How is the effectiveness of immune surveillance impacted by the spatial distribution of spreading infections?

Abstract

What effect does the spatial distribution of infected cells have on the efficiency of their removal by immune cells, such as cytotoxic T lymphocytes (CTL)? If infected cells spread in clusters, CTL may initially be slow to locate them but subsequently kill more rapidly than in diffuse infections. We address this question using stochastic, spatially explicit models of CTL interacting with different patterns of infection. Rather than the effector : target ratio, we show that the relevant quantity is the ratio of a CTL's expected time to locate its next target (search time) to the average time it spends conjugated with a target that it is killing (handling time). For inefficient (slow) CTL, when the search time is always limiting, the critical density of CTL (that required to control 50% of infections, C(*)) is independent of the spatial distribution and derives from simple mass-action kinetics. For more efficient CTL such that handling time becomes limiting, mass-action underestimates C(*), and the more clustered an infection the greater is C(*). If CTL migrate chemotactically towards targets the converse holds-C(*) falls, and clustered infections are controlled most efficiently. Real infections are likely to spread patchily; this combined with even weak chemotaxis means that sterilizing immunity may be achieved with substantially lower numbers of CTL than standard models predict.

Keywords: computational immunology; cytotoxic T cells; spatial modelling.

Figure 1.

Kinetics of infected cells spreading diffusely and being killed by CTL. Infected cells (vertical axes, logarithmic scale) were seeded in the simulation and allowed to grow to 1000 cells, appearing at random within the tissue, before CTL were introduced randomly among them, at a time denoted day 0. We varied the ratio of CTL to effector cells at day 0 (horizontally) and the ratio of search time to handling time (vertically) to explore the ability of the simple mass-action (dashed blue line) and extended mass-action (dotted red line) ODE models to explain the simulated data. Note how the extended model better fits the data when the handling time is greater than the search time (lower left and lower centre figures).

Figure 2.

Probabilities of extinction as a function of CTL density in units of the mass-action estimate C*= r/k. (a–e) Slow CTL surveillance such that S : H > 1 even within packed clusters of infected cells; (f–j) more rapid CTL migration, such that S : H < 1 within clusters. Blue curves, diffuse infections; orange curves, clustered infections. S : H ratios are approximate and calculated as follows: for diffuse infections, S is the expected time for a CTL to locate its first target following the appearance of CTL when infected cell numbers have reached I₀, S ≃ N/kI₀, where I₀/N is the proportion of all susceptible cells infected. For clustered infections, the search time is quoted as the time taken to move between adjacent cells within a packed cluster of infected cells, and so is a strong lower bound on the average search time. E : T ratios are quoted as the ratio of total effectors to total targets at the critical CTL density C*. PI, post-infection.

Figure 3.

Critical CTL densities (in units of the basic mass-action estimate C*= r/k) for different levels of chemotaxis. (a–d) Slow CTL surveillance and (e–h) rapid surveillance. Blue triangles, diffuse infection; orange circles, clustered infection. Note logarithmic scale on y-axes.