Reviewer's report 1
Mikhail Blagosklonny, Cancer Center, Ordway Research Institute, Albany, NY, USA
This reviewer provided no comments for publication.
Reviewer's report 2
David Krakauer, Santa Fe Institute, Santa Fe, NM, USA
The appreciation of cancer as an evolutionary and ecological dynamic within the context of a multicellular organism has allowed many ideas developed in evolution and ecology to migrate across into the study of disease. Best known among these ideas are studies of virus dynamics and immune regulation.
Cancer cells share many dynamic properties with viruses including a rapid rate of proliferation, a high rate of mutation and an elicitation of immune responses. Both deterministic and stochastic evo-eco models have been developed to describe the time course of cancer emergence and cancer cell proliferation.
A number of researchers have thought of coupling these two dynamics (virus and cancer) in an effort to use some of the pathological properties of cancer to mitigate the pathological properties of cancer cells. In particular, encourage viruses to infect cancer cells (oncolytic virus) in order to promote cancer cell clearance through several routes including cytotoxicity and immune activation.
This paper represents an extension of existing work by Wodarz and colleagues to include an alternative non-linear functional response (Holling-type) for infection rate and a more exhaustive analysis of the parametric dependencies of the steady states. The parameters of focal interest are the coefficient of infected cancer cell death and the coefficient of infection.
The analysis is thorough and the extensions are interesting. The appearance of several different outcomes according to parameter values is valuable as it makes clear the potential complexity of behavior in a very simple dynamical system describing disease. The value of such a paper is to use quantitative models to make a qualitative point, that the outcome of biologically based therapies can be diverse and require model-based thinking as a guide to experiment.
The paper does however make a number of very strong assumptions, which limit its use as a therapeutic tool and make comparisons to extant data sets somewhat difficult. I list these assumptions below. I think it worth pointing these out, as they are significant issues for any comparable research into this area.
1. Cancer is an inherently stochastic process dependent on random mutation accumulation and a very noisy (sampling noise) process of cell proliferation when cancer cell numbers are low. Both mutation and extinction near the zero cell boundary condition are left out of this model. This has been treated extensively in some recent cancer models that the authors should probably read and cite. I am thinking of the early papers by Moolgavkar and the more recent models by Murray et al, Speer et al and Nowak et al.
Author response: Cancer is certainly a stochastic process affected by random mutations and cell proliferation. We added to the text some recent references on stochastic modeling of tumorigenesis. However, we do believe that deterministic models are of significant importance in our understanding of cancer. A number of references to deterministic models were also added. Our model does not contain mutation process, but our basic goal was to construct a mathematical model, as simple as possible, that could produce all the regimes which were observed in experiments. Also, it has to be kept in mind that what is being modeled here is the interaction between a tumorolytic virus and the tumor, rather than the process of tumor growth per se, so not all stochastic aspects of the latter are necessarily relevant.
2. Cancer tissue exists within a larger context of healthy tissue and density dependent growth does not only depend on the density of cancer cells as is assumed in this model but the density of healthy cells. While the model treats cell death in the usual phenomenological framework of logistic growth, some consideration of the mechanisms at work such as limitations in blood supply would be valuable, including research into the effects of angiotensin.
Author response: Certainly, healthy tissues are important for tumor growth. Indirectly, this is implied in the logistic model of tumor growth, which is adopted here, as the growth may be limited by the effects of the healthy tissues. We do not believe that including further details into a simple model considered here would be productive.
3. The separation of time scales which allows the free virion to be treated algebraically is perhaps less acceptable in the case of cancer than in virus infection of normal cells. The reason for this is that the cancer cell can have an elevated rate of proliferation, which would render this assumption inappropriate.
Author response: This definitely might be the case, but we think that our description can still be applied to many real situations. Besides, incorporating virus population dynamics explicitly would yield a considerably more complex system of differential equations, which could hamper the comprehensive mathematical analysis of nonanalytical vector field. We agree that this suggestion is a reasonable next step to improve the model.
