Optimal regulation of tumour-associated neutrophils in cancer progression

In a tumour microenvironment, tumour-associated neutrophils could display two opposing differential phenotypes: anti-tumour (N1) and pro-tumour (N2) effector cells. Converting N2 to N1 neutrophils provides innovative therapies for cancer treatment. In this study, a mathematical model for N1-N2 dynamics describing the cancer survival and immune inhibition in response to TGF-β and IFN-β is considered. The effects of exogenous intervention of TGF-β inhibitor and IFN-β are examined in order to enhance N1 recruitment to combat tumour progression. Our approach employs optimal control theory to determine drug infusion protocols that could minimize tumour volume with least administration cost possible. Four optimal control scenarios corresponding to different therapeutic strategies are explored, namely, TGF-β inhibitor control only, IFN-β control only, concomitant TGF-β inhibitor and IFN-β controls, and alternating TGF-β inhibitor and IFN-β controls. For each scheme, different initial conditions are varied to depict different pathophysiological condition of a cancer patient, leading to adaptive treatment schedule. TGF-β inhibitor and IFN-β drug dosages, total drug amount, infusion times and relative cost of drug administrations are obtained under various circumstances. The control strategies achieved could guide in designing individualized therapeutic protocols.

1, The authors define Q1 to Q4 in the Results and Discussion section, but how do you apply them to individual cases in the actual clinical settings? How do you measure the level of N1 and N2 component? Is it by a blood test or histopathological examination such as surgical specimens or biopsy specimens? 2, Have you verified the results of this mathematical model in actual clinical cases or in vitro? If it can be shown that the results of actual biological research and mathematical models match, I think it will be a more meaningful paper in the clinical settings. However, it does not force the authors to do biological research when it is difficult to research.

Decision letter (RSOS-210705.R0)
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Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) Dear Editor, The article presents a novel model of antitumor and protumor neutrophils, which they call N1 and N2 neutrophils, respectively. Then they formulate an optimal control problem in which they administer cytokine therapy to shift the balance of neutrophils towards the antitumor N1 state to help treat the tumor.
I found the work interesting and well organized. They present the results of four optimal control scenarios systematically and thoroughly. It is a firm basis for people, including the authors themselves, to build on to further study this area.
Minor comments: In Figure 1, the authors need to define L and S earlier, so that Figure 1 makes sense. They cannot only define L and S long after presenting Figure 1. In addition to defining these variables int eh text, they ought to also define L and S (and G, C, I) in the figure caption for easy referencing.
On p. 4 at model (1), they authors should describe the model thoroughly and explain the terms even if the model already explained in reference [54]. The reader should not be required to look up [54] to understand the model.
In Figure 2 and elsewhere, make sure all the fonts in the figures re at least 10 point.
On p. 6, in Equations (5) to (8), why did the authors pick these functions to optimize? The authors ought to include an explanation of the reasons?
Reviewer: 2 Comments to the Author(s) In this manuscript entitled "Optimal regulation of tumor-associated neutrophils in cancer progression," based on the report that TAN (tumor-associated neutrophils) status in the tumor microenvironment regulates tumor growth, the authors developed a mathematical model that TGF-β inhibitor and/or IFN-β administration promotes N2 to N1 polarization of TAN and suppresses tumor growth. Then, the authors derived optimal therapeutic strategy for cancer using TGF-β inhibitor and/or IFN-β administration using optimal control theory.
The manuscript demonstrates a large amount of simulation data but seems to lack a clear interpretation of the findings of this study. The authors should clearly show the significance and limitation of this study. Listed below are my specific comments.
1. This manuscript, especially in the Introduction and Conclusion sections, covers a wide range of topics with many references. As a result, this reviewer could not clearly understand the conclusions of this study and thinks this paper should be more concise and focused only on the main point.
2. Since TGF-β inhibitor and IFN-β affect a broader range of cell types in the tumor microenvironment other than the TAN, Treg, CD8+, and Th17 cells that considered in the mathematical model, the simulations in this study may be focused on the limited effects of TGFβ inhibitor and IFN-β in the regulation of tumor growth. Of course, it is difficult to take all factors into account in a mathematical model. Still, the limitation should be clarified, for example, that this mathematical model ignores the effects of TGF-β inhibitor and IFN-βon NK cells, tumor-associated fibroblasts, and so on.
