Application of neural networks and finite element computation for multiscale simulation of bone remodeling

In this paper, a novel multiscale hierarchical model based on finite element analysis and neural network computation was developed to link mesoscopic and macroscopic scales to simulate the bone remodeling process. The finite element calculation is performed at the macroscopic level, and trained neural networks are employed as numerical devices for substituting the finite element computation needed for the mesoscale prediction. Based on a set of mesoscale simulations of representative volume elements of bones taken from different bone sites, a neural network is trained to approximate the responses at the meso level and transferred at the macro level.     

Measuring elastic properties of cells by evaluation of scanning acoustic microscopy V(Z) values using simplex algorithm

In this paper a new technique is proposed to determine the acoustic properties as well as the thickness (and volume) of biological cells. Variations of thickness, density, acoustic wave velocity, stiffness, and attenuation coefficient of a living or dead cell are obtained by scanning the cell by an acoustic microscope. The distance between the cell and the microscope lens is varied and several voltage curves are thus obtained. These curves are then inverted by simplex optimization technique to obtain the cell parameters. The spatial resolution of the method is limited to the resolution of the scanning acoustic microscope. It allows to take advantage of the full range of frequencies and amplification of the microscope. Characteristic distributions of stiffness are exemplified with an endothelial cell in culture. The main part of the thin, lamellar cytoplasm has high stiffness, which drops close to the lamella/cell body transition region and only slightly increases again through the central part of the cell. Acoustic attenuation seems to be related to two factors, cytoplasm accumulation (in the lamellar parts) and scattering in the central part rich in organelles.     

Application of the multiscale FEM to the modeling of cancellous bone

This paper considers the application of multiscale finite element method (FEM) to the modeling of cancellous bone as an alternative for Biot’s model, the main intention of which is to decrease the extent of the necessary laboratory tests. At the beginning, the paper gives a brief explanation of the multiscale concept and thereafter focuses on the modeling of the representative volume element and on the calculation of the effective material parameters, including an analysis of their change with respect to increasing porosity. The latter part of the paper concentrates on the macroscopic calculations, which is illustrated by the simulation of ultrasonic testing and a study of the attenuation dependency on material parameters and excitation frequency. The results endorse conclusions drawn from the experiments: increasing excitation frequency and material density cause increasing attenuation.     

Comparison of acoustic characteristics predicted by Biot's theory and the modified Biot-Attenborough model in cancellous bone

Biot's theory and the modified Biot-Attenborough (MBA) model are applied to predict the dependences of acoustic characteristics on frequency and on porosity in cancellous bone. The phase velocities and the attenuation coefficients predicted by both theories are compared with the experimental data of bovine cancellous bone specimens published in the literature. Biot's theory successfully predicts the dependences of the phase velocity on frequency and on porosity in cancellous bone, whereas a significant discrepancy is observed between the predicted and the measured attenuation coefficients. The MBA model agrees well with the frequency and the porosity dependences of the phase velocity and the attenuation coefficient experimentally measured in bovine bones. Although the MBA model relies on phenomenological parameters derived from the experimental data, its approach to cancellous bone can be usefully employed in the field of clinical ultrasonic bone assessment.     

Scattering of ultrasound in cancellous bone: predictions from a theoretical model

An understanding of the interaction between acoustic waves and cancellous bone is needed in order to realize the full clinical potential of ultrasonic bone measurements. Scattering is likely to be of central importance but has received little attention to date. In this study, we adopted a theoretical model from the literature in which scattering was assumed to be proportional to the mean fluctuation in sound speed, and bone was considered to be a random continuum containing identical scatterers. The model required knowledge only of sound speeds in bone and marrow, porosity, and scatter size. Predicted attenuation, broadband ultrasonic attenuation (BUA) and backscatter coefficient were obtained for a range of porosities and scatterer sizes, and were found to be comparable to published values for cancellous bone. Trends in predicted BUA with porosity agreed with previous experimental observations. All three predicted acoustic parameters showed a non-linear dependence on scatterer size which was independent of porosity. These data confirm the value of the scattering approach and provide the first quantitative predictions of the independent influence of structure and porosity on bone acoustic properties.     

