Effects of CCN1 and Macrophage Content on Glioma Virotherapy: A Mathematical Model

Abstract

Oncolytic virus (OV) is a genetically engineered virus that can selectively replicate in and kill tumor cells while not harming normal cells. OV therapy has been explored as a treatment for numerous cancers including glioblastoma, an aggressive and devastating brain tumor. Experiments show that extracellular matrix protein CCN1 limits OV therapy of glioma by orchestrating an antiviral response and enhancing the proinflammatory activation and migration of macrophages. Neutralizing CCN1 by antibody has been demonstrated to improve OV spread and tends to increase the time to disease progression. In this paper, we develop a mathematical model to investigate the effects of CCN1 on the treatment of glioma with oncolytic herpes simplex virus. We show that numerical simulations of the model are in agreement with the experimental results and then use the model to explore the anti-tumor effects of combining antibodies with OV therapy. Model simulations suggest that the macrophage content of the tumor is a critical factor to the success of OV therapy and to the reduction in tumor volume gained with the CCN1 antibody.

Appendix

1.1 Parameter Estimation for the Dimensionalized Model

1.1.1 Diffusion and Chemotactic Coefficients

We assume that the diffusion coefficients of all tumor and immune cells are equal and take them to be $8.64\times 10^{-5}$ $\hbox {mm}^2\hbox {day}^{-1}$ (Hao and Friedman 2014; Kim et al. 2011). We assume that the diffusion coefficient, $D_\mathrm{p}$, for a protein $p$ can be calculated from its molecular weight, $M_\mathrm{p}$, by the formula $D_\mathrm{p}=KM_\mathrm{p}^{2/3}$ where $K=0.325~\hbox {mm}^2\hbox {day}^{-1}\hbox {dalton}^{-2/3}$, as explained in Hao and Friedman (2014). The molecular weight of CCN1 is 42027 daltons (Jay et al. 1997). Hence, the diffusion coefficient of CCN1 is estimated by

$$\begin{aligned} D_\mathrm{C} = K\times M_{\mathrm{CCN1}}^{2/3} = 0.325 \times (42027)^{2/3} = 392.9 ~\hbox {mm}^2\hbox {day}^{-1}. \end{aligned}$$

The sensitivity of macrophage to chemoattractants is difficult to estimate and values in the literature include the range from $8.64 \times 10^{4}$ to $1.73\times 10^9$ $\hbox {mm}^5\hbox {g}^{-1}\hbox {day}^{-1}$ (Hao and Friedman 2014; Kim and Friedman 2010). We choose an intermediate value of $\chi _\mathrm{C}=\chi _\mathrm{P}=10^6\,\hbox {mm}^5\hbox {g}^{-1}\hbox {day}^{-1}$.

1.1.2 Estimate of $d_\mathrm{C}$, $\lambda _\mathrm{C},\mu _\mathrm{C}$ from Eq. (7)

We assume that the half-life of extracellular CCN1 is 30 h (Yang and Lau 1991) and accordingly take

$$\begin{aligned} d_\mathrm{c}=\frac{\ln 2}{30} \,h^{-1}=0.55\,\hbox {day}^{-1}. \end{aligned}$$

CCN1 is highly overexpressed in glioma so we consider a typical concentration of CCN1 in tumor to be $C_0=10^{-9}\hbox {g}\,\hbox {mm}^{-3}$ (Zhang et al. 2012). The steady-state of Eq. (7) is

$$\begin{aligned} \sigma _\mathrm{c}+\lambda _\mathrm{c}y-d_cC=0. \end{aligned}$$

(18)

We assume that with no virus present, CCN1 reaches the steady-state concentration $C_0$; therefore, we take

$$\begin{aligned} \sigma _\mathrm{c}=d_cC_0=5.5\times 10^{-10}\,\hbox {g}\,\hbox {mm}^{-3}\,\hbox {day}^{-1}. \end{aligned}$$

