Chemo‐mechanistic multi‐scale model of a three‐dimensional tumor microenvironment to quantify the chemotherapy response of cancer

Abstract

Exploring efficient chemotherapy would benefit from a deeper understanding of the tumor microenvironment (TME) and its role in tumor progression. As in vivo experimental methods are unable to isolate or control individual factors of the TME, and in vitro models often cannot include all the contributing factors, some questions are best addressed with mathematical models of systems biology. In this study, we establish a multi‐scale mathematical model of the TME to simulate three‐dimensional tumor growth and angiogenesis and then implement the model for an array of chemotherapy approaches to elucidate the effect of TME conditions and drug scheduling on controlling tumor progression. The hyperglycemic condition as the most common disorder for cancer patients is considered to evaluate its impact on cancer response to chemotherapy. We show that combining antiangiogenic and anticancer drugs improves the outcome of treatment and can decrease accumulation of the drug in normal tissue and enhance drug delivery to the tumor. Our results demonstrate that although both concurrent and neoadjuvant combination therapies can increase intratumoral drug exposure and therapeutic accuracy, neoadjuvant therapy surpasses this, especially against hyperglycemia. Our model provides mechanistic explanations for clinical observations of tumor progression and response to treatment and establishes a computational framework for exploring better treatment strategies.

Full text

Biotechnol Bioeng. 2021;1–17. wileyonlinelibrary.com/journal/bit © 2021 Wiley Periodicals LLC
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1
Received: 15 January 2021
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Revised: 2 June 2021
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Accepted: 10 June 2021
DOI: 10.1002/bit.27863
ARTICLE
Chemo‐mechanistic multi‐scale model of a three‐dimensional
tumor microenvironment to quantify the chemotherapy
response of cancer
Mohammad R. Nikmaneshi |Bahar Firoozabadi |Aliasghar Mozafari
Department of Mechanical Engineering, Sharif
University of Technology, Tehran, Iran
Correspondence
Bahar Firoozabadi, Department of Mechanical
Engineering, Sharif University of Technology,
Tehran, Iran.
Email: firoozabadi@sharif.edu
Abstract
Exploring efficient chemotherapy would benefit from a deeper understanding of the
tumor microenvironment (TME) and its role in tumor progression. As in vivo ex-
perimental methods are unable to isolate or control individual factors of the TME,
and in vitro models often cannot include all the contributing factors, some questions
are best addressed with mathematical models of systems biology. In this study, we
establish a multi‐scale mathematical model of the TME to simulate three‐
dimensional tumor growth and angiogenesis and then implement the model for an
array of chemotherapy approaches to elucidate the effect of TME conditions and
drug scheduling on controlling tumor progression. The hyperglycemic condition as
the most common disorder for cancer patients is considered to evaluate its impact
on cancer response to chemotherapy. We show that combining antiangiogenic and
anticancer drugs improves the outcome of treatment and can decrease accumula-
tion of the drug in normal tissue and enhance drug delivery to the tumor. Our
results demonstrate that although both concurrent and neoadjuvant combination
therapies can increase intratumoral drug exposure and therapeutic accuracy,
neoadjuvant therapy surpasses this, especially against hyperglycemia. Our model
provides mechanistic explanations for clinical observations of tumor progression
and response to treatment and establishes a computational framework for exploring
better treatment strategies.
KEYWORDS
chemotherapy, combination therapy, hyperglycemia, IFP, tumor microenvironment, vascular
normalization
1|INTRODUCTION
The search for efficient treatments of cancer needs a deep under-
standing of the tumor microenvironment (TME) and its role in tumor
progression. Abnormalities of tumor vasculature can make it difficult
to deliver systemically‐administered drugs to cancer cells (Baish
et al., 2011; Goel et al., 2011; R. K. Jain et al., 2002; Martin
et al., 2019; Stylianopoulos et al., 2018). Angiogenic tumor vessels
are hyperpermeable and nonuniformly distributed, and lymphatic
vessels are either absent or made dysfunctional, and thus unable to
remove interstitial fluid, resulting in a high interstitial fluid pressure
(IFP) in solid tumors (Stylianopoulos et al., 2018). As the high in-
tratumoral IFP is approximately equal to tumor microvascular
pressure (MVP), there is insufficient transvascular pressure gradient
to drive convection of drugs from the blood vessels into the tumor
interstitium. Generally, IFP is high level and approximately uniform
within the tumor but quickly falls to normal levels near the tumor
boundary (Baxter & Jain, 1989; Jain et al., 2007; Netti et al., 1995;

