In Chapter 4, the moving boundary partial differential equation (MBPDE) system of eqs. (1.2) through (1.6) with one PDE (eq. [1.2]) is extended to three PDEs to model the growth of a tumor. The outer boundary of the tumor moves as the tumor grows, which is analyzed with the MBPDE algorithm of Chapter 2.

The three PDE tumor growth model is first stated in coordinate-free format as eq.(4.1). The three PDE dependent variables are: (1) density of cancer cells, (2) concentration of the matrix degrading enzyme (MDE), and (3) surrounding extra-cellular matrix (ECM). PDE (1) includes non-linear chemotactic diffusion with sensitivity,1, and non-linear haptotactic diffusion with sensitivity, χ2.

The three PDEs are then stated in 1D spherical coordinates for which distance, r, is the radial coordinate for the tumor. Homogeneous Neumann boundary conditions (BCs) are specified for PDEs (1) and (2). PDE (3) does not have partial derivatives in r (only t) and therefore does not require BCs.

Numerical method of lines (MOL) solutions are computed for three cases: (1) no boundary movement, (2) movement at a constant velocity, and (3) movement according to the velocity d r u d t = k r u u 1 ( r   =   r u , t ) ( 1 − u 1 ( r   =   r u , t ) − u 3 ( r   =   r u , t ) ) u 3 ( r   =   r u , t ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429275128/bc1acef7-dd43-4bcb-9532-bbbe7bb88ee1/content/equ_1.tif"/> For the three cases, the graphical output includes the boundary location, r u(t), and boundary velocity, d r u ( t ) d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429275128/bc1acef7-dd43-4bcb-9532-bbbe7bb88ee1/content/equ_2.tif"/> . The solutions for the three dependent variables are complicated and demonstrate the requirement for a computer-based numerical solution.