We analyzed selected parts of the mandible and the head of the femur (Figure 1) acquired from three mature 2.5-year-old domestic female pigs. The pigs were provided by Lahav C.R.O (Kibbutz Lahav, Israel). Pigs are considered to be a good non-human model for better understanding human skeletal and dental questions such as osteonecrosis of the femoral head, bone fractures, bone growth, and development, as well as for evaluating new dental implant designs (Buser et al., 1991; An and Freidman, 1998; Terheyden et al., 1999; Nkenke et al., 2003). While porcine bones in general have a denser trabecular network than human trabecular networks (Mosekilde et al., 1993), porcine bone shows similarities to human bone in terms of bone biology (modeling and remodeling), as well as bone mineral density (Mosekilde et al., 1993). The study was approved by Institutional Animal Care and Use Committee at the Weizmann Institute of Science. We also analyzed one cork specimen as an example of natural porous solid of non-animal origin that functions under different mechanical constraints and does not remodel. As examples of engineered 3D structures, we used two types of 3D-printed metal lattices. A rhombic dodecahedron honeycomb structure (here referred to as the D-lattice) in the form of a cylinder 15 mm in diameter and 15 mm in height, a strut thickness of 0.9–1.0 mm and the node connectedness was strictly configured by the design (unit cell of rhombic dodecahedron with three intersecting twofold axes) as 4-N, 6-N, and 12-N in an infinite array. This D-lattice contained about 1000 struts connected at 250 nodes, of which approximately 50% were partially truncated at the surface. The stochastic lattice (here referred to as the S-lattice) was also manufactured in a cylindrical shape of 15 mm in diameter and 15 mm in height, but contained a higher number (around 20,000) of fine connected elements of 150–250-µm thick. In the stochastic lattice, the nodes were randomly positioned in 3D and were stochastically interconnected to form a partially connected network with the average number of neighbors given as 4, as described by Ghouse et al. (2017).

The fresh porcine mandibles were cut in half at the symphyseal region using a water cooled rotary diamond saw (Buehler, IsoMet 1000 Precision saw, USA). Each hemi-mandible was sectioned further to obtain three anatomically distinct areas, namely the mandibular condyle, the angle, and the body (Figure 1). For specimens P1 and P2 both the right and the left hemi-mandibles were analyzed. In the case of P3, only the right hemi-mandible was analyzed (Table 1). The proximal part of the femur from each pig was also examined.

A cylindrical cork specimen of 5 mm in diameter was collected by trephine drilling of the outer bark layer (phloem) of a cork oak, Quercus suber, in the direction perpendicular to the trunk natural surface. The cork specimen was not further modified. The engineered lattices were created using computer-aided design software Rhinoceros 5.0 (McNeel Europe, Barcelona, Spain) and 3D-printed using selective laser sintering of commercially pure titanium (Renishaw AM250, Renishaw Additive Manufacturing, UK) for an unrelated study.

The results of the validation of ITA analysis show that for a reticulate network comprised of rod-shaped elements (“trabeculae”) with known ITAs, the software accurately calculates the abundance of node types, accurately detects the mean ITA angles incorporated by design and realistically estimates the number of nodes truncated at the edges and their 3D configurations. Furthermore a stochastic reticulate structure produces distributions totally different from those reported for trabecular bone by Reznikov et al. (2016), showing that the software does not generate the same result regardless of structure.

In natural structures, as opposed to engineered structures the mean ITA values tend to be very close to angles in symmetrical geometric motifs, i.e., 120° for 3-N nodes and 109° for 4-N nodes. In these symmetrical motifs, the connecting edges are offset as far as possible from each other. In this way, the trabeculae can optimally cover 3D space with the least number of connecting elements. Together with the prevalence of simple nodes in the partially connected network, this results in an optimized combination of minimal metabolic cost of the structure and maximal stability to multidirectional loading. The width of the ITA distributions accounts for inherent irregularity of biological structures. Interestingly, both living animal tissue (bone) and dead plant tissue (cork) follow the same principles of network simplicity and maximal volume spanning, despite the different size scale of their networks. This observation stems from the fact that the ITA application analyzes topological determinants of the structure, uncoupled from their size, shape, and global orientation. Since the results obtained from the cork sample were close to the ITA parameters observed in bone, one sample of cork was deemed sufficient for validation. This observation, however, requires a thorough study of plant tissues and other similar biological lattices.

We noticed that as the number of node neighbors increases, the ITA distributions become skewed: the 3-N distribution can be accounted for with a single Gaussian curve, as is expected from a natural continuous distribution of independent values. The distributions of 4-N and 5-N nodes, however, require a minor Gaussian curve in the region of high ITA values, for a satisfactory coefficient of determination (Figure 11). The contribution of the minor Gaussian curve is higher in the case of the 5-N distribution (the height of the minor peak is 25% from the height of the 4-N general distribution, and the height of the minor peak in the 5-N distribution is around 40%). A probable explanation for the presence of the minor Gaussian curves could be that the application detects not only the angles between the near branches of a node but also between remote branches in the same node of high connectivity. Indeed, in Figures 4B, the plot of the D-lattice demonstrates higher values in the region 150–180°. This is a plot of 6-N nodes. In a 6-N node of perfect threefold symmetry all 6 edges are oriented at 90° to one another and form 12 right angles between the near edges. However, there are also 3 angles of 180° between remote edges of the same node. Nodes of three neighbors can have only three near-edge angles; 4-N nodes also can have only near-edge angles (as it has six possible pairs of near edges between which an angle can be measured). However, a 5-N node necessarily incorporates one angle between remote edges, along with nine pairs of near edges. Therefore, starting from the node complexity N-5 or higher, a bimodal ITA distribution can be expected, in which the major peak corresponds to the mean angle for pairs of near edges, and the minor peak corresponds to the angle for the only pair of remote edges. The minor peak position can be expected close to 180°. The fact that in trabecular bone ITA distribution the minor Gaussian peak is present to some extent in the 4-N ITA distribution can be possibly explained by life-long bone remodeling, when some nodes may lose one of their original branches.

