Figure 1.
Two classes of virus growth captured by the mathematical models.
(a) According to class I or fast virus growth, virus growth is exponential as long as the number of uninfected cells is significantly larger than the number of infected cells. This can correspond to a high degree of mixing between infected and uninfected cells. As the virus population grows, the number of cells that contribute to virus spread remains constant because most infected cells will have an uninfected cell in their vicinity. (b) According to class II or slow virus growth, virus growth slows down and saturates as the virus population increase in size, even if the number of uninfected cells is relatively large. This can correspond to spatial clustering of the infected cells. Only infected cells at the surface have uninfected cells in their neighborhood and can thus contribute to virus transmission. As the number of infected cells rises, the number of “active” cells that can contribute to virus transmission declines.
Table 1.
Examples of different virus spread terms, G(x,y).
Table 2.
Examples of different tumor growth terms, F(x+y).
Figure 2.
The effect of a carrying capacity.
The function G(x,y(x)) is plotted for two particular choices of the virus spread law and three different laws of cancer growth: exponential, surface growth and linear growth. (a) Fast virus spread, G(x,y) = x/(x+y+1) and (b) slow virus spread, G = x/(x+1)/(y+1). The solid lines correspond to the unlimited cancer growth; the dotted lines - to a growth up to a given size, W. The parameters are: a = 1, η = 10 and W = 104.
Figure 3.
(a) The equilibrium number of uninfected cancer cells as a function of the viral replication rate β for fast virus growth. There is a threshold viral replication rate at which the number of cancer cells drops sharply from relatively high values to values of the order of one. This can be considered a tumor extinction threshold. (b) Dynamics of the uninfected cancer cells if the viral replication rate lies below (left) and above (right) this threshold. If the viral replication rate lies below the threshold, limited oscillations are observed that dampen out quickly. If the viral replication rate lies above the threshold, extensive oscillations are observed that reduce the cancer cell population to very low levels, and that dampen out very slowly (dampening not observed on time scale shown here). These plots were made by using a specific model from the fast virus growth category, that is G = (ε+1) x/(x+y+ε). Note that the transition in oscillations is not a universal feature of all models in this class. Parameters were chosen as follows: r1 = 1; a = 0.1; ε = 10; η = 108; x0 = 100; y0 = 10; For (b), β = 0.07 and β = 0.13, respectively.
Figure 4.
The phase portrait for a system with a slow virus propagation term.
(a) The intermediate equilibrium, EI, is stable (the basin of attraction is shaded), (b) EI is unstable.
Figure 5.
Dynamics in fast virus growth models assuming that the internal equilibrium EI is (a) stable and (b) unstable.
(a) If the internal equilibrium is stable, then the dynamics can converge to this equilibrium via damped oscillation if the initial number of cancer cells is relatively low. On the other hand, if the initial number of cancer cells is relatively high, then uncontrolled cancer growth is observed. (b) If the internal equilibrium is unstable, then diverging oscillations are observed. Eventually, these diverging oscillations take the populations beyond the saddle node equilibrium, leading to unlimited cancer growth. Before that occurs, however, it is most likely that the cancer has been driven extinct in a stochastic setting because the diverging oscillations drive the tumor size to ever decreasing values. These plots were obtained from a specific model that belongs to the slow virus growth class, i.e. . Parameters were chosen as follows: (a) r = 1; β = 0.8; a = 0.5; ε1 = 20; ε2 = 10; η = 108; x0 = 100 and 10,000, respectively; y0 = 10. For (b) r = 1; β = 1; a = 0.5; ε1 = 10; ε2 = 11$; η = 108; x0 = 10; y0 = 1.
Figure 6.
(a) Data on the growth of A549 human lung cancer nude mouse xenografts in the absence of the virus [28]. Different tumor growth models were fitted, see Table 2. The parameter values and the root mean square values are summarized in the Supporting Information S1. The graph on the right plots the predicted long-term growth curves. (b) Growth dynamics in the presence of the wild-type virus Ad309, which was injected into an established tumor. Both a slow model and a fast model were fitted. For the slow model, G = x/(x y1/3+ε). For the fast model, G = x/(x+y+ε). Tumor growth was assumed to be logistic, F = 1−(x+y)/W. For the slow model, different parameter combinations are shown that fit the data to a similar degree (slow1, slow2). The graph on the right shows the predicted long term dynamics. Parameter values and root mean square values are given in the Supporting Information S1.
Figure 7.
(a) Data on the growth of A549 human lung cancer nude mouse xenografts in the presence of the wild-type virus Ad309, assuming that infected and uninfected cells were mixed before the tumor cells were injected into the mouse [28]. A slow and a fast model were fitted. For each model, different parameter combinations were found that fit the model comparably (slow1, slow2, fast1, fast2).The graph on the right side shows the predicted long term dynamics for the different models and parameter combinations. (b) Infection with the mutant As337 virus, where again the infected and uninfected cells were mixed before the tumor was injected into the mice. Again, a slow and a fast model were fitted, and with each model different parameter combinations were found that provided a comparable fit to the data. As before, the graph on the right hand side shows the predicted long-term dynamics. For the slow model, G = x/(x y1/3+ε). For the fast model, G = x/(x+y+ε). Tumor growth was assumed to be logistic, F = 1−(x+y)/W. Parameter values and root mean square values are given in the Supporting Information S1.