A size and space structured model of tumor growth describes a key role for protumor immune cells in breaking equilibrium states in tumorigenesis

Switching from the healthy stage to the uncontrolled development of tumors relies on complicated mechanisms and the activation of antagonistic immune responses, that can ultimately favor the tumor growth. We introduce here a mathematical model intended to describe the interactions between the immune system and tumors. The model is based on partial differential equations, describing the displacement of immune cells subjected to both diffusion and chemotactic mechanisms, the strength of which is driven by the development of the tumors. The model takes into account the dual nature of the immune response, with the activation of both antitumor and protumor mechanisms. The competition between these antagonistic effects leads to either equilibrium or escape phases, which reproduces features of tumor development observed in experimental and clinical settings. Next, we consider on numerical grounds the efficacy of treatments: the numerical study brings out interesting hints on immunotherapy strategies, concerning the role of the administered dose, the role of the administration time and the interest in combining treatments acting on different aspects of the immune response. Such mathematical model can shed light on the conditions where the tumor can be maintained in a viable state and also provide useful hints for personalized, efficient, therapeutic strategies, boosting the antitumor immune response, and reducing the protumor actions.

Mathematical modeling can help in capturing the complexity of the tumor microenvironment (TME) and its role in driving tumor growth and dissemination. The success of cancer immunotherapies has highlighted the key role of the immune system. Mathematical modeling requires to simplify assumptions and we have focused here on the interaction of tumor and immune cells. We understand the frustration of the reviewer that other components are not explicitly included in the model but it is not realistic at this stage to include too many parameters. In the type of modeling adopted here, angiogenesis can be described very roughly, just by tuning some parameters, those linked to tumor cell growth and division and to protumor immune cells. Thus, discussing the effect of therapeutic combinations is highly relevant. An additional difficulty is to reach quantitative assessment as it relies on the calibration of parameters which are very poorly known. This step requires a specific work of investigation in itself. Such challenges will certainly be the object of future works.
Reviewer #2: The authors introduce a complex PDE model for anti-tumour/pro-tumour immune responses to cancer. Then, they consider a simplified version of this model (described by some simple ODEs) and for this model they investigate the existence and stability of various steady states. After that, the authors focus (I assume…) on the full model and try to find the equilibrium states. Then, the authors perform various numerical simulations showing what happens with model dynamics (although it is not clear whether they focus on the PDE or the ODE models) when various treatments are incorporated into the model.
Overall, the results have the potential to be very interesting. But the way they are presented makes it difficult to follow the flow of the paper. The authors should re-structure the paper and add more details, so that they "lead" the reader through their paper.
We acknowledge the difficulty of reading (and writing !) such an interdisciplinary article, that is expected to be of interest to several scientific communities. We made several trials of organization, considering seriously reviewer's recommendation, but did not find it easier to read, so we have kept the original organization. Going through the building of the mathematical modeling is required before we challenge the model with numerical simulations in the result section. We have revised the manuscript paying attention to clarify when PDE and ODE models are used, to make appear as neatly as possible the aims and scopes of each section and several sentences have been added to guide the reader throughout the article.
Below are some of the issues identified (which need to be addressed): On page 11 the authors consider the case where "V and a are constant". How is this possible when on the previous pages the authors show that "V" and "a" depend on "z" (see equations (1), (2) and (3))? There is no contradiction at all, a constant is just a specific case of a function of the variable z. Nevertheless, we fully agree that the original version could be a bit misleading. We have modified the paper in order to clarify the assumptions on the coefficients, and when we restrict to the constant case, which has the advantage, as indicated, to provide explicitly known equilibrium solutions for the PDE case. Concerning the discussion in p. 11 (original version), this assumption allows us to simplify the PDE model into an ODE system, which can be analyzed more easily. The ambition of this section is more clearly stated in the revised version. It is also indicated that the simplifying assumptions apply in this section only.
Page 14: I assume that the "healthy state (H)" is the one given by the zero vector on page 12? This needs to be made clear, since the notation (H) is never used on pages 12 and 13 ; only on page 14. Same comment about the other states: (NP) and (P).
The formula for the equilibria of the ODE system have now been labelled (H), (NP) and (N), with due reference within the text. It seems that the reviewer has been confused by the lack of captions and comments. We apologize for that. All captions have been modified accordingly. In fact, the equilibrium establishes only in fig. 3-(a): the concentration of immune cells shows damped oscillations around the expected equilibrium value. In the other cases, fig. 3 We understand the possible confusion. We have added several comments to clarify the organization and which model is dealt with in all sections. The results in [35] considers the celldivision equation without any coupling: it analyses the eigenvalue problem. The results in [24] are restricted to a coupling with the antitumor cells. We address the extension of such a statement when adding protumor activities. This is also further commented in the revised version. Page 22: is "Emergent qualitative features …" a sub-section of the section "Results"? Or a parallel section? As discussed above, there is no explanation/flow for how the results connect with each other, so that the reader can follow easily this manuscript.
It is indeed unfortunate that the PlosOne template does not label sections and subsections. We have tried nevertheless to address this issue by adding announcements and comments intended to clarify the flow. Indeed the curves represent space-averaged quantities. The captions have been revised and completed to make the figures as unambiguous as possible.
Are the Sections on pages 31, 34, 36 actually sub-sections of the "Results" section? It would be easier if sections/sub-sections/sub-subsections would be labelled.
Absolutely: the section "Results" is made of several subsections. It is unfortunate that the PlosOne template does not label sections and subsections. We have added a paragraph, at the beginning of the Results section which announces the organization.