Parametric finite element analysis of vertebral bodies affected by tumors

The vertebral column is the most frequent site of metastatic involvement of the skeleton. Due to the proximity to the spinal cord, from 5% to 10% of all cancer patients develop neurologic manifestations. As a consequence, fracture risk prediction has significant clinical importance. In this study, we model the metastatically involved vertebra so as to parametrically investigate the effects of tumor size, material properties and compressive loading rate on vertebral strength. A two-dimensional axisymmetric finite element model of a spinal motion segment consisting of the first lumbar vertebral body (no posterior elements) and adjacent intervertebral disc was developed to allow the inclusion of a centrally located tumor in the vertebral body. After evaluating elastic, mixed, and poroelastic formulations, we concluded that the poroelastic representation was most suitable for modeling the metastatically involved vertebra's response to compressive load. Maximum principal strains were used to localize regions of potential vertebral trabecular bone failure. Radial and axial vertebral body displacements were used as relative indicators of spinal canal encroachment and endplate failure. Increased tumor size and loading rate, and reduced trabecular bone density all elevated axial and radial displacements and maximum tensile strains. The results of this parametric study suggest that vertebral tumor size and bone density contribute significantly to a patients risk for vertebral fracture and should be incorporated in clinical assessment paradigms.     

Quantification of a rat tail vertebra model for trabecular bone adaptation studies

A feedback controlled loading apparatus for the rat tail vertebra was developed to deliver precise mechanical loads to the eighth caudal vertebra (C8) via pins inserted into adjacent vertebrae. Cortical bone strains were recorded using strain gages while subjecting the C8 in four cadaveric rats to mechanical loads ranging from 25 to 100 N at 1 Hz with a sinusoidal waveform. Finite element (FE) models, based on micro computed tomography, were constructed for all four C8 for calculations of cortical and trabecular bone tissue strains. The cortical bone strains predicted by FE models agreed with strain gage measurements, thus validating the FE models. The average measured cortical bone strain during 25-100 N loading was between 298 +/- 105 and 1210 +/- 297 microstrain (muepsilon). The models predicted average trabecular bone tissue strains ranging between 135 +/- 35 and 538 +/- 138 mu epsilon in the proximal region, 77 +/- 23-307 +/- 91 muepsilon in the central region, and 155 +/- 36-621 +/- 143 muepsilon in the distal region for 25-100 N loading range. Although these average strains were compressive, it is also interesting that the trabecular bone tissue strain can range from compressive to tensile strains (-1994 to 380 mu epsilon for a 100 N load). With this novel approach that combines an animal model with computational techniques, it could be possible to establish a quantitative relationship between the microscopic stress/strain environment in trabecular bone tissue, and the biosynthetic response and gene expression of bone cells, thereby study bone adaptation.     

Mechanisms of uniformity of yield strains for trabecular bone

Variations in yield strains for trabecular bone within a specific anatomic site are only a small fraction of the substantial variations that exist for elastic modulus and strength, and yet the source of this uniformity is not known. Our goal was to investigate the underlying mechanisms by using high-resolution, materially nonlinear finite element models of 12 human femoral neck trabecular bone specimens. The finite element models, used to obtain apparent yield strains in both tension and compression, assumed that the tissue-level yield strains were the same across all specimens. Comparison of the model predictions with the experimental data therefore enabled us to isolate the combined roles of volume fraction and architecture from the role of tissue material properties. Results indicated that, for both tensile and compressive loading, natural variations in volume fraction and architecture produced a negligible coefficient of variation (less than 3%) in apparent yield strains. Analysis of tissue-level strains showed that while bending of individual trabeculae played only a minor role in the apparent elastic behavior, the combined effects of this bending and tissue-level strength asymmetry produced apparent-level failure strains in compression that were 14% lower than those at the tissue level. By contrast, tissue and apparent-level yield strains were equivalent for tensile loading. We conclude that the uniformity of apparent yield strains is primarily the result of the highly oriented architecture that minimizes bending. Most of the variation that does occur is the result of the non-uniformity of the tissue-level yield strains.     

