How changes in interconnectivity affect the bulk properties of articular cartilage: a fibre network study

Abstract

The remarkable compressive strength of articular cartilage arises from the mechanical interactions between the tension-resisting collagen fibrils and swelling proteoglycan proteins within the tissue. These interactions are facilitated by a significant level of interconnectivity between neighbouring collagen fibrils within the extracellular matrix. A reduction in interconnectivity is suspected to occur during the early stages of osteoarthritic degeneration. However, the relative contribution of these interconnections towards the bulk mechanical properties of articular cartilage has remained an open question. In this study, we present a simple 2D fibre network model which explicitly represents the microstructure of articular cartilage as collection of discrete nodes and linear springs. The transverse stiffness and swelling properties of this fibre network are studied, and a semi-analytic relationship which relates these two macroscopic properties via microscopic interconnectivity is derived. By comparing this derived expression to previously published experimental data, we show that although a reduction in network interconnectivity accounts for some of the observed changes in the mechanical properties of articular cartilage as degeneration occurs, a decrease in matrix interconnectivity alone do not provide a full account of this process.

Appendix: Derivation of non-affinity coefficients

Consider a 3D network composed of $N_{x} \times N_{y} \times N_{z}$ nodes, where $N_{x}$, $N_{y}$ and $N_{z}$ are the number of nodes in the x (transverse), y (radial) and z (out-of-plane) direction, respectively. We will assume that the nodes are regularly ordered in a repeating cubic lattice; nodes in the x, y and z direction are equally separated from one another by a distance of $L_{x}$, $L_{y}$ and $L_{z}$ $\mu$m, respectively. We shall further assume that nodes immediately adjacent to one another are connected by springs of stiffness $k_{x}$ $\upmu \hbox {N} \times \hbox {strain}^{-1}$ (not $\upmu$N $\times$ displacement$^{-1}$) if they are orientated in the x direction, $k_{y}$ $\upmu \hbox {N} \times \hbox {strain}^{-1}$ if they are orientated in the y direction, or $k_{z}$ $\upmu \hbox {N} \times \hbox {strain}^{-1}$ if they are orientated in the z direction. For the purposes of visualisation, this hypothetical network is schematically shown in Fig. 13.

Fig. 13

A three-dimensional cubic nodal network showing the orientation of the x, y and z coordinate axes. Given that both the x and z directions are parallel to the articulating surface, both are described as transverse directions. Conversely, the y coordinate axis is described as radial

To derive expressions for the transverse stiffness and swelling potential of this network, we will utilise two results. First, given the regular cubic configuration of the network, the network will deform affinely during the mechanical tests being considered. Hence, the entire network can be treated as an effective spring composed of smaller springs in series and parallel. Second, given that the stiffness of each spring is in units $\upmu \hbox {N} \times \hbox {strain}^{-1}$, we can deduce that the total stiffness of n identical springs of stiffness k $\upmu \hbox {N} \times \hbox {strain}^{-1}$ in series and parallel is k $\upmu \hbox {N} \times \hbox {strain}^{-1}$ and $n\times k$ $\upmu \hbox {N} \times \hbox {strain}^{-1}$, respectively. Consequently, the bulk effective spring constant K (i.e. the force exerted by the entire network per unit strain) associated with this regular cubic network in each direction is:

$$\begin{aligned} K^{\text {Affine}}_{x} = N_{y}\,N_{z}\,k_{x} \end{aligned}$$

(29)

$$\begin{aligned} K^{\text {Affine}}_{y} = N_{x}\,N_{z}\,k_{y} \end{aligned}$$

(30)

$$\begin{aligned} K^{\text {Affine}}_{z} = N_{x}\,N_{y}\,k_{z} \end{aligned}$$

(31)

These bulk spring constants give the force exerted by the network per unit strain in a given direction. To determine the transverse Young’s modulus associated with the network $E^{\text {Affine}}_\mathrm{T}$, we must divide these bulk spring constants by the cross-sectional surface area of the network perpendicular to the x direction $A_{x}$:

$$\begin{aligned} E^{\text {Affine}}_\mathrm{T} = \frac{K^{\text {Affine}}_{x}}{A_{x}} = \frac{N_{y}\,N_{z}\,k_{x}}{L_{y}(N_{y} - 1)\times L_{z}(N_{z} - 1)} \end{aligned}$$

(32)

To account for non-affine interactions, we shall modify Eq. (29) by introducing the transverse non-affinity coefficient $\alpha _\mathrm{T}$ such that:

$$\begin{aligned} K_{x} = \alpha _\mathrm{T} \times N_{y}\,N_{z}\,k_{x} = \alpha _\mathrm{T} \times K^{\text {Affine}}_{x} \end{aligned}$$

(33)

where $K_{x}$ is the bulk spring constant associated with an equivalent non-affine network.