4. The assumption of perfect viral specificity is unlikely to be possible and it is therefore important to quantify the cost of tumor therapy in terms of the reduction in the healthy population of cells in relation to the reduction in cancerous cells.
Author response: Yes, this is a simplification, we cannot argue that, but it is one that is necessary to construct a model that can be not only written down but also analyzed in detail.
5. Elimination might not be the desired outcome of therapy but control. We have no real way of knowing whether small micro-metastatic tumors have serious implications on individual morbidity and so alternative treatments that focus on protracted control could prove very interesting.
Author response: Actually, it seems like elimination of a tumor always is the most desirable outcome. It is quite another matter that, this not being practical in a particular situation, control might be effectively as good. What we show here, is that, under certain regimes of virus-tumor interaction, complete, deterministic elimination of the tumor cell population is a possibility.
6. Finally and perhaps most difficult for a model is the whole question of a quantitative statement of "disease". We can model individual cell populations but we do not really understand the relationship of the cell level to the individual clinical symptoms. How might we go about relating these two spatial scales within a common theoretical framework? I think this is an important question that deserves more attention from the research community and certainly goes beyond the realistic ambition of the present study.
Author response: We just must agree and confirm that, perhaps, sadly, this is beyond our ambitions in the foreseeable future.
Reviewer's report 3
Erik van Nimwegen, Division of Bioinformatics, Biozentrum, University of Basel, Basel, Switzerland
Novozhilov et al. study the phase portrait of a simple mathematical model for the dynamics of an oncolytic virus and tumor cells. This model is a particular example from the class of Lotka-Volterra like models and is an extension of a model proposed in the same context by Wodarz. The main feature of interest is that, as a function of the small set of parameters, the model exhibits qualitative different kinds of dynamical behavior such as coexistence of tumor and infection, disappearance of infection, disappearance of all tumor cells, etcetera.
First, a remark as to the novelty of the research here proposed. I do not know to the literature well enough to be able to point to a specific reference in which the model (10) has been studied but I would be quite surprised if this model, and its phase-portrait, had not already been described in the literature on Lotka-Volterra type models. I therefore expect that the main addition to the literature is the application of this model to the oncolytic virus system, and a discussion of its phase portrait in this context.
Author response: This model is not exactly in the class of Lotka-Volterra models. The main difference (considering this model as a model of interaction of two species) is that the functional response is ratio-dependent, and this was the main topic of the work of Arditi and coauthors who emphasized the qualitative differences between these models and the classical Volterra models.
This brings me to the main criticism of the manuscript. The presentation is rather mathematical throughout and will be completely inaccessible to most biomedical researchers. Even I had some difficulties with some of the terminology. For example I had to look up what an "elliptic sector" is (the definition in the paper didn't help me very much) and I also had to look up the Bendixson-Dulac theorem to understand how Lemma 3 follows. I bring this up mainly because I don't believe it is necessary to make the manuscript so technical. I think the main interest of the results lies in discussing the meaning of the phase-portrait for tumor therapy. The discussion does a reasonable job of this but I think it could be made even clearer to the point that the main conclusions would be easily accessible for biomedical researchers.
Author response: The existence of the elliptic sector is the crucial aspect of the model presented in this paper. This mathematical notion is not just a technical detail but rather has important biological implications, i.e., in this particular case, the possibility of deterministic elimination of the tumor. Therefore we felt that it was necessary to include the required proofs.
Some weight is put in the discussion of the mathematics on the fact that "The important mathematical peculiarity of system (7) is that the origin is a nonanalytical complicated equilibrium point". It seems to me that such mathematical subtleties could be easily avoided by expressing the equations in terms of the total number of tumor cells n = x + y and the fraction ρ = y/(x + y) of tumor cells that is infected. In terms of these variables the equations (7) become:
First, these equations are now analytic everywhere in the strip n ≥ 0, 0 ≤ ρ ≤ 1. Furthermore, the second equation does not depend on n and can be directly integrated to obtain explicit analytic expressions for n(t) and ρ(t). Even more simply, it is immediately apparent from the second equation that the sign of β - δ + γ - 1 determines if the system will flow to ρ → 1 as t → ∞ or to ρ → 0. And once we know what values ρ will limit to, we can fill its value in the first equation to see that the tumor will only disappear if β - δ + γ - 1 > 0 and δ > γ. Obviously the authors reach the same conclusions but I think the derivation just sketched is much simpler and more easily accessible to the readership of Biology Direct than the rather technical derivation that the authors present.
