Optimal Control and Tumour Elimination by Maximisation of Patient Life Expectancy

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

• Include some important numeric outcomes in the abstract and also percentage increament.
• There are some typo mistakes must be carefully removed from the whole article.
• The introduction section should be rearranged as well. The references list should be updated like as https://doi.org/10.1038/s41598-025-01161-5, https://doi.org/10.3390/math12182920
• How does the LEP framework fundamentally advance beyond traditional objectives (e.g., tumor size minimization) in reconciling stochastic events (cure/relapse/TRM) with deterministic optimal control?
• Does the TCP approximation (Eq. 9–12) hold for small tumors (N≪10³), where stochastic extinction dynamics dominate deterministic means?
• Why are intermediate SAEs (e.g., grade 3–4 events causing treatment cessation) omitted, and how would their inclusion alter optimal strategies?
• Can the "treatable/untreatable" dichotomy (Figs. 5,12) be translated to clinical biomarkers, and what misclassification risks arise?
• What steps are planned to validate predictions (e.g., treat-and-stop efficacy) against real-world data or clinical trials?
• How do LEP-derived protocols compare to adaptive therapy (dose modulation) or Pareto-optimal solutions in balancing toxicity/tumor control?
• The conclusions follow from the presented results. I suggest that it is better if authors can provide the future research prospective at the end of article.
• References are not cited as required by the journal, correct as per journal requirement.

Author Response

Comment 1: Include some important numeric outcomes in the abstract and also percentage increment. 

Response 1: these numbers would be dependent on chosen parameters, including them into the abstract could be misleading

Comment 2: There are some typo mistakes must be carefully removed from the whole article.

Response 2: Thank you for pointing this out. We have carefully reviewed the manuscript and corrected all identified typographical errors.

Comment 3: The introduction section should be rearranged as well. The references list should be updated like as https://doi.org/10.1038/s41598-025-01161-5, https://doi.org/10.3390/math12182920.

Response 3: Not sure what the reviewer would like to be rearranged. 

Comment 4: How does the LEP framework fundamentally advance beyond traditional objectives (e.g., tumor size minimization) in reconciling stochastic events (cure/relapse/TRM) with deterministic optimal control?

Response 4: We are introducing a new concept in LEP, where post-horizon outcomes are optimised over that depend on key events in the patient/tumour history. This is totally different from minimising tumour size at the horizon time. For instance for very large horizon times, the average tumour size on a LEP optimal solution would be very large, as there is a probability of no elimination, and so it grows after drug application is ended. Traditional objectives would likely control tumour size, possibly with periods where the tumour gets very large. Thus, the solutions are strongly dependent on the objective to be minimisise/maxmised, and the issue is which captures best outcome for the patient

Comment 5: Does the TCP approximation (Eq. 9–12) hold for small tumors (N≪10³), where stochastic extinction dynamics dominate deterministic means?

Response 5: Thank you for the insightful comment. The large tumour approximation is used to simplify the objective. This does not need to be used, and the TCP can be used valid for small tumour sizes. This is now discussed in lines 208-212.

Comment 6: Why are intermediate SAEs (e.g., grade 3–4 events causing treatment cessation) omitted, and how would their inclusion alter optimal strategies?

Response 6: Modeling intermediate SAEs would require a recursive optimization framework that dynamically accounts for expected lifespan conditioned on a SAE occurring. This gives a complex optimisation problem beyond the scope of the current study.

Comment 7: Can the "treatable/untreatable" dichotomy (Figs. 5,12) be translated to clinical biomarkers, and what misclassification risks arise?

Response 7: It is certainly the case that the treatable/untreatable classification will be dependent on the cancer subtype (and thus clinical biomarkers). Analysis of this is beyond the scope of the present manuscript.

Comment 8: What steps are planned to validate predictions (e.g., treat-and-stop efficacy) against real-world data or clinical trials?

Response 8: We are looking at specific cancers where we hope to parametrise our models and ultimately validate our method. This is ongoing.

