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J Biomech Eng. Author manuscript; available in PMC 2010 April 13.

Published in final edited form as:

PMCID: PMC2854001

NIHMSID: NIHMS166636

The publisher's final edited version of this article is available at J Biomech Eng

See other articles in PMC that cite the published article.

A primary mechanism of solute transport in articular cartilage is believed to occur through passive diffusion across the articular surface, but cyclical loading has been shown experimentally to enhance the transport of large solutes. The objective of this study was to examine the effect of dynamic loading within a theoretical context, and to investigate the circumstances under which convective transport induced by dynamic loading might supplement diffusive transport. The theory of incompressible mixtures was used to model the tissue (gel) as a mixture of a gel solid matrix (extracellular matrix/scaffold), and two fluid phases (interstitial fluid solvent and neutral solute), to solve the problem of solute transport through the lateral surface of a cylindrical sample loaded dynamically in unconfined compression with frictionless impermeable platens in a bathing solution containing an excess of solute. The resulting equations are governed by non-dimensional parameters, the most significant of which are the ratio of the diffusive velocity of the interstitial fluid in the gel to the solute diffusivity in the gel (*R _{g}*), the ratio of actual to ideal solute diffusive velocities inside the gel (

Solute transport in biological tissues is a fundamental process of life, providing nutrients to cells and carrying away waste products. In avascular adult articular cartilage, solute transport occurs primarily across the articular surface, with synovial fluid mediating exchanges with the synovium lining the joint capsule [1]. A primary mechanism of solute transport is through diffusion, which has led many investigators to measure the diffusivity of various small and large solutes in articular cartilage (e.g., [2–11], see Table 1). The mechanism of passive diffusion has been shown experimentally to be enhanced by cyclical loading of cartilage, in the case of a large solute such as serum albumin [12], and by electroosmotic flow [13], both of which mechanisms lead to convective flow within the tissue. However, for smaller solutes, dynamic loading does not enhance transport [12, 14, 15], and convective flow enhances it less significantly than with larger solutes [16]. Urban and co-workers have attributed the disparate outcome under dynamic loading to the magnitude of the ratio of the 'fluid transport coefficient' [17] to the solute diffusion coefficient in the tissue [12, 14], thus defining a non-dimensional governing parameter for this problem. Garcia et al. [13] determined that electroosmotic flow was regulated by the Peclet number, which is a different but related non-dimensional parameter.

Molecular weights (MW) and diffusion coefficients for various solutes in aqueous solution (*D*_{0}) or in a tissue/gel environment (*D*). *R*_{d} is calculated with volumetric water fractions (_{w}) obtained from the literature. When not explicitly given, water **...**

Studies of articular cartilage metabolism have demonstrated that static loading as well as loading below a characteristic frequency of 0.001 Hz leads to biosynthetic inhibition, whereas dynamic loading stimulates tissue synthesis [18–22]. However, whether this enhanced biosynthetic response results from an enhanced nutritional supply remains an unresolved question. Kim et al. have suggested that the stimulation of tissue biosynthetic response under dynamic loading is most likely the result of enhanced fluid flow or changes in cell shape, rather than enhanced nutrient transport [19]. Static compression of articular cartilage has been shown to reduce the diffusivity of various solutes within the tissue, and has been implicated in the altered biosynthetic response of the tissue to static loading [7, 10, 23]. Growth factors, which have been shown to regulate the biosynthetic response of articular cartilage, are generally large solutes with molecular weights on the order of tens of kilodaltons. A further complication of growth factor uptake, their binding to specific protein complexes, was analyzed by Schneiderman et al., who demonstrated that IGF-I binding complexes in normal human articular cartilage are largely excluded from the tissue [8]. Bonassar et al. have shown that dynamic compression accelerates the biosynthetic response of cartilage to free IGF-I and increases the rate of transport of free IGF-I into the matrix, suggesting that cyclic compression may improve the access of soluble growth factors [24]. In a similar fashion, convective diffusion likely aids in the removal of metabolic waste products, creating an environment more suitable for cellular metabolism and matrix biosynthesis [25], as well as influencing the distribution and rate of loss of matrix products from tissue constructs [12, 26].

In studies related to cartilage tissue engineering, it has been suggested that cell growth rates are diffusionally limited [27]. In studies analogous to those carried out on cartilage explants, Buschmann et al. examined the response of chondrocyte-seeded agarose gels and suggested that cell matrix interactions and physicochemical effects may be more important than matrix-independent cell deformation and transport limitations [28]. They also observed that “for dynamic compression, fluid flow, streaming potentials, and cell-matrix interactions appeared to be more significant as stimuli than the small increase in fluid pressure, altered molecular transport, and matrix-independent cell deformation [28].” In our own studies of cartilage tissue engineering, we have found that a dynamic loading regimen applied to chondrocyte-seeded agarose gels over a 28-day period considerably enhances extracellular matrix synthesis as measured by mechanical properties and biochemical composition [29]. As observed in the biosynthetic response of cartilage explants in the study by Bonassar et al. [24], we have also found that the stimulatory effect of dynamic loading is synergistically enhanced by the addition of TGF-β1 or IGF-1 to the culture media [30]. Whether the enhanced extracellular matrix synthesis resulted from the increased nutrient transport alone, cell-matrix interactions alone, or a combination thereof remains unresolved, although the literature findings described above suggest that it is likely that both effects play a role.

The purpose of the current study is to examine the effect of dynamic loading on solute transport within a theoretical context, and to investigate the circumstances under which convective transport induced by dynamic loading might supplement diffusive transport in a porous hydrated tissue or tissue engineered construct. From a theoretical perspective, there is a need to identify the non-dimensional parameters governing this problem, and investigate the response as these parameters are varied over their physiological ranges. The motivation for such a study is the need to estimate, a priori, whether the transport of a particular solute (large or small, such as a growth factor or oxygen) might be enhanced by dynamic loading, and the time constant for the transport of that solute within the tissue. Such theoretical estimates can help determine whether or not the results of a particular experiment might be attributed in part to enhanced nutritional transport. Furthermore, these theoretical results might help to optimize the loading regimen of engineered tissue constructs.

