Mathematical Modeling Shows That the Response of a Solid Tumor to Antiangiogenic Therapy Depends on the Type of Growth
Round 1
Reviewer 1 Report
In this manuscript, the author presents a 4 variable, coupled partial differential equation model of generic tumor growth. With this model, utilizing various assumptions, they simulate different types of growth including invasive, compact, and mixed. The stated goal of the paper was to investigate the differential response to anti-angiogenic therapy under the different growth conditions. Broadly, I believe the paper should be greatly strengthened in the following ways: 1) the paper needs more focus, 2) the significance of the model needs to be made more evident as in its current form it seems to be more of a mathematical exercise, and 3) related to 2, a stronger connection to data to in any way validate the model is necessary. More specifics below.
More focus The information content and results of this paper do not require 19 pages of manuscript. Many portions of this manuscript could be shortened. By focusing the text, it will also be easier for readers to follow the points the author is trying to convey. The introduction would benefit from being shortened. The model description could be shortened and would also benefit from additional subsection titles. Some of the derivation of the individual models (compact, invasive, mixed) could be moved to a supplement. The derivation ultimately takes away from the main story being told by the text.
Highlight significance of the model In the discussion and the introduction, more emphasis should be placed on why this model and these results matter. What might change based on these results. How could these results be used by someone studying cancer?
Connection to Data In the discussion, there is one sentence where the authors indicate their results are in agreement with clinical and pre-clinical data. These comparisons should not be left for the reader to completely go and verify. What specifically is in agreement? If it is just “agreeing” what does this model add the field of knowledge beyond the data.
Other: Graphs need to have labels on the axes
Author Response
Point 1:
More focus The information content and results of this paper do not require 19 pages of manuscript. Many portions of this manuscript could be shortened. By focusing the text, it will also be easier for readers to follow the points the author is trying to convey. The introduction would benefit from being shortened. The model description could be shortened and would also benefit from additional subsection titles. Some of the derivation of the individual models (compact, invasive, mixed) could be moved to a supplement. The derivation ultimately takes away from the main story being told by the text.
Response 1:
I have shortened the manuscript trying to remove its parts, less significant for the present study. The Introduction is now shortened by one-third, and the Equations subsection – by one-sixth, being also divided by subsubsections. The Results section has also been shortened as much as possible, with the most massive part – analytical estimation of compact tumor growth speed – moved to Appendix. The main text of the manuscript now occupies 14,5 pages.
Point 2:
Highlight significance of the model In the discussion and the introduction, more emphasis should be placed on why this model and these results matter. What might change based on these results. How could these results be used by someone studying cancer?
Response 2:
I have significantly rewritten the Discussion in accordance with this remark as well as the remarks by two other reviewers. The crucial phrases with regard to this remark are as follows. “To the best of my knowledge, this effect (see response below) was not reproduced before via a continuous model of tumor growth, while this ability of the model should be crucial for investigation of AAT.” “The account for convective component of tumor growth should be as well crucial for modelling of other types of antitumor therapy, since the decrease in the number of tumor cells, caused by any treatment, in reaction-diffusion models should lead to underestimated decrease of tumor growth speed.” Also the model allows to make several predictions, regarding the aspects of solid tumor growth, which are listed in the response below.
Point 3:
Connection to Data In the discussion, there is one sentence where the authors indicate their results are in agreement with clinical and pre-clinical data. These comparisons should not be left for the reader to completely go and verify. What specifically is in agreement?
Response 3:
It is the conclusion that “the tumors, which initially have invasive phenotype, should possess intrinsic resistance to AAT, while compactly growing tumors should be more susceptible to it”, that is in agreement to the clinical and pre-clinical data. This issue has been discussed in the Introduction, the corresponding part being now left unaltered as the most crucial one. I have tried to rephrase the corresponding part in the Discussion in order to make it more clear, and have repeated there the references from the Introduction to the papers, which analyze a large amount of pre-clinical and clinical data, eventually indicating the corresponding hypothesis [Ebos, John ML, and Robert S. Kerbel. "Antiangiogenic therapy: impact on invasion, disease progression, and metastasis." Nature reviews Clinical oncology 8.4 (2011): 210.] [Bergers, Gabriele, and Douglas Hanahan. "Modes of resistance to anti-angiogenic therapy." Nature Reviews Cancer 8.8 (2008): 592-603.].
Point 4:
If it is just “agreeing” what does this model add the field of knowledge beyond the data.