Furthermore, we take the density of tumor cells and macrophages (Eq. (8) to be $\theta = 10^6\,\hbox {cells}/\hbox {mm}^3$ (Friedman et al. 2006) and assume that approximately one-tenth of the tumor cells are infected by the virus (Fulci et al. 2006). Hence, we take a typical concentration of infected tumor cells to be $y_0 = 10^5\, \hbox {cells}/\hbox {mm}^3$. Experimental evidence shows that in the presence of oncolytic virus, CCN1 gene expression is approximately three times higher than without virus (Haseley et al. 2012; Fig. 1a). Accordingly, we let $C_1=3C_0$ and from Eq. (18), we deduce that

$$\begin{aligned} \lambda _\mathrm{c}=\dfrac{d_\mathrm{c}C_1-\sigma _\mathrm{c}}{y_0}=1.1\times 10^{-14}\,\hbox {g}\,\hbox {day}^{-1}\,\hbox {cell}^{-1}. \end{aligned}$$

1.1.3 Estimate of $b,d_\mathrm{y},k_\mathrm{y}, d_\mathrm{v},\delta _\mathrm{y},\delta _\mathrm{v}$ from Eqs. (2) and (5)

The lytic cycle of HSV-1 is approximately 12–16 h (Kurozumi et al. 2008). Accordingly, we take $d_\mathrm{y}$, the lysis rate of infected tumor cells, to be $1.5\,\hbox {day}^{-1}$. CCN1 has been shown to induce a cellular antiviral response that reduces viral replication; experimental data measuring luciferase-encoded virus indicates that the quantity of infectious particles is significantly reduced in the presence of CCN1 (Haseley et al. 2012; Figs. 2, 3). Therefore, we estimate that $k_\mathrm{y}=0.1/C_0=10^8\,\hbox {mm}^3\hbox {g}^{-1}$. For oncolytic HSV-1, the burst size ranges from 10 to 100 (Friedman et al. 2006). We choose the burst size, $b$, to be $50\, \hbox {viruses}/\hbox {cell}$. Friedman et al. (2006) estimated the clearance rate of free HSV-1 to be $0.6\,\hbox {day}^{-1}$ in a model that studied the affects of cyclophosphamide on virotherapy of glioma. We take the clearance rate of virus, $\delta _\mathrm{v}$, to be $0.5\,\hbox {day}^{-1}$.

Friedman et al. (2006) did not consider particular immune cell populations and estimated the average immune killing rate of infected cells to be $4.8\times 10^{-7}\,\hbox {mm}^3\,\hbox {cell}^{-1}\,\hbox {day}^{-1}$. We take the killing rate of infected cells by macrophage, $\delta _\mathrm{y}$, to be $4.8\times 10^{-8}\,\hbox {mm}^3\,\hbox {cell}^{-1}\,\hbox {day}^{-1}$. We make the assumption that the rate of macrophage-mediated killing is proportional to the surface area of the target. We assume the tumor cells to be spherical with a diameter of $40\mu m$ (Tönjes et al. 2013). HSV-1 is an icosahedral virus with a diameter of approximately $200\,nm$ (Rochat et al. 2011). Therefore, the ratio of the surface area of a tumor cell to surface area of a virus is approximately $4\times 10^4$, and we accordingly take

$$\begin{aligned} \delta _\mathrm{v} =\dfrac{\delta _\mathrm{y}}{4\times 10^4} = 1.2\times 10^{-12}\,\hbox {mm}^3\,\hbox {cell}^{-1}\,\hbox {day}^{-1}. \end{aligned}$$

1.1.4 Estimate of $\alpha _\mathrm{C},k_\mathrm{C}$ from Eqs. (2) and (5) and $m$

As explained in Sect. 3, we take the default value of the macrophage content, $m$, to be 0.10 and, according to Eq. (15), take the density of macrophage in a typical tumor to be $M_0= 10^5\,\hbox {cells}\,\hbox {mm}^{-3}$. As supported by experimental evidence (Thorne et al. 2014, Figure 3b), we assume the macrophage density under OV treatment, $M_1$, is 1.5 times higher than in a typical tumor. Cytotoxicity experiments by Haseley et al. (unpublished) suggest that macrophage-mediated killing resulted in 1.2 times more cell death in glioma cells overexpressing CCN1 compared to control glioma cells. Therefore, considering the term for macrophage killing of infected cells in Eq. (2), we estimate that

$$\begin{aligned} M_1\delta _\mathrm{y}y_1\left( 1+\dfrac{\alpha _\mathrm{C}C_1}{k_\mathrm{C}+C_1}\right) = 1.2M_0\delta _\mathrm{y}y_0\left( 1+\dfrac{\alpha _\mathrm{C}C_0}{k_\mathrm{C}+C_0}\right) \!. \end{aligned}$$

