Permeability of Bone and Cartilage, and Stiffness of Collagen within Cartilage, Influence Osteochondral Fluid Transport During Cyclic Compression: A Study in Finite Elements

Abstract

Osteochondral fluid transport likely plays a critical role in joint health and disease, yet the mechanical factors influencing this transport remain incompletely understood. This study established a finite element model of osteochondral fluid transport under cyclic compression, incorporating depth-dependent material properties and osmotic swelling. Using biphasic constitutive models for bone and cartilage, we simulated fluid flux across the osteochondral interface and performed a parametric sensitivity analysis of seven different mechanical properties. Results demonstrate that bone and cartilage permeability, as well as the stiffness of the collagen fiber network within cartilage, significantly affect net osteochondral fluid transport. Specifically, decreased cartilage permeability resulted in increased bone-to-cartilage ostechondral flow, and decreased collagen stiffness resulted in decreased net cartilage-to-bone fluid flow. Conversely, relatively high bone permeability reversed the direction of osteochondral flow. Other parameters, including bone modulus, bone solid volume fraction, cartilage shear modulus, and fixed charge density, had negligible effects. These findings highlight the importance of specific mechanical properties of both bone and cartilage in regulating osteochondral fluid transport and suggest that future studies should consider the complete osteochondral unit to better understand joint mechanobiology and osteoarthritis progression.

Graphical Abstract

1. Introduction

Low-friction articulation between opposing articular cartilage surfaces enables motion of synovial joints. Deterioration of articular cartilage is a hallmark of osteoarthritis, the most common degenerative joint disease. Articular cartilage is a multi-phase tissue containing substantial fluid, type II collagen, and proteoglycans. Underlying cartilage is subchondral bone containing type I collagen, hydroxyapatite mineral, and interstitial fluid. Joint loading induces cartilage deformation [1], and the mechanics (and mechanobiology) of both cartilage and bone depend on the flow of interstitial fluid.

Fluid transport within our joints plays an important role in joint mechanics and health [2–6]. In avascular articular cartilage, fluid transport provides nutrients to chondrocytes (cartilage cells), and the proteoglycan matrix slows this transport while supporting the large compressive loads seen during daily joint loading [7]. High proteoglycan content within the cartilage matrix confers low hydraulic permeability and high osmotic swelling pressures, slowing fluid movement through cartilage [8]. Load induces fluid flow in bone [9] and this flow is important for bone cell mechanotransduction [10]. While many studies examine fluid flow in either cartilage or bone, relatively few studies examine fluid flow across the cartilage-bone interface.

A tidemark separates the deep zone of articular cartilage from calcified cartilage, and the cement line separates calcified cartilage from the subchondral bone. Fluid flow across this interface has potential to carry important biological signals between bone and cartilage. Experimental studies find that osteochondral fluid flux occurs in healthy adult joint tissue [2–4, 11]. Previous simulations support these results [5] finding that osteochondral fluid flux increases with osteoarthritic changes. Simulations also suggest that cartilage mechanics are minimally affected by the subchondral bone changes observed in early osteoarthritis [12]. Yet there is little data regarding how changes in subchondral bone affect fluid flow between bone and cartilage. Our previous computational model [13] demonstrated that fluid pressure gradients develop at the osteochondral interface, and that bone likely plays the role of a pressure sink for the high fluid pressure that develops in articular cartilage during mechanical loading. This combination of experimental and computational results demonstrates the importance of fluid flow across the osteochondral interface. However, how this osteochondral fluid flow changes as a function of health and disease, i.e., changes in material parameters, remains unknown.

The objectives of this study are to 1) establish a osteochondral fluid transport model incorporating cyclic compression and 2) quantify net osteochondral fluid transport as a function of changes in material parameters of bone and cartilage. Because these parameters change in osteoarthritis, our approach captures how osteoarthritis progression may impact osteochondral fluid transport. Using well-established biphasic models of bone and cartilage, we simulated published cyclical loading of osteochondral explants [2]. To determine the sensitivity of osteochondral flow, we performed parametric sweeps of seven key cartilage and bone properties. Results show that the permeability of bone and cartilage, and the stiffness of collagen within cartilage, affect osteochondral fluid transport. When extended to the area of a whole tibia [14], these results suggest that every 1000 cycles of load (e.g. walking) induces approximately 28 μL of osteochondral fluid transport, fluid moving from the underlying bone into the cartilage.

