Biomedical Engineering Reference
In-Depth Information
have been studied [
15
,
34
,
35
], which focus on the dynamics between host immune
cells and tumors. These types of studies have their origin in the Lotka-Volterra
models established almost 100 years ago.
2.1
Lotka-Volterra Models for Tumor-Immune Interactions
The idea to use the qualitative theory of ordinary differential equations in
mathematical biology reaches back to 1920s when Lotka and Volterra formulated
a simple mathematical model in population dynamics theory. A good summary
published in 1997 by Adam and Bellomo [
1
] presents a summary of early work
regarding this approach to tumor-immune dynamics, and much of the original work
on this was done by Kutznetsov [
31
] and colleagues. We review it briefly.
Let
y
is population of its prey
(for example, one can imagine populations of wolves and rabbits in a forest).
Assuming that numbers
x
(
t
)
be the population of predator and
x
(
t
)
are big enough and that the predator and prey
populations are homogeneous, one can view them as continuous functions of time.
Let
(
t
)
,
y
(
t
)
Δ
x
(
t
)=
x
(
t
+
Δ
t
)
−
x
(
t
)
and
Δ
y
(
t
)=
y
(
t
+
Δ
t
)
−
y
(
t
)
be small variations of
populations during a certain period of time
Δ
t
.Taking
Δ
t
=
1 (for example 1
day) one can replace
Δ
x
(
t
)
,
Δ
y
(
t
)
by their derivatives, i.e. write
x
(
t
)
,
y
(
t
)
instead.
The Lotka-Volterra equations are given by
x
=
ax
−
bxy
,
(1)
y
=
−
cy
+
dyx
,
where
a
d
are some positive numbers.
The linear positive term
ax
in the first equation (prey) corresponds to exponential
growth; the negative predation term,
,
b
,
c
,
bxy
, describes the rate prey are lost and is
proportional to number of prey and predators in mass action form. In the second
equation (predator), the negative linear term
−
−
cy
corresponds to natural death, as
prey will not survive without prey, and
dxy
describes the growth of the predator
population proportional to prey and number of predators. The simple form of Lotka-
Volterra (LV) system is remarkable. It allows for investigation of the quantitative
and qualitative behavior for all of its solutions both analytically and numerically.
First, no chaotic behavior is possible according to Poincare-Bendixon theorem,
and, asymptotically, every non-periodic solution either goes to a fixed point or
approaches a limit cycle. Simple analysis shows that most solutions of LV system
are periodic, i.e. the population numbers
x
+
(
t
)
,
y
(
t
)
are oscillating around a certain
y
∗
. The stable, stationary solution is (0,0).
In 1994 Kuznetsov et al. [
31
] applied Lotka-Volterra ideas to cancer modeling,
where
E
x
∗
,
equilibrium state
x
(
t
)=
y
(
t
)=
the tumor cells
(prey). The equations, which are similar to the LV system, are written as follows:
(
t
)
represents the effector immune cells (predators) and
T
(
t
)
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