Bone, the material, exists at the organ level as whole bones. Whilst bone may seem relatively inert compared with other structures in the bone it is in fact an adaptive material which responds to its environment. The fundamental results of Wolff’s Law hold true but his explanation and understanding of the underlying mechanisms were misunderstood (Currey, 2002). The nature of the biological and micro-mechanical mechanisms that drive bone remodeling is still not fully understood. At its material level, bone is a multiphase composite material formed of both organic and inorganic constituents. It has a hierarchical structure that ranges from the sub-nano level of the collagen-mineral composite through to the macro-structure of cortical and cancellous bone.

The density and structure of bone are important characteristics that determine its mechanical behavior in everyday life. An understanding of these underpinning properties is crucial in the investigation of bone as a structural material. Density can be defined in a number of ways ranging from the micro- to the macro- or organ level. The two generally accepted versions of density are the “apparent” and “material” ones. Apparent density (D_app) is the mass of the mineralized tissue over the total volume occupied by the tissue with the inclusion of its voids. The most common representation of D_app used in respect to bone is bone mineral density (BMD_a) which, when measured by dual energy X-ray diffraction (DEXA), is an area assessment of this characteristic. Material density (D_mat) is defined as the same mass as for apparent density divided by the volume the solid mineralized tissue occupies with the exclusion of any voids that may exist within the structure. The most popular use of this is often referred to as tissue mineral density. These definitions highlight that the difference between these properties is the consideration of mass with respect to the micro-structure of the tissue, such as: voids, osteocyte lacunae, osteonal canals, and analogous non-mineralized architectural features: (1)Dapp=Bone mass/Total Volume (2)Dmat=Bone mass/Bone Volume (3)BV/TV=Dapp/Dmat where BV is the bone volume and TV is the total volume. The assessment of densities within bone tissue is considered to be important as it will impact upon the resultant mechanical properties and remodeling characteristics of bone (Martin, 1984; Fyhrie et al., 1993; Zioupos et al., 2008). The derivation of D_app is not contested because it is simply the wet bone mass of a sample over the Cartesian geometric volume occupied by the same bone sample. Different methods for material density, however, have caused some debate (Schileo et al., 2008; Zioupos et al., 2008). The most conventional technique employed for this assessment relies on the Archimedes principle (usually via water displacement). Application of this method relies critically on ensuring that pores must be fully flushed and refilled (Zou et al., 1997). This flushing and refilling are particularly difficult in cases where there are cells of closed geometry within the trabecular architecture (Rho et al., 1995). Comparisons of DEXA and the Archimedes technique have previously reported substantial differences (Keenan et al., 1997) whilst fractional quantitative and cone beam computed tomography has been shown to be in closer agreement with Archimedes (Lee et al., 2004; Ahlowalia et al., 2013). When investigating D_app and D_mat, consideration must be given to the volume of bone or BV/TV (dimensionless ratio of actual bone volume to the total volume of the sample). This can be calculated with the Archimedes principle using Eq. 3. Calculating BV/TV with the Archimedes method depends therefore in ensuring that the displaced suspending medium (water, a solution of known density, or a gas) infiltrates all the pores and thus derives the true BV for the sample. BV/TV has also been calculated/measured by using a series of histological slices (Martin, 1984). This technique can also carry an inherent error due to the limitation of physical slice thickness which requires interpolation between each slice; in the addition to sample destruction.

The D_app is often considered to be one of the primary characteristics of bone that influence its mechanical properties at the macro-mechanical level and has been shown to influence not only the compressive but also the fracture toughness properties of bone (Rice et al., 1988; Cook and Zioupos, 2009). D_mat determines material behavior primarily at the trabecular level. However, due to the fact that D_app is the product of D_mat × BV/TV, it determines properties at the structural level too. A previous study has shown, in elephant bone samples, that the relationship between D_app and D_mat are interdependent and that D_mat is at its highest (~2.3g cm⁻³) value at the extremes of porosity, as BV/TV tends toward 1 and 0, and exhibits minimum values at intermediate levels of BV/TV of 0.4–0.7 (Zioupos et al., 2008).

This relationship has, however, been brought into question when it was suggested that the actual shape of it may be due to limitations of the Archimedes method in the assessment of bone tissue material density (Schileo et al., 2009) because, as commented earlier, the method depends on ensuring that the displaced suspending medium infiltrates all the pores and thus derives the true BV of a sample. To overcome this limitation, μ-CT can be used as it gives information on the internal structure, it is non-destructive and it can penetrate throughout the material so that marrow-filled spaces and more enclosed cells are accessible and thus these will not affect the results. Some previous work has looked at the density relationship applied at cortical and cancellous regions using μ-CT (Schileo et al., 2008), but it has not considered bone densities throughout the entire range with particular attention to the intermediate range of densities. The present study, therefore, aims to test the hypothesis that, when applied to the same bone tissue samples, Archimedes and μ-CT can produce effectively the same results and also to explore the implications of μ-CT derived data in mechanobiology studies.

