Effect of structural distortions on articular cartilage permeability under large deformations

Abstract

The permeability of articular cartilage has a key role in load support and lubrication in diarthrodial joints. The microstructural rearrangement and consequent alteration in permeability caused by the large deformations undergone by cartilage have been previously modelled with a multi-scale approach. At the microscopic scale, the tissue is regarded as a homogeneous fluid-filled proteoglycan matrix reinforced by collagen fibres. A material point is described by a representative element of volume (REV), comprising a collagen fibre surrounded by a jacket of fluid-saturated proteoglycan matrix. At the macroscopic scale, the statistical orientation of the fibres is accounted for via averaging of the REV over all possible directions. The previous models accounted for volumetric deformation and fibre reorientation, but did not consider the cross-sectional distortion of the REV, which changes the widths of the fluid channels in different directions. We account for REV cross-sectional distortion and demonstrate its effects by simulating confined compression tests for the superficial, middle and deep zones of articular cartilage. The proposed model captures published experimental results that were not reproduced correctly by the previous models, and shows that each factor (volumetric deformation, fibre reorientation, REV cross-sectional distortion) can be dominant, depending on fibre orientation and amount of compression, implying that all three factors should be accounted for when modelling cartilage permeability.

Appendices

Appendix 1: Covariant formalism

In the covariant formalism of Continuum Mechanics, both the body ${\mathcal{B}}$ and the space ${\mathcal{S}}$ are equipped with metric tensors, G and g, respectively, which define the scalar products ${\varvec{W}} .{\varvec{Y}} \equiv {\varvec{W}} \, {\varvec{G}} \, {\varvec{Y}} = W^I \, G_{IJ} \, Y^J$ in ${\mathcal{B}}$ and ${\varvec{w}} .{\varvec{y}} \equiv {\varvec{w}} \, {\varvec{g}}\, {\varvec{y}} = w^i \, g_{ij} \, y^j$ in ${\mathcal{S}}$. If Cartesian coordinates are used in the body and the space, the covariant components of the metric tensors are $G_{IJ} = \delta _{IJ}$ and $g_{ij} = \delta _{ij}$, respectively, and the contravariant components of the inverse metric tensors ${\varvec{G}}^{-1}$ and ${\varvec{g}}^{-1}$ are $G^{IJ} = \delta ^{IJ}$ and $g^{ij} = \delta ^{ij}$, respectively.

Thus, in all instances in which we had the “contravariant” identity tensor i with components $\delta ^{ij}$, the covariant formalism prescribes the inverse metric tensor ${\varvec{g}}^{-1}$ with components $g^{ij}$. Specifically, in the definition (2b) of the transverse operator t, we would have ${\varvec{t}} = {\varvec{g}}^{-1} - {\varvec{a}} = {\varvec{g}}^{-1} - {\varvec{m}} \otimes {\varvec{m}}$, with components $t^{ij} = g^{ij} - a^{ij} = g^{ij} - m_i m_j$, and a spherical permeability tensor, such as that of the assumed proteoglycan permeability, would be expressed as ${\varvec{k}}_{{\rm PG}} = k_{{\rm PG}} \, {\varvec{g}}^{-1}$.

More importantly, the metric has a fundamental role in the spectral decomposition of a “covariant” tensor, such as the right Cauchy–Green deformation tensor C or its cross-sectional counterpart ${\varvec{C}}_{{\rm c}}$. Indeed, by looking at Eq. (27), one may deduce that the symbols ${\varvec{Q}}_\alpha$ indicate covectors since ${\varvec{C}}_{{\rm c}}$ is, like C, defined as a tensor with covariant components, but then would think that the push-forward in Eq. (28) is incorrect because it uses the transformation law for vectors and not for covectors (the push-forward of a vector W is ${\varvec{F}} {\varvec{W}}$, with components $F^i{}_J W^J$, but the push-forward of a covector ${\varvec{\Pi}}$ is ${\varvec{F}}^{-T} {\varvec{\Pi}}$, with components $({\varvec{F}}^{-T})_i{}^J \Pi _J$). This misunderstanding can be avoided using the covariant formalism and writing Eq. (27) more rigorously as

$${\varvec{C}}_{{\rm c}} = \sum ^{2}_{\alpha =1} \xi _\alpha ^2 \, ({\varvec{G}} \, {\varvec{Q}}_\alpha ) \otimes ({\varvec{G}} \, {\varvec{Q}}_\alpha ), $$

