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Evolution of Proliferating Cells under Different Decaying
Behaviors of the Total Tumor Cells During a Course of Treatment
Mitra Shojania Feizabadi
Seton Hall University
Department of Physics
400 South Orange Ave, South Orange, NJ
U.S.A
[email protected]
Abstract: The two compartment model of tumor growth proposed by Gyllenberg and Webb can be modified in the
presence of anti mitotic agents. The evolution of proliferating cells is investigated considering this modified model
and the different decaying behaviors of the total cell population.
Key–Words: Cell Kinetics, Tumor Growth
1
2
Introduction
The kinetic of proliferating, P , cells and quiescent,
Q, cells can be expressed by the following interaction
diffusion ordinary equations [2, 3, 4]:
Several theoretical models have been developed to describe the evolution of untreated tumor cells or under
interaction with different chemotherapeutic drugs [1][11]. The population of tumor cells can be considered
as a construction of two subpopulations comprised of
the proliferating and the quiescent. Proliferating cells
are those which are in the active phase of a cell cycle.
Quiescent cells are found in the resting phase. Monitoring the growth evolution of the total number of tumor cells, and also considering the kinetics of each
subpopulation, leads to an analytical expression of the
evolution of each of the subpopulations [1]. Modeling of the dynamic of the proliferating cells improves
our understanding of the inner structure of a tumor
as a complex system. Furthermore, studying the evolution of proliferating cells during a course of therapy
helps in constructing a better therapeutic protocol and,
therefore a better understanding of the therapeutic results.
dP
= [β − µp − ro (N ) − ap (1 − exp(−u))] P +ri (N )Q
dt
(1)
dQ
= ro (N )P −[ri (N ) + µq + aq (1 − exp(−u))] Q,
dt
(2)
N =P +Q
(3)
In the above equations, (ri (N ), ro (N )): the transition rate functions, µp and µq are natural cell decay rates for each compartment, β represents the reproduction rate of proliferating cells, and N is the total number of cell populations during the course of
therapy. In these equations aq (1 − exp(−u)) and
ap (1 − exp(−u)) are the Q and P cells response
function to the drug. The diffusion of the drug is
also considered to exist exponentially in the form of
u(t) = u0 exp(−d · t), while u0 is the initial drug
concentration and d is the per capita decay rate. The
explicit expression for the proliferating cell subpopulation is obtained after some mathematical calculations as follows:
While an untreated tumor growth curve can be expressed by a logistic or a Gompertizan function, no
specific mathematical function can represent the decaying behavior of the total number of tumor cells
during a course of chemotherapy. In this study, three
behaviors, linear decay, exponential decay and a bell
shaped decay are considered for the evolution of the
total tumor cell population. Considering the above assumption, and equipped with the existing model, Feizabadi et al. [2] presents the evolution of the proliferating cell populations. In the following, behavior of the
proliferating cells is analyzed through a model simulation.
ISSN: 1790-5125
Theoretical Model
P2 (t) =
30
dN/dt + [µq + aq (1 − exp(−u))] N
S + (aq − ap )(1 − exp(−u))
(4)
ISBN: 978-960-474-141-0
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where
S = −(aq − ap )(1 − exp(−u0 ))
+
dN0 /dt
N0
[µq + aq (1 − exp(−u0 ))] +
P0
P0
In the above expression, N0 and P0 are the size
of P and Q cells at the beginning of the therapy and
N expresses the behavior of the total number of cell
populations during the course of therapy [2]. The behavior of P cells is simulated and analyzed when N
follows a different decaying behavior.
