Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 132

a b

x x

y y

z z

ε ε

=.25

=1

Fig. 6 Illustrating the superellipsoid. Panels ( a )and( b ) show the boundary of a superellipsoid,
defined as the surface S ( x ; ε )= 1, while the shape parameter ε is set to 1 and 0.25, respectively

where the parameter controls the shape of the superellipsoid. One advantage of

this shape model is that it allows for the flexibility of probing and traversing tortuous
vessels and branches with a softer ellipsoidal shape, while enhancing accuracy when
vessels are more tubular (Fig. 6 ).
It is assumed that the image data denoted I

ε

3 , contain a
vessel segment characterized by two homogeneous regions separated by a boundary
that is well modeled by the superellipsoid. Hence, a piecewise constant vessel model
is defined as

( x ) , with domain Ω ⊂ R

V ( x ; β , I F , I B )= I B − ( I B − I F ) × H ( 1 − S ( x ; β )) (26)

where I F and I B correspond to the foreground and background intensities and S ( x ; β )

has been generalized from ( 25 ) to express local pose via the parameter set β =

μ

, σ , φ , ε , which includes 10 degrees of freedom in 3-D, i.e., 3 rotation

T

( φ ) ,

3 scale is the Heaviside

function, which evaluates to unity when the argument is positive and zero otherwise.

( σ ) ,3shift ( μ ) , and 1 shape parameter ( ε ) . Finally, H ( · )

B.2 Estimation of the Superellipsoid Vessel Model

A maximum likelihood approach is used to estimate the superellipsoid model
parameters. The log-likelihood equation is given by

L ( β , I F , I B )= log f ( I ( x ) − I F ) d x + log f ( I ( x ) − I B ) d x (27)

R ( β ) Ω \R ( β )