Padera et al., 2002). The lack of intratumor convection limits the
delivery of large nanoparticle drugs that rely on convective trans-
port. Therefore, intratumor IFP causes a severe barrier for drug
delivery and further limits transport through the vasculature to
deeper regions of the tumor.
It has been shown in animal models (Chauhan et al., 2011;Claes
et al., 2008; Fukumura & Jain, 2007;Tongetal.,2004), mathematical
models (Jain et al., 2007; Stylianopoulos & Jain, 2013; Stylianopoulos
et al., 2018) and clinical studies (Chauhan et al., 2011;Claesetal.,2008)
that decreasing vascular permeability by restoring the balance between
pro‐and antiangiogenic signaling can normalize the interstitial fluid of
TME, restoring hydrostatic pressure gradients and increasing drug de-
livery. Such vascular normalization can be induced by antiangiogenic
drugs that block signaling pathways primarily involving vascular en-
dothelial growth factor (VEGF) and its receptors; these drugs reduce the
vessel wall permeability and vessel density, resulting in IFP reduction
(Jain, 2001,2005,2013) and increased convection of drugs from tumor
vessels into tumor interstitium (Böckelmann & Schumacher, 2019;
Dewhirst & Secomb, 2017;Ozturketal.,2015).
In addition to the therapeutic problems caused by heterogeneity
and abnormality of TME, cancer patients are faced with systematic
abnormalities arising from either cancer‐induced metabolism im-
pairments or chemotherapy‐induced disorders (Hershey, 2017;
Jacob & Chowdhury, 2015). Hyperglycemia as the main feature of
diabetes is the most common disorder for cancer patients. Clinical
studies show that the probability of diabetes increases up to 30% in
cancer patients that gets even worse during chemotherapy
(Hershey, 2017). On the other hand, diabetes (primarily type 2) is
associated with increased risk of some cancers including liver, pan-
creas, endometrium, colon and rectum, breast, and bladder
(Giovannucci et al., 2010). Some clinical studies have shown that
glycometabolism disorders induced by chemotherapy cause blood
glucose significantly high or even diabetes mellitus (Ahn et al., 2020;
Hershey, 2017). Hyperglycemia through which glucose‐hungry can-
cer cells are fed with an excess amount of glucose contributes to
oncogenesis, chemoresistance, and apoptosis resistance (Duan
et al., 2014).
Pothuraju et al. (2018) demonstrated that type 2 diabetes con-
tributes to pancreatic cancer development with altered metabolic
pathways. Vasconcelos‐Dos‐Santos et al. (2017) showed that hy-
perglycemia induces aberrant glycosylation, increased cell pro-
liferation, invasion, and tumor progression of colon cancer. Duan
et al. (2014) described the role of hyperglycemia whch contributes to
a more malignant phenotype of cancer cells and leads to drug re-
sistance. The clinical results of Attili et al. (2007) for 119 diabetic
cancer patients support the concept that diabetes is associated with
an increase in mortality and poor response rates to chemotherapy.
Through a clinical study on 1105 patients, of whom 257 (23%) were
diabetic, Kleeff et al. (2016) showed that diabetes is associated with
increased tumor size and reduced survival following pancreatic
cancer resection and adjuvant chemotherapy. Therefore, there is a
close relationship between the treatment response of cancer and
diabetes that needs to be considered in the modeling of TME.
Computational recapitulation of tumor therapy to properly
analyze drug and nutrient distributions requires a comprehensive
mathematical model able to simulate important aspects of the TME,
including spatiotemporal distributions of biochemical and bio-
mechanical factors, relationships between different TME scales in-
cluding molecules, cells, and tissue, morphological heterogeneity of
tumor growth and vasculature. Continuous, discrete and hybrid
continuous‐discrete models have been previously developed to si-
mulate the TME (Dogra et al., 2019; Nikmaneshi et al., 2020). Con-
tinuous models can predict spatiotemporal distributions of drugs and
biomolecules within the TME but ignore the morphological hetero-
geneity (Jain et al., 2007; Kashkooli et al., 2019; Voutouri et al., 2019;
Xu et al., 2016; Yonucu et al., 2017). Discrete models can be used to
analyze tumor vascularization and growth but don't explicitly con-
sider transport of drugs or biomolecules, or their dynamic distribu-
tions (Norton & Popel, 2016; Soltani & Chen, 2013; Zhao
et al., 2007). Hybrid models combine the advantages of discrete and
continuous models and are able to accurately recapitulate many
aspects of TME dynamics and heterogeneities (Nikmaneshi
et al., 2020; Shamsi et al., 2018; Stéphanou et al., 2017; Tang
et al., 2014; Vavourakis et al., 2017; Xu et al., 2016).
In this study, we present a three‐dimensional multi‐scale math-
ematical model of the TME for simulating the dynamics of tumor
growth, angiogenesis, and transport; we use the model to analyze the
chemotherapy response of cancer with hyperglycemia compared to
normoglycemic condition. The model includes coupled molecular,
cellular, and tissue‐size scales, thus recapitulating TME hetero-
geneity and biology during treatment with anticancer and anti-
angiogenic drugs. Specifically, we simulate tumor growth and
angiogenesis and then implement the model for an array of treat-
ment regimens including chemotherapy alone, concurrent combina-
tion therapy of anticancer and antiangiogenic drugs, and neoadjuvant
combination therapy with antiangiogenic drug injected before an-
ticancer drugs.
2|MATERIALS AND METHODS
2.1 |Model definition
We selected an 8 mm cube with 160 × 160 × 160 lattice nodes for
the computational domain of TME. The double hybrid continuous‐
discrete (DHCD) method defined in our previous TME model
(Nikmaneshi et al., 2020) was applied to solve the mathematical
equations of the model. A schematic figure of the multi‐scale TME
including molecular, cellular, and tissue scales is shown in Figure 1.
Two separate three‐dimensional cubic lattices are selected to de-
termine the location of each tumor and endothelial cell in TME, and a
finite‐difference mesh with the same size as the lattices is selected to
determine the biochemical and biomechanical distributions of TME.
Tumor tissue starts growing at the center of the computational do-
main during avascular tumor growth. Sprouts initiate from the prone
areas on primary vessels to start angiogenesis. The angiogenic
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NIKMANESHI ET AL.

vessels penetrate tumor tissue to switch from avascular to vascular
tumor growth. Finally, different chemotherapy approaches are ad-
ministered to control tumor progression.
2.2 |Computational implementation
The computational block diagram of the model is presented in
Figure 2. This chart shows different compartments for the length
scales of the model and the relationships between these compart-
ments. The molecular, cellular, and tissue scales are respectively
shown in green, yellow, and blue. According to this chart, the model
setup steps are as below.
(1) Set initial and boundary conditions. (2) Update molecular
agents on finite difference mesh (O
2
,glucose,andCO
2
fields, ECM
and MMPs fields, VEGF and VEGFR‐2 fields, Ang‐1, Ang‐2 and Tie‐2
fields, anticancer and antiangiogenic drug fields [after treatment]).
(3) Update cellular features on finite difference mesh (cellular vi-
tality and energy fields, update phenotypes of tumor and en-
dothelial cells). (4) Update tissue scale on finite difference mesh, for
hemodynamics, interstitial fluid flow, tumor‐induced solid stress,
and vessel growth and remodeling variables. (5) Update tissue scale
on lattice of tumor cells (TCs) for tumor growth. (6) Update
tissue scale on lattice of endothelia cells (ECs) for angiogenesis. (7)
Update the molecular and then cellular scales based on the updated
tissue scale information.
2.3 |Mathematical model
1 Molecular scale
The TME is extremely complex and includes many biochemical
agents. Through all the TME agents, we focused on the species that
play essential roles to control tumor growth, angiogenesis, and, in
consequence, the dynamics of TME. One of the main criteria for
selecting the current model components is the answer to the ques-
tion of at least how many parameters can complete the signaling
pathway between the components of such a dynamic environment as
TME. In addition, the experimental evidence is helpful to find the best
decision for the contributing parameters. As a computational model,
another main criterion for selecting the model parameters is the
simplification as much as possible to optimize the computational cost.
Therefore, instead of all the TME species, we selected representatives
of the main TME components including growth factors, angiopoietins,
as well as ECM and its degrading enzymes. As a major factor to cause
signaling pathway between tumor and vessels during angiogenesis
and lumenogenesis as well as vascular remodeling, VEGF is selected
to be a representative of tumor‐induced growth factors
(Carmeliet, 2005; Hicklin & Ellis, 2005;Jośko et al., 2000;Kerr,2004;
Melincovici et al., 2018; Rajabi & Mousa, 2017). As representatives of
angiopoietins, Ang‐1andAng‐2 are chosen to elucidate the vascular
maturation and dynamics in response to angiopoietins (Baffert
et al., 2004; Korhonen et al., 2016;Lobovetal.,2002; Reiss, 2010).
MMPsasthemajordegradingenzymesofECMareconsideredtobe
FIGURE 1 Schematic of the three‐dimensional multi‐scale model of tumor microenvironment (TME) including molecular, cellular, and tissue
scales. Different computational approaches are used for the various species of TME. Finite difference method (FDM) to determine
biochemical and biomechanical distributions of TME. Two lattices with the same grid as finite difference mesh to determine the location of each
tumor and endothelial cell in TME. Tumor tissue starts growing at the center of the computational domain (avascular tumor growth),
sprouts initiate from the location of primary vessels (angiogenesis), the angiogenic vessels penetrate tumor tissue (vascular tumor growth).
Finally, the grown tumor is treated with different chemotherapy approaches [Color figure can be viewed at wileyonlinelibrary.com]
NIKMANESHI ET AL.
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3

coupled with ECM (Jabłońska‐Trypućet al., 2016;Shapiro,1998). To
model the tumor response to chemotherapy, an antiangiogenic agent
and an Anticancer drug are chosen as representatives of che-
motherapy drugs.
The time‐dependent concentration distribution of species in the
TME,
c
i
,isgovernedbyEquation(1), which includes convection by
interstitial fluid flow, molecular diffusion, and a reaction term, R
i
.The
vascular compartment can be a source or sink for a given soluble
species, represented as S
i
:
u
c
trcDcRS.( )
,
iiiiiifins 2
∂
∂+∇ = ∇ + +
(1)
FIGURE 2 Numerical chart of the tumor microenvironment (TME) model [Color figure can be viewed at wileyonlinelibrary.com]
4
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NIKMANESHI ET AL.

