A mathematical model of angiogenesis and tumor growth: analysis and application in anti-angiogenesis therapy

Abstract

The purpose of this paper is to develop a new coupled mathematical model of angiogenesis (new blood vessel growth) and tumor growth to study cancer development and anti-angiogenesis therapy. The angiogenesis part assumes the capillary to be a viscoelastic continuum whose stress depends on cell proliferation or death, and the tumor part is a Darcy’s law model regarding the tumor mass as an incompressible fluid where the nutrient-dependent growth elicits volume change. For the coupled model, we provide both an inviscid analysis and a parameter sensitivity analysis of the angiogenesis model in response to a stationary hypoxic tumor, and a steady state analysis of the tumor growth in response to a fixed and long blood capillary. The analysis shows that the stable steady state tumor with an invading blood capillary exists if and only if the nutrient release rate divided by the decay rate is less than the tumor viable limit, and the full tumor encloses one part of the capillary in this steady state. Afterwards, we use the coupled model to simulate vascularized tumor growth and anti-angiogenesis therapy. The simulations show that the tumor tends to maximize the nutrient transfer by blood vessel co-option and the anti-angiogenesis treatment by using growth factor neutralizing antibodies would regress the neovasculature and shrink the tumor size. However, the shrunken tumor mass could survive by feeding on mature blood vessels that resist the treatment. This implies the limited efficacy of the anti-angiogenesis monotherapy and its effect on vessel normalization.

Appendices

Appendices

1.1 Appendix A. Proof of Theorem 1

Proof

It suffices to prove for $\beta =1$. First, introduce $w(X,t)=X+u(X,t)$. Second, perform the odd extension of w from $[0,L_V]$ to $[-L_V,L_V]$, that is, letting

$$\begin{aligned} {\tilde{w}}(X,t) = \left\{ \begin{array}{lc} w(X,t) &{}\quad X\in [0,L_V],\\ -w(-X,t) &{}\quad X\in [-L_V,0]. \end{array} \right. \end{aligned}$$

(64)

Similarly $\psi$ is extended to ${\tilde{\psi }}$. Third, define $\phi ={\tilde{w}}_X=1+\frac{\partial u}{\partial X}$ and denote ${\tilde{\Omega }}=[-L_V,L_V]$. Then the Eq. (28) leads to ${\tilde{w}}_t = \frac{{\tilde{w}}_{XX} }{{\tilde{w}}_{X}}$. Differentiate w.r.t. X and notice that $\phi ={\tilde{w}}_X$. Then we get

$$\begin{aligned} \phi _t = \frac{\phi _{XX} \phi - \phi _{X}^2}{\phi ^2}, \end{aligned}$$

(65)

with the initial condition $\phi (X,0)=1+\frac{\partial {\tilde{\psi }}}{\partial X}$ and the boundary condition $\phi (\pm L_V)=e^{\beta ^*_{ec}t}+g$. Using the maximum principle [e.g., Lemma 2.3 in Lieberman (1996)], we obtain

$$\begin{aligned} \min _{\partial '{\tilde{\Omega }}_T} \phi \le \phi (X,t) \quad \text { for } (X,t)\in {\tilde{\Omega }}_T, \end{aligned}$$

(66)

where ${\tilde{\Omega }}_T=[-L_V, L_V]\times [0,T]$ and $\partial '{\tilde{\Omega }}_T= \partial '{\tilde{\Omega }}_T \backslash \{(x,T): -L_V< X < L_V \}$. It is easy to check $\frac{\partial {\tilde{\psi }}}{\partial X}\ge 0$. Thus, $\phi (X,0)\ge 1$. Furthermore, $\phi (\pm L_V,t)=e^{\beta ^*_{ec}t} +g>0$, therefore, $\phi \ge 0$ in ${\tilde{\Omega }}_T$. The global existence of a unique solution has the same proof as in that of Theorem 1(a) of Zheng and **e (2014). $\square$

1.2 Appendix B. Details of parameter study of angiogenesis model

As the VEGF diffusion rate $D_c$ is doubled, the VEGF concentration is higher (compare Figs. 2b, 13b), which promotes faster extension of the capillary. The capillary reaches $x=1$ around $t=4.2$. As $D_c$ is halved, the VEGF concentration is only slightly greater than $c_0=0.2$ near the root (Fig. 13d). Thus, the EC proliferation activity is mostly suppressed, and the capillary grows very slow. At $t=7$, the tip only extends to $x=0.44$ (Fig. 13c).

