The growth of drug-sensitive and drug-resistant mutants in the (a) absence and (b) presence of an oncolytic virus. In the absence of the virus, both cancer cell populations grow, first exponentially and then with a slower rate as the number of cells reaches higher levels. In the presence of the virus, the drug-sensitive cancer cells approach an equilibrium because the virus prevents unlimited growth of the cancer cells. The virus, however, fails to eradicate the cancer. The drug-resistant cancer cells go extinct owing to apparent competition mediated by the oncolytic virus. This simulation does not take into account the generation of resistant cells by mutation. Instead it assumes that a certain number of resistant cells are present at the beginning of the simulation. Parameters were chosen for the purpose of illustration only, since they are currently unknown. The dependence of the outcome on parameter values is discussed in the text. The following parameter values were used: r₁=7, r₂=5, β₁=0.1, β₂=0.1, a₁=0.5, a₂=0.5, ϵ=10, η=10⁹, μ=0. Solid line, wild-type; dashed line, drug resistant.

Apparent competition between wild-type (solid line) and drug-resistant (dashed line) cancer cells in the presence of an oncolytic virus. It is assumed that drug-sensitive cancer cells can give rise to drug-resistant cells by mutation. The two graphs differ in the effectiveness of the oncolytic virus. (a) The virus is less effective and the wild-type cancer cell population reaches a higher equilibrium. (b) The virus is more effective and the drug-sensitive cancer cell population reaches a lower equilibrium. In both cases, however, the virus fails to drive the cancer cells extinct. In the presence of mutations, the resistant cells do not go extinct any more, but persist at low levels, determined by the mutation–selection balance. This is determined by the mutation rate, the selective disadvantage of resistant cells and the number of drug-sensitive cells. The mutation–selection balance is such that the resistant cells are likely to be present in (a) and extinct (on average less than one cell) in (b). Parameters were chosen for the purpose of illustration only, since they are currently unknown. The dependence of the outcome on parameter values is discussed in the text. The following parameter values were used: r₁=7, r₂=5, a₁=0.5, a₂=0.5, ϵ=10, η=10⁹, μ=10⁻⁹. For (a) β₁=0.1, β₂=0.1. For (b) β₁=0.3, β₂=0.3.

Simulation of the sequential combination of oncolytic virus therapy and drug therapy, assuming that drug-sensitive cancer cells can produce drug-resistant cells by mutation. Oncolytic virus therapy results in a selective disadvantage for drug-resistant cells through apparent competition. In the presence of the virus, the resistant cells will be present at a level determined by the mutation–selection balance, as explained in figure 2. (a) Mutation–selection balance maintains the drug-resistant cell population at relatively high numbers, which is likely to correlate with the persistence of these cells. Consequently, when drug therapy is applied, the resistant cells will grow and drug therapy fails. (b) Mutation–selection balance maintains the drug-resistant cell population at low levels: on average less than one cell persists. This corresponds to the extinction of the drug-resistant cell population. Consequently, the subsequent drug therapy is successful at eliminating the cancer because there are no resistant cells left. Parameters were chosen for the purpose of illustration only, since they are currently unknown. The dependence of the outcome on parameter values is discussed in the text. The following parameter values were used: r₁=7, r₂=5, a₁=0.5, a₂=0.5, ϵ=10, η=10⁹, μ=10⁻⁹. For (a) β₁=0.1, β₂=0.1. For (b) β₁=0.3, β₂=0.3. Drug therapy was modelled by assuming that the drug-sensitive cancer cells have an additional death rate d=5, such that the drug can induce a decline of this cell population to extinction. The resistant cells do not receive the additional death rate, as the drug has no effect in this case. Solid line, wild-type; dashed line, drug resistant.