Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 125

velocity wall shear VEGF dilation

1

100

200

300

Fig. 3 Maintaining a stable bifurcation. Shown, from left to right, are the fluid velocity, wall shear,
VEGF level, and the map of instantaneous wall velocities (which determine dilation or contraction
locally). Flow is from left to right. As opposed to models that consider only shear-based remodeling
(and consequently regress to a single outlet channel in this configuration), the system recovers from
the initial perturbation that causes the drop in flow in the bottom channel (iterations 200-300).
Upstream signaling from the low-flow, hypoxic vessel effectively prevents its upstream region
from “pinching off”

5.10 Model Performance and Stability

Figure 3 shows remodeling on a bifurcation that initially displays asymmetric flow
rates. Without any external metabolic signaling, the lower branch will regress.
However, as the VEGF level increases, the lower branch has begun dilating by
iteration 100. Continued dilation by iteration 200 increases the flow in the lower
branch and begins raising the shear rate along the segment. Finally, by iteration
300, the shear rate in both branches is nominal and the simulation has converged.
Interestingly, the final shape after convergence differs from the initial symmetric
configuration.
Figure 4 demonstrates the time course of adaptation in an example network.
Initially the network is a dense mesh (white outline). By iteration 25, the network
has adapted in response to shear forces and local VEGF level. After an initial influx
of VEGF driven by hypoxia, the vessels dilate to bring in more oxygen, and in
the process, the network is subjected to shear-based remodeling. Note that patches
of high VEGF, such as seen at iteration 20, effectively protect vessels in hypoxic
regions from being pruned.
Vascular remodeling and normalization are a central feature of tumor net-
works treated with anti-angiogenic therapy. By including the correct feedback
mechanisms, the present model is capable of reproducing the appropriate network
dynamics and predict transport of soluble species such as oxygen, growth factors,
and drugs. With appropriate calibration of the various thresholds for adaptation for
individual tumor types, this model will enable detailed, time-resolved analyses of
network changes and drug and nutrient delivery to tissue.