Fig. 13

Modulation on VEGF diffusion rate $D_c$. a, b $D_c$ is doubled. From $t=0,1,\ldots$4, u in a goes from bottom to top, and c in b goes from top to bottom. c, d $D_c$ is halved. In c, the curve of u at $t=0,1,\ldots$7 goes from bottom to top

As the VEGF decay rate $\gamma _1$ is doubled, the VEGF concentration is lower (Fig. 14b). The capillary grows very slow and reaches only $x=0.44$ at $t=7$ (Fig. 14a). As $\gamma _1$ is halved, the VEGF concentration is much higher (Fig. 14d). Thus, the EC proliferation activity is highly activated, and the capillary grows very fast. At $t=4.3$, the tip extends to $x=1$ ((Fig. 14c).

Fig. 14

Modulation on VEGF decay rate $\gamma _1$. a, b $\gamma _1$ is doubled, u in a and c in b, $t=0,1,\ldots ,7$. c, d $\gamma _1$ is halved. From $t=0,1,\ldots$5, u in c goes from bottom to top and c in d goes from top to bottom

Fig. 15

Modulation on EC proliferation rate $\beta _{EC}$. a, b $\beta _{EC}$ is doubled. From $t=0,1,\ldots$4, u in a and $\rho$ in b both go from bottom to top. c, d $\beta _{EC}$ is halved. From $t=0,1,\ldots$7, u in c and $\rho$ in d both go from bottom to top

When the EC proliferation rate $\beta _{EC}$ is doubled, the EC density $\rho$ is much higher near the tip than the control case (compare Figs. 3a, 15b), and the capillary extends much faster and reaches the VEGF source at $x=1$ around $t=3.5$ (Fig. 15a). When $\beta _{EC}$ is halved, the capillary grows very slow and reaches only $x=0.54$ at $t=7$.

When the capillary tip pulling force g is doubled, the capillary extends to $x=1$ at $t=4$ as shown in Fig. 16a. However, the Eulerian EC density is lower compared with the control (compare Figs. 3c, 16b). This is due to, on the one hand, the larger pulling force elongating the cells thus reducing density, and on the other hand, the faster capillary extension ($t=4$ compared with $t=7$ in control) decreasing the time duration for mass increase. When g is halved, the capillary extends to $x=0.67$ at $t=7$ (Fig. 16c), and the Eulerian EC density is higher (Fig. 16d).

Fig. 16

Modulation on the pulling force g. a, b g is doubled, for $t=0,1,\ldots ,4$. c, d g is halved, for $t=0,1,\ldots ,7$

Fig. 17

Modulation on EC proliferation threshold $c_0$. a–c $c_0$ is doubled, $t=0,1,\ldots ,7$. d–f $c_0$ is halved, $t=0,1,\ldots ,5$

If the VEGF proliferation threshold $c_0$ is doubled, then VEGF concentration in the region $x<0.4$ are merely slightly greater than $c_0$ (Fig. 17b). Thus, the ECs do not proliferate much (Fig. 17c). The displacement of the capillary tip by $t=7$ is only $u=0.3$, approximately. This value is mainly contributed by the pulling force because the steady state of u for a constant EC density $\rho =1$ at the tip is 0.3 when $g=3$. On the other hand, when $c_0$ is halved, the VEGF available for EC proliferation is increased and the EC density is larger than the control (compare at $t=5$ Figs. 3a, 17f). The capillary reaches $x=1$ at $t=5$.

1.3 Appendix C. Steady state analysis of tumor growth without capillary or with a whole-space capillary

Consider the following equations,

$$\begin{aligned} 0= & {} D_n n_{xx} - \gamma _n n + \beta _n, \quad x\in (x_L, x_R), \end{aligned}$$

(67)

$$\begin{aligned} n(x_L)= & {} n_L, \quad n(x_R)=n_R, \end{aligned}$$

(68)

$$\begin{aligned} v= & {} - K p_x, \quad x\in (x_L, x_R), \end{aligned}$$

(69)

$$\begin{aligned} v_x= & {} \beta _{tc} (n-n_g), \quad x\in (x_L, x_R), \end{aligned}$$

(70)

$$\begin{aligned} p(x_L)= & {} p_L, \quad p(x_R)=p_R, \end{aligned}$$

(71)

$$\begin{aligned} \frac{d x_L(t)}{dt}= & {} v(x_L,t),\quad \frac{d x_R(t)}{dt}=v(x_R,t). \end{aligned}$$

(72)

When $\beta _n=0$, the above equations represent the tumor growth without the effects of the blood capillary. When $\beta _n>0$, it represents the blood capillary occupying the whole space $\mathbb {R}$.