Here we assume that the number of infected cells is inversely proportional to the number of macrophages, so that $y_1=\frac{y_0}{1.5}$ since $M_1=1.5M_0$. We take $k_\mathrm{C}=2C_0$ and thus solve for $\alpha _\mathrm{C}$ to get $\alpha _\mathrm{C}=1$.

1.1.5 Estimate of $\lambda _\mathrm{x}$, $k_\mathrm{x}$, and $\beta _\mathrm{x}$ from Eq. (1)

Since high expression of CCN1 correlates with poor prognosis, we assume that the inhibition of tumor growth due to the IFN response orchestrated by CCN1 is not as strong as the inhibition of viral replication in Eq. (5). Therefore, we estimate $k_\mathrm{x}=0.01/C_0=10^7\,\hbox {mm}^3\hbox {g}^{-1}$, so that $k_\mathrm{x}=0.1k_\mathrm{y}$. We take the proliferation rate of uninfected glioma cells to be $\lambda _\mathrm{x}=0.2\,\hbox {day}^{-1}$ based on data measuring the growth of subcutaneous glioma tumors in a control group of mice treated with phosphate-buffered saline (Yoo et al. 2012). We choose the infection rate of cells by virus to be $\beta _\mathrm{x}=1.7\times 10^{-8}\,\hbox {mm}^3\hbox {day}^{-1}\hbox {virus}^{-1}$ as estimated by Friedman et al. (2006).

1.1.6 Estimate of $\lambda _\mathrm{M}, \alpha _\mathrm{MC}, k_\mathrm{MC}$ from Eq. (4)

Experimental results by Thorne et al. Thorne et al. (2014) indicate that OV-induced CCN1 significantly increases macrophage migration and enhances the proinflammatory activation of macrophages. Accordingly, we take $\alpha _\mathrm{MC}=2$. We also take $k_\mathrm{MC}=C_0$. The death rate of macrophages, $d_\mathrm{M}$, is taken to be $0.015\,\hbox {day}^{-1}$ (Hao and Friedman 2014; Friedman et al. 2008).

To estimate the constant source of macrophages, $\lambda _\mathrm{M}$, we consider the dynamics of the macrophage density at a spatially homogeneous steady-state without virus. In that case, there is a nonnegative advection velocity, $u$, and Eq. (1) implies that

$$\begin{aligned} \dfrac{1}{r^2}\dfrac{\partial {}}{\partial {r}}(r^2u) = \dfrac{\lambda _\mathrm{x}}{1+k_\mathrm{x}C}. \end{aligned}$$

Substituting this expression into the homogeneous steady state of Eq. (4) gives

$$\begin{aligned} M\left( \dfrac{\lambda _\mathrm{x}}{1+k_\mathrm{x}C}\right) = \lambda _\mathrm{M}\left( 1+\dfrac{\alpha _\mathrm{MC}C}{k_\mathrm{MC}+C}\right) -d_\mathrm{M}M, \end{aligned}$$

so that

$$\begin{aligned} \lambda _\mathrm{M}=\dfrac{\left( \dfrac{\lambda _\mathrm{x}}{1+k_\mathrm{x}C_0}+d_\mathrm{M}\right) M_0}{1+\dfrac{\alpha _\mathrm{MC}C_0}{k_\mathrm{MC}+C_0}}=1.07\times 10^4\,\hbox {cells}\,\hbox {mm}^{-3}\,\hbox {day}^{-1}. \end{aligned}$$

(19)