3. The half-life of circulating neutrophils is reported to be 19 hours (Lahoz-Beneytez et al., Blood, 2016). Therefore, this reviewer speculates that all TANs would be replaced in a few days, although the tumor microenvironment may prolong the life of TANs as the authors described in the Discussion. In Figure 5, for example, the N2/N1 ratio is kept around 1.0 under the conditions that IFN-β remains at zero concentration for several days after degradation. However, this reviewer concerns that the N2/N1 ratio would return to the initial condition during the IFN-β-free period due to the turnover of TANs. 4.The differential equations (2) consist of 6 variables (L, G, C, S, I, and T). If this reviewer understands correctly, the time-course of L, G, C, S, and I are shown according to the four optimal control scenarios (Figures 3, 5, 7, and 10), but "T" does not appear in any figures. Therefore, the authors should show the time-course of T, how the tumor size is regulated in response to the treatment of TGF-β inhibitor and/or IFN-β.
I have a few additional minor comments, explained below. 5. In the third paragraph of the Introduction, the authors say, "TGF-β inhibitor such as gefitinib." However, gefitinib is an EGFR inhibitor, not a TGF-β inhibitor.
6. The authors assumed that TGF-β inhibitor and IFN-β can be administrated every other day in the simulations, but what is the evidence?
Reviewer: 3 Comments to the Author(s) In the paper the authors consider a mathematical model of tumor-associated neutrophils in lung cancer progression. This model uses a mathematical model to explain the optimal control problems of TGFβ and INFβ. The part of the mathematical model has been revised in detail in a previous review of another Journal. However, there are some questions about the contents of the paper. Below are these comments: 1, The authors define Q1 to Q4 in the Results and Discussion section, but how do you apply them to individual cases in the actual clinical settings? How do you measure the level of N1 and N2 component? Is it by a blood test or histopathological examination such as surgical specimens or biopsy specimens? 2, Have you verified the results of this mathematical model in actual clinical cases or in vitro? If it can be shown that the results of actual biological research and mathematical models match, I think it will be a more meaningful paper in the clinical settings. However, it does not force the authors to do biological research when it is difficult to research.

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Recommendation?
Accept as is  (1), should say something like "in the equation for C" rather than "in Eq of `C' " and do the same for I, T, and N. Also, the authors should go through the manuscript and fix up the English phrasing and grammar.

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Comments to the Author(s)
The authors have responded to all the concerns raised by this reviewer.

Decision letter (RSOS-210705.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Dr Kim,
It is a pleasure to accept your manuscript entitled "Optimal regulation of tumor-associated neutrophils in cancer progression" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Also, the authors should go through the manuscript and fix up the English phrasing and grammar. The article presents a novel model of antitumor and protumor neutrophils, which they call N1 and N2 neutrophils, respectively. Then they formulate an optimal control problem in which they administer cytokine therapy to shift the balance of neutrophils towards the antitumor N1 state to help treat the tumor.
I found the work interesting and well organized. They present the results of four optimal control scenarios systematically and thoroughly. It is a firm basis for people, including the authors themselves, to build on to further study this area.
Minor comments: • In Figure 1, the authors need to define L and S earlier, so that Figure 1 makes sense. They cannot only define L and S long after presenting Figure 1. In addition to defining these variables int eh text, they ought to also define L and S (and G, C, I) in the figure caption for easy referencing.
(Response) Thank you for careful reading and suggestions. We now defined the main variables at the end of Introduction Section so that it can naturally appear before Fig 1. So, we added the following phrase : "In this work, we consider the following variables in a mathematical model: In addition, in order to make it consistent in the main text, we removed the definition of L, S, G by replacing "dimensionless form where TGF-β inhibitor and TGF-β are denoted by L and G, respectively; N1 and N2 complexes are grouped as module C and I, respectively; and IFN-β is represented by S." with "dimensionless form; N1 and N2 complexes are grouped as module C and I, respectively." [page 6, 1st paragraph, line 7, in the (marked) revised manuscript ] • On p.4 at model (1), they authors should describe the model thoroughly and explain the terms even if the model already explained in reference [54]. The reader should not be required to look up [54] to understand the model.