Let ${\hat{\gamma }}=\sqrt{\gamma _n/D_n}$, ${\hat{\beta }}=\beta _n/\gamma _n$. The nutrient solution of (67) and (68) is

$$\begin{aligned} n=\frac{(n_R-{\hat{\beta }}) \sinh ({{\hat{\gamma }}} (x-x_L)) - (n_L-{\hat{\beta }}) \sinh ({{\hat{\gamma }}}(x-x_R)) }{\sinh ({{\hat{\gamma }}} (x_R-x_L))} + {\hat{\beta }}. \end{aligned}$$

(73)

Inserting (69) into (70), we get

$$\begin{aligned} -p_{xx}= & {} \frac{\beta _{tc}}{K} (n-n_g) , \quad \text { in } \Omega _T. \end{aligned}$$

(74)

The above equation equipped with (71) is an elliptic problem. The pressure solution is

$$\begin{aligned} p(x)= & {} \frac{-\beta _{tc}}{K} \left( \frac{ (n_R-{\hat{\beta }}) \sinh ({{\hat{\gamma }}} (x-x_L)) - (n_L-{\hat{\beta }}) \sinh ({{\hat{\gamma }}}(x-x_R)) }{{\hat{\gamma }}^2 \sinh ({{\hat{\gamma }}}(x_R-x_L))}\right. \nonumber \\&\quad \left. - \frac{(n_g-{\hat{\beta }}) x^2}{2} + c_1 x + c_2 \right) , \end{aligned}$$

(75)

where

$$\begin{aligned} c_1= -\frac{K}{\beta _{tc} } \frac{p_R-p_L}{x_R-x_L} - \frac{n_R-n_L}{{\hat{\gamma }}^2 (x_R-x_L)} + \frac{(n_g-{\hat{\beta }}) (x_R+x_L)}{2}. \end{aligned}$$

(76)

The velocities at the tumor boundary are computed as $v=-K p_x$, and they are equal to

$$\begin{aligned} v(x_L)= & {} \beta _{tc} \frac{(n_R-{\hat{\beta }}) - (n_L-{\hat{\beta }}) \cosh (\hat{V})) }{{\hat{\gamma }} \sinh (\hat{V})} -\frac{\beta _{tc}(n_R-n_L) }{{{\hat{\gamma }}} \hat{V}} \nonumber \\&+ \frac{\beta _{tc} (n_g-{\hat{\beta }}) \hat{V}}{2{{\hat{\gamma }}}} - K \frac{p_R-p_L}{x_R-x_L}, \end{aligned}$$

(77)

$$\begin{aligned} v(x_R)= & {} \beta _{tc} \frac{(n_R-{\hat{\beta }}) \cosh (\hat{V}) -(n_L-{\hat{\beta }}) }{{\hat{\gamma }} \sinh (\hat{V})} -\frac{\beta _{tc}(n_R-n_L) }{{{\hat{\gamma }}} \hat{V}}\nonumber \\&-\frac{\beta _{tc} (n_g-{\hat{\beta }}) \hat{V}}{2{{\hat{\gamma }}}} - K \frac{p_R-p_L}{x_R-x_L}, \end{aligned}$$

(78)

where ${\hat{V}}={\hat{\gamma }} (x_R-x_L)$ is the tumor size. Since $\frac{d{\hat{V}}}{dt} = {\hat{\gamma }}\left( \frac{d x_R(t)}{dt} - \frac{d x_L(t)}{dt} \right) = {\hat{\gamma }}(v(x_R,t) - v(x_L,t))$, we get

$$\begin{aligned} \frac{d\hat{V}}{d\hat{t}} = \hat{V} \left( 2(1-\beta ^*) \frac{\cosh (\hat{V})-1}{\hat{V} \sinh (\hat{V})} - (n_g^*-\beta ^*) \right) , \end{aligned}$$

(79)

where $\hat{t}=\beta _{tc} \frac{(n_L+n_R)}{2} t$,

$$\begin{aligned} \beta ^*=\frac{2{\hat{\beta }}}{n_L+n_R}=\frac{2\beta _n}{\gamma _n(n_L+n_R)}, \quad n_g^*=\frac{2n_g}{n_L+n_R}. \end{aligned}$$

(80)

If $\beta ^*=1$, then ${\hat{V}}={\hat{V}}_0 e^{(1-n_g^*)t}$. In this case, there is a unique steady state ${\hat{V}}=0$, which is stable if $1\le n_g^*$ and unstable if $1>n_g^*$.