1.1.7 Estimate of $\lambda _\mathrm{P}, \alpha _\mathrm{M}, k_\mathrm{M}$ from Eq. (6)

We assume that in normal healthy tissue, the concentration of MCP-1 is $P_0=3\times 10^{-13}\hbox {g}\,\hbox {mm}^{-3}$ (Rhodes et al. 2009) and that the degradation rate of MCP-1 is $d_\mathrm{P}=1.7$ $\hbox {day}^{-1}$ (Chen et al. 2012). According to Eq. (6), the steady-state concentration of MCP-1 satisfies the equation

$$\begin{aligned} \sigma _\mathrm{P}+\lambda _\mathrm{P}y\left( 1+\dfrac{\alpha _\mathrm{M}M}{k_\mathrm{M}+M}\right) -d_\mathrm{P}P=0 \end{aligned}$$

(20)

and, therefore, with $y=0$ and $P=P_0$,

$$\begin{aligned} \sigma _\mathrm{P}=d_\mathrm{P}P_0=5.1\times 10^{-13}\,\hbox {g}\,\hbox {mm}^{-3}\,\hbox {day}^{-1}. \end{aligned}$$

Experimental data show that, compared to uninfected glioma cells, MCP-1 expression is approximately 2.5 times larger in infected glioma cells without macrophages present and about 13 times higher in infected glioma cells cultured with macrophages (Thorne et al. 2014, Figure 5a). Accordingly, we let $P_1=2.5P_0$ and $P_2=13P_0$. By Eq. (20), we can estimate $\lambda _\mathrm{P}$, the rate of MCP-1 production by infected glioma cells, by

$$\begin{aligned} \lambda _\mathrm{P} = \dfrac{d_\mathrm{P}P_1-\sigma _\mathrm{P}}{y_0} =7.65\times 10^{-18}\,\hbox {g}\,\hbox {cell}^{-1}\,\hbox {day}^{-1} \end{aligned}$$

and $\alpha _\mathrm{M}$, corresponding to further production of MCP-1 due to macrophage signaling, by

$$\begin{aligned} \alpha _\mathrm{M} = \dfrac{k_\mathrm{M}+M_1}{M_1}\left( \dfrac{d_\mathrm{P}P_2-\sigma _\mathrm{P}}{\lambda _\mathrm{P}y_0}-1\right) = 11.7 \end{aligned}$$

where we assumed that

$$\begin{aligned} k_\mathrm{M} = M_0. \end{aligned}$$

(21)

1.1.8 Estimate of $\tilde{\alpha },k_\mathrm{P},\tilde{M}$ from Boundary Condition Eq. (14)

The density of macrophages in the blood ranges from $2\times 10^4$ to $10^5$ $\hbox {cells}/\hbox {mL}$ (Tietz 1995). Accordingly, we choose $\tilde{M} = 50\,\hbox {cells}\,\hbox {mm}^{-3}$. We also take $\tilde{\alpha } = 1$ and $k_\mathrm{P}=P_0$.

1.2 Nondimensionalized Model with Fixed Boundary

We nondimensionalize the cell and virus densities by $\overline{x} = x/\theta$, $\overline{y} = y/\theta$, $\overline{n} = n/\theta$, $\overline{M} = M/\theta$, $\overline{v}=\frac{v}{b\theta }$. The protein concentrations are nondimensionalized by $\overline{P}=P/P_0$ and $\overline{C}=C/C_0$ where $P_0$ and $C_0$ are the reference concentrations given in Table 4. Correspondingly, other parameters are nondimensionalized:

$$\begin{aligned} \overline{\beta _\mathrm{x}}= & {} b\theta \beta _\mathrm{x}, \overline{\beta _\mathrm{v}}=\beta _\mathrm{v}\theta , \overline{\delta _\mathrm{y}}=\delta _\mathrm{y}\theta , \overline{\delta _\mathrm{v}}=\delta _\mathrm{v}\theta ,\\ \overline{\chi _\mathrm{P}}= & {} \chi _\mathrm{P}P_0, \overline{\chi _\mathrm{C}}=\chi _\mathrm{C}C_0, \overline{k_\mathrm{x}} = k_\mathrm{x}C_0, \overline{k_\mathrm{y}} = k_\mathrm{y}C_0,\\ \overline{\lambda _\mathrm{M}}= & {} \dfrac{\lambda _\mathrm{M}}{\theta }, \overline{\alpha _\mathrm{MC}}=\dfrac{\alpha _\mathrm{MC}}{\theta }, \overline{k_\mathrm{M}}=\dfrac{k_\mathrm{M}}{\theta }, \overline{\tilde{M}}=\dfrac{\tilde{M}}{\theta },\\ \overline{\sigma _\mathrm{p}}= & {} \dfrac{\sigma _\mathrm{p}}{P_0}, \overline{\lambda _\mathrm{p}}=\dfrac{\lambda _\mathrm{p}\theta }{P_0}, \overline{k_\mathrm{P}}=\dfrac{k_\mathrm{P}}{P_0},\\ \overline{\sigma _\mathrm{C}}= & {} \dfrac{\sigma _\mathrm{C}}{C_0}, \overline{\lambda _\mathrm{C}}=\dfrac{\lambda _\mathrm{C}\theta }{C_0}, \overline{k_\mathrm{C}}=\dfrac{k_\mathrm{C}}{C_0}, \overline{k_\mathrm{MC}}=\dfrac{k_\mathrm{MC}}{C_0}. \end{aligned}$$

We eliminate the equation for dead cells by Eq. (8) and fix the moving boundary by making the transformation $\overline{r}=r/R(t)$. Dropping the bar notation for simplicity, we obtain the following nondimensionalized model for $r \in (0,1)$ and $t > 0$:

$$\begin{aligned} \dfrac{\partial {x}}{\partial {t}}&=\dfrac{D}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {x}}{\partial {r}}\right) -\dfrac{1}{r^2}\dfrac{\partial }{\partial r}\left( r^2\dfrac{ux}{R}\right) +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {x}}{\partial {r}} + \dfrac{\lambda _x}{1+k_xC} x-\beta _\mathrm{x} x v, \end{aligned}$$

(22)

$$\begin{aligned} \dfrac{\partial {y}}{\partial {t}}&= \dfrac{D}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {y}}{\partial {r}}\right) -\dfrac{1}{r^2}\dfrac{\partial }{\partial r}\left( r^2\dfrac{uy}{R}\right) \nonumber \\&\qquad +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {y}}{\partial {r}} + \beta _\mathrm{x} x v-\delta _\mathrm{y}y M\left( 1+\dfrac{\alpha _\mathrm{C}C}{k_\mathrm{C}+C}\right) -\dfrac{d_y}{1+k_yC} y, \end{aligned}$$

(23)

$$\begin{aligned} \dfrac{\partial {M}}{\partial {t}}&= \dfrac{D}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {M}}{\partial {r}}\right) - \dfrac{1}{r^2}\dfrac{\partial }{\partial r}\left( r^2\left( \dfrac{uM}{R}+\dfrac{\chi _\mathrm{P}M}{R^2}\dfrac{\partial {P}}{\partial {r}}+\dfrac{\chi _\mathrm{C}M}{R^2}\dfrac{\partial {C}}{\partial {r}} \right) \right) \nonumber \\&\qquad +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {M}}{\partial {r}} +\lambda _\mathrm{M}\left( 1+\dfrac{\alpha _\mathrm{MC}C}{k_\mathrm{MC}+C}\right) -d_\mathrm{M} M, \end{aligned}$$

(24)

$$\begin{aligned} \dfrac{\partial {v}}{\partial {t}}&= \dfrac{D_\mathrm{v}}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {v}}{\partial {r}}\right) +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {v}}{\partial {r}} +\dfrac{d_y}{1+k_yC} y\nonumber \\&\qquad -\delta _\mathrm{v}vM\left( 1+\dfrac{\alpha _\mathrm{C}C}{k_\mathrm{C}+C}\right) -\beta _\mathrm{v} xv-d_\mathrm{v} v, \end{aligned}$$

