1. Introduction

2. Background

3. Mathematical Formulation

Figure 3.2. Schematic diagram of a control volume with cross flow.

Figure 3.2. Schematic diagram of a control volume with cross flow.

3.5. Dimensionalizing Tumour Concentration Equation

The focus is to solve the concentration equation for tumour concentration $C\left(x,y,t\right)$ . In dimensionless variables, the governing equation, Landau and Lifshitz (1959) is:

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{1}{Pe}\left(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}\right)$$

(3.1)

where$u\left(x,y,t\right)$ and $v\left(x,y,t\right)$ are the given divergence-free advective velocity vector fields along $x$and$y$ directions respectively, $t$ is time and $C\left(x,y,t\right)$ is the tumour concentration levels in human body tissues.

But the Peclet number is given by

$$Pe=\frac{UL}{D}$$

(3.2)

Pe is the Peclet number, where $U$is an axial velocity, $L$is arterial length, and $D$is the diffusivity of the tumour cells.

Substituting (3.2) into (3.1) gives

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=Pe\left(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}\right)$$

(3.3)

3.6 Discretization of Concentration Equation

The partial differential equation for concentration was discretized to form a central Crank Nicolson scheme which was eventually solved using the finite difference method. In application, equation (3.3) is discretized to study the effects of Peclet number for tumour concentration levels. Using a central difference numerical scheme, $C_t$is replaced by forward difference scheme while$C_{xx}$ is the average of central difference approximation at $j$th and$j+1$ th level, and $C_{yy}$ is replaced by central difference approximation. When these approximations are substituted into equation (3.3), and taking $u=v=1$, the second derivative in concentration equation at node could be represented as follows:

The effect of Pe number was investigated on the tumour concentration levels. Taking $\phi =\frac{\Delta t}{\left(\Delta x\right)}=\frac{\Delta t}{\left(\Delta y\right)}$, and $\mu =\frac{\Delta t}{{\left(\Delta x\right)}^2}=\frac{\Delta t}{{\left(\Delta y\right)}^2}$, $\Delta x=\Delta y$and multiplying by $2\Delta t$throughout equation (3.4) and re-arranging, the scheme below is obtained

$$\left(\phi -\mu Pe\right)C_{i+1,j}^n-\left(\phi +\mu Pe\right)C_{i-1,j}^n+\left(6\mu Pe-2\right)C_{i,j}^n=-\phi C_{i,j+1}^n+\mu Pe{C}_{i-1,j+1}^n+\left(2\mu Pe+\phi \right)C_{i,j-1}^n+\mu Pe{C}_{i+1,j+1}^n-2C_{i,j}^{n+1}$$

(3.5)

Taking$\Delta x=\Delta y=0.1$, and $\Delta t=0.01$, $\Rightarrow \phi =0.1$and$\mu =2$ results to central difference scheme below

$$\left(0.1-2Pe\right)C_{i+1,j}^n-\left(0.1+2Pe\right)C_{i-1,j}^n+\left(12Pe-2\right)C_{i,j}^n =-0.1C_{i,j+1}^n+2Pe{C}_{i-1,j+1}^n+\left(4Pe+0.1\right)C_{i,j-1}^n+2Pe{C}_{i+1,j+1}^n-2C_{i,j}^{n+1}$$

(3.6)

The above central difference scheme can be in form of six algebraic equations when i = 1, 2, …,6 with j = 1 and n = 0 as

the following initial and boundary conditions were applied to equation (3.7)

$$C\left(\mathrm{x},\mathrm{y},1\right)=0,\mathrm{t}=0$$

(3.8)

$$\mathrm{C}\left(0,\mathrm{y},\mathrm{t}\right)=10,\mathrm{C}\left(\mathrm{x},0,\mathrm{t}\right)=\mathrm{C}\left(\mathrm{x},2,\mathrm{t}\right)=0,\mathrm{t}>0,\mathrm{x}\ne \mathrm{y}$$

(3.9)

The above algebraic equations in (3.7) can be written in matrix form as

Solving the matrix system (3.10) using MATLAB, solutions were obtained for varying values of Pe. The results were provided in table 4.3.

