Osteoporosis is characterized by the impairment of bone mass, strength, and microarchitecture, which increases the propensity of fragility fractures (1). The mechanical properties of trabecular bone are mainly determined by the architectural arrangement of the network. Typically, the calculated trabecular 3D microstructure includes bone volume fraction (BV/TV), trabecular number (Tb. N), trabecular thickness (Tb. Th), trabecular spacing (Tb. Sp), connectivity density (Conn. D), structure model index (SMI) and the degree of anisotropy (DA) (2). Although trabecular networks are estimated by these different structural parameters, BV/TV is the main determinant of trabecular bone mechanical properties. An estimated 70-80% of the variances in the apparent tissue modulus of trabecular bone can be explained by BV/TV (3–6). The apparent modulus and collapse stress of regular foams (simulating trabecular tissue) are predicted to be quadratic functions of BV/TV (7), especially when accounting for the side-artifacts of cored trabecular specimens (8, 9). There are different regression relations between BV/TV and the tissue modulus of fractured and nonfractured proximal femurs, indicating that the spatial arrangement and attenuation of trabeculae could jeopardize the mechanical properties of trabecular bone (10, 11).

Trabecular samples tested in these studies were obtained from a set of specimens that were pooled from multiple individuals collectively spanning a wide range of densities and BV/TV; when tissue modulus and BV/TV were correlated regardless of side-artifact correction, the goodness of fit was better with the addition of other structural indices, e.g., DA (4), Tb. Sp and mean intercept length (MIL) (3). Since no two trabecular structures are exactly the same and osteoporosis is not necessarily systemic (12), it is almost impossible to control the specific architecture of each trabecular specimen and to correlate their structure to its function. Furthermore, even with similar BV/TV values, healthy and osteoporotic trabeculae may yield significantly different mechanical properties (13). To date, the deterioration of trabecular structure (e.g., from a healthy state to osteoporosis) and the resulting changes in mechanical properties in individuals remain to be better clarified.

The finite element method based on micro-CT (μFEM) has been widely used to study trabecular bone mechanics (14–17). The ability to test each trabecular sample multiple times is greatly beneficial, either along with different loading modes or from different loading directions. For example, Maquer (4) analyzed 743 micro-CT reconstructions of cubic trabecular samples using μFEM to confirm the significant role of both BV/TV and DA in determining apparent elastic properties. Using μFEM, Liu (18) loaded the same trabecular cubes from three principal directions (along the X, Y and Z axes) to suggest that (1) BV/TV alone has a significant effect on the tissue modulus of the trabecular bone, while tissue properties demonstrated little effect on it; and (2) under the same BV/TV value, microarchitecture has superior mechanical properties along the principal direction of loading. Through μFEM, the stress and strain within trabeculae under loading can be estimated, as well as bone formation/resorption (19). However, μFEM simulation needs to be validated with in vitro mechanical testing to confirm that accurate mechanical properties are predicted. This requirement is impossible to attain in μFEM simulations that manipulate an existing trabecular structure, such as bone remodeling or osteoporosis, as the derived trabecular structures do not physically exist (20).

Recently, 3D printed (3DP) trabecular bone has been used to demonstrate the effect of structure deterioration on stiffness and strength. With the help of a high-resolution 3D printer, the 3DP trabecular sample is highly consistent with the actual trabecular structure scanned by micro-CT (21). A decrease of approximately 8% was calculated in the BV/TV of 3DP trabecular bones yielded a corresponding reduction in structural stiffness (17%) and strength (24%), which agrees with in vitro studies that were performed using actual trabecular samples (20). Although there is little information about the true values of strength and stiffness of actual trabecular bone, the decline rate of 3DP trabecular geometries still effectively explains part of the structure−function relationship.

Because μFEM and 3D printing show advantages in replicating the three-dimensional microstructure of trabeculae and reflecting the mechanical behavior, the overall goal of this study is to establish a validated method of given 3DP trabecular geometries simulating bone loss during osteoporosis to evaluate the mechanical properties and simulate them using μFEM, accounting for side-artifact correction. Specifically, our objectives are to (1) 3D print a cohort of trabecular samples (scaled up 20 times, from a rat’s distal femur) sharing the same underlying architecture but with a series of different BV/TV. This was achieved by changing the global threshold during the segmentation process. (2) Then, the 3DP trabecular samples were tested in compression until failure to calculate the tissue modulus and strength. (3) Next, μFEM models of the segmented trabecular samples (1:1 size) were developed, and the effective tissue modulus was calculated by dividing the applied stress by the resultant strain. (4) Finally, the BV/TV values were correlated to the corrected mechanical properties (corrected by side-artifact correction factor) acquired from compressive tests and μFEM simulation. This study is novel in that we combine 3D printing and uFEM to quantify the relation between the mechanical properties and the given continuously deteriorated trabecular bone while taking side artifacts into account.