D
iwhere D
i
is the diffusion coefficient of species i,u
ins
is the
interstitial fluid flow velocity (IFV), and r
f
is the retardation factor
defined by the ratio of the solute velocity to the interstittial fluid
velocity. Species: i= ac (anticancer drug), ag (antiangiogenic agent), G
(glocuse), O
2
(oxygen), CO
2
(carbon dioxid), v (VEGF), a1 (ang‐1), a2
(ang‐2).
A modified pore model was developed to consider the convec-
tional and diffusional transmigration of particles across the angio-
genic vessels (Baxter & Jain, 1989,1990,1991a,1991b; Jain, 1990),
()
SSLppππc
Sc c
(1 ) ( ( ))
().
idi
d
di
iii
P
e
, V p lum ins v lum ins p,
Vp, 1
i
Pi
v
c
σσ
κ
=− ⎡
⎣−− − ⎤
⎦
+−
−
(2)
In this model,
,id
σ
defined in Equation (3) as a function of particle
size of solute,
d
i
, and pore size of porous media,
dp
, is the colloid
osmotic (oncotic) reflection coefficient for solute into plasma and v
σ
is the average oncotic reflection coefficient of plasma proteins,
dv
is
angiogenic vessel diameter,
d
cis the vessel characteristic diameter,
S
V
surface area per unit volume for transvascular exchanges,
p
lum
is
intravascular blood pressure,
p
in
s
is interstitial fluid pressure (IFP),
π
lu
m
and
πin
s
are, respectively, oncotic pressures of the intravascular
plasma and interstitial fluid,
i
κ
is permeability coefficient of the
vessel wall,
L
p
is the hydraulic conductivity of the neo‐vessel wall,
which is defined as a function of the TME growth factors
(Equation 11S of Supporting Information). P
i
defined in Equation (4)
is the transvascular Peclet number to present the ratio of convection
to diffusion across the vessel wall.
In Equation (2), cip, is plasma concentration of species that is
assumed to be constant due to the ratio of venous to arterial plasma
concentration of species is close to 1 (Baxter & Jain, 1989). However,
the plasma concentration of drug is modified by Equation (5)to
model the physiological drug delivery under the influence of drug
clearance. The drug clearance occurs due to blood recirculation into
kidney and liver, modeled by the exponential function of Equation
(5). In Equation (5),
c
p,ac
0
and cp,ag
0
are initial concentrations of the
anticancer and antiangiogenic drugs, and
a
c
τ
and ag
τare the mean
lifetimes of anticancer and antiangiogenic drugs, respectively:
d
d
11
,
ii
d
p
22
σ
⎜⎟
=⎛
⎝
⎜−⎛
⎝−⎞
⎠
⎞
⎠
⎟(3)
P
Lpp ππ(1 ) ( ( ))
,
ii
p
i
vd, lum ins lum ins
σκσ=− ⎡
⎣
⎢−− − ⎤
⎦
⎥(4)
cc t
cc t
exp( / ),
exp( / ).
p,ac p,ac0 ac
p,ag p,ag0 ag
τ
τ
=−
=− (5)
Cellular respiration species are oxygen, glucose, and CO
2
.
Assumed to be a function of cellular vitality, the consumption of
glucose and oxygen as well as the production of CO
2
by TCs are
related by the stoichiometry of the cellular respiration reaction
according to
RR
R
Cellular respiration: C H O 6O 6CO 6H O ATP
6
,
G
612 6 2 2 2
O0
consumption by
TCs
CO 0
production by
TCs
2
2
γυ
γυ
⏟
⏟
+→ + +
==−
=
(6)
0
γ
is the maximum consumption or production rate of the cellular
respiration species and
υ
is cellular vitality function defined in
Equation (19).
VEGF and VEGFR‐2: the hypoxic TCs exposed to oxygen con-
centration below a threshold level, cO
ch
2, secrete VEGF to stimulate
ECs of nearby vessels to sprout. The VEGF also affects vascular
hydraulic conductivity/pore size (Equation 11S of Supporting In-
formation). Once ECs become associated with the tumor tissue, their
VEGFR‐2 receptors are expressed (Gevertz & Torquato, 2006).
Binding to and un‐binding from VEGFR‐2 act as a sink and source for
free VEGF, respectively. VEGF naturally decays in the interstitium.
Therefore, the reaction model of VEGF coupled with VEGFR‐2,
oxygen, and antiangiogenic drug is governed by the set of Equations
(7)–(9). The antiangiogenic drug is assumed to be of tyrosine kinase
inhibitor (TKI) type that binds to VEGFR‐2:
R
Rc
cHkrckrc1
,
cc
vvegf
O
O
ch
production by hypoxic
tumor cells
vv
fv
bound
toVEGFR 2
vv
a
unbound
fromVEGFR 2
vv
natura ldecay
2
2O2
ch O2
ε
⏟⏟
=⎛
⎝
⎜−⎞
⎠
⎟−+ −
⎛
⎝−⎞
⎠
+
‐
−
−


(7a)
R
krc kr
,
v
bvv
fv
bound
to VEGFR 2
vv
a
unbound
from VEGFR 2
⏟
=−
+
−
−
−
 (7b)
dr
dt krc kr k rc
,
f
r
v
f
vv vvv
a
ag v
fag
v
f
=− + −
+−
−(8)
dr
dt krc kr
,
v
a
vv
fvvv
a
=−
+−
(9)
R
v
and
R
v
bare, respectively, reaction rates of free VEGF and VEGF
bound to VEGFR‐2;
c
v
and cv
bare concentrations of free and bound
VEGF, respectively.
R
veg
f
is the rate of production of VEGF by TCs; kv
+
is the binding rate of VEGF to VEGFR‐2;
k
v
−
is the dissociation rate of
VEGF from VEGFR‐2; krag
v
f−is the binding rate of antiangiogenic
agent to VEGFR‐2;
v
ε
is the natural decay rate of VEGF;
r
v
f
is the
concentration of free‐VEGFR‐2, and
r
v
a
is the concentration of active‐
VEGFR‐2 bound to VEGF. As such,
(
)
H
cc
O2
ch O2
−is a Heaviside function
to activate VEFG secretion when oxygen concentration, cO2, falls
below the characteristic value, cO
ch
2.
Ang‐1 and Ang‐2 and their common receptor, Tie‐2: Ang‐1 is se-
creted by ECs, and Ang‐2 is secreted by both ECs associated with
tumor tissue and hypoxic tumor cells (Baffert et al., 2004;
Carmeliet, 2003; Gevertz & Torquato, 2006). There is a competition
between Ang‐1 and Ang‐2 to bind to their common Tie‐2 receptor.
Thus, competitive binding and unbinding of Tie‐2 depletes or pro-
duces, respectively, Ang‐1 and Ang‐2. The reaction models of Ang‐1
and Ang‐2 coupled with Tie‐2 also include natural decay, and are
mathematically modeled through the set of following equations:
NIKMANESHI ET AL.
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5

