Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 134

Algorithm 1 Vessel Tracing
Extract a set of candidate seeds S
Estimate and prioritize seeds: Q ← S
for all s Q do

Initialize from seed:
( β ,

∈

I F , I B ) 0 ← ( β , I F , I B ) s

0 ◦ , 180 ◦ ] for seed direction ∈ [ do

for k do

Shift the model:
˜

= 1: ∞

μ k = μ k − 1 + Δ k
Estimate superellipsoid parameters:
β k ← ( β k − 1 , μ k )

Update intensity estimates
(

˜

I F , I B ) k ← ( I F , I B ) k − 1

break if LRT < τ or intersection

end for
end for
end for

the direction of shift Δ k is determined by the model orientation. The remaining

parameters are initialized directly using the previous estimates. The traversal
terminates when the LRT falls below a threshold, or the vessel intersects with the
image domain or a previously traced vessel.

C Active Contours

Active contours are deformable shape descriptors that evolve a boundary or front
dynamically in response to equations of motion derived either by the minimization
of an energy functional using variational methods [ 46 ] or based on principles
of geometric flow [ 13 , 57 ]. The first type uses an explicit parameterization, e.g.,
tensor product splines, to represent the boundary of the active contour, and is
commonly referred to as snakes . The second type is geodesic active contours ,
which incorporates notions of curve shortening geodesics, while using an intrinsic,
parameter-free representation of the boundary curve.

C.1 Geodesic Active Contours

The concept of a geodesic curve relates to the minimum distance between points.
To see the relation to active contours consider the following functional:

C (

L ( C )= | u ) | d u (30)