Table 1.

Summary of variables and parameters used in the model.

Variable	Definition
v(x, y, t)	Concentration of VEGF (μM)
a 1(x, y, t)	Concentration of Ang-1 (μM)
a 2(x, y, t)	Concentration of Ang-2 (μM)
r v0(x, y, t)	Concentration of unbound VEGFR-2 (μM)
r a0(x, y, t)	Concentration of unbound Tie-2 (μM)
r v(x, y, t)	Concentration of VEGFR-2 bound by VEGF (μM)
r a1(x, y, t)	Concentration of Tie-2 bound by Ang-1 (μM)
r a2(x, y, t)	Concentration of Tie-2 bound by Ang-2 (μM)
e i(x, y, t)	EC indicator function
h i(x, y, t)	Hypoxic cell indicator function
p i(x, y, t)	Proliferative cell indicator function
n i(x, y, t)	Necrotic cell indicator function
h(x, y, t)	Concentration of hypoxic cells (μM)
PDE parameters	Definition	Value
D v	Diffusion coefficient of VEGF	D v = 3.6 × 10−4 mm2/h
D a2	Diffusion coefficient of Ang-2	D a2 = 3.6 × 10−4 mm2/h
b v	Production rate of VEGF by hypoxic cells	b v = 0.05 h−1
b a1	Production rate of Ang-1 by ECs	b a1 = 0.01 h−1
b a2	Production rate of Ang-2 by ECs	b a2 = 0.08 h−1
${\overline{b}}_{a2}$	Production rate of Ang-2 by hypoxic cells	${\overline{b}}_{a2}=0.05$ h−1
μ v	Decay rate of VEGF	μ v = 0.001 h−1
μ a1	Decay rate of Ang-1	μ a1 = 0.003 h−1
μ a2	Decay rate of Ang-2	μ a2 = 0.002 h−1
k 0	Association rate of VEGF/VEGFR-2	k 0 = 46.8 μM−1·h−1
k −0	Dissociation rate of VEGF/VEGFR-2	k −0 = 0.2268 h−1
k 1	Association rate of Ang-1/Tie-2	k 1 = 36 μM−1·h−1
k −1	Dissociation rate of Ang-1/Tie-2	k −1 = 0.1332 h−1
k 2	Association rate of Ang-2/Tie-2	k 2 = 41.7 μM−1·h−1
k −2	Dissociation rate of Ang-2/Tie-2	k −2 = 0.108 h−1
K v	Carrying capacity of VEGF	K v = 10−2  μM
K a	Carrying capacity of angiopoietins	K a = 1.5 × 10−2  μM
e 0	Endothelial cell concentration per blood vessel	e 0 = 10−4  μM
Treatment parameters	Definition	Value
T 1	AI treatment parameter is b v/T 1	T 1 = 10,100,1000
T 2	Fraction of proliferative cells killed by cytotoxic agent	T 2 = 0.24,0.34,0.44
T 3	Fraction of angiogenic vessels destroyed by VDA	T 3 = 0.1,0.3,0.6,0.9