A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

Abstract

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.

Appendices

Appendix 1

To evaluate integrals over the unit sphere, as shown, e.g., in (8) and (10), we apply the numerical method suggested by Bažant and Oh (1986) using $m$ distinct direction vectors $\mathbf{M}^i,\,i=1,2,\ldots , m$. Symmetry of our method allows us to use only half the number of direction vectors, and subsequently double the integration weights $q^i$. With this approach we numerically estimate integrals as

$$\begin{aligned} \begin{aligned} \int _{\Omega } A(\mathbf{M})\mathrm{d}{\Omega }&= \int _{\theta = 0}^{2 \pi } \int _{\phi = 0}^{\pi } A(\theta , \phi ) \sin \theta \, \mathrm{d}\phi \mathrm{d}\theta \\&\approx 4\pi \sum _{i=1}^m A(\mathbf{M}^i) 2 q^i, \end{aligned} \end{aligned}$$

(19)

where $A$ is a function taking tensor arguments. A table of the direction vectors and associated integration weights can be found in Table 1 of Bažant and Oh (1986).

Appendix 2

To obtain solutions of nonlinear problems in computational finite (visco)elasticity via incremental solution techniques of Newton’s type, we solve a series of linearized problems and thus require the linearized constitutive equations. Specifically, we require elasticity tensors for both the isotropic matrix and fiber network contributions to the solid extra stress, as well as the derivative of the filtration velocity $n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}$ with respect to both the deformation gradient $\mathbf {F}_{\mathrm{S}}$ and the material gradient of the interstitial fluid pressure ${\mathrm{grad}}\,{p}$. We write the filtration velocity in the current configuration as (16) and (17) [cf. Pierce et al. (2013a, b)]. In the reference configuration, we write the filtration velocity $n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0}$ as

$$\begin{aligned} \begin{aligned} n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0}&= ~{k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\,\\&\times {\mathbf{H}}_{0} \left( -J_\mathrm{S}\,{\mathrm{grad}}\,{p} + J_\mathrm{S}\,{\rho ^{\mathrm{FR}}}\,\mathbf{F}^\mathrm{T}_\mathrm{S}\,{\mathbf{b}}\right) , \end{aligned} \end{aligned}$$

(20)

with,

$$\begin{aligned} {\mathbf{H}}_{0} = \frac{1}{4\pi } \int _{\Omega } \frac{\rho (\mathbf{M})}{I_{4}(\mathbf{M})} \mathbf{M}\otimes \mathbf{M}\,\mathrm{d}\Omega . \end{aligned}$$

(21)

We write the derivative of the filtration velocity in the reference configuration with respect to the deformation gradient of the solid, i.e., $\partial (n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0})/\partial \mathbf {F}_{\mathrm{S}}$, as

$$\begin{aligned}&\partial \left( n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0}\right) /\partial \mathbf{{F}_{\mathrm{S}}} = \partial /\partial \mathbf{{F}_{\mathrm{S}}} \Bigg [{k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\,\nonumber \\&\quad {\mathbf{H}}_{0} \left( -J_\mathrm{S}{\mathrm{grad}}\,{p} + J_\mathrm{S}{\rho ^{\mathrm{FR}}}\mathbf{F}^\mathrm{T}_\mathrm{S}\,{\mathbf{b}}\right) \Bigg ] \nonumber \\&\quad = \displaystyle {k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\, \Bigg [- \frac{\partial {\mathbf{H}}_{0}}{\partial \mathbf{{F}_{\mathrm{S}}}} J_\mathrm{S}{\mathrm{grad}}\,{p}\nonumber \\&\quad - {\mathbf{H}}_{0} \otimes \frac{\partial J_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}}{\mathrm{grad}}\,{p} + {\rho ^{\mathrm{FR}}}\,\frac{\partial {\mathbf{H}}_{0}}{\partial \mathbf{{F}_{\mathrm{S}}}} J_\mathrm{S}\mathbf{F}^\mathrm{T}_\mathrm{S}\,{\mathbf{b}} \nonumber \\&\quad + {\rho ^{\mathrm{FR}}}\,{\mathbf{H}}_{0} \frac{\partial J_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}} \otimes \mathbf{F}^\mathrm{T}_\mathrm{S}\,{\mathbf{b}} + {\rho ^{\mathrm{FR}}}\,{\mathbf{H}}_{0} J_\mathrm{S} \frac{\partial \mathbf{F}^\mathrm{T}_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}}\,{\mathbf{b}} \Bigg ], \end{aligned}$$

