V

Intermediate viral cytotoxicity i

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| Time sale (arbitrary units)

Start of virus therapy

Fig. 12.3 Simulation of therapy using an oncolytic viruses in the absence of immunity.

(a) The growth rate of infected tumor cells is significantly slower than that of uninfected tumor cells. A non-cytotoxic virus now results in tumor eradication. A more cytotoxic virus results in tumor persistence. Parameters were chosen as follows.' k=10; r=0.5; s=0; ¡3=1; d=0.1; a = 0.1 for the non-cytotoxic virus, and a = 0.5 for the more cytotoxic virus.

(b) The growth rate of infected tumor cells is not significantly reduced relative to that of uninfected cells. An intermediate level of cytotoxicity results in tumor eradication. Weaker or stronger levels of cytotoxicity result in tumor persistence. Parameters were chosen as follows: k=10; r=0.5; s=0; ¡3=1; d=0.1; a = 0.2 for the weakly cytotoxic virus, a = 0.55 for intermediate cytotoxicity, and a = 3 for strong cytotoxicity.

size (Figure 12.3b). If viral cytotoxicity is too weak, the tumor persists. However, if the viral cytotoxicity is too high, the tumor also persists because infected cells die too fast for the virus to spread efficiently (Figure 12.3b). In general, the faster the replication rate of the virus, the higher the optimal level of cytotoxicity.

12.2 Effect of virus-specific CTL

This section expands the above model to include a population of virus-specific CTL, zv. The CTL recognize viral antigen on infected tumor cells. Upon antigenic encounter, the CTL proliferate with a rate cvyzv and kill the infected tumor cells with a rate pvyzv. In the absence of antigenic stimulation the CTL die with a rate bzv. The model is given by the following set of differential equations [Wodarz (2001)].

x = rx I 1---- ) — dx ~ ßxy, x y y = ßxy + sy I 1--:— ) - ay - pvyzv, z — cvyzv bzv.

First, we define the conditions under which an anti-viral CTL response is established. This condition is different depending on whether the virus attains 100% prevalence in the tumor cell population in the absence of the CTL. The strength of the CTL response, or CTL responsiveness, is denoted by cv. If the virus has attained 100% prevalence in the absence of CTL, the CTL become established cv > bs/[k(s — a)]. On the other hand, if the virus is not 100% prevalent in the tumor cell population in the absence of CTL, the CTL invade if cv > b(3{(3k + r- s)/[r((3k - a) - d{(3k - s)].

In the presence of the CTL, we again observe two basic equilibria: either 100% virus prevalence in the tumor cell population, or the coexistence of infected and uninfected tumor cells. Hundred percent virus prevalence in the tumor cell population is described by equilibrium El:

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