A novel fibre-ensemble level constitutive model for exogenous cross-linked collagenous tissues

Abstract

Exogenous cross-linking of soft collagenous tissues is a common method for biomaterial development and medical therapies. To enable improved applications through computational methods, physically realistic constitutive models are required. Yet, despite decades of research, development and clinical use, no such model exists. In this study, we develop the first rigorous full structural model (i.e. explicitly incorporating various features of the collagen fibre architecture) for exogenously cross-linked soft tissues. This was made possible, in-part, with the use of native to cross-linked matched experimental datasets and an extension to the collagenous structural constitutive model so that the uncross-linked collagen fibre responses could be mapped to the cross-linked configuration. This allowed us to separate the effects of cross-linking from kinematic changes induced in the cross-linking process, which in turn allowed the non-fibrous tissue matrix component and the interaction effects to be identified. It was determined that the matrix could be modelled as an isotropic material using a modified Yeoh model. The most novel findings of this study were that: (i) the effective collagen fibre modulus was unaffected by cross-linking and (ii) fibre-ensemble interactions played a large role in stress development, often dominating the total tissue response (depending on the stress component and loading path considered). An important utility of the present model is its ability to separate the effects of exogenous cross-linking on the fibres from changes due to the matrix. Applications of this approach include the utilization in the design of novel chemical treatments to produce specific mechanical responses and the study of fatigue damage in bioprosthetic heart valve biomaterials.

Keywords: collagenous tissues; constitutive model; cross-linking.

Figure 1.

(a) Photomicrograph of native bovine pericardium showing the undulated collagen fibres (adapted from [8]). (b) TEM image also of native bovine pericardium clearly showing interrelationships between the undulated collagen fibres and the underlying fibril structures (magnification 4150×; adapted from [13]).

Figure 2.

A diagram showing the interaction of tropocollagen molecules with glutaraldehyde and how cross-links can form. As the concentration of GLUT increases, the number of activation sites and chain length increases, and a limited number of cross-links will form between such molecules (magnification 4150×; adapted from [13]).

Figure 3.

(a)(i) Pericardial test specimen showing a high degree of fibre orientation and uniformity in preferred fibre directions, with the PD = x₁ and XD = x₂ axes defined, and (ii) a typical biaxial test specimen mounted on the device. (b) A schematic of the biaxial test specimen geometry changes with cross-linking and the corresponding mean deformation gradient tensor components in native state β₀ and EXL state state β₁. Here, cross-linking induced a 6% contraction in the PD and and 7% direction in the XD, with some small shearing.

Figure 4.

An example of the bicubic Hermite surface interpolation of the S₂₂ biaxial test responses to allow interpolation of an equi-biaxial strain path, shown here in red. The blue path defines the span of the strain.

Figure 5.

A representative fibre-ensemble stress–strain (in P_ens−λ_ens) response for (a) native and (b) cross-linked bovine pericardium illustrating a well-defined post-transition fibre recruitment point wherein the response becomes linear. While the native pericardium demonstrated a very low initial modulus (approx. 75 kPa; table 2), the EXLs demonstrate a significantly stiffer modulus. (Online version in colour.)

Figure 6.

(a) A representative fibre-ensemble stress–strain (in P_ens−λ_ens) response for EXL treated bovine periardium, and (b) a close-up of the low stress region. A careful examination revealed that the toe region suggests that a modified Yeoh model was necessary to accurately capture its response (equation (4.8)) due to the convexity of the response. (Online version in colour.)

Figure 7.

A schematic of two collagen fibre ensembles with respective orientations α and β (not restricted to be symmetric about the x₁-axis) with associated orientation vector n₀ and m₀ in the reference configuration. (Online version in colour.)

Figure 8.

Representative final model results (equation (5.3)) for a single fibre-ensemble stress–strain (in S_ens−λ_ens) response for EXL treated bovine pericardium. All model components contributed signficantly to the total stress. Surprisingly, while the collagen phase produced the greatest contribution, the interaction term was of comparable magnitude. (Online version in colour.)

Figure 9.

Representative complete collagen fibre recruitment for the pericardial tissue, showing a rapid recruitment near the upper bound, suggesting the collagen fibres have similar crimp geometries in the native state. (Online version in colour.)

Figure 10.

Complete model (equation (5.3)) results for the S₁₁ and S₂₂ stress components for three protocols. Interestingly for S₁₁ the interactions actually produced the largest contributions, followed by the matrix and collagen fibres. By contrast, for S₂₂ the contributions were much more dependent on the particular loading path, with the collagen phase dominating when λ₂ > λ₁. When λ₁ > λ₂ the matrix phase dominated S₂₂. We further note here that the contribution of the matrix was much less loading path sensitive, owing to its near-linear, isotropic behaviour. (Online version in colour.)

Figure 11.

Simulation results using the unmodified native tissue fibre-ensemble model (equation (4.4)) on both the native and cross-linked data from figure 5, showing that one can increase the fibre modulus determined from the native state to match the post-EXL data (a). However, this will induce a parallel increase in the MTM of approximately 75% (b), which is inconsistent with the experimental findings (table 2). This is the case even when compensating for the effects of tissue contraction. (Online version in colour.)