Mathematical Modeling of Cellular Cross-Talk Between Endothelial and Tumor Cells Highlights Counterintuitive Effects of VEGF-Targeted Therapies

Abstract

Tumor growth and progression are critically dependent on the establishment of a vascular support system. This is often accomplished via the expression of pro-angiogenic growth factors, including members of the vascular endothelial growth factor (VEGF) family of ligands. VEGF ligands are overexpressed in a wide variety of solid tumors and therefore have inspired optimism that inhibition of the different axes of the VEGF pathway—alone or in combination—would represent powerful anti-angiogenic therapies for most cancer types. When considering treatments that target VEGF and its receptors, it is difficult to tease out the differential anti-angiogenic and anti-tumor effects of all combinations experimentally because tumor cells and vascular endothelial cells are engaged in a dynamic cross-talk that impacts key aspects of tumorigenesis, independent of angiogenesis. Here we develop a mathematical model that connects intracellular signaling responsible for both endothelial and tumor cell proliferation and death to population-level cancer growth and angiogenesis. We use this model to investigate the effect of bidirectional communication between endothelial cells and tumor cells on treatments targeting VEGF and its receptors both in vitro and in vivo. Our results underscore the fact that in vitro therapeutic outcomes do not always translate to the in vivo situation. For example, our model predicts that certain therapeutic combinations result in antagonism in vivo that is not observed in vitro. Mathematical modeling in this direction can shed light on the mechanisms behind experimental observations that manipulating VEGF and its receptors is successful in some cases but disappointing in others.

Appendix

Table 4 Complete model variables

1.1 In Vitro Treatment Equations

The complete set of model equations used for simulating in vitro, pre-treatment tumor growth is described in the main text. For the targeted therapies we are considering, the tumor cell, endothelial cell and Bcl-2 equations do not change. Below we describe the modifications to the model that arise due to treatment.

VEGF-Binding Equations: The equations below describe VEGF binding, and they are identical to those presented in the main text, except for the addition of the last two terms in Eqs. (20)–(23), which represent the binding dynamics of anti-VEGF, VEGFR1 or VEGFR2 antibodies.

$$\begin{aligned}&\frac{\hbox {d}A}{\hbox {d}t} = \underbrace{-\,k_{E1}^f\,R_{E1}\,A \,+\, k_{E1}^r\,D_{E1} \,-\,k_{E2}^f\,R_{E2}\,A \,+\, k_{E2}^r\,D_{E2}}_{\text {Reaction with VEGFR1/VEGFR2 on endothelial cells}} \,-\, \underbrace{k_{T1}^f\,R_{T1}\,A \,+\, k_{T1}^r\,D_{T1}}_{\begin{array}{c} {\text {Reaction with VEGFR1}}\\ {\text {on tumor cells}} \end{array}} \nonumber \\&\quad \,-\, \underbrace{\lambda _A\,A}_{\begin{array}{c} {\text {Natural}}\\ {\text {decay}} \end{array}} \,+\, \underbrace{\frac{\mu _A\,B_E}{\beta _A + B_E}\,E}_{\begin{array}{c} {\text {Bcl-2-mediated}}\\ {\text {production by}}\\ {\text {endothelial cells}} \end{array}} \,+\, \underbrace{\nu _A\,T}_{\begin{array}{c} {\text {Production by}}\\ {\text {tumor cells}} \end{array}} \,-\, \underbrace{k_{VAb}^f\,X_V\,A \,+\, k_{VAb}^r\,C_{AV}}_{\begin{array}{c} {\text {Reaction with anti-VEGF}}\\ {\text {antibody}} \end{array}}, \end{aligned}$$

(20)

$$\begin{aligned}&\frac{\hbox {d}R_{E1}}{\hbox {d}t} = \underbrace{-\,k_{E1}^f\,R_{E1}\,A \,+\, k_{E1}^r\,D_{E1} \,+\, k_{E1}^p\,D_{E1}}_{\text {Reaction with VEGF}} \,+\, \underbrace{R_{E1}^t\,\mu _E\,\phi _{E}\,E\,\left( 1 \,-\, \rho _E\,E\right) }_{\begin{array}{c} {\text {Production due to}}\\ {\text {endothelial cell proliferation}} \end{array}} \nonumber \\&\quad -\, \underbrace{\frac{R_{E1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}} \,-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{E1} \,+\, k_{R1Ab}^r\,C_{E1R1}}_{\text {Reaction with anti-VEGFR1 antibody}} , \end{aligned}$$