3.7. Discretization of Cahn-Hilliard Model

Equation (3.10) was discretized to study the effects of mobility$M$, and density$\rho$, of tumour cells. Using a central difference numerical scheme, ${\phi }_t$was replaced by forward difference scheme. ${\phi }_{xx}$ was considered the average of central difference approximation at $j$th and$j+1$ th level, and ${\phi }_{yy}$ was replaced by central difference approximation. These approximations were substituted into equation (3.11). Letting the superscript $n$indicate the time (temporal variable) while the subscript $i$and $j$indicate the spatial variable $x$and $y$, discretization of equation (2.10) gives

$$\frac{{\phi }_{i,j}^{n+1}-{\phi }_{i,j}^n}{\Delta t}={\rm M}\rho \left(\frac{1}{2}\left(\frac{\left({\phi }_{i+1,j}^n-2{\phi }_{i,j}^n+{\phi }_{i-1,j}^n\right)}{{\left(\Delta x\right)}^2}+\frac{{\phi }_{i+1,j+1}^n-2{\phi }_{i,j+1}^n+{\phi }_{i-1,j+1}^n}{{\left(\Delta x\right)}^2}\right)+\frac{{\phi }_{i,j+1}^n-2{\phi }_{i,j}^n+{\phi }_{i,j-1}^n}{{\left(\Delta y\right)}^2}\right)-v\left(\frac{{\phi }_{i+1,j}^n-{\phi }_{i-1,j}^n}{\left(2\Delta x\right)}+\frac{{\phi }_{i,j+1}^n-{\phi }_{i,j-1}^n}{\left(2\Delta y\right)}\right)+\frac{{\phi }_{i+1,j}^n+{\phi }_{i,j}^n}{2}{S}_T$$

(3.12)

Similarly, in three dimension the subscript $i$ and $k$indicate the spatial variable $x$and $z$.

$$\frac{{\phi }_{i,k}^{n+1}-{\phi }_{i,k}^n}{\Delta t}={\rm M}\rho \left(\frac{1}{2}\left(\frac{\left({\phi }_{i+1,k}^n-2{\phi }_{i,k}^n+{\phi }_{i-1,k}^n\right)}{{\left(\Delta x\right)}^2}+\frac{{\phi }_{i+1,k+1}^n-2{\phi }_{i,k+1}^n+{\phi }_{i-1,k+1}^n}{{\left(\Delta x\right)}^2}\right)+\frac{{\phi }_{i,k+1}^n-2{\phi }_{i,k}^n+{\phi }_{i,k-1}^n}{{\left(\Delta y\right)}^2}\right)-v\left(\frac{{\phi }_{i+1,k}^n-{\phi }_{i-1,k}^n}{\left(2\Delta x\right)}+\frac{{\phi }_{i,k+1}^n-{\phi }_{i,k-1}^n}{\left(2\Delta y\right)}\right)+\frac{{\phi }_{i+1,k}+{\phi }_{i,k}^n}{2}{S}_T$$

(3.13)

Taking constant values of $v$and $S_T$ so that $v=S_T=1$and multiplying by 2$\Delta t$ with square mesh for variables $\Delta x=\Delta y$ yields

$${\phi }_{i,j}^{n+1}-{\phi }_{i,j}^n=\frac{M\rho \Delta t}{{\left(\Delta x\right)}^2}\left({\phi }_{i+1,j}^n-2{\phi }_{i,j}^n+{\phi }_{i-1,j}^n+{\phi }_{i+1,j+1}^n-2{\phi }_{i,j+1}^n+{\phi }_{i-1,j+1}^n+2{\phi }_{i,j+1}^n-4{\phi }_{i,j}^n+2{\phi }_{i,j-1}^n\right)-\frac{\Delta t}{\Delta x}\left({\phi }_{i+1,j}^n-{\phi }_{i-1,j}^n+{\phi }_{i,j+1}^n-{\phi }_{i,j-1}^n\right)+\Delta t\left({\phi }_{i+1,j}^n+{\phi }_{i,j}^n\right)$$