Comment 9:  How do LEP-derived protocols compare to adaptive therapy (dose modulation) or Pareto-optimal solutions in balancing toxicity/tumor control?

Response 9: LEP could be extended to adaptive therapy, ie utilise additional information on patient/tumour during treatment to optimise therapy sequentially. This is currently beyond the scope of this manuscript. We also have not analysed Pareto optimal criteria within the LEP approach.

Comment 10: The conclusions follow from the presented results. I suggest that it is better if authors can provide the future research prospective at the end of article.

Response 10: Thank you for your suggestion. We added a paragraph (lines 729-753) relative to potential real-world applications of LEP

Comment 11: References are not cited as required by the journal

Response 11: We have now correct as per journal requirement.

Reviewer 2 Report

Comments and Suggestions for Authors

The study develops this framework within a deterministic control theory context, using ordinary differential equations (ODEs) to model tumor dynamics. By applying Pontryagin's Maximum Principle, the authors demonstrate that for the simplified models considered, the optimal treatment strategies are "bang-bang," meaning the therapy should be administered at the maximum tolerated dose or not at all. This leads to three potential therapeutic paths: continuous MTD, no treatment, or a "treat-and-stop" strategy. A key finding is that patients can be stratified into "treatable" and "untreatable" classes based on a combination of tumor characteristics, patient demographics, and drug toxicity.

Despite its strengths, the paper is not without limitations. The most significant of these is the reliance on highly simplified models of tumor growth and drug action. The use of a single-compartment logistic growth model, for instance, does not account for the complex spatial heterogeneity, microenvironment interactions, and cellular diversity that characterize most solid tumors. While the authors acknowledge this and suggest that the framework can be extended, the current analysis may not fully capture the dynamics of real-world cancers.

The authors should expand the introductory sections to provide a richer context for their work. It's crucial to discuss the evolution of optimal control theory and the various numerical approaches derived from the PMP. To address the lack of context, I would recommend that the authors mention the paper "Sixty Years of the Maximum Principle in Optimal Control: Historical Roots and Content Classification" (2024) doi:10.3390/sym16101398 . Incorporating this reference would allow the authors to frame their work within the historical evolution of the Maximum Principle and demonstrate a deeper understanding of the theoretical landscape.

Another major challenge is the practical implementation of the model, which would require the accurate estimation of numerous parameters. Determining patient-specific values for tumor growth rates, drug efficacy, and the probabilities of various outcomes would be a formidable task in a clinical setting. The paper does not fully address the methodologies for obtaining these parameters, which would be crucial for the framework's translation from theory to practice.

In conclusion, this paper makes a valuable and thought-provoking contribution to the field of mathematical oncology. The introduction of the life-expectancy pay-off function is a significant conceptual advance that has the potential to reshape how we think about optimizing cancer therapy. The study's strengths lie in its clinical relevance, potential for personalization, and mathematical rigor.

I would also suggest to check if there is a typo in the name of the last author.

Author Response

Comment 1: Despite its strengths, the paper is not without limitations. The most significant of these is the reliance on highly simplified models of tumor growth and drug action. The use of a single-compartment logistic growth model, for instance, does not account for the complex spatial heterogeneity, microenvironment interactions, and cellular diversity that characterize most solid tumors. While the authors acknowledge this and suggest that the framework can be extended, the current analysis may not fully capture the dynamics of real-world cancers.

Response 1: We agree. Extending the framework to multi compartment models is underway but beyond the scope of the current paper.

Comment 2:The authors should expand the introductory sections to provide a richer context for their work. It's crucial to discuss the evolution of optimal control theory and the various numerical approaches derived from the PMP. To address the lack of context, I would recommend that the authors mention the paper "Sixty Years of the Maximum Principle in Optimal Control: Historical Roots and Content Classification" (2024) doi:10.3390/sym16101398 . Incorporating this reference would allow the authors to frame their work within the historical evolution of the Maximum Principle and demonstrate a deeper understanding of the theoretical landscape. 