(21)

$$\begin{aligned}&\frac{\hbox {d}R_{E2}}{\hbox {d}t} = \underbrace{-\,k_{E2}^f\,R_{E2}\,A \,+\, k_{E2}^r\,D_{E2} \,+\, k_{E2}^p\,D_{E2}}_{\text {Reaction with VEGF}} \,+\, \underbrace{R_{E2}^t\,\mu _E\,\phi _{E}\,E\,\left( 1 \,-\, \rho _E\,E\right) }_{\begin{array}{c} {\text {Production due to}}\\ {\text {endothelial cell proliferation}} \end{array}} \nonumber \\&\quad -\, \underbrace{\frac{R_{E2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}} \,-\, \underbrace{k_{R2Ab}^f\,X_{R2}\,R_{E2} \,+\, k_{R2Ab}^r\,C_{E2R2}}_{\text {Reaction with anti-VEGFR2 antibody}} , \end{aligned}$$

(22)

$$\begin{aligned}&\frac{\hbox {d}R_{T1}}{\hbox {d}t} = \underbrace{-\,k_{T1}^f\,R_{T1}\,A \,+\, k_{T1}^r\,D_{T1} \,+\, k_{T1}^p\,D_{T1}}_{\text {Reaction with VEGF}} \,+\, \underbrace{R_{T1}^t\,\left( \mu _T \,+\, \nu _T\,\phi _T\right) \,T}_{\begin{array}{c} {\text {Production due to}}\\ {\text {tumor cell proliferation}} \end{array}} \nonumber \\&\quad -\, \underbrace{\frac{R_{T1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,T}{\beta _T + B_T^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}} \,-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,+\, k_{R1Ab}^r\,C_{T1R1}}_{\text {Reaction with anti-VEGFR1 antibody}} , \end{aligned}$$

(23)

$$\begin{aligned}&\frac{\hbox {d}D_{E1}}{\hbox {d}t} = \underbrace{k_{E1}^f\,R_{E1}\,A \,-\, k_{E1}^r\,D_{E1} \,-\, k_{E1}^p\,D_{E1}}_{\begin{array}{c} {\text {VEGFR1 activation by VEGF}}\\ {\text {on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{D_{E1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(24)

$$\begin{aligned}&\frac{\hbox {d}D_{E2}}{\hbox {d}t} = \underbrace{k_{E2}^f\,R_{E2}\,A \,-\, k_{E2}^r\,D_{E2} \,-\, k_{E2}^p\,D_{E2}}_{\begin{array}{c} {\text {VEGFR2 activation by VEGF}}\\ {\text {on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{D_{E2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(25)

$$\begin{aligned}&\frac{\hbox {d}D_{T1}}{\hbox {d}t} = \underbrace{k_{T1}^f\,R_{T1}\,A \,-\, k_{T1}^r\,D_{T1} \,-\, k_{E1}^p\,D_{E1}}_{\begin{array}{c} {\text {VEGFR1 activation by VEGF}}\\ {\text {on tumor cells}} \end{array}} \,-\, \underbrace{\frac{D_{T1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,T}{\beta _T + B_T^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}}, \end{aligned}$$

(26)

VEGF, VEGFR1, VEGFR2 Therapy Equations: The binding dynamics for an anti-VEGF antibody ($X_V$), an anti-VEGFR1 antibody ($X_{R1}$) and an anti-VEGFR2 antibody ($X_{R2}$) are described below:

$$\begin{aligned} \frac{\hbox {d}X_{V}}{\hbox {d}t}= & {} \underbrace{-\,k_{VAb}^f\,X_V\,A \,+\, k_{VAb}^r\,C_{AV}}_{\text {Reaction with VEGF}}, \end{aligned}$$

(27)

$$\begin{aligned} \frac{\hbox {d}C_{AV}}{\hbox {d}t}= & {} \underbrace{k_{VAb}^f\,X_V\,A \,-\, k_{VAb}^r\,C_{AV}}_{\begin{array}{c} {\text {VEGF--anti-VEGF antibody}}\\ {\text {complex formation}} \end{array}}, \end{aligned}$$

(28)

$$\begin{aligned} \frac{\hbox {d}X_{R1}}{\hbox {d}t}= & {} \underbrace{-\,k_{R1Ab}^f\,X_{R1}\,R_{E1} \,+\, k_{R1Ab}^r\,C_{E1R1}}_{\text {Reaction with VEGFR1 on endothelial cells}} \,-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,+\, k_{R1Ab}^r\,C_{T1R1}}_{\text {Reaction with VEGFR1 on tumor cells}} , \end{aligned}$$

(29)

$$\begin{aligned} \frac{\hbox {d}C_{E1R1}}{\hbox {d}t}= & {} \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{E1} \,-\, k_{R1Ab}^r\,C_{E1R1}}_{\begin{array}{c} {\text {VEGFR1--anti-VEGFR1 antibody}}\\ {\text {complex formation on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{C_{E1R1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(30)

$$\begin{aligned} \frac{\hbox {d}C_{T1R1}}{\hbox {d}t}= & {} \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,-\, k_{R1Ab}^r\,C_{T1R1}}_{\begin{array}{c} {\text {VEGFR1--anti-VEGFR1 antibody}}\\ {\text {complex formation on tumor cells}} \end{array}} \,-\, \underbrace{\frac{C_{T1R1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,T}{\beta _T + B_T^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}}, \end{aligned}$$