The excised right femurs were scanned using a high-resolution micro-CT scanner (SkyScan1172, Bruker, USA) at a resolution of 15 μm, with a voltage of 80 kV, a current of 100 μA, a 0.5 mm aluminum filter, a rotation step of 0.6° and an exposure time of 360 ms. Images were then reconstructed using bundled software Nrecon1.7. Another bundled software, DataViewer1.4.3, was employed to orient the cross-sectional images parallel to the transaxial plane. A cylindrical volume of interest (VOI) with a diameter of 1 mm and height of 1 mm was placed randomly in the distal femoral metaphysis. See an example shown in Figures 1A–C.

Figure 1 VOI trabeculae in distal femoral metaphysis. (A–C): VOI taken from the domain of trabeculae in femur samples; (D): Raw “stl” file format VOI trabeculae cylinder full of noises; (E). VOI trabeculae cylinders after optimization.

Morphometric analysis of the VOI trabecular bone was performed using CTAn1.16 (SkyScan1172, Bruker, USA). A cohort of global threshold lower limits ranging from 65 to 100 at an interval of 5 was chosen to segment out trabeculae under the same VOI region. Different global thresholds will lead to microstructural models exhibiting various degrees of osteoporosis. The higher the global threshold chosen, the higher the degree of simulated osteoporosis (Figure 2A). The trabecular parameters BV/TV, Tb. Th, Tb. Sp, were measured for each VOI. The VOI trabecular bones were also saved in “stl” file format for subsequent 3D model optimization and 3D printing.

Figure 2 3DP VOI trabeculae and mechanical test. (A) An overview of 3DP VOI trabeculae from N group and OVX group. A cohort of global thresholds lower limit from 65 to 100 at an interval of 5 was chosen to segment out trabeculae under the same VOI region. With the global threshold increase, the BV/TV value decrease and the 3DP trabeculae structure deteriorate. (B) A view of universal testing machine used to load the 3DP VOI trabeculae in compression. (C) A typical stress–strain curve of 3DP VOI trabeculae tested along the principal direction.

The raw segmentations derived from CTAn software were noisy (for example, full of spikes and unconnected elements, as Supplementary Figure S1 shows). They were too rough for 3D printing and finite element analysis (Figure 1D), and some smoothing steps were undertaken prior to printing and volume meshing. In Geomagic studio 2013 software (Geomagic, United States), the “Mesh Doctor” option was used to automatically detect and repair defects, and then the “Remove Spikes” function was used (the smoothing level was selected as 10 (the level is 0-100) to remove small spikes, but the microstructure characteristics of the trabeculae were retained as much as possible), and Remesh was used to obtain 2D elements with uniform distribution(the element length was set to 8 μm, which is much smaller than rats’ actual trabeculae sizes of 15-194 μm in our samples). The structural difference between the postsmoothing VOI trabecular sample and the original file can be found in Supplementary Figure S2. These postsmoothing 3D models were saved in “stl” file format again, ready for subsequent 3D printing and μFEM preprocessing (Figure 1E).

3D printing can accurately and precisely replicate trabecular specimens and produce informative mechanical alterations (20). The actual VOI trabecular structure (1 mm in diameter, 1 mm in height) was too small to be fabricated and replicated. Therefore, before printing, the VOI trabecular bone was scaled up 20 times (22) using Materialise Magics21.0 software (Materialise, Belgium). The poly lactic acid (PLA) filament with a layer thickness of 80 μm was selected to use in an HP jet fusion 3D printer (Hewlett Packard, USA). The layer thickness is approximately 3.75-48.5 times the size of magnified VOI trabeculae (Figure 2A). The HP Jet Fusion 3D printer uses a novel technique called Multi-Jet Fusion (MJF), which offers low machining time, competitive part properties and minimal postproduction finishing compared to existing 3D printing technologies (23, 24). PLA is an inexpensive and readily available material. According to the manufacturer, PLA filaments have a melting temperature of 170-230°C and a density of approximately 1.35 g/cm³. Before testing 3DP trabecular samples, it is necessary to confirm that the 3D printer can accurately copy the microstructure. This verification can be carried out by regression between the weight of the 3DP trabecular sample and their BV/TV value since all VOI trabecular bone was printed by the same filament. The 3DP trabecular bones were weighed three times to obtain the average weight, and the least square linear regression was developed according to BV/TV to verify the accuracy of the replicated microstructure (20).

