28 research outputs found
Multiscale modelling of trabecular bone: from micro to macroscale
Trabecular bone has a complex and porous microstructure. This study develops approaches
to determine the mechanical behaviour of this material at the macroscopic
level through the use of homogenisation-based multiscale methods using micro-finite
element simulations. In homogenisation-based finite element methods, a simulation
involving a representative volume element of the microstructure of the considered
material is performed with a specific set of boundary conditions. The macroscopic
stresses and strains are retrieved as averaged quantities defined over this domain. Most
of the homogenisation-based work on trabecular bone has been performed to study
its macroscopic elastic regime, and therefore define its constant macroscopic stiffness
tensor.
The rod and plate-shaped microstructure of trabecular bone can be precisely identified
with advanced scanning tools, such as micro-computed tomography devices. Taking
into account the size requirements to achieve a certain independence of boundary conditions
for trabecular bone in a homogenisation-based multiscale setting, the resulting
stack of images can have around ten million solid voxels after binarisation. Although a
completely linear finite element simulation with such a large system may be feasible
with commercial packages (with the proper time and memory requirements), it is not
possible to perform a nonlinear simulation for such a mesh in a reasonable time frame,
and the amount of required memory may not be available. A highly scalable parallel
driver program which solves finite strain elastoplastic systems was developed within
the framework of the existing parallel code ParaFEM. This code was used throughout
this study to evaluate the yield and post-yield properties of trabecular bone. It
was run on cutting edge high performance computing platforms (BlueGene/Q at the
Hartree Centre, Science and Technology Facilities Council; and ARCHER, UK National
Supercomputing Service, at Edinburgh Parallel Computing Centre).
Micro-finite element simulations require definition of properties at the microscopic
scale and it is unclear how these properties affect the macroscopic response. This
study examines the effect of compressive hydrostatic yield at the microscopic scale on
the macroscopic behaviour. Two different microscopic yield criteria, one permitting
yielding at compressive hydrostatic stresses and the other not, were considered. A
large number of load cases were examined. It was found that these two microscopic
yield criteria only influence macroscopic yield behaviour in load scenarios which are
compression-dominated; for other load cases, macroscopic response is insensitive to
the choice of the microscopic yield criterion, provided it has an appropriate strength
asymmetry. Also, in compression-dominated load cases, high density bone is much
more sensitive as it is more like a continuum, resulting in the microscopic properties
being more directly upscaled.
Only a few previous studies have employed homogenisation to evaluate the macroscopic
yield criterion of trabecular bone. However, they either used a simplified
microscopic yield surface or examined only a small number of load cases. A thorough
multiaxial evaluation of the macroscopic yield surface was performed by applying a
wide range of loading scenarios (160 load cases) on trabecular bone samples. Closed-form
yield surfaces with different symmetries (isotropy, orthotropy and full anisotropy)
were fitted to the numerically obtained macroscopic yield points in strain space, and
the fitting errors were evaluated in detail for different subsets of load cases. Although
orthotropy and full anisotropy showed the smallest fitting errors, they were not significantly
superior to the isotropic fit. Thus, isotropy in strain space presents itself as the
most suitable option due to the simplicity of its implementation. The study showed
that fitting errors do depend on the chosen set of load cases and that shear load cases
are extremely important as it was found that even for these highly aligned samples,
trabecular bone presents some degree of shear asymmetry, i.e. different strength in
clockwise and counter-clockwise shear directions.
There have been no previous attempts to evaluate the post-yield behaviour of
trabecular bone through homogenisation-based studies on detailed micro-finite element
trabecular bone meshes. A damage and plasticity constitutive law for the microscale
based on existing data in the literature was considered. A homogenisation-based
multiscale approach was used to evaluate the hardening and stiffness reduction at the
macroscale when uniaxial load scenarios are applied to trabecular bone samples, for a
small range of plastic strain Euclidean norms. Results show that damage progression at
the macroscale for trabecular bone is not isotropic, which is contrary to what has been
assumed previously, and that both the evolution of the yield surface and damage are
different for tension, compression and shear. Nonetheless, they can be correlated with
plastic strain Euclidean norms by using linear relationships. It was also observed that
macroscopic damage in a specific load case affects differently the on-axis orthotropic
stiffness and the off-axis orthotropic stiffness components.
The findings of this study will permit the use of a more rigorous definition of the post-elastic macroscopic behaviour of trabecular bone in finite element settings
Effect of including damage at the tissue level in the nonlinear homogenisation of trabecular bone
Being able to predict bone fracture or implant stability needs a proper constitutive model of trabecular bone at the macroscale in multiaxial, non-monotonic loading modes. Its macroscopic damage behaviour has been investigated experimentally in the past, mostly with the restriction of uniaxial cyclic loading experiments for different samples, which does not allow for the investigation of several load cases in the same sample as damage in one direction may affect the behaviour in other directions. Homogenised finite element models of whole bones have the potential to assess complicated scenarios and thus improve clinical predictions. The aim of this study is to use a homogenisation-based multiscale procedure to upscale the damage behaviour of bone from an assumed solid phase constitutive law and investigate its multiaxial behaviour for the first time. Twelve cubic specimens were each submitted to nine proportional strain histories by using a parallel code developed in-house. Evolution of post-elastic properties for trabecular bone was assessed for a small range of macroscopic plastic strains in these nine load cases. Damage evolution was found to be non-isotropic, and both damage and hardening were found to depend on the loading mode (tensile, compression or shear); both were characterised by linear laws with relatively high coefficients of determination. It is expected that the knowledge of the macroscopic behaviour of trabecular bone gained in this study will help in creating more precise continuum FE models of whole bones that improve clinical predictions.</p
Characterisation of time-dependent mechanical behaviour of trabecular bone and its constituents
Trabecular bone is a porous composite material which consists of a mineral
phase (mainly hydroxyapatite), organic phase (mostly type I collagen) and water
assembled into a complex, hierarchical structure. In biomechanical modelling,
its mechanical response to loads is generally assumed to be instantaneous,
i.e. it is treated as a time-independent material. It is, however, recognised
that the response of trabecular bone to loads is time-dependent. Study
of this time-dependent behaviour is important in several contexts such as: to
understand energy dissipation ability of bone; to understand the age-related
non-traumatic fractures; to predict implant loosening due to cyclic loading; to
understand progressive vertebral deformity; and for pre-clinical evaluation of
total joint replacement.