2. Methods

2.1. Constitutive Models

We use the theory of porous media to describe both bone and cartilage as a biphasic continuum $\phi ={\phi }^{\mathrm{S}}+{\phi }^{\mathrm{F}}$ which consists of a porous solid phase ${\phi }^{\mathrm{S}}$ saturated with the interstitial fluid phase ${\phi }^{\mathrm{F}}$ [15, 16]. The volume fractions $n^{\alpha }$ refer the volume elements $\mathrm{d}{v}^{\alpha }$ of the individual constituents ${\phi }^{\alpha }$ to the bulk volume element $\mathrm{d}v$ with $n^{\alpha }(\mathbf{x},t)=\mathrm{d}{v}^{\alpha }/\mathrm{d}v,\sum _{\alpha =1}^k{n}^{\alpha }(\mathbf{x},t)=\sum _{\alpha =1}^k{\rho }^{\alpha }/{\rho }^{\alpha \mathrm{R}}=1,\alpha \in \{\mathrm{S},\mathrm{F}\}$, where $\mathbf{x}$ is the position vector of the spatial point (reference position $\mathbf{X}$), $t$ is the time, and S and F denote the solid and fluid, respectively. The partial density ${\rho }^{\alpha }=n^{\alpha }{\rho }^{\alpha \mathrm{R}}$ of the constituent ${\phi }^{\alpha }$ is related to the real density of the materials ${\rho }^{\alpha \mathrm{R}}$ involved via the volume fractions $n^{\alpha }$ (with initial volume fractions $n_{0\mathrm{S}}^{\alpha }$).

We calculate the total Cauchy stress within bone (§2.1.1) or cartilage (§2.1.2) as

$$\mathit{\sigma }=-p\mathbf{I}+{\mathit{\sigma }}_{\mathrm{E}}^{\mathrm{S}}=-p\mathbf{I}+2{\rho }^{\mathrm{S}}{\mathbf{F}}_{\mathrm{S}}\frac{\partial {\mathrm{\Psi }}^{\mathrm{S}}}{\partial {\mathbf{C}}_{\mathrm{S}}}{\mathbf{F}}_{\mathrm{S}}^{\mathrm{T}},$$ (1)

where $p$ is the fluid pore pressure, ${\mathit{\sigma }}_{\mathrm{E}}^{\mathrm{S}}$ is the effective Cauchy stress, $\mathbf{I}$ is the identity tensor, ${\mathbf{F}}_{\mathrm{S}}=\partial {\mathbf{x}}_{\mathrm{S}}/\partial {\mathbf{X}}_{\mathrm{S}}$ is the deformation gradient of the solid, and ${\mathbf{C}}_{\mathrm{S}}={\mathbf{F}}_{\mathrm{S}}^{\mathrm{T}}{\mathbf{F}}_{\mathrm{S}}$ is the right Cauchy-Green tensor.

To model the corresponding permeation of interstitial fluid, we introduce the seepage velocity ${\mathbf{w}}_{\mathrm{F}\mathrm{S}}$, which describes the difference in velocity between the fluid phase ${\mathbf{x}}_{\mathrm{F}}^{\prime }$ and the solid phase ${\mathbf{x}}_{\mathrm{S}}^{\prime }$ as ${\mathbf{w}}_{\mathrm{F}\mathrm{S}}={\mathbf{x}}_{\mathrm{F}}^{\prime }-{\mathbf{x}}_{\mathrm{S}}^{\prime }$. We determine the filtration velocity $n^{\mathrm{F}}{\mathbf{w}}_{\mathrm{F}\mathrm{S}}$ as $n^{\mathrm{F}}{\mathbf{w}}_{\mathrm{F}\mathrm{S}}={\mathbf{K}}_{\mathrm{F}}\left(-\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}p+{\rho }^{\mathrm{F}\mathrm{R}}\mathbf{b}\right)$, where ${\mathbf{K}}_{\mathrm{F}}$ is the intrinsic permeability of the solid matrix of either bone (§2.1.1) or cartilage (§2.1.2), and $\mathbf{b}$ is the body force per unit mass [17].