The full results of the way densities compare with each other, when measured by the mode or the mean value of gray levels (after appropriate and specific thresholding of the voxels), are in Figure 5. The material density calculated by the mean underestimates the value produced by the mode, as expected, but the two are very well linearly correlated with each other (Figure 5). There are also very few outliers where the two values deviate considerably for some difficult samples, but these do not spoil the overall pattern nor do they cast a strong shadow of doubt on the very principle of measuring bone densities across the complete range of porosities. The symbols encircled in red are, as is our practice, those for the cortical bone samples. Once again, as in the original paper using the Archimedes method (Zioupos et al., 2008), there is an underlying regressive behavior of the sample density, which goes down and up as bone goes from dense cortical to porous cancellous between these two extremes and which is can be appreciated by the arrows added manually on the graph.

A comparison of the CT-D_app measured from the mean and Arch-D_app by the laboratory method of Zioupos et al. (2008) is shown in Figure 6. The plot has a slight inflection in the intermediate bone density values. If we consider that Arch-D_app is a more reliable method because it is simply the wet weight over the bone sample volume, then it appears that CT-D_app underestimates apparent bone density in these intermediate value regions as it is based on the absorption of X-rays from mineralized tissue alone. Data for Arch-D_app are produced empirically using a microbalance to measure wet weight of the bone and Vernier calipers to measure volume. Arch-D_app is in essence the method for apparent density that is used in every biomechanical lab globally to produce Modulus of Elasticity = f(Apparent Density) relationships. The data show that there is “bone” matter which is not captured or quantified by the CT-D_app variable, and this matter is most likely be the lower density non-mineralized portions of the bone samples in remodeled areas, practically regions of osteoid tissue in its various stages toward full skeletal maturity.

The comparison of BV/TV measurements shown in Figure 7 demonstrates that the Arch-BV/TV measured in the laboratory is higher in the intermediate regions, most likely due the fact that the Archimedes measurements consider all tissue including the un-mineralized layers on the surface of the tissue. It is most apparent in the intermediate region as it is a surface effect, and in the intermediate regions there is the greatest amount of specific surface available (Martin, 1984; Berli et al., 2017) for the volume of bone.

The results of Figures 6 and 7 are in agreement with each other as to what disparities exist between the methods. These low-density regions on the surfaces of bone are due to the remodeling of bone where the “younger” bone regions are less mineralized. Microscope images displaying localized structural properties of bone are shown in Figure 8 and are widespread in the literature. Figure 9 depicts graphically the “mosaic” of tissue compartments that bone is at any point in time. These compartments are regions of varying mineral contents at various temporal points in the development of the mineralization process that leads from osteoid formation to mature, fully mineralized bone.

Figure 10 shows the “boomerang”-like pattern previously shown by Zioupos et al. (2008) is now apparent in both Archimedes-based and μ-CT-produced data. The μ-CT-produced curves do, however, have shallower inflection points: the lowest Arch-D_mat is ~1.48 g cm⁻³ as opposed to 1.60 g cm⁻³ for CT-D_app. The shallower inflection point is due to higher values for measured D_mat in the intermediate bone porosities when μ-CT is used. In views of the previous arguments and graphs, these higher values for D_mat most likely exist for two possible reasons: (i) the density measured by the Archimedes method is skewed by the presence of approximately closed voids in the cancellous bone matrix, which would overestimate the volume to bone if they are not fully flushed for the marrow they contain (in the calculation of D_mat = weight/volume a higher volume will lead to a reduced D_mat whereas in μ-CT these voids do not impact on the data) and/or (ii) the surface of cancellous bone in Archimedes measurements is thresholded at a density of 1 g cm⁻³ (because it displaces water in the Archimedes tests), thus including low-density epithelial layers and newly forming osteoid, but this same soft organic material does not possess a density high enough to register in the μ-CT measured density. The disparity between the measurements most likely exists due to a combination of these factors. The differences between the measurements are also apparent in Figures 6 and 7, which shows a clear inflection in the intermediate range of bone densities. The apparent density measured via Archimedes is undisputed and seems to carry less error than any other physical method—a statement that is open to interpretation (measurement of actual weight and of physical dimensions with calipers). However, we conclude that it is the μ-CT method, not so much the Archimedes method, that needs particularly cautious attention for potentially misleading technical errors. In both ways for both methods, the “boomerang”-like pattern is strongly evident.

Here, we have furthered the investigation into the basic relationships that exist between apparent and material density values within bone in both its cortical and cancellous forms, and throughout its whole porosity range. Bone densities directly impact upon the mechanical competency of the tissue (Rho et al., 1995; Zioupos et al., 2008). Understanding the behavior of bone density across the full range of porosity values is vitally important for the comprehensive understanding of remodeling behavior and remodeling rates at specific sites within the human body (Martin, 1984; Fyhrie et al., 1993). Such density data will contribute to future development of patient-specific finite element modeling, which depends on accurate assessment of the material properties of the tissue and its structure (Chevalier et al., 2007; Schileo et al., 2008). Conflicting reports have been made on the nature of the density variations across the full porosity range (Schileo et al., 2008, 2009; Zioupos et al., 2008), which, however, used different methods to assess the same property and on different samples. Assessing these bone properties has typically been carried out by means of histological measurements (sectioning) and traditional densitometry techniques such as the Archimedes technique (Martin, 1984; Rho et al., 1995; Zou et al., 1997; Zioupos et al., 2008). These methods are destructive and can be criticized for their limitations in reproducibility, which is most likely due to their inability to guarantee full penetration of the sample using various solvents, which has led some studies to use a gas pycnometer (helium displacement method; Zou et al., 1997).

