{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T07:31:15Z","timestamp":1771659075442,"version":"3.50.1"},"reference-count":35,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2022,4,13]],"date-time":"2022-04-13T00:00:00Z","timestamp":1649808000000},"content-version":"vor","delay-in-days":102,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100002543","name":"Pusan National University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002543","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF-2019R1A2C2007249"],"award-info":[{"award-number":["NRF-2019R1A2C2007249"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:p>Cancer chemotherapy has been the most common cancer treatment. However, it has side effects that kill both tumor cells and immune cells, which can ravage the patient\u2019s immune system. Chemotherapy should be administered depending on the patient\u2019s immunity as well as the level of cancer cells. Thus, we need to design an efficient treatment protocol. In this work, we study a feedback control problem of tumor\u2010immune system to design an optimal chemotherapy strategy. For this, we first propose a mathematical model of tumor\u2010immune interactions and conduct stability analysis of two equilibria. Next, the feedback control is found by solving the Hamilton\u2013Jacobi\u2013Bellman (HJB) equation. Here, we use an upwind finite\u2010difference method for a numerical approximate solution of the HJB equation. Numerical simulations show that the feedback control can help determine the treatment protocol of chemotherapy for tumor and immune cells depending on the side effects.<\/jats:p>","DOI":"10.1155\/2022\/2158052","type":"journal-article","created":{"date-parts":[[2022,4,13]],"date-time":"2022-04-13T13:35:21Z","timestamp":1649856921000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Feedback Control of Cancer Chemotherapy Using Hamilton\u2013Jacobi\u2013Bellman Equation"],"prefix":"10.1155","volume":"2022","author":[{"given":"Yong Dam","family":"Jeong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kwang Su","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunil","family":"Roh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sooyoun","family":"Choi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shingo","family":"Iwami","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6086-5124","authenticated-orcid":false,"given":"Il Hyo","family":"Jung","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,4,13]]},"reference":[{"key":"e_1_2_10_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.bbacli.2015.04.001"},{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1038\/nrc4029"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11538-010-9526-3"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1158\/0008-5472.can-05-0564"},{"key":"e_1_2_10_5_2","doi-asserted-by":"publisher","DOI":"10.1038\/nri1374"},{"key":"e_1_2_10_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00262-007-0387-z"},{"key":"e_1_2_10_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0092-8240(05)80260-5"},{"key":"e_1_2_10_8_2","doi-asserted-by":"publisher","DOI":"10.1142\/s0218202508002796"},{"key":"e_1_2_10_9_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2006.1786"},{"key":"e_1_2_10_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.plrev.2007.04.002"},{"key":"e_1_2_10_11_2","doi-asserted-by":"publisher","DOI":"10.1137\/s0036144504446291"},{"key":"e_1_2_10_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s002850050127"},{"key":"e_1_2_10_13_2","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/982978"},{"key":"e_1_2_10_14_2","doi-asserted-by":"publisher","DOI":"10.4049\/jimmunol.134.4.2748"},{"key":"e_1_2_10_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2019.02.002"},{"key":"e_1_2_10_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0895-7177(03)00133-x"},{"key":"e_1_2_10_17_2","doi-asserted-by":"publisher","DOI":"10.1093\/imammb\/2.3.139"},{"key":"e_1_2_10_18_2","doi-asserted-by":"publisher","DOI":"10.1142\/s0218339008002435"},{"key":"e_1_2_10_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2005.06.037"},{"key":"e_1_2_10_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2009.05.001"},{"key":"e_1_2_10_21_2","doi-asserted-by":"publisher","DOI":"10.4172\/1948-5956.1000423"},{"key":"e_1_2_10_22_2","doi-asserted-by":"publisher","DOI":"10.1080\/10273660108833067"},{"key":"e_1_2_10_23_2","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/608598"},{"key":"e_1_2_10_24_2","volume-title":"Mathematical Theory of Optimal Processes","author":"Pontryagin L. S.","year":"1987"},{"key":"e_1_2_10_25_2","doi-asserted-by":"publisher","DOI":"10.1090\/s0002-9904-1954-09848-8"},{"key":"e_1_2_10_26_2","volume-title":"Generalized Solutions of Hamilton-Jacobi Equations","author":"Lions P.-L.","year":"1982"},{"key":"e_1_2_10_27_2","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/17.2.167"},{"key":"e_1_2_10_28_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1000103"},{"key":"e_1_2_10_29_2","volume-title":"Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations","author":"Bardi M.","year":"2008"},{"key":"e_1_2_10_30_2","doi-asserted-by":"publisher","DOI":"10.1090\/s0002-9947-1984-0732102-x"},{"key":"e_1_2_10_31_2","doi-asserted-by":"publisher","DOI":"10.1090\/s0002-9947-1983-0690039-8"},{"key":"e_1_2_10_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0362-546x(00)85016-6"},{"key":"e_1_2_10_33_2","volume-title":"Computational Methods for Fluid Flow","author":"Peyret R.","year":"2012"},{"key":"e_1_2_10_34_2","doi-asserted-by":"publisher","DOI":"10.1109\/tac.2015.2406976"},{"key":"e_1_2_10_35_2","volume-title":"Differential Equations: Classical to Controlled","author":"Lukes D. L.","year":"1982"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/2158052","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/2022\/2158052","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/2158052","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T21:22:24Z","timestamp":1769116944000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2022\/2158052"}},"subtitle":[],"editor":[{"given":"Guang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["10.1155\/2022\/2158052"],"URL":"https:\/\/doi.org\/10.1155\/2022\/2158052","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1]]},"assertion":[{"value":"2021-11-23","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-02-28","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-04-13","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"2158052"}}