Question

11.19. Avascular tumour with phase-two growth. Consider a tumour that has already reached the phase-two growth stage. Assume that the proliferating layer absorbs oxygen at a constant rate A = A$_0$ ml per second per unit volume of cells, and let $c_p(r)$ be the oxygen concentration here. The governing equation is $\frac{D}{r^2}\frac{d}{dr}(r^2\frac{dc_p}{dr}) - A_0 = 0$. $R_q(t) < r < R(t)$ In the proliferating core, with $c_q(r)$ as the concentration, then $\frac{D}{r^2}\frac{d}{dr}(r^2\frac{dc_q}{dr}) = 0$, $0 < r < R_q(t)$. The boundary conditions are $c_p(R_q) = c_q(R_q) = c_q$, $\frac{dc_q}{dr}(0) = 0$. $\frac{dc_q}{dr}(R_q) = \frac{dc_p}{dr}(R_q)$. (a) Give a brief explanation of each of the boundary conditions. Comment on the number of boundary conditions required for solution of the differential equations. (b) Solve for the concentrations $c_q(r)$ and $c_p(r)$. [Hint: It is suggested that you do not use the boundary condition $c_p(R_q) = c_q$ in your solution. This can be set aside to be used after the solution is obtained to find an expression for $R_q$ given $R$].

          11.19. Avascular tumour with phase-two growth. Consider a tumour that has already reached the phase-two growth stage. Assume that the proliferating layer absorbs oxygen at a constant rate A = A$_0$ ml per second per unit volume of cells, and let $c_p(r)$ be the oxygen concentration here. The governing equation is 
$\frac{D}{r^2}\frac{d}{dr}(r^2\frac{dc_p}{dr}) - A_0 = 0$. 
$R_q(t) < r < R(t)$
In the proliferating core, with $c_q(r)$ as the concentration, then
$\frac{D}{r^2}\frac{d}{dr}(r^2\frac{dc_q}{dr}) = 0$,
$0 < r < R_q(t)$.
The boundary conditions are
$c_p(R_q) = c_q(R_q) = c_q$,
$\frac{dc_q}{dr}(0) = 0$.
$\frac{dc_q}{dr}(R_q) = \frac{dc_p}{dr}(R_q)$.
(a) Give a brief explanation of each of the boundary conditions. Comment on the number of boundary conditions required for solution of the differential equations.
(b) Solve for the concentrations $c_q(r)$ and $c_p(r)$.
[Hint: It is suggested that you do not use the boundary condition $c_p(R_q) = c_q$ in your solution. This can be set aside to be used after the solution is obtained to find an expression for $R_q$ given $R$].

11.19. Avascular tumour with phase-two growth. Consider a tumour that has already reached the phase-two growth stage. Assume that the proliferating layer absorbs oxygen at a constant rate A = A0 ml per second per unit volume of cells, and let cp(r) be the oxygen concentration here. The governing equation is
(D)/(r^2)(d)/(dr)(r^2(dcp)/(dr)) - A0 = 0.
Rq(t) < r < R(t)
In the proliferating core, with cq(r) as the concentration, then
(D)/(r^2)(d)/(dr)(r^2(dcq)/(dr)) = 0,
0 < r < Rq(t).
The boundary conditions are
cp(Rq) = cq(Rq) = cq,
(dcq)/(dr)(0) = 0.
(dcq)/(dr)(Rq) = (dcp)/(dr)(Rq).
(a) Give a brief explanation of each of the boundary conditions. Comment on the number of boundary conditions required for solution of the differential equations.
(b) Solve for the concentrations cq(r) and cp(r).
[Hint: It is suggested that you do not use the boundary condition cp(Rq) = cq in your solution. This can be set aside to be used after the solution is obtained to find an expression for Rq given R].

Added by Jack R.

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