(31)

$$\begin{aligned} \frac{\hbox {d}X_{R2}}{\hbox {d}t}= & {} \underbrace{-\,k_{R2Ab}^f\,X_{R2}\,R_{E2} \,+\, k_{R2Ab}^r\,C_{E2R2}}_{\text {Reaction with VEGFR2 on endothelial cells}} , \end{aligned}$$

(32)

$$\begin{aligned} \frac{\hbox {d}C_{E2R2}}{\hbox {d}t}= & {} \underbrace{k_{R2Ab}^f\,X_{R2}\,R_{E2} \,-\, k_{R2Ab}^r\,C_{E2R2}}_{\begin{array}{c} {\text {VEGFR2--anti-VEGFR2 antibody}}\\ {\text {complex formation on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{C_{E2R2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,E}{\beta _E + B_E^2}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}.\nonumber \\ \end{aligned}$$

(33)

1.2 In Vivo Treatment Equations

The model equations used for simulating tumor cells, endothelial cells and drug pharmacokinetics for anti-VEGF therapy in vivo are described in detail in the main text. Below we present the remaining model equations associated with treatment for the in vivo situation.

Intratumoral VEGF: Intratumoral VEGF (A) is governed by Eq. (34). The first eight terms of the equation, representing binding, decay and production, are described in detail in the main text. The last terms in the VEGF equation represent the response to therapies that target the VEGF.

$$\begin{aligned} \frac{\hbox {d}A}{\hbox {d}t}= & {} \underbrace{-\,k_{E1}^f\,R_{E1}\,A \,+\, k_{E1}^r\,D_{E1} \,-\,k_{E2}^f\,R_{E2}\,A \,+\, k_{E2}^r\,D_{E2}}_{\text {Reaction with VEGFR1/VEGFR2 on endothelial cells}} \,-\, \underbrace{k_{T1}^f\,R_{T1}\,A \,+\, k_{T1}^r\,D_{T1}}_{\begin{array}{c} {\text {Reaction with VEGFR1}}\\ {\text {on tumor cells}} \end{array}} \nonumber \\&\,-\, \underbrace{\lambda _A\,A}_{\begin{array}{c} {\text {Natural}}\\ {\text {decay}} \end{array}} \,+\, \underbrace{\frac{\nu _A}{1+e^{-\kappa _A(N_{h}-N)}}\,T}_{\begin{array}{c} {\text {Hypoxia-mediated production}}\\ {\text {by tumor cells}} \end{array}} \,-\, \underbrace{k_{VAb}^f\,X_V\,A \,+\, k_{VAb}^r\,C_{AV}}_{\begin{array}{c} {\text {Reaction with anti-VEGF}}\\ {\text {antibody}} \end{array}}. \end{aligned}$$

(34)

VEGFR1 and VEGFR2: The equations representing VEGF receptor dynamics on ECs and TCs remain also largely unchanged from the in vitro case described in the main text. The only addition is the therapeutic binding of VEGFR1 and VEGFR2 antibodies, where appropriate. The first three terms in each of the equations below represent the response to VEGF of each variable ($R_i$ for $i = 1, 2 \equiv$ free VEGFR1/VEGFR2 receptors on ECs and/or TCs, and $D_i$ for $i = 1,2 \equiv$ VEGF–VEGFR$_i$ complexes on ECs and TCs). Note that in addition to binding and releasing VEGF, the rate of change of each receptor species is also affected by the creation of new unbound receptors as a result of cell division and the removal of receptors and complexes due to cell death. This ensures conservation of receptors and is described by the fourth term in the $D_i$ equations and the fourth and fifth terms in the $R_i$ equations. The last terms in each of the $R_i$ equations represent the response to therapies that target the VEGF receptors.

$$\begin{aligned} \frac{\hbox {d}R_{E1}}{\hbox {d}t}= & {} \underbrace{-\,k_{E1}^f\,R_{E1}\,A \,+\, k_{E1}^r\,D_{E1} \,+\, k_{E1}^p\,D_{E1}}_{\text {Reaction with VEGF}} \,+\, \underbrace{R_{E1}^t\,\mu _E\,\frac{\phi _E^{\rho _3}}{\alpha _E^{\rho _3} + \phi _E^{\rho _3}}\,E}_{\begin{array}{c} {\text {Production due to}}\\ {\text {endothelial cell proliferation}} \end{array}} \nonumber \\&-\, \underbrace{\frac{R_{E1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}} \,-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{E1} \,+\, k_{R1Ab}^r\,C_{E1R1}}_{\text {Reaction with anti-VEGFR1 antibody}} , \end{aligned}$$