()
R
ReK c
Kkrc kr c
()
,
a
a1 ang 1 0a a1
2
a
production by endothelial cells
a1 a
fa1
bound
to Tie 2
a1 a1
unbound
from active Tie 2
a1 a1
natural decay
ε=−−+ −
−+
‐
−
−
  


(10a)
R
krc kr
,
fa
a1
ba1 a a1
bound
to Tie 2
a1 a1
unbound
from active Tie 2
=−
+
−
−
−
 


(10b)
() ()
RR R
krc kr c ,
eK c
K
hK c
K
a2 ang 2
v()
production by
Endothelial cells associated with tumor tissue
ang 2
h()
production by
hypoxic tumor cells
a2 a
fa2
bound
to Tie 2
a2 a2
a
unbound
from active Tie 2
a2 a2
natural decay
0aa2 2
a
0aa2 2
a
ε
=+
−+ −
−−−−
+
−
−
−

  


(11a)
R
krc kr
,
a2
ba2 a
fa2
bound
to Tie 2
a2 a2
a
unbound
from active Tie 2
=−
+
−
−
−
 


(11b)
dr
dt krc kr krc kr
,
f
f
aa1 a a1 a1 a1
aa2 a
fa2 a2 a2
a
=− + − +
+−+− (12)
dr
dt krc kr
,
a1
a
a1 a
fa1 a1 a1
a
=−
+− (13)
dr
dt krc kr
,
a2
a
a2 a
fa2 a2 a2
a
=−
+− (14)
R
a
1
,
R
a2,
R
a
1
b
, and
R
a
2
bare, respectively, reaction rates of free Ang‐1
and Ang‐2, and bound Ang‐1 and Ang‐2 to Tie‐2. ca
1
,
c
a
2
,
c
a
1
b
, and ca
2
b
are, respectively, concentrations of free Ang‐1 and Ang‐2, and bound
Ang‐1 and Ang‐2 to Tie‐2.
R
ang
1
−,
R
ang
2
v−, and
R
ang 2
h−are respectively
the secretion rates of Ang‐1 by ECs, secretion rate of Ang‐2 by ECs
associated with tumor tissue, and secretion rate of Ang‐2 by hypoxic
TCs.
e0
and h
0
are, respectively, the characteristic concentration of
ECs in each blood vessel, and the characteristic concentration of TCs,
Kais the carrying capacity coefficient of angiopoietins,
k
a1
+
and ka1
−are,
respectively, Ang‐1 binding rate to and unbinding rate from Tie‐2, ka2
+
and ka
2
−are, respectively, Ang‐2 binding rate to and unbinding rate
from Tie‐2,
a
1
ε
and a
2
εare the natural decay rates of Ang‐1 and Ang‐
2, respectively. ra
fis the concentration of free Tie‐2,
r
a
1
a
and
r
a
2
a
are
concentrations of active Tie‐2 bound to Ang‐1 and Ang‐2, respec-
tively.
To consider fibronectin concentration in the ECM coupled with
matrix metalloproteins (MMPs) secreted by ECs and TCs, we applied
the reaction terms of our previous model (Nikmaneshi et al., 2020).
Anticancer and antiangiogenic chemotherapy drugs: the anticancer
drug is consumed by TCs and ECs and has cytotoxic effects on both
cell types. In this study, the antiangiogenic drug blocks VEGFR‐2; its
concentration decreases as it binds irreversibly to VEGFR‐2. The
drugs are assumed to naturally decay in the interstitium. The reac-
tion models of anticancer and antiangiogenic drugs are presented in
Eqs. 17 and 18, respectively:
R
kc kc c
,
ac ac
TC ac
Uptake
by tumor cells
ac
EC ac
Uptake
by endothelial cells
ac ac
natural decay
υε=− − −
  


(17)
R
krc c
,
r
ag ag v
fag
bound
to VEGFR 2
ag ag
natural decay
v
fε=− −
−
−
 (18)
kac
T
C
and kac
E
C
are uptake rates of anticancer drug by TCs and ECs,
respectively, a
c
ε
and a
g
ε
are the natural decay rates of anticancer and
antiangiogenic drugs, respectively.
c
ac
and
c
ag
are concentrations of
anticancer and antiangiogenic drugs, respectively.
2 Cellular scale
Tumor cells: We implement a modified cellular vitality
(
υ
)/cellular energy (
ψ
) model to consider the effects of oxygen,
glucose, and CO
2
on TC phenotypes. Cellular vitality is increased
with oxygen and glucose and decreased with CO
2
. Cellular energy
representing available units of ATP determines the bioactivity of
TCs (Berk et al., 2000; Nikmaneshi et al., 2020). The mathematical
model of coupled cellular vitality and cellular energy is presented
in the following equations:
()
()
()
c
cc kc
cc
c
cH
.
exp 5 1 ,
cc
O
OO
ch W
g
gg
ch
CO
CO
ch
4
2
22
2
2
CO2CO2
ch
υφ=+++
⎛
⎝
⎜−− ⎞
⎠
⎟
−(19)
()
d
dt kk kc H k H
11 1
.
qa
pa
cac ac ()c()
ch ch
ψυυ
υ
υ
υ
υ
υ
υυ υ υ
=−
+−+−+
−−
(20)
In Equation (19), φis a proportionality coefficient, cO
ch
2,cg
c
h
, and
cCO
ch
2are oxygen, glucose, and carbon dioxide characteristic con-
centrations, respectively (Buchwald, 2009; Nikmaneshi et al., 2020).
(
)
H
cc
CO2CO2
ch
−is a Heaviside function to ensure that CO
2
reduces cel-
lular vitality when it's concentration,
c
CO
2
, exceeds the characteristic
value, cCO
ch
2.
k
W
is a constant to reproduce the Warburg effect of TCs,
which tend to favor metabolism via glycolysis rather than the oxi-
dative phosphorylation, which is the preference of most other cells in
the body. Therefore, if the oxygen concentration of TCs approaches
zero, they can still survive but with very low vitality.
In this model, the TCs with
υ
below c
h
υ
are assumed to be
quiescent and those with
υ
above c
h
υ
are active (Nikmaneshi
et al., 2020). The active TCs need to achieve characteristic energy,
ch
ψ
, before they can proliferate into two new TCs (DeBerardinis
et al., 2008; Nikmaneshi et al., 2020; Skog et al., 1982).
The active TCs produce ATP at a linear rate related to cellular
vitality with a proportional coefficient,
k
a
p
, and also consume cellular
energy based on an M‐M model with maximum rate
k
a
c
and M‐M
constant 1 (Nikmaneshi et al., 2020; Tang et al., 2014). The quiescent
TCs consume ATP according to a M‐M model with maximum rate
k
q
c
and M‐M constant 1. The quiescent TCs with zero cellular energy are
converted to a necrotic phenotype. Indeed, quiescent TCs can be
converted to an active or necrotic state based on cellular vitality and
energy, and active TCs can become quiescent; however, necrotic TCs
cannot be converted to the other phenotypes. In this model, the
anticancer drug is assumed to slow or stop the topo isomerase 2 of
proliferating TCs, thereby reducing cellular energy of the active TCs.
6
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NIKMANESHI ET AL.

