A Hybrid Model of Cartilage Regeneration Capturing the Interactions Between Cellular Dynamics and Porosity

Abstract

To accelerate the development of strategies for cartilage tissue engineering, models are necessary to investigate the interactions between cellular dynamics and the local microenvironment. We use a discrete framework to capture the individual behavior of cells, modeling experiments where cells are seeded in a porous scaffold or hydrogel and over the time course of a month, the scaffold slowly degrades while cells divide and synthesize extracellular matrix constituents. The movement of cells and the ability to proliferate is a function of the local porosity, defined as the volume fraction of fluid in the surrounding region. A phenomenological approach is used to capture a continuous profile for the degrading scaffold and accumulating matrix, which will then change the local porosity throughout the construct. We parameterize the model by first matching total cell counts in the construct to chondrocytes seeded in a polyglycolic acid scaffold (Freed et al. in Biotechnol Bioeng 43:597–604, 1994). We investigate the influence of initial scaffold porosity on the total cell count and spatial profiles of cell and ECM in the construct. Cell counts were higher at day 30 in scaffolds of lower initial porosity, and similar cell counts were obtained using different models of scaffold degradation and matrix accumulation (either uniform or cell-specific). Using this modeling framework, we study the interplay between a phenomenological representation of scaffold architecture and porosity as well as the potential continuous application of growth factors. We determine parameter regimes where large cellular aggregates occur, which can hinder matrix accumulation and cellular proliferation. The developed modeling framework can easily be extended and can be used to identify optimal scaffolds and culture conditions that lead to a desired distribution of extracellular matrix and cell counts throughout the construct.

Appendices

Appendix

Details on Simulation Algorithm

1.1 Cell Discretization, Initialization and Boundary Conditions

Cells are discrete entities described as circles with constant diameter $c_d$ with a surface covering 48 grid points spread around the cell center. At time zero, cells are randomly positioned on the domain $\varOmega$ to avoid overlap, and cells are initialized with a random age in the interval [0, 3] days. Note that we use the same cell initialization for all simulations and that cell locations are independent of the computational grid for the continuous variables.

In Fig. 1b, the computational domain $\varOmega$ is highlighted in dark gray. Since cells tend to stay within the bio-construct, we assume a no-flux boundary condition on all edges of the domain. If the chosen direction would place a cell outside of the domain, the trajectory is corrected before the cell moves to keep the cell inside the domain. The correction is performed on the coordinate that would exceed the computational domain by reversing the chosen movement direction and reducing its magnitude by a factor of ten. The boundary of the computational domain is padded with several layers of ghost grid points initialized with a value of $p=0$ for porosity. Therefore, when a cell is close to the boundary, if part of the region depicted in Fig. 12 covers any ghost node, that specific direction will be automatically penalized by the direction rules, (7) and (8), to minimize the chances that a cell will exit the domain or will divide outside the domain.

1.2 Moving Cells

For each cell that is in the moving state, i.e., $S_i^{j+1}={\mathcal {M}}$, the angle $\theta _m$ in Eq. (1) needs to be chosen to update the new cell center location. As shown in Fig. 12, the cell will move in nine possible directions denoted by: $\mathbf {\varnothing }$, E, NE, N, NW, W, SW, S, SE. The first direction corresponds to the small probability of staying in the same exact position if the cell is in the moving state whereas the other directions correspond to the cardinal and intercardinal directions.

Fig. 12

Schematic of a cell (shaded gray circle in center) on the computational domain where the eight cardinal and intercardinal directions (E, NE, N, NW, W, SW, S, SE) are shown. The circle on each grid box corresponds to the node value tracking the average porosity and nutrient concentration on that particular volume element. The porosity values are assessed in the direction of no movement $\mathbf {\varnothing }$, on the nodes included in the shaded gray area, and in the eight cardinal and intercardinal directions, on the nodes contained in the annulus surrounding the cell. The inner diameter of the annulus corresponds to $c_d$ while the outer diameter of the annulus $a_{d,\mathrm{mov}}$ is reported in Table 1 (Color figure online)

Fig. 13

Example of direction vector for cell movement. The width of each bin is computed following (7)

In order to determine the cell’s movement direction, the average porosity $p_{\mathrm{avg},k}$ is calculated on the nodes of the computational grid covered by the cell (direction $k=\mathbf {\varnothing }$) and in the eight neighborhoods exterior to the cell, as depicted in Fig. 12. The outer diameter of the annulus $a_{d,\mathrm{mov}}$ represents how far the cell is able to detect gradients in porosity, and it is chosen in this work to be six times the baseline cell movement ($6d_\mathrm{cell}=3c_d$) in one iteration. If the region k contains a grid point with porosity less than $\epsilon _{p}$, which corresponds to an unfavorable region since there is not enough room for the cell, we set $p_{\mathrm{avg},k}=0$.

