Validation of subject-specific automated p-FE analysis of the proximal femur

Background: The use of subject-specific finite element (FE) models in clinical practice requires a high level of automation and validation. In Yosibash et al. [2007a. Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations. J. Biomechanics 40, 3688–3699] a novel method for generating high-order finite element (p-FE) models from CT scans was presented and validated by experimental observations on two fresh frozen femurs (harvested from a 30 year old male and 21 year old female). Herein, we substantiate the validation process by enlarging the experimental database (54 year old female femur), improving the method and examine its robustness under different CT scan conditions.Approach: A fresh frozen femur of a 54 year old female was scanned under two different environments: in air and immersed in water (dry and wet CT). Thereafter, the proximal femur was quasi-statically loaded in vitro by a 1000 N load. The two QCT scans were manipulated to generate p-FE models that mimic the experimental conditions. We compared p-FE displacements and strains of the wet CT model to the dry CT model and to the experimental results. In addition, the material assignment strategy was reinvestigated. The inhomogeneous Young's modulus was represented in the FE model using two different methods, directly extracted from the CT data and using continuous spatial functions as in Yosibash et al. [2007a. Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations. J. Biomechanics 40, 3688–3699].Results: Excellent agreement between dry and wet FE models was found for both displacements and strains, i.e. the method is insensitive to CT conditions and may be used in vivo. Good agreement was also found between FE results and experimental observations. The spatial functions representing Young's modulus are local and do not influence strains and displacements prediction. Finally, the p-FE results of all three fresh frozen human femurs compare very well to experimental observations exemplifying that the presented method may be in a mature stage to be used in clinical computer-aided decision making.     

A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments

The prediction of patient-specific proximal femur mechanical response to various load conditions is of major clinical importance in orthopaedics. This paper presents a novel, empirically validated high-order finite element method (FEM) for simulating the bone response to loads. A model of the bone geometry was constructed from a quantitative computerized tomography (QCT) scan using smooth surfaces for both the cortical and trabecular regions. Inhomogeneous isotropic elastic properties were assigned to the finite element model using distinct continuous spatial fields for each region. The Young's modulus was represented as a continuous function computed by a least mean squares method. p-FEMs were used to bound the simulation numerical error and to quantify the modeling assumptions. We validated the FE results with in-vitro experiments on a fresh-frozen femur loaded by a quasi-static force of up to 1500 N at four different angles. We measured the vertical displacement and strains at various locations and investigated the sensitivity of the simulation. Good agreement was found for the displacements, and a fair agreement found in the measured strain in some of the locations. The presented study is a first step toward a reliable p-FEM simulation of human femurs based on QCT data for clinical computer aided decision making.     

Patient-specific finite-element analyses of the proximal femur with orthotropic material properties validated by experiments

Patient-specific high order finite-element (FE) models of human femurs based on quantitative computer tomography (QCT) with inhomogeneous orthotropic and isotropic material properties are addressed. The point-wise orthotropic properties are determined by a micromechanics (MM) based approach in conjunction with experimental observations at the osteon level, and two methods for determining the material trajectories are proposed (along organs outer surface, or along principal strains). QCT scans on four fresh-frozen human femurs were performed and high-order FE models were generated with either inhomogeneous MM-based orthotropic or empirically determined isotropic properties. In vitro experiments were conducted on the femurs by applying a simple stance position load on their head, recording strains on femurs' surface and head's displacements. After verifying the FE linear elastic analyses that mimic the experimental setting for numerical accuracy, we compared the FE results to the experimental observations to identify the influence of material properties on models' predictions. The strains and displacements computed by FE models having MM-based inhomogeneous orthotropic properties match the FE-results having empirically based isotropic properties well, and both are in close agreement with the experimental results. When only the strains in the femoral neck are being compared a more pronounced difference is noticed between the isotropic and orthotropic FE result. These results lay the foundation for applying more realistic inhomogeneous orthotropic material properties in FEA of femurs.     

