Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 124

hypoxic region. The signal must therefore be discretized for use in our model, which
involves simple thresholding of the signal. Ultimately, our remodeling rules should
lead to a convergence state, whereby all vessels exhibit nominal shear, as well as
being exposed to nominal VEGF levels.

5.9 Stability and Convergence Behavior

To be sure, efforts to achieve this, or similar convergence states, have met some diffi-
culty in the literature. As described in Hacking et al. shear-based remodeling alone is
fundamentally unstable, leading to arteriovenous shunting, complete regression, or
even infinite expansion [ 31 ]. Following previous approaches in the literature, we use
additional metabolic signaling to promote stability during the remodeling process
[ 65 , 66 , 68 ]. However, we have still found it necessary to impose some additional
rules to achieve satisfactory convergence results.
First, our simple model does not include specific delineation of branches and
segments. There is no knowledge of vascular structure or topology, and all remod-
eling is fully local. As a result, an ill-timed contraction of even the smallest region
of a vessel can potentially shut down an entire downstream vascular bed before
VEGF signaling can counteract regression. Hence, we maintain a minimum vessel
radius for a fixed number of remodeling iterations, thereby delaying complete vessel
regression. This is biologically feasible, since endothelial cell reorganization takes
time. Further, as our modeling domain is grid-based, it exhibits a finite resolution
that inherently introduces small artifacts into the flow field. As a result, our shear-
based remodeling rule must be modified to remain insensitive to disturbances that
lie within a narrow band around the nominal shear rate. However, we've also found
it necessary at times to relax this band further in order to achieve good convergence
behavior.
Another important factor in achieving good stability characteristics is the choice
of flow inlet/outlet boundary conditions. As demonstrated by Hacking et al. a
constant-pressure inlet and outlet boundary condition is fundamentally unstable.
The other extreme, a constant-flow inlet boundary condition, does admit stable
flow configurations. However, unlike constant-pressure, the constant-flow boundary
condition causes tight coupling between sibling branches, meaning any change in
one branch will generally impact the flow in other branches. This can potentially
lead to stability issues. Also, this clearly precludes the ability to draw any additional
flow. An intermediate boundary condition involves using pressure boundaries
with an internal series resistance. This is the preferred approach, but requires
some care to properly match the internal resistance to the specific domain being
investigated.