Module-based multiscale simulation of angiogenesis in skeletal muscle

Article (PDF Available)inTheoretical Biology and Medical Modelling 8(1):6 · April 2011with36 Reads
DOI: 10.1186/1742-4682-8-6 · Source: PubMed
Abstract
Mathematical modeling of angiogenesis has been gaining momentum as a means to shed new light on the biological complexity underlying blood vessel growth. A variety of computational models have been developed, each focusing on different aspects of the angiogenesis process and occurring at different biological scales, ranging from the molecular to the tissue levels. Integration of models at different scales is a challenging and currently unsolved problem. We present an object-oriented module-based computational integration strategy to build a multiscale model of angiogenesis that links currently available models. As an example case, we use this approach to integrate modules representing microvascular blood flow, oxygen transport, vascular endothelial growth factor transport and endothelial cell behavior (sensing, migration and proliferation). Modeling methodologies in these modules include algebraic equations, partial differential equations and agent-based models with complex logical rules. We apply this integrated model to simulate exercise-induced angiogenesis in skeletal muscle. The simulation results compare capillary growth patterns between different exercise conditions for a single bout of exercise. Results demonstrate how the computational infrastructure can effectively integrate multiple modules by coordinating their connectivity and data exchange. Model parameterization offers simulation flexibility and a platform for performing sensitivity analysis. This systems biology strategy can be applied to larger scale integration of computational models of angiogenesis in skeletal muscle, or other complex processes in other tissues under physiological and pathological conditions.
RESEARCH Open Access
Module-based multiscale simulation of
angiogenesis in skeletal muscle
Gang Liu
1*
, Amina A Qutub
2
, Prakash Vempati
1
, Feilim Mac Gabhann
3
and Aleksander S Popel
1
* Correspondence: gangliu@jhmi.
edu
1
Systems Biology Laboratory,
Department of Biomedical
Engineering, School of Medicine,
Johns Hopkins University,
Baltimore, MD 21205, USA
Full list of author information is
available at the end of the article
Abstract
Background: Mathematical modeling of angiogenesis has been gaining momentum
as a means to shed new light on the biological complexity underlying blood vessel
growth. A variety of computational models have been developed, each focusing on
different aspects of the angiogenesis process and occurring at different biological
scales, ranging from the molecular to the tissue levels. Integration of models at
different scales is a challenging and currently unsolved problem.
Results: We present an object-oriented module-based computational integration
strategy to build a multiscale model of angiogenesis that links currently available
models. As an example case, we use this approach to integrate modules
representing microvascular blood flow, oxygen transport, vascular endothelial growth
factor transport and endothelial cell behavior (sensing, migration and proliferation).
Modeling methodologies in these modules include algebraic equations, partial
differential equations and agent-based models with complex logical rules. We apply
this integrated model to simulate exercise-induced angiogenesis in skeletal muscle.
The simulation results compare capillary growth patterns betw een different exercise
conditions for a single bout of exercise. Results demonstrate how the computational
infrastructure can effectively integrate multiple modules by coordinating their
connectivity and data exchange. Model parameterization offers simulation flexibility
and a platfor m for performing sensitivity analysis.
Conclusions: This systems biology strategy can be applied to larger scale integration
of computationa l models of angiogenesis in skeletal muscle, or other complex
processes in other tissues under physiolog ical and pathological conditions.
Background
Angiogenesis is a complex process whereby new capillaries are fo rmed from pre-exist-
ing micro vasculature. It p lays important roles in many physiological processes includ-
ing embry onic development, wound healing and exercise-induced vascular adaptation.
In such processes, robust control of capillary growth leads to new healthy pattern of
physiological vessel network that matches the metabolic demands of development,
wound repair, or exercise [1]. In contrast, excessive or insufficient growth of blood ves-
sels is associated w ith an array of pathophysiological processes a nd diseases, among
which are malignant tumor growth, peripheral artery disease, diabetic retinopathy, and
rheumatoid arthritis [1].
Systems-level studies of angiogenesis in physiological and pathophysiological condi-
tions improve our quantitative understanding of the process and hence aid in
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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© 2011 Liu et al; licensee BioMed Central Ltd. This is an Open Access article distribut ed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
therapeutic design. Extensive experimental studies of angiogenesis over the past two
decades have revealed that the angiogen esis process is comprised of a series of e vents
at multiple biological organization levels from molecules to cells, tissues, a nd organs.
For examp le, as a first approximation, exercise-induced angiogenesis can be described
as a sequence of the following events: i) Exercise increa ses oxygen consumption in
tissue, followed by increased blood flow in the vasculature, thus affecting convection-
diffusion oxygen transport processes [2]; ii) As exercise continues, insufficient oxygen
delivery t o the tissue leads to tissue cellular hypoxia, which results in activation of the
transcription factor hypoxia-inducible factor 1a (HIF1a) [3] and the transcription
coactivator peroxisome-proliferator-activated-receptor-gamma coactivator 1a (PGC1a)
[4]; iii) These factors induce the upregula tion of vascular endothelial growth factor
(VEGF) expression [5]. VEGF is secreted from myocytes (and possibly stromal cells),
diffuses through the interstitial space, and binds to VEGF receptors (VEGFRs) on
microvascular endothelial cells; concomitantly, endothelial cell expression of VEGFRs
is also altered [6]; iv) The increase in VEGF and VEGFR concentration and possibly
VEGF gradients results in activation of endothelial cells and cause capillary sprouting.
Thus new capillaries and anastomoses form and new capillary network patterns
develop [7]; v) After exercise, VEGF and VEGFR ex pression remain elevated for a lim-
ited time a nd thereafter return to basal le vels [6]. The signaling set in motion causes
blood vessel remodeling to continue after exercise. Thus, the time scales of individual
events range from seconds in oxygen convection-diffusion processes to hours in VEGF
reaction-diffusion p rocesse s, to days or weeks in capillary s prouting processes. Spatial
scales vary from nanometers at the molecular level to microns at the cellular level, to
millimetres or centimetres at the tissue level.
The complexity of angiogenesis is a function not only of the multiscale characteris-
tics in temporal and spatial domains, but also o f the combinatorial i nteractions
between key biological components across organizational levels. At the molecular level,
multiple HIF-associated molecules and hundreds of genes activated by HIF form a
complex transcriptional regulatory network [8]. Six isoforms of VEGF-A
(VEGF
121
,
145
,
165
,
183
,
189
,
206
), three VEGFRs (VEGFR-1, -2, -3) and two coreceptors (neu-
ropilin-1 and -2) constitute a complex ligand-receptor interaction network, regulating
intracellular signaling and determining cellular response [9]. In addition, other VEGF
proteins, like placental growth factor (PlGF) and VEGF-B, -C, -D, compete with
VEGF-A for some of the same receptors. Matrix meta lloproteinases (MMPs) also form
a key molecular family with a pproxi mately 30 mem bers; MMPs are capable of proteo-
lyzing components of the extracellular matrix (ECM) thus decreasing the physic al bar-
riers encountered by a tip endothelial cell leading a nascent capillary sprout [10]. At
the cellular level, endothelial cell activation, migration and proliferation are driven by
local growt h factor concentrations and gradients. Capillary sprouting is also governed
by the interaction between a tip cell and its following stalk cells, and by ce ll adhesion
to the ECM [1 1]. In addition, parenchymal cells, precursor cells and stromal cells, as
well as the ECM, constitute the nascent sprout micro environment, influencing
endothelial cell signaling, adhesion, proliferation and migration.