The average values of porosity are then normalized in a probability direction vector where higher porosity will correspond to a higher probability in the porosity direction vector. The width of each bin $\omega _{p,k}$, as shown in Fig. 13, corresponds to the probability of moving in each of the directions and is computed as

$$\begin{aligned} \omega _{p,k}=\displaystyle \dfrac{p_{\mathrm{avg},k}}{\sum _{\ell =\mathbf {\varnothing }}^{SE} p_{\mathrm{avg},\ell }} \qquad k=\mathbf {\varnothing },E,\ldots ,SE. \end{aligned}$$

(7)

The movement direction of each cell is determined by choosing a distinct random variable X from a uniform distribution U[0, 1] and determining which direction bin contains $\texttt {X}$. We note that wide bins ${\omega }_{p,k}$ have a higher probability of containing X and have larger width due to a more favorable porosity (larger $p_{\mathrm{avg},k}$). As an example of how the direction is chosen, if X=0.35 and the bin sizes are ${\omega }_{\mathbf {\varnothing }}=0.1$, ${\omega }_E=0.1$, and ${\omega }_{NE}=0.2$, X will fall in the NE bin (since $0.1+0.1+0.2=0.4$), therefore the corresponding cell will move in the NE direction or $\theta _m=\pi /4$. We use the convention that the chosen cell direction k is the first direction to satisfy the following inequality: X$<\left( \sum _{j=\varnothing }^{k}{\omega }_j\right)$ for $k={\mathbf {\varnothing }},E,\ldots ,SE$. We note that if X$=1$, this will correspond to $k=SE$. At a given time step, all cells will simultaneously move in the chosen direction k by the corresponding angle $\theta _m$ and a fixed distance $d_\mathrm{cell}$ (reported in Table 1). Note that if cell movement results in a local porosity with $p>1$, which may occur on occasion when multiple cells have moved into the same location, this issue is resolved by randomly choosing to undo the movement of half of these cells, until the situation is resolved.

1.3 Dividing Cells

To identify the cells that could potentially divide, we first determine the average porosity and nutrient concentration, $p_\mathrm{avg}$ and $c_\mathrm{avg}$, of the nodes in between the cell and the outer annulus with radius of $a_{d,div}$ (refer to Fig. 12 for a schematic). The cells that have the potential to divide (passing conditions listed in Sect. 2.1.1, i.e., $p_\mathrm{avg}>\epsilon _p$ and $c_\mathrm{avg}>c_\mathrm{div}$) undergo an overcrowding check to ensure that there is room for the new daughter cell. If an area is not overcrowded, the cell will divide. A crowded region is defined as dividing cells with center-to-center distance less than $d_\mathrm{div}$ apart. For the cells in crowded regions, half are chosen at random and are allowed to undergo cell division where the new daughter cells are the same size as the dividing cell. This check prevents too many cells that are close to each other from dividing during the same iteration. The new cell is placed adjacent to the mother cell along a direction $\theta _d$ that is a biased random function of the local porosity. The procedure used to determine the division direction vector is similar to the one followed to generate the porosity direction vector. The number of possible directions is reduced from 8 to 6 (E, NE, NW, W, SW, SE, each with an arc of $\pi /3$). We emphasize that this reduced number of regions is utilized based on a spatial argument; the six regions are able to contain the full surface of the new cell and this reduces overlap of daughter cells from nearby cells that are proliferating. The width of the six bins is

$$\begin{aligned} \omega _{d,k}=\displaystyle \dfrac{\exp (100p_{\mathrm{avg},k})-1}{\sum _{\ell =E}^{SE} \exp (100p_{\mathrm{avg},\ell })-1} \qquad k=E,NE,\ldots ,SE, \end{aligned}$$

(8)

and the width of a bin is set to zero if the corresponding region contains a grid point with porosity such that $p<\epsilon _{p}$. Similarly, we follow the same procedure used for cell movement where we choose a uniform variable X for each mother cell and find the bin and direction that will determine the position of the new (daughter) cell. The region where the average porosity is calculated, similar to the one depicted in Fig. 12, has outer diameter $a_{d,div}$. We choose $a_{d,div}=3.75c_d$ (i.e., outer radius of $1.875c_d$) since a gap of one grid point between the mother cell and the new cell is needed to avoid overlaps during cell division. Thus, the new daughter cell will be located at a radial distance of $5c_d/4$ in the chosen direction where the 5/4 ensures that these cells are close but will not automatically go into the collision state. We emphasize that Eq. (8) provides a stronger bias than (7) resulting in a high probability that the new cell will be placed in a region of high porosity.

