Integrating qPLM and biomechanical test data with an anisotropic fiber distribution model and predictions of TGF-$\upbeta$1 and IGF-1 regulation of articular cartilage fiber modulus

Abstract

A continuum mixture model with distinct collagen (COL) and glycosaminoglycan elastic constituents was developed for the solid matrix of immature bovine articular cartilage. A continuous COL fiber volume fraction distribution function and a true COL fiber elastic modulus ($E^\mathrm{f})$ were used. Quantitative polarized light microscopy (qPLM) methods were developed to account for the relatively high cell density of immature articular cartilage and used with a novel algorithm that constructs a 3D distribution function from 2D qPLM data. For specimens untreated and cultured in vitro, most model parameters were specified from qPLM analysis and biochemical assay results; consequently, $E^\mathrm{f}$ was predicted using an optimization to measured mechanical properties in uniaxial tension and unconfined compression. Analysis of qPLM data revealed a highly anisotropic fiber distribution, with principal fiber orientation parallel to the surface layer. For untreated samples, predicted $E^\mathrm{f}$ values were 175 and 422 MPa for superficial (S) and middle (M) zone layers, respectively. TGF-$\upbeta$1 treatment was predicted to increase and decrease $E^\mathrm{f}$ values for the S and M layers to 281 and 309 MPa, respectively. IGF-1 treatment was predicted to decrease $E^\mathrm{f}$ values for the S and M layers to 22 and 26 MPa, respectively. A novel finding was that distinct native depth-dependent fiber modulus properties were modulated to nearly homogeneous values by TGF-$\upbeta$1 and IGF-1 treatments, with modulated values strongly dependent on treatment.

Appendix

In this appendix, details of the SM tangent stiffness matrix required in the UMAT by Abaqus are presented. This stiffness matrix relates the stress increment to the strain increment. Here, the superscript SM is omitted when referring to SM quantities and indicial notation is used.

Abaqus requires the Jacobian $\mathbb C _{ijkl}^\mathrm{Jac}$ defined through the variation of Kirchhoff stress $\tau _{ij}$ as

$$\begin{aligned} J\mathbb C _{ijkl}^\mathrm{Jac} D_{kl} =\delta ^{J}(\tau _{ij} )=\delta \tau _{ij} -W_{ik} \tau _{kj} -\tau _{ik} W_{jk} \end{aligned}$$

(38)

where $\delta ^{J}(\tau _{ij})$ is the Jaumann stress rate and $D_{ij}$ and $W_{ij}$ are the symmetric and skew parts of the displacement increment gradient $L_{ij}$

$$\begin{aligned} L_{ij} =\frac{\partial \delta u_i }{\delta x_j}. \end{aligned}$$

(39)

Recalling (6), the increment in Kirchhoff stress $\delta \tau _{ ij}$ is derived as

$$\begin{aligned} \delta \tau _{ ij} =\delta F_{ iA} S_{ AB} F_{ jB} +F_{ iA} \delta S_{ AB} F_{ jB} +F_{ iA} S_{ AB} \delta F_{ jB} \end{aligned}$$

(40)

where the increment in the deformation gradient tensor $\delta F_{ iA}$ is

$$\begin{aligned} \delta F_{ iA} =L_{ ij} F_{ jA}. \end{aligned}$$

(41)

Recalling (5), the elasticity tensor $\mathbb C _{ ABCD}$ relates the increments in second Piola–Kirchhoff stress ($\delta S_{ AB}$) and Cauchy-Green deformation tensors ($\delta C_{ CD}$) as

$$\begin{aligned} \delta S_{ AB} =\frac{1}{2}\mathbb C _{ ABCD} \delta C_{ CD} \end{aligned}$$

(42)

which, using (2), (39), and (41) leads to

$$\begin{aligned} \delta S_{ AB} =\mathbb C _{ ABCD} F_{ iC} D_{ ij} F_{ jD}. \end{aligned}$$

(43)

Substituting (41) and (43) into (40) and recalling (6), a straightforward derivation leads to

$$\begin{aligned} \delta \tau _{ ij}&= \frac{1}{2}(\delta _{ il} \tau _{ jk} +\delta _{ jk} \tau _{ il} +\delta _{ ik} \tau _{ jl} +\delta _{ jl} \tau _{ ik} )D_{ kl}\nonumber \\&+\, \mathbb C _{ ABCD} F_{ iA} F_{ jB} F_{ lC} F_{ mD} D_{ lm} +W_{ ik} \tau _{ kj} +\tau _{ ik} W_{ jk}\nonumber \\ \end{aligned}$$

(44)

Thus, comparison of (38) and (44) yields the definition of the tangent stiffness matrix required by Abaqus

$$\begin{aligned} \mathbb C _{ijkl}^{Jac}&= \frac{1}{J}\left\{ \frac{1}{2}(\delta _{il} \tau _{jk} +\delta _{jk} \tau _{il} +\delta _{ik} \tau _{jl} +\delta _{jl} \tau _{ik} )\right.\nonumber \\&\left.+\,\mathbb C _{ ABCD} F_{ iA} F_{ jB} F_{ kC} F_{ lD} {} \right\} \end{aligned}$$

(45)

For implementation with the model presented here, the tangent stiffness matrix defined by (45) was derived for each constituent, and these were summed to obtain the total SM tangent stiffness matrix. In addition to coding the constituent stress equations, the elasticity tensors were derived for each constituent using (5). These derivations are straightforward but lengthy, here only the final results are shown as follows:

$$\begin{aligned}&\mathbb C _{ ABCD}^\mathrm{COL}\nonumber \\&\quad =(1/4\pi )\!\int \limits _{\varPhi =0}^{2\pi } {\!\int \limits _{\varTheta =0}^\pi {\phi _\mathrm{N}^\mathrm{f} \hat{{H}}(E_\mathrm{N} )E^\mathrm{f}N_A N_B N_C N_D \sin \varTheta \text{ d}\varTheta \text{ d}\varPhi } } \nonumber \\ \end{aligned}$$

(46)

$$\begin{aligned}&\mathbb C _{ABCD}^\mathrm{GAG}= \left\{ \alpha _1 \left(\rho _0^\mathrm{GAG} \right)^{\alpha _2}\left(C_{AC} C_{BD}\!+\!C_{AD} C_{BC} \right) \right.\nonumber \\&\quad \left.+ \alpha _2 \alpha _1 \left(\rho _0^\mathrm{GAG}\right)^{\alpha _2}C_{DC}^{-1} C_{AB}^{-1} \right\} \left\{ \left(\det C^\mathrm{GAG}\right)^{-\alpha _2 /2}\right\} \end{aligned}$$

(47)

$$\begin{aligned} \mathbb C _{ABCD}^\mathrm{MAT}&= \mu \left(C_{AC}^{-1} C_{BD}^{-1} +C_{AD}^{-1} C_{BC}^{-1}\right). \end{aligned}$$

(48)