Global Dynamics of a Breast Cancer Competition Model
In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body’s natural immune response on the aforementioned cell population...
Saved in:
| Published in | Differential equations and dynamical systems Vol. 28; no. 4; pp. 791 - 805 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
New Delhi
Springer India
01.10.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0971-3514 0974-6870 |
| DOI | 10.1007/s12591-017-0346-x |
Cover
| Abstract | In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body’s natural immune response on the aforementioned cell populations. Employing a variety of analytical methods, we study the global dynamics of the full system, along with various submodels. We find sufficient conditions on parameter values to ensure cancer persistence in the absence of immune cells, and cancer eradication when an immune response is included. We conclude with a discussion on the biological implications of the resulting global dynamics. |
|---|---|
| AbstractList | In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body’s natural immune response on the aforementioned cell populations. Employing a variety of analytical methods, we study the global dynamics of the full system, along with various submodels. We find sufficient conditions on parameter values to ensure cancer persistence in the absence of immune cells, and cancer eradication when an immune response is included. We conclude with a discussion on the biological implications of the resulting global dynamics. In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body's natural immune response on the aforementioned cell populations. Employing a variety of analytical methods, we study the global dynamics of the full system, along with various submodels. We find sufficient conditions on parameter values to ensure cancer persistence in the absence of immune cells, and cancer eradication when an immune response is included. We conclude with a discussion on the biological implications of the resulting global dynamics.In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy cells. Additionally, we consider the effects of excess estrogen and the body's natural immune response on the aforementioned cell populations. Employing a variety of analytical methods, we study the global dynamics of the full system, along with various submodels. We find sufficient conditions on parameter values to ensure cancer persistence in the absence of immune cells, and cancer eradication when an immune response is included. We conclude with a discussion on the biological implications of the resulting global dynamics. |
| Author | Abernathy, Kristen Abernathy, Zachary Baxter, Arden Stevens, Meghan |
| Author_xml | – sequence: 1 givenname: Kristen surname: Abernathy fullname: Abernathy, Kristen organization: Winthrop University – sequence: 2 givenname: Zachary orcidid: 0000-0003-1149-2939 surname: Abernathy fullname: Abernathy, Zachary email: abernathyz@winthrop.edu organization: Winthrop University – sequence: 3 givenname: Arden surname: Baxter fullname: Baxter, Arden organization: Rollins College – sequence: 4 givenname: Meghan surname: Stevens fullname: Stevens, Meghan organization: Drake University |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/33487925$$D View this record in MEDLINE/PubMed |
