Optimization of an amplification protocol for misfolded proteins by using relaxed control

详细信息    查看全文

• 作者：Jean-Michel Coron (1)
  Pierre Gabriel (2)
  Peipei Shang (3)
  
  1. Laboratoire Jacques-Louis Lions ; UPMC Univ Paris 06 ; UMR 7598 ; Sorbonne Universit茅s ; 75005聽 ; Paris ; France
  2. Laboratoire de Math茅matiques de Versailles ; CNRS UMR 8100 ; Universit茅 de Versailles Saint-Quentin-en-Yvelines ; 45 Avenue des 脡tats-Unis ; 78035聽 ; Versailles ; France
  3. Department of Mathematics ; Tongji University ; Shanghai聽 ; 200092 ; China
  
• 关键词：Optimal control ; Relaxed control ; Turnpike ; Pontryagin maximum principle ; Perron eigenvalue ; Floquet eigenvalue ; Structured populations ; 49J15 ; 35Q92 ; 37N25
• 刊名：Journal of Mathematical Biology
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：70
• 期：1-2
• 页码：289-327
• 全文大小：693 KB
• 参考文献：1. Baca毛r N, Abdurahman X (2008) Resonance of the epidemic threshold in a periodic environment. J Math Biol 57(5):649鈥?73. doi:10.1007/s00285-008-0183-1 . ISSN 0303-6812
  2. Baca毛r N, Ouifki R (2007) Growth rate and basic reproduction number for population models with a simple periodic factor. Math Biosci 210(2):647鈥?58. doi:10.1016/j.mbs.2007.07.005 . ISSN 0025-5564
  3. Calvez V, Doumic M, Gabriel P (2012) Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis. Journal de Mathmatiques Pures et Appliques 98(1):1鈥?7. doi:10.1016/j.matpur.2012.01.004 . ISSN 0021-7824
  4. Calvez V, Gabriel P (2012) Optimal growth for linear processes with affine control. (Preprint). arXiv:1203.5189
  5. Calvez V, Lenuzza N, Doumic M, Deslys J-P, Mouthon F, Perthame B (2010) Prion dynamic with size dependency鈥攕train phenomena. J Biol Dyn 4(1):28鈥?2 CrossRef
  6. Calvez V, Lenuzza N, Oelz D, Deslys J-P, Laurent P, Mouthon F, Perthame B (2009) Size distribution dependence of prion aggregates infectivity. Math Biosci 1:88鈥?9 CrossRef
  7. Castilla J, Sa谩 P, Hetz C, Soto C (2005) In vitro generation of infectious scrapie prions. Cell 121(2):195鈥?06 CrossRef
  8. Clairambault J, Gaubert S, Lepoutre T (2009) Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models. Math Model Nat Phenom 4(3):183鈥?09 CrossRef
  9. Doumic M, Gabriel P (2010) Eigenelements of a general aggregation鈥揻ragmentation model. Math Models Methods Appl Sci 20(5):757鈥?83. doi:10.1142/S021820251000443X CrossRef
  10. Doumic M, Goudon T, Lepoutre T (2009) Scaling limit of a discrete prion dynamics model. Comm Math Sci 7(4):839鈥?65 CrossRef
  11. Doumic M, Tine LM (2013) Estimating the division rate for the growth-fragmentation equation. J Math Biol 67(1):69鈥?03. doi:10.1007/s00285-012-0553-6 . ISSN 0303-6812
  12. Gabriel P (2011) 脡quations de transport-fragmentation et applications aux maladies 脿 prions [Transport-fragmentation equations and applications to prion diseases]. PhD thesis, Paris
  13. Gabriel P (2011) The shape of the polymerization rate in the prion equation. Math Comput Model 53(7鈥?):1451鈥?456. doi:10.1016/j.mcm.2010.03.032 . ISSN 0895-7177
  14. Gabriel P (2012) Long-time asymptotics for nonlinear growth-fragmentation equations. Commun Math Sci 10(3):787鈥?20. ISSN 1539-6746
  15. Greer ML, Pujo-Menjouet L, Webb GF (2006) A mathematical analysis of the dynamics of prion proliferation. J Theoret Biol 242(3):598鈥?06. ISSN 0022-5193
  16. Griffith JS (1967) Nature of the scrapie agent: self-replication and scrapie. Nature 215(5105):1043鈥?044 CrossRef
  17. Jarrett JT, Lansbury PT (1993) Seeding 鈥渙ne-dimensional crystallization鈥?of amyloid: a pathogenic mechanism in Alzheimer鈥檚 disease and scrapie? Cell 73(6):1055鈥?058. doi:10.1016/0092-8674(93)90635-4 . ISSN 0092-8674
  18. Lauren莽ot P, Walker C (2007) Well-posedness for a model of prion proliferation dynamics. J Evol Equ 7(2):241鈥?64. ISSN 1424-3199
  19. Ledzewicz U, Sch盲ttler H (2002) Analysis of a cell-cycle specific model for cancer chemotherapy. J Biol Syst 10(03):183鈥?06. doi:10.1142/S0218339002000597 CrossRef
