Mathematical and Computational Modeling for Tumor Virotherapy with Mediated Immunity

Abstract

We propose a new mathematical modeling framework based on partial differential equations to study tumor virotherapy with mediated immunity. The model incorporates both innate and adaptive immune responses and represents the complex interaction among tumor cells, oncolytic viruses, and immune systems on a domain with a moving boundary. Using carefully designed computational methods, we conduct extensive numerical simulation to the model. The results allow us to examine tumor development under a wide range of settings and provide insight into several important aspects of the virotherapy, including the dependence of the efficacy on a few key parameters and the delay in the adaptive immunity. Our findings also suggest possible ways to improve the virotherapy for tumor treatment.

Appendix A: Numerical Methods

A difficulty in the design of numerical algorithms for the model (2) is the presence of a moving boundary, R(t), which describes the growth of the tumor with time and which has to be determined as part of the solution. To partially overcome this challenge, we choose to map the original domain with the moving boundary into a fixed domain, by introducing a new coordinate:

$$\begin{aligned} r= \frac{\rho }{R(t)} \,. \end{aligned}$$

(7)

Obviously, Eq. (7) transforms the original domain, $0 \le \rho \le R(t)$, into a regular interval, $0 \le r \le 1$. Based on such a mapping, we are able to develop accurate and robust numerical methods.

Equation (1) enables us to drop one variable, N. Using the change of coordinates, the original equations in (2) can be written as

$$\begin{aligned} \frac{\partial X}{\partial t} + \frac{U-rR^\prime }{R} \frac{\partial X}{\partial r}&= F_1 , \end{aligned}$$

(8a)

$$\begin{aligned} \frac{\partial Y}{\partial t} + \frac{U-rR^\prime }{R} \frac{\partial Y}{\partial r}&= F_2 , \end{aligned}$$

(8b)

$$\begin{aligned} \frac{\partial Z_1}{\partial t} + \frac{U-rR^\prime }{R} \frac{\partial Z_1}{\partial r}&= F_{3a} , \end{aligned}$$

(8c)

$$\begin{aligned} \frac{\partial Z_2}{\partial t} + \frac{U-rR^\prime }{R} \frac{\partial Z_2}{\partial r}&= F_{3b} , \end{aligned}$$

(8d)

$$\begin{aligned} \frac{\partial V}{\partial t} - \Big ( \frac{rR^\prime }{R} + \frac{2D}{R^2r} \Big ) \frac{\partial V}{\partial r} - \frac{D}{R^2}\frac{\partial ^2 V}{\partial r^2}&= F_4 , \end{aligned}$$

(8e)

$$\begin{aligned} \frac{1}{Rr^2} \frac{\partial }{\partial \rho } (r^2 U )&= F_5, \end{aligned}$$

(8f)

for $0\leqslant r\leqslant 1$ and $t> 0$, where $R^\prime = \frac{\hbox {d}R}{\hbox {d}t}$. The terms on the right-hand sides are:

$$\begin{aligned} F_1= & {} \lambda X- \beta XV - k_2 X Z_2 - FX , \\ F_2= & {} \beta XV - k_1YZ_1 - \delta Y -F Y , \\ F_{3a}= & {} s_1 Y Z_1- c_1 Z_1 - F Z_1 , \\ F_{3b}= & {} s_2 Y( t) Z_2(t)- c_2 Z_2 -F Z_2 , \\ F_4= & {} b \delta Y - k_0 Z_1 V - \gamma V , \\ F_5= & {} F , \end{aligned}$$

where

$$\begin{aligned} F&= \lambda X + s_1 Y Z_1 + s_2 Y(t) Z_2(t) - c_1 Z_1-c_2 Z_2 -\mu (1-X-Y-Z_1-Z_2) . \end{aligned}$$

The kinematic condition in Eq. (3) and boundary conditions in Eq. (4) become

$$\begin{aligned} \frac{\hbox {d} R}{\hbox {d} t}(t)&=U(1,t) , \end{aligned}$$

(9a)

$$\begin{aligned} \frac{\partial V}{\partial r}(1,t)&=0 , \end{aligned}$$

(9b)

$$\begin{aligned} U(0,t)&=0, \end{aligned}$$

(9c)

$$\begin{aligned} \frac{\partial V}{\partial r}(0,t)&=0 . \end{aligned}$$

(9d)

The initial conditions are given below. In particular, the initial values for R and V are taken from Friedman et al. (2005), based on laboratory experiments conducted on rats with brain tumors. When the virotherapy starts, the oncolytic viruses are injected into the center of the tumor, which is measured at 2 mm at that time. Meanwhile, a Gaussian distribution centered at 0 is used to represent the initial profile of the viruses inside the tumor. Additionally, we note that all the cell densities have been normalized so that the total is $\theta = 1\,\hbox {cell mm}^{-3}$.

