A cartilage growth mixture model for infinitesimal strains: solutions of boundary-value problems related to in vitro growth experiments

Abstract

A cartilage growth mixture (CGM) model is linearized for infinitesimal elastic and growth strains. Parametric studies for equilibrium and nonequilibrium boundary-value problems representing the in vitro growth of cylindrical cartilage constructs are solved. The results show that the CGM model is capable of describing the main biomechanical features of cartilage growth. The solutions to the equilibrium problems reveal that tissue composition, constituent pre-stresses, and geometry depend on collagen remodeling activity, growth symmetry, and differential growth. Also, nonhomogeneous growth leads to nonhomogeneous tissue composition and constituent pre-stresses. The solution to the nonequilibrium problem reveals that the tissue is nearly in equilibrium at all time points. The results suggest that the CGM model may be used in the design of tissue engineered cartilage constructs for the repair of cartilage defects; for example, to predict how dynamic mechanical loading affects the development of nonuniform properties during in vitro growth. Furthermore, the results lay the foundation for future analyses with nonlinear models that are needed to develop realistic models of cartilage growth.

Appendix

In this appendix, we outline a cartilage growth mixture (CGM) model for finite deformations obtained from the general growth mixture theory presented in (Klisch et al. 2000; Klisch and Hoger 2003) with the simplifying assumptions presented in the Methods section.

1.1 Kinematics

The deformation gradient tensors ${\mathbf{F}}^\alpha$ [superscript α=p (proteoglycan), c (collagen), oth (others)] for the growing solid matrix constituents are decomposed as

$${\mathbf{F}}^\alpha = {\mathbf{M}}_{\text{e}}^\alpha {\mathbf{M}}_{\text{g}}^\alpha . $$

(52)

The tensor ${\mathbf{M}}_{\text{e}}^\alpha {\mathbf{M}}_{\text{g}}^\alpha$ describes the deformation due to growth relative to a fixed reference configuration, where the amount and orientation of mass deposition are described by ${\mathbf{M}}_{\text{g}}^\alpha .$ The tensor ${\mathbf{M}}_{\text{e}}^\alpha$ is the elastic accommodation tensor that ensures continuity of the growing body, and may include a contribution arising from a superposed elastic deformation. The diffusive velocity a is defined as

$${\mathbf{a}} = {\mathbf{v}}^{\text{w}} - {\mathbf{v}}^{\text{s}} , $$

(53)

where v^w is the fluid velocity and v^s is the solid velocity. The constraint that the growing solid matrix constituents experience the same overall motion requires their displacement vectors (u^α) and velocity vectors (v^α) must equal the solid matrix displacement (u^s) and velocity (v^s) vectors

$${\mathbf{u}}^{\text{s}} = {\mathbf{u}}^{\text{p}} = {\mathbf{u}}^{\text{c}} = {\mathbf{u}}^{{\text{oth}}} ,\quad {\mathbf{v}}^{\text{s}} = {\mathbf{v}}^{\text{p}} = {\mathbf{v}}^{\text{c}} = {\mathbf{v}}^{{\text{oth}}} . $$

(54)

Consequently, the deformation gradient tensors F^α of the growing solid matrix constituents must equal the solid matrix deformation gradient tensor F^s :

$$\mathbf{F}^{\text{s}}=\mathbf{F}^{\text{p}}=\mathbf{F}^{\text{c}}=\mathbf{F}^{\text{oth}} \Rightarrow \mathbf{F}^{\text{s}}=\mathbf{M}_{\text{e}}^{\text{p}} \mathbf{M}_{\text{g}}^{\text{p}}=\mathbf{M}_{\text{e}}^{\text{c}} \mathbf{M}_{\text{g}}^{\text{c}}=\mathbf{M}_{\text{e}}^{{\text{oth}}} \mathbf{M}_{\text{g}}^{{\text{oth}}}. $$

(55)

The constraint of intrinsic incompressibility states that the mixture is fully saturated with constant true densities:

$$\phi ^{\text{p}} + \phi ^{\text{c}} + \phi ^{{\text{oth}}} + \phi ^{\text{w}} = 1, $$

(56)

where $\phi ^\alpha = \rho ^\alpha /\rho ^{\alpha {\text{T}}}$ is the volume fraction, $\rho^\alpha$ the apparent density (mass/tissue volume), $\rho^{\alpha{\rm T}}$ the true density (mass/constituent volume), and superscript w=water.

1.2 Balance of mass

The standard balance of mass equations are generalized by introducing mass growth functions (c^α) that quantify the rate of mass deposition per unit current mass for the growing solid matrix constituents. Due to the constraint (54), we obtain

$$\dot \rho ^\alpha + \rho ^\alpha {\text{div}}\,{\mathbf{v}}^{\text{s}} = \rho ^\alpha c^\alpha ,\quad \dot \rho ^{\text{w}} + \rho ^{\text{w}} {\text{div}}\,{\mathbf{v}}^{\text{w}} = 0, $$

(57)

where a superposed dot indicates the material time derivative following the appropriate constituent and div (·) is the divergence operator. The balance of mass equation for each growing solid matrix constituent is decomposed into two equations by assuming that the apparent density changes only because of the elastic part of the deformation. Thus, the balance of mass equations for the growing solid matrix constituents become

$$\rho ^\alpha {\text{det}}\,{\mathbf{M}}_{\text{e}}^\alpha = \rho _{\text{R}}^\alpha , $$