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.     

Interaction of electromagnetic radiation with a plasma with fully developed ion acoustic turbulence

A study is made of the characteristic features of the reflection and absorption of a monochromatic wave by a plasma with axisymmetric ion acoustic turbulence over a broad frequency range. The absorption anisotropy and the related conversion of an incident, linearly polarized wave into a reflected, elliptically polarized wave are described. The absorption coefficient and the difference in the phase shifts occurring in the reflection of different field components are obtained as explicit functions of the turbulent plasma parameters.     

Cell property determination from the acoustic microscope generated voltage versus frequency curves

Among the methods for the determination of mechanical properties of living cells acoustic microscopy provides some extraordinary advantages. It is relatively fast, of excellent spatial resolution and of minimal invasiveness. Sound velocity is a measure of the stiffness or Young's modulus of the cell. Attenuation of cytoplasm is a measure of supramolecular interactions. These parameters are of crucial interest for studies of cell motility, volume regulations and to establish the functional role of the various elements of the cytoskeleton. Using a phase and amplitude sensitive modulation of a scanning acoustic microscope (Hillman et al., 1994, J. Alloys Compounds. 211/212:625-627) longitudinal wave speed, attenuation and thickness profile of a biological cell are obtained from the voltage versus frequency or V(f) curves. A series of pictures, for instance in the frequency range 980-1100 MHz with an increment of 20 MHz, allows the experimental generation of V(f) curves for each pixel while keeping the lens-specimen distance unchanged. Both amplitude and phase values of the V(f) curves are used for obtaining the cell properties and the cell thickness profile. The theoretical analysis shows that the thin liquid layer, between the cell and the substrate, has a strong influence on the reflection coefficient and should not be ignored during the analysis. Cell properties, cell profile and the thickness of the thin liquid layer are obtained from the V(f) curves by the simplex inversion algorithm. The main advantages of this new method are that imaging can be done near the focal plane, therefore an optimal signal to noise ratio is achieved, no interference with Rayleigh waves occurs, and the method requires only an approximate estimate of the material properties of the solid substratum where the cells are growing on.     

Determination of the geometry of the RVE for cancellous bone by using the effective complex shear modulus

This contribution deals with the application of the inverse homogenization method to the determination of geometrical properties of cancellous bone. The approach represents a combination of an extended version of the Marquardt–Levenberg method with the multiscale finite element method. The former belongs to the group of gradient-based optimization strategies, while the latter is a numerical homogenization method, suitable for the modeling of materials with a highly heterogeneous microstructure. The extension of the Marquardt–Levenberg method is concerned with the selection strategy for distinguishing the global minimum from the plethora of local minima. Within the numerical examples, the bone is modeled as a biphasic viscoelastic medium and three different representative volume elements are taken into consideration. Different models enable the simulation of the bone either as a purely isotropic or as a transversally anisotropic medium. Main geometrical properties of trabeculae are determined from data on effective shear modulus but alternative schemes are also possible.     

Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale

This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties.     

Acoustic wave based MEMS devices for biosensing applications

This paper presents a review of acoustic-wave based MEMS devices that offer a promising technology platform for the development of sensitive, portable, real-time biosensors. MEMS fabrication of acoustic wave based biosensors enables device miniaturization, power consumption reduction and integration with electronic circuits. For biological applications, the biosensors are integrated in a microfluidic system and the sensing area is coated with a biospecific layer. When a bioanalyte interacts with the sensing layer, mass and viscosity variations of the biospecific layer can be detected by monitoring changes in the acoustic wave properties such as velocity, attenuation, resonant frequency and delay time. Few types of acoustic wave devices could be integrated in microfluidic systems without significant degradation of the quality factor. The acoustic wave based MEMS devices reported in the literature as biosensors and presented in this review are film bulk acoustic wave resonators (FBAR), surface acoustic waves (SAW) resonators and SAW delay lines. Different approaches to the realization of FBARs, SAW resonators and SAW delay lines for various biochemical applications are presented. Methods of integration of the acoustic wave MEMS devices in the microfluidic systems and functionalization strategies will be also discussed.     

