On the issue of the bilinear model, deformation mechanisms and parameters of the elastin-collagen transition in biological tissues

Introduction. In general, the physiological response of tissues to external stresses is not linear in accordance with Hooke's law. Biological tissues are known to exhibit a nonlinear relationship between stress and strain, although the origins of this nonlinearity remain incompletely understood. In the literature, authors over the years have proposed many forms of σ-ε correspondence, but this question is still open.

The aim of the study. The article formalizes the characteristics of the elastic properties of soft biological tissues. Elastic behavior is considered within the framework of a two-phase elastin-collagen bilinear model with a variable elastic modulus and its detailed statistical analysis is performed.

Methods. Calculations were performed using the computer algebra system Mathcad 15.0 using the built-in fitting functions linfit, genfit, pwrfit and corr. The residual parameters of the experimental and model data were assessed using descriptive statistics.

Results. The correlation range of empirical and model data was R=0.8696–0.9845, the median of the correlation coefficient was 0.9616, the coefficient of variation was CV=0.03, which indicates a strong connection between the studied random variables and the homogeneity of the calculation procedures. The critical value of deformation ε=ε_cr, at which the mechanism of tissue deformation changes from elastin to collagen in the tissues of living organisms (median 0.38) is quite variable (CV 1.06). The median of the bilinear model parameters E₁ and E₂ were set to 0.02 MPa and 8.0 MPa, the CV was 0.35 and 0.79, respectively. Data outlier points have been identified. Based on the calculations performed, it was concluded that the Young’s modulus of collagen fibers in the tissues of the studied organs is a significant number of times (greater than the elastic modulus of elastin structures; average value 3071.36, median 160.00, CV 0.29), which is consistent with the literature data. The stress-strain relationships of phenomenological models alternative to the bilinear one are obtained. As a numerical example, the tissues of the periodontal ligament of the human central incisor are considered.

Conclusion. To describe the mechanism of evolution of the deformation properties of biological tissues, the terminology of the physics of phase transitions, accompanied by a change in symmetry, is used. It is proposed to consider as an order parameter the measure of «involvement» of collagen structures in the overall resistance of tissues to deformation, which is determined by the degree of «linearization» of the branched collagen network in the direction of the action of uniaxial load.