Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 115

reproduced vessel collapse due to reduced blood flow and mechanical compression.
Welter and coworkers developed a model to analyze the vascular remodeling
process of an arterio-venous vessel network during tumor growth and were able
to reproduce complex vascular geometry with necrotic zones and “hot spots” of
increased vascular density and blood flow of varying size [ 95 ]. Szczerba and
Szekely presented a simple computational model of intussusceptive angiogenesis
and remodeling, predicting bifurcation formation and micro-vessel separation in a
porous cellular medium [ 87 ].
Because shear-based remodeling is, at its foundation, built on a very elegant
mathematical mechanism, many computational models have attempted to describe
it. Hacking et al. performed one of the first detailed analyses of network adaptation,
showing that a system with single inlet driven only by shear-driven diameter
changes is destined to degenerate to a single flow path from inlet to outlet [ 31 ].
Using similar models and experimental systems in vivo, Pries and Secomb have
addressed angioadaptation in normal physiology and ischemia. They conclude that
four parameters are necessary to drive the adaptation to a stable steady state:
endothelial wall shear stress, intravascular pressure, a flow-dependent metabolic
stimulus, and a stimulus conducted from distal to proximal segments along vascular
walls [ 65 - 68 ].
Directly addressing the problem of combination treatment, Ledzewicz et al.
applied optimal control theory to determine desired scheduling of anti-angiogenic
and cytotoxic drugs [ 54 , 72 ]. Other models have been used to analyze perturbations
in blood flow, such as caused by vascular obstructions. Gruionu et al. used
morphometric data from vascular casts to simulate the changes in blood flow caused
by obstructions and the subsequent adaptive remodeling. They found that vascular
arcades can partially maintain blood flow after vascular blockage and that structural
adaptation is important for modifying vessel diameters and controlling flow in this
system [ 30 ].

4 Modeling Tumor Physiology

For many years we have applied the lattice-Boltzmann approach to study interac-
tions between red and white blood cells in dynamic flow [ 59 , 84 ] and the effects of
vessel geometry [ 83 ] and RBC aggregation [ 85 ] on blood rheology. In addition, this
approach has allowed us to analyze the relationship between plasma, intravascular
hematocrit, and blood velocities in simple vessel networks [ 81 ] and blood forces
on digitized vessel walls [ 82 ]. More recently, we have adapted the model to
analyze flowing blood cells with realistic rest shapes and mechanical properties in
three-dimensions [ 20 - 22 , 61 ]. In other modeling work, we have used deterministic
models to analyze fluid convection and pressure distribution in tumors during
normalization [ 40 ]. But this approach is limited, as it does not explicitly account
for blood vessel architecture or adaptive remodeling. The current model, therefore,
extends our lattice-Boltzmann approach to larger domains and more complicated