Passing to the limit 2D–1D in a model for metastatic growth

Figures & data

Figure 1. Phase plan of the systems (2) and (3). The solution is zero out of the stared characteristics coming from points of the boundary (1, θ) with θ∈[θ₀−ϵ, θ₀+ϵ]. The values of the parameters are chosen for illustrative purposes and are not realistic ones: a=2, c=5.85, d=0.1, θ₀=200, ϵ=100.

Figure 2. Convergence of the numerical scheme for the 2D model (problem (1)), with ϵ=100. The values of the parameters for the growth velocity field G are from [11] and correspond to mice data: a=0.192, c=5.85, d=0.00873, θ ₀=625. For the metastasis parameters, we used: m=0.001 and α=2/3. Total simulation time T=50. (a) Relative error in function of M, for various timesteps dt. (b) Relative error in function of dt, for various values of the discretization parameter of the boundary M.

Figure 3. Relative difference between the 1D simulation and the 2D one, for 5 values of ϵ: 100, 50, 10, 1 and 0.1. The values of the parameters are the same than in Figure 2 and the timestep used is dt=0.1. (a) Convergence when ϵ goes to zero, for T=15 and T=100. The value of M used for the 2D simulations is M=10. (b) Convergence when ϵ goes to zero, with respect to M (M=10, 50, 100), for T=50. The three curves are almost all the same.

Table 1. Computational times on a personal computer of various simulations in 1D and 2D.

	2D	1D
T=15 days, dt=0.1	67 s	10.69 s
T=15 days, dt=0.01	1 h\ 42 min	11 min
T=100 days, dt=0.1	46 min	4.7 min