On the elastic properties of mineralized turkey leg tendon tissue: multiscale model and experiment

Abstract

The key parameters influencing the elastic properties of the mineralized turkey leg tendon (MTLT) were investigated. Two structurally different tissue types appearing in the MTLT were considered: circumferential and interstitial tissue. These differ in their amount of micropores and their average diameter of the mineralized collagen fibril bundles. A multiscale model representing the apparent elastic stiffness tensor of MTLT tissue was developed using the Mori–Tanaka and the self-consistent homogenization schemes. The volume fraction of mineral (hydroxyapatite) in the fibril bundle, $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}$, and the tissue microporosity are the variables of the model. The MTLT model was analyzed performing a global sensitivity analysis (Elementary Effects method) and a parametric study. The stiffnesses parallel (axial) and perpendicular (transverse) to the MTLT long axis were the only significantly sensitive components of the apparent stiffness tensor of MTLT tissue. The most important parameters influencing these apparent stiffnesses are $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}$, tissue microporosity, as well as shape and distribution of the minerals in the fibril bundle (intra- vs. interfibrillar). The predicted apparent stiffness was converted to acoustic impedance for model validation. From measurements on embedded MTLT samples, including 50- and 200-MHz scanning acoustic microscopy as well as synchrotron radiation micro-computed tomography, we obtained site-matched acoustic impedance and $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}$ data of circumferential and interstitial tissue. The experimental and the model data compare very well for both tissue types (relative error 6–8 %).

Appendix

1.1 Phase volume fractions

We give below the complete set of formulas that determine the phase volume fractions of our model sequence.

1.1.1 MTLT tissue

The phase volume fractions of the composite MTLT (CIR or INT tissue) are given by:

$$\begin{aligned}&\hbox {vf}^\mathrm{MTLT}_\mathrm{mp} = {\left\{ \begin{array}{ll} \mathrm{Const }, &{} \text {for }\mathrm{Const } \in [0,0.2], \qquad \text {case (C)}\, ,\\ a\cdot \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}+ b, &{} \text {for }\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}\in [0.25,0.35],\\ ~ &{} a<0\qquad \text {case (MD)}. \end{array}\right. }\\&{\hbox {vf}_{{\text {MCFB}}}^{\mathrm{MTLT}}} = 1- {\hbox {vf}_{{\text {mp}}}^{\mathrm{MTLT}}}. \end{aligned}$$

1.1.2 MCFB

The phase volume fractions of the MCFB are given by:

$$\begin{aligned} \hbox {vf}_{{\text {MCF}}}^{{\text {MCFB}}}&= \alpha _{{\text {MCF}}}\cdot \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}+ \hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}\\&= \alpha _{{\text {MCF}}}\cdot \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}+ h\left( \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}\right) ,\\ \hbox {vf}_{{\text {ES}}}^{{\text {MCFB}}}&= 1 - \hbox {vf}_{{\text {MCF}}}^{{\text {MCFB}}}. \end{aligned}$$

The function $h$ is derived based on an empirical formula of (Raum et al. (2006a), Eq. (10), p. 750), which connects the collagen volume fraction $\hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}$, the mineral volume fraction $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}$ and the nanopores volume fraction $\hbox {vf}_{{\text {np}}}^{{\text {MCFB}}}$ to each other:

$$\begin{aligned} \frac{\hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}}{\hbox {vf}_{{\text {np}}}^{{\text {MCFB}}}} = 0.36 + 0.084 \cdot \exp \left( 6.7 \cdot \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}\right) :=\gamma . \end{aligned}$$

(17)

Since ha, col and np are the sole basic constituents of the MCFB, their corresponding volume fractions $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}, \hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}$ and $\hbox {vf}_{{\text {np}}}^{{\text {MCFB}}}$ sum up to one. We then eliminate $\hbox {vf}_{{\text {np}}}^{{\text {MCFB}}}$ from Eq. (17) using $\hbox {vf}_{{\text {np}}}^{{\text {MCFB}}} = 1-\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}- \hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}$ and obtain

$$\begin{aligned}&\frac{\hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}}{1-\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}- \hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}} = 0.36 + 0.084 \cdot \exp \left( 6.7 \cdot \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}\right) \\&\quad \quad \Leftrightarrow \hbox {vf}_{{\text {col}}}^{{\text {MCFB}}}= (1 - \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}) \cdot \frac{\gamma }{(1 + \gamma )}:=h\left( \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}\right) . \end{aligned}$$

The function $h$ is defined by the right-hand side of the equation.

The mineral volume fraction $\hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}$ is given by

$$\begin{aligned} \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}=\frac{\mathrm{{\mathrm{DMB}}}}{\rho _{{\text {ha}}}\cdot (1-\beta )} \end{aligned}$$

where $\beta \in [0,\widetilde{\hbox {vf}}_{{\text {mp}}}]$ is the amount of microporosity which influences the DMB measurements. $\rho _{{\text {ha}}}$ is the mass density of ha and $\mathrm {{\mathrm{DMB}}}$ is the experimental DMB value.

1.1.3 MCF

The phase volume fractions of the MCF are given by:

$$\begin{aligned} \hbox {vf}_{{\text {ha}}}^{{\text {MCF}}}&= \alpha _{{\text {MCF}}}\cdot \frac{ \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}}{ \hbox {vf}_{{\text {MCF}}}^{{\text {MCFB}}}},\\ \hbox {vf}_{{\text {col}}}^{{\text {MCF}}}&= 1 - \hbox {vf}_{{\text {ha}}}^{{\text {MCF}}}. \end{aligned}$$

1.1.4 ES

The phase volume fractions of the ES are given by:

$$\begin{aligned} \hbox {vf}_{{\text {ha}}}^{{\text {ES}}}&= \left( 1 - \alpha _{{\text {MCF}}}\right) \cdot \frac{ \hbox {vf}_{{\text {ha}}}^{{\text {MCFB}}}}{ 1 - \hbox {vf}_{{\text {MCF}}}^{{\text {MCFB}}}},\\ \hbox {vf}_{{\text {np}}}^{{\text {ES}}}&= 1 - \hbox {vf}_{{\text {ha}}}^{{\text {ES}}}. \end{aligned}$$