2.1.1. Bone

We model bone as a 3-D biphasic orthotropic, elastic continuum with relatively low permeability. We use the solid Helmholtz free-energy function for orthotropic elasticity, an extension of the St. Venant-Kirchhoff model [18] provided in FEBio [19], as

$${\mathrm{\Psi }}^{\mathrm{S}}=\frac{1}{{\rho }_{0\mathrm{S}}^{\mathrm{S}}}\left[\sum _{a=1}^3{\mu }_a{\mathbf{A}}_a^0:{\mathbf{E}}^2+\frac{1}{2}\sum _{b=1}^3{\lambda }_{ab}\left({\mathbf{A}}_a^0:\mathbf{E}\right)\left({\mathbf{A}}_b^0:\mathbf{E}\right)\right],$$ (2)

where ${\mu }_a$ are three shear moduli, ${\lambda }_{ab}$ are six moduli with ${\lambda }_{ab}={\lambda }_{ba},\mathbf{E}$ is the Green Lagrange strain-tensor, and ${\mathbf{A}}_a^0={\mathbf{a}}_a^0\otimes {\mathbf{a}}_a^0$ is a structure tensor representing mutually orthogonal planes of symmetry. Note that ${\mu }_a$ and ${\lambda }_{ab}$ relate directly to the Young’s moduli $E_a$, shear moduli $G_{ab}$, and Poisson’s ratios ${\nu }_{ab}$ via a simple transformation [19]. Since the strains in the bone during mechanical loading are very small, we define ${\mathbf{K}}_{\mathrm{F}}$ using Darcy’s law as

$${\mathbf{K}}_{\mathrm{F}}=k_{0\mathrm{S}}\mathbf{I},$$ (3)

where $k_{0\mathrm{S}}\left[{\mathrm{m}}^4/\mathrm{N}\mathrm{s}\right]$ is the initial Darcy permeability.

We provide model parameters for healthy subchondral, cortical bone in Table 1 [20, 21].

Table 1:

Model parameters for healthy bone [20, 21].

Parameter	Value	Unit
${\rho }^{\mathrm{F}\mathrm{R}}$	9.9e-7	kg/mm3
${\rho }^{\mathrm{S}\mathrm{R}}$	1.5e-6	kg/mm3
$E_1$	17.7	GPa
$E_2$	$0.574E_1$	GPa
$E_3$	$0.577E_1$	GPa
$G_{12}$	$0.216E_1$	GPa
$G_{23}$	$0.195E_1$	GPa
$G_{31}$	$0.265E_1$	GPa
$n_{0\mathrm{S}}^S$	0.945	—
$k_{0\mathrm{S}}$	1.0e-5	mm4/N s

2.1.2. Cartilage

We model cartilage as a 3-D biphasic anisotropic and heterogeneous elastic continuum incorporating the mechanical effects of an isotropic proteoglycan matrix with osmotic swelling arising from a negative fixed charge and an enmeshed network of reinforcing collagen fibers, all together generating a low and anisotropic permeability. We use an additive decomposition of the superimposed solid Helmholtz free-energy function ${\mathrm{\Psi }}^{\mathrm{S}}$ into a Donnan osmotic pressure part ${\mathrm{\Psi }}_{\mathrm{O}\mathrm{P}}^{\mathrm{S}}$, an isotropic matrix part ${\mathrm{\Psi }}_{\mathrm{I}\mathrm{M}}^{\mathrm{S}}$, and a fiber network part ${\mathrm{\Psi }}_{\mathrm{F}\mathrm{N}}^{\mathrm{S}}$ as

$${\mathrm{\Psi }}^{\mathrm{S}}={\mathrm{\Psi }}_{\mathrm{O}\mathrm{P}}^{\mathrm{S}}\left(J_{\mathrm{S}}\right)+(1-\nu ){\mathrm{\Psi }}_{\mathrm{I}\mathrm{M}}^{\mathrm{S}}\left(J_{\mathrm{S}},I_1\right)+\nu {\mathrm{\Psi }}_{\mathrm{F}\mathrm{N}}^{\mathrm{S}}\left({\mathbf{C}}_{\mathrm{S}}\right),$$ (4)

where $\nu$ is the volume fraction of collagen to the total solid, $J_{\mathrm{S}}=\mathrm{d}\mathrm{e}\mathrm{t}{\mathbf{F}}_{\mathrm{S}}$ is the Jacobian and $I_1=\mathrm{t}\mathrm{r}{\mathbf{C}}_{\mathrm{S}}$ is the first invariant of ${\mathbf{C}}_{\mathrm{S}}$.

