Catastrophic shifts and lethal thresholds in a propagating front model of unstable tumor progression

Unstable dynamics characterizes the evolution of most solid tumors. Because of an increased failure of maintaining genome integrity, a cumulative increase in the levels of gene mutation and loss is observed. Previous work suggests that instability thresholds to cancer progression exist, defining phase transition phenomena separating tumor-winning scenarios from tumor extinction or coexistence phases. Here we present an integral equation approach to the quasispecies dynamics of unstable cancer. The model exhibits two main phases, characterized by either the success or failure of cancer tissue. Moreover, the model predicts that tumor failure can be due to either a reduced selective advantage over healthy cells or excessive instability. We also derive an approximate, analytical solution that predicts the front speed of aggressive tumor populations on the instability space.

Figure 1

Optimality and lethality in unstable cell populations. The vertical axis indicates the cancer replication rate against instability, as predicted from Eq. (5). The replication rate of normal cells is $r_0=0.01$ (in arbitrary units). The cancer population is assumed to be homogeneous. At low instability rates, competition between the two cell populations is symmetric and cancer coexists or slowly grows. The peak at the optimal rate ${\mu }_o$ is associated with the fastest potential growth of cancer. The gray area indicates the lethal phase, where excessive instability leads to a reduced proliferation of cancer cells, which are overcompeted by normal cells.

Figure 2

Linear model of competition between normal cells ($H$) and a heterogeneous population of cancer cells, indicated as $C_1,C_2,...,C_i$, which replicate with increasing rates $f_i$ and mutate also at faster rates ${\mu }_i$, as highlighted by the increasingly thick arrows. The effective replication rate of a given $C_k$ compartment is $f_k(1-{\mu }_k)$.

Figure 3

The three major dynamical patterns of dynamical behavior displayed by our mathematical model. Here the population density for different instability levels is plotted against instability and time. In (a), unstable tumors expand, evolving towards a stable, high instability rate. Time evolution for $r=0.25,\alpha =20$, ${\mu }_c=0.08$, and ${\mu }_{\mathrm{disp}}=3\times {10}^{-4}$. Each time step is equivalent to a generation of cells. Cancer cells diffuse through the instability space as a wave. At early stages ($t<200$), the fraction of cancer cells in the system is low. However, when cancer cells reach high enough instability (slightly above $\mu =0.015$ in this example), a rapid increase in cancer population density is produced. (b) Tumor fails to get established. The following parameter values have been used: $r=0.25$, $\alpha =50$, ${\mu }_c=0.08$, and ${\mu }_{\mathrm{disp}}={10}^{-2}.$ (c) Population collapse. Here expansion is followed by collapse after a long transient, as shown in (d). Here we have used $r=0.25$, $\alpha =50$, ${\mu }_c=0.08$, and ${\mu }_{\mathrm{disp}}=1\times {10}^{-3}$.

Figure 4

Phases in the tumor growth model. The main plot (a) shows the two phases associated with the propagation (white) or extinction (gray) of the cancer cell population. The transition separating the two phases can be characterized by the transient dynamics exhibited by the model. The inset (b) displays the number of time steps (or cancer cell generations) to reach the corresponding final state represented in (a). Darker (lighter) zones are associated with longer (shorter) transients. As expected from a phase transition phenomenon, long transients are observed close to the boundary between both phases.

Figure 5

Front speed of the cancer population traveling on the instability space, as a function of the characteristic dispersal distance ${\mu }_{\mathrm{disp}}$. The line and the circles stand for the numerical results and the approximate analytical speed $v_{*}$, respectively. The rest of the parameters used to compute the front speed are $r_n=0.25$, $\alpha =20$, $p_e=0.85$, and ${\mu }_c=0.08.$