Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System
Tumor angiogenesis, the formation of new blood vessels from pre-existing vasculature, is a crucial process in cancer growth and metastasis. Mathematical modeling through partial differential equations helps to understand this complex biological phenomenon. Here, we provide a conservation properties...
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MDPI AG
2024-12-01
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| author | Pasquale De Luca Livia Marcellino |
| author_facet | Pasquale De Luca Livia Marcellino |
| author_sort | Pasquale De Luca |
| collection | DOAJ |
| description | Tumor angiogenesis, the formation of new blood vessels from pre-existing vasculature, is a crucial process in cancer growth and metastasis. Mathematical modeling through partial differential equations helps to understand this complex biological phenomenon. Here, we provide a conservation properties analysis in a tumor angiogenesis model describing the evolution of endothelial cells, proteases, inhibitors, and extracellular matrix. The adopted approach introduces a numerical framework that combines spatial and time discretization techniques. Here, we focus on maintaining solution accuracy while preserving physical quantities during the simulation process. The method achieved second-order accuracy in both space and time discretizations, with conservation errors showing consistent convergence as the mesh was refined. The numerical schema demonstrates stable wave propagation patterns, in agreement with experimental observations. Numerical experiments validate the approach and demonstrate its reliability for long-term angiogenesis simulations. |
| format | Article |
| id | doaj-art-2f5b0002469641b7bcf51c4f9c4fc98b |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-2f5b0002469641b7bcf51c4f9c4fc98b2025-01-10T13:18:00ZengMDPI AGMathematics2227-73902024-12-011312810.3390/math13010028Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE SystemPasquale De Luca0Livia Marcellino1International PhD Programme/UNESCO Chair “Environment, Resources and Sustainable Development”, Department of Science and Technology, Parthenope University of Naples, Centro Direzionale Isola C4, 80143 Naples, ItalyInternational PhD Programme/UNESCO Chair “Environment, Resources and Sustainable Development”, Department of Science and Technology, Parthenope University of Naples, Centro Direzionale Isola C4, 80143 Naples, ItalyTumor angiogenesis, the formation of new blood vessels from pre-existing vasculature, is a crucial process in cancer growth and metastasis. Mathematical modeling through partial differential equations helps to understand this complex biological phenomenon. Here, we provide a conservation properties analysis in a tumor angiogenesis model describing the evolution of endothelial cells, proteases, inhibitors, and extracellular matrix. The adopted approach introduces a numerical framework that combines spatial and time discretization techniques. Here, we focus on maintaining solution accuracy while preserving physical quantities during the simulation process. The method achieved second-order accuracy in both space and time discretizations, with conservation errors showing consistent convergence as the mesh was refined. The numerical schema demonstrates stable wave propagation patterns, in agreement with experimental observations. Numerical experiments validate the approach and demonstrate its reliability for long-term angiogenesis simulations.https://www.mdpi.com/2227-7390/13/1/28tumor angiogenesispartial differential equationsnumerical methodsnumerical conservation |
| spellingShingle | Pasquale De Luca Livia Marcellino Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System Mathematics tumor angiogenesis partial differential equations numerical methods numerical conservation |
| title | Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System |
| title_full | Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System |
| title_fullStr | Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System |
| title_full_unstemmed | Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System |
| title_short | Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System |
| title_sort | conservation law analysis in numerical schema for a tumor angiogenesis pde system |
| topic | tumor angiogenesis partial differential equations numerical methods numerical conservation |
| url | https://www.mdpi.com/2227-7390/13/1/28 |
| work_keys_str_mv | AT pasqualedeluca conservationlawanalysisinnumericalschemaforatumorangiogenesispdesystem AT liviamarcellino conservationlawanalysisinnumericalschemaforatumorangiogenesispdesystem |