Response 4:
First of all, as it already was stated in response above, this model highlights the importance for account for both components of tumor growth in simulations of different types of antitumor therapy by means of continuous spatially-distributed models. Furthermore, the model allows to make several predictions regarding the aspects of solid tumor growth:
- it indicates that there exists a limit of growth speed of compact tumors with no capillaries inside them, which would be achieved under infinite increase in the number of capillaries and/or in their permeability, however, the estimations with physiologically based values of parameters suggest, that achieving of more than 80% of the limit value of speed is unlikely for real tumors;
- it suggest, that in terms of tumor speed reduction, maximum possible angiogenic effect for highly invasive monoclonal tumors should be around 10-15%;
- it provides insights into the mechanism of interaction between two types of growth, highlighting their non-additive character, and in particular suggests, that a low-invasive tumor may even grow slower, than the same tumor with immobilized cells.
Point 5:
Other: Graphs need to have labels on the axes
Response 5:
I have to admit, that I don't quite understand this remark. The graphs do have labels. Probably, the graphs of the distributions of model variables are meant, in which the labels of the variables are placed near the corresponding distribution curves, since stacking them near the vertical axes would lead to their overcrowding and to inability of matching them with the curves in black-and-white regime. Therefore, I doubt that such rearrangement would be useful, unless it is required by the specifications of the Journal.
Reviewer 2 Report
Author stated “… Implicit Crank-Nicholson scheme was used for glucose diffusion equation. Since glucose diffusion term provides maximum rate of local change among all the variables and thus requires a sufficiently small time step even for implicit solving, it was decided to solve cell migration equation by a simpler explicit forward Euler scheme, and all kinetic equations by explicit Euler method.”
It must be noted that the forward Euler (explicit Euler) differentiation scheme is very poor in terms of numerical stability. So, it is expected that the accuracy of the results presented in Section 3 may eventually become very poor, if the step sizes are not sufficiently small. Is the author aware of this aspect?
Author Response
Point 1:
Author stated “… Implicit Crank-Nicholson scheme was used for glucose diffusion equation. Since glucose diffusion term provides maximum rate of local change among all the variables and thus requires a sufficiently small time step even for implicit solving, it was decided to solve cell migration equation by a simpler explicit forward Euler scheme, and all kinetic equations by explicit Euler method.”
It must be noted that the forward Euler (explicit Euler) differentiation scheme is very poor in terms of numerical stability. So, it is expected that the accuracy of the results presented in Section 3 may eventually become very poor, if the step sizes are not sufficiently small. Is the author aware of this aspect?
Response 1:
I am aware of this aspect, and in particular for this purpose I have performed the set of simulations with increasingly fine discretization, which are discussed in subsections 3.1 and 3.2. These simulations were to answer, can these numerical methods yield sufficiently close estimations of tumor growth speeds to the analytical ones, which, as it turned out, they indeed can. Since the model describes biological objects, for which significant variability is natural, there is, to my opinion, no need to pursue greater accuracy by spending more computational time, i.e., an error of no more than 5% is absolutely satisfactory. It is the qualitative behavior of the model that is of much greater importance, and it cannot be affected by the used methods as long as the discretization is tuned appropriately.
Reviewer 3 Report
This paper presents a cancer tumour growth model that considers four variables: tumour cells, necrotic tissue, glucose, and capillaries. The model also incorporates the effect of antiangiogenic therapy (AAT). Then, this model is used to simulate three different kinds of growth: compact, invasive, and a mix of both. An analytical estimation is developed for all cases. The results show correspondence between the numerical results, analytical estimations, and experimental results.
The topic is within the scope of Mathematics-MDPI, and interesting to the audience of the journal. However, the paper is hard to read and does not state clearly the contributions. Therefore, I recommend a minor revision of the paper before publication.
Comments:
General: The paper is hard to read. Please consider revising the style to ease the comprehension of the paper.
Introduction: The review of main tumour growth models and the features (or hallmarks) that induce tumour growth is well written. A revision of AAT history and effects is also well written. However, a comparison of the model presented in the paper with other models is lacking. I wonder about the differences between the model developed by the author's research group and other models. Please, add some background and comparison of the model presented in the paper.
Line 126-147. In this paragraph, the authors describe the effect of the antiangiogenic therapy (AAT). AAT effect is the focus of the paper. Therefore, this paragraph is of paramount importance. It is hard to read (I needed to read it at least four times), so please, take a conscientious effort to rewrite it.