Response 2: We have added a paragraph (lines 337–347) providing brief background on PMP to help contextualize its role in our framework. Additionally, we have included remarks on numerical approaches used to solve complex optimal control problems, including PMP-based methods, in the conclusion section (lines 720–727). A review of the evolution of PMP is inappropriate for this paper.

Comment 3 : Another major challenge is the practical implementation of the model, which would require the accurate estimation of numerous parameters. Determining patient-specific values for tumor growth rates, drug efficacy, and the probabilities of various outcomes would be a formidable task in a clinical setting. The paper does not fully address the methodologies for obtaining these parameters, which would be crucial for the framework's translation from theory to practice.

Response 3 : We agree and have added a discussion of this translation to the discussion (lines:736-753). In brief, predictive models for these risks and the tumour response need to be developed, which is an active and growing field.  

Reviewer 3 Report

Comments and Suggestions for Authors

Title: Optimal Control and Tumour Elimination by Maximisation of Patient Life Expectancy

 

Manuscript Number: mathematics-3789361-peer-review-v1

 

Review Report

This interesting and well-written manuscript introduces a novel and thoughtful approach to optimizing cancer treatment by maximizing a patient’s expected lifespan. The concept of a life-expectancy payoff (LEP) is innovative and addresses limitations in existing cancer treatment optimization methods. The use of deterministic models with branching process elements and Pontryagin’s Maximum Principle is technically robust. The study is highly relevant and can potentially influence mathematical oncology and personalized medicine.

 

However, several aspects could be clarified or simplified to improve comprehension and make the work more accessible to a broader modeling audience:

 

1. The abstract is quite technical and dense. Consider simplifying the language to make it more accessible to non-mathematicians or general readers.
2. The manuscript would benefit from one or two sentences explicitly explaining how this method could be applied in real-world cancer therapy, such as:

Personalized dose scheduling;

Avoiding over-treatment in fragile patients;

Supporting clinical decision-making.

3. For readers who may be unfamiliar with control theory, provide a brief explanation of “bang-bang control” when it is first introduced.
4. Ensure consistency in citation formatting, especially on page 2, lines 54 and 56.
5. Improve the use of abbreviations for better presentation and clarity throughout the manuscript.
6. The terms DLP and SAP are introduced somewhat abruptly, so consider defining them more clearly at the beginning.
7. Including a simple diagram or flowchart illustrating how LEP works (e.g., from tumor growth → therapy → outcomes → life expectancy estimation) would help readers better visualize the approach.
8. A small summary table listing key parameters (e.g., tumor size, toxicity rates, event probabilities) and their influence on treatment outcomes would enhance clarity.
9. Proofread the manuscript carefully to correct minor grammatical and formatting issues.
10. Ensure that all references are formatted consistently and by the journal's guidelines.

 

Therefore, I recommend that it be published in your journal only after Minor revision.

 

 

Author Response

Comment 1: The abstract is quite technical and dense. Consider simplifying the language to make it more accessible to non-mathematicians or general readers

Response 1: Thank you for the feedback. We have revised the abstract to simplify the language and improve accessibility for non-specialist readers, while retaining the essential technical content.

Comment 2: The manuscript would benefit from one or two sentences explicitly explaining how this method could be applied in real-world cancer therapy, such as: Personalized dose scheduling; Avoiding over-treatment in fragile; Supporting clinical decision-making patients; 

Response 2: Thank you for this suggestion. In response, we have added a paragraph (lines 729–753) explicitly outlining how the LEP framework could be applied in real-world cancer therapy. 

Comment 3 :For readers who may be unfamiliar with control theory, provide a brief explanation of “bang-bang control” when it is first introduced

Response 3 : We have added a brief explanation of “bang-bang control” when it is first introduced (lines 340–346) to aid readers who may be unfamiliar with control theory.

Comment 4 : Ensure consistency in citation formatting, especially on page 2, lines 54 and 56

Response 4 : Citation formatting is now consistent

Comment 5: Improve the use of abbreviations for better presentation and clarity throughout the manuscript

Response 5: We are not sure what the reviewer is referring to. However, we have gone through the manuscript and ensured abbreviations are defined on first use and are consistently used.