According to Equation (20), the effect of anticancer drug toxicity on
the active TCs is imposed by using an M‐M model with a drug‐
dependent maximum rate, kc
ac a
c
, and M‐M constant equal to 1.
Endothelial cells: During angiogenesis, endothelial tip cells (tECs) mi-
grate toward positive gradients of VEGF (del Toro et al., 2010; Jakobsson
et al., 2010;Kimetal.,2018; Nikmaneshi et al., 2020)andstalken-
dothelial cells (sECs) migrate into the tECs‐generated conduits in the
ECM and also proliferate to create lumens of the angiogenic neo‐vessels
(del Toro et al., 2010;Nikmaneshietal.,2020; Wong et al., 2017;Wood
et al., 2012).Thedeathstateisalsoconsidered for sECs based on VEGF
concentration (see Equation 2S of Supporting Information). Moreover,
the sECs can differentiate into tECs in response to high VEGF con-
centration and high ratio of Ang‐2toAng‐1, and thus generate bifurca-
tion branches from the neo‐vessel wall (del Toro et al., 2010;Eichmann&
Simons, 2012). The branching probability function is presented as
( ( ( )))
P
mn ,
ckcc
Br Br Br
log 1 /
v
bBr,a a2
ba1
b
=
+
(21)
where mBr and
n
Br
are constants, and kBr,
a
is a positive constant to
control the impact of the Ang‐2/Ang‐1 ratio on branching.
3 Tissue scale
Development of tumor tissue and neo‐vessel pathways: The tumor
growth and angiogenesis are determined with cellular density of TCs
and tECs, which respectively rely on random‐walk/ECM‐induced
haptotaxis and random‐walk/ECM‐induced haptotaxis/VEGF‐
induced chemotaxis mechanisms. For this aim, we use the same
cellular density equations and their discretization methods as
Nikmaneshi et al. (2020). In addition to biochemical agents, tumor‐
induced solid pressure presents a resistance to the migration of TCs
and tECs (Haessler et al., 2011; Nikmaneshi et al., 2020; Polacheck
et al., 2011; Tang et al., 2014). As the main assumptions to the dy-
namics of TCs and ECs; (I) mother TCs can displace viable cells, but
not necrotic cells, to accommodate their daughters, (II) tECs migrate
toward high VEGF concentration regions and low solid pressure, (III)
tECs cannot penetrate the regions occupied by necrotic TCs. The
fibronectin gradient in the ECM caused by TC‐and tEC‐induced
MMPs supports haptotactic migration of TCs and tECs (Nikmaneshi
et al., 2020).
The other parts of tissue scale including vessel growth and re-
modeling and fluid dynamics of TME are formulated in the Sup-
porting Information file. The model parameters table (Table S) have
been presented in the Supporting Information file.
2.4 |Initial and boundary conditions
The initial concentrations of glucose, oxygen, and carbon‐dioxide
were assumed to be homogeneous. The initial concentrations of
VEGF, Ang‐1, and Ang‐2 were set to zero. The free and active
VEGFR‐2 and Tie‐2 were initialized to zero. At the boundaries of the
computational domain of the TME, a Dirichlet boundary condition
was used for each agent with a value equal to its initial concentra-
tion. The TME was seeded with five tumor cells located at the center
of the computational cube and a hypothetical circular primary
vascular network with a radius of approximately 4 mm. The locations
of initial sprouts on the circle of primary vessels are determined
based on VEGF concentration but spaced randomly according to
NOTCH induction more than 50 µm apart.
The biomechanical factors, including IFV and IFP, intravascular
blood flow velocity and pressure, and WSS were initially set to zero
in the entire computational domain and boundaries. For these
parameters, a Dirichlet boundary condition with zero value was set
on all boundaries of the TME domain. To allow for increased blood
supply and vascular maturation with tumor growth, we assume the
surrounding supply vessels grow and their pressure increase as the
tumor grows: at the inlet of the neo‐vessels where they connect to
the primary vessel, we developed a M‐M model to calculate the inlet
pressure as a function of tumor size;
p
pV V V(/( )
)
inlet m T T
mT
=+
, where
V
T
is tumor volume,
V
T
mis M‐M constant, and
p
m
is the maximum
pressure in the primary vessels, consistent with the range reported
in the literature (Shirinifard et al., 2009; Stéphanou et al., 2017; Zhao
et al., 2007).
3|RESULTS AND DISCUSSION
We started the simulations with a few cells at the center of TME do-
main that are surrounded by primary vessels and kept the tumor and
angiogenic vessels growing for 34 days. The angiogenic vessels grow
toward and penetrate tumor tissue through which a heterogenous and
abnormal state is made. By receiving high levels of growth factors inside
the tumor, the leakiness of tumor vessels increases that results in a high
level of tumor IFP (see Fluid dynamics of TME in the Supporting In-
formation file). In Figure 3, the vascular tumor growth and IFP contours
of simulated TME for Days 26, 30, and 34 are shown. This simulated 3D
tumor is considered as the untreated control case to implement the
chemotherapy schedules.
To elucidate the improvement in anticancer drug delivery, we
considered the treatment regimens shown in Figure 4. Multiple‐
dosage regimens are utilized to maintain the plasma concentration of
drug within a normal therapeutic window. The dimensionless dose
value of anticancer drug is chosen based on the studies of Yonucu
et al. (2017) and Sengupta et al. (2005) for the encapsulated dox-
orubicin. The antiangiogenic regimen is designed based on clinical
administration of antiangiogenic agent injected every 2 days (Wang
& Tang, 2018; Yonucu et al., 2017). As shown in Figure 4, the ther-
apeutic schedule includes administration of anticancer drug started
on Day 26, concurrent therapy of anticancer and antiangiogenic
drugs, and neoadjuvant therapy with antiangiogenic drug on Day 22
and anticancer drug on Day 26. The therapeutic regimens are se-
parately applied on the computational models of tumors with and
without hyperglycemia as the most common secondary disorder for
cancer patients. We assumed that the blood glucose levels at the
hyperglycemia and normoglycemia conditions are 15 and 5.5 mM,
respectively.
The simulation results of tumor size reduction for the cancer
patients with and without hyperglycemia treated with
NIKMANESHI ET AL.
|
7

