Characterisation of trabecular bone behaviour under impact

Abstract

Increasing ageing population around the world implies deteriorated bone quality due to diseases such as osteoporosis. Osteoporosis results in reduced bone density and strength and increases the risk of bone fractures due to impact loads such as those caused by a fall. While numerous studies have been undertaken to evaluate the mechanical properties of bone using experimental or computational approaches most of these have focused on static or cyclic loads. Little is known about the behaviour of bone under dynamic loading, such as that caused by a fall. This study focuses on characterising the impact behaviour of spongy bone which is also known as trabecular or cancellous bone using drop tower tests. A closed-form solution of drop test on an elastic material is developed in detail to serve as a benchmark. Sequential compressive stress wave equations are derived then coded in programming language to obtain desired number of equations, leading to the resulting force history at the end struck. The number of required intervals to obtain the whole pulse depends on the mass ratio; the developed code can be readily expanded to include a wide range of these ratios. A parametric analysis is undertaken to evaluate the influence of parameters such as elastic modulus and drop height. This evaluation defines the key outputs of the force response, namely peak load, pulse width, interval frequency, number of stress intervals and loading rate; hence it establishes the output variable-parameter relationships, e.g. peak load-elastic modulus relation. The force response from the closed-form solution is used to verify the finite element (FE) models. The Poisson's ratio is found to have insignificant effect on the resulting peak load and pulse width. Stiffer response is indicated by an increase in peak load or a decrease in pulse width for the same impact. Fast Fourier transform (FFT) is applied to the pulses obtained from the closed-form solution and FE analysis; the returned frequency values reveal that the discrepancy observed in FE results can be attributed to its solver rather than the FFT technique. The time-independent elastic FE model is extended to include the time-dependent viscoelastic behaviour; the latter provides a stiffer response. The output variableparameter relations from FE analysis are verified by similar expressions derived from the closed-form solution. Drop tests are conducted in the lab on trabecular bone samples with varying bone volume fraction (bone volume to total volume ratio or BV/TV) to investigate the resulting force response. These responses are categorised on the basis of BV/TV of the tested samples. Samples with large BV/TV are shown to behave in an apparently elastic manner. Low BV/TV samples respond inelastically which the study attempts to simulate using plasticity. Behaviour of samples with growth plates shows inelastic behaviour even though they have large BV/TV. Regression analysis is carried out on the outputs of pulses obtained from higher BV/TV samples to find the output variable-parameter relations, e.g. inverse relation of pulse width-BV/TV. Inverse modelling using FE analysis is also performed to estimate a representable elastic modulus by matching the peak load and pulse width of the resulting force response. The elastic moduli are found to be in the range of 650 to 1400 MPa for samples with BV/TV ranging between 35 to 53%. Elastic modulus is also evaluated using the initial load-response curve for both lower and higher BV/TV samples to establish linear and power-law loading rate versus BV/TV relationships. The inelastic behaviour of low BV/TV samples consistently shows initial peak, followed by a drop to a finite non-zero value in the post-elastic regime which is maintained for a considerable prolonged time before the load returns to zero. This pattern has not been previously demonstrated. The duration of plasticity is found to be larger for lower BV/TV samples, while shorter drop time is observed exhibiting brittle breakage after the peak load is attained. The study also considers simulation of the post-yield inelastic response using strain-softening plasticity. It is shown that strain-softening is capable of approximately replicating the post-yield impact response pattern by assuming appropriate value of tangent modulus. Four dynamic elastic modulus expressions as functions of BV/TV are derived by correlating the relationships regressed experimentally with their correspondences from the closed-form solution. The performance of these linear and power law equations are compared to several relations from the literature and the response from tests conducted at an apparent strain rate of 0.01 /sec on similar samples. It is found that the dynamic elastic moduli found in this study are higher than their quasi-static and monotonic loading counterparts; a conclusion showing the effect of strain rate magnitude on the stiffness of the trabecular bone. The study evaluates the strain rate experienced by bone due to impact loading considered in this study to be a maximum of 44.3 /sec at the initial stage with secant value of 30 /sec at the peak strain. Assuming the high BV/TV samples behave fully elastic, the apparent strain due to impact loads can be as high as 3% in compression.