3DP trabecular bones were uniaxially compressed using a universal testing machine equipped with a 10 kN load cell (Jingzhuo Machinery Factory, Yangzhou, China), with an error in the indicating value of 0.3%. 3DP trabecular bones were loaded at a rate of 2 mm/min (i.e., strain rate of 0.1/min). A small compression preload of 5 N was applied at the beginning of each experiment (Figure 2B). The measurements were recorded at 10 Hz (every 100 milliseconds). The tissue modulus and strength were determined where the tissue modulus was defined as the slope of the stress−strain curve in the linear region and the strength was defined as the peak of the stress−strain curve (Figure 2C).

The μFEM was developed to obtain the effective tissue modulus (denoted as Ez) of VOI trabecular bone (18). The postoptimized VOI trabecular bones were meshed to four-node tetrahedral elements (C3D4) using Hypermesh 14.0 software (Altair, USA), with an average size of 8 μm, which is smaller than the rats’ actual trabecular dimensions so that the details of the 3D architectures were well maintained. Our sensitivity analysis suggests that 8 μm is sufficient to converge (Supplementary Figure S3). As the global threshold increases, the number of μFEM model elements decreases (Figure 3).

Figure 3 Number of μFEM models’ element.

The μFEM model was considered linear elastic with a Poisson’s ratio of 0.3 (25). The Young’s modulus of the VOI trabecular bone was assigned according to the relation derived from the rat’s distal femoral trabeculae (26):

where E represents Young’s modulus. In this study, we attempted to investigate the effect of microarchitecture deterioration on the mechanical properties of trabecular tissue. Therefore, the modulus assigned to the μFEM trabeculae element will be consistent within each group. The average BV/TV values of the N and OVX groups were 42.19% and 25.25%, respectively, and their Young’s moduli were approximately 3000 MPa and 900 MPa, respectively.

A small uniaxial compressive force (F=1 N) (27) was applied along the Z axis (i.e., the principal direction), and the resultant displacement was therefore computed (Figure 4). We denoted r, h, and Δh as the radius, original height, and maximal displacement of VOI trabecular bone, respectively, and F as the applied force. The stress was acquired by F/πr², and the strain by Δh/h. Specifically, the cutoff values for the upper 95th percentile of those nodal displacements (the top 5% of nodal displacement) along the Z axis in each VOI trabecular bone were defined to represent the maximal displacement (28, 29). The maximal displacement was calculated by averaging the top 5% of nodal displacements. Therefore, the Ez of VOI trabecular bone can be derived by the following equation:

Figure 4 μFEM model of VOI trabeculae. Black arrows represent load by means of loading control, black triangles represent boundary condition.

With this approach, Ez was obtained.

The cored bone specimen can be considered unloaded near its sides, resulting in errors in the calculated tissue modulus and strength compared to the in situ values in which all trabeculae would carry load. This experimental artifact is called ‘side-artifact’ (8, 9). Due to trabeculae on the sample border losing connectivity and therefore load-bearing capacity, the true modulus and strength need to be corrected by a side-artifact correction factor. The correction factor tends to markedly increase as BV/TV decreases, especially when using a small trabecular sample (8). Both 3DP and computational models in this study will suffer from side-artifacts. To assess the radial distribution of stresses for identification of the region affected by the side-artifact, we examined von-mise stresses on radial rings of thickness 0.05 mm on 10 equally spaced transverse cross sections for each trabecular model. The presence of the side-artifact was characterized by a statistically significant reduction in stresses for the outer radial rings relative to the inner radial rings. The corresponding thickness (t) of outer radial rings was used to calculate the side-artifacts correction factor according the following equation (8):

Where D is the diameter of trabecular sample. BV/TV value and correction factor of each sample can be found in Supplementary Table S1. The corrected tissue modulus, strength and Ez were respectively denoted as tissue modulus corrected, strength corrected and Ez corrected.

A previous study theoretically predicted that the tissue modulus and strength of isotropic foams (similar to trabecular bone regardless of anisotropic mineral substance and collagen distribution) are quadratic functions of BV/TV (7). Given the 3DP trabecular bones corrected by the side-artifact correction factor, in the present study, a nonlinear regression function (SPSS20.0, IBM, USA) was established to determine the effect of deteriorated BV/TV in the form:

where Y is one of three measured dependent variables (i.e., tissue modulus corrected, strength corrected and Ez corrected), A and B are the derived coefficients.

