A structural, kinetic model of soft tissue thermomechanics

Abstract

A structure-based kinetic model was developed to predict the thermomechanical response of collagenous soft tissues. The collagen fibril was represented as an ensemble of molecular arrays with cross-links connecting the collagen molecules within the same array. A two-state kinetic model for protein folding was employed to represent the native and the denatured states of the collagen molecule. The Monte Carlo method was used to determine the state of the collagen molecule when subjected to thermal and mechanical loads. The model predictions were compared to existing experimental data for New Zealand white rabbit patellar tendons. The model predictions for one-dimensional tissue shrinkage and the corresponding mechanical property degradation agreed well with the experimental data, showing that the gross tissue behavior is dictated by molecular-level phenomena.

Figure 1

(A) Native and denatured states of the collagen molecule based on Harris and Humphrey . (B) Energy states of the collagen. There are two energy levels corresponding to the two states, with the native state being at the lower energy level. The activation Gibbs free energy isΔG^o Under tensile force, F, applied on the molecule, an extra term accounting for the mechanical work, Fx_n, contributes to the energy balance. x_n, distance between the native and the transition states; x_d, distance between the denatured and the transition states; ΔG_N-D, energy of denaturation between the native and denatured states; Δ(ΔG_N-D), contribution of the mechanical work to the free energy of denaturation.

Figure 2

(A) Structure of the collagen fibril. The native state of the collagen molecule is shown. Molecules form a staggered pattern and are bonded together by cross-links. Molecular arrays are grouped together to form the fibril. Thick line shows the in-series arrangement of the molecules. (B) Model representation of the collagen fibril. Molecular arrays are aligned parallel to each other.

Figure 3

Schematic of the solution algorithm.

Figure 4

Shrinkage versus thermal damage for computed (solid line) and experimental (squares) results. The value of x_n is 0.2 nm, and the modulus E_d was calculated as 132.6 MPa. Specimen 1, test load 10 N, calculated force/molecular array 218 pN, R² = 0.987. Specimen 3, test load 7.5 N, calculated force/molecular array 185 pN, R² = 0.967. Specimen 7, test load 5 N, calculated force/molecular array 144 pN, R² = 0.989. Specimen 8, test load 2.5 N, calculated force/molecular array 102 pN, R² = 0.967.

Figure 5

Shrinkage versus thermal damage for computed (solid line) and experimental (squares) results. The value of x_n is 1.7 nm, and the modulus E_d was calculated as 17.7 MPa. Specimen 1, test load 10 N, calculated force/molecular array 26 pN, R² = 0.994. Specimen 3, test load 7.5 N, calculated force/molecular array 22 pN, R² = 0.990. Specimen 7, test load 5 N, calculated force/molecular array 17 pN, R² = 0.993. Specimen 8, test load 2.5 N, calculated force/molecular array 12 pN, R² = 0.950.

Figure 6

Variation of calculated E_d and F_i (for specimen 8) with x_n.

Figure 7

Predicted denaturation kinetics for (A) F_i = 15 pN while the temperature is increased from 64 to 72°C, and (B) T = 66°C while the force varies from 10 to 18 pN. The value of x_n is 1.7 nm.

Figure 8

Master curve (solid line) showing the predicted time-temperature-load equivalency. The experimental measurements are also shown: specimen 1 (×), specimen 3 (+), specimen 7 (▴), and specimen 8 (●). Dimensionless time is the time divided by the time at 50% denaturation.

Figure 9

Decrease in the normalized tangent modulus of the fibril, E/E_o, with shrinkage (1 − Λ) for the experimental (solid line) and the computed results for the four studied conditions: specimen 1 (×), specimen 3 (+), specimen 7 (▴), and specimen 8 (●). The value of x_n is 1.7 nm.