(35)

$$\begin{aligned} \frac{\hbox {d}R_{E2}}{\hbox {d}t}= & {} \underbrace{-\,k_{E2}^f\,R_{E2}\,A \,+\, k_{E2}^r\,D_{E2} \,+\, k_{E2}^p\,D_{E2}}_{\text {Reaction with VEGF}} \,+\, \underbrace{R_{E2}^t\,\mu _E\,\frac{\phi _E^{\rho _3}}{\alpha _E^{\rho _3} + \phi _E^{\rho _3}}\,E}_{\begin{array}{c} {\text {Production due to}}\\ {\text {endothelial cell proliferation}} \end{array}} \nonumber \\&-\, \underbrace{\frac{R_{E2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}} \,-\, \underbrace{k_{R2Ab}^f\,X_{R2}\,R_{E2} \,+\, k_{R2Ab}^r\,C_{E2R2}}_{\text {Reaction with anti-VEGFR2 antibody}} , \end{aligned}$$

(36)

$$\begin{aligned} \frac{\hbox {d}R_{T1}}{\hbox {d}t}= & {} \underbrace{-\,k_{T1}^f\,R_{T1}\,A \,+\, k_{T1}^r\,D_{T1} \,+\, k_{T1}^p\,D_{T1}}_{\text {Reaction with VEGF}}\nonumber \\&+ \underbrace{R_{T1}^t\,\left[ \mu _T \,+\, \nu _T\,\frac{\phi _T^{\rho _1}}{\alpha _T^{\rho _1} + \phi _T^{\rho _1}}\right] \,T\,\mathcal {H}(N - N_\mathrm{p})}_{\begin{array}{c} {\text {Production due to}}\\ {\text {tumor cell proliferation}} \end{array}} \nonumber \\&-\, \underbrace{\frac{R_{T1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,\beta _T^{\rho _2}\,[1 + \mathcal {H}(N_\mathrm{d} - N)]\,T}{\beta _T^{\rho _2} + B_T^{\rho _2}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}} \,\nonumber \\&-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,+\, k_{R1Ab}^r\,C_{T1R1}}_{\text {Reaction with anti-VEGFR1 antibody}} , \end{aligned}$$

(37)

$$\begin{aligned} \frac{\hbox {d}D_{E1}}{\hbox {d}t}= & {} \underbrace{k_{E1}^f\,R_{E1}\,A \,-\, k_{E1}^r\,D_{E1} \,-\, k_{E1}^p\,D_{E1}}_{\begin{array}{c} {\text {VEGFR1 activation by VEGF}}\\ {\text {on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{D_{E1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(38)

$$\begin{aligned} \frac{\hbox {d}D_{E2}}{\hbox {d}t}= & {} \underbrace{k_{E2}^f\,R_{E2}\,A \,-\, k_{E2}^r\,D_{E2} \,-\, k_{E2}^p\,D_{E2}}_{\begin{array}{c} {\text {VEGFR2 activation by VEGF}}\\ {\text {on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{D_{E2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(39)

$$\begin{aligned} \frac{\hbox {d}D_{T1}}{\hbox {d}t}= & {} \underbrace{k_{T1}^f\,R_{T1}\,A \,-\, k_{T1}^r\,D_{T1} \,-\, k_{E1}^p\,D_{E1}}_{\begin{array}{c} {\text {VEGFR1 activation by VEGF}}\\ {\text {on tumor cells}} \end{array}}\nonumber \\&- \underbrace{\frac{D_{T1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,\beta _T^{\rho _2}\,[1 + \mathcal {H}(N_\mathrm{d} - N)]\,T}{\beta _T^{\rho _2} + B_T^{\rho _2}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}}.\nonumber \\ \end{aligned}$$

(40)

1.2.1 Intracellular Bcl-2

Bcl-2 mRNA is constitutively expressed within both cell types and undergoes natural degradation. Activation of its receptors by VEGF leads to additional Bcl-2 synthesis in each cell type. Combining these processes, we obtain the following equations for Bcl-2 mRNA expression in ECs and TCs, as described in the main text:

$$\begin{aligned} \frac{\hbox {d}B_E}{\hbox {d}t}= & {} \underbrace{\chi _E}_{\begin{array}{c} {\text {Constitutive}}\\ {\text {expression}} \end{array}} \,+\, \underbrace{\eta _E\,\frac{\phi _E^{\omega }}{\gamma _E^{\omega } + \phi _E^{\omega }}}_{\begin{array}{c} {\text {Activated VEGFR2-}}\\ {\text {mediated production}} \end{array}} \,-\, \underbrace{\lambda _B\,B_E}_{\begin{array}{c} {\text {Natural}}\\ {\text {decay}} \end{array}}, \end{aligned}$$

(41)

$$\begin{aligned} \frac{\hbox {d}B_T}{\hbox {d}t}= & {} \underbrace{\chi _T}_{\begin{array}{c} {\text {Constitutive}}\\ {\text {expression}} \end{array}} \,+\, \underbrace{\eta _T\,\frac{\phi _T^{\theta }}{\gamma _T^{\theta } + \phi _T^{\theta }}}_{\begin{array}{c} {\text {Activated VEGFR1-}}\\ {\text {mediated production}} \end{array}} \,-\, \underbrace{\lambda _B\,B_T}_{\begin{array}{c} {\text {Natural}}\\ {\text {decay}} \end{array}}. \end{aligned}$$

(42)

1.2.2 Molecular-Level Variables Associated with Treatment

Below, we present the model equations for the effects of an anti-VEGF antibody, like avastin, an anti-VEGFR1 antibody-like IMC-18F1, and an anti-VEGFR2 antibody.

