Vertebral stress of a cervical spine model under dynamic load.

The objective of this study is to develop cervical spine models that predict the stresses in each vertebra by taking account of the biodynamic characteristics of the neck. The loads and the moments at the head point (Occipital Condyle) used for the models were determined by the rigid body dynamic response of the head due to G-z acceleration. The experimental data used were collected from the biodynamic responses of human volunteers during an acceleration in the z direction on the drop tower facility at Armstrong Laboratory at Wright Patterson Air Force Base (WPAFB). Three finite element models were developed: an elastic local model, viscoelastic local model and complete viscoelastic model. I-DEAS software was used to create the solid models, the loadings and the boundary conditions. Then, ABAQUS finite element software was employed to solve the models, and thus the stresses on each vertebral level were determined. Beam elements with different properties were employed to simulate the ligaments, articular facets and muscles. The complete viscoelastic model was subjected to 11 cases of loadings ranging from 8 G-z to 20 G-z accelerations. The von Mises and Maximum Principal stress fields, which are good indicators of bone failure, were calculated for all the cases. The results indicated that the maximum stress in all cases increased as the magnitude of the acceleration increased. The stresses in the 10 to 12 G-z cases were comfortably below the injury threshold level. The majority of the maximum stresses occurred in C6 and C4 regions.

The effect of surface roughness on the stress adaptation of trabecular architecture around a cylindrical implant.

The effect of implant-bone bonding and the effect of implant surface roughness on bone remodeling near the bone-implant interface were studied by using a surface remodeling theory and the boundary element method. The study has shown that implant attachment plays an important role in bone remodeling near the implant. It has been observed in animal experiments and in clinical situations that the remodeled trabecular bone architecture around a cylindrical implant could vary, on one hand, from a hub surrounding the implant with a set of external spokes to, on the other hand, a hubless situation in which a set of spokes attach directly to the implant. It is shown here that the difference in these structures may be attributed to differences in implant attachment. The results show that the bone with perfect bonding or roller boundary condition without a gap remodeled to a hubless spoke trabecular bone architecture. On the other hand, the roller boundary condition with a specified gap yielded a spoke trabecular architecture with a hub or ring surrounding the implant. These quantitative results mirror the experimental and clinical observations. It is concluded that the hub is a consequence of the gap and not a consequence of the lack of friction between the implant and the bone.

Implementation of strain rate as a bone remodeling stimulus.

Strain rate is implemented as a stimulus for surface bone remodeling. Using idealized models for trabecular bone structures, the surface remodeling predictions using the strain rate as the stimulus are compared with the predictions using the peak strain magnitude as the stimulus. For a uniaxially loaded cruciform shape, the comparison shows that the two surface remodeling stimuli predict the same final shape under a periodic compressive load, but the two evolutionary paths to final shapes are different. Two biaxially loaded regular grid models of trabecular structure were considered, one a grid of square diamond shaped elements and the other a brick wall patterned grid. For both of these idealized trabecular structures, the comparison shows that the two surface remodeling stimuli predict the same final shape under a periodic compressive load, even from these distinctly different initial grid patterns, and the evolutionary paths to final shapes are quite different. In general the two stimuli do not predict the same remodeling and the conditions under which they do are derived. The models developed are also applied to the data from the animal experiments reported in Goldstein et al. (1991), and it is shown that the strain rate stimulus predicts bone remodeling similar to what was experimentally observed.

On the sufficiency conditions for the stability of bone remodeling equilibrium.

In this technical note a sufficiency condition is established for the stability of a strain-energy-based bone remodeling theory in the special case of a beam loaded by an axial force and a bending moment. In a previous report the same condition was shown to be a necessary condition for stability in the same situation. The remodeling scheme is one characterized by a remodeling stimulus equal to the strain energy density divided by the bulk or apparent density raised to an exponent m as well an elastic modulus proportional to bulk or apparent density raised to an exponent n. In order for a remodeling scheme to be stable for an elastic beam loaded by an axial force and a bending moment, it is established that the condition that m must be greater than n is not only necessary, but also sufficient.