To capture Donnan osmotic pressure we define the molar concentrations of dissolved ions and fixed charges as $c_{\mathrm{m}}^{\gamma }:=\mathrm{d}{N}^{\gamma }/\mathrm{d}{v}^{\mathrm{F}}$, with $\gamma =\{+,-,\mathrm{f}\mathrm{c}\}$, where ${\bar{c}}_{\mathrm{m}}^{-}={\bar{c}}_{\mathrm{m}}^{+}=:{\bar{c}}_{\mathrm{m}}$ is the ion concentration of the external solution (here treated as a model parameter and not a boundary condition), $c_{\mathrm{m}}^{-}=c_{\mathrm{m}}^{+}-c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}$ corresponds to the internal solution. Furthermore, the concentration of the fixed charges depends on the deformation, i.e., $c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}=c_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}\left(1-n_{0\mathrm{S}}^{\mathrm{S}}\right)/\left(\mathrm{d}\mathrm{e}\mathrm{t}{\mathbf{F}}_{\mathrm{S}}-\right.\left.n_{0\mathrm{S}}^{\mathrm{S}}\right)$ where $n_{0\mathrm{S}}^{\alpha }$ are initial volume fractions and $c_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}$ is the initial concentration of fixed charges. Thus, we capture the osmotic strain energy as [22, 23]

$${\mathrm{\Psi }}_{\mathrm{O}\mathrm{P}}^{\mathrm{S}}=R\mathrm{\Theta }{c}_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}{n}_{0\mathrm{S}}^{\mathrm{F}}\left[\frac{2{\bar{c}}_{\mathrm{m}}}{c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}}-\frac{\sqrt{4{\left({\bar{c}}_{\mathrm{m}}\right)}^2+{\left(c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}\right)}^2}}{c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}}+\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{c_{\mathrm{m}}^{\mathrm{f}\mathrm{c}}}{2{\bar{c}}_{\mathrm{m}}}\right)\right],$$ (5)

where $R=8.314\left({\mathrm{M}\mathrm{P}\mathrm{a}\mathrm{m}\mathrm{m}}^3\right)/(\mathrm{K}\mathrm{m}\mathrm{o}\mathrm{l})$ is the universal gas constant and $\mathrm{\Theta }$ is the absolute temperature. We capture the strain energy of the (largely) proteoglycan solid matrix using a neo-Hookean strain-energy function for ${\mathrm{\Psi }}_{\text{IM}}^{\mathrm{S}}$ [24] with ${\mu }^{\mathrm{S}}$ corresponding to the shear modulus of the underlying matrix in the reference configuration (Lamé’s second parameter), which we extend to include compaction effects [17, 25, 26] (not shown).

We capture the anisotropic, nonlinear response of the dispersed network of collagen using $\rho (\mathbf{M})$ as the angular density of fibers (the Orientation Distribution Function or ODF) so that [27, 28] ${\int }_{\mathrm{\Omega }}\rho (\mathbf{M})\mathrm{d}\mathrm{\Omega }/(4\pi )=1$, where $\mathrm{\Omega }=\mathbf{M}\in {\mathbb{R}}^3:\left|\mathbf{M}\right|=1$ is the unit sphere. For a single fiber of reference orientation $\mathbf{M}$ the fourth pseudo-invariant $I_4$ is the square of the stretch of this fiber in the direction $\mathbf{m}={\mathbf{F}}_{\mathrm{S}}\mathbf{M}$, i.e. $I_4(\mathbf{M})={\lambda }^2(\mathbf{M})=\mathbf{M}\cdot {\mathbf{C}}_{\mathrm{S}}\mathbf{M}$. We capture the strain energy of the dispersed collagen network as

$${\mathrm{\Psi }}_{\mathrm{F}\mathrm{N}}\left(\mathbf{C},\mathbf{M}\right)=\frac{1}{{\rho }_{0\mathrm{S}}^{\mathrm{S}}}{\int }_{\mathrm{\Omega }}\rho \left(\mathbf{M}\right)\frac{k_1}{2k_2}\left\{\exp \left[k_2{\left(I_4-1\right)}^2\right]-1\right\}\mathscr{H}\left(I_4-1\right)\mathrm{d}\mathrm{\Omega },$$ (6)

where $k_1>0$ is a material parameter, $k_2>0$ is a dimensionless parameter, and $\mathscr{H}$ is a Heaviside step function evaluated at $\left(I_4-1\right)$, i.e., collagen fibers engage only under tension $(\lambda >1)$. Since permeation of interstitial fluid within cartilage is least restricted in the direction parallel to the fibers [29, 30] we define ${\mathbf{K}}_{\mathrm{F}}$ as [17, 23]