Drug Pharmacokinetics: Experimental evidence suggests a biphasic plasma concentration–time curve for avastin Lin et al. (1999). Consequently, a two-compartment model is proposed to govern the pharmacokinetics of all of the drugs we consider. Systemic circulation (together with well-vascularized organs) is taken to be the first compartment from where the drug may be eliminated or enter, in a reversible process, a second compartment which consists of tissues and organs with poor vascular perfusion. Combining these effects, the equations governing drug pharmacokinetics can be written as follows:

$$\begin{aligned} \frac{\hbox {d}X_{is}}{\hbox {d}t}= & {} \underbrace{-\, k_{i12}\,X_{is} \,+\, k_{i21}\,X_{ip}}_{\text {Pharmacokinetics}} \,-\, \underbrace{k_{iel}\,X_{is}}_{\text {Elimination}} \,+\, \text {Dosing}, \end{aligned}$$

(43)

$$\begin{aligned} \frac{\hbox {d}X_{ip}}{\hbox {d}t}= & {} \underbrace{k_{i12}\,X_{is} \,-\, k_{i21}\,X_{ip}}_{\text {Pharmacokinetics}}, \end{aligned}$$

(44)

where $X_{is}$ and $X_{ip}$ ($i = V, R1, R2$ for anti-VEGF, anti-VEGFR1, and anti-VEGFR2 therapy) represent the amount of drug in systemic circulation and the peripheral compartment, respectively. The dosing function in Eq. (43) represents the periodic administration of avastin and is taken to have the following form:

$$\begin{aligned} \text {Dosing}(t)= & {} \sum _{i=1}^n D_0\,\mathcal {H}(t-t_i)\,\mathcal {H}(t_i+t_a-t), \end{aligned}$$

(45)

where $D_0$ is the amount of drug delivered in each dose; $t_i$, the timing of each dose; $t_a$, the length of each injection; and n, the total number of doses.

Intratumoral Drug Dynamics: The tumor is assumed to reside in a pharmacokinetic compartment of its own. For simplicity, the rates of transfer of drug into and out of the tumor are taken to be the same as those for the peripheral compartment [Eq. (44)]. Further, the tumor is assumed to occupy a negligible volume compared to the volume of the animal; therefore, the extravasation of drug from the tumor into systemic circulation will not affect the amount of drug in the central compartment. Finally, avastin–VEGF complexes are assumed to undergo the same degradation as free VEGF in tissue. Combined together, these processes yield the following equations governing drug dynamics in the tumor space. Recall, $X_{i}$ ($i = V, R1, R2$ for the anti-VEGF, anti-VEGFR1 and anti-VEGFR2 therapy) represent free forms of each drug and $C_{j}$ ($j = AV, E1R1, T1R1, E2R2$) represent the VEGFR1–anti-VEGFR1 antibody complexes on ECs/TCs and the VEGFR2–anti-VEGFR2 antibody complexes on ECs.

$$\begin{aligned} \frac{\hbox {d}X_{V}}{\hbox {d}t}= & {} \underbrace{-\,k_{VAb}^f\,X_V\,A \,+\, k_{VAb}^r\,C_{AV}}_{\text {Reaction with VEGF}} \,+\, \underbrace{k_{V12}\,X_{Vs} \,-\, k_{V21}\,X_{V}}_{\text {Pharmacokinetics}}, \end{aligned}$$

(46)

$$\begin{aligned} \frac{\hbox {d}C_{AV}}{\hbox {d}t}= & {} \underbrace{k_{VAb}^f\,X_V\,A \,-\, k_{VAb}^r\,C_{AV}}_{\begin{array}{c} {\text {VEGF--anti-VEGF antibody}}\\ {\text {complex formation}} \end{array}} \,-\, \underbrace{\lambda _A\,C_{AV}}_{\begin{array}{c} {\text {Natural}}\\ {\text {decay}} \end{array}}, \end{aligned}$$

(47)

$$\begin{aligned} \frac{\hbox {d}X_{R1}}{\hbox {d}t}= & {} \underbrace{-\,k_{R1Ab}^f\,X_{R1}\,R_{E1} \,+\, k_{R1Ab}^r\,C_{E1R1}}_{\text {Reaction with VEGFR1 on endothelial cells}} \,-\, \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,+\, k_{R1Ab}^r\,C_{T1R1}}_{\text {Reaction with VEGFR1 on tumor cells}} \nonumber \\&+\, \underbrace{k_{V12}\,X_{R1s} \,-\, k_{V21}\,X_{R1}}_{\text {Pharmacokinetics}}, \end{aligned}$$