To formulate the governing equations for this problem, we employ the theory of incompressible mixtures [31–38] to model the tissue as a mixture of a solid matrix (representing the extracellular matrix/scaffold), a fluid phase representing the solvent (water), and a fluid phase representing the solute. Throughout the text, this mixture model is generically referred to as a 'gel', with the understanding that this gel can represent a biological tissue, an engineered tissue construct, a hydrogel, or any other deformable porous media. The mixture model is useful in this case because it allows us to model transport of a solute in a dynamically loaded gel, and subsumes the classical transport relations, such as Fick's laws, and generalizes them to include the role of gel deformation. For simplicity in this first analysis, the effects of charges in the mixture (e.g., the fixed charge density contributed by proteoglycans and the ionic charges in the interstitial fluid) are neglected. The governing equations of mixture theory are used to solve the problem of solute transport through the lateral surface of a cylindrical gel sample loaded dynamically in unconfined compression, using frictionless impermeable platens. This configuration is employed because it has been frequently used in experimental studies of the biosynthetic response of cartilage explants and chondrocyte-seeded scaffolds [18, 19, 24, 28, 29] as well as in the experimental study of diffusion under cyclical loading [12].

Only one solute is considered in the analysis of this study, and each of the components of the mixture are considered to be of neutral valence. In mixture theory, given that the mass, volume, and number of moles of phase α are represented by *m*^{α}, *V*^{α}, and *n*^{α}, respectively, and that *V* = Σ_{α}*V*^{α}, the following definitions apply:

$$\begin{array}{cc}{\phi}^{\alpha}=\frac{{\mathit{\text{dV}}}^{\alpha}}{\mathit{\text{dV}}}\hfill & \text{volume fraction}\hfill \\ {\rho}^{\alpha}=\frac{{\mathit{\text{dm}}}^{\alpha}}{\mathit{\text{dV}}}\hfill & \text{apparent density}\hfill \\ {\rho}_{T}^{\alpha}=\frac{{\mathit{\text{dm}}}^{\alpha}}{{\mathit{\text{dV}}}^{\alpha}}\hfill & \text{true density}\hfill \\ {\tilde{c}}^{\alpha}=\frac{{\mathit{\text{dn}}}^{\alpha}}{\mathit{\text{dV}}}\hfill & \text{mixture volume - based concentration}\hfill \\ {c}^{\alpha}=\frac{{\mathit{\text{dn}}}^{\alpha}}{{\mathit{\text{dV}}}^{w}}\hfill & \text{solvent volume - based concentration}\hfill \\ {M}_{\alpha}=\frac{{\mathit{\text{dm}}}^{\alpha}}{{\mathit{\text{dn}}}^{\alpha}}\hfill & \text{molecular weight},\hfill \end{array}.$$

(1)

where the superscript ‘*w*’ indicates the solvent phase. These relations are given in differential form because the variables are defined locally in this continuum model. The molecular weight is constant for a given phase α; furthermore, since each phase is assumed to be intrinsically incompressible, its true density is also constant. From these definitions, it follows that

$$\sum _{\alpha}{\phi}^{\alpha}}=1,\phantom{\rule{thinmathspace}{0ex}}{\rho}^{\alpha}={\phi}^{\alpha}{\rho}_{T}^{\alpha},\phantom{\rule{thinmathspace}{0ex}}{\tilde{c}}^{\alpha}={\phi}^{w}{c}^{\alpha}=\frac{{\rho}^{\alpha}}{{M}_{\alpha}},$$

(2)

thus, concentrations and apparent densities can be interchanged using these relations.

For a cylindrical specimen placed in a bathing solution which contains a higher (or lower) concentration of the solute phase, and loaded between impermeable frictionless platens, the objective of the analysis is to describe the transient response as the solute transports through the tissue, in the presence of a dynamic axial load or strain (Figure 1). The governing equations for this problem are based on the theory of mixtures [31–38]; using the notation of Gu et al. [38], the equations are given by

$$\text{div}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{\sigma}}_{I}=\mathbf{0}\phantom{\rule{thinmathspace}{0ex}},$$

(3)

$$-{\rho}^{w}\phantom{\rule{thinmathspace}{0ex}}\text{grad}\phantom{\rule{thinmathspace}{0ex}}{\tilde{\mu}}^{w}+{f}_{\mathit{\text{ws}}}\phantom{\rule{thinmathspace}{0ex}}\left({\mathbf{v}}^{s}-{\mathbf{v}}^{w}\right)+{f}_{\mathit{\text{wf}}}\phantom{\rule{thinmathspace}{0ex}}\left({\mathbf{v}}^{f}-{\mathbf{v}}^{w}\right)=\mathbf{0}\phantom{\rule{thinmathspace}{0ex}},$$

(4)

$$-{\rho}^{f}\phantom{\rule{thinmathspace}{0ex}}\text{grad}\phantom{\rule{thinmathspace}{0ex}}{\tilde{\mu}}^{f}+{f}_{\mathit{\text{fs}}}\left({\mathbf{v}}^{s}-{\mathbf{v}}^{f}\right)+{f}_{\mathit{\text{wf}}}\left({\mathbf{v}}^{w}-{\mathbf{v}}^{f}\right)=\mathbf{0}\phantom{\rule{thinmathspace}{0ex}},$$

(5)

$$\text{div}\left({\phi}^{s}{\mathbf{v}}^{s}+{\phi}^{w}{\mathbf{v}}^{w}+{\phi}^{f}{\mathbf{v}}^{f}\right)=0\phantom{\rule{thinmathspace}{0ex}}.$$

(6)

$$\frac{\partial {\rho}^{\alpha}}{\partial t}+\text{div}\left({\rho}^{\alpha}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{v}}^{\alpha}\right)=0,\phantom{\rule{thinmathspace}{0ex}}\alpha =w,f\phantom{\rule{thinmathspace}{0ex}}.$$

(7)

Schematic of dynamic unconfined compression of gel construct between frictionless impermeable platens in a bathing solution containing an excess of solute.