| BookMark | eNp9kUtP3DAUhS00CIYpP4BNFYkNm1A_ktjZINHhVQnEpl1bN56bqVFiD3amgn-Pw0B5SO3qWrrfOTq-Z49MnHdIyAGjx4xS-S0yXtYsp0zmVBRV_rBFprSWRV4pSSfPb5aLkhW7ZC_GO0orWRdyh-wKUShZ83JKisvON9BlZ48Oemti5tsMsu8BIQ7ZHJzBkM19v8LBDta77MYvsPtCtlvoIu6_zBn5dXH-c36VX99e_pifXuemkHTIK0ZrwwE552rRKDQMlalEq9pGgZAoq0YpAaosqVqUWKdE1CA0HABYLUsxIycb39W66XFh0A0BOr0KtofwqD1Y_XHj7G-99H-0VJzVlUgGRy8Gwd-vMQ66t9Fg14FDv46aF4pKMeZM6OEn9M6vg0vfS5QUivJKjYZf3yf6G-X1oAmQG8AEH2PAVhs7wHi6FNB2mlE9Vqc31elUnR6r0w9JyT4pX83_p-EbTUysW2J4C_1v0RPJr6p_ |
| CitedBy_id | crossref_primary_10_3390_math11112410 crossref_primary_10_1109_ACCESS_2019_2940060 crossref_primary_10_1016_j_cnsns_2021_105997 crossref_primary_10_1007_s11042_023_16245_5 crossref_primary_10_1109_ACCESS_2019_2941990 crossref_primary_10_1016_j_health_2024_100306 crossref_primary_10_1088_1402_4896_ad2753 crossref_primary_10_1515_nleng_2022_0364 crossref_primary_10_1142_S0218127422502388 crossref_primary_10_1016_j_aej_2023_05_071 crossref_primary_10_3390_jpm11101031 crossref_primary_10_1007_s12190_024_02102_7 crossref_primary_10_1038_s41598_023_50045_z crossref_primary_10_19053_01217488_v15_n2_2024_15522 crossref_primary_10_31083_j_fbl2808174 crossref_primary_10_3390_fractalfract7030237 crossref_primary_10_1089_cmb_2024_0618 crossref_primary_10_3390_fractalfract6040184 crossref_primary_10_1002_mma_7057 crossref_primary_10_3390_math11204275 crossref_primary_10_1186_s13662_020_02869_6 crossref_primary_10_3390_computation12080160 crossref_primary_10_1002_adts_202401425 crossref_primary_10_2200_S00976ED1V01Y201912MAS031 crossref_primary_10_1016_j_rico_2024_100437 |
| Cites_doi | 10.1016/j.cnsns.2013.09.023 10.1142/S0218127414500205 10.1016/S0895-7177(02)00227-3 10.1016/j.jtbi.2015.06.007 10.1007/s00285-003-0211-0 10.1016/j.jtbi.2005.06.037 10.1155/2012/473572 10.1073/pnas.0530291100 10.1016/j.physleta.2005.12.104 10.1210/jcem-43-2-436 10.1155/2016/1239861 10.1080/10273660108833067 10.1093/carcin/18.2.251 10.1007/s10625-006-0003-6 10.3934/mbe.2016030 10.1016/j.jtbi.2006.12.010 10.1016/j.jtbi.2011.10.027 10.1016/j.mbs.2006.05.003 10.1016/j.cnsns.2015.10.001 10.1016/j.nonrwa.2012.10.006 10.1016/j.cnsns.2016.04.025 10.1002/mma.3023 10.1158/0008-5472.CAN-05-0564 10.1142/S021812741350020X |
| ContentType | Journal Article |
| Copyright | Foundation for Scientific Research and Technological Innovation 2017 Foundation for Scientific Research and Technological Innovation 2017. |
| Copyright_xml | – notice: Foundation for Scientific Research and Technological Innovation 2017 – notice: Foundation for Scientific Research and Technological Innovation 2017. |
| DBID | AAYXX CITATION NPM 7X8 5PM |
| DOI | 10.1007/s12591-017-0346-x |
| DatabaseName | CrossRef PubMed MEDLINE - Academic PubMed Central (Full Participant titles) |
| DatabaseTitle | CrossRef PubMed MEDLINE - Academic |
| DatabaseTitleList | PubMed MEDLINE - Academic |
| Database_xml | – sequence: 1 dbid: NPM name: PubMed url: https://proxy.k.utb.cz/login?url=http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed sourceTypes: Index Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Mathematics Computer Science |
| EISSN | 0974-6870 |
| EndPage | 805 |
| ExternalDocumentID | PMC7821963 33487925 10_1007_s12591_017_0346_x |
| Genre | Journal Article |
| GrantInformation_xml | – fundername: National Science Foundation grantid: DMS-1358534 funderid: http://dx.doi.org/10.13039/100000001 – fundername: National Institutes of Health grantid: 2 P20 GM103499 15 funderid: http://dx.doi.org/10.13039/100000002 – fundername: NIGMS NIH HHS grantid: P20 GM103499 |