(48)

$$\begin{aligned} \frac{\hbox {d}C_{E1R1}}{\hbox {d}t}= & {} \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{E1} \,-\, k_{R1Ab}^r\,C_{E1R1}}_{\begin{array}{c} {\text {VEGFR1--anti-VEGFR1 antibody}}\\ {\text {complex formation on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{C_{E1R1}\,R_{E1}^t}{R_{E1}+D_{E1}+C_{E1R1}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}, \end{aligned}$$

(49)

$$\begin{aligned} \frac{\hbox {d}C_{T1R1}}{\hbox {d}t}= & {} \underbrace{k_{R1Ab}^f\,X_{R1}\,R_{T1} \,-\, k_{R1Ab}^r\,C_{T1R1}}_{\begin{array}{c} {\text {VEGFR1--anti-VEGFR1 antibody}}\\ {\text {complex formation on tumor cells}} \end{array}} \,\nonumber \\&-\, \underbrace{\frac{C_{T1R1}\,R_{T1}^t}{R_{T1}+D_{T1}+C_{T1R1}}\,\frac{\delta _T\,\beta _T^{\rho _2}\,[1 + \mathcal {H}(N_\mathrm{d} - N)]\,T}{\beta _T^{\rho _2} + B_T^{\rho _2}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {tumor cell apoptosis}} \end{array}}, \end{aligned}$$

(50)

$$\begin{aligned} \frac{\hbox {d}X_{R2}}{\hbox {d}t}= & {} \underbrace{-\,k_{R2Ab}^f\,X_{R2}\,R_{E2} \,+\, k_{R2Ab}^r\,C_{E2R2}}_{\text {Reaction with VEGFR2 on endothelial cells}}\,+\, \underbrace{k_{V12}\,X_{R2s} \,-\, k_{V21}\,X_{R2}}_{\text {Pharmacokinetics}} , \end{aligned}$$

(51)

$$\begin{aligned} \frac{\hbox {d}C_{E2R2}}{\hbox {d}t}= & {} \underbrace{k_{R2Ab}^f\,X_{R2}\,R_{E2} \,-\, k_{R2Ab}^r\,C_{E2R2}}_{\begin{array}{c} {\text {VEGFR2--anti-VEGFR2 antibody}}\\ {\text {complex formation on endothelial cells}} \end{array}} \,-\, \underbrace{\frac{C_{E2R2}\,R_{E2}^t}{R_{E2}+D_{E2}+C_{E2R2}}\,\frac{\delta _E\,\beta _E^{\rho _4}\,E}{\beta _E^{\rho _4} + B_E^{\rho _4}}}_{\begin{array}{c} {\text {Loss due to}}\\ {\text {endothelial cell apoptosis}} \end{array}}.\nonumber \\ \end{aligned}$$

(52)

1.2.3 Receptor Conservation

In the above equations, $R_{Ei}^t$, $i=1,2$ and $R_{T1}^t$ represent the VEGFRi and VEGFR1 receptor expression levels on endothelial and tumor cells, respectively. From equations (13), (35), (38) and (49), it follows that $R_{E1}^tE \,=\, R_{E1}\,+\,D_{E1} \,+\, C_{E1R1}$, that is, the number of VEGFR1 per endothelial cell is conserved. Likewise, $R_{E2}^tE \,=\, R_{E2}\,+\,D_{E2} \,+\, C_{E2R2}$ and $R_{T1}^tT \,=\, R_{T1}\,+\,D_{T1} \,+\, C_{T1R1}$.

1.3 Parameters

A list of parameter values estimated from the literature for the in vitro and in vivo treatment models is provided in Table 5. We remark that the rates of forward reaction are typically expressed in units of “per time, per concentration.” However, in our model, all chemical species are expressed in terms of total quantity (fmol in the in vitro case and pmol in the in vivo case). Therefore, to preserve dimensional accuracy, the rates of forward reaction listed in Table  4 (and Table 1 of the main text) have been scaled by the volume of the experiment being simulated. For instance, the rate of forward reaction between avastin and VEGF has been determined experimentally to be 2.9794 (fmol/ml)$^{-1}$ day$^{-1}$ (or (pmol/$\upmu$l)$^{-1}$ day$^{-1}$). Consequently, $k_{VAb}^f$ is taken to be 2.9794/Vol$_C$ fmol$^{-1}$ day$^{-1}$ when simulating in vitro experiments and 2.9794/Vol$_T$ pmol$^{-1}$ day$^{-1}$ in the in vivo case. Here, $Vol_C$ is the volume of the cell culture experiments in ml and $\hbox {Vol}_T$, the volume of the tumor in $\mu$l. Note that $Vol_T$ $=$ (volume of 1 million ECs)$\times E$ $+$ (volume of 1 million TCs)$\times T$ and the volume of 1 EC is 2.2$\times 10^{-6}$ $\mu$l King et al. (2004) and that of 1 TC is 1$\times 10^{-6}$ $\mu$l Monte (2009).