  20. Ledzewicz U, Sch盲ttler H (2002) Optimal bang鈥揵ang controls for a two-compartment model in cancer chemotherapy. J Optim Theory Appl 114(3):609鈥?37. doi:10.1023/A:1016027113579 . ISSN 0022-3239
  21. Ledzewicz U, Sch盲ttler H (2006) Analysis of models for evolving drug resistance in cancer chemotherapy. Dyn Contin Discrete Impuls Syst Ser A Math Anal 13B(suppl.): 291鈥?04. ISSN 1201-3390
  22. Ledzewicz U, Sch盲ttler H (2006) Drug resistance in cancer chemotherapy as an optimal control problem. Discret Contin Dyn Syst Ser B 6(1):129鈥?50. ISSN 1531-3492
  23. Lee E, Markus L (1986) Foundations of optimal control theory, 2nd edn. Robert E. Krieger Publishing Co., Inc, Melbourne. ISBN 0-89874-807-0
  24. Masel J, Jansen VAA, Nowak MA (1999) Quantifying the kinetic parameters of prion replication. Biophys Chem 77(2鈥?): 139鈥?52. doi:10.1016/S0301-4622(99)00016-2 . ISSN 0301-4622
  25. Michel P (2006) Optimal proliferation rate in a cell division model. Math Model Nat Phenom 1(2):23鈥?4. ISSN 0973-5348
  26. Perthame B (2007) Transport equations in biology. In: Frontiers in mathematics. Birkh盲user Verlag, Basel. ISBN 978-3-7643-7841-7; 3-7643-7841-7
  27. Prusiner SB (1982) Novel proteinaceous infectious particles cause scrapie. Science 216(4542):136鈥?44 CrossRef
  28. Pr眉ss J, Webb LGF, Zacher R (2006) Analysis of a model for the dynamics of prions. Discret Contin Dyn Syst Ser B 6(1): 225鈥?35. ISSN 1531-3492
  29. Sa谩 P, Castilla J, Soto C (2005) Cyclic amplification of protein misfolding and aggregation. Methods Mol Biol 299:53鈥?5
  30. Saborio GP, Permanne B, Soto C (2001) Sensitive detection of pathological prion protein by cyclic amplification of protein misfolding. Nature 411:810鈥?13 CrossRef
  31. Serre D (2002) Matrices. Theory and applications. In: Graduate texts in mathematics, vol 216. Springer, New York. ISBN 0-387-95460-0. Translated from the 2001 French original
  32. Simonett G, Walker C (2006) On the solvability of a mathematical model for prion proliferation. J Math Anal Appl 324(1):580鈥?03. ISSN 0022-247X doi:10.1016/j.jmaa.2005.12.036
  33. 艢wierniak A, Ledzewicz U, Sch盲ttler H (2003) Optimal control for a class of compartmental models in cancer chemotherapy. Int J Appl Math Comput Sci 13(3):357鈥?68. ISSN 1641-876X (Cancer growth and progression, mathematical problems and computer simulations (Bedlewo, 2002))
  34. Tao T (2008) When are eigenvalues stable? http://terrytao.wordpress.com/2008/10/28/
  35. Tr茅lat E, Zuazua E (2014) Turnpike in finite-dimensional nonlinear optimal control. Preprint
  36. Walker C (2007) Prion proliferation with unbounded polymerization rates. In Proceedings of the sixth Mississippi State-UBA conference on differential equations and computational simulations. Electronic Journal of Differential Equations: Conference, vol 15 , pp 387鈥?97, San Marcos, TX, Southwest Texas State Univ
  37. Zaslavski AJ (2006) Turnpike properties in the calculus of variations and optimal control. Nonconvex optimization and its applications, vol 80. Springer, New York. ISBN 978-0-387-28155-1; 0-387-28155-X
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Biology
  Applications of Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1416

文摘

We investigate an optimal control problem which arises in the optimization of an amplification technique for misfolded proteins. The improvement of this technique may play a role in the detection of prion diseases. The model consists in a linear system of differential equations with a nonlinear control. The appearance of oscillations in the numerical simulations is understood by using the Perron and Floquet eigenvalue theory for nonnegative irreducible matrices. Then to overcome the unsolvability of the optimal control, we relax the problem. In the two dimensional case, we solve explicitly the optimal relaxed control problem when the final time is large enough.