$${\mathbf{K}}_{\mathrm{F}}=\frac{k_{0\mathrm{S}}}{4\pi }{\left(\frac{n^{\mathrm{F}}}{1-n_{0\mathrm{S}}^{\mathrm{S}}}\right)}^m{\int }_{\mathrm{\Omega }}\frac{\rho (\mathbf{M})}{I_4(\mathbf{m})}\mathbf{m}\otimes \mathbf{m}\mathrm{d}\mathrm{\Omega },$$ (7)

where $m$ is a dimensionless parameter controlling the general isotropic deformation dependence of the permeability.

We provide model parameters for healthy cartilage in Table 2 [17, 23]. We specified that collagen fibers within the superficial zone of cartilage (between the surface and transition to middle zone, i.e., 1 to 0.85 normalize depth) were well-aligned and parallel to the articular surface, within the middle zone (between 0.85 and 0.30) were oriented randomly (isotropic), and within the deep zone (between 0.3 and 0, the interface of the subchondral bone) were well-aligned and oriented normal to the subchondral bone. We implemented depth-dependent model parameters using a custom Python script.

Table 2:

Model parameters for healthy cartilage [17, 23]. The parameter z* ∈ [0, 1] is the normalized tissue thickness from the osteochondral interface (0) to articular surface (1).

Parameter	Value	Unit
${\rho }^{\mathrm{F}\mathrm{R}}$	9.9e-7	kg/mm3
${\rho }^{\mathrm{S}\mathrm{R}}$	1.5e-6	kg/mm3
${\mu }^{\mathrm{S}}$	0.23	MPa
$k_1$	3.0	MPa
$k_2$	8.0	—
$n_{0\mathrm{S}}^{\mathrm{S}}$	0.3 – 0.15z*	—
$\nu$	0.43z*2 − 0.26z* + 0.68	—
$J_{\text{cp}}^{\mathrm{S}}$	0.41	—
$k_{0\mathrm{S}}$	0.1e-3 + (0.9e-3) z*	mm4/N s
$m$	8.0 – 5.0z*	—
$\mathbf{D}$	(see text)	—
$c_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}$	2.0e-7	mol/mm3
${\bar{c}}_{\mathrm{m}}$	1.5e-7	mol/mm3

2.2. Boundary Value Problem

2.2.1. Model Geometry

We created a symmetry-slice model of the cylindrical, osteochondral explant from a femoral condyle using 20-node hexahedral elements in FEBio [19], see Fig. 1. Specifically, we established a 1◦ symmetry-slice of a cylinder with height 2 mm (1 mm each of cartilage and bone) and with a radius of 4 mm [2]. We assumed that the osteochondral interface was smooth due to the scale difference between the surface roughness of the interface (~30 μm) and the overall model dimensions. To ensure reliable results we completed standard mesh convergence studies to verify the axial fluid pressure throughout the model and the axial fluid flux across the cartilage-bone interface. Our final finite element mesh comprised one element in the circumferential direction (with collapsed nodes at the radial center), twenty elements in the axial direction within both bone and cartilage (0.9 bias towards the bone interface within bone, no bias within cartilage), and 64 elements in the radial direction (0.93 bias towards the radial free edge).

Figure 1:

Model geometry and boundary conditions.

2.2.2. Boundary Conditions and Solution Routine

We simulated 120 minutes of unconfined cyclic compression and calculated the net fluid flux across the osteochondral interface. We applied symmetry boundary conditions to the two symmetry “cut” slices in the axial-radial planes, i.e., no displacement and no fluid flux normal to those surfaces. We fixed all nodes on the bottom surface of the subchondral bone in all degrees of freedom and set no fluid flux normal to this surface. On the radial surface of the model we set no displacement constraints and set the fluid pressure to zero ($p=0$), allowing for radial fluid flux (Fig. 1).

We assumed that the initial configuration was stress-free and allowed the model to osmotically swell for the first second of our simulation and then allowed the model to come to equilibrium, with no other additional loading, in the following 999 seconds [31]. Next, we compressed our model to 5% strain in one second, after which we applied compressive displacements to the nodes of the articular surface equal to strains oscillating between 3% and 7% (5% ± 2%) sinusoidal compression at 1.1 Hz for 120 minutes to match our previous work [2]. We used the full Newton nonlinear implicit solver in quasi-static mode within FEBio in conjunction with adaptive time stepping via the auto time stepper option.