Damage in trabecular bone at small strains

Evidence that damage decreases bone quality, increases fracture susceptibility, and serves as a remodeling stimulus motivates further study of what loading magnitudes induce damage in trabecular bone. In particular, whether damage occurs at the smaller strains characteristic of habitual, as opposed to traumatic, loading is not known. The overall goal of this study was to characterize damage accumulation in trabecular bone at small strains (0.20 - 0.45% strain). A continuum damage mechanics approach was taken whereby damage was quantified by changes in modulus and residual strain. Human vertebral specimens (n = 7) were tested in compression using a multi-cycle load - unload protocol in which the maximum applied strain for each cycle, epsilonmax, was increased incrementally from epsilonmax = 0.20% on the first loading cycle to epsilonmax = 0.45% on the last cycle. Modulus and residual strain were measured for each cycle. Both changes in modulus and residual strains commenced at small strains, beginning as early as 0.24 and 0.20% strain, respectively. Strong correlations between changes in modulus and residual strains were observed (r = 0.51 - 0.98). Fully nonlinear, high-resolution finite element analyses indicated that even at small apparent strains, tissue-level strains were sufficiently high to cause local yielding. These results demonstrate that damage in trabecular bone occurs at apparent strains less than half the apparent compressive yield strain reported previously for human vertebral trabecular bone. Further, these findings imply that, as a consequence of the highly porous trabecular structure, tissue yielding can initiate at very low apparent strains and that this local failure has detectable and negative consequences on the apparent mechanical properties of trabecular bone.     

Dependence of yield strain of human trabecular bone on anatomic site

Understanding the dependence of human trabecular bone strength behavior on anatomic site provides insight into structure-function relationships and is essential to the increased success of site-specific finite element models of whole bones. To investigate the hypothesis that the yield strains of human trabecular bone depend on anatomic site, the uniaxial tensile and compressive yield properties were compared for cylindrical specimens from the vertebra (n=61), proximal tibia (n=31), femoral greater trochanter (n=23), and femoral neck (n=27) taken from 61 donors (67+/-15years). Test protocols were used that minimized end artifacts and loaded specimens along the main trabecular orientation. Yield strains by site (mean+/-S.D.) ranged from 0.70+/-0.05% for the trochanter to 0.85+/-0.10% for the femoral neck in compression, from 0.61+/-0.05% for the trochanter to 0.70+/-0.05% for the vertebra in tension, and were always higher in compression than tension (p<0.001). The compressive yield strain was higher for the femoral neck than for all other sites (p<0.001), as was the tensile yield strain for the vertebra (p<0.007). Analysis of covariance, with apparent density as the covariate, indicated that inter-site differences existed in yield stress even after adjusting statistically for density (p<0.035). Coefficients of variation in yield strain within each site ranged from only 5-12%, consistent with the strong linear correlations (r(2)=0.94-0.98) found between yield stress and modulus. These results establish that the yield strains of human trabecular bone can differ across sites, but that yield strain may be considered uniform within a given site despite substantial variation in elastic modulus and yield stress.     

Incorporating tissue anisotropy and heterogeneity in finite element models of trabecular bone altered predicted local stress distributions

Trabecular bone is composed of organized mineralized collagen fibrils, which results in heterogeneous and anisotropic mechanical properties at the tissue level. Recently, biomechanical models computing stresses and strains in trabecular bone have indicated a significant effect of tissue heterogeneity on predicted stresses and strains. However, the effect of the tissue-level mechanical anisotropy on the trabecular bone biomechanical response is unknown. Here, a computational method was established to automatically impose physiologically relevant orientation inherent in trabecular bone tissue on a trabecular bone microscale finite element model. Spatially varying tissue-level anisotropic elastic properties were then applied according to the bone mineral density and the local tissue orientation. The model was used to test the hypothesis that anisotropy in both homogeneous and heterogeneous models alters the predicted distribution of stress invariants. Linear elastic finite element computations were performed on a 3 mm cube model isolated from a microcomputed tomography scan of human trabecular bone from the distal femur. Hydrostatic stress and von Mises equivalent stress were recorded at every element, and the distributions of these values were analyzed. Anisotropy reduced the range of hydrostatic stress in both tension and compression more strongly than the associated increase in von Mises equivalent stress. The effect of anisotropy was independent of the spatial redistribution high compressive stresses due to tissue elastic heterogeneity. Tissue anisotropy and heterogeneity are likely important mechanisms to protect bone from failure and should be included for stress analyses in trabecular bone.     