(3.14)

Taking the mesh sizes $r=\frac{\Delta t}{{\left(\Delta x\right)}^2},\tau =\frac{\Delta t}{\Delta x},\xi =\Delta t$

$${\phi }_{i,j}^{n+1}-{\phi }_{i,j}^n=M\rho r\left({\phi }_{i+1,j}^n-2{\phi }_{i,j}^n+{\phi }_{i-1,j}^n+{\phi }_{i+1,j+1}^n-2{\phi }_{i,j+1}^n+{\phi }_{i-1,j+1}^n+2{\phi }_{i,j+1}^n-4{\phi }_{i,j}^n+2{\phi }_{i-1,j-1}^n\right)-\tau \left({\phi }_{i+1,j}^n-{\phi }_{i-1,j}^n+{\phi }_{i,j+1}^n-{\phi }_{i,j-1}^n\right)+\xi \left({\phi }_{i+1,j}^n+{\phi }_{i,j}^n\right)$$

(3.15)

But if $\Delta t=0.01,\Delta x=0.1$,$r=\frac{0.01}{{\left(0.1\right)}^2}=1,\tau =\frac{0.01}{\left(0.1\right)}=0.1,\xi =\frac{0.01}{1}=0.01$,

Equation (3.15) becomes

$${\phi }_{i,j}^{n+1}-{\phi }_{i,j}^n=M\rho \left({\phi }_{i+1,j}^n-2{\phi }_{i,j}^n+{\phi }_{i-1,j}^n+{\phi }_{i+1,j+1}^n-2{\phi }_{i,j+1}^n+{\phi }_{i-1,j+1}^n+2{\phi }_{i,j+1}^n-4{\phi }_{i,j}^n+2{\phi }_{i,j-1}^n\right)-0.1\left({\phi }_{i+1,j}^n-{\phi }_{i-1,j}^n+{\phi }_{i,j+1}^n-{\phi }_{i,j-1}^n\right)+0.01\left({\phi }_{i+1,j}^n+{\phi }_{i,j}^n\right)$$

(3.16)

multiplying both sides of equation (3.16) by 100 gives

$$100{\phi }_{i,j}^{n+1}-100{\phi }_{i,j}^n=100M\rho \left({\phi }_{i+1,j}^n-2{\phi }_{i,j}^n+{\phi }_{i-1,j}^n+{\phi }_{i+1,j+1}^n-2{\phi }_{i,j+1}^n-{\phi }_{i-1,j+1}^n+2{\phi }_{i,j+1}^n-4{\phi }_{i,j}^n+2{\phi }_{i,j-1}^n\right)+100{\phi }_{i+1,j}^n-100{\phi }_{i-1,j}^n+100{\phi }_{i,j+1}^n-100{\phi }_{i,j-1}^n+{\phi }_{i+1,j}^n+{\phi }_{i,j}^n$$

(3.17)

Putting the $j+1$level on LHS of equation (3.17) yields

$$-20M\rho {\phi }^n{}_{i+1,j+1}+\left(1-50M\rho \right){\phi }^n{}_{i,j+1}-20M\rho {\phi }^n{}_{i-1,j+1}=\left(10.01-40M\rho \right){\phi }^n{}_{i,j}+\left(10M\rho -0.99\right){\phi }^n{}_{i+1,j}+\left(10M\rho +1\right){\phi }^n{}_{i-1,j}+\left(10M\rho +\right){\phi }_{i,j-1}^n-10{\phi }_{i,j}^{n+1}$$

(3.18)