A Novel 3D Microstructural Model for Trabecular Bone: II. The Relationship Between Fabric and the Yield Surface

A novel 3D microstructural model was proposed and validated in part I of this publication. In part II, the model was used to identify the yield surface of a representative volume element of human trabecular bone as a function of volume fraction and degree of anisotropy. Finite element models of open and closed cells geometries were used to calculate effective yield stresses for a variety of loading cases with periodic boundary conditions. The postyield behaviour of the trabecular tissue was assumed from data available for cortical tissue. The yield stresses defined by a 0.2% offset in the global stress-strain curve were fit to an orthotropic Hill criterion and the parameters of the surface calculated. Similarly to the previous elastic analysis, distinct but strong relationships were obtained between volume fraction, fabric and the yield surface parameters for both the open and closed cell geometries. This finding suggests that volume fraction and fabric may be used to predict the initiation of mechanical damage in human trabecular bone at the continuum level.     

A laser-induced ultrasonic probe of the mechanical properties of aligned lipid multibilayers.

The recently developed laser-induced phonon spectroscopy (LIPS) technique is applied to the determination of dynamic mechanical properties of aligned dilauroylphosphatidylcholine (DLPC) multibilayer arrays containing 2 and 20% water by weight. Sample excitation by two crossed 100-ps laser pulses generates a longitudinal ultrasonic wave whose wavelength depends on the crossing angle. In these experiments, the acoustic wave propagates parallel to the bilayer planes. The ultrasonic velocity and attenuation are monitored through the diffraction of a variably delayed probe pulse by the acoustic grating. The velocity measures the lateral area compressibility of the bilayers, while the attenuation is related to the viscosity. Velocities obtained in the gel and liquid crystal phases are compared with those found previously using Brillouin scattering. The acoustic attenuation is shown to be an order of magnitude more sensitive to the gel-liquid crystal phase transition than the velocity. The lipid area compressibility and viscosity of DLPC-20% water multilayers with and without 100 mM CaCl2 are found to be identical within our experimental error.     

Wave Transmission through an Assembly of Randomly Branching Elastic Tubes

Calculations are presented of the transmission of oscillations through an assembly of randomly branching elastic tubes, as a model of not only the major arteries, but also a peripheral vascular bed. It appears that the viscosity of the arterial wall must be the major source of attenuation in the larger arteries, while the viscosity of the blood plays a significant role only in the smaller vessels. In all situations, variations of cross-sectional area have a considerable effect on wave transmission, causing a general decrease in amplitude and an accentuation of reflection from the terminations. The effects of variation in cross-sectional area are sufficiently great to indicate that they should be included in future models of the arterial system. Finally, it is argued that because of the presence of random branching and elastic nonuniformity, the determination of the reflection coefficient for a system such as the arterial tree may be quite misleading.     

Acoustic microscope study of the elastic properties of fluorapatite and hydroxyapatite, tooth enamel and bone.

Measurements of Rayleigh velocity and attenuation were taken in single mineral crystals of hydroxyapatite and fluorapatite at angular intervals relative to their c axes, using an acoustic microscope. These results are compared with the values that were calculated using the elastic constants of apatite from Yoon and Newnham [(1969) Am. Miner. 54, 1193-1197.] and Katz and Ukraincik [(1971) J. Biomechanics 4, 221-227.]. The slowness curves of various wave modes are plotted and discussed in relation to cross-coupling effects that were found to cause instability in measurements of attenuation for the c axes direction. Velocity measurements were taken in specimens of tooth enamel and bone. Here comparisons are made with the values that were calculated by modelling the 'z' scan response of the microscope, using published data for the elastic and acoustic properties. Comparisons are also made with the measurements on single crystals, since apatite is a major component of enamel and bone.     