| GroupedDBID | -5D -5G -BR -EM -Y2 -~C .VR 06D 0R~ 0VY 1N0 203 29G 2J2 2JN 2JY 2KG 2KM 2LR 2VQ 30V 4.4 406 408 40D 40E 5VS 67Z 6NX 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABDZT ABECU ABFTV ABHQN ABJNI ABJOX ABKCH ABLLD ABMQK ABNWP ABQBU ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACPIV ACSNA ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFGCZ AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN B-. BA0 BAPOH BDATZ BGNMA CAG COF CSCUP DDRTE DNIVK DPUIP EBLON EBS EIOEI EJD ESBYG FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 H13 HF~ HG6 HLICF HMJXF HRMNR HZ~ IKXTQ IWAJR IXD I~X I~Z J-C J0Z JBSCW JZLTJ KOV LLZTM M4Y MA- NPVJJ NQJWS NU0 O9- O93 O9J P9R PF0 PQQKQ PT4 QOS R89 R9I ROL RSV S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TUC U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW WK8 YLTOR Z45 ZMTXR ~A9 AAPKM AAYXX ABBRH ABDBE ABFSG ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION ABRTQ NPM 7X8 5PM |
| ID | FETCH-LOGICAL-c470t-6109c2ae2228db8ec1e8c63f8fb8a37e76b883a85508d5e97920ceab2aaa19753 |
| IEDL.DBID | U2A |
| ISSN | 0971-3514 |
| IngestDate | Thu Aug 21 18:11:21 EDT 2025 Fri Jul 11 15:40:22 EDT 2025 Wed Sep 10 13:40:43 EDT 2025 Mon Jul 21 06:04:52 EDT 2025 Tue Jul 01 00:57:19 EDT 2025 Thu Apr 24 22:52:09 EDT 2025 Fri Feb 21 02:30:02 EST 2025 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | Ordinary differential equations Breast cancer Tumor-immune dynamics Global dynamics Estrogen |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c470t-6109c2ae2228db8ec1e8c63f8fb8a37e76b883a85508d5e97920ceab2aaa19753 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ORCID | 0000-0003-1149-2939 |
| OpenAccessLink | https://www.ncbi.nlm.nih.gov/pmc/articles/7821963 |
| PMID | 33487925 |
| PQID | 2473802683 |
| PQPubID | 2043992 |
| PageCount | 15 |
| ParticipantIDs | pubmedcentral_primary_oai_pubmedcentral_nih_gov_7821963 proquest_miscellaneous_2480736109 proquest_journals_2473802683 pubmed_primary_33487925 crossref_citationtrail_10_1007_s12591_017_0346_x crossref_primary_10_1007_s12591_017_0346_x springer_journals_10_1007_s12591_017_0346_x |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2020-10-01 |
| PublicationDateYYYYMMDD | 2020-10-01 |
| PublicationDate_xml | – month: 10 year: 2020 text: 2020-10-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | New Delhi |
| PublicationPlace_xml | – name: New Delhi – name: India – name: Dordrecht |
| PublicationSubtitle | International Journal for Theory, Real World Modelling and Simulations |
| PublicationTitle | Differential equations and dynamical systems |
| PublicationTitleAbbrev | Differ Equ Dyn Syst |
| PublicationTitleAlternate | Differ Equ Dyn Syst |
| PublicationYear | 2020 |
| Publisher | Springer India Springer Nature B.V |
| Publisher_xml | – name: Springer India – name: Springer Nature B.V |