(40)

where the eigenvectors ${\varvec{Q}}_\alpha$ are righteously vectors and the material metric G maps the ${\varvec{Q}}_\alpha$ into the associated covectors ${\varvec{Q}}_\alpha ^\flat = {\varvec{G}} \, {\varvec{Q}}_\alpha$, with components $Q_{\alpha I} = G_{IJ} \, Q_\alpha ^J$. The explanation for this is given as follows.

Strictly speaking, the usual presentation of the eigenvalue problem applies to “mixed” tensors, i.e. tensors with the first leg being a vector and the second a covector or, equivalently, with the first index being contravariant (high) and the second covariant (low). Indeed, in component-free and component notation, the eigenvalue equation for a mixed (material) tensor T reads

$${\varvec{T}} \, {\varvec{W}} = \lambda \, {\varvec{W}}, \qquad T^I{}_J \, W^J = \lambda \, W^J, $$

(41)

where W is an eigenvector associated with the eigenvalue $\lambda$. The characteristic equation reads, as usual, $\det ({\varvec{T}} -\lambda {\varvec{I}}) =0$. If T is symmetric (with respect to the metric G, i.e. if $G_{IJ} T^J{}_K = T^J{}_I G_{JK}$), its spectral decomposition, rigorously, should be written

$${\varvec{T}} = \sum _p \lambda _p \, {\varvec{N}}_p \otimes ({\varvec{G}} \, {\varvec{N}}_p), $$

(42)

where the (material) metric tensor serves to transform the second leg of the tensor product into a covector.

For the case of the right Cauchy–Green deformation tensor C, which is a “covariant” tensor, the eigenvalues $\lambda _p^2$ are calculated with respect to a metric, in this case with respect to the material metric tensor G, i.e.

$${\varvec{C}} \, {\varvec{W}} = \lambda ^2 \, {\varvec{G}} \, {\varvec{W}}, \qquad C_{IJ} \, W^J = \lambda ^2 \, G_{IJ} W^J, $$

(43)

where both sides of the eigenvalue equation correctly represent covectors, although note that we are always looking for a vector, W. Once the eigenvalues $\lambda _p$ and eigenvectors ${\varvec{N}}_p$ are found via the characteristic equation (which this time reads $\det ({\varvec{C}} -\lambda \, {\varvec{G}}) = 0$), the spectral decomposition is

$${\varvec{C}} = \sum _p \lambda _p^2 \, ({\varvec{G}} \, {\varvec{N}}_p) \otimes ({\varvec{G}} \, {\varvec{N}}_p), $$

(44)

which has the same form that we used for ${\varvec{C}}_{{\rm c}}$ in Eq. (40).

Appendix 2: Cross-sectional principal stretches

Here, we elucidate the procedure for obtaining the principal cross-sectional stretches $\xi _1$ and $\xi _2$. Since ${\varvec{C}}_{{\rm c}}$ is a rank-two, symmetric, second-order tensor, its characteristic equation reduces to the biquadratic expression

$$\xi ^4 - I_1({\varvec{C}}_{{\rm c}}) \, \xi ^2 + I_2({\varvec{C}}_{{\rm c}}) = 0, $$

(45)

which yields two positive roots, the two non-vanishing eigenvalues of ${\varvec{C}}_{{\rm c}}$, i.e.

$$\xi ^2_1= \frac{I_1({\varvec{C}}_{{\rm c}}) + \sqrt{I^2_1({\varvec{C}}_{{\rm c}}) - 4 \, I_2({\varvec{C}}_{{\rm c}})}}{2}, $$

(46)

$$\xi ^2_2= \frac{I_1({\varvec{C}}_{{\rm c}}) - \sqrt{I^2_1({\varvec{C}}_{{\rm c}}) - 4 \, I_2({\varvec{C}}_{{\rm c}})}}{2}. $$

(47)

where

$$I_1({\varvec{C}}_{{\rm c}})= {\rm tr}\, {\varvec{C}}_{{\rm c}} = \xi ^2_1 + \xi ^2_2, $$