Localized strain measurements of the intervertebral disc annulus during biaxial tensile testing

Both inter-lamellar and intra-lamellar failures of the annulus have been described as potential modes of disc herniation. Attempts to characterize initial lamellar failure of the annulus have involved tensile testing of small tissue samples. The purpose of this study was to evaluate a method of measuring local surface strains through image analysis of a tensile test conducted on an isolated sample of annular tissue in order to enhance future studies of intervertebral disc failure. An annulus tissue sample was biaxial strained to 10%. High-resolution images captured the tissue surface throughout testing. Three test conditions were evaluated: submerged, non-submerged and marker. Surface strains were calculated for the two non-marker conditions based on motion of virtual tracking points. Tracking algorithm parameters (grid resolution and template size) were varied to determine the effect on estimated strains. Accuracy of point tracking was assessed through a comparison of the non-marker conditions to a condition involving markers placed on tissue surface. Grid resolution had a larger effect on local strain than template size. Average local strain error ranged from 3% to 9.25% and 0.1% to 2.0%, for the non-submerged and submerged conditions, respectively. Local strain estimation has a relatively high potential for error. Submerging the tissue provided superior strain estimates.     

Role of endplates in contributing to compression behaviors of motion segments and intervertebral discs

The purpose of this study was to gain an improved understanding of the mechanical behavior of the intervertebral disc in the presence and absence of the vertebral endplates. Mechanical behaviors of rat caudal motion segments, vertebrae and isolated disc explants under two different permeability conditions were investigated and viscoelastic behaviors were evaluated using a stretched-exponential function to describe creep and recovery behaviors. The results demonstrated that both vertebrae and discs underwent significant deformations in the motion segment even under relatively low-loading conditions. Secondly, disruption of the collagenous network had minimal impact on equilibrium deformations of disc explants as compared to disc deformations occurring in the motion segments provided that vertebral deformations were accounted for; however, differences in endplate permeability conditions had a significant effect on viscoelastic behaviors. Creep occurred more quickly than recovery for motion segment and explant specimens. In addition, disc explants and motion segments both exhibited non-recoverable deformations under axial compression under low- and high-loading conditions. Results have important implications for interpreting the role of vertebral endplates in contributing to disc mechanical behaviors and direct application to mechanobiology studies involving external loading to rodent tail intervertebral discs.     

Micromechanics of the human vertebral body for forward flexion

To provide mechanistic insight into the etiology of osteoporotic wedge fractures, we investigated the spatial distribution of tissue at the highest risk of initial failure within the human vertebral body for both forward flexion and uniform compression loading conditions. Micro-CT-based linear elastic finite element analysis was used to virtually load 22 human T9 vertebral bodies in either 5° of forward flexion or uniform compression; we also ran analyses replacing the simulated compliant disc (E=8 MPa) with stiff polymethylmethacrylate (PMMA, E=2500 MPa). As expected, we found that, compared to uniform compression, forward flexion increased the overall endplate axial load on the anterior half of the vertebra and shifted the spatial distribution of high-risk tissue within the vertebra towards the anterior aspect of the vertebral body. However, despite that shift, the high-risk tissue remained primarily within the central regions of the trabecular bone and endplates, and forward flexion only slightly altered the ratio of cortical-to-trabecular load sharing at the mid-vertebral level (mean±SD for n=22: 41.3±7.4% compression; 44.1±8.2% forward flexion). When the compliant disc was replaced with PMMA, the anterior shift of high-risk tissue was much more severe. We conclude that, for a compliant disc, a moderate degree of forward flexion does not appreciably alter the spatial distribution of stress within the vertebral body.     

High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone

The ability to predict trabecular failure using microstructure-based computational models would greatly facilitate study of trabecular structure–function relations, multiaxial strength, and tissue remodeling. We hypothesized that high-resolution finite element models of trabecular bone that include cortical-like strength asymmetry at the tissue level, could predict apparent level failure of trabecular bone for multiple loading modes. A bilinear constitutive model with asymmetric tissue yield strains in tension and compression was applied to simulate failure in high-resolution finite element models of seven bovine tibial specimens. Tissue modulus was reduced by 95% when tissue principal strains exceeded the tissue yield strains. Linear models were first calibrated for effective tissue modulus against specimen-specific experimental measures of apparent modulus, producing effective tissue moduli of (mean±S.D.) 18.7±3.4 GPa. Next, a parameter study was performed on a single specimen to estimate the tissue level tensile and compressive yield strains. These values, 0.60% strain in tension and 1.01% strain in compression, were then used in non-linear analyses of all seven specimens to predict failure for apparent tensile, compressive, and shear loading. When compared to apparent yield properties previously measured for the same type of bone, the model predictions of both the stresses and strains at failure were not statistically different for any loading case (p>0.15). Use of symmetric tissue strengths could not match the experimental data. These findings establish that, once effective tissue modulus is calibrated and uniform but asymmetric tissue failure strains are used, the resulting models can capture the apparent strength behavior to an outstanding level of accuracy. As such, these computational models have reached a level of fidelity that qualifies them as surrogates for destructive mechanical testing of real specimens.     