Prediction of the mechanical response of the femur with uncertain elastic properties

A mandatory requirement for any reliable prediction of the mechanical response of bones, based on quantitative computer tomography, is an accurate relationship between material properties (usually Young's modulus E) and bone density ρ. Many such E-ρ relationships are available based on different experiments on femur specimens with a large spread due to uncertainties. The first goal of this study is to pool and analyze the relevant available experimental data and develop a stochasticE-ρ relationship. This analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some "popular" E-ρ relationships are based on data that contains extreme scatter, while others are based on a very limited amount of information. The second goal is to use the newly developed stochastic E-ρ relationship in high-order finite element analyses (FEAs) for the computation of strains and displacements in two human proximal femurs, mimicking in vitro experiments. When compared with the experimental observations, the FEA predictions using the median of the stochastic E-ρ relationship follow the underlying distribution of the stochastic E-ρ relationship. Thus, most deviations of the FEA predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur.     

Relation between subject-specific hip joint loading, stress distribution in the proximal femur and bone mineral density changes after total hip replacement

In the prediction of bone remodelling processes after total hip replacement (THR), modelling of the subject-specific geometry is now state-of-the-art. In this study, we demonstrate that inclusion of subject-specific loading conditions drastically influences the calculated stress distribution, and hence influences the correlation between calculated stress distributions and changes in bone mineral density (BMD) after THR.For two patients who received cementless THR, personalized finite element (FE) models of the proximal femur were generated representing the pre- and post-operative geometry. FE analyses were performed by imposing subject-specific three-dimensional hip joint contact forces as well as muscle forces calculated based on gait analysis data. Average values of the von Mises stress were calculated for relevant zones of the proximal femur. Subsequently, the load cases were interchanged and the effect on the stress distribution was evaluated. Finally, the subject-specific stress distribution was correlated to the changes in BMD at 3 and 6 months after THR.We found subject-specific differences in the stress distribution induced by specific loading conditions, as interchanging of the loading also interchanged the patterns of the stress distribution. The correlation between the calculated stress distribution and the changes in BMD were affected by the two-dimensional nature of the BMD measurement.Our results confirm the hypothesis that inclusion of subject-specific hip contact forces and muscle forces drastically influences the stress distribution in the proximal femur. In addition to patient-specific geometry, inclusion of patient-specific loading is, therefore, essential to obtain accurate input for the analysis of stress distribution after THR.     

Femoral strength is better predicted by finite element models than QCT and DXA.

Clinicians and patients would benefit if accurate methods of predicting and monitoring bone strength in-vivo were available. A group of 51 human femurs (age range 21-93; 23 females, 28 males) were evaluated for bone density and geometry using quantitative computed tomography (QCT) and dual energy X-ray absorptiometry (DXA). Regional bone density and dimensions obtained from QCT and DXA were used to develop statistical models to predict femoral strength ex vivo. The QCT data also formed the basis of a three-dimensional finite element (FE) models to predict structural stiffness. The femurs were separated into two groups; a model training set (n = 25) was used to develop statistical models to predict ultimate load, and a test set (n = 26) was used to validate these models. The main goal of this study was to test the ability of DXA, QCT and FE techniques to predict fracture load non-invasively, in a simple load configuration which produces predominantly femoral neck fractures. The load configuration simulated the single stance phase portion of normal gait; in 87% of the specimens, clinical appearing sub-capital fractures were produced. The training/test study design provided a tool to validate that the predictive models were reliable when used on specimens with "unknown" strength characteristics. The FE method explained at least 20% more of the variance in strength than the DXA models. Planned refinements of the FE technique are expected to further improve these results. Three-dimensional FE models are a promising method for predicting fracture load, and may be useful in monitoring strength changes in vivo.     

Mapping anisotropy of the proximal femur for enhanced image based finite element analysis

Finite element (FE) models of bone derived from quantitative computed tomography (QCT) rely on realistic material properties to accurately predict bone strength. QCT cannot resolve bone microarchitecture, therefore QCT-based FE models lack the anisotropy apparent within the underlying bone tissue. This study proposes a method for mapping femoral anisotropy using high-resolution peripheral quantitative computed tomography (HR-pQCT) scans of human cadaver specimens. Femur HR-pQCT images were sub-divided into numerous overlapping cubic sub-volumes and the local anisotropy was quantified using a ‘direct-mechanics’ method. The resulting directionality reflected all the major stress lines visible within the trabecular lattice, and provided a realistic estimate of the alignment of Harvesian systems within the cortical compartment. QCT-based FE models of the proximal femur were constructed with isotropic and anisotropic material properties, with directionality interpolated from the map of anisotropy. Models were loaded in a sideways fall configuration and the resulting whole bone stiffness was compared to experimental stiffness and ultimate strength. Anisotropic models were consistently less stiff, but no statistically significant differences in correlation were observed between material models against experimental data. The mean difference in whole bone stiffness between model types was approximately 26%, suggesting that anisotropy can still effect considerable change in the mechanics of proximal femur models. The under prediction of whole bone stiffness in anisotropic models suggests that the orthotropic elastic constants require further investigation. The ability to map mechanical anisotropy from high-resolution images and interpolate information into clinical-resolution models will allow testing of new anisotropic material mapping strategies.     