Figure 5 shows that with BV/TV increase, N1 weight increased from 1.517 g to 3.153 g, N2 weight increased from 2.253 g to 3.427 g, N3 weight increased from 2.057 g to 3.413 g; OVX1 weight increased from 1.250 g to 1.907 g, OVX2 weight increased from 1.043 g to 1.767 g, and OVX3 weight increased from 1.297 to 2.000 g. The weight of 3DP trabecular bones was found to significantly correlate with their BV/TV in both the N and OVX groups (R² > 0.99). It is therefore reasonable to assume that the 3DP trabecular bones replicated the actual microstructure well, and the 3D printer we used can accurately print the trabecular model.

Figure 5 3DP accuracy validation. Linear regression between BV/TV and the weight of 3DP VOI trabeculae.

As Figure 6 shows, as the global threshold limit increased from 65 to 100, the BV/TV of N1, N2, and N3 VOI trabecular bone decreased from 51.51%, 53.41%, and 53.94% to 28.64%, 36.92%, and 34.47%, respectively; the BV/TV of OVX1, OVX2, and OVX3 VOI trabecular bone decreased from 31.06%, 28.86%, and 33.33% to 20.61%, 18.40%, and 22.70%, respectively. Similarly, Tb. Th of N1, N2, and N3 decreased from 0.0877 mm, 0.1112 mm, and 0.1003 mm to 0.0642 mm, 0.0879 mm, and 0.0770 mm, respectively. Tb. Th of OVX1, OVX2, and OVX3 decreased from 0.1096 mm, 0.0950 mm, 0.1085 mm to 0.0930 mm, 0.0750 mm, 0.0905 mm, respectively. Tb. Sp of N1, N2, and N3 increased from 0.0782 mm, 0.0908 mm, and 0.0864 mm to 0.1366 mm, 0.1651 mm, and 0.1580 mm, respectively. Tb. Sp of OVX1, OVX2, and OVX3 increased from 0.1268 mm, 0.1428 mm, and 0.1429 mm to 0.2455 mm, 0.2388 mm, and 0.2517 mm, respectively. The weight of 3DP trabecular bones demonstrated decreasing trends similar to those of BV/TV. OVX trabecular samples show lower BV/TV and corresponding weight.

Figure 6 Structural parameter and the weight of VOI trabeculae at different global threshold.

As Figure 7 demonstrates, in the N group, deteriorated BV/TV was found to significantly correlate with tissue modulus corrected as well as strength corrected and Ez corrected in a manner of power law. This significant correlation was also found in the OVX group. Specifically, the normal specimens display a steeper slope than OVX. The detailed coefficients, R² and P value of the nonlinear regression can be found in Tables 1–3.

Figure 7 Nonlinear regression for tissue modulus corrected, strength corrected and Ez corrected as a function of BV/TV.

Table 1 Coefficient of nonlinear regression (Y = A · (BV/TV)^B) for tissue modulus.

Table 2 Coefficient of nonlinear regression (Y = A · (BV/TV)^B) for strength.

Table 3 Coefficient of nonlinear regression (Y = A · (BV/TV)^B) for Ez.

Studies involving structural differentiation between healthy and osteoporotic trabecular samples have shown that the BV/TV of osteoporotic individuals decreases by 10%-60%, depending on the anatomical site, sex and age (20, 30). The method to change the global threshold of scanned images to simulate individual bone resorption and transfer into μFEM for mechanical study has been proposed (31). In our research, through a series of global thresholds applied to micro-CT scanned images, the BV/TV value maximally decreased by 28.64%, 16.50%, 19.46%, 13.46%, 10.46% and 9.75% in the N1, N2, N3, OVX1, OVX2 and OVX3 samples, respectively. These decrements support our attempt to segment out the VOI trabeculae characterized by continuous deterioration.

Previous studies have quantified the effect of BV/TV decrease on mechanical properties. For example, a reduction of 8-10.4% in BV/TV led to a 13-17% decrease in stiffness (31, 32), and the ability of BV/TV to explain the variance of elastic properties was more than 70% (3, 26, 33). However, as mentioned above, no two trabecular structures are the same. When these studies compare normal trabeculae against osteoporotic trabeculae (in terms of BV/TV decrease), they only reveal an “average” mechanical result. It is impossible to quantify the mechanical effect of individual structural deterioration for each actual bone sample simply because we cannot test the same sample in various states (20), not to mention the dynamic deterioration process. 3D printing provides a solution to overcome this issue since the correlation coefficient (R²) of the mechanical properties between 3DP and actual trabecular bone samples can up to 0.94 (34).