Table 5 Binding rates for VEGF, VEGFR1 and VEGFR2 antibodies

A list of parameter values for the in vivo model is provided in Table 6.

Table 6 Parameter estimates for the in vivo model

The best fit of the in vivo treatment model to avastin pharmacokinetic data reported in Lin et al. (1999) is shown in Fig. 9.

Fig. 9

Model fit to avastin pharmacokinetic data reported in Lin et al. (1999)

1.4 Parameter Sensitivity

A local sensitivity analysis is carried out by varying the parameters associated with the vascular growth and response to treatment of in vivo tumor xenografts. These include: the VEGF-dependent rate of tumor cell proliferation, $\nu _T$; the sensitivity of tumor cell proliferation to VEGFR1 receptor occupancy, $\alpha _T$; the Bcl-2-modulated rate of tumor cell death, $\delta _T$; the sensitivity of tumor cell death to Bcl-2 expression, $\beta _T$; the VEGF-dependent rate of endothelial cell proliferation, $\mu _E$; the sensitivity of endothelial cell proliferation to VEGFR2 receptor occupancy, $\alpha _E$; the Bcl-2-modulated rate of endothelial cell death, $\delta _E$; the sensitivity of endothelial cell death to Bcl-2 expression, $\beta _E$; the VEGF expression rate of tumor cells, $\nu _A$; and the sensitivity of tumor cells to hypoxia, $\kappa _A$ [see equations (12)–(14) in the main text]. The two cases—control (no treatment) and treatment—are considered separately. In each case, the residual between simulated and experimental data (tumor volume time course) is computed as each parameter is varied from its baseline estimate, and the resulting residual is plotted as a function of the percent change in the value of parameter being varied. Given that vascular tumor growth is driven by hypoxia in our model, we also test the sensitivity of tumor oxygenation to the various parameters by plotting the maximum partial pressure of oxygen as a function of the percent change in the value of parameter being varied.

Local Sensitivity Analysis–Control

Figure 10a, c, e reveal that the model predictions of tumor xenograft size are sensitive to endothelial and tumor cell growth parameters, tumor cell death parameters, as well as tumor cell VEGF expression and sensitivity to hypoxia, with low values of $\alpha _T$ and high values of $\nu _A$ and $\nu _T$ resulting in the most change. Since these simulations are without treatment, tumor xenograft size is predictably insensitive to parameters relating to endothelial cell death and tumor cell death. Figure 10b, d, f reveal that tumor oxygenation is most sensitive to high values of $\mu _E$ and $\alpha _T$ and low values of $\alpha _E$, $\nu _T$ and $\nu _A$. Once again, cell death has little impact on tumor oxygenation.

Fig. 10

Local sensitivity analysis for parameters relating to tumor xenograft growth in the absence of treatment. a, c, e The residual or Euclidean norm between simulated and experimental data (tumor volume time course, see Fig. 6a in main text) is plotted on the y-axis, and percentage variation of the parameters from their baseline values is plotted on the x-axis. b, d, f The predicted maximum partial pressure in the tumor over the time course of the simulation is plotted on the y-axis, and percentage variation of the parameters from their baseline values is plotted on the x-axis

Local Sensitivity Analysis–Treatment

Figure 11a, c, e reveal that the model predictions of tumor xenograft size under avastin therapy are sensitive to endothelial and tumor cell growth parameters, as well as tumor cell VEGF expression and sensitivity to hypoxia, with low values of $\alpha _T$ and high values of $\nu _A$ and $\nu _T$ resulting in the most change. VEGF expression level per tumor cell ($\nu _A$) now emerges as the most influential parameter. Interestingly, although the simulations are sensitive to tumor cell death parameters, cellular proliferation rates are still more influential in comparison, indicating that the inhibition of cell proliferation—as opposed to inducing cell death—might be what drives the observed changes in xenograft volume in response to treatment with avastin. Figure 11b, d, f reveal that tumor oxygenation is sensitive to high values of $\mu _E$, $\alpha _T$, $\delta _T$ and $\beta _T$ and low values of $\alpha _E$, $\nu _T$ and $\nu _A$. In contrast to the earlier case, the tumor cell death rate $\delta _T$ emerges as the most influential parameter since increased tumor cell death could have the overall effect of increasing tumor oxygen levels.

