{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T10:08:24Z","timestamp":1773310104709,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T00:00:00Z","timestamp":1660521600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration.<\/jats:p>","DOI":"10.3390\/sym14081694","type":"journal-article","created":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T23:44:03Z","timestamp":1660607043000},"page":"1694","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1202-2149","authenticated-orcid":false,"given":"Muhammad","family":"Abaid Ur Rehman","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan"}]},{"given":"Jamshad","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2515-2313","authenticated-orcid":false,"given":"Ali","family":"Hassan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1846-3615","authenticated-orcid":false,"given":"Witold","family":"Pawlowski","sequence":"additional","affiliation":[{"name":"Institute of Machine Tools and Production Engineering, Lodz University of Technology, 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5162-2692","authenticated-orcid":false,"given":"Hanen","family":"Karamti","sequence":"additional","affiliation":[{"name":"Department of Computer Sciences, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Fahad M.","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.apm.2017.03.045","article-title":"Modelling of tumor cells regression in response to chemotherapeutic treatment","volume":"48","author":"Ansarizadeh","year":"2017","journal-title":"Appl. Math. 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