Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 128

a b

Rhodamine BSA Digitized Vessel

Line Scan Position Line Scan Position

100 μ m

c d

300

2000
1800
1600
1400
1200
1000
800
600
400
200
0

250

200

150

100

50

0

Fig. 5 Measuring blood flow velocities at single vessel resolution over a network [ 44 ]. ( a )
To analyze large networks, lines intersect multiple vessels and the velocities and fluxes are
deconvolved using the structural information (i.e., angles between the vessel and scan line). ( b )
Network trace of (a), showing the binary representation of the network. ( c , d ) Velocity and
flux maps, respectively, superimposed on the traced digital network to visualize the relationships
between flow and topology. Image size is 630 × 630 × 50 μ m

f d 3 x d 3 v

1
m f d t

( x + v d t , v + , t + d t ) − f ( x , v , t ) = Ω (14)

The problem to solve is how the collision operator has to be constructed. Here,

it can be mentioned that Boltzmann himself could not solve this problem. However,
he proposed the following thoughts that illustrate the problem.
Ω

Ω

is the number of particles that enter or leave the phase space volume μ =

[ . Following assumptions are used: Every collision results in

either a particle entering or leaving

x , x + d x ] ; [ v , v + d v ]

μ . Further, particles only collide with particles

outside gives

the probability that two particles with velocity v and v 2 collide and end up with
the velocities v 3 and v 4 while all velocities except v are outside

μ or particles from outside μ are collide into μ . W ( v , v 2 , v 3 , v 4 )

μ . Therefore, the