(Response) Thank you for careful reading and suggestion. In order to assist the understanding of the model without looking up Kim et al, 2019 [28] as follows: [page 4, Eq (1), in the (marked) revised manuscript ] In addition, we added the following statements below the equations above in order to explain the model: "Here, two sources (IL-6, TGF-β) of the N2 complex are represented in the first and second terms in Eq of 'C' while the IFN-β-mediated source of the N1 complex is provided in the first term in Eq of 'I'. The third term in Eq of 'C' and second term in Eq of 'I' represent the mutual inhibition between N1 and N2, respectively. Tumor growth and N1-mediated tumor cell killing are represented in the first and second term in Eq of 'T ', respectively. Finally, decay process of the N1 and N2 complexes in the last terms in Eq of 'C' and 'N ', respectively. " [page 4, lines 1-6 starting from Eq (1), in the (marked) revised manuscript ] • In Figure 2 and elsewhere, make sure all the fonts in the figures are at least 10 point.
(Response) Thank you for careful reading. We increased all the fonts in all figures (from Fig 1 to Fig 13). Now, those figures look much better with the larger fonts.
In addition, we made the following changes (blue) for caption of Fig 2 by replacing "Nullclines of model (1)  • On p.6, in Equations (5) to (8), why did the authors pick these functions to optimize?
The authors ought to include an explanation of the reasons?
(Response) Thank you for careful suggestions. Some of reasons are already mentioned in the text. For instance, in equation 5, it is written: "TGF-β inhibitor control only. In this scheme, we want to investigate the anti-tumor effect of TGF-β inhibitor on tumor growth." In addition, following the reviewer's suggestion, we now made the following changes: We replaced "The OCP" with "In particular, we want to minimize the tumor size (T ) and dose of TGF-β inhibitor while keeping the same IFN-β supply. Thus, the OCP" [page 7, 1st paragraph, lines 2-3, in the (marked) revised manuscript ] We replaced "The problem" with "In particular, we want to minimize the tumor size (T ) and dose of IFN-β in the absence of IFN-β inhibitor. Thus, the problem" [page 7, 2nd paragraph, lines 1-3, in the (marked) revised manuscript ] We replaced "The OCP " with "Specifically, we want to minimize the tumor size (T ) and doses of both TGF-β inhibitor and IFN-β. " [page 7, 3rd paragraph, lines 2-3, in the (marked) revised manuscript ] We replaced "The OCP " with "Specifically, we want to minimize the tumor size (T ) and doses of both TGF-β inhibitor and IFN-β in an alternating injection scheme. Thus, the OCP" [page 7, 4th paragraph, lines 2-4, in the (marked) revised manuscript ] In addition to these changes, we made the following minor changes in the (marked) revised manuscript in order to meet the journal requirement (from editorial staffs) (i) "Ohio, " → "OH 43210, " [page 1, author affiliation ]

REFEREE 2:
In this manuscript entitled "Optimal regulation of tumor-associated neutrophils in cancer progression," based on the report that TAN (tumor-associated neutrophils) status in the tumor microenvironment regulates tumor growth, the authors developed a mathematical model that TGF-β inhibitor and/or IFN-β administration promotes N2 to N1 polarization of TAN and suppresses tumor growth. Then, the authors derived optimal therapeutic strategy for cancer using TGF-β inhibitor and/or IFN-β administration using optimal control theory.
The manuscript demonstrates a large amount of simulation data but seems to lack a clear interpretation of the findings of this study. The authors should clearly show the significance and limitation of this study. Listed below are my specific comments.
1. This manuscript, especially in the Introduction and Conclusion sections, covers a wide range of topics with many references. As a result, this reviewer could not clearly un-derstand the conclusions of this study and thinks this paper should be more concise and focused only on the main point.
(Response) In order to make it more structured and focused, we now added the following paragraph in the INTRODUCTION SECTION: "We found therapeutic regimen at regulating anti-and protumoral neutrophil phenotypic states by means of four different administration modalities of a TGF-β inhibitor and IFN-β cytokine in lung cancer. The optimal control strategy predicts that, depending on pathological states, therapies may have to be adjusted accordingly to minimize adverse effects of drugs and its administration cost." [INTRODUCTION SECTION, page 2, 6th paragraph (whole), in the (marked) revised manuscript ] In the same vein, in order to make Conclusion Section more structured and focused, we added the following paragraph: "We identified therapeutic regimen for promoting anti-tumoral TANs and suppressing protumoral neutrophil phenotypic states through four different administration of the TGF-β inhibitor and IFN-β in lung cancer. The optimal control scheme predicts that, depending on relative states of N1 and N2 phenotypes, therapy schedules may have to be adjusted properly to minimize adverse effects of these drugs and its administration cost in addition to maximizing anti-tumor efficacy. " 2. Since TGF-β inhibitor and IFN-β affect a broader range of cell types in the tumor microenvironment other than the TAN, Treg, CD8+, and Th17 cells that considered in the mathematical model, the simulations in this study may be focused on the limited effects of TGF-β inhibitor and IFN-β in the regulation of tumor growth. Of course, it is difficult to take all factors into account in a mathematical model. Still, the limitation should be clarified, for example, that this mathematical model ignores the effects of TGF-β inhibitor and IFN-β on NK cells, tumor-associated fibroblasts, and so on.
