Modeling Tumor Blood Vessel Dynamics - Mathematical Methods and Models in Biomedicine - page 122

where R

= ∑ i R i represents the reaction rate. Key parameters like the diffusion rate
and solubility of oxygen in blood/tissue, as well as the oxygen consumption rate,
are reported in [ 52 ], based on studies in a rat model. Data related to the reaction rate
of oxygen and hemoglobin are reported in [ 23 , 47 , 77 ].

5.6 Simulating VEGF Production and Drug Delivery

Simulating VEGF production and drug delivery: Adding a passive reactive solute to
the model is straightforward. The difference between VEGF and oxygen is primarily
in the incorporation of different boundary conditions and release mechanism.
Sources of VEGF are identified as regions deemed sufficiently hypoxic within the
interstitium (

1mm Hg). The source is added as a local increase in concentration
flux of an additional distribution corresponding to VEGF. The transport of VEGF
follows the same advection-diffusion through the interstitium as oxygen, but with
its own, lower coefficient of diffusion. Drugs are handled in a similar way, but the
source is restricted to the inlet vessels, with concentrations varying according to an
imposed function. Drugs or particles of various sizes can be included by assigning
different vessel permeabilities and different interstitial diffusivities (set via bounce-
back rules). Figure 2 e demonstrates the ability of the model to simulate varying
interstitial permeabilities by adjusting bounce-back conditions for the extra-vascular
nodes. In this way, we can simulate delivery of drugs of various sizes.

<

5.7 Distal to Proximal Conductive Signaling

Though the precise mechanism is not well understood, Pries and Secomb make
a compelling argument that conductive signaling along vessel walls is biologically
plausible and more importantly, critical to the stabilization of their theoretical model
[ 65 , 66 , 68 ]. Similarly, we have found it necessary to incorporate a mechanism
to conduct an upstream signal from distal, hypoxic vessels to proximal feeding
arterioles. The approach is relatively simple. Regions of the vasculature that are
identified as being hypoxic are first segmented. Each distinct region is able to
generate an upstream signal with strength proportional to the region size and with
an exponential decay based on distance upstream as measured along the vessel wall.
Finally, signals from distinct regions are summed:

)= ∑ e − d i ( x ) / c S ( x log ( Area i ) × (12)

where d i is the distance to a particular region and c is decay constant.

Propagation is achieved by using an active contour technique adopted from
image processing [ 75 ]. A typical application of active contours in image processing
involves monotonically growing a small initial region boundary outward with a

( x )