflect callus porosity changes, while the propagation of guided waves is sensitive to pores' distribution for higher frequencies.

Phase velocity and attenuation predictions of waves in cancellous bone using an iterative effective medium approximation.

The quantitative determination of wave dispersion and attenuation in bone is an open research area as the factors responsible for ultrasound absorption and scattering in composite biological tissues have not been completely explained. In this study, we use the iterative effective medium approximation (IEMA) proposed in [1] so as to calculate phase velocity and attenuation in media with properties similar to those of cancellous bones. Calculations are performed for a frequency range of 0.4-0.8 MHz and for different inclusions' volume concentrations and sizes. Our numerical results are compared with previous experimental findings so as to assess the effectiveness of IEMA. It was made clear that attenuation and phase velocity estimations could provide supplementary information for cancellous bone characterization.

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.

Theoretical study of bone's microstructural effects on Rayleigh wave propagation.

The linear theory of classical elasticity cannot effectively describe bone's mechanical behavior since only homogeneous media and local stresses are assumed. Additionally, it cannot predict the dispersive nature of Rayleigh wave which has been experimental observed. By adopting Mindlin Form II gradient elastic theory and performing Boundary Element (BEM) simulations we also recently demonstrated Rayleigh dispersion. In this work we use this theory to analytically determine the dispersion of Rayleigh wave. We assume an isotropic semi-infinite space with mechanical properties equal to those of bone and microstructure and microstructural effects. Calculations are performed for various combinations between the internal constants l(1), l(2), h(1), h(2) which corresponded to a) values from closed form relations derived from a realistic model and b) values close to the osteon's size. Comparisons are made with the corresponding computational results as well as with the classical elastic case. The agreement between the computational and the analytical results was perfect demonstrating the effectiveness of Mindlin's Form II gradient theory of elasticity to predict the dispersive nature of Rayleigh wave. This study could be regarded as a step towards the ultrasonic characterization of bone.

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 Mindlin's 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 Mindlin's 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.

BEM simulations of Rayleigh wave propagation in media with microstructural effects: Application to long bones.

Bone is a strongly heterogeneous natural composite with microstructure. Although the classical theory of linear elasticity has been largely used in bone ultrasonic studies, it cannot sufficiently describe the mechanical behavior of materials with microstructure. Furthermore, this theory predicts non-dispersive behavior of Rayleigh waves, which is in conflict with experimental observations. By using the simplest theory of gradient elasticity we recently demonstrated that bone's microstructure significantly affects the dispersion of classical Lamb modes. In this work, we investigate the effect of bone's microstructure on the propagation of Rayleigh waves by using the Boundary Element Method (BEM). We assume an isotropic semi-infinite space with mechanical properties equal to those of bone and microstructure. Microstructural effects are taken into account by introducing in the stress analysis the internal length scale parameters l(1), l(2), h(1), h(2). BEM computations are performed for various combinations of these parameters with values empirically chosen close to the osteon's size. The constants' values are also compared to those derived from closed form relations. The results made clear that bone's microstructure significantly affects Rayleigh wave dispersion.

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

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 bone's surfaces are assumed free.