(25)

$$\begin{aligned} \dfrac{\partial {P}}{\partial {t}}&=\dfrac{D_\mathrm{P}}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {P}}{\partial {r}}\right) +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {P}}{\partial {r}} +\sigma _\mathrm{P}+\lambda _\mathrm{P}y\left( 1+\dfrac{\alpha _\mathrm{M}M}{k_\mathrm{M}+M}\right) -d_\mathrm{P} P, \end{aligned}$$

(26)

$$\begin{aligned} \dfrac{\partial {C}}{\partial {t}}&= \dfrac{D_\mathrm{C}}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{r^2}{R^2}\dfrac{\partial {C}}{\partial {r}}\right) +\dfrac{r}{R}\dfrac{dR}{dt}\dfrac{\partial {C}}{\partial {r}} +\sigma _\mathrm{C}+\lambda _\mathrm{C}y-d_\mathrm{C} C, \end{aligned}$$

(27)

$$\begin{aligned} u&= \dfrac{R}{r^2}\int _0^r s^2\left( \dfrac{\lambda _x}{1+k_xC} x-d_\mathrm{n}(1-x-y-M)\right. \nonumber \\&\qquad \quad \left. +\, \lambda _\mathrm{M}\left( 1+\dfrac{\alpha _\mathrm{MC}C}{k_\mathrm{MC}+C}\right) -d_\mathrm{M}M\right) \,ds\nonumber \\&\qquad -\dfrac{\chi _\mathrm{P}}{R}M\dfrac{\partial {P}}{\partial {r}}-\dfrac{\chi _\mathrm{C}}{R}M\dfrac{\partial {C}}{\partial {r}}, \end{aligned}$$

(28)

$$\begin{aligned} \dfrac{dR}{dt}&= u(1,t). \end{aligned}$$

(29)

With this transformation, the moving boundary condition (Eq. (10)) becomes Eq. (29). The boundary conditions at the tumor center remain unchanged so that

$$\begin{aligned} \dfrac{\partial {x}}{\partial {r}} = \dfrac{\partial {y}}{\partial {r}} = \dfrac{\partial {n}}{\partial {r}} = \dfrac{\partial {M}}{\partial {r}} = \dfrac{\partial {v}}{\partial {r}} = \dfrac{\partial {P}}{\partial {r}} = \dfrac{\partial {C}}{\partial {r}} = 0 \qquad \text {at } r=0, \end{aligned}$$

(30)

while the boundary conditions at the boundary of the tumor, Eqs. (13), (14), become

$$\begin{aligned} \dfrac{\partial {x}}{\partial {r}}=\dfrac{\partial {y}}{\partial {r}}=\dfrac{\partial {n}}{\partial {r}}=\dfrac{\partial {v}}{\partial {r}}=\dfrac{\partial {P}}{\partial {r}} =\dfrac{\partial {C}}{\partial {r}}&=0 \qquad \text {at } r=1, \end{aligned}$$

(31)

$$\begin{aligned} \dfrac{1}{R}\dfrac{\partial {M}}{\partial {r}}+\tilde{\alpha }\dfrac{P}{k_\mathrm{P}+P}(M-\tilde{M})&= 0 \qquad \text {at } r=1. \end{aligned}$$

(32)

1.3 Numerical Scheme

We rewrite the nondimensionalized model (22)–(29) as

$$\begin{aligned} \dfrac{\partial {z}}{\partial {t}} + A_z\dfrac{\partial {z}}{\partial {r}}- \dfrac{D_z}{R^2}\dfrac{\partial {^2z}}{\partial {r^2}}&= F_z, \qquad \qquad z = x,y,M,v,P,C \end{aligned}$$

(33)

$$\begin{aligned} \dfrac{1}{Rr^2}\dfrac{\partial {}}{\partial {r}}(r^2u)&= F, \end{aligned}$$