The first equation represents the balance of linear momentum for the mixture, neglecting inertial effects (which are significant only in wave propagation problems); the second and third equations represent the balance of linear momentum for the solvent phase (α = *w*) and the solute phase (α = *f*), respectively (also neglecting inertial effects, along with fluid viscosity). The fourth equation is the balance of mass for the mixture, taking into account that each phase is intrinsically incompressible; and the fifth equation represents the balance of mass equations for each of the solvent and solute phases. In these expressions **v**^{α} represents the velocity of phase α and *f*_{αβ} represents the diffusive drag coefficient for the momentum exchanged between phases α and β as a result of molecular collisions (*f*_{αβ} = *f*_{βα}). The constitutive relations for the mixture stress **σ*** _{I}* and the solvent and solute electrochemical potentials,

$${\mathbf{\sigma}}_{I}=-p\mathbf{I}+{\lambda}_{s}\left(\text{tr}\mathbf{E}\right)\mathbf{I}+2{\mu}_{s}\mathbf{E},$$

(8)

$${\tilde{\mu}}^{w}={\mu}_{0}^{w}(\theta )+\frac{1}{{\rho}_{T}^{w}}(p-R\theta \mathrm{\Phi}{c}^{f}),$$

(9)

$${\tilde{\mu}}^{f}={\mu}_{0}^{f}(\theta )+\frac{R\theta}{{M}_{f}}\text{ln}\phantom{\rule{thinmathspace}{0ex}}{a}^{f},$$

(10)

where we assume that the solute volume fraction is negligible, ^{f} 1. This assumption is justified for most circumstances as illustrated by the two following examples: For a small solute such as glucose $({M}_{f}=180\mathrm{g}/\text{mole},\phantom{\rule{thinmathspace}{0ex}}{\rho}_{T}^{f}=\mathrm{1,500}\mathrm{g}/\mathrm{L})$ with a typical concentration in media of 1g/L (* ^{f}* = 1/180 mole/L), the volume fraction according to Eq.(1) is given by ${\phi}^{f}={M}_{f}{\tilde{c}}_{f}/{\rho}_{T}^{f}=0.0007$, and thus negligible compared to unity. With larger solutes such as growth factors, a representative concentration in supplemented media is on the order of 10µg/L for TGFβ-1 (

The gel solid matrix is assumed to behave as a linearly elastic isotropic material, with Lamé constants λ_{s},μ_{s}, infinitesimal strain tensor **E** = (grad**u** + grad^{T} **u**)/2, and solid displacement **u**, similarly to the biphasic analysis of unconfined compression by Armstrong et al. [39]. The electrochemical potentials for the solvent and solute have the same form as in classical physical chemistry [40, 41]. In these expressions, *R* is the universal gas constant, θ is the absolute temperature, *p* is the fluid pressure (inclusive of osmotic effects), ${\mu}_{0}^{\alpha}\phantom{\rule{thinmathspace}{0ex}}(\alpha =w,f)$ are the chemical potentials in a reference configuration (standard state), and *a ^{f}* is the activity of the solute. The model assumes isothermal conditions, which is appropriate for physiological systems. The reference configuration is achieved when

$$-\text{grad}\phantom{\rule{thinmathspace}{0ex}}p+({\lambda}_{s}+{\mu}_{s})\text{grad}(\text{div}\phantom{\rule{thinmathspace}{0ex}}\mathbf{u})+{\mu}_{s}{\nabla}^{2}\mathbf{u}=\mathbf{0},$$

(11)

$$-{\phi}^{w}\text{grad}\phantom{\rule{thinmathspace}{0ex}}p+{\phi}^{w}R\theta \text{grad}{c}^{f}+{f}_{\mathit{\text{ws}}}({\mathbf{v}}^{s}-{\mathbf{v}}^{w})+{f}_{\mathit{\text{wf}}}({\mathbf{v}}^{f}-{\mathbf{v}}^{w})=\mathbf{0},$$

(12)

$$-{\phi}^{w}R\theta \text{grad}{c}^{f}+{f}_{\mathit{\text{sf}}}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{v}}^{s}-{\mathbf{v}}^{f})+{f}_{\mathit{\text{wf}}}\phantom{\rule{thinmathspace}{0ex}}({\mathbf{v}}^{w}-{\mathbf{v}}^{f})=\mathbf{0}.$$

(13)

The unconfined compression problem has been shown to be one-dimensional when the loading platens are assumed frictionless, because the axial normal strain is uniform [39]. Thus, it can be shown that dependent variables in this problem exhibit the dependencies *p* = *p*(*r,t*), *u _{r}* =

$$-\frac{\partial p}{\partial r}+{H}_{A}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial r}(r{u}_{r})\right)=0,$$

(14)

$$-{\phi}^{w}\frac{\partial p}{\partial r}+{\phi}^{w}R\theta \frac{\partial {c}^{f}}{\partial r}+{f}_{\mathit{\text{ws}}}({v}_{r}^{s}-{v}_{r}^{w})+{f}_{\mathit{\text{wf}}}({v}_{r}^{f}-{v}_{r}^{w})=0,$$

(15)

$$-{\phi}^{w}R\theta \frac{\partial {c}^{f}}{\partial r}+{f}_{\mathit{\text{sf}}}\phantom{\rule{thinmathspace}{0ex}}({v}_{r}^{s}-{v}_{r}^{f})+{f}_{\mathit{\text{wf}}}\phantom{\rule{thinmathspace}{0ex}}({v}_{r}^{w}-{v}_{r}^{f})=0,$$