Mathematical and computational models of angiogenesis have become useful tools to
represent this level of biological complexity and s hed new light on key control
mechanisms. In particular, computational mode ling of tumor-induced angiogenesis has
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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been an active area of research over the past two decades and has also been extensively
reviewed [12-15]. Here we giv e a brief overview of the angiogenesis models relevant to
building a multiscale model of angiogenesis in skeletal muscle using different modeling
methodologies. The models can be classified into continuous, discrete and hybrid cate-
gories. Continuum models of growth factor activity often applied molecular-detailed
reaction and reaction-diffusion differen tial equations. These models have been used to
describe many aspects of angiogenesis, e.g., host tissue distrib ution of a chemotactic
factor following its secretion from a tumor [16], VEGF-VEGFR interactions [17], a
fibroblast growth factor-binding network [18], whole-bo dy compartmental distribution
of VEGF under exercise and peri pheral artery di sease conditions [19,20], the contribu-
tion of endothelial progenitor cells to circulation of VEGF in organs and their effects
on tumor growth and angiogenesis [21], and a VEGF reaction-transport model in ske-
letal muscle [22]. Models of other angiogenesis-associated pro teins such as MMP2 and
MMP9 also have been developed [23,24]. By describing capillary networks in terms of
endothelial cell densities, continuum mo dels have also been developed to represent
tumor-induced capillary growth [25-27] and the wound healing process [28]. Discrete
models such as cellular automata [29], cellular-Potts model [30], and agent-based mod-
els [31-33] have been developed to describe tissue behavior stemming from the interac-
tion between cells, extracellular proteins and the microenvironment. These cell-based
models offer unique capability of representing a nd interpreting blood vessel growth
pattern as an emergent property of the interactions of many individual cells and their
local microenvironment. By combining the continuum approach with the cell-based
modeling approach, hybrid modeling can be used to describe the in vivo vascular
structure along with detailed molecular distributions [34-37], providing appropriate
computational resolution across various scales.
With a number of computational models currently avail able to describe different
aspects of angiogenesis, integration of existing models along with new biological infor-
mation is a promising strategy to build a complex multiscale model [38,39]. While cur-
rent advances mainly focus on the representation format of molecular interaction
models (e.g., XML-based notation) and dynamic integration of these models (e.g.,
Cytosolve [40]), few strategies exist to combine existing models at multiple scales with
mixed methodologies. Here we describe our developm ent of a novel com putational
infrastructure to coordinate and integrate modules of angiogenesis across various scales
of biological organization and spatial resolution. Using this approach can significantly
reduce model development time an d avoid repetitive development efforts. These mod-
ules can be adapted from p reviously-developed mathematical models. Our laboratory
has developed a number of angiogenesis models including: oxygen transport [41];
VEGF reaction-diffusion [22]; capillary sprouting [33]; FGF-FGFR ligand-receptor bind-
ing kinetics [18]; MMP proteolysis [23,24,42]; and MMP-mediated VEGF releas e from
the ECM [43]. We show results of a test case , which integrated a blood flow model, an
oxygen transport model, a VEGF transport model and a cell-based capillary sprouting
model. With the use of Java Native Interface functions, previously-developed angiogen-
esis models were redesigned as pluggable modules and integrated into the angiogen-
esis modeling environment. Another advantage of this simulation infrastructure is its
flexibility, allowing integra tion of models written using differen t simulation techniques
and different programming languages. Note that the primary aim of this study i s
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building methodology for multiscale modeling, rather than obtaining novel physio logi-
cal results; detailed simulations of skeletal muscle angiogenesis and comparison to
experimental data will be presented elsewhere.
The computational scheme presented here fits into the Physiome Project defined as a
computational framework allowing the integration of models and databases that
intends to enhance the descriptive, integrative and quantitative understanding of the
functions of cells, tissues and organs in human bod y [44-46]. Integral parts of the Phy-
siomeProjectaretheCardiacPhysiome[47],theMicrocirculatonPhysiome[48,49],
and the EuHeart project http://www.euheart.eu/, which are aimed at sp ecific organs or
physiological systems. The Virtual Physiological Human project is also aimed at a
quantitative description of the entire human [50-52]. To achieve the goals of these pro-
jects, it is essential to share computational models between a variety of modeling
methodologies, computational platforms, and computer languages and incorporate
them into integrative models. One approach in the past decade is to develop XML-
based m arkup languages to facilitate model representation and exchange. The two
most-accepted formats, SBML [53] and CellML [54], are designed to describe bio-
chemical reaction networks in compartmental systems expressed by ordinary differen-
tial equations (i.e., they have no spatial description). FieldML [55] allowing for spatial
description is under development. Alternatively, the o bject-oriented modeling metho-
dology provides a strategy to describe the biological organizations and flexible solution
to integrate curre ntly available models. For example, universal modeling language
(UML) [56,57] and other meta-languages such as E-cell [58] have been proposed. How-
ever, the robustness of the integration of external models is dependent on the interface
of these meta-languages. In the current study, we propose to use a natural object-
oriented language, Java, as a modeling language to design the integration controller
and link currently available modules at different scales.
Systems and Methods
We describe a computational platform capable of linking any number of modules. In
the particular example of s keletal muscle angiogenesis, we integrate four modules:
microvascular blood flow; oxygen transport; VEGF ligand-receptor interactions and
transport; and a cell module describing capillary sprout formation. These four modules
use diverse modelin g metho dologies: algebraic equations ( blood flow), partial differ en-
tial eq uations (PDEs, oxygen and VEGF t ransport) and agent-base d m odeling (ABM,
cell model). An overview of the simulation scheme is shown i n Figure 1A. Brie fly, the
model initiates with the input of a three-dimensional (3D) muscle tissue geometry that
includes muscle fibers and a microvascular network; rat extensor digitorum longus
(EDL) muscle is used as a prototype, as previously described [22]. This tissue geometry
is first used to calculate blood flow in the vascular network, and then in the computa-
tion of oxygen distribution in the vascular and extravascular space, followed by simula-
tion of VEGF distribution in the interstitial space and on the endothelial surface, and
finally simulation of capillary sp routing and remodeling of the vascular network. Blood
flow and hematocr it are simulated using the two-phase continuum model proposed b y
Pries et al [59] . Oxygen transport model [60] is used to calculate the spatial distrib u-
tion of oxygen tension throughout the tissue. VEGF secretion from myocytes through
an o xygen-dependent pathway is described by an experiment-based oxygen-dependent
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Angiogenesis Process
O2
Ļ
HIF&PGC1Į
Ĺ
VEGF
Ĺ
Capillary
Formation
Flow
Ĺ
Geometr
y
Stimulus
A
B
Shared Object library (SO) (in Linux)
Java Class
(Flow Module)
JNI to C
Wrapper
Fortran Codes
(Flow Module)
C to Fortran
Wrapper
JNI Func.
SO library
Java Class
(O2/VEGF Module)
C/C++ Codes
(O2/VEGF Module)
JNI to C
Wrapper
JNI Func.
Initial Geometry
File
New Geometry
File
Parameter
Database File
Angiogenesis Modeling Controller (Java-based)
Flow Module
in Fortran
VEGF Module
in C/C++
Oxygen Module
in C/C++
Process
JNI PLUGINS
Exception
IO Bios
y
stem
Cell Module
In Java
Algebraic Equations PDEs
PDEs Agent-based modeling
Figure 1 Schematics of Mod ule-based Mulitscale Angiogenesis Modeling Methodology. A) Skeletal
muscle angiogenesis is modeled as a multi-step process. It starts with a blood flow simulation followed by
a simulation of oxygen convection-transport process. Using O
2
tissue distribution, VEGF secretion by
myocytes is computed as a function of oxygen-dependent transcription factors HIF1a and PGC1a; then a
VEGF reaction-transport process is computed. Lastly, capillary formation is simulated based on VEGF
concentration and gradients. Feedback loops increase the complexity of the model since a new geometry
with nascent vessels will affect blood flow conditions, tissue hypoxia, and VEGF secretion and distributions.
All four processes are simulated using a variety of modeling techniques and languages. We use Java as the
language for modeling the controller, and apply JNI plugins to link these modules together. The controller
is composed of four sub-packages, including Process, Biosystems, IO and Exceptions. B) Communications
between different modules and Java codes in core package are implemented by transferring each module
into a shared object library (SO file in Linux). Upper panel shows that two wrapper files (includes Java-to-C
and C-to-Fortran wrapper) are written to communicate between the flow Java class defined in the
controller and the Fortran flow module, to call the flow module in Fortran. Lower panel shows that a JNI C
wrapper is required to transfer the data between the modeling controller (in Java) and the Oxygen/VEGF
module (in C/C++).
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transfer function dependent on the factors HIF1a and PGC1a (details are below).
A modified VEGF reaction-diffusion model [22,61] is used to predict the spat ial VEGF
distribution in tissue interstitial space and at the surface of the endothelium. Using our
agent-based model with this VEGF concentration profile defined as input [33], we
further compute elongation, proliferation and migration of endothelial cells forming
capillary sprouts. The result is a new capillary network. In turn, this new structure
feeds back into the integrated model as an updated vascular geometry, and starts a
new cycle with the flow model, oxygen transport model and VEGF reaction-diffusion
model, thus simulating the dynamics of the angi ogenesis process. Governing equations
and a brief description of each individual module are given below.