Hence, the transverse Young’s modulus of a network which deforms non-affinely $E_\mathrm{T}$ can be written as:

$$\begin{aligned} E_\mathrm{T} = \alpha _{x} \times \frac{N_{y}\,N_{z}\,k_{x}}{L_{y}(N_{y} - 1)\times L_{z}(N_{z} - 1)} \end{aligned}$$

(34)

Suppose that the bulk transverse spring constant of a 2D fibre network is computed to be $K^{\text {2D}}_\mathrm{T}$ $\upmu \hbox {N} \times \hbox {strain}^{-1}$. If we assume that this 2D bulk spring constant is equal to the bulk spring stiffness of an equivalent 3D network averaged over each layer of nodes in the z direction (i.e. $K^{\text {2D}}_\mathrm{T}$ = $K_\mathrm{T}$/$N_{z}$), then the non-affinity coefficient $\alpha$ can be evaluated from this arbitrary 2D network by rearranging Eq. (33) for $\alpha _\mathrm{T}$:

$$\begin{aligned} \alpha _\mathrm{T} = \frac{K^{\text {2D}}_\mathrm{T}}{N_{z}}\times \frac{1}{N_{y}\,k_{x}} = \frac{K^{\text {2D}}_\mathrm{T}}{N_{y}\,k_{x}} \end{aligned}$$

(35)

Furthermore, by considering the limit $N_{z} \rightarrow \infty$ (i.e. the network is semi-infinite in the out-of-plane direction), Eq. (34) become:

$$\begin{aligned} E_\mathrm{T} = \alpha _\mathrm{T} \times \frac{N_{y}\,k_{x}}{L_{y}(N_{y} - 1)\times L_{z}} \end{aligned}$$

(36)

From this expression, the transverse Young’s modulus of a 2D network which deforms non-affinely may be calculated.

With regard to the free swelling test, we can write both the initial $V_{0}$ and the post-swelling volume V of the network as:

$$\begin{aligned} V_{0}= & {} L_{x}\,L_{y}\,L_{z} \end{aligned}$$

(37)

$$\begin{aligned} V= & {} L_{x}(1+\epsilon _{x})\,L_{y}(1+\epsilon _{y})\,L_{z}(1+\epsilon _{z}) \end{aligned}$$

(38)

where $\epsilon _{x}$, $\epsilon _{y}$ and $\epsilon _{z}$ are the strains experienced by the network in the x, y and z direction in response to the swelling pressure, respectively. These strains can be evaluated by dividing the applied swelling force by the bulk spring constant of the network in the appropriate direction:

$$\begin{aligned} \epsilon _{x} = \frac{F^{\text {Swell}}_{x}}{K^{\text {Affine}}_{x}} = \frac{F^{\text {Swell}}_{x}}{N_{y}\,N_{z}\,k_{x}} \end{aligned}$$

(39)

$$\begin{aligned} \epsilon _{y} = \frac{F^{\text {Swell}}_{y}}{K^{\text {Affine}}_{y}} = \frac{F^{\text {Swell}}_{y}}{N_{x}\,N_{z}\,k_{y}} \end{aligned}$$

(40)

$$\begin{aligned} \epsilon _{z} = \frac{F^{\text {Swell}}_{z}}{K^{\text {Affine}}_{z}} = \frac{F^{\text {Swell}}_{z}}{N_{x}\,N_{y}\,k_{z}} \end{aligned}$$

(41)

where $F^{\text {Swell}}_{x}$, $F^{\text {Swell}}_{y}$ and $F^{\text {Swell}}_{z}$ are the swelling forces in the x, y and z directions, respectively. These forces are simply the swelling pressure multiplied by the surface area of the network normal to the appropriate coordinate axis:

$$\begin{aligned} F^{\text {Swell}}_{x} = P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1) \end{aligned}$$

(42)

$$\begin{aligned} F^{\text {Swell}}_{y} = P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1) \end{aligned}$$

(43)

$$\begin{aligned} F^{\text {Swell}}_{z} = P\,L_{x}(N_{x}-1)\,L_{y}(N_{y} - 1) \end{aligned}$$

(44)

Substituting Eqs. (42), (43) and (44) into Eqs. (39), (40) and (41), respectively, yields:

$$\begin{aligned} \epsilon _{x} = \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}} \end{aligned}$$

(45)

$$\begin{aligned} \epsilon _{y} = \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}} \end{aligned}$$

(46)

$$\begin{aligned} \epsilon _{z} = \frac{P\,L_{x}(N_{x}-1)\,L_{y}(N_{y} - 1)}{N_{x}\,N_{y}\,k_{z}} \end{aligned}$$

(47)

Hence, the post-swelling volume of the network V becomes:

$$\begin{aligned} V^{\text {Affine}}= & {} L_{x}\bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \nonumber \\&\times \, L_{y}\bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \nonumber \\&\times \, L_{z}\bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \end{aligned}$$

(48)