(16)

$$\frac{1}{r}\frac{\partial}{\partial r}[r({\phi}^{s}{v}_{r}^{s}+{\phi}^{w}{v}_{r}^{w})]+\dot{\epsilon}(t)=0,$$

(17)

$$\frac{\partial ({\phi}^{w}{c}^{f})}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}(r{\phi}^{w}{c}^{f}{v}_{r}^{f})=0,$$

(18)

where *H _{A}* = λ

$${\phi}^{s}{v}_{r}^{s}+{\phi}^{w}{v}_{r}^{w}+\frac{r}{2}\dot{\epsilon}(t)=0.$$

(19)

Now Eqs.(15), (16) and (19) can be solved for $\partial p/\partial r,{v}_{r}^{w},{v}_{r}^{f}$:

$$\frac{\partial p}{\partial r}=\left[\frac{1}{k}+\left(1-\frac{D}{{D}^{\mathit{\text{wf}}}}\right)\frac{R\theta {c}^{f}}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\right]\left({v}_{r}^{s}+\frac{r}{2}\dot{\epsilon}\left(t\right)\right)+\left(1-\frac{D}{{D}^{\mathit{\text{wf}}}}\right)R\theta \frac{\partial {c}^{f}}{\partial r},$$

(20)

$${v}_{r}^{w}=-\frac{1}{{\phi}^{w}}\left({\phi}^{s}{v}_{r}^{s}+\frac{r}{2}\dot{\epsilon}(t)\right),$$

(21)

$${v}_{r}^{f}=-\frac{D}{{c}^{f}}\frac{\partial {c}^{f}}{\partial r}+\left(1-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\right){v}_{r}^{s}-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\dot{\epsilon}(t)\frac{r}{2},$$

(22)

where the diffusion coefficients *D ^{wf}*,

$${D}^{\mathit{\text{wf}}}=\frac{{\phi}^{w}R\theta {c}^{f}}{{f}_{\mathit{\text{wf}}}},\phantom{\rule{thinmathspace}{0ex}}{D}^{\mathit{\text{sf}}}=\frac{{\phi}^{w}R\theta {c}^{f}}{{f}_{\mathit{\text{sf}}}},\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}k=\frac{{\phi}^{{w}^{2}}}{{f}_{\mathit{\text{ws}}}},$$

(23)

and

$$\frac{1}{D}=\frac{1}{{D}^{\mathit{\text{sf}}}}+\frac{1}{{D}^{\mathit{\text{wf}}}}.$$

(24)

The definition of *D ^{wf}*, the diffusivity of the solute in the solvent, is consistent with the classical treatment of diffusion [44] as will be shown below. The definition of permeability

$$\frac{\partial {c}^{f}}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left\{r\phantom{\rule{thinmathspace}{0ex}}\left[D\frac{\partial {c}^{f}}{\partial r}-\left(1-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\right){c}^{f}\frac{\partial {u}_{r}}{\partial t}+\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\frac{r}{2}{c}^{f}\dot{\epsilon}(t)\right]\right\}=0,$$

(25)

where we have substituted ${v}_{r}^{s}=\partial {u}_{r}/\partial t$ and assumed that ^{w} is constant, to produce a generalized Fick's equation which only depends on solute concentration and solid displacement. Equation (22) can also be rearranged to provide the molar flux (per unit solvent area) of the solute relative to the solid phase,

$${c}^{f}({v}_{r}^{f}-{v}_{r}^{s})=-D\frac{\partial {c}^{f}}{\partial r}-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}{c}^{f}\left({v}_{r}^{s}+\frac{r}{2}\dot{\epsilon}(t)\right).$$

(26)

To get the relative molar flux per unit mixture area, ${\tilde{c}}^{f}({v}_{r}^{f}-{v}_{r}^{s})$, multiply this equation by ^{w}. Using Eq.(20) in Eq.(14) similarly produces a differential equation which only depends on solute concentration and solid displacement,

$$\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial r}({\mathit{\text{ru}}}_{r})\right)-\left[\frac{1}{{H}_{A}k}+\left(1-\frac{D}{{D}^{\mathit{\text{wf}}}}\right)\frac{R\theta}{{H}_{A}}\frac{{c}^{f}}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\right]\left(\frac{\partial {u}_{r}}{\partial t}+\frac{r}{2}\dot{\epsilon}(t)\right)-\left(1-\frac{D}{{D}^{\mathit{\text{wf}}}}\right)\frac{R\theta}{{H}_{A}}\frac{\partial {c}^{f}}{\partial r}=0.$$

(27)

Thus, to obtain a final solution for this problem, the coupled system of differential equations in Eqs.(25) and (27) for *u _{r}*(

The boundary conditions for this problem need to be determined at *r* = 0 and *r* = *r*_{0}. Because of the axisymmetric conditions at *r* = 0, all velocities are zero at that location, and according to Eqs. (20)–(22), it can be noted that

$${u}_{r}(0,t)=0,\phantom{\rule{thinmathspace}{0ex}}{\frac{\partial p}{\partial r}|}_{r=0}=0,\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\frac{\partial {c}^{f}}{\partial r}|}_{r=0}=0.$$

(28)

At *r* = *r*_{0}, jump interface (or boundary) conditions must be satisfied to enforce continuity of mass, momentum and energy at the interface with the external solution [46–48], [[σ_{rr}]] = 0, [[^{w}]] = 0, and [[^{f}]] = 0, which lead to the relations

$${H}_{A}{\frac{\partial {u}_{r}}{\partial r}|}_{r={r}_{0}}+{\lambda}_{s}\left[\frac{{u}_{r}({r}_{0},t)}{{r}_{0}}+\epsilon (t)\right]=0,$$