The finding of highly conserved ITA properties may seem to clash with the well-established connection between trabecular bone texture and function. This connection was originally described in the work of Julius Wolff, who suggested that individual trabeculae of the proximal human femur are aligned along the principal stress directions (Wolff, 1892). Many studies examined this premise (Enlow, 1968; Bertram and Swartz, 1991; Currey, 2012), including several experimental studies that directly demonstrate this connection. Studies of trabecular bone in the knee joint of guinea fowl showed that the fine trabecular bone in the distal femur has a high degree of correspondence between the changes in joint angle and trabecular orientation (Pontzer et al., 2006). A study of the distal radius in a sheep model also supported the finding that trabecular bone adjusts and realigns in response to change in loading direction (Barak et al., 2011). The study of the calcaneus in normal and deformed human feet shows that consistent long-term loading results in the formation of a reproducible anisotropy pattern of trabecular bone. However, there is no real conflict between the fact that bone is capable of functional adaptation and the invariance of the ITA parameters in the analyzed samples. The difference between the traditionally analyzed morphometric parameters and the topological parameters reported here is that the ITA analysis is blind to the local thickness and form of the elements and only considers the trabecular network determinants. The universal topology of a trabecular network can be viewed as the 3D archetype on which various changes in size and shape can be superimposed. Thus, the topology observed in trabecular bone can adopt a multitude of different texture parameters such as increase in Tb.Th, without negating the principles of network simplicity and maximal spanning of the 3D space.

A detailed comparison of the nodes within the condylar head and neck did, however, reveal an alternative form of adaptation that does not involve trabecular shape, but also does not breach the topology. A higher DA was identified in all five condylar heads analyzed, in comparison with the condylar neck. Along with that the average Tb.Th and its SD are consistently lower in the condylar head, and the trabecular spacing and its SD are also lower. These observations are in accord with a finer and denser network. In addition, the total number of nodes and the proportions of 4-N and 5-N nodes were higher in the condylar head than in the condylar neck, further indicating that the fine trabecular network of the condylar head is densely connected. One way to achieve a higher DA is by network coarsening and thickening of the trabeculae that are co-oriented with the principal stress trajectories (Ryan and Krovitz, 2006). But the opposite is observed in the fine and densely connected network of the condylar head. Significantly, the ratio of planar 3-N nodes to non-planar 3-N nodes is consistently larger in the head than in the neck. Moreover, the orientation of the planes of these flat 3-N nodes was found along the physiological stress trajectory. Therefore, the higher anisotropy can be achieved by local transformation of the network elements that does not amend the general topology, but also allows retaining the scale and shape of the network elements.

The mandibular condyle is one of the components of the TMJ. The condylar head is subjected to forces in well-defined directions (Liu and Herring, 2000a,b; Herring et al., 2002; Cornish et al., 2006) as this is a paired joint that has limited freedom around the sagittal or vertical axes, and primarily allows movement around the transverse axis. Since consistent and repetitive loading of bones results in increasing their anisotropy, the structure of trabecular bone in the condylar head is expected to adapt accordingly, consistent with our results and with previous studies (Gosen, 1974; Hekneby, 1974; Teng and Herring, 1995; Kim et al., 2013). Interestingly, the higher anisotropy in the condylar head is achieved by transformation of the network motifs (preferred orientation of the planar 3-N nodes), and the fine scale of the network remains unaltered. We speculate that the biological rationale for that might be the preservation of a higher compliance of the fine trabecular tissue. Indeed, increased anisotropy combined with higher bone volume density provides the highest stiffness of trabecular bone (Maquer et al., 2015). On the other hand, it was shown that higher stiffness of trabecular bone is associated with degenerative disease of articular cartilage (Li and Aspden, 1997a,b; Hurwitz et al., 2001). We also speculate that increasing trabecular bone anisotropy by network transformation (and not by trabecular coarsening and stiffening in certain orientations) might be Nature’s strategy to preserve articular cartilage from untimely wear and tear. This observation opens new insights into two important problems in bone research, namely joint degeneration (perhaps, adaptation to stereotypical loading by increased anisotropy with trabecular network coarsening, rather than co-alignment of some topological elements) and bone fragility (i.e., inability to stably cope with loads low in magnitude, but unusual in direction).

The analysis presented here certainly pertains to rod-shaped trabeculae; its validity with regard to plate-shaped trabeculae needs to be studied, as the skeletonization procedure currently used in our ITA analysis code may have some difficulties in properly identifying and separately representing plate-shaped trabeculae. Analysis results will, therefore, be more valid and mechanically significant in those instances where the majority of the trabeculae are rod shaped, like the bodies of vertebrae (Parkinson and Fazzalari, 2013).