(48)

$$I_2({\varvec{C}}_{{\rm c}})= \frac{1}{2} (I^2_1({\varvec{C}}_{{\rm c}})-{\varvec{C}}_{{\rm c}} : {\varvec{C}}_{{\rm c}}) = \xi ^2_1 \, \xi ^2_2 $$

(49)

are the two (in general) non-vanishing principal invariants of ${\varvec{C}}_{{\rm c}}$. From the eigenvalue expressions (48, 49), it is straightforward to check that $I^2_1({\varvec{C}}_{{\rm c}}) - 4 \, I_2({\varvec{C}}_{{\rm c}})>0$ which is necessary for $\xi ^2_1$ and $\xi ^2_2$ to be real valued.

In order to evaluate the principal cross-sectional stretches $\xi _1$ and $\xi _2$ with Eq. (46, 47), it is necessary to express $I_1({\varvec{C}}_{{\rm c}})$ and $I_2({\varvec{C}}_{{\rm c}})$ in terms of the right Cauchy–Green tensor C and the unit vector M. Using (26) and (48) yields

$$I_1({\varvec{C}}_{{\rm c}}) = I_1({\varvec{C}}) - \lambda _{\varvec{M}}^{-2} \, {\varvec{C}}^2 : ({\varvec{M}} \otimes {\varvec{M}}). $$

(50)

Using (49) and (50) results in

$$I_2({\varvec{C}}_{{\rm c}}) = I_2({\varvec{C}}) - I_1({\varvec{C}}) \, \lambda _{\varvec{M}}^{-2} \, {\varvec{C}}^2 : ({\varvec{M}} \otimes {\varvec{M}}) + \lambda _{\varvec{M}}^{-2} \, {\varvec{C}}^3 : ({\varvec{M}} \otimes {\varvec{M}}). $$

(51)

Appendix 3: Volumetric deformation

To prove the expression $J = \xi _1 \, \xi _2 \, \lambda _{\varvec{M}}$ of Eq. (29), one may use the Cayley–Hamilton theorem for tensor C as

$${\varvec{C}}^3 - I_1({\varvec{C}}) \, {\varvec{C}}^2 + I_2({\varvec{C}}) \, {\varvec{C}} - I_3({\varvec{C}}) \, {\varvec{I}} = {\varvec{0}}. $$

(52)

Double-contraction of Eq. (52) with ${\varvec{M}} \otimes {\varvec{M}}$ on the right and using the properties $I_3({\varvec{C}})=J^2$, ${\varvec{I}} : ({\varvec{M}} \otimes {\varvec{M}}) = 1$ and of Eq. (20) yield

$$I_2({\varvec{C}}) \, \lambda _{\varvec{M}}^2 - I_1({\varvec{C}}) \, {\varvec{C}}^2 : ({\varvec{M}} \otimes {\varvec{M}})+{\varvec{C}}^3 : ({\varvec{M}} \otimes {\varvec{M}})=J^2. $$

(53)

Then, we multiply both sides of Eq. (51) by $\lambda _{\varvec{M}}^2$ to obtain

$$I_2({\varvec{C}}_{{\rm c}}) \lambda _{\varvec{M}}^2= I_2({\varvec{C}}) \lambda _{\varvec{M}}^2- I_1({\varvec{C}}) \, {\varvec{C}}^2 : ({\varvec{M}} \otimes {\varvec{M}}) + {\varvec{C}}^3 : ({\varvec{M}} \otimes {\varvec{M}}). $$

(54)

Lastly, upon comparing Eqs. (54) and (53), and using Eq. (49), we reach the expression

$$J^2= \, \xi ^2_1 \, \xi ^2_2 \, \lambda _{\varvec{M}}^2, $$

(55)

which is equivalent to Eq. (29). Note that, in the covariant formalism, the identity I in Eq. (52) should be replaced by the material metric tensor G, so that the contraction with ${\varvec{M}} \otimes {\varvec{M}}$ reads ${\varvec{G}} : ({\varvec{M}} \otimes {\varvec{M}}) = G_{IJ} \, M^I \, M^J = \Vert {\varvec{M}} \Vert ^2 = 1$.