Fig. 11

Local sensitivity analysis for parameters relating to tumor xenograft growth under treatment with avastin. a, c, e The residual or Euclidean norm between simulated and experimental data (tumor volume time course, see Fig. 6a in main text) is plotted on the y-axis, and percentage variation of the parameters from their baseline values is plotted on the x-axis. b, d, f The predicted maximum partial pressure in the tumor over the time course of the simulation is plotted on the y-axis, and percentage variation of the parameters from their baseline values is plotted on the x-axis

Global Sensitivity Analysis

Although the above analysis provides some information on which parameters are more influential in our model, this analysis is local, with each parameter varied individually, keeping all others fixed. This approach suffers from the limitation that only a small fraction of parameter space is explored and does not take into account the simultaneous variation, and thus possible interactions, of model parameters. We therefore also conduct a global sensitivity analysis on the same set of parameters using the elementary effects method developed by Morris (1991) and implemented in MATLAB by Pianosi et al. (2015). An elementary effect is computed as follows. Consider a model with n independent input parameters $x_i$, $i=1,\ldots ,n$ with each parameter uncertainty interval divided into p equal intervals resulting in an n-dimensional p-level hyperspace. For any given set of parameter values, the elementary effect of the ith input factor on the output y is defined as:

$$\begin{aligned} EE_i= & {} [y(x_1,x_2,\ldots x_i+\Delta ,x_{i+1}\ldots ,x_n) \,-\, y(x_1,x_2,\ldots ,x_n)]/\Delta , \end{aligned}$$

(53)

Fig. 12

Global sensitivity analysis for parameters relating to tumor xenograft growth in the absence of treatment. a Phase plane showing the mean versus standard deviation of elementary effects (EEs). b Bar graphs showing the corresponding global sensitivity indices for the various parameters when the output function is taken to be the residual or Euclidean norm between simulated and experimental data (tumor volume time course, see Fig. 6a in main text). c Phase plane showing the mean versus standard deviation of elementary effects (EEs). d Bar graphs showing the corresponding global sensitivity indices for the various parameters when the output function is taken to be the maximum partial pressure in the tumor over the time course of the simulation

where $\Delta = p/(2(p-1))$. The distribution of elementary effects associated with $x_i$ is then obtained by randomly sampling r different sets of parameters from the prescribed hyperspace. Thus, the Morris method depends not only on p, but also on the sampling number r. The sensitivity measures proposed by Morris are $\mu _i$ and $\sigma _i$—the mean and the standard deviation, respectively, of the distribution of elementary effects for each $x_i$. The mean $\mu _i$ assesses the overall influence of one parameter on the output while the standard deviation $\sigma _i$ assesses the parameter’s higher-order effects, such as interactions with other parameters or nonlinear effects on the output. For instance, a low value of $\sigma _i$ indicates that the effect of $x_i$ is almost independent of the values taken by the other factors. Elementary effects are typically represented graphically in the $(\mu _i,\sigma _i)$ plane, and the most influential parameters appear on the upper right corner.

An alternative and more quantitative measure to determine a parameter $x_i$’s sensitivity is its global index (Ciric et al. 2012), defined as:

$$\begin{aligned} \hbox {GI}_i= & {} \sqrt{(\mu _i)^2 + (\sigma _i)^2}. \end{aligned}$$

(54)

Figure 12a, b reveal that in the control case, the output y—defined as the residual between simulated and experimental data (tumor volume time course)—is most sensitive to $\alpha _T$ (tumor cell proliferation rate sensitivity to activated VEGFR1) and $\mu _E$ (activated VEGFR2-mediated endothelial cell proliferation rate), followed by $\nu _T$ (activated VEGFR1-mediated tumor cell proliferation rate), $\nu _A$ (tumor cell VEGF expression level) and $\alpha _E$ (endothelial cell proliferation rate sensitivity to activated VEGFR2). However, as shown in Fig. 12c, d, when the output is chosen as the maximum level of tumor oxygenation, the most influential parameters emerge as $\alpha _T$ and $\nu _A$, followed by $\delta _T$ (tumor cell death rate), $\alpha _E$ and $\mu _E$. We remark that $\delta _T$ has a high global sensitivity index even though locally the model is insensitive to it.

In the treatment case, Fig. 13a, b reveal that when the output is taken as the residual between simulated and experimental data, the most influential parameters remain $\alpha _T$, $\mu _E$ and $\nu _T$ followed by $\nu _A$ and $\alpha _E$. However, as shown in Fig. 13c, d, when the output is chosen as the maximum level of tumor oxygenation, the most influential parameters emerge as $\alpha _T$, $\delta _T$ and $\alpha _E$, followed by $\nu _A$, $\mu _E$, $\nu _T$ and $\delta _E$.

Fig. 13

Global sensitivity analysis for parameters relating to tumor xenograft growth under treatment with avastin. a Phase plane showing the mean versus standard deviation of elementary effects (EEs). b Bar graphs showing the corresponding global sensitivity indices for the various parameters when the output function is taken to be the residual or Euclidean norm between simulated and experimental data (tumor volume time course, see Fig. 6a in main text). c Phase plane showing the mean versus standard deviation of elementary effects (EEs). d Bar graphs showing the corresponding global sensitivity indices for the various parameters when the output function is taken to be the maximum partial pressure in the tumor over the time course of the simulation