{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T00:02:11Z","timestamp":1769558531441,"version":"3.49.0"},"reference-count":36,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MBE"],"published-print":{"date-parts":[[2026]]},"DOI":"10.3934\/mbe.2026023","type":"journal-article","created":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T10:03:42Z","timestamp":1769508222000},"page":"594-618","source":"Crossref","is-referenced-by-count":0,"title":["Evolutionary dynamics of acid resistance in tumors: A mathematical model"],"prefix":"10.3934","volume":"23","author":[{"given":"Prithvi","family":"Anickode","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Arizona State University, 1031 Palm Walk, Tempe, AZ 85287, USA","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabio Augusto","family":"Milner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"School of Mathematical and Statistical Sciences, Arizona State University, 901 Palm Walk Room 216, Tempe, AZ 85281 USA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/mbe.2026023-1","doi-asserted-by":"publisher","unstructured":"A. Anemone, L. Consolino, F. Arena, M. Capozza, D. L. Longo, Imaging tumor acidosis: A survey of the available techniques for mapping in vivo tumor pH, <i>Cancer Metastasis Rev.<\/i>, <b>38<\/b> (2019), 25\u201349. https:\/\/doi.org\/10.1007\/s10555-019-09782-9","DOI":"10.1007\/s10555-019-09782-9"},{"key":"key-10.3934\/mbe.2026023-2","doi-asserted-by":"publisher","unstructured":"M. G. Vander Heiden, L. C. Cantley, C. B. Thompson, Understanding the Warburg effect: The metabolic requirements of cell proliferation, <i>Science<\/i>, <b>324<\/b> (2009), 1029\u20131033. https:\/\/doi.org\/10.1126\/science.1160809","DOI":"10.1126\/science.1160809"},{"key":"key-10.3934\/mbe.2026023-3","doi-asserted-by":"publisher","unstructured":"H. F. Dvorak, Why are tumor blood vessels abnormal and why is it important to know?, <i>Br. J. Cancer<\/i>, <b>100<\/b> (2009), 865\u2013869. https:\/\/doi.org\/10.1038\/sj.bjc.6604967","DOI":"10.1038\/sj.bjc.6604967"},{"key":"key-10.3934\/mbe.2026023-4","doi-asserted-by":"publisher","unstructured":"L. Aubert, N. Nandagopal, Z. Steinhart, C. Guilbaud, K. \u00d6zkan, D. Brusa, et al., Tumor acidosis-induced DNA damage response and tetraploidy enhance sensitivity to ATM and ATR inhibitors, <i>EMBO Rep.<\/i>, <b>25<\/b> (2024), 1469\u20131489. https:\/\/doi.org\/10.1038\/s44319-024-00089-7","DOI":"10.1038\/s44319-024-00089-7"},{"key":"key-10.3934\/mbe.2026023-5","doi-asserted-by":"publisher","unstructured":"W.-Y. Shie, Y.-H. Chu, Y.-F. Chen, H.-W. Chen, M.-T. Lin, X.-J. Su, et al., Acidosis promotes the metastatic colonization of lung cancer via remodeling of the extracellular matrix and vasculogenic mimicry, <i>Int. J. Oncol.<\/i>, <b>63<\/b> (2023), 136. https:\/\/doi.org\/10.3892\/ijo.2023.5584","DOI":"10.3892\/ijo.2023.5584"},{"key":"key-10.3934\/mbe.2026023-6","doi-asserted-by":"publisher","unstructured":"N. Raghunand, R. Martinez-Zaguilan, S. H. Wright, R. J. Gillies, pH and drug resistance. \u2161. Turnover of acidic vesicles and resistance to weakly basic chemotherapeutic drugs, <i>Biochem. Pharmacol.<\/i>, <b>57<\/b> (1999), 1047\u20131058. https:\/\/doi.org\/10.1016\/S0006-2952(99)00021-0","DOI":"10.1016\/S0006-2952(99)00021-0"},{"key":"key-10.3934\/mbe.2026023-7","doi-asserted-by":"publisher","unstructured":"S. R. Pilon-Thomas, K. N. Kodumudi, A. E. El-Kenawi, S. Russell, A. M. Weber, K. Luddy, et al., Neutralization of tumor acidity improves antitumor responses to immunotherapy, <i>Cancer Res.