The nanoscale dimension known as D-spacing describes the staggering of collagen molecules, which are fundamental to the biphasic makeup of bone tissue. This dimension was long assumed to be constant, but recent studies have shown that the periodicity of collagen is variable. Given that the arrangement of collagen molecules is closely related to the degree of bone mineralization, recent studies have begun to look at D-spacing as a potential factor in the ongoing effort to battle postmenopausal osteoporosis. The theoretical models presented by previous studies have only opted to model a single collagen-hydroxyapatite period, so the creation of an intricate computational approach that more exhaustively models a network of collagen and mineral is well-warranted. Sheep present an excellent opportunity to examine metabolic disorders, as their bone structure similar to that of the human skeleton. Six Rambouillet-cross ewes were used for the purpose of gathering experimental data. Three ewes underwent a sham surgery (controls), while an ovariectomy (OVX) was performed on the remaining three sheep. Each sheep was sacrificed after 12 months and their radius and ulna were harvested for atomic force microscopy and mechanical testing. Each sheep bone produced up to 25 beam samples that were available for analysis, and two were randomly selected from each test sheep. The cranial anatomical sector was selected for testing as it replicates the tensile loading condition characteristically experienced by collagen molecules and its exclusive examination removes any unintended variation due to bone section. Experimental D-spacing measurements were used in a finite element software, Abaqus, to create the ``Complex Model'': a large-scale, 2-D staggered array representation of collagen and hydroxyapatite periodicity. D-spacings intrinsic variability was mimicked through a Gaussian distribution that randomly determined periodic lengths based on provided experimental data. The model was generated with these random conditions for 2 x 100 units. Safeguards were implemented to ensure appropriate ratios of collagen to hydroxyapatite throughout the randomization. Collagen was assigned viscoelastic material properties originally developed by Dr. Frank Richter and modified by Miguel Mendoza. Hydroxyapatite was modeled as an elastic isotropic material. Four models were created using randomized D-spacings from control sheep and four separate models were created based on OVX sheep. Tangent delta--a damping characteristic--was recorded to evaluate bone viscoelasticity across four test frequencies: 1, 3, 9, and 15 Hz. Results strongly suggest that the Complex Model matches experimental findings more accurately than previous computational approaches. These results indicate the complicated network of many collagen units is an essential parameter of adequate modeling. A repeated measures analysis of variance was performed to examine the differences between control and OVX sheep. After adjusting for all other predictors, at the 1% significance level, after adjusting for all other variables, there is not enough evidence to convince this study that the Surgical Treatment alone has a significant impact on output tangent delta. This finding leads this study to conclude that OVX is fully accounted for within the Complex Model through the inclusion of its D-spacing, and the answers to bone's complicated mechanical properties during estrogen loss may lie in how OVX changes collagen viscoelasticity. Significant interactions were found between the Model Type and the Test Frequency. A Tukey-Kramer pairwise comparison was performed between Complex and Experimental data, which determined the Complex Model did not behave statistically differently from experimental findings at 15 Hz. This result suggests the Complex Model may begin to be validated to experimental results in a statistically meaningfully way that is a first for this style of FEA approach. The flexibility implemented in the randomization of the Complex Model welcomes refinement primarily in modeling viscoelasticity and fine-tuning the representation of mineralization. Through adjusting these material characteristics, the Complex Model may become an even more powerful tool in examining bone viscoelasticity and metabolic disorders.

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.

The D-spacing that is characteristic of collagen and its structural arrangement was previously thought to be a constant value. Much research is revealing that it is actually a distribution of values in biological tissues. Recent ovine experimentation has also shown that the D-spacing distribution is significantly altered following estrogen depletion. While ewes contain some major biological differences between their human counterparts, they are an economical and robust large animal model for postmenopausal osteoporosis. So, the exploration of the possible implications that D-spacing has on the mechanical properties of the whole bone utilizing animal models and computational methods is warranted. Six Warhill ewes were used in this experiment and were either ovariectomized or underwent a sham surgery. The animals were sacrificed after 3 years and the radius and ulna bone were harvested for further analysis. Rectangular beams of compact bone tissue were machined from six different sectors in the whole bone and dynamic mechanical analysis tests were performed on the 24 specimens. The viscoelastic property, tangent delta, was measured from each test at varying frequencies. A composite arrangement of collagen and hydroxyapatite were then computationally modeled utilizing finite element analysis to observe the effects of altered D-spacing on the mechanical properties. Jager and Fratzl’s staggered array model allowed the inclusion of a D-spacing configuration as well as the simplified 2 dimensional plane strain analysis. Hydroxyapatite was modeled as a perfectly elastic material, while the hydrated collagen component was linear viscoelastic through the use of the standard linear solid model. The main finding of the work is that D-spacing only significantly altered the tangent delta of the computational model when the mineral volume fraction changed. Since the composite model analyzed the structural arrangement of compact bone at such a small scale, the change in mineral volume fraction could only be realistically attributed to intrafibrillar mineral. The results of this preliminary analysis are promising and warrant the continued investigation of D-spacing and mineral content and their significance in the osteoporotic condition.