Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 17

a T cell to become infected. This probability depends on the distance to a virus
producing cell. So, in contrast to Zorzenon dos Santos and Coutinho [ 130 ], the
target cell does not have to be in the direct neighborhood of a virus producing cell
in order to become infected. In the model, the authors considered the spread of the
infection to be mainly driven by diffusion of free viral particles rather than cell-to-
cell transmission as assumed in [ 130 ]. The initialization of the infection with one
infected cell in the center of the lattice results in a radial wave of infection moving
to the boundaries of the grid. Dependent on the replacement rate of dead cells, the
long-term spatial pattern of the infection differs. If the replacement rate is high,
virus from the wave front can diffuse back leading to a chaotic steady state with
coexistence of target, infected, and dead cells. On the other hand, if the replacement
rate is low, a single wavefront moves to the boundaries of the lattice and the system
eventually recovers to a lattice full of target cells. In their model virus released in
a burst would only spread in the direct neighborhood of the bursting cell. Because
of this, infectivity depends on the concentration of T cells as neighboring cells can
become infected more easily if they are tightly packed. Compared to their model,
the authors found that a spatially averaged model, such as an ODE model [Eq. ( 1 )],
would overestimate viral infectiousness by more than an order of magnitude [ 120 ].
In the infected steady state of their cellular automaton, the authors found that for
most parameter values about half of the CD4 + T cells were infected, which is higher
than actual observations from clinical data suggest [ 41 , 116 ]. However, this might
be due to the fact that their model neglects any type of immune response, such as
those generated by CD8 + T cells, which might interfere with the spatio-temporal
dynamics of the infection. Furthermore, they assume the CD4 + T cells, i.e., the
target cells for HIV, are fixed in space. This is not a realistic assumption as T cells
are highly motile [ 73 , 77 - 80 ]. Clearly, if both virus and infected T cells move the
infection dynamics will be affected in ways that have not been studied.
Cellular Automata and the Modeling of Infections in Solid Tissue
Beauchemin [ 8 ] developed a two-dimensional cellular automaton to study the effect
of viral infection spread in a solid tissue [ 8 , 9 ]. Instead of HIV, the Beauchemin
model was applied to influenza A virus. Each site of the grid represented a target
cell that can become infected by influenza at a rate dependent on the number of
infected cells in the direct neighborhood of the site. Additionally, immune cells,
which are able to recognize and kill infected cells, move through the modeled tissue
environment. While in the two studies presented above, infection was initialized
by a single infected cell, Beauchemin [ 8 ] investigated how different distributions
of infected cells affect the spread of the infection in the tissue. Beauchemin [ 8 ]
distinguished between infected cells randomly distributed over the grid, arranged
in small isolated clusters or as a single large cluster. As infected cells can only
infect their immediate neighbors, the initial distribution of infected cells has a large
effect on the dynamics of the infection. If an infected cell is already part of a large
cluster of infected cells then most of its neighbors are already infected. This leads to