Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 123

fixed speed until the boundary intersects with an image feature like an edge. This
approach requires an external rule that is then used to locally stop growth, for
example, when the active contour hits a vessel wall. Refer to Appendix ( C.2 )foran
overview of the theory and a brief mathematical development of active contours. In
particular, the development of geodesic active contours with level sets is described.
Using the concepts outlined in Appendix ( C.2 ), we incorporate a classic balloon
force into a speed function that drives a curve C according to the following equation
of motion:

∂ t C =[ B ( C ) × H ( u · n )] ( − ν + c κ ) n (13)

where is the boundary curvature and encourages smoothness with a rate propor-

tional to the constant c , n is the unit boundary normal (inward pointing),

κ

ν is a

constant speed (e.g., unity), and the nonlinear function actsasa

stopping force on the evolving curve. Here, the stopping function is used to enforce
two separate constraints. The first is that the curve is confined to evolve within fluid
nodes. This is achieved by choosing B as a binary image that is unity if a location
in the computational domain is a fluid site and zero otherwise. Hence, B

[ B ( C ) × H ( u · n )]

will be
zero if the curve intersects with a non-fluid site in the domain. The second constraint
is that the curve can only evolve in a direction opposite the flow field u . With H

( C )

( · )
as the Heaviside function and the convention that the unit normal is inward pointing,
H 90 ◦ to the direction of flow, thus ensuring

that the curve only evolves upstream. By initializing the active contour to lie within
a vessel, the evolution equation given above, by virtue of the balloon force, will
drive the contour through branches and other convex regions, eventually settling
against every point on the vessel wall that is upstream from the initial region. By
further registering the epoch that each point on the active contour first reaches a
vessel wall, the “distance” from the initial region can be calculated.

( u · n )= 0 if the boundary is oriented >

5.8 Vessel Remodeling

The algorithm uses a set of key parameters to control the remodeling process. Fun-
damentally, this includes two distinct regimes: one where remodeling is dominated
by wall shear stress and the other where VEGF causes an interruption in the shear-
based remodeling. In the shear-based regime, vessels will dilate/contract in an effort
to achieve a nominal shear rate. As our model is based on a discrete domain, i.e.,
a binary image with discrete pixels, shear-based remodeling essentially follows a
discrete rule . Similar to Hacking et al. this rule is designed to

achieve constant shear in all vessels, which follows directly from Murray's law [ 31 ].
In contrast, when in a moderate to high-VEGF regime, vessels will undergo dilation
in an effort to promote an increase in flow. The rate of dilation is constant, e.g., 1
pixel (1 lattice-Boltzmann unit) per iteration. In the highest VEGF regime, upstream
signaling will cause additional dilations in proximal vessels. As described above, the
signal varies continuously in strength as a function of distance and initial size of the

δ r = sign ( τ − τ 0 )