In ref. (1), “A validated mathematical model of cell-mediated immune response to tumor growth”, de Pillis et al. present a mathematical model describing the interactions between the immune cells, CD8+ T-lymphocytes and natural killer (NK) cells. Differential equations are used to describe tumor-immune growth, response, and interaction rates. The parameters reflect general biological quantities: most values were provided by biology literature, whereas some of them were estimated using a least-squares method and validated on mice and human real data. The authors use the results of their simulations to conclude that tumor regression is more influenced by CD8+ T cell activity than by NK cells.

We tried to reproduce the results in ref. (1) with the same constants, but obtained different results. We explain this by the choice of initial conditions of the model. We use, as initial conditions, the healthy steady state (i.e., absence of tumor). In contrast, we could reproduce the results in ref. (1) only by taking initial conditions that were not biologically plausible. This changes the conclusion about the efficacy of NK cells.

The model in ref. (1) is:

\[\begin{array}{ccc}\frac{\mathrm{d}T}{\mathrm{d}t}&=&aT(1{-}bT){-}cNT{-}D\\\frac{\mathrm{d}N}{\mathrm{d}t}&=&{\sigma}{-}fN+\frac{gT^{2}}{h+T^{2}}N{-}pNT\\\frac{\mathrm{d}L}{\mathrm{d}t}&=&{-}mL+\frac{jD^{2}}{k+D^{2}}L{-}qLT+rNT\\D&=&d\frac{(L/T)^{{\lambda}}}{s+(L/T)^{{\lambda}}}T\end{array}\]

with T(t) the tumor cell population at time t, N(t) the total level of NK cell effectiveness, and L(t) the level of tumor-specific CD8+ T cell effectiveness. Details about constants are discussed in ref. (1).

A natural choice for the initial conditions is the healthy state, which we now analyze. Assume that no tumor is present [T(0) = 0], thus

\[\begin{array}{ccc}\frac{\mathrm{d}T}{\mathrm{d}t}&=&0\\\frac{\mathrm{d}N}{\mathrm{d}t}&=&{\sigma}{-}fN\\\frac{\mathrm{d}L}{\mathrm{d}t}&=&{-}mL\end{array}\]

Hence, N(t) →

\(\frac{{\nu}{\sigma}}{f}\)
and N(t) → 0 when t → ∞. This shows that without a tumor, the system converges to a stable state. Simulations launched with these initial conditions produce different results than those in ref. (1). The only way we can reproduce the original results is to choose N(0) = 0 and L(0) = 0 as initial conditions. This is not biologically plausible because NK cells are known to be always present and immediately effective against tumor (innate immune system response). The results with our initial conditions are shown in Fig. 1, in which we also compare the original results in ref. (1).

Figure 1.

Results of the same simulations as in ref. (1) with corrected initial conditions. A, response by NK and CD8+ cells to ligand-transduced vaccine using our initial conditions; B, same as in (A) but with the initial conditions in Fig. 4 (top right) from ref. (1); C, simulation of mouse model with anti-CD8+ antibody challenged with ligand-transduced cells; D, same as (C) but with the initial conditions in Fig. 4 (bottom right) from ref. (1).

Figure 1.

Results of the same simulations as in ref. (1) with corrected initial conditions. A, response by NK and CD8+ cells to ligand-transduced vaccine using our initial conditions; B, same as in (A) but with the initial conditions in Fig. 4 (top right) from ref. (1); C, simulation of mouse model with anti-CD8+ antibody challenged with ligand-transduced cells; D, same as (C) but with the initial conditions in Fig. 4 (bottom right) from ref. (1).

Close modal

In Fig. 1A, we see that NK cells alone are able to control the tumor size before CD8+ cells start destroying the tumor, contrary to what is shown in Fig. 1B. Similarly, Fig. 1C shows that, without CD8+ cells, NK cells are able to control tumors up to a size of 106 cells, whereas in Fig. 1D, they can only control tumors with a size of up to 104 cells.

We showed that applying biologically plausible initial conditions to the model presented in ref. (1) leads to a different conclusion: NK cells are by far more effective. Any potential ineffectiveness of NK cells must come from causes not present in the model.

1
de Pillis LG, Radunskaya AE, Wiseman CL. A validated mathematical model of cell-mediated immune response to tumor growth.
Cancer Res
2005
;
65
:
7950
–8.