2.3. Sensitivity Analyses

We systematically varied the model parameters of both bone and cartilage, specifically the parameters known to vary with the progression of OA [32—34], to investigate the impact on net fluid transport across the osteochondral interface. Specifically, we investigated three bone parameters (moduli, initial solid volume fraction, and permeability) and four cartilage parameters (shear modulus, permeability, and fixed charge of the matrix; and fiber stiffness), see Table 3. Each parameter range included six parameter values, centered from the baseline. We adjusted the functions $k_{0\mathrm{S}}$ (initial permeability) and $c_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}$ (initial concentration of fixed charges) by shifting the depth-dependent profiles. We adjusted each parameter independently, and did not consider the effects of interactions because of the large combinatorial space (7⁷ models). Note that cartilage shear modulus decreases with age and OA and that subcondral cortical bone elastic modulus increases with age [32].

Table 3:

Range of model parameters considered in our sensitivity analyses.

	Parameter	Baseline	Values	Unit
bone	$E_1$	17.7	5.0, 10, 15, 20, 25, 30	GPa
	$n_{0\mathrm{S}}^{\mathrm{S}}$	0.945	0.91, 0.92, 0.93, 0.96, 0.97, 0.98	-
	$k_{0\mathrm{S}}$	1e-5	1e-7, 5e-7, 1e-6, 1e-4, 5e-4, 1e-3	mm4/N s
cartilage	${\mu }^{\mathrm{S}}$	0.23	0.080, 0.15, 0.20, 0.30, 0.35, 0.40	MPa
	$k_1$	3.0	0.50, 1.0, 2.0, 4.0, 5.0, 5.5	MPa
	$k_{0\mathrm{S}}$	1.00e-4	4.75e-5, 7.25e-5, 9.75e-5, 1.48e-4, 1.73e-4, 1.98e-4	mm4/N s
	$c_{0\mathrm{S}}^{\mathrm{f}\mathrm{c}}$	2.00e-7	0.920e-7, 1.17e-7, 1.42e-7, 1.92e-7, 2.17e-7, 2.42e-7	mol/mm3

2.4. Analyses of Osteochondral Fluid Transport

We extracted element-area fluid fluxes from each of the 64 radial (one circumferential row) elements at the bone and cartilage interface (instantaneous fluid fluxes) and summed these to determine the total fluid flux across the bone-cartilage interface at each time step. To determine the cumulative fluid transfer over time at the osteochondral interface (specifically at 15, 30, 60, and 120 minutes) we completed numerical integration using a custom Python script. To better understand the distributions of maximum (7% compression)/minimum (3%) fluid pressures and axial displacements during cyclic loading we plotted the distributions of z-displacements and fluid pressures using FEBio Studio [19]. We defined positive fluid flux as that moving from bone into cartilage.

3. Results

In Fig. 2 we provide distributions in fluid pressures for our baseline model representing the normal, healthy osteochondral explant.

Figure 2:

Distributions of fluid pressures within the normal, healthy osteochondral explant: A) after osmotic swelling and after equilibrium, B) at maximum compressive strain (7% compression) during cyclic compression, and C) at minimum compressive strain (3% compression) during cyclic compression. D) Bulk strain versus time with labels referring to time points A, B, and C.

In Fig. 3 we provide the net fluid flux across the osteochondral interface for 15, 30, 60, and 120 minutes of cyclic compression (5% ± 2% sinusoidal compression at 1.1 Hz) at a function of the seven model parameters considered in our sensitivity analysis.

Figure 3:

Cumulative fluid transfer over time across the osteochondral interface for 15, 30, 60, and 120 minutes of cyclic compression (5% ± 2% sinusoidal compression at 1.1 Hz) as a function of: A) bone modulus, B) bone initial solid volume fraction, C) bone permeability, D) cartilage shear modulus, E) cartilage fiber modulus, F) shifts in cartilage initial permeability, and G) shifts in cartilage fixed charge density, and H) sign convention for osteochondral flow.

