ODE models for oncolytic virus dynamics

Abstract

Replicating oncolytic viruses are able to infect and lyse cancer cells and spread through the tumor, while leaving normal cells largely unharmed. This makes them potentially useful in cancer therapy, and a variety of viruses have shown promising results in clinical trials. Nevertheless, consistent success remains elusive and the correlates of success have been the subject of investigation, both from an experimental and a mathematical point of view. Mathematical modeling of oncolytic virus therapy is often limited by the fact that the predicted dynamics depend strongly on particular mathematical terms in the model, the nature of which remains uncertain. We aim to address this issue in the context of ODE modeling, by formulating a general computational framework that is independent of particular mathematical expressions. By analyzing this framework, we find some new insights into the conditions for successful virus therapy. We find that depending on our assumptions about the virus spread, there can be two distinct types of dynamics. In models of the first type (the "fast spread" models), we predict that the viruses can eliminate the tumor if the viral replication rate is sufficiently high. The second type of models is characterized by a suboptimal spread (the "slow spread" models). For such models, the simulated treatment may fail, even for very high viral replication rates. Our methodology can be used to study the dynamics of many biological systems, and thus has implications beyond the study of virus therapy of cancers.

Figure 1

The shape of the function G_exp(x) and the number of equilibria as a function of β. (a,b) Fast virus spread. (c,d) Slow virus spread.

Figure 2

Fast virus spread. The function G(x, y(x)) (equations (3-4)) for the particular choice of the virus spread law (G(x, y) = x/(x + y + 1)) and three different laws of cancer growth: exponential, surface growth and linear growth. 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

The dependence of the equilibrium on β and s_t. We use the model with F = η/(η + x + y) (thus, s_t is defined by η) and G = x/(x + y + ε). The function G(x, y(x)) is plotted vs x for different values of η. The dashed vertical lines indicate the scales of interest: the leftmost such line corresponds to x ∼ s_v, and the rest of the lines to x ∼ s_t for different values of η. The other parameters are: a = 4, ε = 1.

Figure 4

Slow virus spread. The function G(x, y(x)) (equations (3-4)) for two particular choices of the virus spread law: (a) G(x, y) = x/(x + 1)/(y + 1) and (b) $G(x,y)=x/(x+1+\sqrt{x}(y+1))$. Different laws of cancer growth are implemented: exponential, surface growth and linear growth (in (a), with two values of η, η = 5 and η = 1). The solid lines correspond to the unlimited cancer growth; the dotted lines - to a growth up to a given size, W. The other parameters are: a = 1, W = 10⁴ in (a) and W = 10⁵ in (b).

Figure 5

The threshold-type dependence of the equilibrium number of cells on the infectivity. The steady-state value, x₀, is plotted as a function of β, for the model with G = x/(x + y + ε) and F = x/(x + y + η). The parameters are a = 10², ε = 1. The two solid curves correspond to the values η = 20 and η = 10⁷. the dashed curve represents the approximation x₀ ≈ a/(β − a).

Figure 6

Bifurcation diagram for the case ε ≪ 1. The gray lines correspond to ε = 0 and the black lines to ε = 0.05. In the former case, the quantities x̄ and 0 are plotted vs β; in the latter case, the values x₁ and x₂ are plotted as functions of β. The solid lines represent stable solutions and dashed lines represent unstable solutions. The parameters are a = 0.5 and w = 10.

Figure 7

The phase portrait for system with F = r and G = x(1+ε₁)(1+ε₂)/[(x+ε₁)(y+ε₂)], a schematic. (a) The intermediate equilibrium is stable, (b) it is unstable.