μ-CT imaging has its own limitations which interfere with density measurements, too. In μ-CT imaging, it is important to ensure that image resolution is suitable for the size and structures being assessed. In this study, the imaging resolution was sufficient for determination of cancellous micro-architecture but not for further assessment of the vascular nano-micro-architecture, which is at a range <1 μm (Yan et al., 2011). This is an important consideration when looking at the specific surface of bone, because when looking at cellular sites for bone remodeling the cortical bone may be more porous than the results here would suggest and consequently the reported remodeling rates would be also affected. Additionally, the densities presented in this work were calculated in grams per cubic centimeter to provide comparison between the two methods. In contrast, for clinical relevance HU would be of greater value, which have been shown to be suitable when using cone beam micro-computed tomography (Mah et al., 2010). Moreover, in μ-CT imaging consideration must also be given to the methods of density determination; be it by the “mean” or “mode” values of the gray value distribution (or other approaches as noted above in Section “Materials and Methods”); as in highly porous samples determination of both can be problematic, and as shown by Figure 4 there is a deviation in the produced values in such cases of very porous bone material.

This study’s findings have confirmed the trend of data from experiments on the same collection of samples used by Zioupos et al. (2008). These samples when using the Archimedes density determination method showed a highly non-linear relationship between D_mat and D_app for bone across all porosities and showed an inflection in the data in the intermediate regions between cortical and cancellous bone (Figure 1). The results from both methods (Figure 10) agree in the shape of the curve but differ in the magnitude of the non-linearity. The difference is understandable in that the two methods (one mechanical; one physical) use different physical principles and measure bone density differently. The limitation of a μ-CT scan is that it assesses density indirectly through the absorption of X-rays by the hard matter of bone (Schileo et al., 2008; Mah et al., 2010) and thus it certainly ignores the soft organic matrix. The limitations of the Archimedes method is that it requires repeated measurements and very careful preparation for cancellous samples with more closed cell architecture where the suspending medium (usually water) does not penetrates fully the entire space. In spite of these differences, both data sets are in agreement of a “boomerang” like effect which is prominent (density variation between high and low values of at least 37%) and is certainly not a constant value for D_mat across the whole range as claimed by Schileo et al. (2008, 2009).

In bone histology, less dense regions of bone are typically considered to be younger bone which in turn suggests that the intermediate porosity regions, which here are shown to contain bone of lower material density, do so because they experience higher rates of remodeling. The work of Berli et al. (2017) has attributed this to a process which is a surface-moderated effect, whereby the greater the specific surface area available, the higher the remodeling rate and the lower the mineral density due to the increased formation and presence of osteoid. The results also confirm that μ-CT based measurements of density, which are pursued for the purpose of scanning bones for micro-FEA, ought to use relationships between gray level/density/modulus of elasticity that are produced by micro-mechanics (e.g., nano-indentation tests) at the same magnification level and not the commonly provided ones produced at a macro-mechanical level (Morgan et al., 2003). Micro-mechanical tests are needed for micro-mechanical level data for micro-FEA because the bone material density fluctuates with porosity and does not maintain a constant value as has been assumed in past studies. This is shown by Figure 11. Erroneous material data values can be assigned if elastic modulus vs. density relationships from the literature [E = f(D)], which have been produced at a macro-mechanical level, are assigned to bone at the voxel size level for micro-FEA. When the voxel size is smaller than the bone pore size, for instance, if the voxel is at a void the E value is zero, if it is where bone mass is, the modulus value should be a function of the bone mineralization level and of only this.

This study has shown that bone material density varies non-linearly with bone apparent density across the full spectrum of bone porosities. We have provided further evidence in favor of density-dependent material models for the future development of patient-specific finite element models. Additional care must be taken when setting thresholds and sampling the material density—it is recommended that further work be carried out into the impact of setting μ-CT sampling thresholds on the material data. More investigation is needed into the source of the disparity, where it exists, between data obtained from the Archimedes and μ-CT methods. Additionally, the micro-architectural properties of bone across all porosities should be investigated more carefully because this may allow more profound inferences into the development of remodeling models.

The study was approved by institutional Cranfield University Research & Ethics committee.

GA, JH, RC, and PZ design the study. GA performed the study. GA, RC, and PZ analyzed the results. GA and PZ wrote the manuscript. GA, RC, JH, and PZ had equal contribution to the paper.

The authors have no commercial or any other financial relationship or any conflict of interest to declare in conjunction with the work presented in this paper.