An ultrasound elastography method to determine the local stiffness of arteries with guided circumferential waves

Arterial stiffness is highly correlated with the functions of the artery and may serve as an important diagnostic criterion for some cardiovascular diseases. To date, it remains a challenge to quantitatively assess local arterial stiffness in a non-invasive manner. To address this challenge, we investigated the possibility of determining arterial stiffness using the guided circumferential wave (GCW) induced in the arterial wall by a focused acoustic radiation force. The theoretical model for the dispersion analysis of the GCW is presented, and a finite element model has been established to calculate the dispersion curve. Our results show that under described conditions, the dispersion relations of the GCW are basically independent of the curvature of the arterial wall and can be well-described using the Lamb wave (LW) model. Based on this conclusion, an inverse method is proposed to characterize the elastic modulus of artery. Both numerical experiments and phantom experiments had been performed to validate the proposed method. We show that our method can be applied to the cases in which the artery has local stenosis and/or the geometry of the artery cross-section is irregular; therefore, this method holds great potential for clinical use.     

Influence of the outer-magnetospheric magnetohydrodynamic waveguide on the reflection of hydromagnetic waves from a shear flow at the magnetopause

The coefficient of reflection of a fast magnetosonic wave incident on the magnetosphere from the solar wind is studied analytically in the framework of a plane-stratified model of the medium with allowance for the transverse inhomogeneity of the magnetosphere and a jump of the plasma parameters at the magnetopause. Three factors decisively affecting the properties of reflection are taken into account: the shear flow of the solar wind plasma relative to the magnetosphere; the presence of a magnetospheric magnetohydrodynamic waveguide caused by the transverse plasma inhomogeneity; and the presence of an Alfvén resonance deep in the magnetosphere, where the oscillation energy dissipates. If the solar wind velocity exceeds the wave phase velocity along the magnetopause, then the wave energy in the solar wind is negative and such a wave experiences overreflection. In the opposite case, the wave energy is positive and the wave is reflected only partially. The wave reflection has a pronounced resonant character: the reflection coefficient has deep narrow minima or high narrow maxima at the eigenfrequencies of the magnetospheric waveguide. For other frequencies, the reflection coefficient only slightly differs from unity. The wave energy influx into the magnetosphere is positive for waves with both positive and negative energies. For waves with a negative energy, this is a consequence of their overreflection, because the flux of negative energy carried away by the reflected wave exceeds the incident flux of negative energy.     

Network thermodynamic analysis and stimulation of isotonic solute-coupled volume flow in leaky epithelia: an example of the use of network theory to provide the qualitative aspects of a complex system and its verification by stimulation

A network thermodynamic model was developed to provide insights into the nature of isotonic solute-coupled volume flow in "leaky" epithelia, where the transepithelial volume flow is assumed to be primarily through the cellular pathway. The coupled flows of solute and volume at each membrane in this four membrane model are described by the practical phenomenological equations as developed by Kedem & Katchalsky (1958). The model contains one permeable non-electrolyte solute (s) and a fixed amount of an impermeable non-electrolyte (i) inside the cell. The cell is assumed to be capable of volume regulation under the steady-state experimental conditions simulated. A solute-pump, located in the basolateral membrane, uses feedback regulation to adjust Cs in the cell in order to maintain cell volume at or near control levels in all simulations. Model behavior is, in general, very consistent with experimental observations with respect to tonicity and magnitude of volume flow over a wide range of experimental conditions. Examination of the parameter space suggests the following important features when isotonic solute-coupled volume flow moves primarily through the cellular pathway: (1) the apical membrane reflection coefficient must be less than that of the basolateral membrane; (2) the basement membrane reflection coefficient must be small; (3) the apical membrane solute permeability and reflection coefficient are the two most "sensitive" parameters and need to vary in an inverse manner in order to maintain isotonicity when both solute and volume flows increase; and (4) relationships (1) and (3) above imply the need for at least two separate solute pathways in the apical membrane, one that is shared with volume flow and one that is not.     

A mathematical model for bone tissue regeneration inside a specific type of scaffold

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.