Development of a parametric finite element model of the proximal femur using statistical shape and density modelling

Skeletal fractures associated with bone mass loss are a major clinical problem and economic burden, and lead to significant morbidity and mortality in the ageing population. Clinical image-based measures of bone mass show only moderate correlative strength with bone strength. However, engineering models derived from clinical image data predict bone strength with significantly greater accuracy. Currently, image-based finite element (FE) models are time consuming to construct and are non-parametric. The goal of this study was to develop a parametric proximal femur FE model based on a statistical shape and density model (SSDM) derived from clinical image data. A small number of independent SSDM parameters described the shape and bone density distribution of a set of cadaver femurs and captured the variability affecting proximal femur FE strength predictions. Finally, a three-dimensional FE model of an 'unknown' femur was reconstructed from the SSDM with an average spatial error of 0.016 mm and an average bone density error of 0.037 g/cm(3).     

Effect of boundary conditions,impact loading and hydraulic stiffening on femoral fracture strength

Patient specific quantitative CT (QCT) imaging data together with the finite element (FE) method may provide an accurate prediction of a patient's femoral strength and fracture risk. Although numerous FE models investigating femoral fracture strength have been published, there is little consent on the effect of boundary conditions, dynamic loading and hydraulic strengthening due to intra-medullary pressure on the predicted fracture strength. We developed a QCT-derived FE model of a proximal femur that included node-specific modulus assigned based on the local bone density. The effect of three commonly used boundary conditions published in literature were investigated by comparing the resulting strain field due to an applied fracture load. The models were also augmented with viscoelastic material properties and subject to a realistic impact load profile to determine the effect of dynamic loads on the strain field. Finally, the effect of hydraulic strengthening was investigated by including node specific permeability and performing a coupled pore diffusion and stress analysis of the FE model. Results showed that all boundary conditions yield the same strain field patterns, but peak strains were 22% lower and fracture load was 18% higher when loaded at the greater trochanter than when loaded at the femoral head. Comparison of the dynamic models showed that material viscoelasticity was important, but inertial effects (vibration and shock) were not. Finally, pore pressure changes did not cause significant hydraulic strengthening of bone under fall impact loading.     

QCT/FEA predictions of femoral stiffness are strongly affected by boundary condition modeling

Quantitative computed tomography-based finite element models of proximal femora must be validated with cadaveric experiments before using them to assess fracture risk in osteoporotic patients. During validation, it is essential to carefully assess whether the boundary condition (BC) modeling matches the experimental conditions. This study evaluated proximal femur stiffness results predicted by six different BC methods on a sample of 30 cadaveric femora and compared the predictions with experimental data. The average stiffness varied by 280% among the six BCs. Compared with experimental data, the predictions ranged from overestimating the average stiffness by 65% to underestimating it by 41%. In addition, we found that the BC that distributed the load to the contact surfaces similar to the expected contact mechanics predictions had the best agreement with experimental stiffness. We concluded that BC modeling introduced large variations in proximal femora stiffness predictions.     

Bone Load Estimation for the Proximal Femur Using Single Energy Quantitative CT Data

A density-based load estimation method was applied to determine femoral load patterns. Two-dimensional finite element models were constructed using single energy quantitative computed tomography (QCT) data from two femora. Basic load cases included parabolic pressure joint loads and constant tractions on the greater trochanter. An optimization procedure adjusted magnitudes of the basic load cases, such that the applied mechanical stimulus approached the ideal stimulus throughout each model. Dominant estimated load directions were generally consistent with published experimental data for gait. Other estimated loads suggested that loads at extreme joint orientations may be important to maintenance of bone structure. Remodeling simulations with the estimated loads produced density distributions qualitatively similar to the QCT data sets. Average nodal density errors between QCT data and predictions were 0.24 g/cm(3) and 0.28 g/cm(3). The results indicate that density-based load estimation could improve understanding of loading patterns on bones.     