(34)

$$\begin{aligned} \dfrac{dR}{dt}&= u(1,t) \end{aligned}$$

(35)

where $D_z = D$ for $z = x,y,M$. The advection coefficients are given by

$$\begin{aligned} A_z&= \dfrac{u}{R}-\dfrac{r}{R}\dfrac{dR}{dt}-\dfrac{2D_z}{R^2r},&\qquad \quad z = x,y,M \end{aligned}$$

(36)

$$\begin{aligned} A_z&= -\dfrac{r}{R}\dfrac{dR}{dt} -\dfrac{2D_z}{R^2r},&\qquad \quad z = v,P,C \end{aligned}$$

(37)

and the reaction terms are given by

$$\begin{aligned} F&= \dfrac{\lambda _x}{1+k_xC} x-d_\mathrm{n}(1-x-y-M)+\lambda _\mathrm{M}\left( 1+\dfrac{\alpha _\mathrm{MC}C}{k_\mathrm{MC}+C}\right) -d_\mathrm{M}M \end{aligned}$$

(38)

$$\begin{aligned}&\qquad \qquad -\dfrac{1}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{\chi _\mathrm{P}}{R^2}r^2M\dfrac{\partial {P}}{\partial {r}}\right) -\dfrac{1}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{\chi _\mathrm{C}}{R^2}r^2M\dfrac{\partial {C}}{\partial {r}}\right) , \nonumber \\ F_\mathrm{x}&= \dfrac{\lambda _x}{1+k_xC} x-\beta _\mathrm{x} x v-xF, \end{aligned}$$

(39)

$$\begin{aligned} F_\mathrm{y}&= \beta _\mathrm{x} x v-\delta _\mathrm{y}y M\left( 1+\dfrac{\alpha _\mathrm{C}C}{k_\mathrm{C}+C}\right) -\dfrac{d_y}{1+k_yC} y-yF,\end{aligned}$$

(40)

$$\begin{aligned} F_\mathrm{M}&=\lambda _\mathrm{M}\left( 1+\dfrac{\alpha _\mathrm{MC}C}{k_\mathrm{MC}+C}\right) -d_\mathrm{M} M-\dfrac{1}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{\chi _\mathrm{P}}{R^2}r^2M\dfrac{\partial {P}}{\partial {r}}\right) \nonumber \\&\qquad -\dfrac{1}{r^2}\dfrac{\partial {}}{\partial {r}}\left( \dfrac{\chi _\mathrm{C}}{R^2}r^2M\dfrac{\partial {C}}{\partial {r}}\right) -MF,\end{aligned}$$

(41)

$$\begin{aligned} F_\mathrm{v}&= \dfrac{d_y}{1+k_yC} y-\delta _\mathrm{v}vM\left( 1+\dfrac{\alpha _\mathrm{C}C}{k_\mathrm{C}+C}\right) -\beta _\mathrm{v} xv-d_\mathrm{v} v,\end{aligned}$$

(42)

$$\begin{aligned} F_\mathrm{P}&= \sigma _\mathrm{P}+\lambda _\mathrm{P}y\left( 1+\dfrac{\alpha _\mathrm{M}M}{k_\mathrm{M}+M}\right) -d_\mathrm{P} P,\end{aligned}$$

(43)

$$\begin{aligned} F_\mathrm{C}&= \sigma _\mathrm{C}+\lambda _\mathrm{C}y-d_\mathrm{C} C. \end{aligned}$$

(44)

We formulate a finite difference upwind scheme to solve (30)–(35). Determining an upper bound for $A_z$ in order to calculate the CFL condition (Gustafsson et al. 2013) is not analytically feasible. Thus, we use an adaptive scheme to ensure stability of the numerical method.

The mesh is defined as follows. Let

$$\begin{aligned} r_i = (i-1)\Delta r \end{aligned}$$

for $i=1,\ldots ,N$ where $\varDelta r = \frac{1}{N-1}$; thus $r_1=0$ and $r_\mathrm{n}=1$. We take $N=50$. Let $t_1=0$ and

$$\begin{aligned} t_{n+1} = t_{n}+\varDelta t_{n}, \end{aligned}$$

for $n = 1,2,\ldots$ where $\varDelta t_{n}$ is chosen according to Eq. (47) below. Let $z^n_i$ denote the numerical solution approximating $z(r_i,t_\mathrm{n})$ for $z = x,y,M,v,P,C,u$ and let $R^n$ denote $R(t_\mathrm{n})$.