(Response) Thank you for careful reading and suggestions. We now organized limitations in three items with more detailed discussions on limitations of our work. In particular, we included detailed discussions on effect of TGF-β inhibitor and IFN-β on NK cells and tumor-associated fibroblasts. So, we replaced "It has been reported that a more a more realistic clinical situation employs linear cost functions. These type of problems have been used in several works devoted to the administration of single and combination therapies to treat different types of tumors [19,34,50,51,55]. On the contrary, a study by Glick and Mastroberardino [15] concluded that quadratic control yields continuous, low doses of the therapeutic drug producing a better outcome in the eradication of the solid tumor. Several researches on optimal control approaches for cancer treatment still favor quadratic controls [41,52,58]. Under certain circumstances, both linear and quadratic controls obtain qualitatively similar results [9,12,33]. It is true that profiles of treatment strategies depend on the landscape of the cost functionals. These differences show the importance of carefully defining an objective functional that most accurately reflects the toxicities of a particular drug along with the objective of the treatment strategy, for instance, decreasing the tumor mass at the end of the treatment interval, reducing the overall tumor burden over the treatment interval, and some other clinically relevant criteria [14]. It is therefore suggested that further model iterations should include exhaustive investigations on linear controls to have a holistic treatment strategies for cancer treatment. Further analysis of models with other types of objective functionals should also be pursued. Varying results should then be presented to the medical practitioner for them to decide which solutions best fit the biological situation and can be used for practical implementation, if any [33]. The current work does not consider the role of other factors in TME such as intracellular pathways such as STAT1/STAT3 in response to IFN-β, TGF-β, and other stimuli [18], neutrophil elastase (NE) in the presence [20] and absence [43,16] of LPS, tumor-promoting or -suppressive immune response of NK cells [32,1], ECM remodeling [11,35,27,29], angiogenesis from blood vessels [49,48,61], or growth factors such as EGF [59,40], and CSF-1 [26,47]. NK cells were shown to mediate dual roles of neutrophils in metastatic colonization [38,60]. For example, neutrophils can enhance extravasation of circulating tumor cells by suppressing NK cells [60]. Depletion or adjuvant therapy of NK cells was shown to increase anti-tumor efficacy in a combination therapy (OV-bortezomib-NK) relative to control, showing nonlinear behavior of immune system [32]. These factors may play major roles in regulation of N1→N2 transition, thus cancer progression. For example, NE and NET were shown to promote tumor growth and invasion [37] by turning on the multiple signaling pathways including PI3K in lung cancer cells [24]. In particular, up-regulated NET activities near tumour sections induce the transformation of B cells [54], contributing to cancer progression [65]. However, NET was also suggested to suppress tumor growth in colonic adenocarcinoma [2]. These factors and unforeseen microenvironmental factors may limit bi-lateral switches between N1 and N2 TANs. In our study, optimal control was applied only to IFN-β and TGF-β inhibitors. We plan to investigate the specific role and optimization of these stimuli in TME for better understanding of role of TANs (either promotion [7,10,66,64] or suppression of tumor progression [2]). A new optimal control method has to be developed for a possible triple combination therapy, i.e., TGF-β+TGF-β inhibitor+immune agents [3]. This work, however, provides a general framework of optimal control approach for the fundamental transition from N2 and N1 TANs, thus inhibiting tu-mor growth, in response to known key players. We plan to develop an optimal control of the TAN's plasticity of the possibly continuous spectrum of the N1→N2 switches, which requires better understanding and experiments of the biological system as well as advanced optimal control theory." with "Our study has four limitations: • The current work does not consider the role of other factors in TME such as intracellular pathways, for instance, STAT1/STAT3/JAK [36] in response to IFN-β, TGF-β, and other stimuli [18], neutrophil elastase (NE) in the presence [20] and absence [43,16] of LPS, tumor-promoting or -suppressive immune response of NK cells [32,1], tumor-associated fibroblasts (TAFs) [69,53,31,30], ECM remodeling [11,35,27,29], angiogenesis from blood vessels [49,48,61], or growth factors such as EGF [59,40], and CSF-1 [26,47]. NK cells were shown to mediate dual roles of neutrophils in metastatic colonization [38,60]. For example, neutrophils can enhance extravasation of