Statistical shape and appearance models for fast and automated estimation of proximal femur fracture load using 2D finite element models

Estimating the risk of osteoporotic fractures is an important diagnostic step that needs to be taken before medicinal treatment. Densitometry-based criteria are normally used in clinical practice for this purpose. However, densitometry-based techniques could not explain all low-energy fractures. As patient-specific finite element (FE) models allow for consideration of other parameters (e.g. load conditions) that are known to be associated with fracture, they are considered promising candidates for more accurate fracture risk estimation. Nevertheless, they are often time consuming, expensive, and complex to build and may need the type of expertise that is not normally available in clinical settings. In this study, we report the development of an automated platform for estimating proximal femur fracture loads using patient-specific 2D FE models generated using dual-energy x-ray absorptiometry (DXA) scans. First, a statistical shape and appearance model (SSAM) is built using DXA scans of patients screened for osteoporosis following a low energy fracture. SSAM is then used together with Active Appearance Models (AAM) for automated segmentation of the proximal femur from new unseen DXA scans. The mean point-to-curve error of the automated procedure, i.e. 1.2–1.4 mm, is shown to be only slightly larger than the intra-observer variability of manual segmentation, i.e. 1.0 mm. Moreover, the developed platform automatically meshes the segmented shape, assigns density-based mechanical properties, assigns loads and boundary conditions, submits the 2D FE model for solution, and performs post-processing of the 2D FE simulation data to determine fracture loads. The fracture loads predicted using the manually generated and automatically generated 2D FE models are shown to be very close with a mean difference of around 8.8%. Repeated measures ANOVA showed no significant differences between the fracture loads calculated using FE models manually generated by three independent observers and those calculated using the automatically generated FE models (p>0.05).     

The use of sparse CT datasets for auto-generating accurate FE models of the femur and pelvis

The finite element (FE) method when coupled with computed tomography (CT) is a powerful tool in orthopaedic biomechanics. However, substantial data is required for patient-specific modelling. Here we present a new method for generating a FE model with a minimum amount of patient data. Our method uses high order cubic Hermite basis functions for mesh generation and least-square fits the mesh to the dataset. We have tested our method on seven patient data sets obtained from CT assisted osteodensitometry of the proximal femur. Using only 12 CT slices we generated smooth and accurate meshes of the proximal femur with a geometric root mean square (RMS) error of less than 1 mm and peak errors less than 8 mm. To model the complex geometry of the pelvis we developed a hybrid method which supplements sparse patient data with data from the visible human data set. We tested this method on three patient data sets, generating FE meshes of the pelvis using only 10 CT slices with an overall RMS error less than 3 mm. Although we have peak errors about 12 mm in these meshes, they occur relatively far from the region of interest (the acetabulum) and will have minimal effects on the performance of the model. Considering that linear meshes usually require about 70-100 pelvic CT slices (in axial mode) to generate FE models, our method has brought a significant data reduction to the automatic mesh generation step. The method, that is fully automated except for a semi-automatic bone/tissue boundary extraction part, will bring the benefits of FE methods to the clinical environment with much reduced radiation risks and data requirement.     

Effect of varus/valgus malalignment on bone strains in the proximal tibia after TKR: an explicit finite element study