<\/i>, <b>76<\/b> (2016), 1381\u20131390. https:\/\/doi.org\/10.1158\/0008-5472.CAN-15-1743","DOI":"10.1158\/0008-5472.CAN-15-1743"},{"key":"key-10.3934\/mbe.2026023-8","doi-asserted-by":"publisher","unstructured":"C. Corbet, O. Feron, Tumour acidosis: From the passenger to the driver's seat, <i>Nat. Rev. Cancer<\/i>, <b>17<\/b> (2017), 577\u2013593. https:\/\/doi.org\/10.1038\/nrc.2017.77","DOI":"10.1038\/nrc.2017.77"},{"key":"key-10.3934\/mbe.2026023-9","doi-asserted-by":"publisher","unstructured":"V. Estrella, T. Chen, M. Lloyd, J. Wojtkowiak, H. H. Cornnell, A. Ibrahim-Hashim, et al., Acidity generated by the tumor microenvironment drives local invasion, <i>Cancer Res.<\/i>, <b>73<\/b> (2013), 1524\u20131535. https:\/\/doi.org\/10.1158\/0008-5472.CAN-12-2796","DOI":"10.1158\/0008-5472.CAN-12-2796"},{"key":"key-10.3934\/mbe.2026023-10","doi-asserted-by":"publisher","unstructured":"E. Boedtkjer, S. F. Pedersen, The acidic tumor microenvironment as a driver of cancer, <i>Annu. Rev. Physiol.<\/i>, <b>82<\/b> (2020), 103\u2013126. https:\/\/doi.org\/10.1146\/annurev-physiol-021119-034627","DOI":"10.1146\/annurev-physiol-021119-034627"},{"key":"key-10.3934\/mbe.2026023-11","doi-asserted-by":"crossref","unstructured":"M. Greaves, C. C. Maley, Clonal evolution in cancer, <i>Nature<\/i>, <b>481<\/b> (2012), 306\u2013313. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1038\/nature10762\">https:\/\/doi.org\/10.1038\/nature10762<\/ext-link>","DOI":"10.1038\/nature10762"},{"key":"key-10.3934\/mbe.2026023-12","doi-asserted-by":"publisher","unstructured":"S. Liao, G. Wu, Z. Xie, X. Y. Lei, X. Y. Yang, S. Huang, et al., pH regulators and their inhibitors in tumor microenvironment, <i>Eur. J. Med. Chem.<\/i>, <b>267<\/b> (2024), 116170. https:\/\/doi.org\/10.1016\/j.ejmech.2024.116170","DOI":"10.1016\/j.ejmech.2024.116170"},{"key":"key-10.3934\/mbe.2026023-13","doi-asserted-by":"publisher","unstructured":"S. R. Pillai, M. Damaghi, Y. Marunaka, E. P. Spugnini, S. Fais, R. J. Gillies, Causes, consequences, and therapy of tumors acidosis, <i>Cancer Metast. Rev.<\/i>, <b>38<\/b> (2019), 205\u2013222. https:\/\/doi.org\/10.1007\/s10555-019-09792-7","DOI":"10.1007\/s10555-019-09792-7"},{"key":"key-10.3934\/mbe.2026023-14","doi-asserted-by":"publisher","unstructured":"S. Yuan, R. J. Norgard, B. Z. Stanger, Cellular plasticity in cancer, <i>Cancer Discov.<\/i>, <b>9<\/b> (2019), 837\u2013851. https:\/\/doi.org\/10.1158\/2159-8290.CD-19-0015","DOI":"10.1158\/2159-8290.CD-19-0015"},{"key":"key-10.3934\/mbe.2026023-15","doi-asserted-by":"publisher","unstructured":"C. E. Meacham, S. J. Morrison, Tumour heterogeneity and cancer cell plasticity, <i>Nature<\/i>, <b>501<\/b> (2013), 328\u2013337. https:\/\/doi.org\/10.1038\/nature12624","DOI":"10.1038\/nature12624"},{"key":"key-10.3934\/mbe.2026023-16","doi-asserted-by":"publisher","unstructured":"Z.-D. Shi, K. Pang, Z.-X. Wu, Y. Dong, L. Hao, J.-X. Qin, et al., Tumor cell plasticity in targeted therapy-induced resistance: Mechanisms and new strategies, <i>Signal Transduct. Target. Ther.<\/i>, <b>8<\/b> (2023), 113. https:\/\/doi.org\/10.1038\/s41392-023-01383-x","DOI":"10.1038\/s41392-023-01383-x"},{"key":"key-10.3934\/mbe.2026023-17","doi-asserted-by":"publisher","unstructured":"K. S. Hoek, C. R. Goding, Phenotype switching: Tumor cell plasticity as a resistance mechanism and target for therapy, <i>Cancer Res.