Chaos in the discrete-time algorithm for bone-density remodeling rate equations.

We compare the predictions of the differential equation form of a class of bone-density stress adaptation models with their associated discrete-time computational algorithms. Although our considerations apply to the class of adaptation models based on bulk or apparent bone-density remodeling, we focus attention on a particular model in this class, a model employed by Weinans et al. [Trans. Orthop. Res. Soc. 14, 310 (1989); Trans. First World Congress of Biomechanics, Vol. II, p. 75 (1990)]. We show that the discrete-time computational algorithm of that stress adaptation model has a well-known chaos mechanism for stress values of practical interest. Further, we obtain a condition on the discrete-time step that prevents the transition to chaos, and conditions that insure monotonic convergence. This chaos mechanism is only present in the discrete-time computational algorithm; we show that the corresponding differential equation form of the bone-density stress adaptation model is smooth, monotonic and nonchaotic.

Bone ingrowth: an application of the boundary element method to bone remodeling at the implant interface.

Surface bone remodeling theory and the boundary element method are employed to investigate the microstructural remodeling of bone at the bone-implant interface. Three situations are considered: remodeling-induced penetration between the screw threads of an implanted screw, penetration of bone tissue into a slot or cavity in an implant, and the interaction of individual trabeculae in the remodeling processes near an implant. For each case the bone ingrowth is determined as a function of the geometry and the applied load.

An evolutionary Wolff's law for trabecular architecture.

A continuum model is proposed to describe the temporal evolution of both the density changes and the reorientation of the trabecular architecture given the applied stress state in the bone and certain material parameters of the bone. The data upon which the proposed model is to be based consist of experimentally determined remodeling rate coefficients and quantitative stereological and anisotropic elastic constant measurements of cancellous bone. The model shows that the system of differential equations governing the temporal changes in architecture is necessarily nonlinear. This nonlinearity is fundamental in that it stems from the fact that, during remodeling, the relationship between stress and strain is changing as the stress and strain variables themselves are changing. In order to preserve the remodeling property of the model, terms that are of the order strain times the changes in density and/or microstructural properties must be retained. If these terms were dropped, there would be no feedback mechanism for architectural adaptation and no adaptation of the trabecular architecture. There is, therefore, no linearized version of the model of the temporal evolution of trabecular architecture. An application of the model is illustrated by an example problem in which the temporal evolution of homogeneous trabecular architecture is predicted. A limitation of the proposed continuum model is the length scale below which it cannot be applied. The model cannot be applied in regions of cancellous bone where the trabecular bone architecture is relatively inhomogeneous or at a bone-implant interface.

Correction formulae for the misalignment of axes in the measurement of the orthotropic elastic constants.

In the experimental determination of the orthotropic elastic constants, one often encounters the situation in which the symmetry axes of the material are not coincident with specimen axes along which the material testing is accomplished. The problem of calculating the compliance coefficients in the symmetry coordinate system from measurements of the compliance coefficients made in an arbitrary, specimen fixed, coordinate system is considered here.

Non-interacting modes for stress, strain and energy in anisotropic hard tissue.

The six non-interacting modes for stress, strain and energy in an orthotropic elastic model of human femoral cortical bone tissue are discussed and illustrated. The stress and strain modes are illustrated using the representation of the stress and strain fields around a circular hole in a flat plate of cortical bone subjected to a uniaxial field of tension as the example. The six modes play a role in the stress analysis of orthotropic elastic materials similar to the roles played by the hydrostatic and deviatoric non-interacting stress, strain and energy modes in isotropic elasticity. The biomechanical significance of the six non-interacting modes for stress, strain and energy in hard tissue is both practical and suggestive. The modes suggest a practical scheme for the representation of stress and strain fields in hard tissue. The existence of the modes suggests physical insights, for example, possible failure mechanisms or adaptation strategies possessed by the hard tissues.