(22)

with,

$$\begin{aligned} \displaystyle \partial {\mathbf{H}}_{0} / \partial \mathbf{{F}_{\mathrm{S}}}= & {} \displaystyle \partial / \partial \mathbf{{F}_{\mathrm{S}}} \left[ \frac{1}{4\pi } \int _{\Omega } \frac{\rho (\mathbf{M})}{I_{4}(\mathbf{M})} \mathbf{M}\otimes \mathbf{M}\,\mathrm{d}{\Omega } \right] \nonumber \\= & {} \displaystyle \frac{1}{4\pi } \int _{\Omega } \rho (\mathbf{M})\frac{\partial I_{4}(\mathbf{M})^{-1}}{\partial \mathbf{{F}_{\mathrm{S}}}} \otimes \mathbf{M}\otimes \mathbf{M}\,\mathrm{d}{\Omega }, \end{aligned}$$

(23)

and

$$\begin{aligned} \displaystyle \partial I_{4}(\mathbf{M})^{-1}/ \partial \mathbf{{F}_{\mathrm{S}}}= & {} \displaystyle \frac{\partial I_{4}(\mathbf{M})^{-1}}{\partial I_{4}(\mathbf{M})} \frac{\partial I_{4}(\mathbf{M})}{\partial \mathbf{C}_\mathrm{S}} : \frac{\partial \mathbf{C}_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}} \nonumber \\= & {} - \displaystyle I_{4}(\mathbf{M})^{-2} \mathbf{M}\otimes \mathbf{M}\ : \left( \frac{\partial \mathbf{F}^\mathrm{T}_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}} \mathbf{{F}_{\mathrm{S}}}\right. \nonumber \\&\left. + \mathbf{F}^\mathrm{T}_\mathrm{S} \frac{\partial \mathbf{{F}_{\mathrm{S}}}}{\partial \mathbf{{F}_{\mathrm{S}}}} \right) . \end{aligned}$$

(24)

In index notation, we write the results from (22)$_2$–(24)$_2$ as

$$\begin{aligned}&\displaystyle (\partial (n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0})/\partial \mathbf{{F}_{\mathrm{S}}})_{{ IJK }} = \displaystyle {k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\, \nonumber \\&\quad \displaystyle \times \Bigg [ - \left( \frac{\partial {\mathbf{H}}_{0}}{\partial \mathbf{{F}_{\mathrm{S}}}}\right) _{ IJKL } J_\mathrm{S} ({\mathrm{grad}}\,{p})_{L} \nonumber \\&\quad \displaystyle - ({\mathbf{H}}_{0})_{IJ} \left( \frac{\partial J_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}} \right) _{ KL } ({\mathrm{grad}}\,{p})_{L} \nonumber \\&\quad \displaystyle + {\rho ^{\mathrm{FR}}}\, \left( \frac{\partial {\mathbf{H}}_{0}}{\partial \mathbf{{F}_{\mathrm{S}}}}\right) _{ IJKM } J_\mathrm{S} (\mathbf{F}^\mathrm{T}_\mathrm{S})_{ ML } \,({\mathbf{b}})_{L} \nonumber \\&\quad \displaystyle + {\rho ^{\mathrm{FR}}}\,({\mathbf{H}}_{0})_{ IM } \left( \frac{\partial J_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}}\right) _{ MJ } (\mathbf{F}^\mathrm{T}_\mathrm{S})_{KL}\,({\mathbf{b}})_{L} \nonumber \\&\quad \displaystyle + {\rho ^{\mathrm{FR}}}\,({\mathbf{H}}_{0})_{ IM } J_\mathrm{S} \left( \frac{\partial \mathbf{F}^\mathrm{T}_\mathrm{S}}{\partial \mathbf{{F}_{\mathrm{S}}}}\right) _{ MJKL }\,({\mathbf{b}})_{L} \Bigg ], \end{aligned}$$