(29)

$${c}^{f}({r}_{0},t)={\kappa}^{f}{c}^{{f}^{*}},$$

(30)

$$p={p}^{*}-R\theta (1-{\kappa}^{f}){c}^{{f}^{*}},$$

(31)

where *p*^{*} is the ambient pressure and *c*^{f*} is the external bath solute concentration. Since the mixture is assumed to be ideal, the solute activity coefficients are the same inside and outside the tissue, and thus the solubility is equal to the *partition factor* [2, 49]. In this study, it is assumed that the external bath is well mixed and that *c*^{f*} remains constant. The initial conditions can be derived in a similar manner, assuming that the tissue is at equilibrium with its initial environment, consisting of an external solute concentration of ${c}_{0}^{f}$ and ambient pressure *p*_{0}:

$${u}_{r}(r,0)=0,\phantom{\rule{thinmathspace}{0ex}}{c}^{f}(r,0)={\kappa}^{f}{c}_{0}^{f},\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}p(r,0)={p}_{0}.$$

(32)

If ${c}_{0}^{f}<{c}^{{f}^{*}}$, we would expect a net transport of solute into the tissue over time, whereas the converse would occur when ${c}_{0}^{f}>{c}^{{f}^{*}}$. Finally, the axial normal traction component and the total axial load also need to be evaluated,

$$\begin{array}{c}\hfill {\sigma}_{\mathit{\text{zz}}}(r,t)=-p(r,t)+{\lambda}_{s}\left(\frac{\partial {u}_{r}}{\partial r}+\frac{{u}_{r}}{r}\right)+{H}_{A}\epsilon (t),\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{\mathit{\text{zz}}}^{*}(r,t)=-{p}^{*}(r,t)\hfill \\ \hfill \text{and}\phantom{\rule{thinmathspace}{0ex}}W(t)=2\pi {\displaystyle {\int}_{0}^{{r}_{0}}r\left[{\sigma}_{\mathit{\text{zz}}}(r,t)-{\sigma}_{\mathit{\text{zz}}}^{*}\right]\mathit{\text{dr}}.}\hfill \end{array}$$

(33)

Classically, diffusion problems have been analyzed using Fick's first and second laws of diffusion. The simplest case of diffusion is represented by a two-phase mixture, consisting of single neutral solute diffusing in a solvent phase. Under these circumstances, since there is no gel solid matrix (^{s} = 0, ^{w} ≈ 1), *D* reduces to *D ^{wf}* and ${v}_{r}^{s}$ and (

$${c}^{f}{v}_{r}^{f}=-{D}^{\mathit{\text{wf}}}\frac{\partial {c}^{f}}{\partial r},$$

(34)

whereas Eq.(25) yields Fick's second law of diffusion (in cylindrical coordinates),

$$\frac{\partial {c}^{f}}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left(r{D}^{\mathit{\text{wf}}}\frac{\partial {c}^{f}}{\partial r}\right)=0.$$

(35)

This result confirms that mixture theory is consistent with classical laws.

If we next consider diffusion of a solute in a mixture with *rigid* porous solid matrix (^{s} ≠ 0), then ε(*t*) = 0 since no solid matrix deformation can take place. In general, ${v}_{r}^{s}$ need not be equal to zero if there is a rigid body motion, however Fick's laws are formulated for problems with no net convective motion, i.e., with ${v}_{r}^{s}=0$, in which case Eq.(26) reduces to

$${c}^{f}{v}_{r}^{f}=-D\frac{\partial {c}^{f}}{\partial r},$$

(36)

while Eq.(25) yields

$$\frac{\partial {c}^{f}}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left(\mathit{\text{rD}}\frac{\partial {c}^{f}}{\partial r}\right)=0.$$

(37)

In this case, Fick's laws are also recovered, with a diffusivity *D* instead of *D ^{wf}* to account for the presence of the porous solid matrix. From Eq.(24), and given that

Finally, if the gel solid matrix is deformable, but in the absence of dynamic loading ((*t*) = 0), the governing equations are found to differ in general from Fick's classical laws, with Eqs. (25)–(27) reducing to

$${c}^{f}({v}_{r}^{f}-{v}_{r}^{s})=-D\frac{\partial {c}^{f}}{\partial r}-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}{c}^{f}{v}_{r}^{s},$$

(38)

$$\frac{\partial {c}^{f}}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left(r\left[D\frac{\partial {c}^{f}}{\partial r}-\left(1-\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}\right){c}^{f}{v}_{r}^{s}\right]\right)=0,$$

(39)

$$\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial r}({\mathit{\text{ru}}}_{r})\right)-\frac{1}{{H}_{A}k}\frac{\partial {u}_{r}}{\partial t}+\frac{R\theta}{{H}_{A}{D}^{\mathit{\text{sf}}}}{c}^{f}({v}_{r}^{f}-{v}_{r}^{s})=0.$$

(40)

It is noteworthy that for the special case when *D* = ^{w} *D ^{wf}*, Eqs. (38)–(39) become uncoupled from the solid displacement and velocity, and we recover Fick's first and second laws as in Eqs. (36)–(37). For this special category of problems (most applicable to small solutes, as discussed below), the classical closed-form solution of the diffusion equation can be obtained for

In order to determine the dimensionless parameters governing the general solution of this problem, it is now necessary to non-dimensionalize the governing equations. If the following non-dimensional variables are proposed,

$$\widehat{r}=\frac{r}{{r}_{0}},\phantom{\rule{thinmathspace}{0ex}}\widehat{t}=\frac{D}{{r}_{0}^{2}}t,\phantom{\rule{thinmathspace}{0ex}}{\widehat{u}}_{r}=\frac{{u}_{r}}{{r}_{0}},\phantom{\rule{thinmathspace}{0ex}}{\widehat{c}}^{f}=\frac{R\theta D}{{H}_{A}{D}^{sf}}{c}^{f},\phantom{\rule{thinmathspace}{0ex}}\widehat{p}=\frac{p}{{H}_{A}},\phantom{\rule{thinmathspace}{0ex}}{\widehat{v}}_{r}^{\alpha}=\frac{{r}_{0}}{D}{v}_{r}^{\alpha},\phantom{\rule{thinmathspace}{0ex}}\widehat{W}=\frac{W}{{H}_{A}{r}_{0}^{2}},$$