Also, the conclusion (AAT decreases P to the normal tissue value) is surprisingly simple. Is this simplification enough? How many hours are needed for the AAT effect to show up? How does this simplification stand compared to more complex models such as the one presented in:
Colli, P., Gomez, H., Lorenzo, G., Marinoschi, G., Reali, A., & Rocca, E. (2019). Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. arXiv preprint arXiv:1907.11618.
Author Response
Point 1:
General: The paper is hard to read. Please consider revising the style to ease the comprehension of the paper.
Response 1:
I have tried to simplify the seemingly difficult parts of the paper. Moreover, due to the remark of another reviewer, the paper has been now shortened, which should also ease the comprehension.
Point 2:
Introduction: The review of main tumour growth models and the features (or hallmarks) that induce tumour growth is well written. A revision of AAT history and effects is also well written. However, a comparison of the model presented in the paper with other models is lacking. I wonder about the differences between the model developed by the author's research group and other models. Please, add some background and comparison of the model presented in the paper.
Response 2:
I have significantly rewritten the Discussion in accordance with this remark as well as the remarks by other two reviewers.
Point 3:
Line 126-147. In this paragraph, the authors describe the effect of the antiangiogenic therapy (AAT). AAT effect is the focus of the paper. Therefore, this paragraph is of paramount importance. It is hard to read (I needed to read it at least four times), so please, take a conscientious effort to rewrite it.
Response 3:
I have tried my best to make it more comprehensible.
Point 4:
Also, the conclusion (AAT decreases P to the normal tissue value) is surprisingly simple. Is this simplification enough? How many hours are needed for the AAT effect to show up? How does this simplification stand compared to more complex models such as the one presented in: Colli, P., Gomez, H., Lorenzo, G., Marinoschi, G., Reali, A., & Rocca, E. (2019). Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. arXiv preprint arXiv:1907.11618.
Response 4:
This simplification allows to consider the two major effects of AAT on a qualitative level, allowing to perform analytical estimations. Therefore, it is definitely enough for the purposes of the work. Under more straightforward consideration of angiogenesis, i.e., introduction of a separate variable for VEGF, which would affect the capillaries, one would obtain a distribution of capillaries with non-uniform alterations in their density and permeability. Such approach (but usually without account for alterations in capillaries permeability) is typical for reaction-diffusion models [Swanson, Kristin R., et al. "Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology." Cancer research 71.24 (2011): 7366-7375.][Szomolay, Barbara, et al. "Modeling the inhibition of breast cancer growth by GM-CSF." Journal of theoretical biology 303 (2012): 141-151.][Alfonso, J. C. L., et al. "Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights." Scientific reports 6 (2016): 37283.]
However, for every simulation of such extended model, there will exist some value of P, lying in the range between its normal value and the maximum local product of the capillary density and their effective permeability, which in the simplified model would provide the same increase in tumor growth speed. Moreover, the expression for “a limit of the tumor growth speed under infinite increase in the number of capillaries and/or in their permeability” for a compact tumor with no capillaries inside it will remain the same in the extended model. Now I have added such reasoning in the discussion.
The effect of AAT shows up gradually at a range of several days – see, e.g., the following experimental papers [Dings, Ruud PM, et al. "Scheduling of radiation with angiogenesis inhibitors anginex and Avastin improves therapeutic outcome via vessel normalization." Clinical Cancer Research 13.11 (2007): 3395-3402.] [Abdollahi, Amir, et al. "Combined therapy with direct and indirect angiogenesis inhibition results in enhanced antiangiogenic and antitumor effects." Cancer research 63.24 (2003): 8890-8898.]
At a high level, the model effect of AAT in my model corresponds to the one in the abovementioned paper by Colli et al., since therein “antiangiogenic therapy is modeled as a reduction in intratumoral nutrient supply“. In my work both considered effects, induced by VEGF, affect the inflow of glucose from the capillaries, thus, the action of AAT as well comes down to the reduction in tumor nutrient supply. However, in their work there is no explicit account for the microvessels, which is overcome by an assumption of two qualitatively different spatially separate types of nutrient sources for tumor and healthy tissue. On the one hand, the shortcoming of such approach is the presence of nutrient sources in the central part of the tumor, which should be devoid of capillaries or, at least, have few of them compared to the normal tissue. Moreover, the necrotic zones are as a rule located in the centers of the tumors with characteristic length of several millimeters. However, it seems that in their simulations there is always maximum tumor cell density at the center of a sufficiently large tumor, at which cell proliferation stops. So effectively this aspect may cancel out the mentioned shortcoming, at least for the consideration of the tumor surface dynamics (i.e., for it, it does not matter what exactly is located inside the tumor, as long as it is inactive). However, it is hard to judge about the degree of correspondence of such method to the one, mentioned above, without direct numerical comparison of the approaches.