Skeletal Muscle Tissue Geometry
A 3D representation of muscle tissue structure is constructed using a previously-
described algorithm [22]; it includes cylindrical fibers arranged in regular arrays and a
network of c apillaries, small precapillary arterioles and postcapillary venules. The
dimensions of the tissue studied are 200 μm width (x-axis), 208 μm heig ht (y-axis) and
800 μm length (z-a xis). The fiber and v ascular geome try can be specified using differ-
ent methods, includi ng tissue-specific geometries with irregular- shaped fibers obtained
from in vivo imaging; tissue dimensions can also be extended.
Flow Module
The in vivo hemorheological model [59,60] is applied to calculate the distribution of
blood flow rate ( Q) and discharge hematocrit (H
D
), among all the capillary segments
under steady state conditions during exercise. The governing equations are derived
from the mass conservation law for volumetric blood flow rate and red blood cell flow
rate at the j
th
node (vascular bifurcation), as follows:
i
Q
i
j
=
0
(1)
i
H
Di
j
Q
i
j
=
0
(2)
Here Q
ij
is the volumetric flow rate in vessel segment ij, a cylinder whose ends are
the i
th
and j
th
nodes of the network. Flow rate is calculated as Q
ij
= πR
4
( p
i
-p
j
)/(8hL),
where p
j
is the hydrodynamic pressure at node j, R and L are the radius and length of
the segment, and h is the apparent viscosity which is a function of R and H
D
(Fah-
raeus-Lindqvist effect). These equations are supplemented by the empirical equations
governing r ed blood cell-pl asma separation at vascular bifurcations. The system of
nonlinear algebraic equations for all N segments is sol ved with respect to pressure and
discharge hematocrit, from which flow in each segment is calculated.
Oxygen Module
Oxygen delivery from the microvasculature to ske letal muscle myocytes is one of the
key functions of microcirculation. During exercise, oxygen consumption may increase
many folds compared to resting state, affecting both extravascular and intravascular
oxygen transport. The oxygen model consists of two partial differential equations,
Eqns. 3 and 4, governing extravascular and intravascular oxygen transport, respectively,
assuming muscle fibers and interstitial space are a single tissue phase [41,60].
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Oxygen tension in the tissue, P
O2
= P(x,y,z,t), is governed by free oxygen diffusion,
myoglobin-facilitated diffusion, and oxygen consumption by tissue cells:
P
t
1+
C
Mb
bind
α
tis
S
Mb
P
= D
O
2
2
P+
1
α
tis
D
Mb
C
Mb
bind
(
S
Mb
P
P)
1
α
tis
M
c
P
(
P + P
crit
)
(3)
Here
D
O
2
and D
Mb
are the diffusivities of oxygen and myoglobin in tissue respec-
tively; S
Mb
is the oxygen-myoglobin saturation; a
tis
is the oxygen solubility in tissue;
C
Mb
bind
is th e binding capacity of myoglobin with oxygen; M
c
is th e oxygen consumption
rate coefficient for Michaelis-Menten kinet ics; P
crit
is the critical P
O2
at which oxygen
consumption equals to 50% of M
c
;andS
Mb
is defined as P/(P+P
50,Mb
)assumingthe
local binding equilibrium between oxygen and myoglobin, where P
50,Mb
is the P
O2
necessary for 50% myoglobin oxygen saturation.
Oxygen transport in the blood vessels is governed by:
v
b
[H
D
C
RBC
bind
S
RB
C
Hb
P
b
+ H
T
α
RBC
+(1 H
T
)α
pl
]
P
b
ξ
+
2
R
J
wall
=
0
(4)
Here
S
RB
C
H
b
is the oxygen-hemoglobin saturation in blood vessel; a
RBC
and a
pl
are
oxygen solubility in red blood cell and plasma, respectively; P
b
is the oxygen tension in
blood plasma; ν
b
is the mean blood velocity (ν
b
= Q/(πR
2
)); H
T
and H
D
are the tube
and discharge hematocrit, calculated from blood flow model;
C
RB
C
bind
is binding capacity
of hemoglobin w ith oxygen; ξisthedistancealongavessels longitudinal axis; J
wall
is
the capillary wall flux; and
S
RB
C
H
b
is defined as
P
n
h
/(P
n
h
+ P
n
h
50
,
Hb
)
assuming the b inding
equilibrium between oxygen and hemoglobin, where P
50,Hb
is the P
O2
necessary for
50% hemoglobin oxygen saturation.
In addition, continuity of oxygen flux at the interf ace between blood vessels and tis-
sue yields:
J
wall
= (α
tis
D
O
2
+ D
Mb
C
Mb
bind
S
Mb
P
b
)
P
b
n
= k
0
(
P
b
P
wall
)
(5)
where n is the unit normal vector, P
wall
is the local P
O2
at the vessel wall and k
0
is
the mass transfer coefficient estimated from an empirical equation k
0
= 3.15+3. 26H
T
-
9.71
S
RB
C
H
b
+9.74(H
T
)
2
+ 8.54(
S
RB
C
H
b
)
2
. The system of nonlinear partial differential equations
was solved using the finite difference method, with a grid size of 1 micron as described
in [60].
VEGF module
VEGF is the most-studied molecular factor involved in angiogenesis, including exer-
cise-induced angiogenesis. Among several splice isoforms in the VEGF family, VEGF
120
and VEGF
164
(in rodents; human isoforms are VEGF
121
and VEGF
165
) are considered
to be the major pro-angiogenic cytokines that induce proliferation and migration of
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endothelial cells. The molecular weights of VEGF
164
and VEGF
120
are 45 and 36 kDa
respectively and thus their diffusion coefficients are slightly different; in addition,
VEGF
164
binds the heparan sulfate proteoglycans (HSPGs) while VEGF
120
does not
and thus the shorter isoform diffuses more freely through the ECM.
A reaction-diffusion model [22] is used to predict molecular distribution in the inter-
stitial space and on the endothelial surface. The governing equati ons for VEGF
164
and
VEGF
120
are:
C
V164
t
= D
V164
2
C
V164
+
k
o
ff
,V164,H
C
V164
H
k
on,V164,H
C
V164
C
H
(6)
C
V120
t
= D
V120
2
C
V12
0
(7)
where C
V164
,C
V120
, C
H
and C
V164H
are the concentrations of VEGF
164
,VEGF
120
,
HSPG and VEGF
164
·HSPG complex; D
V164
and D
V120
are the diffusion coefficients of
VEGF
164
and VEGF
120
; and k
on,V164,H
and k
off,V164,H
are the association and dissociation
rate consta nts b etween VEGF
164
and HSPG. The boundary conditions for VEGF
164
and VEGF
120
at the surfaces of muscle fibers and endothelial cells, and the complete
details of ligand-receptor interactions, were described in [61].
The model describes the secretion of two VEGF is oforms from the muscle fibers,
molecular transport of each isoform in the interstitial space, binding of VEGF
164
to
HSPG in the ECM, VEGF
164/120
binding to VEGFR2 at the endothelial cell surface and
internalization of these ligand-receptor complexes. The model also consi ders VEGFR1
and neuropilin-1 (NRP1) coreceptor binding with VEGF ligands.
We previously applied an empirical equation to describe the relationship between
P
O2
and VEGF secretio n rate to estimate local fiber VEGF secretion [22]. The empiri-
cal relationship was d erived by combining exper imentally-based r elat ionships between
intracellular HIF1a concentration and P
O2
in vitro,andHIF1a concentration and
VEGF secretion in vivo in skeletal muscle. However, PGC1a has rec ently been found
to be another important regulator of VEGF in exercise [4,62]. It was first discovered as
a cold-inducible transcriptional coactivator for nuclear hormone receptors in brown fat
and an enhancer of mitochondria l metabolism and function [63,64]. Recently, a series
of experimental studies [65-67] have shown that VEGF gene expression and protein
levels are highly dependent on the presence and concen tration of PGC1a,through
HIF-independent and HIF-dependent pathways. Thus, w e have modified t he equation
by incorporating the effect of PGC1a:
S
VEGF
=
S
B,VEGF
,forP
O2
20 mmHg
S
B,VEGF
×
1+(α
max
HIF
1) ×
20 P
O2
19
3
S
B
,
VEGF
× α
max
HIF
,forP
O2
< 1 mmHg
(8)
Here S
VEGF
is the VEGF secretion rate, S
B,VEGF
is basal secretion rate at normoxic levels
of [HIF1a]. It is defined as a function of [PGC1a], written as a sigmoidal form,
S
B,VEGF
= S
0,VEGF
× [A(
[PGC1α]
n
p
k
h
+
[
PGC1α
]
n
p
)+B
]
,where[PGC1a]isthePGC1a concentration
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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normalized relative to normoxic expression for wild type skeletal muscle, n
p
is the Hill con-
stant, and k
h
, A, and B are empirical constants. S
0,VEGF
is defined as basal VEGF secretion
rate at normoxic levels of [HIF1a] and [PGC1a] for wild type skeletal muscle. Equations
for oxygen-dependent [PGC1a] under wild type, knockout and over-expression conditions
are shown in additional file 1 (Eqns. S4-S6). Data fitting based on an array of experimental
data [4,66,67] results in the following parameters: A = 2.3167, B = 0.35, k
h
= 2.5641, n
p
=
1.086,
α
max
HIF
=3.
