Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology
Numerical analysis and computational simulation of partial differentialequation models in mathematical biology are now an integral partof the research in this field. Increasingly we are seeing the development ofpartial differential equation models in more than one space dimension, and itis therefore...
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| Format: | Article |
| Language: | English |
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AIMS Press
2006-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.571 |
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| author | Heiko Enderling Alexander R.A. Anderson Mark A.J. Chaplain Glenn W.A. Rowe |
| author_facet | Heiko Enderling Alexander R.A. Anderson Mark A.J. Chaplain Glenn W.A. Rowe |
| author_sort | Heiko Enderling |
| collection | DOAJ |
| description | Numerical analysis and computational simulation of partial differentialequation models in mathematical biology are now an integral partof the research in this field. Increasingly we are seeing the development ofpartial differential equation models in more than one space dimension, and itis therefore necessary to generate a clear and effective visualisation platformbetween the mathematicians and biologists to communicate the results. Themathematical extension of models to three spatial dimensions from one or twois often a trivial task, whereas the visualisation of the results is more complicated.The scope of this paper is to apply the established marching cubesvolume rendering technique to the study of solid tumour growth and invasion,and present an adaptation of the algorithm to speed up the surface renderingfrom numerical simulation data. As a specific example, in this paper we examinethe computational solutions arising from numerical simulation resultsof a mathematical model of malignant solid tumour growth and invasion in anirregular heterogeneous three-dimensional domain, i.e., the female breast. Dueto the different variables that interact with each other, more than one data setmay have to be displayed simultaneously, which can be realized through transparencyblending. The usefulness of the proposed method for visualisation ina more general context will also be discussed. |
| format | Article |
| id | doaj-art-46546e170f954878a0c2479484f91882 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2006-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-46546e170f954878a0c2479484f918822025-01-24T01:52:27ZengAIMS PressMathematical Biosciences and Engineering1551-00182006-07-013457158210.3934/mbe.2006.3.571Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biologyHeiko Enderling0Alexander R.A. Anderson1Mark A.J. Chaplain2Glenn W.A. Rowe3Division of Mathematics, University of Dundee, 23 Perth Road, Dundee, DD1 4HNDivision of Mathematics, University of Dundee, 23 Perth Road, Dundee, DD1 4HNDivision of Mathematics, University of Dundee, 23 Perth Road, Dundee, DD1 4HNDivision of Mathematics, University of Dundee, 23 Perth Road, Dundee, DD1 4HNNumerical analysis and computational simulation of partial differentialequation models in mathematical biology are now an integral partof the research in this field. Increasingly we are seeing the development ofpartial differential equation models in more than one space dimension, and itis therefore necessary to generate a clear and effective visualisation platformbetween the mathematicians and biologists to communicate the results. Themathematical extension of models to three spatial dimensions from one or twois often a trivial task, whereas the visualisation of the results is more complicated.The scope of this paper is to apply the established marching cubesvolume rendering technique to the study of solid tumour growth and invasion,and present an adaptation of the algorithm to speed up the surface renderingfrom numerical simulation data. As a specific example, in this paper we examinethe computational solutions arising from numerical simulation resultsof a mathematical model of malignant solid tumour growth and invasion in anirregular heterogeneous three-dimensional domain, i.e., the female breast. Dueto the different variables that interact with each other, more than one data setmay have to be displayed simultaneously, which can be realized through transparencyblending. The usefulness of the proposed method for visualisation ina more general context will also be discussed.https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.571partial differential equationsvisualizationadaptive marching cubesmathematical modeltumour growth and invasion.transparency blendingvisualisation |
| spellingShingle | Heiko Enderling Alexander R.A. Anderson Mark A.J. Chaplain Glenn W.A. Rowe Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology Mathematical Biosciences and Engineering partial differential equations visualization adaptive marching cubes mathematical model tumour growth and invasion. transparency blending visualisation |
| title | Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| title_full | Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| title_fullStr | Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| title_full_unstemmed | Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| title_short | Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| title_sort | visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology |
| topic | partial differential equations visualization adaptive marching cubes mathematical model tumour growth and invasion. transparency blending visualisation |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.571 |
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