Integrating Mathematical Modeling and Oncology: A Improvised Cubic B-Spline Collocation Approach for Prediction of Chemotherapy Response

Integrating Mathematical Modeling and Oncology: A Improvised Cubic B-Spline Collocation Approach for Prediction of Chemotherapy Response | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Integrating Mathematical Modeling and Oncology: A Improvised Cubic B-Spline Collocation Approach for Prediction of Chemotherapy Response Sumita Dahiya, Priyanka Yadav, Nadeem Malik This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8133919/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Cancer is one of the major global health challenges and predicting the response of malignant tumors to chemotherapy is particularly difficult due to the complex interaction between malignant cell growth and drug effects across time and space. Mathematical modeling plays an important role in understanding these dynamics, as it provides a systematic way to simulate malignant tumor behavior under treatment and to evaluate potential outcomes of different therapeutic strategies. In this work, a Reaction–Diffusion model is considered to analyze the dynamics of malignant tumors during chemotherapy treatment. The temporal derivative of the model equation is discretized using the Crank–Nicolson scheme, while the spatial derivatives are approximated using an improvised cubic B– spline collocation approach. Nonlinearities in the governing model are treated with the Rubin–Graves method, which transforms the nonlinear terms into a tractable linear form. This approach maintains diagonal dominance of the system matrix while ensuring consistent enforcement of boundary conditions. To evaluate stability, we apply the Spectral Fourier (von Neumann) analysis to the proposed scheme. Four numerical experiments are conducted to test the effectiveness of the scheme under different treatment parameters. The calculated numerical solutions and graphical representations are of high precision and are in very good agreement with previously reported results. Our approach is both easy to implement and computationally efficient, making it a highly promising option for biomedical simulations of tumor dynamics. Applied Mathematics Oncology Reaction–Diffusion model Malignant tumor dynamics Improvised cubic B–spline collocation approach Crank–Nicolson scheme Rubin–Graves method Spectral Fourier analysis Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8133919","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":546205016,"identity":"148bac6d-d219-4cbc-a0d4-93c0b031e657","order_by":0,"name":"Sumita Dahiya","email":"","orcid":"","institution":"Netaji Subhas University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Sumita","middleName":"","lastName":"Dahiya","suffix":""},{"id":546205017,"identity":"46e2263f-2be9-4c6d-955d-26d256373db8","order_by":1,"name":"Priyanka Yadav","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFUlEQVRIiWNgGAWjYFACxgYgkQDEPAwSH37YwMV5iNIiObMnDchgJqQFDCBapHnYDsO14ATm0ofbJH7uSJNnkO49eIOH53ye/Iz8Aww/ahhkzHFosexLbJPsPZNj2CBzLtlCwuJ2scGNZAbGnmMMPJYN2LUYnGFsk+Btq2BskMgxkzDguZ24QSKZgYG3gYHH4ABuLZJ/2yrswVoS2M4lzp8BtOUvAS3SvG05iWAtB9gOJDYAHcaMzxbLHsZma9m2tOQ2mTPGlo09yYkbzjw2OCxzTAKnFnMe9oc337Yl2/ZL9xje/vPDLnF+e+LDh29qbOxxOoyBgUUCxGCTQBIFKpbArh6ihfkDmIVbzSgYBaNgFIx0AAA9t1cgIkbuEwAAAABJRU5ErkJggg==","orcid":"","institution":"Netaji Subhas University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Priyanka","middleName":"","lastName":"Yadav","suffix":""},{"id":546205018,"identity":"d0d40bc3-48f6-47e2-9a1c-92c950ecb40c","order_by":2,"name":"Nadeem Malik","email":"","orcid":"","institution":"Netaji Subhas University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Nadeem","middleName":"","lastName":"Malik","suffix":""}],"badges":[],"createdAt":"2025-11-17 10:21:18","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8133919/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8133919/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":96249529,"identity":"66a1f785-0e72-42a9-8317-4ff6434f2a3e","added_by":"auto","created_at":"2025-11-19 07:33:51","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":954330,"visible":true,"origin":"","legend":"","description":"","filename":"DahiyaYadavMalik.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8133919/v1_covered_eca51f82-9865-4a3f-a1ab-aa3375eefbeb.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eIntegrating Mathematical Modeling and Oncology: A Improvised Cubic B-Spline Collocation Approach for Prediction of Chemotherapy Response\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Netaji Subhas University of Technology","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Oncology, Reaction–Diffusion model, Malignant tumor dynamics, Improvised cubic B–spline collocation approach, Crank–Nicolson scheme, Rubin–Graves method, Spectral Fourier analysis","lastPublishedDoi":"10.21203/rs.3.rs-8133919/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8133919/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCancer is one of the major global health challenges and predicting the response of malignant tumors to chemotherapy is particularly difficult due to the complex interaction between malignant cell growth and drug effects across time and space. Mathematical modeling plays an important role in understanding these dynamics, as it provides a systematic way to simulate malignant tumor behavior under treatment and to evaluate potential outcomes of different therapeutic strategies. In this work, a Reaction–Diffusion model is considered to analyze the dynamics of malignant tumors during chemotherapy treatment. The temporal derivative of the model equation is discretized using the Crank–Nicolson scheme, while the spatial derivatives are approximated using an improvised cubic B– spline collocation approach. Nonlinearities in the governing model are treated with the Rubin–Graves method, which transforms the nonlinear terms into a tractable linear form. This approach maintains diagonal dominance of the system matrix while ensuring consistent enforcement of boundary conditions. To evaluate stability, we apply the Spectral Fourier (von Neumann) analysis to the proposed scheme. Four numerical experiments are conducted to test the effectiveness of the scheme under different treatment parameters. The calculated numerical solutions and graphical representations are of high precision and are in very good agreement with previously reported results. Our approach is both easy to implement and computationally efficient, making it a highly promising option for biomedical simulations of tumor dynamics.\u003c/p\u003e","manuscriptTitle":"Integrating Mathematical Modeling and Oncology: A Improvised Cubic B-Spline Collocation Approach for Prediction of Chemotherapy Response","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-18 07:52:04","doi":"10.21203/rs.3.rs-8133919/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"8933e90e-4b97-4fea-ad9b-53a7ee50dbd0","owner":[],"postedDate":"November 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":58164553,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2025-11-18T07:52:04+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-18 07:52:04","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8133919","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8133919","identity":"rs-8133919","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}