Cell Module
The Cell module is adapted from our 3D agent-based in vitro m odel [33] to describe
how capillary endothelial cells resp ond to stimuli, specifically VEGF concentration and
gradients, during the time course of sprout formation. The model applies logical rules
to define cell activation, elongation, migration and proliferation events, based on exten-
sive published experimental data [33]. The model makes predictions of how single-cell
events contribute to vessel formation and patterns through the interaction of various
cell types and their microenvironment.
The primary rules used in the in vitro model [33] are specified as follows:
i) Endothelial cells are activated at an initial time point and the number of activated
cells is constrained by a specified maximu m number per unit capillary length. This
activation initiates development of a tip cell segment of a sprout, later follo wed by the
formation of a first stalk cell segment. ii) Following invasion of new sprouts into the
tissue by extending the leading tip and stalk cell segments, the tip cell continues to
migrate in the interstitial space f ollowing VEGF gradients and moves towards higher
VEGF concentration. In addition, the tip cell can also proliferate with a certain prob-
ability, and the stalk cell can elongate and proliferate in a specified fashion; note that
the probability of tip cell proliferation is much smaller than that of the stalk cells. The
combined effect of these two cell phenotypes can be simulated as a biophysical push-
pull system. iii) Branching occurs with a specified probability after a designated time
threshold has elapsed at either a stalk or tip cell. The branching angle is selected
stochastically and is less than 120 degrees. In the original model [33] the frequency of
branching events during the spouting process was a function of the expression of
ligand Dll4 and receptor No tch on the endothelial cells. Details of other rules and the
parameters were described in [33].
To simulate in vivo conditions in the skeletal muscle vessel network, we modified
some of the previously-defined rules and introduced additional rules to the model.
Since muscle fibers and vasculature occupy respe ctively 79.7% and 2.5% of the ti ssue
volume, the interstitial space totals 17.8%. Hence the freedom of tip endothelial cell
migration during s prout formation is constr ained to occur in a small volume of inter-
stitial space. Note th at in the model the endoth elial cells consist of cylindrical cell seg-
ments (10 μmlengthand6μm diameter per capillary segment; 4 segments per cell
defined in this study); the rules are formulated for these segments rather than for
whole cells. This part of the model can be readily modified. The additional rules
imposed in this study are as fo llows: i) E longation or migration of cell segments fol-
lows the original rules as developed and defined in [33], except when the cell may
encounter a fiber by following the g rowth factor gradient, we assume that the tip cell
filopodia will sense the fiber and instead the cell follows the second largest VEGF
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gradient direction alternati vely to elongate or migrate. ii) Anastomoses are formed
when the tip cell senses an existing capillary or a sprout within 5 microns. iii) Since
the function of Dll4-Notch is not clearly defined in skeletal muscle, their effects are
not taken into account in the current simulation s, but this effect can be readily added.
For model simplification and demonstration purpose, tip cell elongation and the
branching are not allowed in the present study.
Integration of computational modules
The development of an anatomically-, biophysically- and molecular-detailed spatio-
temporal model by integration of different modules is a novel and challenging task.
One of the main objectives of this study is t o c reate a platform for integration of dif-
ferent modules written in different programming languages and using mixed modeling
methodologies. The component modules may be created in the same or different
laboratories, and could also be selected from a public model database. The difficulty of
this task stems from the fact tha t few stan dards and open-source software/libraries for
PDE solvers and ABMs e xist. As a result, mo dules are dependent on their native lan-
guages an d on d ifferentia l equation solvers , making the integration difficult. Another
problem facing the integration of modules is how to define and implement the connec-
tivity b etween them, i.e., the exchange of data between the modules. Here we solve
these two problems using a novel computational infrastructure and object-oriented
design as described below.
Computational Infrastructure
To ov ercome the language barrier between the four modules selected in this study
(Fortran for the Flow module, C/C++ for the Oxygen and VEGF modules and Java for
Cell Module, Figure 1A), we choose Java to design the controller, which provides a
flexible high-level interface and object-oriented facilities. Instea d of rewriting the codes
in each module in Java, we use a mixed-language programming environment to link
the modules and save repetitive effort. The native codes in Fortran and C run faster
than Java, and this compromise solution can also inherit advantages of these two lan-
guages. Another important technical aspect that renders this hybrid system feasible is
the existence of Java Na tive Interf ace (JNI) API to c onvert functions and data type
from native codes (Fortran and C) automatically to Java. To fulfil this purpose, we
redesigned th e native codes for the Flow, Oxygen, and VEGF modules to the fo rmat of
functions and subroutines, and compiled these codes into the Java-readable libraries,
turning all four modules into pluggable libr aries that can be called by the controller
coded in Java (Figure 1B). Furthermore, these libraries can be dynamically linked, mak-
ing the simulation of dynamic angiogenesis processes feasible. Thus, using the control-
ler as a bridge between each module, communication between different modules is
relatively easy to implement. Last, it is easy to use Java to implement the connection
between core codes and a new parameter database file used by the four selected
modules.
To achieve high performance of native codes, paralle l computing is implemented in
the Oxygen and VEGF modules, as they r equire extensive computing resources. The
current ver sion of the modules adds OPENMP (open multi-processing) sup port, an
industry standard for memory-shared parallel systems, to shorten the simulation time
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when the PDE solver is called by the controller. Numerical simulation time for the
Oxygen and VEGF PDEs has been sped up ~5 fold using an 8 quad-core processor.
Object-oriented design
As a starting point to integrate all four modules, we focused on robust design of the
controller, providing the connectivity between the modules, rather than providing sol-
vers/software for mathematical models (i.e., rather than focusing on the capability of
solving equations or agent-based models numerically). This will not impose constraints
on the modeling methodology used in the modules, and it will allow a wide choice of
modules to be integrated, providing more flexibility. More detail on the procedure for
integration of modules at multiple time scales using object-oriented classes is given
below in the section Integration of modules at multiple time scales.
Regarding design at the upper level (i.e., package design), we devised four sub-sys-
tems: Biosystem, Process, IO (Input/Output) and Exception as shown in lower panel in
Figure 1A. The Biosystem is a repository of hierarchical structure of biological and bio-
physical information in the tissue as shown in Figure 2. The Process is composed of
angiogenesis subproce sses occurring at various stages in growth process. The IO is
used to provide an interface for the interaction between the system and the user, for
example, to read parameters used for the simulation of the Oxygen and VEGF mod-
ules. The Exception provides the capability to debug and handle runtime errors.
At the lower level (i.e., class design), we applied object-oriented co ncepts to describe
a hierarchical structure of sk eletal muscle (EDL) and events in the angiogenesis pro-
cess. The class diagram shown in Figure 2 dep icts the relationships among the major
Figure 2 Object-orie nted design for the angiogenesis modeling package. Major classes across tissue
and cell scales in the modeling controller are shown. They include SkeletalMuscle, Myofiber, Vessel, Grid,
Segment and Node classes in the Biosystems subpackage, and BloodFlow, O2Diffusion, VEGFRxnDiffusion,
CellSprouting, and StartAngio classes in the Process subpackage. The hierarchical structure of relationships
between the classes is represented by arrows.
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classes propose d. In the Biosyste m sub-package, six classes are defined to describe the
entities composing the tissue of interest: SkeletalMuscle, Myofiber, Vessel, Grid, Node
and Segment. In a ddition, angiogenesis event classes defined in the sub-package Pro-
cess include BloodFlow, O2Diffusion, VEGFRxnDiffusion, CellSprouting and
StartAngio.