| References | Al-HajjMWichaMSBenito-HernandezAMorrisonSJClarkeMFProspective identification of tumorigenic breast cancer cellsProc. Natl. Acad. Sci. USA200310073983398810.1073/pnas.0530291100 PinhoSTRFreedmanHINaniFA chemotherapy model for the treatment of cancer with metastasisMath. Comput. Model.200236773803195073310.1016/S0895-7177(02)00227-3 De PillisLGRadunskayaAA mathematical tumor model with immune resistance and drug therapy: an optimal control approachJ. Theor. Med.2001327910010.1080/10273660108833067 Robertson-TessiMEl-KarehAGoriellyAA mathematical model of tumorimmune interactionsJ. Theor. Biol.2012294567310.1016/j.jtbi.2011.10.027 StarkovKECoriaLNGlobal dynamics of the Kirschner–Panetta model for the tumor immunotherapyJ. Nonlinear Anal. Ser. B Real World Appl.20131414251433300451010.1016/j.nonrwa.2012.10.006 KangK-SMoritaIAngelaCJeonYJTroskoJEChangC-CExpression of estrogen receptors in a normal human breast epithelial cell type with luminal and stem cell characteristics and its neoplastically transformed cell linesCarcinogenisis199718225125710.1093/carcin/18.2.251 KrishchenkoAPLocalization of invariant compact sets of dynamical systemsDiffer. Equ.20054116691676224345610.1007/s10625-006-0003-6 KarevaIBerezovskayaFCancer immunoediting: a process driven by metabolic competition as a predator–prey–shared resource type modelJ. Theor. Biol.201538046347210.1016/j.jtbi.2015.06.007 De PillisLGGuWRadunskayaAEMixed immunotherapy and chemotherapy of tumors: modeling, applications and biological implicationsJ. Theor. Biol.2006238484186210.1016/j.jtbi.2005.06.037 Nordqvist, C.: Breast Cancer: Causes, Symptoms and Treatments. Medical News Today. MediLexicon, Intl., 5 May. 2016, Web. 18 Jan. 2017 De LisettePRadunskayaAWisemanCA validated mathematical model of cell-mediated immune response to tumor growthCancer Res.200565177950795810.1158/0008-5472.CAN-05-0564 Mufudza, C., Sorofa, W., Chiyaka, E.: Assessing the effects of estrogen on the dynamics of breast cancer. Comput. Math. Methods Med. (2012). doi:10.1155/2012/473572 StarkovKEPlata-AnteCOn the global dynamics of the cancer AIDS-related mathematical modelKybernetika20145056357932750851310.34056 KrishchenkoAPStarkovKELocalization of compact invariant sets of the Lorenz systemPhys. Lett. A2006353383388222180310.1016/j.physleta.2005.12.104 StarkovKEKrishchenkoAPOn the global dynamics of one cancer tumour growth modelCommun. Nonlinear Sci. Numer. Simul.20141914861495312867610.1016/j.cnsns.2013.09.023 EnderlingHChaplainMAJAndersonARAVaidyaJSA mathematical model of breast cancer development, local treatment and recurrenceJ. Theor. Biol.20072462245259230689510.1016/j.jtbi.2006.12.010 Abernathy, K., Burke, J.: Modeling the treatment of glioblastoma multiforme and cancer stem cells with ordinary differential equations. Comput. Math. Methods Med. (2016). doi:10.1155/2016/1239861 StarkovKEGomboaDLocalization of compact invariant sets and global stability in analysis of one tumor growth modelMath. Methods Appl. Sci.20143728542863328353010.1002/mma.3023 VallePAStarkovKECoriaLNGlobal stability and tumor clearance conditions for a cancer chemotherapy systemCommun. Nonlinear Sci. Numer. Simul.201640206215350593510.1016/j.cnsns.2016.04.025 McDuffieMA hormone therapy model for breast cancer using linear cancer networksRose-Hulman Undergrad. Math. J.201415114215632162251398.92122 VillasanaMRadunskayaAA delay differential equation model for tumor growthJ. Math. Biol.2003473270294202438210.1007/s00285-003-0211-0 ValentineBWischhusenJCancer stem cell immunology: key to understanding tumorigenesis and tumor immune escape?Front. Immunol.20145113 Starkov, K.E., Villegas, A.: On some dynamical properties of a seven-dimensional cancer model with immunotherapy. Int. J. Bifurc. Chaos Appl. Sci. Eng. (2014). doi:10.1142/S0218127414500205 