(25)

where $(\partial \mathbf{F}^\mathrm{T}_\mathrm{S}/ \partial \mathbf{{F}_{\mathrm{S}}})_{ IJKL } = \delta _{ JK }\delta _{ IL }$, with $\delta _{ IJ } = \mathbf{e}_{I}\cdot \mathbf{e}_{J}$ the Kronecker delta, and with

$$\begin{aligned}&\displaystyle (\partial {\mathbf{H}}_{0} / \partial \mathbf{{F}_{\mathrm{S}}})_{ IJKL } \nonumber \\&\quad =\displaystyle \frac{1}{4\pi } \int _{\Omega } \rho (\mathbf{M}) \displaystyle \left( \frac{\partial I_{4}(\mathbf{M})^{-1}}{\partial \mathbf{{F}_{\mathrm{S}}}} \right) _{IJ} \left( \mathbf{M}\otimes \mathbf{M}\right) _{ KL }\,\mathrm{d}{\Omega }, \end{aligned}$$

(26)

and

$$\begin{aligned} \begin{aligned}&\displaystyle \left( \partial I_{4}(\mathbf{M})^{-1}/ \partial \mathbf{{F}_{\mathrm{S}}}\right) _{ IJ } \\&\quad = - \displaystyle I_{4}(\mathbf{M})^{-2} (\mathbf{M}\otimes \mathbf{M})_{ KL } \left[ \delta _{KJ}(\mathbf{{F}_{\mathrm{S}}})_{ IL } + (\mathbf{{F}_{\mathrm{S}}})_{ IK }\delta _{ LJ } \right] . \end{aligned} \end{aligned}$$

(27)

To continue, we write the derivative of the filtration velocity in the reference configuration with respect to the material gradient of the interstitial fluid pressure, i.e., $\partial (n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0})/\partial {\mathrm{grad}}\,{p}$, as

$$\begin{aligned}&\partial \left( n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0}\right) \Big /\partial {\mathrm{grad}}\,{p}\nonumber \\&\quad = - {k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\,{\mathbf{H}}_{0} J_\mathrm{S}\,\frac{\partial {\mathrm{grad}}\,{p}}{\partial {\mathrm{grad}}\,{p}}. \end{aligned}$$

(28)

In index notation, we write (28) as

$$\begin{aligned}&\left( \partial \left( n^\mathrm{F}\,\mathbf{{w}_{\mathrm{FS}}}_{0}\right) /\partial {\mathrm{grad}}\,{p}\right) _{IJ}\nonumber \\&\quad = - {k_\mathrm{0S}}\,\left( \frac{n^\mathrm{F}}{\mathrm{1} - {n^\mathrm{S}_\mathrm{0S}}}\right) ^m\,J_\mathrm{S}\,({\mathbf{H}}_{0})_{ IK } \delta _{ KJ }. \end{aligned}$$

(29)

Appendix 3

In a ‘classic’ DT-MRI experiment, the tensor data is reconstructed from six or more diffusion-weighted images under the modeling assumption of a single Gaussian diffusion compartment per voxel (Tuch 2004). In Cartesian coordinates the anisotropic Gaussian probability distribution function (PDF) for a single voxel can be stated according to (1), where therein $({\varvec{\xi }}) = \left( x, y, z\right) ^{\mathrm{T}}$ denotes the relative displacement of water molecules. Since we are only interested in the orientation density function (ODF) we drop the diffusion time by setting $\delta = 0.5\,\mathrm{s}$ and, without loss of generality, drop the $b$-value by setting $b = 1.0\,\mathrm{s/mm^2}$, yielding the familiar form of the multivariate normal distribution (with zero mean)

$$\begin{aligned} P({\varvec{\xi }}) = (2\pi )^{-3/2} \left| \mathbf{D}\right| ^{-1/2} \exp \left\{ - \tfrac{1}{2} {\varvec{\xi }}^\mathrm{T} \mathbf{D}^{-1} {\varvec{\xi }} \right\} . \end{aligned}$$

