Biomedical Engineering Reference
In-Depth Information
the site of inflammation. It is important to note that the white blood cell may release vaso-
active compounds to increase the diameter of the intercellular cleft.
Many groups try to model the adhesion of white blood cells to endothelial cells using
the empirical data discussed previously. Binding association and dissociation constants as
well as the forces acting on the cell are relatively easy to model. However, the selectin den-
sity and actual binding of the white blood cell to the endothelial cell wall are more prob-
lematic. Selectin density can be assumed to have some initial value, but this can change
drastically based on the conditions described above (shear stress and cytokines, among
others). This level of involvement within the model can greatly increase the number of
simultaneous equations to solve. Also, probability functions are needed to model the asso-
ciation and dissociation of bonds along with the likelihood for contact. Probability func-
tions are never that accurate and typically depend on the selectin density and the distance
between the white blood cell and the endothelial cell. As the reader can imagine, as the
selectin density increases and the distance decreases, there should be a higher likelihood
for the selectin adhesion to occur within the model. However, the exact probabilities are
not known and assumptions have to be made to model this phenomenon. Therefore, the
accuracy of these assumptions limits the predictive capability of the model.
It is typical in models of this kind to initiate the adhesion through one receptor-ligand
binding event. This event is governed by the receptor density and/or distance between
cells, as discussed above. All remaining association and dissociation events are then
described by receptor kinetics and the density of the receptors/ligands present. This prin-
cipal can be used to define the receptor association/dissociation because it is assumed that
if one bond can be formed, then the other receptors are close enough to bind to other
expressed ligands. Also, it is typical to make the time steps in the model small enough
that the quantity of receptors-ligand events can only increase by one, decrease by one, or
remain the same during each subsequent time step (this significantly simplifies the model-
ing). In equation form, the quantity of bonds in a time step would be the summation of
the amount of bonds in the previous time step, plus the association of a new bond (which
is dependent on kinetics) and the dissociation of any one bond (again based on kinetics).
This type of modeling will be discussed in more detail in Chapter 13.
END OF CHAPTER SUMMARY
7.1
To model red blood cell oxygenation within the pulmonary capillaries, there are five main
diffusion events that we should consider. The first, alveolar diffusion, is governed by Fick's
laws of diffusion:
-
52
Dr
-
and
2
C
@x
2
Solving this differential equation, we can obtain a measure for the time needed to reach
equilibrium. Using this formulation, we can show that alveolar diffusion occurs rapidly and
@
C
@t
5
D
@
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