StarkovKEBunimovich-MendrazitskySDynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapyMath. Biosci. Eng.20161310591075355718310.3934/mbe.2016030 WuC-HMotohashiTAbdel-RahmanHFlickingerGMikhailGFree and protein-bound plasma estradiol-17 β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upbeta $$\end{document} during the menstrual cycleJ. Clin. Endocrinol.200943243644510.1210/jcem-43-2-436 StarkovKEPogromskyAYOn the global dynamics of the Owen–Sherratt model describing the tumor–macrophage interactionsInt. J. Bifurc. Chaos2013231350020304147010.1142/S021812741350020X KrishchenkoAPStarkovKEOn the global dynamics of a chronic myelogenous leukemia modelCommun. Nonlinear Sci. Numer. Simul.201633174183341717710.1016/j.cnsns.2015.10.001 De PillisLGGuWFisterKHeadTMaplesKMuruganANealTYoshidaKChemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controlsMath. Biosci.20062091292315235087910.1016/j.mbs.2006.05.003 M Robertson-Tessi (346_CR17) 2012; 294 M Villasana (346_CR27) 2003; 47 LG De Pillis (346_CR4) 2006; 209 KE Starkov (346_CR20) 2014; 19 PA Valle (346_CR26) 2016; 40 K-S Kang (346_CR8) 1997; 18 P De Lisette (346_CR6) 2005; 65 I Kareva (346_CR9) 2015; 380 LG De Pillis (346_CR3) 2001; 3 M Al-Hajj (346_CR2) 2003; 100 346_CR23 C-H Wu (346_CR28) 2009; 43 AP Krishchenko (346_CR12) 2016; 33 KE Starkov (346_CR19) 2013; 14 B Valentine (346_CR25) 2014; 5 346_CR1 AP Krishchenko (346_CR11) 2006; 353 KE Starkov (346_CR18) 2016; 13 KE Starkov (346_CR21) 2014; 50 KE Starkov (346_CR22) 2013; 23 KE Starkov (346_CR24) 2014; 37 LG De Pillis (346_CR5) 2006; 238 H Enderling (346_CR7) 2007; 246 M McDuffie (346_CR13) 2014; 15 STR Pinho (346_CR16) 2002; 36 346_CR14 346_CR15 AP Krishchenko (346_CR10) 2005; 41 |
| References_xml | – reference: StarkovKEGomboaDLocalization of compact invariant sets and global stability in analysis of one tumor growth modelMath. Methods Appl. Sci.20143728542863328353010.1002/mma.3023 – reference: Abernathy, K., Burke, J.: Modeling the treatment of glioblastoma multiforme and cancer stem cells with ordinary differential equations. Comput. Math. Methods Med. (2016). doi:10.1155/2016/1239861 – reference: EnderlingHChaplainMAJAndersonARAVaidyaJSA mathematical model of breast cancer development, local treatment and recurrenceJ. Theor. Biol.20072462245259230689510.1016/j.jtbi.2006.12.010 – reference: VallePAStarkovKECoriaLNGlobal stability and tumor clearance conditions for a cancer chemotherapy systemCommun. Nonlinear Sci. Numer. Simul.201640206215350593510.1016/j.cnsns.2016.04.025 – reference: KrishchenkoAPLocalization of invariant compact sets of dynamical systemsDiffer. Equ.20054116691676224345610.1007/s10625-006-0003-6 – reference: KrishchenkoAPStarkovKELocalization of compact invariant sets of the Lorenz systemPhys. Lett. A2006353383388222180310.1016/j.physleta.2005.12.104 – reference: KrishchenkoAPStarkovKEOn the global dynamics of a chronic myelogenous leukemia modelCommun. Nonlinear Sci. Numer. Simul.201633174183341717710.1016/j.cnsns.2015.10.001 – reference: StarkovKEKrishchenkoAPOn the global dynamics of one cancer tumour growth modelCommun. Nonlinear Sci. Numer. Simul.20141914861495312867610.1016/j.cnsns.2013.09.023 – reference: StarkovKEPlata-AnteCOn the global dynamics of the cancer AIDS-related mathematical modelKybernetika20145056357932750851310.34056 – reference: De PillisLGGuWRadunskayaAEMixed immunotherapy and chemotherapy of tumors: modeling, applications and biological