<\/i>, <b>74<\/b> (2014), 5937\u20135941. https:\/\/doi.org\/10.1158\/0008-5472.CAN-14-1174","DOI":"10.1158\/0008-5472.CAN-14-1174"},{"key":"key-10.3934\/mbe.2026023-18","doi-asserted-by":"publisher","unstructured":"M. Archetti, Evolutionary dynamics of the Warburg effect: Glycolysis as a collective action problem among cancer cells, <i>J. Theor. Biol.<\/i>, <b>341<\/b> (2014), 1\u20138. https:\/\/doi.org\/10.1016\/j.jtbi.2013.09.017","DOI":"10.1016\/j.jtbi.2013.09.017"},{"key":"key-10.3934\/mbe.2026023-19","doi-asserted-by":"publisher","unstructured":"A. Kaznatcheev, R. Vander Velde, J. G. Scott, D. Basanta, Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature, <i>Br. J. Cancer<\/i>, <b>116<\/b> (2017), 785\u2013792. https:\/\/doi.org\/10.1038\/bjc.2017.5","DOI":"10.1038\/bjc.2017.5"},{"key":"key-10.3934\/mbe.2026023-20","doi-asserted-by":"publisher","unstructured":"G. Fiandaca, M. Delitala, T. Lorenzi, A mathematical study of the influence of hypoxia and acidity on the evolutionary dynamics of cancer, <i>Bull. Math. Biol.<\/i>, <b>83<\/b> (2021), 83. https:\/\/doi.org\/10.1007\/s11538-021-00914-3","DOI":"10.1007\/s11538-021-00914-3"},{"key":"key-10.3934\/mbe.2026023-21","doi-asserted-by":"publisher","unstructured":"H. J. Park, J. C. Lyons, T. Ohtsubo, C. W. Song, Acidic environment causes apoptosis by increasing caspase activity, <i>Br. J. Cancer<\/i>, <b>80<\/b> (1999), 1892\u20131897. https:\/\/doi.org\/10.1038\/sj.bjc.6690617","DOI":"10.1038\/sj.bjc.6690617"},{"key":"key-10.3934\/mbe.2026023-22","doi-asserted-by":"publisher","unstructured":"C. M. Worsley, R. B. Veale, E. S. Mayne, Inducing apoptosis using chemical treatment and acidic pH, and detecting it using the Annexin V flow cytometric assay, <i>PLoS ONE<\/i>, <b>17<\/b> (2022), e0270599. https:\/\/doi.org\/10.1371\/journal.pone.0270599","DOI":"10.1371\/journal.pone.0270599"},{"key":"key-10.3934\/mbe.2026023-23","doi-asserted-by":"publisher","unstructured":"M. Salvioli, J. Dubbeldam, K. Sta\u0148kov\u00e1, J. S. Brown, The contribution of evolutionary game theory to understanding and treating cancer, <i>Dyn. Games Appl.<\/i>, <b>12<\/b> (2022), 313\u2013342. https:\/\/doi.org\/10.1007\/s13235-021-00397-w","DOI":"10.1007\/s13235-021-00397-w"},{"key":"key-10.3934\/mbe.2026023-24","doi-asserted-by":"publisher","unstructured":"S. Jamali, M. Klier, S. Ames, L. F. Barros, R. McKenna, J. W. Deitmer, et al., Hypoxia-induced carbonic anhydrase IX facilitates lactate flux in human breast cancer cells by non-catalytic function, <i>Sci. Rep.<\/i>, <b>5<\/b> (2015), 13605. https:\/\/doi.org\/10.1038\/srep13605","DOI":"10.1038\/srep13605"},{"key":"key-10.3934\/mbe.2026023-25","doi-asserted-by":"publisher","unstructured":"E. M. England, S. K. Matarneh, T. L. Scheffler, C. Wachet, D. E. Gerrard, pH inactivation of phosphofructokinase arrests postmortem glycolysis, <i>Meat Sci.<\/i>, <b>98<\/b> (2014), 850\u2013857. https:\/\/doi.org\/10.1016\/j.meatsci.2014.07.019","DOI":"10.1016\/j.meatsci.2014.07.019"},{"key":"key-10.3934\/mbe.2026023-26","unstructured":"L. Perko, Differential Equations and Dynamical Systems, 3rd edition, Springer-Verlag, New York, 2001. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1007\/978-1-4613-0003-8\">https:\/\/doi.org\/10.1007\/978-1-4613-0003-8<\/ext-link>"},{"key":"key-10.3934\/mbe.2026023-27","doi-asserted-by":"publisher","unstructured":"S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, <i>J. Theor. Biol.<\/i>, <b>254<\/b> (2008), 178\u2013196. https:\/\/doi.org\/10.1016\/j.jtbi.2008.04.011","DOI":"10.1016\/j.jtbi.2008.04.011"},{"key":"key-10.3934\/mbe.2026023-28","doi-asserted-by":"publisher","unstructured":"R. K. Jain, Normalizing tumor microenvironment to treat cancer: bench to bedside to biomarkers, <i>J. Clin. Oncol.