3.1. Parameters Impacting Osteochondral Fluid Transport

Bone permeability profoundly impacts net osteochondral fluid flux, with fluxes differing from the baseline model within the first 15 minutes of cyclic compression, see Fig. 3A. We also see that if the bone permeability decreases below 1 ×10⁻⁵ mm⁴/N s, net osteochondral fluid flux remains negligible even after 120 minutes of cyclic compression. The stiffness of collagen fibers within the cartilage also impacts fluid flux, especially after 120 minutes of cyclic compression, see Fig. 3B. Osteochondral fluid transport was net cartilage-to-bone for all stiffnesses of the collagen fibers considered in this analysis. Finally, the initial Darcy permeability of cartilage impacts fluid flux, see Fig. 3C.

In Fig. 4 we provide the distributions of fluid pressure at maximum and minimum strains during cyclic compression for values of the bone permeability within our sensitivity analyses, cf. Table 3. We found that for bone permeabilities <1 ×10⁻⁵ mm⁴/N s, the maximum fluid pressure varied from 0.863 to 0.864 to 1.1 MPa, but the increased pressure did not did not lead to increased net fluid flux.

Figure 4:

Distributions of fluid pressures from sensitivity analyses of bone permeability at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate bone permeability with the fraction of the baseline value in parentheses

In Fig. 5 we provide the distributions of fluid pressure at maximum and minimum strains during cyclic compression for values of the collagen fiber stiffness in cartilage within our sensitivity analyses, cf. Table 3. We found drops in fluid pressurization within cartilage with decreasing collagen fiber network stiffness (e.g., from 0.628 to 0.274 MPa), likely driving decreased ostoechondral fluid flux.

Figure 5:

Distributions of fluid pressures from sensitivity analyses of collagen fiber stiffness within cartilage at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate collagen fiber stiffness with the fraction of the baseline value in parentheses

In Fig. 6 we provide the distributions of fluid pressure at maximum and minimum strains during cyclic compression for values of the permeability in cartilage within our sensitivity analyses, cf. Table 3. We found that decreasing the initial Darcy permeability of cartilage led to increased fluid flux from cartilage-to-bone with initial increases around 60 minutes and larger increases after 120 minutes of cyclic compression. We also note that increasing the permeability of cartilage decreased cartilage fluid pressurization, whereas decreasing permeability significantly increased fluid pressurization within cartilage.

Figure 6:

Distributions of fluid pressures from sensitivity analyses of permeability in cartilage at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate cartilage permeability with the fraction of the baseline value in parentheses

3.2. Parameters Not Impacting Osteochondral Fluid Transport

Modulus and initial solid volume fraction of bone, and shear modulus and fixed charge density of cartilage, showed a negligible impact on net osteochondral fluid transport within the range of values used in our sensitivity analyses. In Figs. A1–A4 we provide the distributions of fluid pressure at maximum and minimum strains during cyclic compression for values of the bone modulus, and bone initial solid volume fraction, cartilage shear modulus, cartilage fixed charge density, within our sensitivity analyses, cf. Table 3.

4. Discussion

This model provides novel data on how cartilage and bone mechanical properties affect osteochondral fluid transport during cyclic compression. Osteochondral fluid transport may drive progression of OA [2–5, 13], but its role in healthy adult joints is likely underappreciated. However, several recent studies [2–5], suggest that osteochondral fluid transport occurs in healthy adult joints, not just during the development and progression of OA. Previous computational modeling demonstrates that gradients in fluid pressure develop at the osteochondral interfaces [13], and simulated mechanical changes associated with OA lead to increases in osteochondral fluid flux [5]. We chose strain oscillating between 3% and 7% to match our previous experimental work [2].

Our present model provides several advances, including: 1) incorporating osmotic swelling and depth-dependent material parameters and 2) accounting for temporal changes in fluid flux during cyclic compression. The data show that cartilage fiber network moduli and cartilage and bone permeability impact osteochondral fluid transport, and that osteochondral fluid transport is net cartilage-to-bone for all models except for those with the highest values of bone permeability. Our current results are consistent with previous work and demonstrate the importance of several mechanical properties that directly impact osteochondral fluid transport [2, 3, 5].

4.1. Parameters Impacting Osteochondral Fluid Transport

Bone permeability, cartilage permeability, and stiffness of the collagen network within cartilage all affected osteochondral fluid transport. Previous experimental studies show that bone stiffness increases with OA [35] and that bone initial solid volume fraction and the thickness of the subchondral bone plate impact bone-cartilage fluid transport [11]. However, bone permeability is distinct from volume fraction because of many factors including vasculature and porosity within the lacunar-canulicular system, and more research is needed to understand the role of bone permeability in osteochondral flow. Previous modeling studies show that fluid pressure gradients develop at the osteochondral interface [13] and that osteochondral fluid transport increases in a simulated OA joint [5]. Cartilage permeability directly influences net osteochondral fluid flux during cyclic compression, see Fig. 3F. Decreased cartilage permeability leads to increased fluid pressurization, see Fig. 6, and thus increased net cartilage-to-bone fluid flux, see Fig. 3.