Comment 6: The terms DLP and SAP are introduced somewhat abruptly, so consider defining them more clearly at the beginning. 

Response 6 : We have revised the text where these terms are first introduced (lines 128–132) to improve clarity and ensure their definitions are more immediately accessible to the reader.

Comment 7 : including a simple diagram or flowchart illustrating how LEP works (e.g., from tumor growth → therapy → outcomes → life expectancy estimation) would help readers better visualize the approach.

Response 7 : We have revised Fig 1 to improve understanding.

Comment 8 :A small summary table listing key parameters (e.g., tumor size, toxicity rates, event probabilities) and their influence on treatment outcomes would enhance clarity

Response 8 : We have added two summary tables for the parameters to imprive clarity: one (above line 484) listing the tumour evolution and DLP parameters, and another (above line 561) listing the SAP parameters.

Comment 9 : Proofread the manuscript carefully to correct minor grammatical and formatting issues.

Response 9: Manuscript proofread carefully grammatical and formatting issues corrected

Comment 10: Ensure that all references are formatted consistently and by the journal's guidelines

Response 10: All references are now formatted consistently and by the journal's guidelines

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Address the comments properly, and still some are unattended.
Still have mistakes in the reference styles somewhere, you mentioned Doi numbers double and single for example (see lines 962, 991, 999, 1007, 1014, 1088, 1097, 1104). So correct these and cite the suggested reading that has a deficiency.

Author Response

Comment 1 (round 2). Address the comments properly, and still some are unattended.

Response. We have revised our responses to all the previous comments as below, and made some additional changes to the manuscript (highlighted in purple). The round 1 comments are still retained in blue for clarity. We are not sure which comment we failed to address, so hope that this second pass over the comments attends to them all.

Revised round 1 comments

Comment 1: Include some important numeric outcomes in the abstract and also percentage increment.

Response 1: The abstract has been rewritten to make it more accessible and readable. Including percentage increment would be dependent on chosen parameters and model, therefore including them into the abstract could be misleading, so we have not included these.

Comment 2: There are some typo mistakes must be carefully removed from the whole article.

Response 2: Thank you for pointing this out. We have carefully reviewed the manuscript and corrected all identified typographical errors.

Comment 3: The introduction section should be rearranged as well. The references list should be updated like as https://doi.org/10.1038/s41598-025-01161-5, https://doi.org/10.3390/math12182920.

Response 3: We are not sure what the reviewer would like to be rearranged. However, we have already editted the introduction along the lines suggested by the other reviewers. We have revised the reference list and removed multiple doi’s as detailed in comment 2 (2^nd round), below. 

Comment 4: How does the LEP framework fundamentally advance beyond traditional objectives (e.g., tumor size minimization) in reconciling stochastic events (cure/relapse/TRM) with deterministic optimal control?

Response 4: Traditional objectives do not incorporate stochastic events. Therefore by devising a payoff that is weighted by the probabilities of these stochastic events we are introducing a new concept in LEP. Its fundamental advance is that it models events that substantially affect outcome, and lifespan post treatment, and partitions by outcome at the end of treatment. The fact that therapy is independent of the given horizon time is an unexpected bonus, thereby giving definitive optimal solutions valid for any sufficiently long horizon time. These points are discussed in the manuscript (lines 664-695) 

Comment 5: Does the TCP approximation (Eq. 9–12) hold for small tumors (N≪10³), where stochastic extinction dynamics dominate deterministic means?

Response 5: Thank you for the insightful comment. The large tumour approximation is used to simplify the objective. This approximation however does not need to be used, and the TCP expression (8) can be used valid for small tumour sizes. This is now discussed in lines 208-212.

Comment 6: Why are intermediate SAEs (e.g., grade 3–4 events causing treatment cessation) omitted, and how would their inclusion alter optimal strategies?