To investigate time-dependent behaviour, bovine trabecular bone samples
were subjected to compressive loading, creep, unloading and recovery at multiple
load levels (corresponding to apparent strain of 2,000-25,000 με). The
results show that: the time-dependent behaviour of trabecular bone comprises
of both recoverable and irrecoverable strains; the strain response is nonlinearly
related to applied load levels; and the response is associated with bone volume
fraction. It was found that bone with low porosity demonstrates elastic
stiffening followed by elastic softening, while elastic softening is demonstrated
by porous bone at relatively low loads. Linear, nonlinear viscoelastic and nonlinear
viscoelastic-viscoplastic constitutive models were developed to predict
trabecular bone’s time-dependent behaviour. Nonlinear viscoelastic constitutive model was found to predict the recovery behaviour well, while nonlinear
viscoelastic-viscoplastic model predicts the full creep-recovery behaviour reasonably
well. Depending on the requirements all these models can be used to
incorporate time-dependent behaviour in finite element models.
To evaluate the contribution of the key constituents of trabecular bone and
its microstructure, tests were conducted on demineralised and deproteinised
samples. Reversed cyclic loading experiments (tension to compression) were
conducted on demineralised trabecular bone samples. It was found that demineralised
bone exhibits asymmetric mechanical response - elastic stiffening
in tension and softening in compression. This tension to compression transition
was found to be smooth. Tensile multiple-load-creep-unload-recovery experiments
on demineralised trabecular samples show irrecoverable strain (or
residual strain) even at the low stress levels. Demineralised trabecular bone
samples demonstrate elastic stiffening with increasing load levels in tension,
and their time-dependent behaviour is nonlinear with respect to applied loads .
Nonlinear viscoelastic constitutive model was developed which can predict its
recovery behaviour well. Experiments on deproteinised samples showed that
their modulus and strength are reasonably well related to bone volume fraction.
The study considers an application of time-dependent behaviour of trabecular
bone. Time-dependent properties are assigned to trabecular bone in a
bone-screw system, in which the screw is subjected to cyclic loading. It is
found that separation between bone and the screw at the interface can increase
with increasing number of cycles which can accentuate loosening. The
relative larger deformation occurs when this system to be loaded at the higher
loading frequency. The deformation at the bone-screw interface is related to
trabecular bone’s bone volume fraction; screws in a more porous bone are at
a higher risk of loosening
Nonlinear viscoelastic characterization of bovine trabecular bone
The time-independent elastic properties of trabecular bone have been extensively investigated, and several stiffness–density relations have been proposed. Although it is recognized that trabecular bone exhibits time-dependent mechanical behaviour, a property of viscoelastic materials, the characterization of this behaviour has received limited attention. The objective of the present study was to investigate the time-dependent behaviour of bovine trabecular bone through a series of compressive creep–recovery experiments and to identify its nonlinear constitutive viscoelastic material parameters. Uniaxial compressive creep and recovery experiments at multiple loads were performed on cylindrical bovine trabecular bone samples ([Formula: see text] ). Creep response was found to be significant and always comprised of recoverable and irrecoverable strains, even at low stress/strain levels. This response was also found to vary nonlinearly with applied stress. A systematic methodology was developed to separate recoverable (nonlinear viscoelastic) and irrecoverable (permanent) strains from the total experimental strain response. We found that Schapery’s nonlinear viscoelastic constitutive model describes the viscoelastic response of the trabecular bone, and parameters associated with this model were estimated from the multiple load creep–recovery (MLCR) experiments. Nonlinear viscoelastic recovery compliance was found to have a decreasing and then increasing trend with increasing stress level, indicating possible stiffening and softening behaviour of trabecular bone due to creep. The obtained parameters from MLCR tests, expressed as second-order polynomial functions of stress, showed a similar trend for all the samples, and also demonstrate stiffening–softening behaviour with increasing stress
Linear viscoelasticity - bone volume fraction relationships of bovine trabecular bone
Trabecular bone has been previously recognized as time-dependent (viscoelastic) material, but the relationships of its viscoelastic behaviour with bone volume fraction (BV/TV) have not been investigated so far. Therefore, the aim of the present study was to quantify the time-dependent viscoelastic behaviour of trabecular bone and relate it to BV/TV. Uniaxial compressive creep experiments were performed on cylindrical bovine trabecular bone samples ([Formula: see text] ) at loads corresponding to physiological strain level of 2000 [Formula: see text] . We assumed that the bone behaves in a linear viscoelastic manner at this low strain level and the corresponding linear viscoelastic parameters were estimated by fitting a generalized Kelvin–Voigt rheological model to the experimental creep strain response. Strong and significant power law relationships ([Formula: see text] ) were found between time-dependent creep compliance function and BV/TV of the bone. These BV/TV-based material properties can be used in finite element models involving trabecular bone to predict time-dependent response. For users’ convenience, the creep compliance functions were also converted to relaxation functions by using numerical interconversion methods and similar power law relationships were reported between time-dependent relaxation modulus function and BV/TV