Malalignment is the main cause of tibial component loosening. Implants that migrate rapidly in the first two post-operative years are likely to present aseptic loosening. It has been suggested that cancellous bone stresses can be correlated with tibial component migration. A recent study has shown that patient-specific finite element (FE) models have the power to predict the short-term behavior of tibial trays. The stresses generated within the implanted tibia are dependent on the kinematics of the joint; however, previous studies have ignored the kinematics and only applied static loads. Using explicit FE, it is possible to simultaneously predict the kinematics and stresses during a gait cycle. The aim of this study was to examine the cancellous bone strains during the stance phase of the gait cycle, for varying degrees of varus/valgus eccentric loading using explicit FE. A patient-specific model of a proximal tibia was created from CT scan images, including heterogeneous bone properties. The proximal tibia was implanted with a commercial total knee replacement (TKR) model. The stance phase of gait was simulated and the applied loads and boundary conditions were based on those used for the Stanmore knee simulator. Eccentric loading was simulated. As well as examining the tibial bone strains (minimum and maximum principal strain), the kinematics of the bone-implant construct are also reported. The maximum anterior-posterior displacements and internal-external rotations were produced by the model with 20 mm offset. The peak minimum and maximum principal strain values increased as the load was shifted laterally, reaching a maximum magnitude for -20 mm offset. This suggests that when in varus, the load transferred to the bone is shifted medially, and as the bone supporting this load is stiffer, the resulting peak bone strains are lower than when the load is shifted laterally (valgus). For this particular patient, the TKR design analyzed produced the highest cancellous bone strains when in valgus. This study has provided an insight in the variations produced in bone strain distribution when the axial load is applied eccentrically. To the authors' knowledge, this is the first time that the bone strain distribution of a proximal implanted tibia has been examined, also accounting for the kinematics of the tibio-femoral joint as part of the simulation. This approach gives greater insight into the overall performance of TKR.     

Densitometric, morphometric and mechanical distributions in the human proximal femur

With the prevalent use of DXA-measured BMD to assess pathologic hip fractures and its recently reported lack of reliability to predict fracture or account for efficacy of anti-resorptive therapy, it is reasonable to assess whether variations in the primary and secondary tensile and compressive trabecular microstructure can account for variations in proximal femur strength in comparison to DXA-measured BMD. To that end, microstructural and densitometric measures of trabecular bone specimens, from discrete sites within the proximal femur, were correlated with their mechanical properties. We hypothesize that accounting for regional variations in trabecular microstructure will improve predictions of proximal femur strength and stiffness compared to bone density measured by DXA. Forty-seven samples (seven donors) from seven distinct sites of human proximal femur underwent DXA and muCT imaging and mechanical testing. The results revealed significant variations in BMC, morphometric indices and mechanical properties within the proximal femur. This work has demonstrated that the mechanical performance of each sub-region is highly dependent on the corresponding trabecular microstructure. BMD measured by DXA at standard regions of interest cannot resolve the variations in trabecular density and microstructure that govern the mechanical behavior of the proximal femur. This work suggests that a quantitative Singh index that uses high resolution QCT to monitor the trabecular microstructure at specific sub-regions of the proximal femur may allow better predictions of hip fracture risk in individual patients and an improved assessment of changing bone structure in response to pharmacological interventions.     

Finite element analyses of human vertebral bodies embedded in polymethylmethalcrylate or loaded via the hyperelastic intervertebral disc models provide equivalent predictions of experimental strength

Quantitative computer tomography (QCT)-based finite element (FE) models of vertebral body provide better prediction of vertebral strength than dual energy X-ray absorptiometry. However, most models were validated against compression of vertebral bodies with endplates embedded in polymethylmethalcrylate (PMMA). Yet, loading being as important as bone density, the absence of intervertebral disc (IVD) affects the strength. Accordingly, the aim was to assess the strength predictions of the classic FE models (vertebral body embedded) against the in vitro and in silico strengths of vertebral bodies loaded via IVDs. High resolution peripheral QCT (HR-pQCT) were performed on 13 segments (T11/T12/L1). T11 and L1 were augmented with PMMA and the samples were tested under a 4° wedge compression until failure of T12. Specimen-specific model was generated for each T12 from the HR-pQCT data. Two FE sets were created: FE-PMMA refers to the classical vertebral body embedded model under axial compression; FE-IVD to their loading via hyperelastic IVD model under the wedge compression as conducted experimentally. Results showed that FE-PMMA models overestimated the experimental strength and their strength prediction was satisfactory considering the different experimental set-up. On the other hand, the FE-IVD models did not prove significantly better (Exp/FE-PMMA: R²=0.68; Exp/FE-IVD: R²=0.71, p=0.84). In conclusion, FE-PMMA correlates well with in vitro strength of human vertebral bodies loaded via real IVDs and FE-IVD with hyperelastic IVDs do not significantly improve this correlation. Therefore, it seems not worth adding the IVDs to vertebral body models until fully validated patient-specific IVD models become available.     