1.4 Discretization, Initialization and Boundaries for Continuous Variables

The porosity and nutrient are continuous variables that are calculated on a computational grid with a step size $\triangle x=\triangle y=c_d/8$, i.e., the grid is considerably smaller than the cellular diameter to guarantee an accurate description of the continuous quantities and convergence of numerical results. This ensures that cells are able to bias motion and cell division in regions where there is a rapid transition in the magnitude of the porosity and/or nutrient. Since the porosity is $p=1-\varPhi$, we must compute each of the solid volume fractions on the grid. Figure 14a shows a cell volume calculation where there is partial overlap of cells on the grid and 14b shows how the final solid volume fraction is determined, accounting for scaffold and ECM solid volume fractions. As described in Sect. 2.2.1, we assume a nutrient profile representative of a fixed nutrient concentration at the right and bottom portion of the construct that is exposed to the nutrient-rich medium and no-flux boundary conditions are assumed at the top and left side of the construct. The initial nutrient profile is high through the entire construct. For the porosity, we are solving differential equations for ECM and scaffold as given in Sect. 2.2.2 at each of the grid points and hence we do not apply boundary conditions. We assume that the scaffold is a given solid volume fraction at time zero, $\varPhi _{SC}^0$, with values based on experiments (Freed et al. 1994; Erickson et al. 2009). We also assume zero ECM is formed at the start of the experiment, $\varPhi _{ECM}^0=0$.

Fig. 14

Examples of solid volume fractions for different components in the model. a The cellular solid volume fraction $\varPhi _\mathrm{cell}$ is shown for the case where we assume each cell contributes ${\widehat{\varPhi }}=0.3$ to the solid volume fraction. b Total solid volume fraction ${\varPhi =\varPhi _\mathrm{cell}+\varPhi _{SC}+\varPhi _{ECM}}$ for the same cellular locations with $\varPhi _{SC}+\varPhi _{ECM}=0.1$. The domain $\varOmega$ assumes a thickness that allows for partial overlap of cells and for scaffold and/or extracellular matrix around the cells

Cell Aggregates

We use the built in function cluster in MATLAB to determine the size and distribution of cell aggregates within the construct. An example of a small- and large-cell aggregate is given in Fig. 15. In this work, a cell aggregate is defined as a group of three or more cells where the center-to-center distance to a subset of cells in the aggregate is less than $1.75c_d$. This threshold value is selected to account for cells shapes that might not be perfectly spherical (Buxton et al. 2007; Hauselmann et al. 1992; Kino-oka et al. 2005). This choice of distance and the minimum count of three cells per aggregate also avoids that a cell and its daughter cell will be considered as an aggregate.

Fig. 15

Examples of the shape of small (a, 4 cells) and large (b, 11 cells) cellular aggregates within the construct (Color figure online)

Aleatory Uncertainty Analysis

The model contains several stochastic elements that introduce aleatory uncertainty in each simulation. It is therefore necessary to perform an uncertainty quantification procedure to determine the minimum number of simulations to account for the full variation of the system (Alden et al. 2013; Cosgrove et al. 2015; Read et al. 2012). First, 20 sets of k simulations each with the same parameters are gathered. Then, sets 2–20 are compared with set 1 using the A-Test developed by Vargha and Delaney (2000). The A-Test returns an A-score for each comparison which represents the probability that a randomly selected sample from the first population is larger than a random sample from the second population. The A-score for the comparison between set a and b, $A_{a,b}$ is computed following the approximation proposed in Vargha and Delaney (2000) as:

$$\begin{aligned} A_{a,b}=\dfrac{\#(X_i>Y_j)}{k_a k_b} \, + \, 0.5\, \dfrac{\#(X_i=Y_j)}{k_a k_b}, \qquad i=1,\ldots ,k_a; \, j=1,\ldots ,k_b \end{aligned}$$

where $\#(X_i>Y_j)$ counts the number of times the event $(X_i>Y_j)$ happens for all the $X_i$ in set a and $Y_j$ in set b and similarly $\#(X_i=Y_j)$ counts the number of times the event $(X_i=Y_j)$ happens, and $k_a$ and $k_b$ are the dimension of set a and b, respectively. The A-score is used to determine statistical significance as follows: a score below 0.29 or above 0.71 indicates a large effect of sample size k on the model results; a score of 0.36 or 0.64 indicates a medium effect of k on model results; a score within 0.44 and 0.56 indicates a small effect of k on model results, and a score of 0.5 indicates no effect of k on model results. Figure 16 shows the results obtained for sample size $k=300$ considering the final number of cells as the variable of interest, for simulations with the baseline parameters in Table 1 with $\varPhi _{SC}^0=0.01$. Twenty groups with $k=$300 distinct simulations are obtained and then the A-score $A_{1,j}$ is computed for $j=1,\ldots ,20$ and reported in Fig. 16. The results obtained, with all of the A-scores within the [0.44, 0.56] interval, show that a value of 300 simulations is sufficient to mitigate the uncertainty introduced by the stochasticity of the model. The correct application of the A-Test is validated by the first point in Fig. 16, where the comparison of sample 1 with itself results in an A-score $A_{1,1}=0.5$ as expected from the literature.

Fig. 16

A-score $A_{1,j}$ for comparison of set 1 with set $j=1,\ldots ,20$ on the variable final number of cells in simulations with $\varPhi _{SC}^0=0.01$ and parameters from Table 1 (Color figure online)