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Analysis of control for a free boundary problem of steady plaques in the artery. (English) Zbl 1427.49003

Summary: In the earlier paper of A. Friedman et al. [J. Differ. Equations 259, No. 4, 1227–1255 (2015; Zbl 1322.35181)] a simplified model of plaque growth involving LDL and HDL cholesterols, macrophages and foam cells is considered and they satisfy a coupled system of PDEs with a free boundary. The paper adds some control function to that model, allowing the controlled growth of LDL, HDL and plaque. Next, the new dual dynamic programming approach for free boundary problem is developed to formulate sufficient optimality conditions for the optimal steering of drugs. Finally an approximate optimality and numerical calculations are presented.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35Q93 PDEs in connection with control and optimization
35Q35 PDEs in connection with fluid mechanics
76Z05 Physiological flows
92C35 Physiological flow
92C30 Physiology (general)
35K51 Initial-boundary value problems for second-order parabolic systems
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35R35 Free boundary problems for PDEs

Citations:

Zbl 1322.35181
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References:

[1] Calvez, V.; Ebde, A.; Meunier, N.; Raoult, A., Mathematical modeling of the atherosclerotic plaque formation, ESAIM: Proceedings, Vol. 28, 1-12 (2009) · Zbl 1176.76145
[2] Cobbold, C. A.; Sherratt, J. A.; Maxwell, S. R.J., Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach, Bull. Math. Biol., 64, 65-95 (2002) · Zbl 1334.92131
[3] Cohen, A.; Myerscough, M. R.; Thompson, R. S., Athero-protective effects of high density lipoproteins (HDL): an ODE model of the early stages of atherosclerosis, Bull. Math. Biol., 76, 1117-1142 (2014) · Zbl 1297.92041
[4] Little, M. P.; Gola, A.; Tzoulaki, I., A model of cardiovascular disease giving a plausible mechanism for the effect of fractionated low-dose ionizing radiation exposure, PLoS Comput. Biol., 5, e1000539 (2009)
[5] C. McKay, S. McKee, N. Mottram, T. Mulholland, S. Wilson, S. Kennedy, R. Wadsworth, Towards a model of atherosclerosis, University of Strathclyde research report, 2005, 1-29.; C. McKay, S. McKee, N. Mottram, T. Mulholland, S. Wilson, S. Kennedy, R. Wadsworth, Towards a model of atherosclerosis, University of Strathclyde research report, 2005, 1-29.
[6] Zhang, S.; Ritter, L. R.; Ibragimov, A. I., Foam cell formation in atherosclerosis: HDL and macrophage reverse cholesterol transport, Discret. Contin. Dyn. Syst., 825-835 (2013) · Zbl 1304.92055
[7] Friedman, A.; Hao, W., A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors, Bull. Math. Biol., 1-24 (2014)
[8] Friedman, A.; Hao, W., The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model, PLoS ONE, 9, e90497 (2014)
[9] A. Friedman, W. Hao, B. Hu, A free boundary problem for steady small plaques in the artery and their stability, J. Differ. Equ.https://doi.org/10.1016/j.jde.2015.02.002; A. Friedman, W. Hao, B. Hu, A free boundary problem for steady small plaques in the artery and their stability, J. Differ. Equ.https://doi.org/10.1016/j.jde.2015.02.002 · Zbl 1322.35181
[10] Hinze, M.; Ziegenbalg, S., Optimal control of the free boundary in a two-phase Stefan problem, J. Comput. Phys., 223, 657-684 (2007) · Zbl 1115.80008
[11] Hinze, M.; Ziegenbalg, S., Optimal control of the free boundary in a two-phase Stefan problem with flow driven by convection, ZAMM Z. Angew. Math. Mech., 87, 430-448 (2007) · Zbl 1123.49029
[12] Antil, H.; Nochetto, R. H.; Sodré, P., Optimal control of a free boundary problem: analysis with second order sufficient conditions, SIAM J. Control Optim., 52, 2771-2799 (2014) · Zbl 1311.49008
[13] Antil, H.; Nochetto, R. H.; Sodré, P., Optimal control of a free boundary problem with surface tension effects: a priori error analysis, SIAM J. Numer. Anal., 53, 2279-2306 (2015) · Zbl 1343.49043
[14] Nowakowski, A., The dual dynamic programming, Proc. Am. Math. Soc., 116, 1089-1096 (1992) · Zbl 0769.49024
[15] A. Miniak-Górecka, Construction of computational method for ε-optimal solutions shape optimization problems, (PhD thesis, IBS Warsaw) 2015.; A. Miniak-Górecka, Construction of computational method for ε-optimal solutions shape optimization problems, (PhD thesis, IBS Warsaw) 2015.
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