According to (16) and (17), the initial conditions are $x_i^1 = 1-\tilde{M}$, $y_i^1=0$, $M_i^1 = \tilde{M}$, $v_i^1 = a\exp (r_i^2/(2\sigma ^2))$, $P_i^1 = C_i^1 = 1$ for $i=1,\ldots ,N$ and $R^1 = 3$.

Given $R^{n}$ and $z^{n}$ for $z = x,y,M,v,P,C$ we calculate $R^{n+1}$ and $z^{n+1}$ according to the following scheme:

• 1. Compute $F^{n}, F^{n}_z$ for $z = x,y,M,v,P,C$ according to Eqs. (38)–(44) where the chemotaxis terms in $F$ and $F_\mathrm{M}$ are approximated by the form:

  $$\begin{aligned}&\dfrac{\chi _\mathrm{P}}{(R^n)^2}\dfrac{1}{r_i^2\varDelta r^2}\left( r_{i+\frac{1}{2}}^2\left( \dfrac{M^n_{i+1}+M^n_{i}}{2}\right) (P^n_{i+1}-P^n_{i})\right. \nonumber \\&\quad \left. -r_{i-\frac{1}{2}}^2\left( \dfrac{M^n_{i}+M^n_{i-1}}{2}\right) (P^n_{i}-P^n_{i-1})\right) \end{aligned}$$

  (45)

  for $i=2,...,N-1$ where $r_{i+\frac{1}{2}} = r_i +\frac{\varDelta r}{2}$.

• 2. Compute the advection velocity $u^{n}$, according to Eq. (34), with the trapezoidal rule:

  $$\begin{aligned} u_i^n = \dfrac{1}{r_i^2}\left( r_{i-1}^2u^n_{i-1}+\dfrac{R^n\varDelta r}{2}(r_i^2F^n_i+r^2_{i-1}F^n_{i-1})\right) \end{aligned}$$

  (46)

  for $i=2,\ldots ,N$ where $u^n_1 = 0$.

• 3. Compute $A_z^{n}$ for $z = x,y,M,v,P,C$ according to Eqs. (36) and (37).

• 4. Compute $\Delta t_{n}$ as follows:

  $$\begin{aligned} \Delta t_\mathrm{n} = \dfrac{0.1\Delta r}{\max (|A_\mathrm{x}^n|,|A_\mathrm{v}^n|,|A_\mathrm{P}^n|,|A_\mathrm{C}^n|)} \end{aligned}$$

  (47)

  in order to satisfy the CFL condition.

• 5. Compute $z_i^{n+1}$ for $z = x,y,M,v,P,C$ and $i=2,\ldots ,N-1$ using the upwind scheme:

  $$\begin{aligned} z_i^{n+1}= & {} z_i^n + \Delta t_\mathrm{n}\left[ \dfrac{D_z}{(R^n)^2}\dfrac{1}{\varDelta r^2}\left( z^n_{i+1}-2z^n_i+z^n_{i-1}\right) \right. \nonumber \\&\quad \left. -\left( [A_z^+]_i^n[z_r^-]_i^n+[A_z^-]_i^n[z_r^+]_i^n\right) + [F_z]_i^n \right] \end{aligned}$$

  (48)

  where $A_z^+=\max (A_z,0)$, $A_z^-=\min (A_z,0)$, $[z_r^-]_i^n = \dfrac{z_i^n-z_{i-1}^n}{\Delta r}$, and $[z_r^+]_i^n = \dfrac{z_{i+1}^n-z_{i}^n}{\Delta r}$. Use the boundary conditions (30)–(32) to calculate $z_i^n$ for $i=1,N$.

• 6. Compute $R^{n+1}$, according to Eq. (29), by

  $$\begin{aligned} R^{n+1} = R^{n} + \Delta t_{n} u_\mathrm{n}^{n}. \end{aligned}$$

  (49)