The Node and Segment classes in the Biosystem sub-package are defined below the
cell sc ale. The Node class represents the circular surface of cylindrical segments at
their ends. I t contains spatial i nformation, including the circle center posit ion and the
circle radius. It may also have biophysical informatio n such as blood pr essure, flow
velocity and hematocrit if the node is cont ained in a blood vessel. The Segment class
represents a cylinder in 3D space, corresponding to either a fraction of a blood vessel
or muscle f iber (a ssuming the fiber and blood vessel are cylindrical in shape). A Seg-
ment object contains two Node objects at its ends and the length of the cylinder. Each
Segment also contains biophysical information such as blood flow, pressure and hema-
tocrit, and VEGF receptor density if the segment type is a blood vessel.
At the tissue scale, SkeletalMuscle, Myofiber, Vessel and Grid are defined in the Bio-
system sub-pa ckage. The Vessel class is used for representation of microvessels includ-
ing capillaries, venules and arterioles. The Myofiber class is used for representation of
skeletal muscle myocyt es. These two classes are each composed of a series of seg-
ments. Interstitial space information including VEGF and HSPG concentrations is
described in the Grid class. The Grid class also contains the local P
O2
value. Ensemble
components of fiberTissue, capillary, venule, arteriole, voxel, tissueSize and gridUnit-
Size render the SkeletalMuscle class, which describes detailed skeletal muscle structure.
In the Process sub-package, four classes including BloodFlow, O2Diffusion,
VEGFRxnDiffusion and CellSprouting provide connections with Java-readable libraries
compiled from their corresponding modules. The calling of these Java classes involves
three steps: i) The skeletal muscle object (realization of SkeletalMuscle class) will be
initialized and then tra nsferred as the input to each module. ii) Each specific module
will compute using their intrinsic numerical solvers and then the results will be trans-
ferred to the Java interface class. iii) Finally, the skeletal muscle object will be updated
with the solutions. The StartAngio class contains methods defined to specifically simu-
late exercise-induced angiogenesis.
Developing the computational environment
The simulation experiments were run on a computer with 64 bit Linux Ubuntu sys-
tem, 8 quad-CPU and 128 Gbyte memory. Eclipse http://www.ecli pse.org is used as an
Integrated Development Environment for coding purposes. JDK (Java Development
Kit) 1.6.16 (Oracle, Redwood Shores, CA) is used as Java compiler, and Intel Fortran/C++
compiler suite (v.11.1) (Intel, Santa Clara, CA) is used as Fortran/C++ compiler. We also
incorporate Java 3D API (Oracle, Redwood Shores, CA) for 3D programming purposes
and the Log4j package http://logging.apache.org/log4j/1.2 to log all the runtime messages
for the purpose of debugging. We use Bazaar http://bazaar.canonical.com as our source
code version control system since as an industry standard for software development it
provides support for a large scale project development by a team of programmers. It has
advantages in terms of branching, merging and keeping revision versions. The GNU Make
tool http://www.gnu.org/software/make/ is chosen for automation of building executable
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programs and libraries from source codes, and running programs from binary codes. In
particular, it is useful for a hybrid system (i.e., mixed programming environment) in a sin-
gle program. The MASON package http://cs.gmu.edu/~eclab/projects/mason/ is used as
the agent-based modeling library. Unit tests are performed using the JUNIT 4.0 package
http://www.junit.org/.
Integration of modules at multiple time scales
We performed the integration of the modules using a sequential method, that is, the
modules are run sequentially rather than in parallel. This is based on a time scale ana-
lysis of each process integ rated into the multiscale angiogenesis model . We computed
the module at the fastest time scale first. The outcome of that module was considered
as a pseudo-steady state and used as an i nput feeding into th e modules at slower time
scales. This continued sequentially until we computed the module at the slowest time
scale. Specifically , the blo od flow re gulation and oxygen distribution modules reach
equilibrium within seconds to minutes; VEGF gradients at time scales of minutes to
hours; and capillary sprouting from hou rs to weeks. Thus we computed the steady-
state flow and oxygen module first, then used their simulation results as an input to
VEGF reactio n-diffusion model and run PDE solver to compute VEGF profile, and
then finally run the agent-based model to simulate angiogenesis patterns for single-
bout exercise. When e ndurance exercise for days or weeks is simulated, t he updated
model geometry will be used as the new input to run flow, oxygen and cell modules
sequentially.
Example run of a simulation of single-bout exercise
Using the object-oriented design concept, we constructed a controller which is capable
of interacting with each individual module defined in our multiscale model. A run of
the simulation starts with input of geometry files and parameter database file (written
in database file format). These inputs initialize an object of SkeletalMuscle clas s with
the 3D coordinate information of segments and nodes for skeletal muscle fiber and
blood vessel network. The parameter database file assigns values to the biochemical
and physiological parameters defined in the model. Following this, the controller calls
the flow module as a dynamic linked library (dll) as described above, using the geo-
metric and biochemical parameters as input. The flow module has its own built-in sol-
ver that returns (to the controller) results including blood pressure, hematocrit,
viscosity and flow rates in the various vessel segments. The information is stored in
the controller SkeletalMuscle object. This object, now with the updated flow infor-
mation and the increased oxygen consumption rate, is passed by the controller to the
oxygen module. This module computes the oxygen tension at each grid point defined
within the skeletal muscle and passes the values back to the controller to be stored in
the voxel field (Grid class) of the SkeletalMuscle object (element P
O2
is defined in
Grid Class). The controller passes the O
2
-updated object to the VEGF module, which
includes O
2
-PGC1-HIF-VEGF empirical equations. Upo n completing its simulation,
theVEGFmodulepassestheVEGF,HSPG,and other concentrations throughout the
tissue back to the controller. This spatial concentration profile information is then
passed to the cell module to simulate capillary growth. During the 8-hour post-exercise
period, due to the cessation of exercise, blood flow rate and oxygen consumption rate
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return to basal level. We thus assume that capillary sprouting is the only active module
during this pe riod; blood flow rate and oxygen distributions are assumed to be at basal
levels.
Results
Using the integration strategy and simulation package described above, we performed a
series of computational experiments to simulate activity-induced angiogenesis in EDL
during a single bout of exercise. First, a 3D simulation was conducted to sequentially
integrate modules from blood flow to oxygen convection-diffusion, to VEGF diff usion-
reaction, to capillar y sprouti ng. Second, se nsitivity analysis was used to examine sensi-
tivities of key biochemical, biophysical and p hysiological parameters to capillary
growth. Third, angiogenesis patterns were compared between different exercise intensi-
ties, and high- or low-inspired oxygen conditions.
Sequential integration of modules from blood flow to capillary sprouting
Single-bout exercise was simulated by dynamic integration of four modules: Flow, Oxy-
gen, VEGF and Cell. The simul ations are based primarily o n physiological param eters
used in our previousl y developed mod els [33,60,61], as shown in Table 1. Most of the
parameters were taken from experimental data for rat skeletal muscle while others
were adopted from measurements in other tissues or theoretical estimates when para-
meters are unavailable from literature (e.g., diffusion coefficients for VEGF
164/120
,
secretion rates for VEGF
164/120
). Here we describe several key parameters used in the
simulation.
Oxygen consumption rate, M
c
, increases to 2- to 50-fold the b asal rate during exer-
cise [68]. Previous models simulated the system under 6- and 12-fold oxygen con-
sumption [22,61]. Here we used oxygen consumption of 9-fold the basal rate,
corresponding to moderate exercise intensity conditions. These moderate exercise
conditions are typical of experimental rat (mouse) aerobic t readmill exercise [69,70].
Another important paramete r of the exercise environment is oxygen saturation of
arterioles feeding into capillaries ( S
O2A
). The model uses values of 0.6 and 0.3 for S
O2A
,
corresponding to a normoxic environment and a low oxygen environment (hypoxic
hypoxia). Hydrodynamic pressure drop between small arteries and venules (ΔP) is spe-
cified at 10 mmHg and i nlet discharge hematocrit is 0.4, consistent with physiological
observations [41,71]. For exercise-induced angiogenesis, experimental observations
show that VEGF mRNA expression is elevated significantly within 1 hour post-
exercise, and it remains el evated until returning to basal levels after 8 to 24 hours
[69,70,72-74]. Due to the lack of experimental measurement of VEGF protein secretion
rate from the myofibers in vivo, w e assume that VEGF protein expression follows the
similartimecourseasmRNA.Thus,weconsider VEGF secretion as a step function
starting at high levels at the onset o f the post-exercise period, remaining constant for
an 8-hour interval, and finally ending after 8 hours. We simulate the system for eight
hours as the period of capillary growth; beyond this time elevated VEGF secretion
ceases and the stimulus to sprout declines.