chemotherapy—case a of Figure 4—are shown in Figure 5.In
agreement with clinical observations, tumor size increases when
hyperglycemia coincides with cancer, Figure 5a. Indeed, hypergly-
cemia elevates the living volume of tumors comprising of pro-
liferative and quiescent TCs, Figure 5b. When a tumor undergoes
chemotherapy, the competition between drug toxicity and hy-
perglycemia through killing and strengthening proliferative TCs
control tumor progression. During chemotherapy, hyperglycemia‐
induced chemoresistance decreases the reduction rate of living TCs,
Figure 5b, and the necrosis rate of TCs, Figure 5c, and, in con-
sequence, increases the progression rate of tumor, Figure 5a. These
results agree with the experimental observations showing that hy-
perglycemia attenuates the antiproliferative effect of chemotherapy
drugs on cancer cells (Li et al., 2019).
To improve drug delivery, combination therapies—cases b and
cofFigure4—have been administered and the results of con-
trolling tumor progression are shown in Figure 6.Accordingto
these results, combination therapies increase the reduction rate of
tumor, which is a consequence of increasing anticancer drug
concentration inside the tumor trough antiangiogenic‐induced
vascular normalization. Based on the results of Figures 6a,d,the
concurrent combination therapy compared with chemotherapy
alone increases the tumor size reduction by around 8% and 5%,
respectively, for normoglycemic and hyperglycemic conditions.
However, the neoadjuvant combination therapy compared with
chemotherapy results in 15% and 25% reductions in the tumor
size, respectively, for normoglycemic and hyperglycemic condi-
tions. It seems that neoadjuvant therapy is more efficient than
concurrent therapy, especially for diabetic cancer patients. In-
deed, compared with the concurrent therapy, the neoadjuvant
therapy with more reduction of angiogenic vessel density can re-
duce the hyperglycemia impact and result in more promising
therapeutic response of diabetic cancer, Figure 6d.
As shown in Figure 6b,e of the living TCs, compared with the
chemotherapy, the concurrent and neoadjuvant therapies decrease
the living TCs through decreasing the proliferative TCs as the target
of anticancer drug. The reduction rates of living TCs in Figure 6b,e
shows that hyperglycemic tumor can benefit sufficiently from
neoadjuvant therapy rather than concurrent therapy. However, in
FIGURE 3 Tumor growth and
angiogenesis of untreated control case with
interstitial fluid pressure (IFP) contours at the
cross‐section of the tumor microenvironment
(TME) domain for Days 26, 30, and 34. Three‐
dimensional images show side and top views
of the simulated vascular tumor. The IFP
values have been normalized [Color figure can
be viewed at wileyonlinelibrary.com]
FIGURE 4 Therapeutic schedule. From top to bottom, regimens
for chemotherapy with anticancer drug (case a), concurrent
combination therapy with antiangiogenic and anticancer drugs (case
b), neoadjuvant combination therapy with antiangiogenic drug
injected before anticancer drug (case c). The dose values of
anticancer and antiangiogenic drugs are nondimensional [Color
figure can be viewed at wileyonlinelibrary.com]
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NIKMANESHI ET AL.

FIGURE 5 Chemotherapy response of cancer under hyperglycemic (HG) and normoglycemic (NG) conditions. (a) total number of tumor
cells, (b) living tumor cells including proliferative and quiescent cells, (c) necrotic tumor cells [Color figure can be viewed at
wileyonlinelibrary.com]
FIGURE 6 Improvement of therapeutic response of cancer with concurrent and neoadjuvant combination therapies for hyperglycemic (HG)
and normoglycemic (NG) conditions. (a) total number of NG tumor, (b) living cells of NG tumor, (c) necrotic cells of NG tumor, (d) total number
of HG tumor, (e) living cells of HG tumor, (f) necrotic cells of HG tumor [Color figure can be viewed at wileyonlinelibrary.com]
NIKMANESHI ET AL.
|
9

terms of normoglycemic condition, neoadjuvant therapy shows just a
typical improvement compared with concurrent therapy.
According to Figure 6c,f, the necrosis rate of TCs is maximized
with concurrent therapy for both hyperglycemic and normoglycemic
conditions. Compared with chemotherapy, the neoadjuvant therapy
shows a very low impact on the necrosis rate of the normoglycemic
tumor, but it significantly enhances the necrosis rate of the hy-
perglycemic tumor.
To explain the reason of differences in cancer response to different
treatments, anticancer drug distributions of the therapeutic cases a–c
for both normoglycemic and hyperglycemic conditions are shown in
Figure 7. Tumor boundaries are indicated with solid lines. For che-
motherapy alone, the drug is significantly distributed outside the tumor,
but slightly penetrates inside the tumor, near the tumor rim where the
IFP is decreased, Figure 7, case a for NG and HG conditions. The drug
barrier effect of high tumor IFP makes for low intratumoral drug ac-
cumulation, Figure 7, case a. In combination therapies, drug extra-
vasation outside the tumor decreases and instead increases inside the
tumor, Figure 7cases b and c. The main reason for this change is the
reduction of tumor IFP through antiangiogenesis‐induced vascular
normalization. The reduction of tumor IFP produces a transvascular
pressure gradient across the tumor vessels restoring drug convection
inside the tumor. Therefore, in the combination therapies, the tumor
size reduction is increased, as can be seen in Figure 6as well, because of
increased drug concentration inside the tumor, Figure 7cases b and c.
As another advantage of combination therapy especially neoadjuvant,
the reduction of angiogenic vessel density and vascular permeability
inhibits drug extravasation in normal tissue, Figure 7,casesbandcfor
both NG and HG conditions.
We defined a “relative intra/extra‐tumoral drug concentration
(INT/EXT‐TD)”parameter that is calculated as the ratio of drug
exposure per volume inside the tumor to outside of it. The spatio-
temporal average values of drug exposure inside and outside the tumor
and INT/EXT‐TD parameter for both normoglycemic and hyperglycemic
conditions are presented in Figure 8. According to these results, the
drug exposure inside the tumor is significantly increased for combina-
tion cases, especially case b; the concurrent combination. It can be the
reason for elevated necrotic TCs of this therapeutic regimen in
Figure 6d,f. As another advantage, the combination cases, especially
neoadjuvant, remarkably decrease the undesirable extratumoral drug
accumulation, Figure 8b. Consequently, the INT/EXT‐TD parameter is
improved with combination therapy, especially case c; the neoadjuvant
combination. By comparing normoglycemic and hyperglycemic condi-
tions, the combination‐induced improvement of INT/EXT‐TD for hy-
perglycemia is more than the normoglycemic condition. It seems that
hyperglycemic tumors with larger volumes (see tumor boundaries in
Figure 7) can take in more vessels through which is supplied a higher
level of drug. Therefore, despite tumor drug exposure of hyperglycemia
being increased, the tumor response to chemotherapy is decreased
through hyperglycemia‐induced chemoresistance (Figure 6).
In Figure 9, the average IFP over whole tumor for both normo-
glycemic (NG) and hyperglycemic (HG) conditions is illustrated. The
values of both NG and HG cases are normalized relative to the max-
imum value of HG condition. For the combination cases, there is a
significant reduction at tumor IFP arising from vascular normalization,
Figure 9. Although the reduction of tumor vessel density and vascular
permeability during vascular normalization reduces the drug supply, the
tumor IFP reduction seems to compensate for drug convection from
vessels into the tumor (Figures 7and 8). In our results, the tumor IFP
reduction of concurrent combination is more than neoadjuvant,
Figure 9, that can increase intratumoral drug extravasation and ex-
posure, Figures 7and 8a. As shown in Figure 9, the chemotherapy alone
FIGURE 7 Drug distribution of tumor microenvironment (TME) for normoglycemic (NG) and hyperglycemic (HG) conditions by the end of
simulation, Day 34. Contours show the normalized concentration of anticancer drug and the black line indicates the tumor boundary in the
cross‐section [Color figure can be viewed at wileyonlinelibrary.com]
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NIKMANESHI ET AL.

