Spatial Aspects of HIV Infection - Mathematical Methods and Models in Biomedicine - page 13

they observed the development of local hot spots for lymphocyte proliferation on
the spatial grid even if the total antigen concentration was below the ODE predicted
threshold for lymphocyte proliferation.

4.2 Partial Differential Equations in Viral Dynamics

Another approach to describing the spatial aspects of viral infection is to use PDE,
which describe the change of cell or virion populations in time and space. Models
based on PDEs eliminate the arbitrary spatial discreteness introduced by the ODE
models. PDE models of viral infection dynamics have been developed by Frank
[ 32 ], Strain et al. [ 120 ], and Graziano et al. [ 38 ].
Strain et al. [ 120 ] modeled the diffusion and binding of virions to target cells
within a host by a system of PDEs given by

∂ V f

∂ t =

D Δ V f − β TV f − cV f ,

∂ V b

∂ t = β

TV f . (4)

( , ) ( , )

Here, V f x t and V b x t denote the concentrations of free and bound virions,

( , )

respectively, and T the concentration of available target cells. D is the diffusion

coefficient of free virions in the system, and

x t

and c the transmission rate and
clearance rate of free virions, respectively. As done by Funk et al. [ 34 ], Strain et al.
[ 120 ] used their model to compare the mean-field expectations generated by this
model to spatially explicit in silico simulations of HIV infection ( see Sect. 5 ).
Graziano et al. [ 38 ] used a finite element (FE) method to study lymphocytes
and viral load as a viscous incompressible fluid occupying a two-dimensional
rectangular area. Their aim was to examine the effect of the spatial distribution of T
cells and the HIV viral load on HIV progression during an infection and to include
the effects of therapy with a reverse transcriptase inhibitor (RTI). This FE-approach
is comparable to the approach proposed by Funk et al. [ 34 ]. The modeled area is
subdivided into a predefined number of subregions, the finite elements, which do
not have to be of regular size. T

β

( , , ) ( , , )

are defined as the solutions
for the CD4 + T cell count and the viral load, respectively, for each position

x y t and V x y t

)
in the modeled area at time point t given specific conditions for the boundaries
of the area. These functions are then discretized to approximate the solution for
each FE-element. The authors compared the solutions of their model to clinical data
and by this determined the robustness of their model. They showed that there was
no significant difference between the predicted half-life of the CD4 + T cell count
and the overall half-life estimated from1,500 patients. Using their approach, they

( ,

x y