In actual bone samples, tissue modulus and strength appear to be linear within a single anatomic site but to be power-law across multiple sites because of the differentiation of apparent density (ρ in g/cm3) and BV/TV (30, 35). However, these in vitro tested trabecular specimens inevitably suffered from the side-artifacts effect, which results from interruption of the trabecular network along the sides of segmented specimens. Side artifacts are the underestimation of the true mechanical properties, particularly when testing small-size and low-density trabecular bone. Side-artifact correction is essential for obtaining accurate mean estimates of mechanical properties for a cohort of specimens.

The modulus and apparent collapse stress (strength) of an intact open-cell foam model (7) were calculated to be quadratic functions of the volume fraction. Given the side-artifact correction factor, the segmented trabecular specimen can be thought of as an in vitro intact foam, and therefore, its mechanical properties should conform to a quadratic function of BV/TV. However, the tissue modulus corrected, strength corrected as well as Ez corrected did not precisely demonstrate a quadratic relation as a function of BV/TV (for example, the exponents B varied between 1.405 and 2.954 in μFEM models). This is because the trabecular tissue is not simply an assembly of regular trabeculae, but a functionalized pore structure with varying trabecular orientation, thickness, length and shape (“rod-like”, “plate-like” and their composite) (4, 33, 36). In fact, the exponents B were not exactly equal, but varied around 2, according to the majority of previous studies involving mechanical tests (30, 35–38). According to the cellular solid theory, exponents B were interpreted as indicators of dominant deformation mechanisms within the trabecular structure. For example, a predominance of axial deformation would lead to a linear relationship, whereas a quadratic relationship would imply that cell wall/strut bending dominates (35, 39). The clustering of exponents for our 3DP models in the range of 1.175–2.970 might therefore be interpreted to suggest that deformation mechanisms are similar across normal and osteoporotic state and involve appreciable bending. The slope of the function in this study is an indicator of geometry attenuation, which suggests that normal trabecular bone is more susceptible to a decrease in bone fraction. This tendency can be verified when we refer to real bone tissue, where bone tissue with a higher bone fraction shows a steeper slope than bone tissue with a lower bone fraction (26).

Few previous studies have used a 3DP trabecular model to simulate the actual trabecular bone (21, 40–43) and test its mechanical properties (20, 44, 45); however, these studies mainly focused on the printing accuracy rather than the structure−function relation, as we did in our study. Dobson et al. (46) used stereolithography 3DP models to validate FE predictions for trabecular bone structures. The 3DP models were tested in compression, and their stiffness values demonstrated a strong correlation with the predictions by the FE analysis. The authors concluded that 3DP models are an important technique to complement the use of FE models for the assessment of the mechanical properties of complex trabecular bone structures. 3DP models are the only way to validate FE models of trabecular bone remodeling, as the derived structures do not physically exist. In this study, we compared the same trabecular structure with a series of BV/TV value reductions in several trabecular samples and established their structure−function relation, which distinguishes ours from previous studies.

This study confirms the long-known relationship measured in trabecular tissue with different volume fractions using 3D printing, suggesting promising applications in skeletal biomechanics (45, 47, 48). However, several potential limitations in our study should be noted. First, we cannot claim that changing the global threshold actually simulates the complex biological process of osteoporosis. In particular, even/global structure deterioration could result in misinterpretations of the influence of trabeculae loss on biomechanical behavior. Simulated bone deterioration offers an opportunity or a tool to investigate some phenomenological aspects of bone loss (49, 50). Second, 3DP filaments are isotropic, while actual bone tissue is hierarchical and anisotropic. This may to some extent reduce the biological significance of our results. Third, 3D-printed trabecular bone may exhibit anisotropy due to the inherently layered construction (e.g., print along the X axis, Y axis and Z axis), but our printed samples were tested orthogonally in the direction of printing. This possible caveat should not affect our results. Last but not least, a broader range of thresholds simulating severe osteoporosis and more parallel trabecular samples (replicated by 3DP) are helpful to prove the structure−function rule more accurately.

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding authors.

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fendo.2023.1061758/full#supplementary-material

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