can also decrease IFP through the toxicity effect of anticancer drug on
endothelial cells and, in consequence, decrease tumor vessel density.
The tumor IFP reduction of chemotherapy alone seems to be unable to
restore drug convection, Figures 7and 8,casea.Assuch,acomparison
between NG and HG conditions in Figure 9a,b shows that HG condition
can slightly increase tumor IFP by about 5% and, in consequence, HG
can probably increase the resistance against drug delivery from tumor
vessels to tumor tissue.
3.1 |Validation of model predictions with the
experimental observations
For validation of tumor progression, we set the tumor type‐
dependent parameters using experimental data to simulate control
cases of special types of tumors. In this study, the tumor type‐
dependent parameters that can control tumor growth include cel-
lular respiratory reaction rate (
0
γ
), cellular phenotype conversion
FIGURE 8 Drug exposure per volume for different therapeutic regimens including case a (chemotherapy alone), case b (concurrent
combination therapy), and case c (neoadjuvant combination therapy) applied on both normoglycemic (NG) and hyperglycemic (HG) tumors. (a)
spatiotemporal average of intratumoral drug exposure per volume, (b) spatiotemporal average of extratumoral drug exposure per volume, and
(c) relative intra/extra‐tumoral drug concentration (INT/EXT‐TD). The values of (a) and (b) are normalized [Color figure can be viewed at
wileyonlinelibrary.com]
FIGURE 9 Spatial average of tumor interstitial fluid pressure (IFP) on normoglycemic (NG) tumor (a) and hyperglycemic (HG) tumor (b)
from Day 20 to Day 34. Solid lines show untreated tumors and dashed lines show treated tumors. The chemotherapy, concurrent and
neoadjuvant combination therapies are respectively shown by dark blue, orange, and brown for NG condition (a), and by green, violet, and light
blue for HG condition (b). The values of average IF in each panel are normalized with the maximum value of the untreated case under HG
condition [Color figure can be viewed at wileyonlinelibrary.com]
NIKMANESHI ET AL.
|
11

threshold ( c
h
υ
), and cellular energy consumption rates (
k
a
c
and
k
q
c
). For
validation of tumor vessel density under antiangiogenic therapy, the
branching probability constant (mBr) that can control the vascular
density was set to reproduce the control (untreated) case.
Our results are compared with experimental results and other
computational models and show good agreement. The calculated
results for a critical hyperglycemic condition—extreme glucose
concentration of plasma up to 25 mM instead of 5.5 mM—predict
the experimental measurements of Bao et al. (2019)forthewell‐
vascularized tumor, Figure 10a. Both the experimental and com-
putational results of Figure 10a show more rapid growth of tu-
mors in diabetic condition with larger tumor size arising from
increased number of proliferating TCs. For the control case of the
validation of tumor growth in response to critical hyperglycemia,
the tumor type‐dependent parameters were set to simulate a NG
tumor in agreement with the normoglycemic glioblastoma tumor
of Bao et al. (2019). To evaluate how much parameter variation
could be underlying tumor growth results, we changed the values
of model parameters including the degree of glycemia (DG), the
fraction of tumor necrosis (FTN), and branching constant (m
Br
)(to
change vessel density). The related parameters were increased by
50% and 80% of the baseline values (see Supporting Information,
Table. S) and also decreased by 50% and 80%. The tumor growth
rate is most sensitive to DG and least sensitive to FTN. Increasing
DGby80%increasesthesizeofcriticalHGtumor(Figure10a)by
18%, and decreasing DG by 80% results in a 13% reduction for NG
tumor size of Figure 10a.
We also predicted the tumor size reduction in response to
doxorubicin that reproduces the experimental measurements of
Sengupta et al. (2005) (Figure 10b). For the validation of cancer
response to anticancer therapy (Figure 10b), we set the tumor type‐
dependent parameters with the results of Sengupta et al. (2005)to
simulate a control (untreated) case for Lewis lung carcinoma. To
evaluate how much parameter variation could be underlying tumor
response to chemotherapy, we changed the values of model para-
meters including dosage parameters (start time of drug administra-
tion [STD], normalized drug concentration [NDC], frequency of drug
administration [FDA]), DG, and FTN. All parameters except STD were
increased and decreased by 50% and 80% of the baseline values (see
Supporting Information, Table S). STD was selected 2 and 4 days
before and after its baseline value. As the result of this parametric
study, tumor size reduction is most sensitive to DG with a 46% re-
duction by 80% increment of DG and a 20% increment by 80% de-
crement of DG. Variations of other parameters cause changes of less
than 20% for tumor size reduction.
Our model also agrees with the antiangiogenic‐induced reduc-
tion of tumor vessel density measured by Zhou et al. (2019) (treat-
ment duration= 7 days; Figure 10c). For this validation, the branching
probability constant (m
Br
) was set so that the simulated tumor for
the untreated control case has the same tumor vessel density as
colon cancer of Zhou et al. (2019). To evaluate how much parameter
variation could be underlying the reduction of tumor vessel density
in response to antiangiogenic therapy, we changed the values of
model parameters including timing of antiangiogenic drug adminis-
tration (TAG), DG, and FTN. DG and FTN parameters were increased
and decreased by 50% and 80% of the baseline values (see Sup-
porting Information, Table S). TAG was selected 2 and 4 days before
and after its baseline value (see Supporting Information, Table S). As
the result of this parametric study, the tumor vessel density is most
sensitive to TAG with an inverse correlation and maximum change
up to 60% reduction.
The results for IFP and IFV are compared with the results of Jain
et al. (2007) model, Figure 10d. The TME results are in good
agreement with the model of Jain et al. (2007), showing that IFP is
uniformly high within the tumor but rapidly decreases in the normal
surrounding stroma. Both models are also show that IFV is the
maximum at the tumor boundary and decreases within stroma,
Figure 10d. These special distributions of the tumor IFP and IFV are
known as characteristics of vascularized solid tumors.
3.2 |The parameter sensitivity analysis
We also performed a parameter sensitivity analysis for the model by
predicting the main outcomes in response to varying model para-
meters. The main model outcomes chosen to analyze the parameter
sensitivity are tumor IFP, INT/EXT‐TD, tumor vessel density, and
tumor size reduction. The model parameters to sensitivity analysis
include the STD, NDC, FDA, the TAG, DG, and FTN. The control case
to compare the results of parametric study is a NG tumor treated by
anticancer drug alone with STD = day 26 (treatment case a). In this
study, NDC, FDA, DG, and FTN parameters were increased by 50%
(marked with ↑) and 80% (marked with ↑↑) of the baseline values (see
Supporting Information, Table S) and also decreased by 50% (marked
with ↓) and 80% (marked with ↓↓). DG equal to the baseline value for
hyperglycemia, 15 mM, is unmarked. To evaluate the sensitivity of
model outcomes to the schedule of anticancer drug administration,
STD parameters were selected for 2 days (marked with ↓) and 4 days
(marked with ↓↓) before the STD of control case (day 26), and also
2 days (marked with ↑) and 4 days (marked with ↑↑) after the STD of
control case. The effect of antiangiogenic drug administration on the
model outcomes was analyzed by selecting TAG parameter for
2 days (marked with ↓) and 4 days (marked with ↓↓) before the STD
of control case (day 26), at the same time as the eSTD of control case
(unmarked), and also 2 days (marked with ↑) and 4 days (marked with
↑↑) after the STD of control case.
In Figure 11, the results of sensitivity analysis of the model outputs
during 34 days has been illustrated. Figure 11a shows the sensitivity of
tumor IFP to the variations of model parameters including STD, NDC,
FDA, TAG, DG, and FTN. According to these results, tumor IFP is most
sensitive to antiangiogenic therapy and decreases with all the values of
the TAG parameter. Other parameters including STD, NDC, FDA, DG,
and FTN have a low effect on tumor IFP. For the STD parameter, there is
a direct correlation between increasing/decreasing amount and incre-
ment/reduction of tumor IFP. However, NDC and FDA with similar
trends have inverse correlations with tumor IFP. According to these
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|
NIKMANESHI ET AL.

