Date of Award


Degree Name

MS in Biomedical Engineering


Biomedical and General Engineering


College of Engineering


Scott Hazelwood

Advisor Department

Biomedical and General Engineering

Advisor College

College of Engineering


Bone is a largely bipartite viscoelastic composite. Its mechanical behavior is determined by strain rate and the relative proportions of its principal constituent elements, hydroxyapatite and collagen, but is also largely dictated by their geometry and topology. Collagen fibrils include many segments of tropocollagen in staggered, parallel sequences. The physical staggering of this tropocollagen allows for gaps known as hole-zones, which serve as nucleation points for apatite mineral. The distance between adjacent repeat units of tropocollagen is known as D-Spacing and can be measured by Atomic Force Microscopy (AFM). This D-Spacing can vary in length slightly within a bundle, but by an additional order of magnitude within the same specimen, and can significantly alter the proportion of hydroxyapatite. Previous researchers have built and refined a Finite Element Analysis “Complex Model” to capture the consequences of adjusting D-Spacing and the viscoelastic parameters. This will ultimately serve to elucidate and perhaps predict the mechanical consequences of biological events that alter these parameters. This study aims to further refine the model’s precision by accounting for crosslinking between fibrils, the presence of which serves to add mechanical strength. This study also looks to refine the currently used rheological models by way of frequency dependent parameters in the hopes of improving model accuracy over a wider frequency range.

Hormonal factors such as estrogen can significantly determine the composition of bone. Menopause marks a significant reduction in circulating estrogen and has been shown to factor heavily in the development of conditions like osteoporosis. Because sheep feature a hormonal cycle and skeletal structure similar to humans, three of six mature Columbia-Rambouillet ewes were randomly selected to undergo an ovariectomy, the remainder serving as sham-operated controls. Twelve months later twenty-five beam samples were harvested from their radius bones for mechanical analysis and other testing, including atomic force microscopy (AFM) and dynamic mechanical analysis (DMA). The data gleaned from these tests provide an experimental basis of comparison with The Complex Model.

A 2-D Finite Element Analysis model in Abaqus was first created by Miguel Mendoza, which enforced viscoelasticity and a realistic proportion and placement of hydroxyapatite and collagen. The viscoelasticity was modeled using a Standard Linear Solid involving springs and a dashpot element. Crosslinks of varying number and location were arranged within the former model configuration as node to surface tie-constraints to explore the treatment of the FEA Model as a more realistic assembly of parts. Frequencies utilized for this model included 1, 3, 9 and 12 Hz. This approach is referred to in this research as the Intermolecular Forces (IMF) Scheme.

The model was subsequently refined by Christopher Ha and Austin Cummings. The model was characterized by 2x100 unit half-cells, the lengths of which were randomly generated by a Python script. This script ingested the mean and standard deviation D-Spacing length to generate a model geometrically similar to a real specimen bearing those dimensions. A frequency dependent value for the dashpot element in the rheological model used for tropocollagen was developed using this latter FEA model, named the Complex Model. Dashpot values explored for this variable dashpot included 0.0125, 0.125, 0.3125, 0.45, 0.5875, 0.725, 0.8625 and 1.25 GPa-s, some values chosen for their high performance in past studies and others to further narrow the search for the best performing dashpot. All dashpot values were investigated over the previously stated frequencies in addition to 2, 5, 7 and 12 Hz. The best fit dashpot values were plotted against the frequencies in which they best performed and a polynomial trend line was fitted to establish an equation, and that equation was used to modify an existing user material subroutine for tropocollagen to provide an automatic frequency dependent dashpot value to Abaqus. This approach is referred to in this research as the Variable Dashpot (VD) Scheme.

Results for the IMF scheme generally performed poorly, with the fully tie-constrained model performing best with 0.77 and 0.024 for R2 and RMSE respectively. Of the randomized crosslink models, that with the lowest number (N=20) of randomly placed non-enzymatic crosslinks performed best with 0.81 and 0.051 for R2 and RMSE respectively. Increasing the number of randomized crosslinks reduced model fit, and the remaining three variants exhibited mean R2 and RMSE values of 0.66-0.67 and 0.052 respectively. For the VD scheme, models running custom modified variable dashpot UMATs yielded R2 and RMSE values of 0.87 and 0.012 for C2207, and 0.89 and 0.008 for C1809. This is a notable fit considering all other material property parameters are held constant throughout each frequency. In the rheological model, this research also found a striking difference between the frequency dependent viscous element values that made each model perform best. This indicates that differences in D-Spacing standard deviations between OVX and control may be associated with distinct strain-rate dependent mechanical responses.

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