Using skeletal muscle geometry (Figure 3A) the blood flow simulation results (Figure 3B)
show heterogeneous distribution of blood flow velocity in the microcirculatory network,
with variation in velocity between blood vessels across the x-y plane. The mean flow
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Table 1 Parameters of the multiscale model*
Parameter Description Value Unit Module
P
in
Inlet pressure 10 mmHg Flow
H
d,in
Inlet hematocrit 0.4 - Flow
a
tis
O
2
solubility in tissue 3.89 × 10
-5
ml O
2
ml
-1
mmHg
-1
Oxygen
a
RBC
O
2
solubility inside RBC 3.38 × 10
-5
ml O
2
ml
-1
mmHg
-1
Oxygen
a
pl
O
2
solubility in plasma 2.82 × 10
-5
ml O
2
ml
-1
mmHg
-1
Oxygen
D
O2
O
2
diffusivity in tissue 2.41 × 10
-5
cm
2
s
-1
Oxygen
D
Mb
Myoglobin diffusivity in tissue 1.73 × 10
-7
cm
2
s
-1
Oxygen
M
c
Mass consumption O
2
by tissue 1.5 × 10
-3
, 1.67 ×
10
-4
ml O
2
ml
-1
s
-1
Oxygen
C
M
b
bind
Myoglobin O
2
-binding capacity 1.016 × 10
-2
ml O
2
(ml tissue)
-1
Oxygen
C
RB
C
bind
Hemoglobin O
2
-binding capacity 0.52 ml O
2
(dl RBC)
-1
Oxygen
P
50,Mb
P
O2
for 50% myoglobin oxygen saturation 5.3 mmHg Oxygen
P
50,Hb
P
O2
for 50% hemoglobin oxygen saturation 37 mmHg Oxygen
n
h
Oxyhemoglobin saturation
Hill coefficients
2.7 - Oxygen
S
O2A
Oxygen saturation for arteriolar inlets 0.6,0.3 - Oxygen
D
V120
V
120
diffusivity in ECM 1.13 × 10
-6
cm
2
s
-1
VEGF
D
V164
V
164
diffusivity in ECM 1.04 × 10
-6
cm
2
s
-1
VEGF
H
ECM
HSPG density in ECM 7.5 × 10
-10
pmol μm
-3
ECM VEGF
k
on,v164,H
V
164
HSPG complex association rate constant 4.2 × 10
8
pmol
-1
μm
3
s
-1
VEGF
K
off,v164,H
V
164
HSPG complex dissociation rate constant 1 × 10
-2
s
-1
VEGF
k
int
R1 and -R2 internalization rate (free and
complex form)
2.8 × 10
-4
s
-1
VEGF
S
R
R1 and -R2 insertion rate 9.2 × 10
-16
pmol μm
-2
s
-1
VEGF
H
EBM
,H
MBM
HSPG density in EBM and MBM 1.3 × 10
-8
pmol μm
-3
BM VEGF
k
on,vegf,R1
V
120
R1 (V
164
R1 ) complex association rate
constant
3×10
10
pmol
-1
μm
3
s
-1
VEGF
k
on,vegf,R2
V
120
R2 (V
164
R2 ) complex association rate
constant
1×10
10
pmol
-1
μm
3
s
-1
VEGF
k
off,vegf,R1,
k
off,
vegf,R2
V
120
R1/R2, V
164
R1/R2 complex dissociation rate
constant
1×10
-3
s
-1
VEGF
k
off,v164,N,
k
off,
R1,N
V
164
N, R1N complex dissociation rate constant 1 × 10
-3
s
-1
VEGF
k
on,v164,N
V
164
N complex association rate constant 3.2 × 10
9
pmol
-1
μm
3
s
-1
VEGF
k
on,R1,N
R1N association rate constant 1 × 10
10
pmol
-1
μm
2
s
-1
VEGF
k
on,V164R2,N
V
164
R
2
·N complex association rate constant 3.1 × 10
9
pmol
-1
μm
2
s
-1
VEGF
k
on,V164N,R2
V
164
N·R
2
complex association rate constant 1 × 10
10
pmol
-1
μm
2
s
-1
VEGF
S
V120,b
V
120
basal secretion rate 0.17 fmol(L tissue)
-1
s
-1
VEGF
S
V164,b
V
164
basal secretion rate 1.97 fmol(L tissue)
-1
s
-1
VEGF
B
tip-prolif
Boolean value of whether tip cells proliferate No - Cell
B
stalk-prolif
Boolean value of whether stalk cells proliferate Yes - Cell
B
tip-branch
Boolean value of whether tip cells branch No - Cell
B
stalk-branch
Boolean value of whether stalk cells branch No - Cell
B
tip-elongation
Boolean value of whether tip cell elongate No - Cell
B
stalk-elongation
Boolean value of whether stalk cells elongate Yes - Cell
*Abbreviations: EBM, Endothelial Basement Membrane; MBM, Myocyte Basement Membrane; V
120/164
, VEGF
120/164
;N:
Neuropilin 1. Note other agent-based rules and the parameters in Cell module can be found in [33].
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Figure 3 3D simulation of blood flow, oxygen, and VEGF distribution during the single-bout exercise: A) 2D cross section of skeletal muscle: gray circles represent fibers and red circles
represent capillaries; B) Blood flow velocity distribution in skeletal muscle microvascular network; C) Oxygen distribution throughout the tissue; D) VEGF secretion level along the muscle fibers; E)
total VEGFR-bound VEGF distribution (including both 120 and 164 isoforms) on vascular surface; and F) free VEGF concentration distribution in interstitial space.
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velocity of all capillaries in the network is 0.11 cm/s and total blood flow is 227.82 ml·(100
g tissue)
-1
min
-1
, consistent with experimental measurements [75-77]. EDL tissue displays
oxygen spatial heterogeneity not only in the x-y plane, but also in the z-direction, as shown
in the 3D oxygen distribution plot (Figure 3C), generated from the Oxygen module simula-
tion results. Most of the hypoxic tissue space has few nearby capillaries, confi rming that
the oxygen tension gradient and distribution is dependent on capillary spatial positions in
the skeleta l muscle space. Calcu lating all oxygen tensions in the tissue, we obtained an
average tissue P
O2
of 13.23 mmHg with a coefficient of variation 0.28.
Using o xygen tension distribution as input to the VEGF simulation parsed by the
integration core package, we calculated V EGF secretion rate (Figure 3D). The average
VEGF secretion level is 2.12-fold with a coefficient of variation of 0.32. The increase in
VEGF secretion levels results in the increase of soluble VEGF concentration in tissue
interstitial space (average free VEGF concentration is 1.9 pM). Simulation results from
the VEGF module (Figure 3E) show that VEGF gradients in the longitudinal direction
(along the fibers) are much smaller than in the cross-s ectional planes, as indica ted by
their mean values (0.56% and 3.93% c hange in VEGF concentration over 10 microns,
respectively). Similarly, VEGF bin ding to the en dothelial surface rece ptor VEGFR2 is
non-uniform across the vascular network (Figure 3F). The mean surf ace concentration
of total receptor-bound VEGF is 1023 × 10
-7
pmol/μm
2
(coefficient of variation of
0.20), of which 8% is VEGF
120
and 92% is VEGF
164
; NRP1-facilitated VEGF
120
-
VEGFR1 and VEGF
164
-VEGFR2 complexes contribute 4.69% and 67.09% of the total
receptor complexes, respectively.
We simulated the capillary activation process using our modifi ed in vivo cell model.
The precondition for an endothelial cell to be considered as activated is that any of its
cell segments is expose d to a VEGF co ncentration greater than a specified VEGF acti-
vation threshold. A random selection algorithm is used to determine the sprouting
position in the activated endothelial cells and the number of activated endothelial cells
is also constrained by the allowed maximum number of activated tip cells per capillary
(5 cells per capillary) and the minimum distance between tip cells (40 μm). These
rules allow us to define activated tip cell and cell segments as the sprouting point for
new c apillary sprouts. Using a VEGF activation threshold of 2.0 pM, 2% percent of
cells are activated and these cells initiate the sprouting process.