(41)

the governing equations in Eqs.(25) and (27) reduce to

$$\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{t}}-\frac{1}{\widehat{r}}\frac{\partial}{\partial \widehat{r}}\left(\widehat{r}\left[\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{r}}-(1-{R}_{d}){\widehat{c}}^{f}\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{t}}+{R}_{d}\frac{\widehat{r}}{2}{\widehat{c}}^{f}\frac{d\epsilon}{d\widehat{t}}\right]\right)=0$$

(42)

$$\frac{\partial}{\partial \widehat{r}}\left(\frac{1}{\widehat{r}}\frac{\partial}{\partial \widehat{r}}(\widehat{r}{\widehat{u}}_{r})\right)-\left(\frac{1}{{R}_{g}}+{R}_{d}{\widehat{c}}^{f}\right)\left(\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{t}}+\frac{\widehat{r}}{2}\frac{d\epsilon}{d\widehat{t}}\right)-\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{r}}=0,$$

(43)

where the non-dimensional parameters *R _{g}* and

$${R}_{g}=\frac{{H}_{A}k}{D}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{R}_{d}=\frac{D}{{\phi}^{w}{D}^{\mathit{\text{wf}}}}.$$

(44)

The non-dimensional parameter *R _{g}* =

The pressure gradient, Eq.(20), and molar flux of solute relative to the solvent, Eq.(26), similarly reduce to

$$\frac{\partial \widehat{p}}{\partial \widehat{r}}=\left(\frac{1}{{R}_{g}}+{R}_{d}{\widehat{c}}^{f}\right)\left(\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{t}}+\frac{\widehat{r}}{2}\frac{d\epsilon}{d\widehat{t}}\right)+\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{r}}.$$

(45)

$${\widehat{c}}^{f}({\widehat{v}}_{r}^{f}-{\widehat{v}}_{r}^{s})=-\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{r}}-{R}_{d}{\widehat{c}}^{f}\left(\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{t}}+\frac{\widehat{r}}{2}\frac{d\epsilon}{d\widehat{t}}\right).$$

(46)

The boundary and initial conditions and the axial normal traction and load, when using non-dimensional variables, remain essentially unchanged,

$${\widehat{u}}_{r}(0,\widehat{t})=0,\phantom{\rule{thinmathspace}{0ex}}{\frac{\partial {\widehat{c}}^{f}}{\partial \widehat{r}}|}_{\widehat{r}=0}=0,\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\frac{\partial \widehat{p}}{\partial \widehat{r}}|}_{\widehat{r}=0}=0.$$

(47)

$${\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{r}}|}_{\widehat{r}=1}+\frac{{\lambda}_{s}}{{H}_{A}}\left[{\widehat{u}}_{r}(1,\widehat{t})+\epsilon (\widehat{t})\right]=0,\phantom{\rule{thinmathspace}{0ex}}{\widehat{c}}^{f}(1,\widehat{t})={\widehat{c}}^{{f}^{*}},\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\widehat{p}(1,\widehat{t})={\widehat{p}}^{*},$$

(48)

$${\widehat{u}}_{r}(\widehat{r},0)=0,\phantom{\rule{thinmathspace}{0ex}}{\widehat{c}}^{f}(\widehat{r},0)={\widehat{c}}_{0}^{f},\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\widehat{p}(\widehat{r},0)={\widehat{p}}_{0}.$$

(49)

$${\widehat{\sigma}}_{\mathit{\text{zz}}}(\widehat{r},\widehat{t})=-\widehat{p}(\widehat{r},\widehat{t})+\frac{{\lambda}_{s}}{{H}_{A}}\left(\frac{\partial {\widehat{u}}_{r}}{\partial \widehat{r}}+\frac{{\widehat{u}}_{r}}{\widehat{r}}\right)+\epsilon (t),\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\widehat{W}(\widehat{t})=2\pi {\displaystyle {\int}_{0}^{1}\widehat{r}{\widehat{\sigma}}_{\mathit{\text{zz}}}(\widehat{r},\widehat{t})d\widehat{r}}.$$

(50)

The parameters, *R _{g}* and

It is evident that the governing equations for this problem, Eqs. (42)–(43), are nonlinear so that a closed-form analytical solution is not available in the most general case. A numerical scheme must be employed to achieve a general solution.

For the general problem considered in this study, a numerical solution can be obtained for the set of nonlinear partial differential equations described above using a finite difference scheme. In this section, results are presented for various choices of the parameters governing the solution. First, a forcing function ε(*t*) is proposed for the case when dynamic loading is applied under displacement control,

$$\epsilon (t)=\frac{{\epsilon}_{0}}{2}(1-\text{cos}2\pi f\phantom{\rule{thinmathspace}{0ex}}t),$$

(51)

where ε_{0} is the peak-to-peak strain amplitude and *f* is the loading frequency. When solving in the non-dimensional domain, this equation reduces to

$$\epsilon \left(\widehat{t}\right)=\frac{{\epsilon}_{0}}{2}\left(1-\text{cos}2\pi \widehat{f}{R}_{g}\widehat{t}\right),$$

(52)

where the non-dimensional loading frequency is given by

$$\widehat{f}=\frac{{r}_{0}^{2}}{{H}_{A}k}f.$$

(53)

Note that is the loading frequency normalized by the characteristic frequency (the inverse of the biphasic theory's *gel time constant* [51]) for the flow of solvent in the porous matrix (e.g., [52, 53]). For the case when no dynamic loading is applied, it suffices to let = 0.