Similar remarks apply to the system (10). If one again transforms to n and ρ one obtains
Here one cannot explicitly solve for n(t) and (t) in closed form but the system is analytic everywhere in the strip and the fixed points, their Jacobians, and the eigenvalues of these Jacobians can be easily determined. Again one would of course reach the same conclusions as the authors1 but I think the mathematics is again substantially simplified in this coordinate system.
Author response: The models (1)-(2) and (3)-(4) proposed by the reviewer are indeed equivalent to our models (7) and (10), respectively, only if the total population size n = x + y is constant or, at least, is bounded from below, i.e., n ≥ Const > 0. Otherwise, however, the systems are not equivalent and exhibit qualitatively different behaviors.
On page 16 it is said "Noting that the main part of (12) coincides with system (7), we can use the results described above for the latter system to obtain the structure of a positive neighborhood of the origin of system (10)".
I presume that the expression "main part of (12)" is a technical term and that the authors are implicitly using a theorem here (that I am unfamiliar with) that proves that the topology of the set of trajectories close to the origin is the same in system (10) and system (7). Some more explanation would be helpful.
Author response: The requisite explanations have been incorporated in the revision.
Reviewer's report 4
Ned S. Wingreen, Department of Molecular Biology, Princeton University, Princeton, NJ, USA
The authors address the question of the possible outcomes of tumor therapy with oncolytic viruses within a simple but appropriate mathematical framework. The model allows for two populations of cells: uninfected tumor cells and infected tumor cells. The concentration of viral particles is not explicitly followed, and the tumor is assumed to be homogeneous in space, i.e. no spatial dependence of cell concentration is considered. As such, the system consists of two coupled, deterministic differential equations allowing for cell reproduction and death, and cell infection. Previous related work by Wodarz (Refs. 12, 26, and 32) assumed an infection rate proportional to the product of infected and uninfected cell concentrations (a la the law of mass action). Within this assumption, the equations never lead to elimination of the tumor. The main result obtained by Novozhilov et al. is that a plausible change in the form of the infection rate can lead, in some range of parameters, to additional outcomes in which all tumor cells, or either infected or uninfected cells can be completely eliminated. This is an important conclusion in terms of the medical implications of oncolytic viral therapy.
The paper is very clearly written, and, while rather technical, the mathematical analysis of the coupled equations for cell concentrations is enriched by some nice qualitative discussion of what the assumptions and results mean. Mathematically, the use of a ratio-dependent infection rate (~ XY/(X+Y), where X and Y are the uninfected and infected cell populations) leads to interesting behavior in the limit of low cell concentrations, and the authors do a good job of deriving and explaining this behavior.
Overall, I found the paper both interesting and informative. The careful study of a simple model is very instructive, and is likely to be helpful in guiding future work on oncolytic viral therapy.
A few minor comments on the manuscript:
1. It would have been helpful to have a more detailed review of the experimental status of oncolytic viral therapy in the introduction. The sentences leading up to citations to Ref. 18–20 give no quantitative sense of how effective this therapy is compared to any control experiments.
Author response: We added one piece of quantitative information. We were reluctant to get deeper into that because the reliability of quantitative assessment of the comparative effects of different therapeutic strategies is a matter of concern.
2. The discussion of the use of a ratio-dependent infection rate is rather terse considering how important this is to the manuscript. It is clear that in the limits X << Y and Y << X, the form used is sensible, but this is never discussed clearly. Similarly, it is never explained why, mathematically, the ratio-dependent and alternative form of the infection rate lead to qualitatively different outcomes. Again, this seems central to the paper, and deserves some more attention.
Author response: In the revision, the reasons for adopting a ratio-dependent infection rate in our model are discussed in greater detail.