3D diffusion model within the collagen apatite porosity: An insight to the nanostructure of human trabecular bone

Abstract

Bone tissue at nanoscale is a composite mainly made of apatite crystals, collagen molecules and water. This work is aimed to study the diffusion within bone nanostructure through Monte-Carlo simulations. To this purpose, an idealized geometric model of the apatite-collagen structure was developed. Gaussian probability distribution functions were employed to design the orientation of the apatite crystals with respect to the axes (length L, width W and thickness T) of a plate-like trabecula. We performed numerical simulations considering the influence of the mineral arrangement on the effective diffusion coefficient of water. To represent the hindrance of the impermeable apatite crystals on the water diffusion process, the effective diffusion coefficient was scaled with the tortuosity, the constrictivity and the porosity factors of the structure. The diffusion phenomenon was investigated in the three main directions of the single trabecula and the introduction of apatite preferential orientation allowed the creation of an anisotropic medium. Thus, different diffusivities values were observed along the axes of the single trabecula. We found good agreement with previous experimental results computed by means of a genetic algorithm.

Fig 1

Model of the mineralized collagen fibril: (a) 3D view of apatite platelets embedded in a collagen matrix, (b) longitudinal plane, (c) cross section and (d) frontal plane.

Fig 2

Schematic representation of collagen molecules staggered arrangement based on Hodge et al. [18]: (a) longitudinal plane and (b) cross section of the nanostructure.

Fig 3

Sketch not drawn to scale of the apatite array considering a staggered arrangement in the axial direction (a) and a distribution of parallel layers in the transverse direction of the fibrils (b, c). In (a) the longitudinal plane of the apatite matrix is shown, indicated as LT plane. In (b) is represented WT plane, while in (c) the frontal plane is illustrated (LW plane). The principal geometric variables are also highlighted.

Fig 4

(a) Sketch not drawn to scale of the representative unit cell within a single trabecula. By hypothesis, the three platelets composing the unit cell (UC) maintain their reciprocal parallelism independently of the cell inclination with respect to global coordinate system. The c-axis of the apatite platelets is aligned with the collagen long axis and points into the longitudinal direction of the single trabecula [23]. The global coordinate system and the three orthogonal planes considered are illustrated. (b) The dashed rectangle represents a zoom of the UC composed of three apatite platelets embedded in the collagen matrix. The local coordinate system is shown.

Fig 5

Two examples of aligned (θ = 0 degrees) and inclined (θ = 20 degrees) apatite platelets configurations are depicted in a) LW plane, b) LT plane and c) WT plane. The blue dashed line (at θ = 0 degrees) and the green dashed line (at θ = 20 degrees) represent the streamlines of water molecule within the apatite network along the investigated direction in order to highlight the primary influence of the tortuosity on the diffusion coefficient. Whilst the red continuous line indicates the Euclidean distance between the path extremes.

Fig 6. Diffusion coefficients D_L versus the standard deviation (σ_{θ_LW}) of the Gaussian PDF that characterizes the apatite platelets inclination in the LW plane.

The color bands represent the Confidence Interval at 95 percent. We reported also two different degrees of mineralization (V_{f_A} = 0.25, and V_{f_A} = 0.43). The continuous black line represent the Diffusion coefficient computed by means of a genetic algorithm [14].

Fig 7. Diffusion coefficients D_W versus the standard deviation (σ_{θ_LW}) of the Gaussian PDF that characterizes the apatite platelets inclination in the LW plane.

The color bands represent the Confidence Interval at 95 percent. We reported also two different degrees of mineralization (V_{f_A} = 0.25, and V_{f_A} = 0.43). The continuous black line represent the Diffusion coefficient computed by means of a genetic algorithm [14].

Fig 8. Diffusion coefficients D_T versus the standard deviation (σ_{θ_LT}) of the Gaussian PDF that characterizes the apatite platelets inclination in the LT plane.

The color bands represent the Confidence Interval at 95 percent. We reported also two different degrees of mineralization (V_{f_A} = 0.25, and V_{f_A} = 0.43). The continuous black line represent the Diffusion coefficient computed by means of a genetic algorithm [14].