We next simulate cellular sprouting and predict dynamic development of nascent
vessels during 8 hours post-exercise, as defined in the timeline scheme in Figure 4A.
Figure 4B-F shows new vascular network formation at 1, 3, 5, 7 and 8 hours following
exercise. One of the assumptions in the model is that new capillary formation does not
affect muscle fiber oxygenation, VEGF secretion or VEGF internalization, due to the
slow ca pillary maturation process. Thus, a feedback loop is not implemented for a sin-
gle bout of exercise, and a static VEGF profile is used as input for the cell model. This
is a condition that can be relaxed in future models. We also assume periodic condi-
tions for capillary growth, i.e., when capillary vessel sprouts encounter the boundary,
they continue to grow beyond the boundary (e.g. arrow 1 at Figur e 4C) and are repre-
sented by a corresponding capillary growing fr om t he opposite face (e.g. arrow 2 at
Figure 4E). As shown in Figure 4D-F, capillary sprouts on the same capillary initially
migrate in a similar direction, while at a lat er time, they may form anastomoses with
adj acent vasculature while migrating (arrows 3, 4, 5). Using para meter values listed in
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Table 1, capillary proliferation and migration rules defined in th e cell module result in
a 12.6% increase in total capillary length compared to the initi al length before exercise
(Figure 4F). However, for the conditions of a single bout of exercise considered in this
study most of these new s prouts are likely to retract; sustained exerci se conditions
with formation of multiple functional anastomoses will be considered in subsequent
studies.
B C
D E
F G
3
4
5
1h 3 h
5h 7h
8h
1
2
A
Figure 4 3D simulation of capillary network growt h during the single-bout exercise. A) Time course
of capillary growth was simulated based on the timeline scheme for exercise-induced angiogenesis. The
simulation leads to new capillary network at: B) 1 h post-exercise; C) 3h;D) 5h;E) 7h;F) 8 h. When new
capillary segments grow out of boundaries (Arrow 1), capillary will grow from corresponding periodic
boundaries (Arrow 2). Arrows 3-5 refer to the capillary anastomoses during growth. Vessels colored in red
represent the pre-existing microvascular vessels, ones in light blue are the newly-developed blind-ended
capillaries, and ones in dark blue are new capillary anastomoses. G) Relative capillary length change vs
time.
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Sensitivity analysis of cellular parameters to define viable parameter space
We further explore the effect of parameters defined in the multiscale model of capil-
lary growth. For this purpose, all biochemical, biophysical, and physiological para-
meters are extracted from each module and are deposited in a parameter database file.
We show an example of how endothelial cell activation affects the capillary growth
pattern and growth rate (t wo key variables related t o capillary growt h). We vary the
threshold level of VEGF for endothelial cell activation within a range of 1.5 to 2.3 pM
and this change significantly affects the capillary growth rate. Figure 5A shows that
above a VEGF threshold of 2.3 pM, none of the endothelial cells are activated and the
capillary network has no growth. Over t he range of 2.3-1.8 pM, capillary growth rate
increases slightly as the threshold value decreases. Below 1.8 pM, capillary growth
increases dramatically since a large number of capillaries are activated. The response
to these threshold levels depends on the mean concentration of VEGF in the tissue
(here, 1.9 pM).
The number of anastomoses is another important variable that indicates vascular
adaptation of the mi crovasculature since the increase o f blood flow r esults from these
newly-formed vessel l oops (and not from blind-ended vessels). We show in Figure 5B
that the number of anastomoses increases in a similar fashion when threshold values
decrease from 2.3 pM to 1.5 pM. Interestingly, more than half of the anastomoses are
predicted to form during the first 4 hours.
Simulation of exercise-induced angiogenesis under different conditions
Varying experiment-based parameters defined in the Oxygen modules, we can simulate
vesse l growth in skeletal muscle during different exercise conditions. For example, dif-
ferent oxygen cons umption rates, M
C,
correspond to different exercise intensities and
arterial oxygen saturation values S
O2A
are associated with different available oxygen
levels. We compare capillary growth patterns b etween three exercise conditions: i) M
C
=1×10
-3
ml O
2
/ml tissue/s an d S
O2A
= 0.6 (light intensity exercise in normoxia;
results not shown); ii) M
C
=1.5×10
-3
and S
O2A
= 0.6 (moderate intensity exercise in
normoxia; Figure 6B and 4F); iii) M
C
=1.5×10
-3
and S
O2A
= 0.3 (moderate intensity
exercise in hypoxic hypoxia; Figure 6A). Simulation results show that during low-
intensity exercise (6-fold basal level oxygen con sumption rate) with normoxic condi-
tions, the capillary network will not change (case i). Compared to exercise condition of
a 9-fold basal oxygen consumption rate in normoxia (case ii), exercise under hypoxic
hypoxia conditions at a 9-fold basal oxygen consumption rate further increases
capillary growth to 125% (case iii).
Discussion and Conclusions
Using a module-based computational platform, we connected four modules, each of
which describes different aspects of the angiogenesis process (i.e, blood flow, oxygen
distribution, VEGF distribution, and capillary growth). This architecture allowed the
individual modules to remain in their native language and required rewriting only of
the input/output routines to handle the passing of common data between the submo-
dules and the integrator. This significantly reduced the time required to integrate the
available modules. JNI plugins are used to link the codes written in different languages,
and they allow ready communication between the core package and each module. As a
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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1.6
1.8
2
2.2
0
2
4
6
8
100
110
120
130
140
150
VEGF Threshold (pM
)
Time (hour)
A
B
Relative Length (%)
Figure 5 Sensitivity of the capillary length and the number of anastomoses formed with respect to
VEGF threshold for EC activation. A) Relative capillary length as a function of time and VEGF threshold;
B) Number of anastomoses formed as a function of VEGF threshold at 4 h and 8 h. Simulation sample size
is five for each VEGF threshold at a given time.
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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Figure 6 Angiogenesis pattern during different exercise conditions at 8 h post-exercise. A) Oxygen
consumption at 9-fold the basal level under hypoxic hypoxia conditions (S
O2A
= 0.3); B) Oxygen
consumption at 9-fold the basal level under normoxic conditions (S
O2A
= 0.6).
Liu et al. Theoretical Biology and Medical Modelling 2011, 8:6
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validation of our simulation strategy and integration package, we have conducted a ser-
ies of computational experiments to simulate dynamic activity-induced a ngiogenesis
during a single bout of exe rcise, using rat EDL skeletal muscle. We illustrated how the
computational design of a core controller module is beneficial to integrating different
aspects of biological knowledge with a variety of numerical simulation s chemes: i)
sequential integration; ii) sensitivity analysis; and iii) simulation of various exercise
conditions.
Object-oriented programming (OOP) is attracting more modelers attention and has
been increasingly used in biomedical computational projects [56,57,78]. We applied
OOP concepts to describe a complex biological system which encompasses biological
components and processes from the molecular level to the tissue level. With our
model architecture, we also depicted the hierarchical relationships among the different
classes abstracted from such systems. For the core packa ge, Java was used since it has
a natural object-oriented d esign a nd method inheritance structure that is intuitively
appealing to describe the biological system.
Our approach is complementary to other ongoing projects in multiscale modeling
fields, e.g. CompuCell [79], the Multiscale Systems Immunology (MSI) simulation fra-
mework [80] and Virtual Cell [81]. These project s share the same objecti ve: the devel-
opment of multiscale modeling software or libraries with a versatile user interface.
CompuCell implements cellular Potts models to simulate cell dynamics, MSI uses
agent-based models and Virtual Cell uses differential equations. While we did not
attempt to provide standards or software for the angiogenesis models, we chose to
focus on the extensibility and generality of an integration methodology that allows
additional individual modules to be integrated at any t ime, and does not impose con-
straints on the modeling techniques employed.
As a starting point, current study shows an example of the simulation by integrating
four modules. We used existing models for the first modules to be able to compare
the results of the integrated model t o published r esults of each submodule. This com-
putational paradigm will allow us to incorporate additional modu les developed in t he
laboratory, e.g., MMP [23,24], HIF1a [82] and FGF2 [18], using an SBML/CellML-
based open source simulation packages to represent the molecular interaction
networks. In a ddition, this platform can incorporate other relevant published models
from the literature or public model repositories, e.g. CellML Models Repository http://
models.cellml.org/ and BioModels Database http://www.ebi.ac.uk/biomodels-main/. It
is also feasible to impro ve on or add to t he individual modules currently comprising
the integrated model. For example, the ge ometry used by the modules can be updated
to that of a d ifferent muscle type without rewriting the other module codes; or more
detailed information such as the fiber type o f each muscle fiber can be included as an
additional data notation, with corresponding changes in local oxygen consumption and
O
2
diffusion coefficient and solubility.