Response 6: Modeling intermediate SAEs would require an iterative optimization framework that accounts for expected lifespan conditioned on a SAE occurring. This gives a complex optimisation problem. We set out the LEP framework in section 5.1. 

Comment 7: Can the "treatable/untreatable" dichotomy (Figs. 5,12) be translated to clinical biomarkers, and what misclassification risks arise?

Response 7: It is expected that the treatable/untreatable classification will be dependent on the cancer subtype (and thus clinical biomarkers). We now mention that treatability may correlate with tumour type and stage which are linked to clinical biomarkers (line 712-714). However, analysis of the misclassification risk is beyond the scope of the present manuscript.  

Comment 8: What steps are planned to validate predictions (e.g., treat-and-stop efficacy) against real-world data or clinical trials?

Response 8: We are looking at specific cancers where we hope to parametrise our models and ultimately validate our method. This is ongoing and mentioned in the Conclusions at lines 766-790 (and related to comment 10).

Comment 9: How do LEP-derived protocols compare to adaptive therapy (dose modulation) or Pareto-optimal solutions in balancing toxicity/tumor control?

Response 9: LEP could be extended to adaptive therapy, ie utilise additional information on patient/tumour during treatment to optimise therapy sequentially. This is currently beyond the scope of this manuscript. We discuss Pareto optimality (lines 684-695). Since the LEP approach intrinsically weights outcomes, to compare to Pareto-optimal solutions we would need to define competing objects in order to carry out a Pareto analysis. This is beyond the scope of the current manuscript.

Comment 10: The conclusions follow from the presented results. I suggest that it is better if authors can provide the future research prospective at the end of article.

Response 10: Thank you for your suggestion. We added a paragraph (lines 766-790) relative to potential real-world applications of LEP

Comment 11: References are not cited as required by the journal

Response 11: We have now correct as per journal requirement.

 

Comment 2 (round 2). Still have mistakes in the reference styles somewhere, you mentioned Doi numbers double and single for example (see lines 962, 991, 999, 1007, 1014, 1088, 1097, 1104). So correct these and cite the suggested reading that has a deficiency.

Response. This has been addressed.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper can be published in the present form.

Author Response

Comment: The paper can be published in the present form

Response: Thank you very much for your positive feedback. This is the latest version of the manuscript, in which we have incorporated all the suggestions and comments provided during the previous rounds of review. We greatly appreciate your time and efforts in evaluating our work.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

Include the suggested works in my report to increase the significance. 
Also provide a table which tells how your work is novel from the previous available literature?

Author Response

Comment 1:  Include the suggested works in my report (round 1) to increase the significance. 

Response 1:  

We have already modified the manuscript with additional context (real world implementation and parametrisation), thus improving its significance. As regards the two suggested references, we do not feel they are appropriate as they are not directly relevant to the focus or methodology of our work, for instance neither relate to cancer therapy or applications of optimal control. We discuss each in turn, specifically,

• Paper 1: Application of interpretable machine learning algorithms to predict macroangiopathy risk in Chinese patients with type 2 diabetes mellitus, N. Zhang et al

  This study applies machine learning techniques to predict macroangiopathy in diabetic patients; it focuses on cardiovascular complications rather than cancer treatment. As such, it falls outside the scope of our study, which is centered on applied optimal control in cancer therapy. 

• Paper 2: Upper Bounds for the Remainder Term in Boole’s Quadrature Rule and Applications to Numerical Analysis,  M.Z. Javed et al.

This work deals with deriving upper bounds for Boole’s inequality in the context of convex and bounded mappings, with applications in numerical analysis. Our methodology does not involve such inequalities, nor do we use numerical quadrature techniques for the maximization of the LEP.  

We appreciate the reviewer’s input but respectfully maintain that these works do not contribute to enhancing the significance or context of the research presented in our manuscript.

 

Comment 2: Provide a table which tells how your work is novel from the previous available literature

Response 2:  Thank you for your suggestion, table added (page 31) and referenced in the text (lines 664-695)