Combined finite element model of human proximal femur behaviour considering remodeling and fracture

The purpose of this work was to develop a combined remodeling-to-fracture finite element model allowing for the combined simulation of human proximal femur remodeling under a given boundary conditions followed by the simulation of its fracture behaviour under quasi-static load. The combination of remodeling and fracture simulation into one unified model consists in considering that the femur properties resulting from the remodeling simulation correspond to the initial state for the fracture prediction. The remodeling model is based on a coupled strain and fatigue damage stimulus approach. The fracture model is based on continuum damage mechanics in order to predict the progressive fracturing process, which allows to predict the fracture pattern and the complete force-displacement curve under quasi-static load. To investigate the potential of the proposed unified remodeling-to-fracture model, we performed remodeling simulations on a 3D proximal femur model for a duration of 365 days followed by a side fall fracture simulation reproducing.     

A novel methodology for generating 3D finite element models of the hip from 2D radiographs

Finite element (FE) modelling has been proposed as a tool for estimating fracture risk and patient-specific FE models are commonly based on computed tomography (CT). Here, we present a novel method to automatically create personalised 3D models from standard 2D hip radiographs. A set of geometrical parameters of the femur were determined from seven a–p hip radiographs and compared to the 3D femoral shape obtained from CT as training material; the error in reconstructing the 3D model from the 2D radiographs was assessed. Using the geometry parameters as the input, the 3D shape of another 21 femora was built and meshed, separating a cortical and trabecular compartment. The material properties were derived from the homogeneity index assessed by texture analysis of the radiographs, with focus on the principal tensile and compressive trabecular systems. The ability of these FE models to predict failure load as determined by experimental biomechanical testing was evaluated and compared to the predictive ability of DXA. The average reconstruction error of the 3D models was 1.77 mm (±1.17 mm), with the error being smallest in the femoral head and neck, and greatest in the trochanter. The correlation of the FE predicted failure load with the experimental failure load was r²=64% for the reconstruction FE model, which was significantly better (p<0.05) than that for DXA (r²=24%). This novel method for automatically constructing a patient-specific 3D finite element model from standard 2D radiographs shows encouraging results in estimating patient-specific failure loads.     

On the importance of modelling organ geometry and boundary conditions for predicting three-dimensional prostate deformation

The use of an ultrasound probe or a needle guide during biopsy deforms both the rectal wall and the prostate, resulting in lesion motion. An accurate patient-specific finite element (FE)-based biomechanical model can be used to predict prostate deformations. In this study, an FE model of a prostate phantom is developed using magnetic resonance images, while soft-tissue elasticity is estimated in vivo using an ultrasound-based acoustic radiation force impulse imaging technique. This study confirms that three-dimensional FE-predicted prostate deformation is predominantly dependent on accurate modelling of prostate geometry and boundary conditions. Upon application of various compressive displacements, our results show that a linear elastic FE model can accurately predict prostate deformations. The maximum global error between FE-predicted simulations and experimental results is 0.76 mm. Moreover, the effect of including the urethra, puboprostatic ligament and urinary bladder on prostate deformations is investigated by a sensitivity study.     

Comparison of micro-level and continuum-level voxel models of the proximal femur

Continuum-level finite element (FE) models became standard computational tools for the evaluation of bone mechanical behavior from in vivo computed tomography scans. Such scans do not account for the anisotropy of the bone. Instead, local mechanical properties in the continuum-level FE models are assumed isotropic and are derived from bone density, using statistical relationships. Micro-FE models, on the other hand, incorporate the anisotropic structure in detail. This study aimed to quantify the effects of assumed isotropy, by comparing continuum-level voxel models of a healthy and a severely osteoporotic proximal femur with recently analyzed micro-FE models of the same bones. The micro-model element size was coarsened to generate continuum FE models with two different element sizes (0.64 and 3.04 mm) and two different density–modulus relationships found in the literature for wet and ash density. All FE models were subjected to the same boundary conditions that simulated a fall to the side, and the stress and strain distributions, model stiffness and yield load were compared. The results indicated that the stress and strain distributions could be reproduced well with the continuum models. The smallest differences between the continuum-level model and micro-level model predictions of the stiffness and yield load were obtained with the coarsest element size. Better results were obtained for both continuum-element sizes when isotropic moduli were based on ash density rather than wet density.