Reviewer 4 Report
The paper developed a PDE-ODE coupled mathematical model to study the growth of tumor by considering various factors. Confirmed by the numerical simulations, analytical estimates of tumor growth are obtained for both compact and invasive tumors. Overall the paper is well represented, and the results look reasonable. Here are a few issues that need to be addressed,
1: It is not very convincing why 1-D model is sufficient to study the system. A more detailed discussion for high dimensional models are needed.
2: For the numerical simulations, the paper only briefly describes the numerical methods by words, while the discretized equations with a table to prove the order of convergence should be presented though C++ code is given in the supplement.
3: It is also questionable for the model validation. Is there any experimental/clinical data available to verify the model prediction?
Author Response
Point 1:
It is not very convincing why 1-D model is sufficient to study the system. A more detailed discussion for high dimensional models are needed.
Response 1:
Consideration of radial symmetry is a very general approach for continuous models of tumor growth, when there is no interest in heterogeneous effects, apart from radially-oriented proliferative heterogeneity. Consideration of higher dimensions may lead to new effects, like the fingering instability of a tumor surface, which can be regarded as a way for a tumor to counteract diffusional limitations of the nutrient inflow. [Franks, S. J., and J. R. King. "Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties." Mathematical medicine and biology 20.1 (2003): 47-89.] [Franks, S. J., and J. R. King. "Interactions between a uniformly proliferating tumour and its surroundings: Stability analysis for variable material properties." International journal of engineering science 47.11-12 (2009): 1182-1192.] However, while this phenomenon may affect the tumor growth speed quantitatively, the main qualitative result of the presented work, obtained under the assumption of radial symmetry, can hardly be affected in higher dimensions. I have added the corresponding reasoning in the discussion, citing the mentioned works.
Point 2:
For the numerical simulations, the paper only briefly describes the numerical methods by words, while the discretized equations with a table to prove the order of convergence should be presented though C++ code is given in the supplement.
Response 2:
In order to check the adequacy and efficiency of the used methods, I have performed the set of simulations with increasingly fine discretization, which are discussed in subsections 3.1 and 3.2. These simulations were to answer, can these numerical methods yield sufficiently close estimations of tumor growth speeds to the analytical ones, which, as it turned out, they indeed can. Since the model describes biological objects, for which significant variability is natural, there is, to my opinion, no need to perform more detailed investigation of the performance of the methods, and to pursue greater accuracy by spending more computational time, i.e., an error of no more than 5% is absolutely satisfactory. It is the qualitative behavior of the model that is of much greater importance, and it cannot be affected by the used methods as long as the discretization is tuned appropriately.
I have to admit, that presenting the discretized equations, in my opinion, would be redundant for this paper, since: 1) the Euler schemes should be very familiar to a broad audience; 2) adequate description of the flux-corrected transport algorithm would require a lot of space in the text, and it will merely repeat a part of the program code.
Point 3:
It is also questionable for the model validation. Is there any experimental/clinical data available to verify the model prediction?
Response 3:
The conclusion that “the tumors, which initially have invasive phenotype, should possess intrinsic resistance to AAT, while compactly growing tumors should be more susceptible to it” is in good agreement to the clinical and pre-clinical data. This issue has been already discussed in the Introduction part, the corresponding part being now left unaltered. I have tried to rephrase the corresponding part in the Discussion in order to make it more clear, and have repeated there the references from the Introduction to the papers, which analyze a large amount of pre-clinical and clinical data, eventually indicating the corresponding hypothesis [Ebos, John ML, and Robert S. Kerbel. "Antiangiogenic therapy: impact on invasion, disease progression, and metastasis." Nature reviews Clinical oncology 8.4 (2011): 210.] [Bergers, Gabriele, and Douglas Hanahan. "Modes of resistance to anti-angiogenic therapy." Nature Reviews Cancer 8.8 (2008): 592-603.].
Round 2
Reviewer 1 Report
The author has addressed my concerns.
I will modify my comment about the graphs, however. I see now that you do have labels. However, they are in quite non-traditional locations, not where a reader would be expecting to see them. I encourage you to move them to more traditional locations (under and beside the graphs) and to modify the graphs as needed so that they are still legible.
