Aperiodic Optimal Chronotherapy in Simple Compartment Tumour Growth Models Under Circadian Drug Toxicity Conditions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please find attached a pdf file with the review report.

Comments for author File: Comments.pdf

Author Response

Comment 1. The authors should justify their choice of applying the optimal strategy on the simplest form of the tumour growth model. In particular, the differences from the previously related models and what are the advantages of a simple model in this case?

Response 1: We now emphasise that general principles of infusion timing in chronotherapy are lacking, L92-93, whilst simple models have not been analysed in a chronotherapy perspective.

Comment 2:  The Theorems should be numbered.

Response 2. Done

Comment 3: In Theorem 1, Page 6, either the definition of the function $\gamma_1$, its domain etc

Response 3: We removed reference to the map $\gamma_1$ in both theorems, simplifying the statement. The map was only relevant when discussing bang-bang solutions meeting the singular arc; we have expanded the comment.

Comment 4: How are the parameters chosen for the numerical simulations (starting from Figure 4 to Figure 16)? Is there any real data that supports these choices?

Response 4: We have now added statements clarifying choices; earlier in the text this was to allow solutions to be seen clearly, and the trajectory structure. In Conclusions, we illustrate with parameters that are obtained from the literature.

Comment 5: Please propose an extension of the proposed mathematical model to cover more complex cancer models, including spatial cancer models or structured growth models that can be considered in the future.

Response 5: This is discussed in the last paragraph on Conclusions. 

Comment 6:  The authors should use the Template of the Journal.

Response 6: Done.

Comment 7: All references must be written in the same style as the journal and should be numbered either inside the text or in the references list.

Response 7: Done

Comment 8: I suggest that the authors revise the manuscript for possible typos.

Response 8: Done.

 

Reviewer 2 Report

Comments and Suggestions for Authors

See the attached file.

Comments for author File: Comments.pdf

Author Response

Comment 1: Specify initial condition in Equation 1

Response 1: Done

Comment 2: Specify a class of admissible controls

Response 2: Done, we specified (in line 151) that admissible controls are piecewise continuous functions. 

Comment 3: The coefficient d in the expression of h can be reduced or  is it a specific example of h ?Please explain. 

Response 3: The cost has no explicit dependence on d, since as observed by the reviewer; h is defined to incorporate d in the manuscript to simplify the calculations. The definition of h is clarified in line 148. For multi compartment models h is defined without the d, clarified in line 512.

Comment 4: Change Gamkrelidze, R. (2019). Selected research papers: Ls pontryagin select works volume by Gamkrelidze, R. (2019). Selected research papers: L. S. Pontryagin select works volume

Response 4:Done