If $\beta ^*\ne 1$, then the nonzero steady state ${\hat{V}}^*$ satisfies

$$\begin{aligned} g({\hat{V}}^*)=\frac{n_g^*-\beta ^*}{1-\beta ^*}, \end{aligned}$$

(81)

where

$$\begin{aligned} g(x)\triangleq 2\frac{ (\cosh (x)-1)}{x\sinh (x)}. \end{aligned}$$

(82)

Direct calculation shows that $\lim \limits _{x\rightarrow 0}g (x)=1$ and $\lim \limits _{x\rightarrow \infty }g(x)=0$ and $g'(x)= \frac{2 (x-\sinh (x))(\cosh (x)-1)}{(x\sinh (x))^2}$. It is easy to check that $g'(x)<0$ when $x>0$. Thus in the domain $(0,\infty )$, g(x) is decreasing from 1 to 0 as x moves from 0 to infinity. We summarize these result in the following lemma.

Lemma 1

The function g(x) defined in (82) (the same as in (53)) is a monotonically decreasing function in the domain $(0,\infty )$ and $\lim \limits _{x\rightarrow 0+}g (x)=1$ and $\lim \limits _{x\rightarrow \infty }g(x)=0$.

The existence of the nonzero steady states requires $0<\frac{n_g^*-\beta ^*}{1-\beta ^*}<1$. If $\beta ^*<1$, then it is equivalent to $\beta ^*<n_g^*<1$. In this case, the steady state $\hat{V}^*=g^{-1}( \frac{n_g^*-\beta ^*}{1-\beta ^*})$ and is stable according to (79). If $\beta ^*>1$, then $0<\frac{n_g^*-\beta ^*}{1-\beta ^*}<1$ is equivalent to $\beta ^*>n_g^*>1$. There also exists a unique nonzero steady state ${\hat{V}}^*$ satisfying (81), which is unstable according to (79).

In the nonzero steady state of $\hat{V}$, we can get $n_g^*-\beta ^* =2(1-\beta ^*) \frac{\cosh (\hat{V})-1}{\hat{V} \sinh (\hat{V})}$ from (79). Inserting this relation to the above velocities at the boundary leads to

$$\begin{aligned} v(x_L)=v(x_R)=\frac{\beta _{tc}}{{{\hat{\gamma }}}} \frac{n_R-n_L}{2} \left( \frac{\cosh (\hat{V}) + 1 }{\sinh (\hat{V}) } - \frac{2}{\hat{V}} \right) - K \frac{p_R-p_L}{x_R-x_L}. \end{aligned}$$

(83)

Remark 4

In the steady states of tumor size, the whole tumor mass moves with a constant speed. In (83), the first term on the right has the same sign as $n_R-n_L$, and the second term has the opposite sign as $p_R-p_L$. This shows the tumor always moves towards the region of higher nutrient concentration and escapes away from the higher pressure.

The analysis in this section can be summarized as follows.

Theorem 3

The following results hold for the tumor growth model (67)–(72).

• (1) If $\beta ^*=1$, then there is a unique steady state ${\hat{V}}=0$, which is stable if $1\le n_g^*$ and unstable if $1>n_g^*$.

• (2) If $\beta ^*<n_g^*<1$, then the tumor size has two steady states, which are $\hat{V}=0$ (unstable) and $\hat{V}^*=g^{-1}( \frac{n_g^*-\beta ^*}{1-\beta ^*})$ (stable).

• (3) If $\beta ^*>n_g^*>1$, then the tumor size has two steady states, which are $\hat{V}=0$ (stable) and $\hat{V}^*=g^{-1}( \frac{n_g^*-\beta ^*}{1-\beta ^*})$ (unstable).

• (4) When the tumor size is in a nonzero steady state, the tumor moves with a constant speed given by (83).

• (5) In any other cases, there are no nonzero steady states of tumor size.