Direct measurement of intervertebral disc maximum shear strain in six degrees of freedom: motions that place disc tissue at risk of injury

Human intervertebral disc specimens were tested to determine the regions of largest maximum shear strain (MSS) experienced by disc tissues in each of three principal displacements and three rotations, and to identify the physiological rotations and displacements that may place the disc at greatest risk for large tissue strains and injury. Tearing of disc annulus may be initiated by large interlamellar shear strains. Nine human lumbar discs were tagged with radiographic markers on the endplates, disc periphery and with a grid of wires in the mid-transverse plane and subjected to each of the six principal displacements and rotations. Stereo-radiographs were taken in each position and digitized for reconstruction of the three-dimensional position of each marker. Maximum tissue shear strains were calculated from relative marker displacements and normalized by the input displacement or rotation. Lateral shear, compression, and lateral bending were the motions that produced the mean (95% confidence interval) largest mean MSS of 9.6 (0.7)%/mm, 9.0 (0.5)%/mm, and 5.8 (1.6)%/ degrees , respectively, and which occurred in the posterior, posterolateral and lateral peripheral regions of the disc. After taking into account the reported maximum physiological range of motion for each degree of freedom, motions producing the highest physiological MSS were lateral bending (57.8 (16.2)%) and flexion (38.3 (3.3)%), followed by lateral shear (14.4 (1.1)%) and compression (12.6 (0.7)%).     

Experimental and finite element analysis of the rat ulnar loading model-correlations between strain and bone formation following fatigue loading

The rat forelimb compression model has been used widely to study bone response to mechanical loading. We used strain gages to assess load sharing between the ulna and radius in the forelimb of adult Fisher rats. We used histology and peripheral quantitative computed tomography (pQCT) to quantify ulnar bone formation 12 days after in vivo fatigue loading. Lastly, we developed a finite element model of the ulna to predict the pattern of surface strains during compression. Our findings indicate that at the mid-shaft the ulna carries 65% of the applied compressive force on the forelimb. We observed large variations in fatigue-induced bone formation over the circumference and length of the ulna. Bone formation was greatest 1-2 mm distal to the mid-shaft. At the mid-shaft, we observed woven bone formation that was greatest medially. Finite element analysis indicated a strain pattern consistent with a compression-bending loading mode, with the greatest strains occurring in compression on the medial surface and lesser tensile strains occurring laterally. A peak strain of -5190 microepsilon (for 13.3N forelimb compression) occurred 1-2 mm distal to the mid-shaft. The pattern of bone formation in the longitudinal direction was highly correlated to the predicted peak compressive axial strains at seven cross-sections (r2 = 0.89, p = 0.014). The in-plane pattern of bone formation was poorly correlated to the predicted magnitude of axial strain at 51 periosteal locations (r2 = 0.21, p < 0.001), because the least bone formation was observed where tensile strains were highest. These findings indicate that the magnitude of bone formation after fatigue loading is greatest in regions of high compressive strain.     

A novel in vivo mouse model for mechanically stimulated bone adaptation--a combined experimental and computational validation study

To facilitate the investigation of bone formation, in vivo, in response to mechanical loading a caudal vertebra axial compression device (CVAD) has been developed to deliver precise mechanical loads to the fifth caudal vertebra (C5) of the C57BL/6 female mouse. A combined experimental and computational approach was used to quantify the micro-mechanical strain induced in trabecular and cortical components following static and dynamic loading using the CVAD. Cortical bone strains were recorded using micro-strain gages. Finite element (FE) models based on micro-computed tomography were constructed for all C5 vertebrae. Both theoretical and experimental cortical strains correlated extremely well (R(2)>0.96) for a Young's modulus of 14.8 GPa, thus validating the FE model. In this study, we have successfully applied mechanical loads to the C5 murine vertebrae, demonstrating the potential of this model to be used for in vivo loading studies aimed at stimulating both trabecular and cortical bone adaptation.     