(58)

where $\rho_{\rm R}^\alpha$ is the reference apparent density. Growth continuity equations are then derived from Eqs. (57)₁ and (58)

$${\text{det}}\,{\mathbf{M}}_{\text{g}}^\alpha = {\text{exp}}\left ( {\int\limits_{\tau = t_0 }^t {c^\alpha {\text{d}}\tau } } \right), $$

(59)

where det (·) is the determinant operator. The balance of mass equation for the mixture requires

$$\rho ^{\text{p}} c^{\text{p}} + \rho ^{\text{c}} c^{\text{c}} + \rho ^{{\text{oth}}} c^{{\text{oth}}} = \rho ^{\text{s}} c^{\text{s}} , $$

(60)

where c^s is the mass growth function for the solid matrix.

1.3 Stresses, diffusive forces, and balance of linear momentum

The solid matrix Cauchy stress tensor T^s and diffusive force ${\varvec{\pi}}^{\rm s}$ are assumed to be the sum of the partial solid matrix constituent stresses T^α and diffusive forces ${\varvec{\pi }}^\alpha$, respectively:

$${\mathbf{T}}^{\text{s}} = {\mathbf{T}}^{\text{p}} + {\mathbf{T}}^{\text{c}} + {\mathbf{T}}^{{\text{oth}}},\quad \varvec{\pi} ^{\text{s}} = \varvec{\pi} ^{\text{p}} + \varvec{\pi} ^{\text{c}} + \varvec{\pi} ^{{\text{oth}}} . $$

(61)

It was shown in (Klisch et al. 2000; Klisch and Hoger 2003) that the constraint (55) produces constraint responses in T^α and ${\varvec{\pi }}^\alpha,$ which cancel upon addition when forming the solid matrix stress tensor and diffusive force vector. Consequently, the balance of linear momentum equations reduce to one for the solid and one for the fluid

$${\text{div}}\,{\mathbf{T}}^{\text{s}} +{\varvec{\pi}}^{\text{s}} = \rho ^{\text{s}} \dot v^{\text{s}} ,\quad {\text{div}}\,{\mathbf{T}}^{\text{w}} + {\varvec{\pi}}^{\text{w}} = \rho ^{\text{w}} \dot v^{\text{w}} . $$

(62)

The balance of linear momentum for the mixture requires

$${\varvec{\pi}} = {\varvec{\pi}}^{\text{s}} = - {\varvec{\pi}}^{\text{w}},$$

(63)

where $\varvec{\pi }$ is the diffusive force.

1.4 Growth laws

To obtain a complete theory, growth response functions that describe the time rate of change of ${\bf M}_{\rm g}^\alpha$ for the growing solid matrix constituents are defined. Mathematically, growth response functions of the general form are represented as

$$\dot{\mathbf{M}}_{\text{g}}^\alpha = \hat G^\alpha {\text{ }} (M^\alpha ), $$

(64)

where $\hat G^\alpha$ is a function of mechanical stimuli M^α that drives the growth process for each growing solid matrix constituent.

1.5 Constitutive equations

Constitutive equations are required for the partial stresses as well as the diffusive force. In Klisch et al. (2000) and Klisch and Hoger (2003), constitutive restrictions were derived that relate the constituent stresses to partial Helmholtz free energy functions that were allowed to depend on the elastic deformation gradient tensors and their gradients, fluid density and its gradient, diffusive velocity, and temperature. Here, we restrict those constitutive equations substantially as discussed in the Methods section. Generally, the assumed stress and diffusive force constitutive equations in the CGM model are

$${\mathbf{T}}^{\text{p}} = - \phi ^{\text{p}} {\text{p}}{\mathbf{I}} + \hat {\text{T}}^{\text{p}} ({\mathbf{M}}_{\text{e}}^{\text{p}} ,{\mathbf{M}}_{\text{e}}^{\text{c}} ,\rho ^{\text{w}} ), $$

(65)

$${\mathbf{T}}^{\text{c}} = - \phi ^{\text{c}} p{\mathbf{I}} + \hat {\mathbf{T}}^{\text{c}} ({\mathbf{M}}_{\text{e}}^{\text{c}} ,\gamma ), $$

(66)

$${\mathbf{T}}^{{\text{oth}}} = - \phi ^{{\text{oth}}} p{\mathbf{I}}, $$

(67)

$${\mathbf{T}}^{\text{w}} = - \phi ^{\text{w}} p{\mathbf{I}}, $$

(68)

$$\varvec{\pi} = \frac{{{\text{grad}}\rho ^{\text{s}} }} {{\rho ^{{\text{sT}}} }}p{\mathbf{I}} + \hat{\varvec{\pi}} ({\mathbf{F}}^{\text{s}} ,{\mathbf{a}}), $$

(69)

where I is the identity tensor, p an arbitrary Lagrange multiplier (i.e., the fluid pore pressure) that arises due to the intrinsic incompressibility constraint, and γ is a collagen remodeling parameter that may model a structural change in collagen network integrity (e.g., collagen crosslink density).