Visualisation of the Numerical Solution of Partial Differential Equation Systems in Three Space Dimensions and its Importance for Mathematical Models in Biology

H Enderling1*, ARA Anderson1, MAJ Chaplain1, GW Rowe2

1Division of Mathematics,   2Department of Applied Computing
University of Dundee,   Dundee, UK

*Correspondence: Heiko Enderling

Mathematical Biosciences and Engineering 3(4), 571-582, 2006.
[pdf]

Abstract:
Numerical analysis and computational simulation of partial differential equation models in mathematical biology is now an integral part of the research in this field. More and more we are seeing the development of partial differential equation models in more than one space dimension and it is therefore necessary to generate a clear and effective visualisation platform between the mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualisation of the results is more complicated. The scope of this paper is to apply the established Marching Cubes volume rendering technique to the study of solid tumour growth and invasion, and present and adaptation of the algorithm to speed-up the surface rendering from numerical simulation data. As a specific example, in this paper we examine the computational solutions arising from numerical simulation results of a mathematical model of malignant solid tumour growth and invasion in an irregular heterogeneous three-dimensional domain, i.e. the female breast. Due to the different variables that interact with each other, more than one data set may have to be displayed simultaneously which can be realized through transparency blending. The usefulness of the proposed method for visualisation in a more general context will also be discussed.

Keywords:
Visualisation, Marching Cubes, transparency blending, partial differential equations, mathematical model, tumour growth and invasion

click on the images/animations to access the video

 

(8.8 MB)
 
Figure 8: Solid tumour growth and invasion into a heterogeneous tissue. As time evolves, the initially spherical tumour (red) evolves to a heterogeneous shape due to haptotaxis towards areas of high tissue density (intense blue).

 
without proliferation
 		 with proliferation
 

(10.3 MB)
(11.5 MB)

(8.8 MB)
(9.0 MB) 		(additional animations)

Figure 9: Three-dimensional tumour growing in heterogeneous tissue without (left) and with proliferation (right). The visualisation of isosurfaces with transparency blending shows huge variations in the internal structure.

 

(10.3 MB)
 
Figure 12: Tumour growth in a heterogeneous, irregularly shaped domain. The tumour shown in red in the top row is initially located close to the nearby domain surface which is shown in dark yellow. Different tissue densities are shown in less intense blue which denotes lower density and intense blue which denotes higher density, respectively. Different tumour densities are shown in the bottom row. High densities are represented by an intense pink colour and lower densities in a less intense pink, respectively.

 
additional videos (not in paper):

 
(18.5MB)(19.4 MB)

Development of tumours in a highly heterogeneous tissue.