Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 121

nearly two orders of magnitude greater than the dissolved oxygen in an equivalent
volume of plasma [ 93 ]. Therefore, to accurately simulate oxygen delivery to tissue,
a mechanism to simulate the oxygen content of hemoglobin in blood must be
included. A particle suspension is a natural choice. Figure 2 c shows oxygen delivery
in a domain containing an idealized 2-D capillary bed: a single arteriole and single
venule span the domain, connected by parallel capillaries. Plasma and RBC particles
flow into the network, delivering oxygen based on local oxygen tension. The oxygen
then convects and diffuses through the tissue (color map). In Fig. 2 d, a 3D network
was acquired from actual tumor tissue, binarized with our vessel tracing algorithm
and perfused with simulated blood.
Our ongoing work has focused on highly accurate modeling of cell-cell and
cell-wall interactions in blood flow. For example, important phenomena such as
inertial lift forces and RBC aggregation evolve naturally from these complex models
based on the underlying physics. In the current simulations, we utilize a simpler
particle model and impose the same physics as external constraints. This allows us
to focus on the primary problem of interest, oxygen delivery and consumption in
the interstitium. Hence, we utilize an approach whereby RBCs simply advect with
the underlying fluid. To describe the position p j of each particle we also include a
Brownian motion along with an inter-particle and wall repulsion force:

t u ))+ k = j F p ( p i ( t ) , p k ( t )) p j ( t + δ t )= p j ( t )+ δ ( p j ( t ))+ w σ + F wall ( p j ( t

(10)

Here, the underlying fluid velocity u provides the advective component, and

w σ is a Gaussian white-noise process whose integration over time is equivalent to
a Brownian motion. The wall force density is a nonlinear function involving the
magnitude and gradient of the Euclidean distance map. This map gives the minimum
distance between any point in the domain and the nearest wall point. The wall force
is tuned to ensure an appropriate cell-free layer. Inter-particle forces are defined
similarly over a small local neighborhood around each particle.

( p j ( t ))

5.5 Metabolism

Accurately modeling the kinetics of oxygen delivery and consumption are a key
aspect of the proposed model. Modeling reactive flows is straightforward using
LBM and in its simplest form amounts to a modification of the collision operator to
include a change in density/concentration at each time step [ 3 , 79 ]:

f i

) +

1
τ

f eq
i

f i ( x + c i δ t , t + δ t )= f i ( x , t ) − ( x , t ) − ( x , t R i (11)