Modeling the effects of resection, radiation and chemotherapy in glioblastoma

Abstract

The standard treatment for newly diagnosed glioblastoma multiforme is surgical resection followed by radiotherapy and chemotherapy. Most studies on these treatments are retrospective clinical data analysis. To integrate these studies, a mathematical model is developed. The model predicts the survival time of patients who undergo resection, radiation, and chemotherapy with different protocols.

Appendix

Consider a radially symmetrical tumor and denote by r the distance from a point to the origin. We denote the boundary of the tumor by r = R(t). Set

$$ \begin{aligned} x=&\hbox{ number density of tumor stem cells, } \\ y=&\hbox{ number density of dead cells. }\\ \end{aligned} $$

The proliferation and removal of cells cause a movement of cells within the tumor, with a convection term, for tumor cells x, in the form $\frac{1}{r}\frac{\partial}{\partial r}[r^{2}u(r,t)x(r,t)]$, where u(r,t) is the radial velocity; u(R _*,t) = 0 since the tumor does not grow inward. By mass conservation law,

$$ \frac{\partial x(r,t)}{\partial t}+\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}u(r,t)x(r,t))=\lambda x(r,t)-\delta x(r,t), $$

(1)

Similarly,

$$ \frac{\partial y(r,t)}{\partial t}+\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}u(r,t)y(r,t))=\delta x(r,t) -\mu y(r,t). $$

(2)

We assume that the total density of tumor and necrotic cells is constant through the tumor, that is, x(r,t) + y(r,t) = const = θ, and θ = 10⁶/mm³ [5]. By adding Eqs. (1) and (2) together, we obtain an equation for the radial velocity:

$$ \frac{\theta}{r^{2}}\left(\frac{\partial}{\partial r}r^{2}u\right)=(\lambda +\mu)x(r,t)-\mu \theta. $$

(3)

The tumor radius evolves according to

$$ \frac{{\hbox{d}}R}{{\hbox{d}}t}=u(R(t),t). $$

(4)

We assume that,

$$ x(r,0)=\frac{9}{10}\theta,\hbox{ for }R_{\ast}\leq r\leq R_{0}, $$

(5)

that is, initially 90% of cells are tumor cells, and 10% are necrotic cells.

We need to solve Eqs. (1), (3) in R _* ≤ r ≤  R(t) with the initial condition (5) and with the tumor growth condition (4).

The above model does not include radiotherapy and chemotherapy. If the standard radiotherapy is administered over a period of 6 weeks during the time period 6 ≤ t ≤ 12 and the temozolomide is given for 40 weeks, Eqs. (1) and (2) are replaced by

$$ \frac{\partial x(r,t)}{\partial t}+\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}u(r,t)x(r,t))=\lambda x(r,t)-\delta x(r,t)-A\rho(t)x(r,t)-B\tau(t)x(r,t), $$

(6)

$$ \frac{\partial y(r,t)}{\partial t}+\frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}u(r,t)y(r,t))=\delta x(r,t)+A\rho(t)x(r,t)+B\tau(t)x(r,t) -\mu y(r,t). $$

(7)

By adding the two equations, we obtain the same Eq. (3), as before, for the velocity u(r,t).

Figures (1)–(7) are based on solving Eqs. (6) and (3) in R _* ≤ r ≤ R(t) together with (4)and (5).