implicationsJ. Theor. Biol.2006238484186210.1016/j.jtbi.2005.06.037 – reference: KarevaIBerezovskayaFCancer immunoediting: a process driven by metabolic competition as a predator–prey–shared resource type modelJ. Theor. Biol.201538046347210.1016/j.jtbi.2015.06.007 – reference: McDuffieMA hormone therapy model for breast cancer using linear cancer networksRose-Hulman Undergrad. Math. J.201415114215632162251398.92122 – reference: VillasanaMRadunskayaAA delay differential equation model for tumor growthJ. Math. Biol.2003473270294202438210.1007/s00285-003-0211-0 – reference: PinhoSTRFreedmanHINaniFA chemotherapy model for the treatment of cancer with metastasisMath. Comput. Model.200236773803195073310.1016/S0895-7177(02)00227-3 – reference: Mufudza, C., Sorofa, W., Chiyaka, E.: Assessing the effects of estrogen on the dynamics of breast cancer. Comput. Math. Methods Med. (2012). doi:10.1155/2012/473572 – reference: KangK-SMoritaIAngelaCJeonYJTroskoJEChangC-CExpression of estrogen receptors in a normal human breast epithelial cell type with luminal and stem cell characteristics and its neoplastically transformed cell linesCarcinogenisis199718225125710.1093/carcin/18.2.251 – reference: Robertson-TessiMEl-KarehAGoriellyAA mathematical model of tumorimmune interactionsJ. Theor. Biol.2012294567310.1016/j.jtbi.2011.10.027 – reference: StarkovKEBunimovich-MendrazitskySDynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapyMath. Biosci. Eng.20161310591075355718310.3934/mbe.2016030 – reference: Al-HajjMWichaMSBenito-HernandezAMorrisonSJClarkeMFProspective identification of tumorigenic breast cancer cellsProc. Natl. Acad. Sci. USA200310073983398810.1073/pnas.0530291100 – reference: Starkov, K.E., Villegas, A.: On some dynamical properties of a seven-dimensional cancer model with immunotherapy. Int. J. Bifurc. Chaos Appl. Sci. Eng. (2014). doi:10.1142/S0218127414500205 – reference: De LisettePRadunskayaAWisemanCA validated mathematical model of cell-mediated immune response to tumor growthCancer Res.200565177950795810.1158/0008-5472.CAN-05-0564 – reference: StarkovKEPogromskyAYOn the global dynamics of the Owen–Sherratt model describing the tumor–macrophage interactionsInt. J. Bifurc. Chaos2013231350020304147010.1142/S021812741350020X – reference: De PillisLGRadunskayaAA mathematical tumor model with immune resistance and drug therapy: an optimal control approachJ. Theor. Med.2001327910010.1080/10273660108833067 – reference: Nordqvist, C.: Breast Cancer: Causes, Symptoms and Treatments. Medical News Today. MediLexicon, Intl., 5 May. 2016, Web. 18 Jan. 2017 – reference: De PillisLGGuWFisterKHeadTMaplesKMuruganANealTYoshidaKChemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controlsMath. Biosci.20062091292315235087910.1016/j.mbs.2006.05.003 – reference: StarkovKECoriaLNGlobal dynamics of the Kirschner–Panetta model for the tumor immunotherapyJ. Nonlinear Anal. Ser. B Real World Appl.20131414251433300451010.1016/j.nonrwa.2012.10.006 – reference: ValentineBWischhusenJCancer stem cell immunology: key to understanding tumorigenesis and tumor immune escape?Front. Immunol.20145113 – reference: WuC-HMotohashiTAbdel-RahmanHFlickingerGMikhailGFree and protein-bound plasma estradiol-17 β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upbeta $$\end{document} during the menstrual cycleJ. Clin. Endocrinol.200943243644510.1210/jcem-43-2-436 – volume: 19 start-page: 1486 year: 