FIGURE 10 Comparison of the model predictions with experimental measurements, (a) tumor growth under critical hyperglycemic (HG)
compared to NG condition for 31 days. (b) normalized tumor size for untreated and doxorubicin treatment for 7 days. (c) Normalized tumor
vessel density (TVD) for untreated and antiangiogenic treatment. (d) relative interstitial fluid pressure (IFP) from the center of tumor to stroma,
1 (in vertical axis) shows the tumor‐stroma interface (left), relative interstitial fluid velocity (IFV) from the center of tumor to stroma, 1 (in
vertical axis) shows the tumor‐stroma interface (right) [Color figure can be viewed at wileyonlinelibrary.com]
results, significant correlations between tumor IFP and FTN were not
found, implying that the tumor IFP is not influenced significantly by the
development of necrosis. In terms of sensitivity of IFP to the degree of
glycemia, the lowest value of DG slightly decreases IFP, however, the
other values of DG representing hyperglycemic condition slightly in-
crease IFP.
The parameter sensitivity analysis for INT/EXT‐TD variable
has been shown in Figure 11b.INT/EXT‐TD is most sensitive to
NIKMANESHI ET AL.
|
13

TAG, and moderately sensitive to FDA and FTN with a direct
correlation. There is no significant correlation between INT/EXT‐
TD and the other parameters including STD, NDC, and DG.
Compared to the control case (NG tumor treated by anticancer
alone), all the schedules of antiangiogenic drug administration
with different values of TAG significantly increase the INT/EXT‐
TD variable.
The results of sensitivity analysis for tumor vessel density has
been shown in Figure 11c. According to these results, tumor vessel
density has the largest sensitivity to antiangiogenic drug
FIGURE 11 Parameter sensitivity analysis for the model by predicting the main outcomes in response to varying model parameters. The
main model outcomes include tumor interstitial fluid pressure (IFP) (ac), INT/EXT‐TD (b), tumor vessel density (c), and tumor size reduction (d).
The model parameters to sensitivity analysis include; start time of drug administration (STD), normalized drug concentration (NDC), frequency
of drug administration (FDA), timing of antiangiogenic drug administration (TAG), degree of glycemia (DG), and fraction of tumor necrosis
(FTN). For NDC, FDA, DG, and FTN parameters, ↑and ↑↑ signs show increment of parameter value by 50% and 80% of the baseline value, ↓and
↓↓ signs show reduction of parameter value by 50% and 80% of the baseline value. For STD and TAG, ↑and ↑↑ signs show 2 and 4 days after
Day 26, ↓and ↓↓ signs show 2 and 4 days before Day 26 [Color figure can be viewed at wileyonlinelibrary.com]
14
|
NIKMANESHI ET AL.

administration with different TAG values. The administration of an-
tiangiogenic drug significantly reduces tumor vessel density. STD and
FTN have a direct correlation to tumor vessel density, but NDC and
FDA have an inverse correlation. In terms of sensitivity of tumor
vessel density to the degree of glycemia, the lowest value of DG
slightly decreases tumor vessel density, and the other values of DG
representing hyperglycemic condition slightly increase tumor vessel
density.
As the major goal of chemotherapy, the sensitivity of tumor size
reduction to the model parameters has been shown in Figure 11d.
tumor size reduction is most sensitive to DG with an inverse cor-
relation, which shows the chemoresistance caused by increasing
blood glucose level. For NDC, FDA, and FTN, there is a direct cor-
relation between increasing/decreasing parameter values and in-
crement/reduction of tumor size reduction. STD has an inverse
correlation with tumor size reduction. All the schedules of anti-
angiogenic drug administration increase tumor size reduction, so that
decreasing the administration TAG improves tumor size reduction.
4|CONCLUSION
Although there have been many models of TME, recapitulating tumor
growth heterogeneity and treatment response of cancer has been a
challenge. The heterogeneous tumor vascularization, tumor dy-
namics and metabolisms all have profound implications for delivery
of blood‐borne drugs, but have not been previously modeled ex-
plicitly. Here, we developed a comprehensive multi‐scale mathema-
tical model that recapitulates the three‐dimensional TME, with
naturally evolving angiogenic vasculature, dynamics and metabolism
of tumor cells, and fluid dynamics to elucidate the effects of com-
bined anticancer/antiangiogenic therapy on drug delivery and tumor
killing. We showed that antiangiogenesis‐induced vascular normal-
ization can decrease tumor IFP through decreased vessel wall per-
meability, and consequently can increase penetration of anticancer
drugs into the tumor. At the same time, this process can decrease the
accumulation of anticancer drugs in the stroma, potentially de-
creasing toxicity. Compared with concurrent combination therapy,
neoadjuvant therapy with a better therapeutic accuracy can improve
tumor killing, especially for the hyperglycemic tumor. The results
show that despite a better drug delivery for hyperglycemic condi-
tions, hyperglycemia induces a chemoresistance to decrease cancer
treatment response. The results show that combination therapy,
especially neoadjuvant, is necessary to resolve the hyperglycemia
chemoresistance. This model can be applied for any type of cancer
under different environmental conditions to simulate the tumor re-
sponse to different drugs either separately or in combination.
AUTHOR CONTRIBUTIONS
Mohammad R. Nikmaneshi and Bahar Firoozabadi contributed to the
design and implementation of the research, to the analysis of the
results and to the writing of the manuscript. Mohammad R.
Nikmaneshi designed and developed the theoretical framework,
made the simulations, analyzed the results and wrote the article with
support from Bahar Firoozabadi. Bahar Firoozabadi supervised the
findings of this study and directed the project. Aliasghar Mozafari
helped supervise the project.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from
the corresponding author upon reasonable request.
ORCID
Bahar Firoozabadi http://orcid.org/0000-0002-4774-0896
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SUPPORTING INFORMATION
Additional Supporting Information may be found online in the
supporting information tab for this article.
How to cite this article: Nikmaneshi, M. R., Firoozab adi, B., &
Mozafari, A. (2021). Chemo‐mechanistic multi‐scale model of
a three‐dimensional tumor microenvironment to quantify the
chemotherapy response of cancer. Biotechnology and
Bioengineering,1–17. https://doi.org/10.1002/bit.27863
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