The tumor is truly stationary (both the tumor size and the location are invariant) only when the boundary velocity given by (83) is zero. This can occur if $n_R=n_L$ and $p_R=p_L$, or these boundary conditions of nutrient and pressure are delicately modulated to make the boundary velocity zero. However, even in the truly stationary state, the inner velocity of the tumor does not vanish. Indeed, by using $v=-Kp_x$ and (75) we can obtain at any moment (including unsteady state)

$$\begin{aligned} v(x)= & {} \beta _{tc} \left( \frac{(n_R-{\hat{\beta }}) \cosh ({{\hat{\gamma }}}(x-x_L))- (n_L-{\hat{\beta }}) \cosh ({{\hat{\gamma }}}(x-x_R)) }{ {{\hat{\gamma }}} \sinh ({\hat{\gamma }} (x_R-x_L))} \right. \nonumber \\&\left. - \frac{ (n_R-n_L) }{{\hat{\gamma }}^2 (x_R-x_L)} + \frac{ (n_g-{\hat{\beta }}) (x_R+x_L-2x)}{2} \right) - K \frac{p_R-p_L}{x_R-x_L}. \end{aligned}$$

(84)

The typical graphs of nutrient, pressure, and velocity when the tumor is in the steady state are plotted in Fig. 18. Note the velocity on both the left and right sides of the tumor region is pointing towards the tumor center.

Fig. 18

One steady state of tumor. $K=1$, $D_n=1$, $\gamma _n=10$, $\beta _n=0$, $\beta _{tc}=1$, $n_g=0.5$, $n_R=n_L=1$, $p_L=p_B=1$. Thus, $\beta ^*=0$ and $n_g^*=0.5$. The tumor region is where the nutrient $n<1$, roughly [0.4, 1.6]. Outside the tumor, we let $n=1$, $p=0$, and $u=0$

From the velocity solution (84), we realize the cell motility constant K only shows up in Darcy’s term, $K\frac{p_R-p_L}{x_R-x_L}$, which is in accordance with Darcy’s law of fluid velocity in porous medium (extracellular matrix in this case). This term provides a uniform background flow advecting the entire tumor volume. If the boundary pressures are equal, i.e., $p_L=p_R$, then Darcy’s advection will vanish, which means the cell motility has no effect on tumor growth. But even if $p_L\ne p_R$, Darcy’s term still has no effect on the rate of change of tumor size [see Eq. (79)]. The fundamental reason for this neglect is the constant density assumption. Indeed, due to the constant density assumption, the rate of change of the tumor volume $\Omega _T$ is given by

$$\begin{aligned} \frac{d}{dt} \int _{\Omega _T} d\vec {x} = \int _{\Omega _T} \nabla \cdot \vec {v} d\vec {x} = \int _{\Omega _T} \beta _{tc} (n-n_g) d\vec {x}. \end{aligned}$$

(85)

Thus, the rate of change of tumor volume solely relies on proliferative parameters, and is irrelevant to cell motility or pressure. But there is one concern when the cell motility is small (e.g., cells are very viscous or permeability is small) but the divergence of velocity remains the same because the pressure magnitude would be very large according to Darcy’s law or solution (75). The high pressure would prevent cells from proliferation or even crush the cells. Therefore, the current model would be valid only for medium to large cell motility.

1.4 Appendix D. Numerical method for the coupled model of angiogenesis and tumor growth

The numerical algorithm is briefly stated below. First, set up the initial values of VEGF, nutrient, blood vessels, and tumor. At each time step $t^k$, $k=0,1,\ldots$, denote the tumor domain as $\Omega _{T_k}=(x_L(t^k), x_R(t^k))$, VEGF as $c(x,t^{k})$, EC density as $\rho (X,t^{k})$, EC displacement as $u(X,t^{k})$, where $x\in (0, L_{tissue})$, $X\in (0, L_V)$, then we do the following.

• (1) Solve the nutrient $n(x,t^k)$ inside the tumor domain $\Omega _T$ with Eqs. (20) and (25).

• (2) Solve the pressure $p(x,t^k)$ from

  $$\begin{aligned} -K p_{xx} = \beta _{tc} (n-n_g), x\in (x_L, x_R) \end{aligned}$$

  (86)

  with the boundary value (24).

• (3) Solve the velocity $v(x,t^k)$ on the tumor boundary $x_L(t^k)$ and $x_R(t^k)$ with Eq. (18).

• (4) Update the tumor boundary points $x_R$ and $x_L$ with (21).

• (5) Solve VEGF $c(x,t^{k+1})$ with Eq. (15).

• (6) Solve EC density $\rho (X,t^{k+1})$ in the capillary domain $(0,L_V)$ with Eq. (16).

• (7) Solve EC displacement $u(X,t^{k+1})$ in the capillary domain $(0,L_V)$ with Eq. (17) and boundary conditions (23).

• (8) Let $k=k+1$, then go to the next time step $t^{k+1}=t^k+\Delta t$.

A second order numerical method is developed to solve the nutrient and pressure equations and the details are in Zheng and Sweidan (2018)