As mentioned in Section 4.3, we focus our discussion on one of the three scenarios that are used in [57] to simulate tumour growth, which is seeding a single cluster of tumour cells in the centre of the computational domain. This scenario is considered on two different computational domains: a square with a regular triangular grid (as shown in Figure 5.3) and a circle with an unstructured triangular grid (as shown in Figure 5.6). The boundary conditions, the initial conditions and the parameter values that are used to simulate this model are as introduced in Section 4.3. Also, selected original results from [57] are presented in Subsection 4.3.2.
The numerical simulations in Figure 7.5 and Figure 7.9 show typical volume fractions of each phase on the square and circle domains respectively. In Subsection 4.3.2 we described the original results physically. The behaviour of our results physically is qualitatively the same as for the original results. It can be noted however the tumour has a different shape depending on the structure of the domains. In particular, the orientation of the triangles in the regular grid on the square domain has a noticeable influence on the spread direction of the tumour.
When the irregular grid is used on the circular domain the tumour has an incomplete circle shape since the grid is not radially symmetric. However, overall the tumour growth has similar behaviour in these two domains. The volume fraction of tumour cells reaches a maximum value in the early time, which is in the range between 0.8 and 0.9 (see Figure 7.4), and then the tumour cells in the centre start dying, leaving a ring of tumour to spread far from the centre of the domain.
Figure 7.6, Figure 7.7 and Figure 7.8 illustrate the evolution of the phase fluxes, pressures and the nutrient concentration on regular square grid, respectively. In Figure 7.10, Figure 7.11 and Figure 7.12 the evolutions of the phase fluxes, pressures and the nutrient concentration on
93 7.2. Further Numerical Results
an unstructured circle grid are presented.
As mentioned in Subsection 4.3.2, the proliferation and death rates of the tumour cells are assumed to be double and half the values of the proliferation and death rates of the healthy cells respectively (see Table 4.1). So, the tumour cells grow faster and die slower than the healthy cells. In this case, the tumour cells absorb the extracellular material during their growth. This leads to a fall in the extracellular material θ4and therefore leads to decreasing the healthy cells’
birth rate. Also, when the tumour cells grow faster than the healthy cells that leads to the tumour cells pushing the healthy cells in front and replacing the volume by the extracellular material (as illustrated in Figure 7.6 on a regular grid and Figure 7.10 on an unstructured grid). Furthermore, the high tumour cells density generate high pressures which leads to the occlusion of the blood vessels and therefore restricts the supply of the nutrient inside the tumour (cf. Figure 7.8 on a regular grid and Figure 7.12 on an unstructured grid).
Figure 7.4: The evolution of the tumour cells on the square domain along y = 0 with increasing time from lift to right, t=100, 200 and 300. The grid size is 33×33 with the number of unknowns in the momentum system equal to 34889.
Figure 7.5: Evolution of the volume fraction for each phase arranged from the top row to the bottom row: healthy cells θ1, tumour cells θ2, blood vessels θ3 and extracellular material θ4, with increasing time from left to right, t=100, 200 and 300. The grid size is 33 × 33 with the number of unknowns in the momentum system equal to 34889.
95 7.2. Further Numerical Results
Figure 7.6: Evolution of the fluxes of phases, arranged from the top row to the bottom row:
healthy cells θ1~u01, tumour cells θ2~u02and extracellular material θ4~u04, with increasing time from left to right, t=100, 200 and 300. The grid size is 33 × 33 with the number of unknowns in the momentum system equal to 34889.
Figure 7.7: Evolution of the pressures arranged from the top row to the bottom row: healthy and tumour cells p1 = p2 and extracellular material (ECM) p4, with increasing time from left to right, t=100, 200 and 300. The blood vessels pressure p3is zero in this model. The grid size is 33 × 33 with the number of unknowns in the momentum system equal to 34889.
Figure 7.8: Evolution of the nutrient concentration c with increasing time from left to right, t=100, 200 and 300. The grid size is 33 × 33 with the number of unknowns in the momentum system equal to 34889.
97 7.2. Further Numerical Results
Figure 7.9: Evolution of the volume fraction for each phase arranged from the top row to the bottom row: healthy cells θ1, tumour cells θ2, blood vessels θ3 and extracellular material θ4, with increasing time from left to right, t=100, 200 and 300. The number of unknowns in the momentum system equal to 76237.
Figure 7.10: Evolution of the fluxes of phases, arranged from the top row to the bottom row:
healthy cells θ1~u01, tumour cells θ2~u02and extracellular material θ4~u04, with increasing the time from left to right, t=100, 200 and 300. The number of unknowns in the momentum system equal to 76237.
99 7.2. Further Numerical Results
Figure 7.11: Evolution of the pressures arranged from the top row to the bottom row: healthy and tumour cells p1 = p2 and extracellular material (ECM) p4, with increasing time from left to right, t=100, 200 and 300. The blood vessels pressure p3 is zero in this model. The number of unknown in the momentum system equal to 76237.
Figure 7.12: Evolution of the nutrient concentration c with increasing time from left to right, t=100, 200 and 300. The number of unknown in the momentum system equal to 76237.