11 A model of cancer volume dynamics
A model for angiogenesis
In the last chapter we looked at a number of different ways we might model cancer dynamics, but did minimal analysis of these. In this chapter we will take a more in-depth look at one particular model where we consider the phsyical growth of a cancerous tumour (see chapter references).
We can model the size of a tumour by the number of cancer cells that make it up. These dynamics may actually be well represented by the logistic growth model we studied in the very first lecture (other forms, such as the Gompertz model, are often used instead, but the logistic model will do just fine). That is because tumours tend to slowly increase in size at first, then rapidly grow and finally saturate to a finite size due to resource limitations such as physical space and blood supply. Therefore, the density of cancer cells, [latex]c[/latex], in a tumour may be expected to obey the dynamics,
[latex]\begin{equation} \frac{dc}{dt}=r_0c\left(1-\frac{c}{K}\right) \end{equation}[/latex]
where [latex]r_0[/latex] is the basic growth rate and [latex]K[/latex] the carrying capacity.
An important feature of tumour growth, however, is that they can change their environment as they grow through angiogenic factors, such that they can both stimulate and inhibit their own growth. For example, as the tumour grows it can physically create more space to grow in to, as well as secrete chemicals that encourage blood vessels to grow. On the other hand as the tumour grows it may cause damage to the existing blood supply. By affecting their environment in this way, cancer cells are changing their own carrying capacity. Therefore, while we previously assumed a fixed carrying capacity, [latex]K[/latex], it now makes sense to treat this is a dynamic variable that may grow or shrink over time depending on these angiogenic processes. The model that is proposed for these dynamics is,
[latex]\begin{equation} \frac{dK}{dt}=\phi c - \theta Kc^{2/3}=c\left[\phi-\theta Kc^{-1/3}\right]. \end{equation}[/latex]
Cells stimulate the growth of blood vessels, and hence the carrying capacity, at individual-level rate [latex]\phi[/latex]. The inhibition rate, with parameter [latex]\theta[/latex], is rather more complicated and stems from the argument we made in the previous chapter about the volume of tumour, but applied to how tumour growth will damage local blood vessel networks.
Anaylsis
As usual, we will proceed by finding possible equilibria, classifying their stability and drawing phase portraits.
We already know from our previous work on the logistic model that,
[latex]\begin{equation} \frac{dc}{dt}=0\implies c=0 \textrm{ or } c=K. \end{equation}[/latex]
Substituting these values in to our second equation, we find that,
- if [latex]c=0[/latex], [latex]dK/dt[/latex]=0 for any [latex]K[/latex], and therefore there are a continuum of equilibria with no tumour but a positive carrying capacity.
- if [latex]c=K[/latex], [latex]dK/dt=0\impli