Modelling biological cell attachment and growth on adherent surfaces

Abstract

A mathematical model, in the form of an integro-partial differential equation, is presented to describe the dynamics of cells being deposited, attaching and growing in the form of a monolayer across an adherent surface. The model takes into account that the cells suspended in the media used for the seeding have a distribution of sizes, and that the attachment of cells restricts further deposition by fragmenting the parts of the domain unoccupied by cells. Once attached the cells are assumed to be able to grow and proliferate over the domain by a process of infilling of the interstitial gaps; it is shown that without cell proliferation there is a slow build up of the monolayer but if the surface is conducive to cell spreading and proliferation then complete coverage of the domain by the monolayer can be achieved more rapidly. Analytical solutions of the model equations are obtained for special cases, and numerical solutions are presented for parameter values derived from experiments of rat mesenchymal stromal cells seeded onto thin layers of collagen-coated polyethylene terephthalate electrospun fibers. The model represents a new approach to describing the deposition, attachment and growth of cells over adherent surfaces, and should prove useful for studying the dynamics of the seeding of biomaterials.

Appendix: Parameter value determination

Here explanation is given of how the values of the model parameters were determined; the resulting values are summarised in Table 2. Data was used from experiments of the culture of rat mesenchymal stromal cells (MSCs), taken from passage 5, seeded onto thin layers ($\lesssim 20 {\,\upmu \text{ m}}$ thick) of collagen-coated PET electrospun fibers lining the bottom of multi-well tissue culture plates. Briefly, aliquots comprising $16 \times 10^3$ cells suspended in 500 $\upmu$L media were placed in three wells containing the fibers and cultured for 50 h at 37$^\circ$C with 5 % $\text{ CO}_2$. At the end of the incubation the media was removed, the cell cytoskeleton stained with phalloidin, and fluorescence light microscopy images each 0.61 mm$^2$ in area were taken at ten uniformly spaced positions across the bottom of each well. The images were analysed using the CellProfiler software package. The MSCs were observed to spread readily over the fibers with cell proliferation confined largely within the plane of the plate as depicted in Fig. 1a. Further details of the experimental methods are provided in Gustafsson et al. (2012).

To measure the size distribution of the rat MSCs when they are suspended in media, cells were stained live using carboxyfluorescein diacetate succinimidyl ester (CDFA-SE) dye and seeded onto the bottom of a culture flask. Before the cells had time to settle and adhere to the flask, they were imaged with fluorescence light microscopy to obtain the two-dimensional projections of approximately $13\times 10^3$ cells. The projected areas of the cells were computed using CellProfiler however objects with areas less than $50\,{\,\upmu \text{ m}}^2$ or greater than $\text{1,400}\,{\,\upmu \text{ m}}^2$ were deemed to be artifacts and excluded from the analysis. The resulting size distribution of cells is plotted in Fig. 7a as a histogram of the number of cells with respect to area of the cells distributed into 100 bins.

Fig. 7

a Histogram of the experimental data showing the area distribution of the rat MSCs, and b the corresponding cumulative distribution of cell width obtained from the histogram together with the corresponding curve of $c(x)$ defined by (10) fitted to the experimental data

The cells were observed to have a mainly circular profile indicating a rounded morphology while in suspension. Since the model considers seeding of cells in one space dimension, the effective diameters of the cells was used as the measure of their initial width at the moment of deposition onto the fibers. Hence for a cell with cross-sectional area $A$ the corresponding cell width is $x=2\pi ^{-1/2}A^{1/2}$. Figure 7b shows the area data transformed into width expressed as a cumulative distribution (solid curve). The parameter values $x_0=22 \upmu$m and $\beta =6.8$ were obtained by using the method of least-squares to fit the function $c(x)$ given by (10) to the experimental data. The graph of the fitted function $c(x)$ appears as the dashed curve in Fig. 7b.

The macroscale distance $L$ was taken to be equal to the diameter of a single well of a 24-well plate i.e. 15.6 mm. Strictly the mesoscale distance, $l$, should be taken to be much larger than the typical distance between cells on the fibers after seeding is complete. However to facilitate comparison with the analytical solutions to the model equations presented in Sect. 4.1.1 a value of $l$ was chosen so that the non-dimensional median cell width, $x_0$, is equal to 1/3 hence $l=65 \upmu$m.

The value of the seeding rate parameter $\alpha$ was derived by calculating the average rate of sedimentation of cells in the media onto the fibers in the wells. There were initially $16\times 10^3$ cells in each well with cross-sectional area 190 mm$^2$ and that it was deemed that all of the cells had settled out of the media and attached to the fibers after 2 h. This enabled the seeding rate to be calculated appropriate for a two-dimensional surface, which was $4.2\times 10^{-5} \upmu \mathrm{m}^{-2}$ h$^{-1}$. The seeding rate appropriate for the one-dimensional domain of the model was derived by considering the rate at which cells strike a straight line lying on the two-dimensional seeded surface. Hence the two-dimensional seeding rate was multiplied by the average cell width determined from the size distribution data, $22 {\,\upmu \text{ m}}$, the result being $\alpha =9.4\times 10^{-4} \upmu \mathrm{m}^{-1}$ h$^{-1}$. Using this approach ensures that for a given cell-size distribution the seeding rate of the one-dimensional domain can be related proportionately to the seeding rate of a two-dimensional surface.

The rate of cell spreading, $s$, was approximated from the experimental data based on the assumption that the confluency of cells remains low. Indeed from three images of the cells on the fibers taken at the end of the 50 h incubation the confluency was determined to have a mean and standard deviation of $(34\pm 23)\,\%$. In Sect. 4.2.2 analytical expressions are derived to predict the total cell width as a function of time both during the seeding phase when cells are being deposited onto the domain, and after the seeding where there is no further cell deposition but the monolayer continues to grow. These are $w_a(t)$, given by (45), and $w_b(t)$ given by (48), respectively. Using these expressions the value of the parameter $\nu$ was determined by solving $w_b(T_f)=0.34$ to give $\nu =0.048$ which using (19) corresponds to a non-dimensional spreading rate of $s=0.19 {\,\upmu \text{ m}}$ h$^{-1}$.