Taking $i=1,2,3,4,5$, …, $n=0$, $j=1$ we get the algebraic equations

$$\begin{gathered} -20M\rho {\phi }^o{}_{2,2}+\left(1-50M\rho \right){\phi }^0{}_{1,2}-20M\rho {\phi }^0{}_{0,2}=\left(10M\rho -0.99\right){\phi }^0{}_{2,1}+\left(10.01-40M\rho \right){\phi }^0{}_{1,1}+\left(10M\rho +1\right){\phi }^0{}_{0,1}+\left(10M\rho +1\right){\phi }^0{}_{1,0}0-10{\phi }^1{}_{1,1} \\ -20M\rho {\phi }^o{}_{32}+\left(1-50M\rho \right){\phi }^0{}_{22}-20M\rho {\phi }^0{}_{1,2}=\left(10M\rho -0.99\right){\phi }^0{}_{3,1}+\left(10.01-40M\rho \right){\phi }^0{}_{2,1}+\left(10M\rho +1\right){\phi }^0{}_{1,1}+\left(10M\rho +1\right){\phi }^0{}_{2,0}0-10{\phi }^1{}_{2,1} \\ -20M\rho {\phi }^o{}_{4,2}+\left(1-50M\rho \right){\phi }^0{}_{3,2}-20M\rho {\phi }^0{}_{2,2}=\left(10M\rho -0.99\right){\phi }^0{}_{4,1}+\left(10.01-40M\rho \right){\phi }^0{}_{3,1}+\left(10M\rho +1\right){\phi }^0{}_{2,1}+\left(10M\rho +1\right){\phi }^0{}_{3,0}0-10{\phi }^1{}_{3,1} \\ -20M\rho {\phi }^o{}_{5,2}+\left(1-50M\rho \right){\phi }^0{}_{4,2}-20M\rho {\phi }^0{}_{3,2}=\left(10M\rho -0.99\right){\phi }^0{}_{5,1}+\left(10.01-40M\rho \right){\phi }^0{}_{4,1}+\left(10M\rho +1\right){\phi }^0{}_{3,1}+\left(10M\rho +1\right){\phi }^0{}_{4,0}0-10{\phi }^1{}_{4,1} \\ -20M\rho {\phi }^o{}_{6,2}+\left(1-50M\rho \right){\phi }^0{}_{5,2}-20M\rho {\phi }^0{}_{4,2}=\left(10M\rho -0.99\right){\phi }^0{}_{6,1}+\left(10.01-40M\rho \right){\phi }^0{}_{5,1}+\left(10M\rho +1\right){\phi }^0{}_{4,1}+\left(10M\rho +1\right){\phi }^0{}_{5,0}0-10{\phi }^1{}_{5,1} \end{gathered}$$

(3.19)

Taking the initial $\phi \left(\mathrm{x},\mathrm{y},0\right)=\phi \left(\mathrm{x},1,0\right)=1$, and boundary conditions, $\phi \left(0,\mathrm{y},\mathrm{t}\right)=\phi \left(\mathrm{x},0,\mathrm{t}\right)=1$, the above system of algebraic equations become

Solving the above matrix equation, the solutions for varying values of $\rho$and M were obtained and provided in table 4.1 and 4.2.

4. Results and Discussions

4.1. Effects of Cell Density on Tumour Growth Rate

Equation (3.27) was solved in MATLAB and to obtain the results of the effects of $\rho$ on tumour growth rate when we hold M constant and get results as shown in table 4.1 below

The above results were presented in figure 4.1 below

Figure 4.1 shows how tumour density impacts tumour growth. Figure 4 shows that tumour growth slows with increasing tumour density and accelerates in the opposite direction as tumours spread from their original location. Tumours formed in high-density cells are suppressed through the application of compressive pressures, leading to smaller tumours than those grown in low-density cells. As a result, an increase in cell density typically results in a smaller tumour.

Figure 4.1. Graph of tumour growth rate against length of tumour spread from source at varying tumour density.

Figure 4.1. Graph of tumour growth rate against length of tumour spread from source at varying tumour density.

5. Summary and Conclusions

6. Recommendations

Nomanclature

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