(30)

Due to the simple analytic form of the diffusion probability density, a closed form ODF can be computed. We first convert (30) to spherical coordinates

$$\begin{aligned} x = r \cos \theta \sin \phi , \quad y = r \sin \theta \sin \phi , \quad z = r \cos \phi , \end{aligned}$$

(31)

with $r \in [0, \infty ),\,\theta \in [0, 2\pi )$, and $\phi \in [0, \pi ]$. Next, we marginalize out the radius $r$, account for the factor of $4\pi$ in (8) and the required ODF results as

$$\begin{aligned} \begin{aligned} \rho (\mathbf{M},\mathbf{D}) =&~ \displaystyle \left( \tfrac{2}{\pi }\right) ^{1/2} \left| \mathbf{D}\right| ^{-1/2} \\&\times \int _0^\infty \! \exp \left\{ - \tfrac{1}{2} r^2 \mathbf{M}^\mathrm{T} \mathbf{D}^{-1} \mathbf{M} \right\} r^2 \sin \theta \, \mathrm{d}r, \end{aligned} \end{aligned}$$

(32)

which gives (18). Note that the Jacobian determinant $r^2 \sin \theta$ is introduced by the change from Cartesian to spherical coordinates to account for the fact that the surface element on the sphere is not uniform across the entire domain. Assuming the symmetric, positive-definite diffusion tensor is given as

$$\begin{aligned} \left( \mathbf {D}\right) = \left( \begin{array}{ccc} D_{xx} &{}\quad D_{xy} &{}\quad D_{xz} \\ D_{xy} &{}\quad D_{yy} &{}\quad D_{yz} \\ D_{xz} &{}\quad D_{yz} &{}\quad D_{zz} \\ \end{array} \right) , \end{aligned}$$

(33)

the ODF can be written as

$$\begin{aligned} \rho (\mathbf{M},\mathbf{D})= & {} \sin \theta \Big / \bigg \{ \bigg (-\left( D_{xz}^2 D_{yy}\right) + 2 D_{xy} D_{xz} D_{yz} \\&- D_{xx} D_{yz}^2 - D_{xy}^2 D_{zz} + D_{xx} D_{yy} D_{zz}\bigg )^{1/2} \\&\bigg (\Big (\left( D_{xy}^2 - D_{xx} D_{yy}\right) \cos ^2\phi \\&+\left( D_{yz}^2 - D_{yy} D_{zz}\right) \cos ^2\theta \sin ^2\phi \\&- D_{xy} D_{xz} \sin 2\phi \sin \theta + D_{xx} D_{yz} \sin 2\phi \sin \theta \\&+ D_{xz}^2 \sin ^2\phi \sin ^2\theta - D_{xx} D_{zz} \sin ^2\phi \sin ^2\theta \\&+ \cos \theta \left[ \left( D_{xz} D_{yy} - D_{xy} D_{yz}\right) \sin 2\phi \right. \\&\left. - 2 D_{xz} D_{yz} \sin ^2\phi \sin \theta \right. ]\\&+ D_{xy} D_{zz} \sin ^2\phi \sin 2\theta \Big )\\&\Big / \left[ D_{xz}^2 D_{yy} - 2 D_{xy} D_{xz} D_{yz} + D_{xy}^2 D_{zz}\right. \\&\left. + D_{xx} (D_{yz}^2 - D_{yy} D_{zz})\right. ]\bigg )^{2/3} \bigg \}. \end{aligned}$$

Note that most trigonometric functions and several sub-expressions could be pre-computed within a FE implementation, in particular when a numerical integration scheme is used and only certain directions are evaluated.