Biomedical Engineering Reference
In-Depth Information
6
Conclusion and Discussion
It is likely that growth factors such as VEGF enable shear stress-driven adap-
tive remodeling of immature tumor vessel networks and that there are optimal
configurations of network architecture and hydraulic conductivity for delivering
drugs to tumors. A better understanding of this complex physiology requires
a comprehensive mathematical model such as presented here, which calculates
blood flow in complex tumor networks. The model incorporates the necessary
and sufficient components for simulating vascular structure and function during
anti-VEGF therapy: (1) lattice-Boltzmann calculations of the full flow field within
the vasculature and within the tissue, (2) diffusion and convection of soluble
species such as oxygen or drugs within vessels and the tissue domain, (3) distinct
and spatially resolved vessel hydraulic conductivities and permeabilities for each
species, (4) erythrocyte particles advecting in the flow and delivering oxygen
with real oxygen release kinetics, (5) hypoxia-driven VEGF production, and (6)
shear stress-mediated vascular remodeling. This model, combined with data from
intravital imaging of tumor vessel structure and function (Fig.
5
), will provide a
rational framework for the development of new therapeutic strategies or more effect
use of existing drugs.
A
Lattice Boltzmann Background
A.1
The Boltzmann Equation
The fundamentals of the lattice-Boltzmann method are the kinetic Boltzmann
equation which is well described in textbooks. It is based on the evolution of the
single particle distribution function
f
. This function indicates how many
particles are present at a point in space
x
with the velocity
v
at time
t
. During the
time d
t
the particles will now move according to their velocities to the new location
x
∗
=
(
x
,
v
,
t
)
v
d
t
. In case of external forces
f
the velocity of the particles will change as
well to
v
∗
=
x
+
m
f
d
t
, with
m
being the mass of a particle. Without particle-particle
collisions they would just move on like
f
v
+
d
3
x
d
3
v
1
m
f
d
t
(
x
+
v
d
t
,
v
+
,
t
+
d
t
)
−
f
(
x
,
v
,
t
)
=
0
but due to intermolecular collisions the particle distribution function
f
changes.
This change can be described by a collision operator
that represents the number
of particles that enter or leave the given phase space volume, so the Boltzmann
equation reads as
Ω
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