<\/i>, <b>31<\/b> (2013), 2205\u20132218. https:\/\/doi.org\/10.1200\/JCO.2012.46.3653","DOI":"10.1200\/JCO.2012.46.3653"},{"key":"key-10.3934\/mbe.2026023-29","doi-asserted-by":"publisher","unstructured":"K. L. Schadler, N. J. Thomas, P. A. Galie, D. H. Bhang, K. C. Roby, P. Addai, et al., Tumor vessel normalization after aerobic exercise enhances chemotherapeutic efficacy, <i>Oncotarget<\/i>, <b>7<\/b> (2016), 65429\u201365440. https:\/\/doi.org\/10.18632\/oncotarget.11748","DOI":"10.18632\/oncotarget.11748"},{"key":"key-10.3934\/mbe.2026023-30","doi-asserted-by":"publisher","unstructured":"I. Moen, L. E. Stuhr, Hyperbaric oxygen therapy and cancer\u2013a review, <i>Target. Oncol.<\/i>, <b>7<\/b> (2012), 233\u2013242. https:\/\/doi.org\/10.1007\/s11523-012-0233-x","DOI":"10.1007\/s11523-012-0233-x"},{"key":"key-10.3934\/mbe.2026023-31","doi-asserted-by":"publisher","unstructured":"M. Beloueche-Babari, S. C. Galobart, T. Delgado-Goni, S. Wantuch, H. G. Parkes, D. Tandy, et al., Monocarboxylate transporter 1 blockade with AZD3965 inhibits lipid biosynthesis and increases tumour immune cell infiltration, <i>Br. J. Cancer<\/i>, <b>122<\/b> (2020), 895\u2013903. https:\/\/doi.org\/10.1038\/s41416-019-0717-x","DOI":"10.1038\/s41416-019-0717-x"},{"key":"key-10.3934\/mbe.2026023-32","doi-asserted-by":"publisher","unstructured":"P. C. McDonald, S. C. Chafe, W. S. Brown, S. Saberi, M. Swayampakula, G. Venkateswaran, et al., Regulation of pH by carbonic anhydrase 9 mediates survival of pancreatic cancer cells with activated KRAS in response to hypoxia, <i>Gastroenterology<\/i>, <b>157<\/b> (2020), 823\u2013837. https:\/\/doi.org\/10.1053\/j.gastro.2019.05.004","DOI":"10.1053\/j.gastro.2019.05.004"},{"key":"key-10.3934\/mbe.2026023-33","doi-asserted-by":"publisher","unstructured":"S. R. Amith, L. Fliegel, Na+\/H+ exchanger-mediated hydrogen ion extrusion as a carcinogenic signal in triple-negative breast cancer etiopathogenesis and prospects for its inhibition in therapeutics, <i>Semin. Cancer Biol.<\/i>, <b>43<\/b> (2017), 35\u201341. https:\/\/doi.org\/10.1016\/j.semcancer.2017.01.004","DOI":"10.1016\/j.semcancer.2017.01.004"},{"key":"key-10.3934\/mbe.2026023-34","unstructured":"R. A. Gatenby, E. T. Gawlinski, A reaction-diffusion model of cancer invasion, <i>Cancer Res.<\/i>, <b>56<\/b> (1996), 5745\u20135753."},{"key":"key-10.3934\/mbe.2026023-35","doi-asserted-by":"publisher","unstructured":"J. B. McGillen, E. A. Gaffney, N. K. Martin, P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, <i>J. Math. Biol.<\/i>, <b>68<\/b> (2014), 1199\u20131224. https:\/\/doi.org\/10.1007\/s00285-013-0665-7","DOI":"10.1007\/s00285-013-0665-7"},{"key":"key-10.3934\/mbe.2026023-36","doi-asserted-by":"publisher","unstructured":"N. Beerenwinkel, R. F. Schwarz, M. Gerstung, F. Markowetz, Cancer evolution: Mathematical models and computational inference, <i>Syst. Biol.<\/i>, <b>64<\/b> (2015), e1\u2013e25. https:\/\/doi.org\/10.1093\/sysbio\/syu081","DOI":"10.1093\/sysbio\/syu081"}],"container-title":["Mathematical Biosciences and Engineering"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/mbe.2026023?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T10:03:45Z","timestamp":1769508225000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/mbe.2026023"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.3934\/mbe.2026023","relation":{},"ISSN":["1551-0018"],"issn-type":[{"value":"1551-0018","type":"print"}],"subject":[],"published":{"date-parts":[[2026]]}}}