Contribution of vertebral [corrected] bodies, endplates, and intervertebral discs to the compression creep of spinal motion segments

Spinal segments show non-linear behavior under axial compression. It is unclear to what extent this behavior is attributable to the different components of the segment. In this study, we quantified the separate contributions of vertebral bodies and intervertebral discs to creep of a segment. Secondly, we investigated the contribution of bone and osteochondral endplate (endplates including cartilage) to the deformation of the vertebral body. From eight porcine spines a motion segment, a disc and a vertebral body were dissected and subjected to mechanical testing. In an additional test, cylindrical samples, machined from the lowest thoracic vertebrae of 11 porcine spines, were used to compare the deformation of vertebral bone and endplate. All specimens were subjected to three loading cycles, each comprising a loading phase (2.0 MPa, 15 min) and a recovery phase (0.001 MPa, 30 min). All specimens displayed substantial time-dependent height changes. Average creep was the largest in motion segments and smallest in vertebral bodies. Bone samples with endplates displayed substantially more creep than samples without. In the early phase, behavior of the vertebra was similar to that of the disc. Visco-elastic deformation of the endplate therefore appeared dominant. In the late creep phase, behavior of the segment was similar to that of isolated discs, suggesting that in this phase the disc dominated creep behavior, possibly by fluid flow from the nucleus. We conclude that creep deformation of vertebral bodies contributes substantially to creep of motion segments and that within a vertebral body endplates play a major role.     

A comparison between porcine,ovine, and bovine intervertebral disc anatomy and single lamella annulus fibrosus tensile properties

This project aimed to compare gross anatomical measures and biomechanical properties of single lamellae from the annulus fibrosus of ovine and porcine lumbar vertebrae, and bovine tail vertebrae. The morphology of the vertebrae of these species differ significantly both from each other and from human, yet how these differences alter biomechanical properties is unknown. Geometric parameters measured in this study included: 1) absolute and relative intervertebral (IVD) and vertebral body height and 2) absolute and relative intervertebral disc (IVD) anterior‐posterior (AP) and medial‐lateral (ML) widths. Single lamella tensile properties included toe‐region stress and stretch ratio, stiffness, and tensile strength. As expected, the bovine tail IVD revealed a more circular shape compared with both the ovine and porcine lumbar IVD. The bovine tail also had the largest IVD to vertebral body height ratio (due to having the highest absolute IVD height). Bovine tail lamellae were also found to be strongest and stiffest (in tension) while ovine lumbar lamellae were weakest and most compliant. Histological analysis revealed the greatest proportion of collagen in the bovine corroborating findings of increased strength and stiffness. The observed differences in anatomical shape, connective tissue composition, and tensile properties need to be considered when choosing an appropriate model for IVD research. J. Morphol. 277:244–251, 2016. © 2015 Wiley Periodicals, Inc.     

Embedding of human vertebral bodies leads to higher ultimate load and altered damage localisation under axial compression

Computer tomography (CT)-based finite element (FE) models of vertebral bodies assess fracture load in vitro better than dual energy X-ray absorptiometry, but boundary conditions affect stress distribution under the endplates that may influence ultimate load and damage localisation under post-yield strains. Therefore, HRpQCT-based homogenised FE models of 12 vertebral bodies were subjected to axial compression with two distinct boundary conditions: embedding in polymethylmethalcrylate (PMMA) and bonding to a healthy intervertebral disc (IVD) with distinct hyperelastic properties for nucleus and annulus. Bone volume fraction and fabric assessed from HRpQCT data were used to determine the elastic, plastic and damage behaviour of bone. Ultimate forces obtained with PMMA were 22% higher than with IVD but correlated highly (R² = 0.99). At ultimate force, distinct fractions of damage were computed in the endplates (PMMA: 6%, IVD: 70%), cortex and trabecular sub-regions, which confirms previous observations that in contrast to PMMA embedding, failure initiated underneath the nuclei in healthy IVDs. In conclusion, axial loading of vertebral bodies via PMMA embedding versus healthy IVD overestimates ultimate load and leads to distinct damage localisation and failure pattern.     