circulating tumor cells by suppressing NK cells [60]. Depletion or adjuvant therapy of NK cells was shown to increase anti-tumor efficacy in a combination therapy (OV-bortezomib-NK) relative to control, showing nonlinear behavior of immune system [32]. These factors may play major roles in regulation of N1→N2 transition, thus cancer progression. In particular, we focused on the effect of TGF-β inhibitor and IFN-β on TANs in this study but it would be important to see the effect of those two inhibitors on NK cells and TAFs in future work. For example, TGF-β and its inhibitor [69,53,17] and IFNs [4,17,39,6] play an important role in regulation of tumor-associated fibroblasts in cancer progression. We plan to include those players in a future optimal-control model. • NET and NE were shown to promote tumor growth and invasion [37] by turning on the multiple signaling pathways including PI3K in lung cancer cells [24]. For example, up-regulated NET activities near tumour sections induce the transformation of B cells [54], contributing to cancer progression [65]. In particular, mathematical models [37] and experimental data [45] suggest that NET can mediate the critical metastatic process [13,8] to stabilize the circulating tumor cells in the blood stream and help extravasation of these cancerous cells. However, NET was also suggested to suppress tumor growth in colonic adenocarcinoma [2]. In our study, we did not take into account these critical influence of NETs in a spatial domain. We plan to develop an optimal control approach in a new framework of the partial differential equations of NETs in order to improve the therapeutic, anti-invasion strategies. • Unforeseen microenvironmental factors may limit bi-lateral switches between N1 and N2 TANs. In our study, optimal control was applied only to IFN-β and TGF-β inhibitors. We plan to investigate the specific role and optimization of these stimuli in TME for better understanding of role of TANs (either promotion [7,10,66,64] or suppression of tumor progression [2]). A new optimal control method has to be developed for a possible triple combination therapy, i.e., TGF-β+TGF-β inhibitor+immune agents [3]. We plan to develop an optimal control of the TAN's plasticity of the possibly continuous spectrum of the N1→N2 switches, which requires better understanding and experiments of the biological system as well as advanced optimal control theory.
• It has been reported that a more a more realistic clinical situation employs linear cost functions. These type of problems have been used in several works devoted to the administration of single and combination therapies to treat different types of tumors [19,34,50,51,55].
On the contrary, a study by Glick and Mastroberardino [15] concluded that quadratic control yields continuous, low doses of the therapeutic drug producing a better outcome in the eradication of the solid tumor. Several researches on optimal control approaches for cancer treatment still favor quadratic controls [41,52,58]. Under certain circumstances, both linear and quadratic controls obtain qualitatively similar results [9,12,33]. It is true that profiles of treatment strategies depend on the landscape of the cost functionals. These differences show the importance of carefully defining an objective functional that most accurately reflects the toxicities of a particular drug along with the objective of the treatment strategy, for instance, decreasing the tumor mass at the end of the treatment interval, reducing the overall tumor burden over the treatment interval, and some other clinically relevant criteria [14]. It is therefore suggested that further model iterations should include exhaustive investigations on linear controls to have a holistic treatment strategies for cancer treatment. Further analysis of models with other types of objective functionals should also be pursued. Varying results should then be presented to the medical practitioner for them to decide which solutions best fit the biological situation and can be used for practical implementation, if any [33].
This work, however, provides a general framework of optimal control approach for the fundamental transition from N2 and N1 TANs, thus inhibiting tumor growth, in response to known key players. " [page 17, last paragraph -page 18, 5th paragraph, in the (marked) revised manuscript ] 3. The half-life of circulating neutrophils is reported to be 19 hours (Lahoz-Beneytez et al., Blood, 2016). Therefore, this reviewer speculates that all TANs would be replaced in a few days, although the tumor microenvironment may prolong the life of TANs as the authors described in the Discussion. In Figure 5, for example, the N2/N1 ratio is kept around 1.0 under the conditions that IFN-β remains at zero concentration for several days after degradation. However, this reviewer concerns that the N2/N1 ratio would return to the initial condition during the IFN-β free period due to the turnover of TANs.
