Analysis of control for a free boundary problem of steady plaques in the artery

Miniak-Górecka, Alicja; Nowakowski, Andrzej

doi:10.1016/j.amc.2018.02.016

Cited by 2 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, since the main focus of this research is on the numerical solution, the features of the model are also very important. As a result, among the available optimal control problems for this disease, the authors consider the following atherosclerosis optimal control problem [21] in which the concentration of LDL, HDL and foam cells are controlled using control variable φ(τ ) as follows with the following boundary and initial conditions…”

Section: Optimal Control Problem Of Atherosclerosismentioning

confidence: 99%

“…So, in this model of optimal control problem, the cost functional is designed to be 1 − R(T ). It is worth to note that the existence of the solution of the optimal control problem (1)-( 11) is presented in [21].…”

Section: Optimal Control Problem Of Atherosclerosismentioning

confidence: 99%

“…In turn, one of the most remarkable models which need to be controlled for treatment purposes is the plaque growth model. For instance, in [21] the authors add some control function to the model of plaque growth involving LDL and HDL and foam cells in [13] which satisfy a coupled system of free boundary PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we intend to solve an optimal control problem governed by a system of nonlinear free boundary partial differential equations which has been modelled to control quantities growth of the plaque in the artery and obtain the minimal value of the radius of the plaque using direct and indirect methods. The motivation for this work is the article [21] in which a control function is added to a simplified model of plaque growth involving LDL 2 and HDL 3 [13], allowing the controlled growth of LDL, HDL and plaque. The optimal control problem contains a system of free boundary equations with mixed boundary conditions which models atherosclerosis and consists of three parabolics, one elliptic and one ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

Nasresfahani¹,

Eslahchi²

2022

Preprint

View full text Add to dashboard Cite

We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL, HDL and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the firstorder optimality conditions. Numerical results are considered to illustrate the efficiency of methods.

show abstract

Section: Optimal Control Problem Of Atherosclerosismentioning

confidence: 99%

Section: Optimal Control Problem Of Atherosclerosismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

Nasresfahani¹,

Eslahchi²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Some hints how to construct such optimal control theory we find in references. 17,18 But the problems considered in those papers are different and in fact easier. We need for problem (1.15)-(1.17) a different dual dynamic programming theory to derive verification theorem-sufficient 𝜀-optimality conditions for approximate solutions of problem (1.15)- (1.16).…”

Section: Optimal Control Problemmentioning

confidence: 99%

Control of tumor growth modeled by system of PDE, numerical analysis of optimality conditions

Nowakowski

Krawczyk

2022

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

We analyze a mathematical model of a tumor that includes the interactions among tumor cells (live or dead), macrophages, endothelial cells, and the cytokines based on a system of partial differential equations which describes inhibition of tumor growth by granulocyte‐macrophage colony‐stimulating factor (GM‐CSF) treatment. The aim of the model is to predict the growth of the tumor volume under partial blocking of hypoxia‐inducible factor (HIF)‐1 or stabilization of HIF‐2, with injection of GM‐CSF. It was assumed that tumor is a spherical and the treatment by GM‐CSF is not optimized. In this paper, we do not assume that tumor is a spherical model and we develop a dual dynamic programming approach to formulate a verification condition of an approximate treatment for an injection of GM‐CSF. Next, we analyze numerically an ε$$ \varepsilon $$‐optimal treatment.

show abstract

Analysis of control for a free boundary problem of steady plaques in the artery

Cited by 2 publications

References 10 publications

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

Control of tumor growth modeled by system of PDE, numerical analysis of optimality conditions

Contact Info

Product

Resources

About