Compressive fatigue behavior of human vertebral trabecular bone

Damage accumulation under compressive fatigue loading is believed to contribute significantly to non-traumatic, age-related vertebral fractures in the human spine. Only few studies have explored trabecular bone fatigue behavior under compressive loading and none examined the influence of trabecular architecture on fatigue life. In this study, trabecular bone samples of human lumbar and thoracic vertebrae (4 donors from age 29 to 86, n=29) were scanned with a microCT system prior to compressive fatigue testing to determine morphology-mechanical relationships for this relevant loading mode. Inspired from previous fabric-based relationships for elastic properties and quasi-static strength of trabecular bone, a simple power relationship between volume fraction, fabric eigenvalue, applied stress and the number of cycles to failure is proposed. The experimental results demonstrate a high correlation for this relationship (R2=0.95) and detect a significant contribution of the degree of anisotropy towards prediction of fatigue life. Step-wise regression for total and residual strains at failure suggested a weak, but significant correlation with volume fraction. From the obtained results, we conclude that the applied stress normalized by volume fraction and axial fabric eigenvalue can estimate fatigue life of human vertebral trabecular bone in axial compressive loading.     

Mechanics of linear microcracking in trabecular bone

Microcracking in trabecular bone is responsible both for the mechanical degradation and remodeling of the trabecular bone tissue. Recent results on trabecular bone mechanics have demonstrated that bone tissue microarchitecture, tissue elastic heterogeneity and tissue-level mechanical anisotropy all should be considered to obtain detailed information on the mechanical stress state. The present study investigated the influence of tissue microarchitecture, tissue heterogeneity in elasticity and material separation properties and tissue-level anisotropy on the microcrack formation process. Microscale bone models were executed with the extended finite element method. It was demonstrated that anisotropy and heterogeneity of the bone tissue contribute significantly to bone tissue toughness and the resistance of trabecular bone to microcrack formation. The compressive strain to microcrack initiation was computed to increase by a factor of four from an assumed homogeneous isotropic tissue to an assumed anisotropic heterogenous tissue.     

Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue

The ability to determine trabecular bone tissue elastic and failure properties has biological and clinical importance. To date, trabecular tissue yield strains remain unknown due to experimental difficulties, and elastic moduli studies have reported controversial results. We hypothesized that the elastic and tensile and compressive yield properties of trabecular tissue are similar to those of cortical tissue. Effective tissue modulus and yield strains were calibrated for cadaveric human femoral neck specimens taken from 11 donors, using a combination of apparent-level mechanical testing and specimen-specific, high-resolution, nonlinear finite element modeling. The trabecular tissue properties were then compared to measured elastic modulus and tensile yield strain of human femoral diaphyseal cortical bone specimens obtained from a similar cohort of 34 donors. Cortical tissue properties were obtained by statistically eliminating the effects of vascular porosity. Results indicated that mean elastic modulus was 10% lower (p<0.05) for the trabecular tissue (18.0+/-2.8 GPa) than for the cortical tissue (19.9+/-1.8 GPa), and the 0.2% offset tensile yield strain was 15% lower for the trabecular tissue (0.62+/-0.04% vs. 0.73+/-0.05%, p<0.001). The tensile-compressive yield strength asymmetry for the trabecular tissue, 0.62 on average, was similar to values reported in the literature for cortical bone. We conclude that while the elastic modulus and yield strains for trabecular tissue are just slightly lower than those of cortical tissue, because of the cumulative effect of these differences, tissue strength is about 25% greater for cortical bone.     

Fatigue of bovine trabecular bone

Fatigue loading of bone, from the activities of daily living in the elderly, or from prolonged exercise in the young, can lead to increased risk of fracture. Elderly patients with osteoporosis are particularly prone to fragility fractures of the vertebrae, where load is carried primarily by trabecular bone. In this study, specimens of bovine trabecular bone were loaded in compressive fatigue at four different normalized stresses to one of six maximum strains. The resulting change in modulus and residual strain accumulation were measured over the life of the fatigue test. The number of cycles to reach a given maximum compressive strain increased with decreasing normalized stress. Modulus reduction and specimen residual strain increased with increasing maximum compressive strain, but few differences were observed between specimens loaded to the same maximum strain at different normalized stresses.