The Mathematical Model

The mathematical model presented here is a nonlinear system of differential equations describing the growth rate of the cell population with concentration $C$, and the growth rate of the extracellular matrix population with concentration $E$. The model assumes that the growth rate of the cell population depends on (1) the type of cells studied, (2) the type of the artificial surface, (3) the amount of cells present at a given time $t$, and (4) the interaction between the cells and the extracellular matrix. As far as point (3) is concerned, our model incorporates the well-known fact that the more cells are present, the slower the cell growth rate. In regard to point (4) the interaction between the cells and ECM is a very complex one. Our simple model, however, uses some general observations based on our experiments and on the results published in [3,1,5], that indicate two things: (1) cells cannot proliferate and generate the extracellular matrix at the same time, and (2) there is a negative feedback mechanism influencing the ECM molecules deposition regulation.

The following system captures the main features of this cell-ECM dynamics:

$$\frac{d C}{dt} = \alpha - \delta C + f(E) C$$ (2.1)

$$\frac{d E}{dt} = - f(E) C$$ (2.2)

where

$$f(E) = \gamma E - \frac{\beta}{h+E}.$$ (2.3)

Here $\alpha$ is the constant growth rate of a given cell population, $\delta$ is a constant that describes the decrease in the cell population production due to the already present cell concentration, and the term $f(E)C$ denotes a function that describes the interaction between the cell and ECM populations. More precisely, if we consider the numerator in the expression for $f(E)$

$$f(E) = \frac{\gamma E^2 + \gamma h E - \beta}{h + E}$$ (2.4)

we see that $f(E) < 0$ for $E \in (0,E_*)$, where $E_*$ is the value of ECM at the positive equilibrium. The positive equilibrium is a nodal sink, given by

$$C_* = \alpha/\delta, \quad E_* = \left(-\gamma h +\sqrt{\gamma^2 h^2 + 4 \gamma \beta}\right)/(2\gamma).$$ (2.5)

Indeed, if we denote by $F$ the right hand-side of (2.1), (2.2):

$$F(C,E) = \left( \begin{array}{r} \alpha - \delta C + f(E) C \\ -f(E) C \end{array} \right)$$ (2.6)

and calculate the Jacobian matrix of $F$:

$$dF(C,E) = \left( \begin{array}{rr} -\delta + f(E)& f'(E) C\\ -f(E) & -f'(E)C \end{array} \right)$$ (2.7)

we see that the determinant, the trace and the discriminant of $dF$ evaluated at the positive equilibrium are

$$\det dF = \delta f'(E_*)C_* > 0, {\rm tr}\ dF = \delta -f'(E_*) C_* < 0, {\rm disc}\ dF = (\delta - f'(E_*) C_*)^2 > 0,$$ (2.8)

implying that $(C_*,E_*)$ is a nodal sink.

Thus, since $f(E) < 0$, the cell production rate is diminished by the need to produce the ECM component. In the expression for $f(E)$, parameters $\gamma$ and $\beta$ describe the cell-matrix interaction that affects proliferation of both $C$ and $E$, and parameter $h$ is the ``treshhold'' parameter describing where the negative feedback control in the production of ECM is ``switched on''. More precisely, the expression $\beta/(h+E)$, which determines, among other things, the generation rate of ECM, will be influenced to the ``leading order'' only after $E$ reaches the treshold value of $h$. A typical graph of $f(E)$ is given in Figure 2.1.

Figure: The graph of $f(E)$ for the low proliferation rate data ().

The non-dimensional form of the above system is obtained by the following scalings of the variables

$$x = \frac{C}{h}, y = \frac{E}{h}, \tau = t \frac{h}{\beta},$$ (2.9)

and of the parameters

$$\alpha' = \frac{\alpha}{\beta}, \gamma' = \gamma \frac{h^2}{\beta}, \delta'=\delta\frac{h}{\beta}.$$ (2.10)

The resulting non-dimensional system is given by

$$\frac{d x}{d\tau} = \alpha' - \delta' x + \gamma' x y - \frac{x}{1+y}$$ (2.11)

$$\frac{d y}{d\tau} = \frac{x}{1+y} - \gamma' x y.$$ (2.12)

This model is similar to that proposed in [5] to study the dynamics of cell and extracellular matrix. The model in [5], however, gives rise to the equilibrium that is a spiral sink for the data of our interest. This would mean, in particular, that the concentrations of chondrocytes and ECM approached the equilibrium in an oscillatory fashion, which was not observed in our experiments. In particular, the model proposed in [5] showed a region of decrease in the concentration of ECM in the intermediate stage of the ECM development, indicating degradation of ECM, not experimentally observed or justified. Never the less, the model in [5] provided a guidance for the development of the model in the present manuscript.

In the next section we show how experimental results compare with the numerical simulations of the model (2.1)-(2.3), showing excellent agreement. Additionally, we shall see that the parameters that determine the cell proliferation rate on a particular artificial surface are $\gamma$ and $\delta$. In fact, it is the ratio $\gamma/\delta$ that determines the artificial surface-dependent cell proliferation rate. High ratio $\gamma/\delta$ means highly condusive artificial surface.