$$\begin{aligned} R(0)= & {} 2 \, \mathrm {mm} \,, \end{aligned}$$

(10a)

$$\begin{aligned} V(r,0)= & {} \alpha e^{- \frac{4r^2}{a^2}}, \quad \mathrm {where} \quad \alpha \int _0^1 r^2e^{- \frac{4r^2}{a^2}} \, \mathrm {d}r = 0.45, \end{aligned}$$

(10b)

$$\begin{aligned} X(r,0)= & {} 0.8 \, \mathrm {cell\,mm^{-3}}, \quad Y(r,0)=0.1\, \mathrm {cell\,mm^{-3}} , \end{aligned}$$

(10c)

$$\begin{aligned} Z_1(r,0)= & {} 0.05 \, \mathrm {cell\,mm^{-3}} ,\quad Z_2(r,0)=0.05 \, \mathrm {cell\,mm^{-3}} . \end{aligned}$$

(10d)

The numerical simulation of Eq. (8) is essentially a time marching problem. We denote the numerical solution at the nth time step by

$$\begin{aligned} \left( R^n, X^n, Y^n, Z_1^n, Z_2^n, U^n, V^n\right) . \end{aligned}$$

We use the second-order Adams–Bashforth method to advance R in time. Applying it to Eq. (10a) yields

$$\begin{aligned} R^{n+1} = R^n + \frac{\Delta t}{2}(3U^n-U^{n-1}). \end{aligned}$$

(11)

The hyperbolic type equations, for example Eq. (8a), is solved by a leapfrog scheme:

$$\begin{aligned} \frac{X_j^{n+1}- X_j^{n-1}}{2 \Delta t}+ A_j^n \frac{X_{j+1}^{n}- X_{j-1}^{n}}{2 \Delta r}=(F_1)^n_j, \end{aligned}$$

(12a)

where

$$\begin{aligned} A = \frac{U-rR^\prime }{R} \,, \end{aligned}$$

(12b)

and the subscript j refers to the jth spatial grid point in the radial direction. Meanwhile, we supplement the leapfrog method with a simple average in time: $X_j^n = \frac{1}{2} (X_j^{n+1} + X_j^{n-1} )$, implemented in our code for every 10 steps. This additional time-average procedure is introduced to overcome potential mesh drifting instability (Press et al. 1996) when applying the leapfrog method to nonlinear equations, yet retaining its second-order accuracy (Wang and Tian 2008). In addition, Y, $Z_1$, and $Z_2$ are updated in a similar way.

Once we have $X^{n+1}$, $Y^{n+1}$, $Z_1^{n+1}$ and $Z_2^{n+1}$ calculated, we compute $U^{n+1}$ using the Trapezoidal Rule for Eq. (8f):

$$\begin{aligned} r^2_{j+1} U_{j+1}^{n+1} - r^2_j U_j^{n+1} = \frac{R^{n+1}}{2} \Delta r \Big ( r^2_{j+1} (F_5)^{n+1}_{j+1} + r^2_j (F_5)^{n+1}_j \Big ) \,. \end{aligned}$$

(13)

Finally, we solve the parabolic Eq. (8e). For convenience, we write the equation in the following form:

$$\begin{aligned} \frac{\partial V}{\partial t} + A_1 \frac{\partial V}{\partial r} + A_2 \frac{\partial ^2 V}{\partial r^2}= F_4, \end{aligned}$$

(14a)

with

$$\begin{aligned} A_1= -\Big ( \frac{rR^\prime }{R} + \frac{2D}{R^2r} \Big ) \quad , \quad A_2= -\frac{D}{R^2} \,. \end{aligned}$$

(14b)

Since $F_4$ is a linear function of $V^{n+1}$, the above equation is a linear parabolic equation at time step $n+1$. We use the second-order Backward Difference Formula (BDF) in time and central differences in space to approximate Eq. (14a):

$$\begin{aligned}&\frac{3V_j^{n+1}-4V_j^n+V_j^{n-1}}{2\Delta t}+ (A_1)^{n+1}_j \frac{V_{j+1}^{n+1}-V_{j-1}^{n+1}}{2\Delta r} \nonumber \\&\quad +\,(A_2)^{n+1}_j \frac{V_{j+1}^{n+1}-2V_j^{n+1}+V_{j-1}^{n+1}}{(\Delta r)^2} = (F_4)^{n+1}_j \,. \end{aligned}$$

(15)

The discretized equation can then be written in a tridiagonal algebraic form which can be computed efficiently (Golub and Loan 2012). Once solved, a full cycle of the time marching is completed, and the procedure repeats at the next cycle.

The methods presented here achieve second-order accuracy in both time and space and possess strong numerical stability, which allows us to conduct a careful numerical study on the complex interaction among the tumor, the viruses, and the immune systems involved in the tumor virotherapy.