2014 ident: 346_CR20 publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2013.09.023 – volume: 5 start-page: 1 year: 2014 ident: 346_CR25 publication-title: Front. Immunol. – ident: 346_CR23 doi: 10.1142/S0218127414500205 – volume: 36 start-page: 773 year: 2002 ident: 346_CR16 publication-title: Math. Comput. Model. doi: 10.1016/S0895-7177(02)00227-3 – volume: 380 start-page: 463 year: 2015 ident: 346_CR9 publication-title: J. Theor. Biol. doi: 10.1016/j.jtbi.2015.06.007 – volume: 47 start-page: 270 issue: 3 year: 2003 ident: 346_CR27 publication-title: J. Math. Biol. doi: 10.1007/s00285-003-0211-0 – ident: 346_CR15 – volume: 50 start-page: 563 year: 2014 ident: 346_CR21 publication-title: Kybernetika – volume: 238 start-page: 841 issue: 4 year: 2006 ident: 346_CR5 publication-title: J. Theor. Biol. doi: 10.1016/j.jtbi.2005.06.037 – ident: 346_CR14 doi: 10.1155/2012/473572 – volume: 15 start-page: 142 issue: 1 year: 2014 ident: 346_CR13 publication-title: Rose-Hulman Undergrad. Math. J. – volume: 100 start-page: 3983 issue: 7 year: 2003 ident: 346_CR2 publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.0530291100 – volume: 353 start-page: 383 year: 2006 ident: 346_CR11 publication-title: Phys. Lett. A doi: 10.1016/j.physleta.2005.12.104 – volume: 43 start-page: 436 issue: 2 year: 2009 ident: 346_CR28 publication-title: J. Clin. Endocrinol. doi: 10.1210/jcem-43-2-436 – ident: 346_CR1 doi: 10.1155/2016/1239861 – volume: 3 start-page: 79 issue: 2 year: 2001 ident: 346_CR3 publication-title: J. Theor. Med. doi: 10.1080/10273660108833067 – volume: 18 start-page: 251 issue: 2 year: 1997 ident: 346_CR8 publication-title: Carcinogenisis doi: 10.1093/carcin/18.2.251 – volume: 41 start-page: 1669 year: 2005 ident: 346_CR10 publication-title: Differ. Equ. doi: 10.1007/s10625-006-0003-6 – volume: 13 start-page: 1059 year: 2016 ident: 346_CR18 publication-title: Math. Biosci. Eng. doi: 10.3934/mbe.2016030 – volume: 246 start-page: 245 issue: 2 year: 2007 ident: 346_CR7 publication-title: J. Theor. Biol. doi: 10.1016/j.jtbi.2006.12.010 – volume: 294 start-page: 56 year: 2012 ident: 346_CR17 publication-title: J. Theor. Biol. doi: 10.1016/j.jtbi.2011.10.027 – volume: 209 start-page: 292 issue: 1 year: 2006 ident: 346_CR4 publication-title: Math. Biosci. doi: 10.1016/j.mbs.2006.05.003 – volume: 33 start-page: 174 year: 2016 ident: 346_CR12 publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2015.10.001 – volume: 14 start-page: 1425 year: 2013 ident: 346_CR19 publication-title: J. Nonlinear Anal. Ser. B Real World Appl. doi: 10.1016/j.nonrwa.2012.10.006 – volume: 40 start-page: 206 year: 2016 ident: 346_CR26 publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2016.04.025 – volume: 37 start-page: 2854 year: 2014 ident: 346_CR24 publication-title: Math. Methods Appl. Sci. doi: 10.1002/mma.3023 – volume: 65 start-page: 7950 issue: 17 year: 2005 ident: 346_CR6 publication-title: Cancer Res. doi: 10.1158/0008-5472.CAN-05-0564 – volume: 23 start-page: 1350020 year: 2013 ident: 346_CR22 publication-title: Int. J. Bifurc. Chaos doi: 10.1142/S021812741350020X |
| SSID | ssj0067947 |
| Score | 2.2949777 |
| Snippet | In this paper, we present a system of five ordinary differential equations which consider population dynamics among cancer stem cells, tumor cells, and healthy... |
| SourceID | pubmedcentral proquest pubmed crossref springer |