3
A
B
N : Total Cells
P : Proliferating Cells
8.  108
5. 107
7.5  108
Simulation and Conclusion
4. 107
7.  108
As indicated in equation 4, together with some parameters, the evolution of the proliferating cells is traceable upon knowing the evolution of the total number
of tumor cells. In this section, the decaying behavior
of the total number of tumor cells is considered to be
best fitted by one of the following categories:
3. 107
6.5  108
Time
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Time
41
42
43
44
43
44
45
1. 107
45
N : Total Cells
P : Proliferating Cells
7.  10 7
8.  108
6.  10 7
1. N = N0 − kl .t
6.  108
2. N = N0 exp(−ke .t)
5.  10 7
4.  10 7
4.  108
3.  10 7
3. N = N0 exp[k1 /k2 .(1 − exp(k1 .t))]
2.  10 7
2.  108
1.  10 7
In the above expressions, N0 is the size of the
tumor cells at the beginning of the therapy. It is
considered that the growth of the tumor follows the
Gompertz function and the therapy starts at t = 40
units of time. Therefore N0 can be expressed as:
N0 = exp[k+ /k− (1 − exp(−k− t(= 40)))], where
k+ and k− are the growth and retardation rate constant during the in-growth phase. Considering these
linear, exponential, and bell shaped decaying behaviors as N , Equation 4 is simulated to different choice
of parameters. The course of therapy is considered to
be started at t = 40 and ended at t = 45.
The tumor growth curve and the corresponding
proliferating subpopulations are shown in figure 13. It is observed that the linearly decaying behavior
of the total tumor cells produces a monotonically decreasing behavior of the proliferating cells. In the case
of exponentially decaying, and the bell shaped decay,
the decreasing the size of the proliferating cells begins with a brief delay and a small growth in size can
be seen in a short period even after the start of the
therapeutic phase. In the next step of simulations, a
sharper decay of the total tumor cells is also examined (dashed lines). In this case, for the first category,
a slight change can be seen in the size of the proliferating cells at the beginning of the therapy while the
reduction in size of proliferating cells becomes more
ISSN: 1790-5125
42
Time
Time
41
42
43
44
45
N : Total Cells
41
42
43
44
45
41
42
43
44
45
P : Proliferting Cells
8.  108
1.2  10 8
7.5  108
1.  10 8
7.  108
8.  10 7
6.5  108
6.  10 7
6.  108
4.  10 7
Time
Time
41
42
43
44
45
Figure 1: A: decaying evolution of the total tumor cells, B:
evolution of the corresponding proliferating cell subpopulations. Common parameters: k+ = 2.76, k− = 0.134,
ap = aq = 0.9, µq = 0.2, u0 = 1, d = 1.5. The first row,
kl = 0.4.103 (solid line), and kl = 0.4.108 (dashed line).
The second row,ke = 1 (solid line), and ke = 1.1 (dashed
line).The last row, k1 = 0.3 and k2 = 15 (solid line), and
k1 = 0.5 and k2 = 15 (dashed line).
31
ISBN: 978-960-474-141-0
P : P roliferating Cells
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6.  10 7
4.  10 7
[5]
2.  10 7
Time
41
42
43
44
45
[6]
Figure 2: Proliferating cells evolution for the three categories of linear, exponential and bell shaped decay.
[7]
obvious at the end (t=45). In the exponentially decaying category, a very slight increase in the decaying rate
causes a noticeable decrease in size of the proliferating cells at the beginning of the therapy, while no great
change is detected at the end. In the last category,
when a shaper decay is considered for the bell shaped
evolution, the size of the proliferating cells shows a
noticeable growth at the beginning of the therapy and
even a slight growth at the end. In summary, the evolution of the proliferating cells is variously constructed
under the different total tumor cells’ decaying behavior.
In Figure 2, the evolution of the proliferating cells
under a linear, exponential or a bell shaped decay of
the tumor cells is compared. It can be seen that for
the exponentially decaying behavior, the population
of the proliferating cells is smaller in size at the end of
the therapy. However, the population of proliferating
cells is relatively large at the first half of the therapy as
compared to other categories. This indicates that the
periodic time of a therapy can be crucial in controlling
the size of the proliferating cells.
In this work, the process of the decaying size of
proliferating cells is investigated by combining the dynamic of inter tumor subpopulations and the evolution
of the total number of tumor cells during the course
of therapy. Monitoring the behavior of proliferating
cells during a course of therapy is informative in order to better implement a more effective approach to
treating a tumor.
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ISBN: 978-960-474-141-0