Second-order finite difference schemes were employed for spatial discretization of the governing equations. The spatial domain 0 ≤ ≤ 1 was discretized into 200 equally-sized increments Δ. In certain challenging cases, such as those with a high *R _{g}* and low

To achieve a solution in the non-dimensional domain requires the specification of ${R}_{g},\phantom{\rule{thinmathspace}{0ex}}{R}_{d},\phantom{\rule{thinmathspace}{0ex}}{\nu}_{s},\phantom{\rule{thinmathspace}{0ex}}{\widehat{c}}_{0}^{f},\phantom{\rule{thinmathspace}{0ex}}{\widehat{c}}^{{f}^{*}},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{0}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\widehat{f}$. To help determine characteristic values for these parameters, we look at some typical properties reported in the literature for articular cartilage and agarose gels, as representative examples for the current analysis. According to Table 2, the equilibrium aggregate modulus of articular cartilage in compression is typically in the range *H _{A}* = 0.1–1.0 MPa, whereas the permeability is on the order of

Aggregate modulus (*H*_{A}) and hydraulic permeability (*k*) for human and bovine articular cartilage and tissue engineered constructs. Values of *R*_{g} are given for a range of typical small (*D* = 1 × 10^{−9} m^{2}/s) and large (*D* = 1 × 10^{−11} **...**

Poisson's ratio, ν_{s} = 0, the initial solute concentration in the tissue, ${\widehat{c}}_{0}^{f}=0$, and the external bath concentration (scaled by the partition factor), κ^{f}*ĉ*^{f*} = 10^{−3}, were kept constant in this analysis. The parameters which were varied were *R _{g}* = 1–100,

Representative results for selected cases are presented first, followed by aggregate results for all tested cases. The solute concentration, *ĉ*^{f}(,), is shown as a function of the radial coordinate at various times in Figure 2a for the case *R _{g}* = 100,

$${\widehat{c}}_{\text{avg}}^{f}\left(\widehat{t}\right)=\frac{{\displaystyle {\int}_{0}^{1}2\pi \widehat{r}{\widehat{c}}^{f}\left(\widehat{r},\widehat{t}\right)d\widehat{r}}}{{\displaystyle {\int}_{0}^{1}2\pi \widehat{r}\phantom{\rule{thinmathspace}{0ex}}d\widehat{r}}},$$

(54)

can be evaluated and plotted as a function of time to indicate the rate at which solute is taken up by the whole tissue, as shown in Figure 3 for select choices of governing parameters. When the *R _{g}* and

Solute concentration, *ĉ*^{f}(, ), at select time points for the case *R*_{g} = 100, *R*_{d} = 0.1, (a) in the absence of dynamic loading (ε_{0} = 0), and (b) when ε_{0} = −0.20 and = 1000. **...**

Average solute concentration normalized by the external bath concentration, ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left(\widehat{t}\right)/{\kappa}^{f}{\widehat{c}}^{{f}^{*}}$, as a function of time, for various choices of governing parameters (*R*_{g} = 1, *R*_{d} = 1, ε_{0} = −0.20 and = 1000; *R*_{g} = 1, **...**

Average solute concentration normalized by the external bath concentration, ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left(\widehat{t}\right)/{\kappa}^{f}{\widehat{c}}^{{f}^{*}}$, as a function of time, for various choices of governing parameters (*R*_{g} = 100, *R*_{d} = 0.1, ε_{0} = −0.20, and = 0; *R*_{g} = **...**

For *R*_{d} = 0.1, 0.5, and 1.0, ε_{0} = −0.20, (a) steady-state value of ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left(\widehat{t}\right)/{\kappa}^{f}{\widehat{c}}^{{f}^{*}}$ (averaged over a loading cycle) as →∞, and (b) time *t*_{e} when ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left({\widehat{t}}_{e}\right)/{\kappa}^{f}{\widehat{c}}^{{f}^{*}}=1-{e}^{-1}$ **...**

For *R*_{d} = 0.1 and *R*_{g} = 100, and = 100, (a) transient value of ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left(\widehat{t}\right)/{\kappa}^{f}{\widehat{c}}^{{f}^{*}}$ versus with increasing strain magnitude (ε_{0} = 0 to −0.20) and (b) steady-state values of ${\widehat{c}}_{\mathit{\text{avg}}}^{f}\left(\widehat{t}\right)/{\kappa}^{}$ **...**

To help understand the solute transport mechanism, the molar flux of solute relative to the solid phase is plotted in Figure 7, for representative loading-free and dynamically loaded cases. In the loading-free case ( = 0), the solute transport is always directed into the tissue as indicated by its negative values, and is greatest in magnitude in the early time response, eventually decreasing to zero at equilibrium. In the dynamically loaded cases ( = 1000), the solute transport is directed into the tissue in the very early time response, with a magnitude much greater than the loading-free case; and the first cycle is seemingly independent of *R _{d}*. In subsequent cycles the solute molar flux relative to the solid phase becomes oscillatory, alternating between influx and efflux from the tissue. Interestingly, with

Solute molar flux relative to solid phase, ${\widehat{c}}^{f}({\widehat{v}}_{\mathrm{r}}^{f}-{\widehat{v}}_{\mathrm{r}}^{s})$, at = 1 over the first five loading cycles for the case *R*_{g} = 100, *R*_{d} = 1, in the absence of dynamic loading (ε_{0} = 0), and when ε_{0} = −0.20, **...**

Figure 8 shows a representative plot of the radial displacement *û _{r}*( = 1,) and the axial load