| SourceType | Open Access Repository Aggregation Database Index Database Enrichment Source Publisher |
| StartPage | 791 |
| SubjectTerms | Breast cancer Cancer Computer Science Differential equations Dynamics Engineering Estrogens Immune system Mathematics Mathematics and Statistics Ordinary differential equations Original Research Stem cells |
| Title | Global Dynamics of a Breast Cancer Competition Model |
| URI | https://link.springer.com/article/10.1007/s12591-017-0346-x https://www.ncbi.nlm.nih.gov/pubmed/33487925 https://www.proquest.com/docview/2473802683 https://www.proquest.com/docview/2480736109 https://pubmed.ncbi.nlm.nih.gov/PMC7821963 |
| Volume | 28 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1LS8QwEB5EL3rw_agvInhSAt30kfS4u7qKoicX9FTSNEVBuuKu4M93Jn2s6ws8lZI0bSbJ5Jtm5huAY9yTAitlxlUkDA8T3-fEOsaNUUILE-k4pgDnm9v4chhe3Uf3dRz3uPF2b44knaaeBrshUifTV3I_CGOOwHEhQtOdVuNQdBv1G-MEczHSiew4N_XmKPOnJmY3o28I87uj5JfTUrcJDVZhuUaPrFsN9xrM2XIdVprMDKxeqOuw9IlmEO9uWm7W8QaEFc0_O6ty0Y_ZqGCa9cg5fcL6NAleWd-haefNxShb2vMmDAfnd_1LXudO4CaU_oQTi7oR2tIPnjxT1nSsMnFQqCJTOpBWxplSgSY6M5VHNpGJ8I3VmdBadyjYdgvmy1Fpd4AJFSfYSJ5LXYSIB9BGKxAk5BbVQxTmmQd-I8TU1MTilN_iOZ1SIpPcU5R7SnJP3z04aR95qVg1_qq834xMWi-wcSpCGSi0H1XgwVFbjEuDzjt0aUdvVEehAiNJeLBdDWT7NgpAxj5HHsiZIW4rEO32bEn59OjotxFTkdry4LSZDNPP-rUTu_-qvQeLgux65zS4D_OT1zd7gOBnkh3CQnfQ693S9eLh-vzQTf4PpwD9OQ |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1Lb9swDCaG7LDtsEfWdt7aVQN2WiHAkR-Sj13WINuSnBIgN0OWZWxA4RRxAvTnl5QfaZa2wI6G5IdIivpkkR8BvuKaFFgpM64iYXiY-D4n1jFujBJamEjHMSU4T2fxeBH-WkbLJo-7aqPd2yNJ56l3yW6I1GnrK7kfhDFH4PgcsYCisgULcdm63xgNzOVIJ3LgwtTbo8yHHrG_GB0gzMNAyX9OS90iNHoLrxv0yC5rdb-DZ7bsw5u2MgNrJmofXt2jGcSracfNWr2HsKb5Zz_qWvQVWxVMs-8UnL5hQzKCNRs6NO2iuRhVS7s-gsXoaj4c86Z2Ajeh9DecWNSN0JZ-8OSZsmZglYmDQhWZ0oG0Ms6UCjTRmak8solMhG-szoTWekDJtsfQK1el_QBMqDjBh-S51EWIeAD3aAWChNyie4jCPPPAb4WYmoZYnOpbXKc7SmSSe4pyT0nu6a0H37pbbmpWjac6n7aaSZsJVqUilIHC_aMKPPjSNePUoPMOXdrVlvoodGAkCQ9OakV2b6MEZBxz5IHcU3HXgWi391vKv38c_TZiKnJbHly0xrD7rEcH8fG_ep_Di_F8OkknP2e_P8FLQXt8F0B4Cr3NemvPEAhtss_O8O8A4Vr9HA |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1ZS8QwEB5EQfTB-6hnBJ-UYDc9kj7q6uKNDy74VtI0RUG64lbw5zvTa10v8LEkPTKZTL5pZr4B2Mc9ybNSJlwFwnA_cl1OrGPcGCW0MIEOQ0pwvrkNz_v-5UPwUNc5HTbR7s2RZJXTQCxNeXH0kmZHo8Q3RO3kBkvuen7IEUROoTXukKL3xXFjikNUtjJfOpKdMmS9Odb86RHjG9M3tPk9aPLLyWm5IfUWYK5Gkuy4mvpFmLD5Esw3VRpYvWiXYPYT5SBe3bQ8rcNl8CvKf3Za1aUfskHGNDuhQPWCdUkhXlm3RNZlZBejymnPK9Dvnd13z3ldR4EbX7oFJ0Z1I7Slnz1poqzpWGVCL1NZorQnrQwTpTxN1GYqDWwkI-EaqxOhte5Q4u0qTOaD3K4DEyqM8CFpKnXmIzZAfy1DwJBaNBWBnyYOuI0QY1OTjFOti-d4RI9Mco9R7jHJPX534KC95aVi2Pir81YzM3G92Iax8KWn0JdUngN7bTMuEzr70LkdvFEfhcaMJOHAWjWR7dsoGRnHHDggx6a47UAU3OMt-dNjScWN-IpMmAOHjTKMPuvXQWz8q_cuTN-d9uLri9urTZgR5O6XsYRbMFm8vtltxERFslPq_QddyQFn |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Global+Dynamics+of+a+Breast+Cancer+Competition+Model&rft.jtitle=Differential+equations+and+dynamical+systems&rft.au=Abernathy%2C+Kristen&rft.au=Abernathy%2C+Zachary&rft.au=Baxter%2C+Arden&rft.au=Stevens%2C+Meghan&rft.date=2020-10-01&rft.issn=0971-3514&rft.eissn=0974-6870&rft.volume=28&rft.issue=4&rft.spage=791&rft.epage=805&rft_id=info:doi/10.1007%2Fs12591-017-0346-x&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s12591_017_0346_x |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0971-3514&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0971-3514&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0971-3514&client=summon |