Controlling the thickness of the atherosclerotic plaque by statin medication

Atherosclerosis, a chronic inflammatory disorder of the arterial wall, is a complex process whose dynamics are affected by multiple factors. The disease control consists of restraining it by administering statins. Slowing down or halting the plaque growth depends on the patient age at which the statin treatment begins and on the thickness of the intima-media (IMT) at that time. In this paper, we propose a mathematical model to estimate the sets of atherosclerosis states, from which the use of statins can restrain the disease. Our model is control-theoretic, and the estimated sets are the viability kernels, in the parlance of viability theory. To our best knowledge, this way of modelling the atherosclerosis progression is original. We compute two viability kernels, each for a different statin-treatment dose. Each kernel is composed of the vector [age, IMT] from which the disease can be restrained. By extension, the disease can’t be restrained from the kernel complements, this being mainly because of the disease and patient-age advancement. The kernels visualise tradeoffs between early and late treatments, which helps the clinician to decide when to start the statin treatment and which statin dose may be sufficient.

[-] we have combined our replies to your and Reviewer #1's comments and queries in this letter. Please find our replies below. Your comments and queries are in italics.
• If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results.
We think this recommendation is not applicable to our paper. Limited parameter estimation was performed and described in our previous paper cited as: [9] Formanowicz D, Krawczyk JB, Perek B, Formanowicz P. 'A Control-Theoretic Model of Atherosclerosis. ' Int J Mol Sci. 2019;20:785. In the current paper, the parameters have been calibrated i.e., their values are proposed on the basis of other authors' research referenced to in our bibliography.
• During our internal evaluation of the manuscript, we found significant text overlap between your submission and the following previously published works: https: //www.mdpi.com/1422-0067/20/3/785/html Please revise the manuscript to rephrase the duplicated text, cite your sources, and provide details as to how the current manuscript advances on previous work. Please note that further consideration is dependent on the submission of a manuscript that addresses these concerns about the overlap in text with published work.
The above mentioned paper has been also authored by us (plus two other coauthors).
While writing the current paper we felt that some repetitions concerning, in particular, the representation of atherosclerosis as a dynamic process and the selection of the model variables, common for both papers, are unavoidable. Action: We have tried and rephrased a significant amount of the overlapping text. See the green-highlighted parts in 'Revised Manuscript with Track Changes'.
• Please clarify in your Data availability statement whether any existing datasets were used, and whether the code for the model has been made available.
Action: Two Matlab files needed for production of Figures 3-5 are available from authors.
• I do not understand how the viability kernels apply to their model. I would like to suggest writing the explicit formulation of viability kernels in the manuscript.
The viability kernel in a controlled model with a target is the locus of states from which the target can be reached, using available controls. A reason for us to use the viability kernel in analysing atherosclerosis is that the medical problem of controlling patient's IMT to a desired thickness can be modelled and solved by computing the viability kernel, for a mathematical model of atherosclerosis progression.
We have said in the paper (in lines 30-31) that the "[..] two viability kernels [..] contain the atherosclerosis states, from which one can slow down the disease process". A more comprehensive description of the viability kernel is provided in the subsection "Viability kernels as 'safety regions' ", see lines 396-415.
We believe that a more mathematically involved description than we have provided could turn some medical specialists off form our model.  Action: We have now included in the paper the yellow highlighted sentence from above, in lines 412-415.

• In eq. 3, how can you guarantee the continuity at t = T ?
There is no need to assure this continuity. Eq (3) describes the derivative of c. We contend that the variable c describes patient's health status. The pace of c -i.e., the derivative -changes after a statin treatment is implemented. The treatment begins at T therefore the pace changes at T . This means that the derivative -dc dt -"jumps" at T .
However, the integral of dc dt is c is continuous. This process is illustrated in Figure 3. There are many statin brands used in therapies, some will be stronger than some others. (3), and so is its integralc(t). Therefore, if the statin dose -constant or variable -is restricted to [40,80], then the corresponding (to this statin dose) atherosclerosis evolution will remain between the evolutions emanating, respectively, from the points A and B in The relationship is the other way around: the kernels result from multiple solutions to the system of two differential equations: (1) and (3).
An atherosclerosis evolution (several such evolutions are shown as lines in Figures 4   and 5), is a solution to the system (1)-(3). The initial points of evolutions, which meet the target x(T * ) ≤ 1.15 [mm], constitute the viability kernels.