The objective of this study was to examine theoretically the effect of dynamic loading of a biological tissue or gel on solute transport, and to identify how the various non-dimensional parameters governing this problem affect its outcome. Using the theory of mixtures has provided a convenient framework to combine transport phenomena with porous media models of biological tissues. Indeed, under the proper limiting conditions, the general governing equations derived in this study reduce to Fick's laws of diffusion [44] or the linear isotropic biphasic theory of Mow and co-workers [34, 39]. A direct outcome of the analysis has been the formulation of familiar dimensional and non-dimensional parameters which govern the response. In particular, the analysis is able to distinguish between the solute diffusivity in the gel (*D*) and its diffusivity in the solvent (*D ^{wf}*) by taking into account frictional effects between solute and solvent (

The theoretical analysis also produces a non-dimensional parameter, *R _{g}* =

It is found that *R _{d}* and

In contrast to loading-free diffusion, under dynamic loading, the values of *R _{d}*,

Conversely, when 0.5 ≤ *R _{d}* ≤ 1, and

To understand why dynamic loading can enhance solute transport in a hydrated gel or soft tissue, it is necessary to examine the governing equations along with their numerical solutions. The molar flux of solute relative to the solid phase is given by Eq.(46), which shows solute transport is driven by the concentration gradient (*ĉ ^{f}*/) as in classical diffusion, as well as an additional term contributed by the dynamic loading of the gel solid matrix,

Based on the results of this study, it may be possible to estimate whether dynamic loading of tissue engineered constructs produces enhanced transport of growth factors that are present in or added to culture media. In our previous studies [29, 30], it was found that the equilibrium aggregate modulus of chondrocyte-seeded agarose gels increases from *H _{A}*~5 kPa to

This novel application of mixture theory to describe solute transport in dynamically loaded porous permeable gels may give new insight into the possible mechanisms by which physical stimuli modulate tissue response and enhance tissue development. The limitations of this first theoretical formulation, however, are in considering both the solute and gel in which it diffuses as neutrally charged. Serum growth factors, such as those from the transforming growth factor (TGF) and insulin like growth factor (IGF)/somatomedin family have isoelectric points that range from acidic to basic [64–72]. Molecules that are neutral in a physiologic environment will move independent of external charge, while those which carry a charge will be effected by the fixed charges of the matrix and the molecules around them [4, 73]. Thus the incorporation of fixed and movable charges into this mixture theory formulation may broaden its general application. Incorporating charge effects would require modeling the interstitial fluid with at least three species (a neutral solvent phase, an anionic solute phase and a cationic solute phase) [36, 37, 74–76], instead of the two employed in this study. Analyses of these more complex models may be easier to interpret given the results of the current study.

In addition to charge effects, the inhomogeneity of material properties in both cartilage and tissue engineered constructs may influence the local distribution of solute transport [77]. Modeling the tension-compression nonlinearity of cartilage, as performed in our earlier studies [78, 79], may also represent an additional refinement of this analysis. Finally, the model proposed herein does not take account of consumption of solute molecules (either by cellular metabolism or binding to the extracellular matrix; e.g., [80–83]), which may influence both transient and steady state solute concentration throughout the gel matrix. If solute binding or consumption is to be considered, equations for chemical reactions must be incorporated, and the balance of mass equations must be amended to incorporate mass generation (source or sink) terms. Incorporating the above modifications into this theory will increase its complexity, but also its utility both in understanding natural mechanotransduction mechanisms in native articular cartilage, as well its ability to predict the optimum mechanical environment for the *in vitro* functional tissue engineering of replacement tissues. This theoretical framework may be further extended to examine the influence of deformational loading on solute transport when engineered constructs are placed *in situ*, in which case transport across the articular surface may be of greater relevance.

This analysis also assumes that the volume fractions of the gel solid matrix and the solvent, the hydraulic permeability, and the solute diffusion coefficient remain constant under the small deformations being applied to the gel. It is possible to incorporate the higher order effect of deformation on these parameters using the conservation of mass relation for the volume fractions [36],

$${\phi}^{w}\approx {\phi}_{0}^{w}+\left(1-{\phi}_{0}^{w}\right)\text{tr}\mathbf{E},\phantom{\rule{thinmathspace}{0ex}}{\phi}^{s}\approx {\phi}_{0}^{s}\left(1-\text{tr}\mathbf{E}\right),$$

(55)

where ${\phi}_{0}^{w}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\phi}_{0}^{s}$ are the solvent and solid volume fractions in the absence of deformation. Similarly, constitutive relations can be formulated for the functional dependence of *k* and *D* on tr **E** [10, 45, 84, 85], for example,

$$k={k}_{0}{e}^{{M}_{k}\phantom{\rule{thinmathspace}{0ex}}\text{tr}\phantom{\rule{thinmathspace}{0ex}}\mathbf{E}},\phantom{\rule{thinmathspace}{0ex}}D={D}_{0}{e}^{{M}_{D}\phantom{\rule{thinmathspace}{0ex}}\text{tr}\phantom{\rule{thinmathspace}{0ex}}\mathbf{E}},$$

(56)

where *k*_{0} and *D*_{0} are the hydraulic permeability and solute gel diffusion coefficient in the absence of strain, and *M _{k}*,

In summary, this study presents an original theoretical analysis on the effect of dynamic loading on neutral solute transport in a neutral gel. It identifies the non-dimensional parameter *R _{d}* as an important governing parameter for this problem. It formulates the non-dimensional number

As a final remark, the theoretical results of this study are not uniquely relevant to natural and tissue engineered articular cartilage but apply to any porous hydrated tissue or gel. Dynamic loading is inherent to a variety of biological tissues (e.g., muscle, tendon, ligaments, cardiovascular system, etc.). Its effects need not be limited to extracellular matrix but may be equally relevant to transport at the cellular level [86]. It is therefore plausible to hypothesize that dynamic loading may serve to enhance solute transport in a variety of physiological processes, particularly in the case of larger molecular weight solutes. Such effects may be investigated using the theoretical framework provided in this study.

This study was supported with funds from the National Institutes of Health (AR46532, AR46568, AR43628) and a pre-doctoral fellowship from the Whitaker Foundation.

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