(Response) Thank you for pointing out this and thoughtful questions. Yes, the half-life of TANs is short and TME tends to prolong the half-life. In Fig  5, green bar does not represent IFN-β concentration rather it represents the rate of injection of IFN-β. The IFN-β concentration is marked in green curve in Fig 5A. The whole dynamics essentially leads to relatively similar population of N1 (pink curve in Fig 5A) and N2 (blue curve in Fig 5A) TANs in response to the fluctuating levels of IFN-β (concentration green curve in Fig 5A) following governing equations (2). Please note that concentrations of N2 and N1 TANs are in the similar ranges in Fig 5A. Please note that the bifurcation diagrams in  Added texts in main text for Fig 4: "Time courses of tumor volume in these four cases illustrate the relatively poor anti-tumor efficacy when the initial condition is in the tumorigenic region Q 4 (Figure 4(C)), despite the highest cost (Figure 4(A))." [page 9, 1st paragraph, lines 4-6, in the (marked) revised manuscript ] I have a few additional minor comments, explained below.

5.
In the third paragraph of the Introduction, the authors say, TGF-β inhibitor such as gefitinib. However, gefitinib is an EGFR inhibitor, not a TGF-β inhibitor.
(Response) Thank you for careful reading. We added the evidences for IFN-β (as well as other references for toxicity) by replacing "Taking into account that IFN-β is a noxious drug, it can be administered only every other day in increasing amount." with "Taking into account that IFN-β is a noxious drug [62,25,63], it can be administered only every other day in increasing amount as illustrated in experimental system [46,5,62]." [page 10, 1st paragraph, lines 2-4, in the (marked) revised manuscript ] In addition to these changes, we made the following minor changes in the (marked) revised manuscript in order to meet the journal requirement (from editorial staffs) (i) "Ohio, " → "OH 43210, " [page 1, author affiliation ] REFEREE 3: In the paper the authors consider a mathematical model of tumor-associated neutrophils in lung cancer progression. This model uses a mathematical model to explain the optimal control problems of TGF-β and IFN-β. The part of the mathematical model has been revised in detail in a previous review of another Journal. However, there are some questions about the contents of the paper. Below are these comments: 1. The authors define Q1 to Q4 in the Results and Discussion section, but how do you apply them to individual cases in the actual clinical settings? How do you measure the level of N1 and N2 component? Is it by a blood test or histopathological examination such as surgical specimens or biopsy specimens?
(Response) As we can see from the analysis of the mathematical model, there are relatively high and low probabilities of moving toward the N2dominant status (high N2 and low N1), i.e, tumorigenic status, depending on the relative proportion of N1 and N2 populations in the given patient as in Fig 4. We acknowledge that it is very challenging to measure the level of N1 and N2 components. The researchers in the field is moving fast for further experiments and etc. For instance, Ohms et al (2020) [44] could measure the relative proportion of N1 and N2 populations. However, it is still challenging to measure and use it in a clinical setting yet. If one can measure individual levels of N2 and N1 in the blood and TME in each patient, not N2/N1 ratio, whether it is by a blood test or histopathological examination such as surgical specimens or biopsy specimens, one can determine their locations (Q1, Q2, Q3, Q4) in the N2-N1 phase plane and apply our strategies for optimal results. As we mentioned in the response to item 2 below, we are vigorously working with experimentalists (professor Jinsu Kim for instance) and clinicians at KIRAMs and Korea Institute of radiological and medical sciences in Seoul, South Korea, in order to develop this particular method to overcome this technical difficulties.

2.
Have you verified the results of this mathematical model in actual clinical cases or in vitro? If it can be shown that the results of actual biological research and mathematical models match, I think it will be a more meaningful paper in the clinical settings. However, it does not force the authors to do biological research when it is difficult to research.
(Response) Thank you very much for nice suggestions. It would be great if we can show that the simulated results from the mathematical model can be integrated or verified in a series of experiments or in a clinical setting. Our colleagues at KIRAMs and Korea Institute of radiological and medical sciences (hospital especially for cancer patients at last stage) are actually trying very hard to do experiments to verify and optimize the outcomes of the treatment. However, practically speaking, this type of experiments requires lots of approval and internal processes and it usually takes much time to get patients and setup the program in the clinical setting. We also tried to integrate the simulating tools to the system at the hospital but we have to overcome a large number of hurdles due to government regulations and ethical issues etc. We hope to do it in near future. However, our paper could contribute to better research environment where experimentalists can actively participate for joint projects so that the system with mathematical models can function altogether.
In addition to these changes, we made the following minor changes in the (marked) revised manuscript in order to meet the journal requirement (from editorial staffs) (i) "Ohio, " → "OH 43210, " [page 1, author affiliation ]