Modelling the mechanics of partially mineralized collagen fibrils, fibres and tissue

Abstract

Progressive stiffening of collagen tissue by bioapatite mineral is important physiologically, but the details of this stiffening are uncertain. Unresolved questions about the details of the accommodation of bioapatite within and upon collagen's hierarchical structure have posed a central hurdle, but recent microscopy data resolve several major questions. These data suggest how collagen accommodates bioapatite at the lowest relevant hierarchical level (collagen fibrils), and suggest several possibilities for the progressive accommodation of bioapatite at higher hierarchical length scales (fibres and tissue). We developed approximations for the stiffening of collagen across spatial hierarchies based upon these data, and connected models across hierarchies levels to estimate mineralization-dependent tissue-level mechanics. In the five possible sequences of mineralization studied, percolation of the bioapatite phase proved to be an important determinant of the degree of stiffening by bioapatite. The models were applied to study one important instance of partially mineralized tissue, which occurs at the attachment of tendon to bone. All sequences of mineralization considered reproduced experimental observations of a region of tissue between tendon and bone that is more compliant than either tendon or bone, but the size and nature of this region depended strongly upon the sequence of mineralization. These models and observations have implications for engineered tissue scaffolds at the attachment of tendon to bone, bone development and graded biomimetic attachment of dissimilar hierarchical materials in general.

Keywords: bone; mechanics of developing tissues; mineralization; mineralized fibrils; nanomechanics; tendon-to-bone attachment.

Figure 1.

Simplified steric model of the arrangements of tropocollagen molecules (grey) and bioapatite platelets (dark-shaded; red in online version) with a partially mineralized collagen fibril, adapted from [3]. Intrafibrillar bioapatite is known to accumulate in the gap regions (length 0.6 D, where D = 0.67 nm is the spacing between gap regions) that form at the termination of tropocollagen molecules (grey circles in cross sections: tropocollagen molecules that pass through the gap region in the cross section shown; hollow circles: tropocollagen molecules that terminate at the cross section shown). Bioapatite platelets can fit within the structure of the gap regions in five distinct orientations. The degree and manner to which bioapatite can accumulate within the remainder of the fibril (overlap regions of length 0.4 D is unknown). (Online version in colour.)

Figure 2.

Five possible models were considered for the sequence of accumulation of bioapatite within and upon collagen fibrils. Model A, the ‘gap region-nucleated model’, involved a filling to capacity of the gap regions followed by extrafibrillar bioapatite nucleating over the gap regions spreading along the fibril. Model B, the ‘external nucleation inhibited model’, involved a filling to capacity of the gap regions followed by extrafibrillar bioapatite emanating from a single nucleation site on each fibril. Model C, the ‘external nucleation promoted model’, involved a filling to capacity of the gap regions, followed by random accumulation of bioapatite on the exterior of fibrils, with no subsequent growth of these accumulations. Model D, the ‘internal nucleation inhibited model’, involved random extrafibrillar accumulations of bioapatite preceding the filling of gap regions. Model E was identical to model A, except that it included mineralization of the overlap regions following the filling to capacity of the gap regions and prior to extrafibrillar mineralization. Bottom panel: homogenization bounds that do not account for collagen and bioapatite architecture are broader than those that do. We used these models to generate bounds and estimates on the mechanics of fibrils. These bounds and estimates were then applied to bounds and estimates of fibre mechanics. (Online version in colour.)

Figure 3.

Estimates of the nonlinear stress–strain response of individual collagen fibrils. (a) Stress–strain behaviour of collagen fibrils, based upon molecular dynamic simulations [44], and the estimated effective continuum constitutive response of collagen (solid line, red in online version). (b,c) These data for fibrils were used to make first-order estimates of the mechanics of fibres. Stiffening of fibres, following the mineralization scheme of model A, increased with increasing bioapatite volume fraction ϕ_m, but does so nonlinearly. As in the linear models, extrafibrillar bioapatite nucleating around the gap regions contributes little to longitudinal stiffness until the extrafibrillar bioapatite sheaths extend beyond the mineralized portions of the gap regions. (d) Tangential modulus of fibres, shown as a function of ϕ_m at several levels of strain, displays rapid stiffening for bioapatite volume fractions beyond the percolation threshold. (Online version in colour.)

Figure 4.

At the fibre level, some transverse and longitudinal moduli were estimated from finite-element simulation of the mechanical response of a representative unit cell. The unit cell shown with approximately 20 000 reduced integration tetrahedral elements was adequate for convergence of simulations of unmineralized collagen fibres. Significantly finer meshes discretized to allow sheaths of mineral were required for simulations of partially mineralized fibres. (Online version in colour.)

Figure 5.

Estimates of the fibre-level linear, longitudinal moduli. Shaded regions correspond to bounds described in the supplemental document. The broadest bounds are the HS bounds (lightly shaded, yellow in online version), truncated here because they extend several decades below the lowest estimate. The darker shaded regions (orange and tan in online version) represent ‘tighter’ bounds that are obtained by considering the details of the structure of collagen and bioapatite. (a) For models A–C, intrafibrillar mineralization precedes extrafibrillar mineralization. As motivated by the results of Monte Carlo simulations (circular markers, model C), the longitudinal moduli for models A and B were taken to follow lower bound estimates. (b) For model D, rapid stiffening occurs owing to extrafibrillar mineralization that precedes intrafibrillar mineralization (numerical results shown). (c) For model E, bioapatite accumulating in the overlap region leads to still greater stiffening of collagen fibrils. Circular markers represent numerical results for randomly distributed extrafibrillar bioapatite, as in model D. (Online version in colour.)

Figure 6.

Linear estimates of transverse moduli of fibres. Shaded regions (yellow in online document) are HS bounds. (a) For models A–C, intrafibrillar mineralization precedes extrafibrillar mineralization, and little stiffening occurs prior to the onset of extrafibrillar mineralization and the associated rapid percolation phenomenon. Percolation occurs more rapidly in the transverse direction than in the longitudinal direction, and some fibrils with intermediate degrees of mineralization may therefore be stiffer in the transverse direction than in the longitudinal direction. (b) For model D, rapid stiffening occurs due to extrafibrillar mineralization that precedes intrafibrillar mineralization (numerical results shown). (c) For model E, as with models A–C, little stiffening occurs prior to percolation. (Online version in colour.)

Figure 7.

The spatially varying mechanical properties of the insertion were modelled based upon competing gradients in (a) organization of collagen fibres, and (b) bioapatite volume fraction. Disorganization of collagen fibres (increasing angular deviation) increases in the material between tendon and bone. The gradient in bioapatite volume fraction begins near the point of greatest disorganization (× symbol). The trendlines are based upon data described elsewhere [45]. (Online version in colour.)

Figure 8.

Tissue-level estimates of linear elastic moduli. The longitudinal modulus of tissue along the axis of the insertion is lower than that of either tendon or bone over a ‘compliant region’ within the insertion. The mineralization schemes most consistent with the literature serve to broaden this compliant region relative to model D. (Online version in colour.)