Correspondingly, the modeling controller module will be further expanded to include
the classes representing other components, which may be relevant to skeletal muscle
angiogenesis when new biological information and models are available. In the long
run, definit ion of unified XML-based notation to describe structure and spatial infor-
mation of physiological systems , ordinary/pa rtial differential equations, agent-based
rules and system parameters will make each module more portable and more easily
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integrated. Pro gress has been initiated in this direction such as insili coML [83]. In
addition, development of the ontology for the field of angiogenesis modeling will also
be beneficial to the sharing, usage and integration of different modules, similar to gene
ontology [84], and the open biological and b iomedical ontologies [85]. These ongoing
and future developments should contribute to the overall efforts of the Systems Biology
and Physiome P rojects [46,86], and could be applied to cardiovascula r disorders such
as coronary and peripheral artery diseases [87].
Along with this framework for modeling the system, there is now the opportunity to
include other useful functionality such as loc al and global parameter sensitivity mod-
ules, or parameter optimization modules. While local parameter sensitivity calculations
can be performed with the current model, the computational expense of the model (in
particular the PDE solver) is such that parameter estimation would be very slow. We
aim to use cluster computing in order to permit these activities in the future.
In summary, we have developed the computational methodology capable of the inte-
gration of b iologically- and computationally-heterogeneous modules and this systems
biology strategy can be applied to larger scale integratio n of computational models o f
angiogenesis in skeletal muscle.
Additional material
Additional file 1: Modified empirical equations for the relationship between P
O2
and VEGF secretion rate.
Acknowledgements
The authors thank the Imaging Science Center at Johns Hopkins University for use of the computer cluster; Anthony
Kolasny, IT Architect, for technical assistance; Nicholas Tan for his contribution to the PGC1a model; Dr. Corban Rivera
for the discussion. This work was supported by the National Institutes of Health grants R01 HL101200, R33 HL087351,
R01 HL079653 and R01 CA138264 (ASP), and R00 HL093219 (FMG).
Author details
1
Systems Biology Laboratory, Department of Biomedical Engineering, School of Medicine, Johns Hopkins University,
Baltimore, MD 21205, USA.
2
Department of Bioengineering, Rice University, Houston, TX 77521, USA.
3
Institute for
Computational Medicine and Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD 21218,
USA.
Authors contributions
Conceived and designed the simulations: GL AAQ PV FMG ASP; Performed the simulations: GL; Analyzed the data: GL
AAQ PV FMG ASP; Wrote the paper: GL AAQ PV FMG ASP. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 3 November 2010 Accepted: 4 April 2011 Published: 4 April 2011
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doi:10.1186/1742-4682-8-6
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    • "Techniques to parameterize [74], coarse-grain, modularize [21], and distribute [75] models have helped advance the field. Recent multiscale models have been applied to characterize angiogenesis in a heterogeneous brain cancer environment [76], effects of exercise on skeletal muscle angiogenesis [21,62], and monocyte trafficking [77]. "
    Full-text · Chapter · Nov 2015 · Frontiers in Oncology
    • "Other stimuli can induce angiogenesis , including shear stress [4] and genetic transformation in tumor cells [3]. Angiogenesis is important for organ development [5] as well as for physiological processes including wound closure and exercise training [6,7]. It is upregulated but disorganized in pathological processes such as diabetic retinopathy and solid organ tumorigenesis8910, where vasculature is needed to supply the tumor's rapid consumption of glucose and oxygen beyond the limits of diffusion. "
    [Show abstract] [Hide abstract] ABSTRACT: The regulation of vascular endothelial growth factor A (VEGF) is critical to neovascularization in numerous tissues under physiological and pathological conditions. VEGF has multiple isoforms, created by alternative splicing or proteolytic cleavage, and characterized by different receptor-binding and matrix-binding properties. These isoforms are known to give rise to a spectrum of angiogenesis patterns marked by differences in branching, which has functional implications for tissues. In this review, we detail the extensive extracellular regulation of VEGF and the ability of VEGF to dictate the vascular phenotype. We explore the role of VEGF-releasing proteases and soluble carrier molecules on VEGF activity. While proteases such as MMP9 can ‘release’ matrix-bound VEGF and promote angiogenesis, for example as a key step in carcinogenesis, proteases can also suppress VEGF's angiogenic effects. We explore what dictates pro- or anti-angiogenic behavior. We also seek to understand the phenomenon of VEGF gradient formation. Strong VEGF gradients are thought to be due to decreased rates of diffusion from reversible matrix binding, however theoretical studies show that this scenario cannot give rise to lasting VEGF gradients in vivo. We propose that gradients are formed through degradation of sequestered VEGF. Finally, we review how different aspects of the VEGF signal, such as its concentration, gradient, matrix-binding, and NRP1-binding can differentially affect angiogenesis. We explore how this allows VEGF to regulate the formation of vascular networks across a spectrum of high to low branching densities, and from normal to pathological angiogenesis. A better understanding of the control of angiogenesis is necessary to improve upon limitations of current angiogenic therapies.
    Full-text · Article · Nov 2013
    • "For decades, mathematical models have been employed to help address some of the pressing questions associated with tumor angiogenesis. As discussed in detail in Jackson and Zheng (2010), Zheng et al. (2013), existing models of tumor-induced angiogenesis can be characterized as continuous approaches (Balding and McElwain, 1985; Byrne and Chaplain, 1995, 1996; Anderson and Chaplain, 1998a,b; Holmes and Sleeman, 2000; Levine et al., 2001; Arakelyan et al., 2002; Sleeman and Wallis, 2002; Manoussaki, 2003; Plank and Sleeman, 2003, 2004; Plank et al., 2004; Levine and Nilsen-Hamilton, 2006; Schugart et al., 2008; Billy et al., 2009; Xue et al., 2009; Travasso et al., 2011), wherein cells are assumed to have a continuous distribution; discrete or hybrid models (Stokes and Lauffenburger, 1991; Anderson and Chaplain, 1998b; Tong and Yuan, 2001; Plank and Sleeman, 2003, 2004; Sun et al., 2005; Bartha and Rieger, 2006; Gevertz and Torquato, 2006; Frieboes et al., 2007; Milde et al., 2008; Capasso and Morale, 2009; Owen et al., 2009; Perfahl et al., 2011), wherein cells are modeled as individual agents and diffusible chemicals are modeled as a continuum; and cell-based formulations (Peirce et al., 2004; Bauer et al., 2007; Bentley et al., 2009; Qutub and Popel, 2009; Wcislo et al., 2009; Jackson and Zheng, 2010; Liu et al., 2011) wherein explicit incorporation of different properties of individual cells allows collective behavior of cell clusters to be predicted from the behavior and interactions of individual cells. Reviews of these models that appeared in or before 2009 can be found in Mantzaris et al. (2004), Peirce (2008), Qutub et al. (2009). "
    [Show abstract] [Hide abstract] ABSTRACT: Vascular endothelial growth factor (VEGF) is the most studied family of soluble, secreted mediators of endothelial cell migration, survival, and proliferation. VEGF exerts its function by binding to specific tyrosine kinase receptors on the cell surface and transducing the effect through downstream signaling. In order to study the influence of VEGF binding on endothelial cell motion, we develop a hybrid model of VEGF-induced angiogenesis, based on the theory of reinforced random walks. The model includes the chemotactic response of endothelial cells to angiogenic factors bound to cell-surface receptors, rather than approximating this as a function of extracellular chemical concentrations. This allows us to capture biologically observed phenomena such as activation and polarization of endothelial cells in response to VEGF gradients across their lengths, as opposed to extracellular gradients throughout the tissue. We also propose a novel and more biologically reasonable functional form for the chemotactic sensitivity of endothelial cells, which is also governed by activated cell-surface receptors. This model is able to predict the threshold level of VEGF required to activate a cell to move in a directed fashion as well as an optimal VEGF concentration for motion. Model validation is achieved by comparison of simulation results directly with experimental data.
    Full-text · Article · May 2013
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