Study of Reversible Platelet Aggregation Model by Nonlinear Dynamics

Round 1

Reviewer 1 Report

The reviewed article is devoted to the analysis of the  of reversible platelet aggregation model. Despite the importance of research in health care and related disciplines, I cannot recommend this manuscript for publication due to the following reasons:

1.It is not worth mentioning DOI of previous articles in the abstract, since this section of the article should contain a short and understandable description of present study without such explanations.

2.For the alternative models mentioned in the introduction, two disadvantages are indicated. The first drawback is the high demands on the computing power used for their numerical integration. What does this mean? What is the dimension of such systems? What is the peculiarity of simulation, for example, are these equations stiff? Is there any research that confirm this claim?

3.As the second disadvantage of the existing models, it is indicated that they have many parameters. One, it is believed that a large number of parameters allows for more flexible customization of the model. And modern optimization methods allow one to successfully choose them. Therefore, this drawback needs clarification.

4.What are the parameters listed in formula (1) responsible for in addition to the state variables themselves?

5. The Authors claim that model (1) describes the experimental data quite accurately (line 75), but this is not confirmed in this article. Are they in your previous article? However, this statement has been left without reference.

6. The statement in line 76 is a little confusing, since the Authors comment on formula (1), which explicitly describes this relationship. If we are talking about correlation with experimental data, it is better to indicate the degree of accuracy.

7.What integration method and the value of the integration step (and fixed or variable step) were used to simulate system (4)? There are several studies that show that the choice of the integration method for identification has a significant impact on the quality of the result.

8.Since the coefficient p0 is introduced into the model, it would make sense to investigate the bifurcation behavior of the system depending on this parameter. One-dimensional and multidimensional bifurcation diagrams for different parameters are also of interest, since bifurcation analysis is the main topic of this study.

9.There is no comparison of model (4) with existing ones. The proposed substitution violates the shortcomings mentioned by the Authors in the introduction - a large number of parameters, as well as complexity and dimension. The proposed modification complies with both points.

10.In the study, there are practically no references to recent work on this topic.

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

The authors analyze a mathematical model of the process of platelet aggregation. The model was previously proposed and verified against the experimental data by the authors. Then, the original model equations are transformed and analyzed by calculating equilibrium points and studying the system’s stability at these points. Finally, the authors study the effects of parameter variations on the course of platelet aggregation process.

 

The motivation of the work is clear and the presentation of the mathematical model is good. However, I have some comments concerning the qualitative analysis of the model.

 

1) From the mathematical point of view, the presented analysis is not complete. In Figure 3, the authors show that the system admits only one positive equilibrium point, but these results are only obtained for parameter values from Table 1. If I modify some parameters values, then I can obtain two positive equilibrium points. For example, for k_1=0.0101, k_3=2.62e-4, k1=0.1233, k2=3.3785e-5, k3=0.0104, I have two equilibrium points (a*,y*)=(0.3639,0.5078) and (a*,y*)=(0.6511,0.5073). Does it make a physical sense that the system has two positive equilibrium points? In my opinion, the authors should start their analysis by determining possible equilibrium points in the system. Of course, this analysis can be restricted to the equilibrium points that only make physical sense, but this must be clearly stated by the authors.

 

2) On page 5, the authors claim that there is a bifurcation point as the singular point is changed from stable node to stable focus. Taking into account the formal definition of the bifurcation, this is not a bifurcation point. The bifurcation occurs if there is no topological equivalence between two systems (for different values of a bifurcation parameter). Take a look at the definition given in (Kuznetsov, 2013), in the book it is even shown that stable node and stable focus are topologically equivalent. This means that this is not a bifurcation point. The bifurcation may occur when there is a change in the stability of the system (e.g., from stable node to unstable node), a change in the number of stationary solutions, or in the case of appearance/disappearance of period solutions in the system. Please, verify the results presented in Figure 4 and define the bifurcation points correctly. Here, I can also recommend a very nice book on bifurcation theory written by Strogatz (1994).

 

3) I would also suggest to improve the quality of Figure 4. I can hardly notice the axis labels. Moreover, I suggest to show only one variable at a time in the bifurcation diagrams to make it more clear.

 

4) In the paper, I could find some references to Figure S1 which does not exist.

 

 

 

Literature

Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory (Vol. 112). Springer Science & Business Media.

S.H. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books Publishing, Cambridge, MA (1994)

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Dear Authors!

Thank you for taking my comments into account. I have only one comment left about the study. For numerical integration, the backward differentiation formula with a large integration step is used. At the same time, it is known that at such step values, multistep methods can lose stability, and the reliability of numerical results decreases. I recommend the Authors, if the problem under consideration is not stiff, repeat the simulations with other multi-step methods, for example, the explicit Adams methods or the Adams-Bashforth-Multon methods, since the accuracy of the solutions obtained when using them is much higher than that of the backward differentiation formulas. In recent articles one can find a comparison of the computational efficiency of these methods (for example, article "Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods" in Mathematics). Since one of the goals of your research is to speed up the simulation, I believe that integration methods should also be selected based on these considerations.

Author Response

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Reviewer 2 Report

The authors have modified the paper and some parts are improved. However, I can still find some unclear statements. For example, on page 6, line 28, the authors mention that there are 4 singular points. Are these points positive? Do they make any physical sense? What Figure 5 (colorful areas) shows us? What do the grey lines in Figure 5d represents? Are they nullclines? This part has to be clarified.

Author Response

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Author Response File: Author Response.docx