Our results are consistent with the known role of permeability in pressurization of the cartilage fluid and provide a critical insight into how subchondral bone sees an increase in fluid flux from cartilage when cartilage is under greater pressure or has lower permeability. Cartilage permeability decreases with GAG loss in OA, emphasizing the importance of this finding. Decreasing the cartilage fiber network moduli decreases the net osteochondral fluid transport from cartilage-to-bone, see Fig. 3E. Mechanistically, decreasing collagen modulus decreases the fluid pressures, see Fig. 5, leading to less flux. Previous studies find large decreases in cartilage fiber modulus in advanced OA [36, 37], while in early OA studies show increases in disorganization of the fiber network, especially in the superficial and deep zones [36–39].

4.2. Parameters Not Impacting Osteochondral Fluid Transport

Osteochondral fluid transport was not affected by variations in bone modulus, bone solid volume fraction, cartilage shear modulus, or cartilage fixed charge density. Prior work suggests that subchondral bone progressively stiffens during the progression of OA [35], and it is well documented that bone initial solid volume fraction increases during the onset and progression of OA [40]. There are depth-dependent decreases in cartilage shear modulus with progression of OA [41, 42]. Loss of proteoglycans in the superficial zone—a hallmark of early progression of OA—leads to a reduction in fixed charge density [34, 43]. Such effects are likely coupled with reduced cartilage shear modulus and potentially a loss of cartilage fiber network integrity. While varying these parameters did not affect osteochondral flow in this study, it is possible that these parameters may interact with other parameters to affect osteochondral flow, which would be of interest in future studies.

4.3. Limitations and Outlook

This study provides insight into how the mechanical properties of bone and cartilage impact osteochondral fluid transport, but important limitations remain. Our model only considers univariate changes in mechanical properties; thus, interactions between parameters remain unknown. We did not consider interactions between parameters because of the large combinatorial space (7⁷ models). Our model doesn’t account for full unloading of the osteochondral unit, which may occur during in vivo joint loading. Direct interpretation of the absolute flux across the osteochondral interface is not possible due to numerical issues at the radial surface of the model. Finally, direct comparison of our model results to our experimental studies is challenging due to measurement limitations in the experimental systems. Our computational model allows us to determine the flux across the bone-cartilage interface directly, whereas we used the cartilage fluid chamber of our bioreactor to measure flux from bone to cartilage.

These data show how variations in mechanical parameters affect osteochondral flow. However, several key gaps remain. To our knowledge, only 3 kDa or smaller neutral molecule sizes have been tested in experimental osteochondral transport studies. Physiologically relevant molecules can be upwards of 500 kDa in size and have different physicochemical properties, so future studies should broaden the range of tracer molecules. Finally, computational models should be expanded to account for diffusion and tracer physicochemical properties.

In summary, we found that specific mechanical properties of bone and cartilage impact osteochondral fluid transport and net cartilage-to-bone fluid transport. These results suggest that future experimental and computational studies of joint fluid transport should consider the integrated osteochondral unit. These results improve our understanding of the mechanical properties driving osteochondral fluid transport. Expanding this knowledge has the potential to advance our understanding of osteochondral flow, which may result in new strategies for preventing the onset and progression of OA.

Appendix A: Additional Distributions of Fluid Pressures from Sensitivity Analyses

Figure A1:

Distributions of fluid pressures from sensitivity analyses of bone modulus at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate bone modulus with the fraction of the baseline value in parentheses

Figure A2:

Distributions of fluid pressures from sensitivity analyses of Bone initial solid volume fraction at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate initial bone solid volume fraction with the fraction of the baseline value in parentheses

Figure A3:

Distributions of fluid pressures from sensitivity analyses of cartilage shear modulus at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate cartilage shear modulus with the fraction of the baseline value in parentheses

Figure A4:

Distributions of fluid pressures from sensitivity analyses of cartilage fixed charge density at minimum strain (3% compression, left column) and maximum strain (7% compression, right column). Subpanel titles indicate cartilage fixed charge density with the fraction of the baseline value in parentheses