The fractional change in volume $\varDelta V$ for an affine network is then:

$$\begin{aligned} \begin{aligned} \varDelta V^{\text {Affine}} =&\,\frac{V^{\text {Affine}} - V_{0} }{V_{0}} \\ =\,&\, (1+\epsilon _{x})(1+\epsilon _{y})(1+\epsilon _{z}) - 1 \\ =\,&\, \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \\&\times \, \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\&\times \, \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) - 1 \end{aligned} \end{aligned}$$

(49)

We shall now introduce a swelling non-affinity coefficient $\alpha _\mathrm{S}$ such that:

$$\begin{aligned} \begin{aligned} \varDelta V + 1 =\,&\, \alpha _\mathrm{S}\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \\&\times \, \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\&\times \, \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\ =\,&\, \alpha _\mathrm{S}\times (\varDelta V^{\text {Affine}} + 1) \end{aligned} \end{aligned}$$

(50)

Dividing both sides of Eq. (50) by the $(1 + \epsilon _{z})$ term, we reach:

$$\begin{aligned} \begin{aligned} \frac{\varDelta V + 1}{1 + \epsilon _{z}} =\,&\, \alpha _\mathrm{S}\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \\&\times \, \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\ = \,&\, \alpha _\mathrm{S}\times \frac{\varDelta V^{\text {Affine}} + 1}{1 + \epsilon _{z}} \end{aligned} \end{aligned}$$

(51)

By rearranging Eq. (49) for $(\varDelta V^\mathrm{Affine} + 1)/(1+\epsilon _{z})$, we observe that:

$$\begin{aligned} \frac{\varDelta V^{\text {Affine}} + 1}{1 + \epsilon _{z}} = (1+\epsilon _{x})(1+\epsilon _{y}) = \varDelta A^{\text {Affine}} \end{aligned}$$

(52)

where $A^{\text {Affine}}$ is the fractional change in the area of the network in the xy plane. Hence, Eq. (51) becomes:

$$\begin{aligned} \begin{aligned} \frac{\varDelta V + 1}{1 + \epsilon _{z}} =\,&\, \alpha _\mathrm{S}\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \\&\times \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\ =\,&\, \alpha _\mathrm{S}\times (\varDelta A^{\text {Affine}} + 1) \end{aligned} \end{aligned}$$

(53)

If we assume that the computed fraction change in area of a 2D fibre network in the xy plane $\varDelta A$ is approximately equal to the fractional volume change of an equivalent 3D network normalised with respect to the fractional change in length in the out-of-plane direction or, equivalently, that the change in area after swelling for each layer of springs in the xy plane for the 3D network are independent of one another (i.e. [$\varDelta V + 1$]/[$1 + \epsilon _{z}$] =   $\varDelta A + 1$), we can write Eq. (53) as:

$$\begin{aligned} \begin{aligned} \varDelta A + 1 =&\, \alpha _\mathrm{S}\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}(N_{z} - 1)}{N_{y}\,N_{z}\,k_{x}}\bigg ) \\&\times \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}(N_{z} - 1)}{N_{x}\,N_{z}\,k_{y}}\bigg ) \\ =&\, \alpha _\mathrm{S}(\varDelta A^{\text {Affine}} + 1) \end{aligned} \end{aligned}$$

(54)

As we did previously, we shall now take the limit $N_{z} \rightarrow \infty$ (i.e. the network is semi-infinite in the out-of-plane direction) to show that:

$$\begin{aligned} \begin{aligned} \varDelta A + 1 =\,&\, \alpha _\mathrm{S}\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}}{N_{y}\,k_{x}}\bigg ) \\&\times \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}}{N_{x}\,k_{y}}\bigg ) \\ =\,&\, \alpha _\mathrm{S}(\varDelta A^{\text {Affine}} + 1) \end{aligned} \end{aligned}$$

(55)

The swelling non-affinity coefficient associated with a 2D network which deforms non-affinely can, therefore, be determined from the swelling potential of the network, which is calculated by the fibre network solver, and the swelling potential of an equivalent 2D affine network, which can be calculated analytically from:

$$\begin{aligned} \varDelta A^{\text {Affine}} =\,&\bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}}{N_{y}\,k_{x}}\bigg ) \end{aligned}$$

(56)

$$\begin{aligned}&\times \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}}{N_{x}\,k_{y}}\bigg ) \end{aligned}$$

(57)

Once both $\varDelta A$ and $\varDelta A^{\text {Affine}}$ are known, the swelling non-affinity coefficient may be evaluated using:

$$\begin{aligned} \begin{aligned} \alpha _\mathrm{S} =\,&\frac{\varDelta A + 1}{\varDelta A^{\text {Affine}} + 1} \\ =\,&(\varDelta A + 1)\times \bigg (1 + \frac{P\,L_{y}(N_{y}-1)\,L_{z}}{N_{y}\,k_{x}}\bigg )^{-1} \\&\times \bigg (1 + \frac{P\,L_{x}(N_{x}-1)\,L_{z}}{N_{x}\,k_{y}}\bigg )^{-1} \end{aligned} \end{aligned}$$

(58)