Figure 1.
Schematic representation of the mathematical model.
The mathematical model considers three levels of the differentiation hierarchy of cells: stem cells, progenitors and differentiated cells. These cell types are present in the system as healthy cells (left), drug-sensitive cancer cells (middle) and drug-resistant cancer cells (right). Stem cells give rise to progenitors which in turn give rise to differentiated cells. Additionally, cancer progenitors may have the ability to dedifferentiate to stem cells. The rate of dedifferentiation is denoted by γ. Drug-sensitive cancer stem cells produce drug-resistant cancer stem cells at rate u per cell division.
Figure 2.
The effect of dedifferentiation on the abundance of differentiated cancer cells and the probability of resistance.
In panel a, we show the abundance of differentiated cancer cells over time since the initiation of therapy. In panel b, we plot the probability of resistance versus time. Growth rates during treatment are
and
, and death rates are
for i = 0,1,2. Other parameters are rx = 0.005, ry = 0.008, d0 = 0.004, d1 = 0.008, d2 = 0.05, ax = 100d1, bx = 100d2, ay = 2ax, by = 2bx, kx = 1.2×106, ky = 6×107, u = 5×10−9, and ω = 0.1. The initial condition for the panels is found by simulating system (1) using the pretreatment parameter values and the initial condition x0(0) = 106, x1(0) = 108, x2(0) = 1010, y0(0) = 1, and y1(0) = y2(0) = 0. We simulate this system until detection time T, i.e., when y2(T) ≥1012, and then simulate the treatment phase by running system (1) with the initial conditions x0(T), x1(T), …, y2(T) and the treatment parameter values.
Table 1.
The effect of different hypothetical treatment strategies on cancer cell populations.
Figure 3.
The effect of different treatment strategies on the abundance of differentiated cancer cells and the probability of resistance.
The figure shows the abundance of differentiated cancer cells, y2, over time since initiation of therapy in panel a and the probability of resistance, P(t), as a function of time in panel b. We display four different treatment types that affect the cancer cell populations differentially. Treatment 1 represents a drug that affects only the production of cancer progenitor and differentiated cells, and
. Treatment 2 is a drug affecting all cancer cell types while not inhibiting cancer stem cells by a substantial amount,
while
, and
and
. Treatment 3 represents a drug that affects all cancer cell types and has a substantial effect on stem cells,
,
and
. Treatment 4 is a drug that decreases only the growth rate of cancer stem cells,
. The pre-treatment parameters are identical to those in Figure 2, and in both panels we set
for i = 0,1,2.
Figure 4.
The effect of different cancer stem cell treatment strategies on the time until disease progression.
The figure shows the time until the disease burden increases despite continuous therapy versus the birth rate (panels a and b) and death rate (panels c and d) of cancer stem cells during therapy. The pre-treatment growth parameters are identical to those in Figure 2, and also and
, lastly we set
. In panel a, we set
for i = 0,1,2, and u = 5×10−9. The parameter
varies along the x-axis and we consider three different values of γ. In panel b, we set
for i = 0,1,2, and γ = 10−4. The parameter
varies along the x-axis and we consider three different values of u. In panel c, we set
for i = 1,2, u = 10−7,
, vary
along the x-axis and consider three different values of γ. In panel d, we set
for i = 1,2, γ = 10−4,
, vary
along the x-axis and consider three different values of u.
Figure 5.
The effect of different cancer treatment strategies on the number of differentiated cancer cells in the presence of resistance.
Panels a and b display the tumor cell population after 500 days of treatment for two different types of treatment. In panel a we consider a treatment that can target all types of cells, and in panel b we consider a treatment that only targets progenitor and differentiated cells. Panels c and d display the tumor cell population after 5000 days of treatment for two different types of treatment. In panel c we consider a treatment that can target all types of cells, and in panel d we consider a treatment that only targets progenitor and differentiated cells. The pre-treatment growth parameters are identical to those in Figure 2 and the growth rate of the resistant cells is identical to that in Figure 4. In all four panels we set u = 5×10−9 and we set for i = 0,1,2. In panels a and c we set
and
, and vary
(i.e., the drug effect on cancer stem cells) along the horizontal axis. In panels b and d, we set
and
, and vary
(i.e., the drug effect on cancer progenitors) along the horizontal axis. In panels a and b, the vertical axis corresponds to the number of differentiated cancer cells after 500 days of treatment, including resistant and sensitive cancer cells. In panels c and d, the vertical axis corresponds to the number of differentiated cancer cells after 5000 days of treatment, including resistant and sensitive cancer cells.
Figure 6.
The relationship between dedifferentiation rate and pre-existing resistance.
Panel a considers the probability of pre-existing resistance versus the dedifferentiation rate γ for several mutation rates. We use the same pre-treatment growth rates as in Figure 2 and the same growth rates for the resistant cells as in Figure 4, and evolve the system until the tumor population hits size 1012 and then evaluate the probability of resistance at that time. Panel b plots the response of a tumor population to a drug, assuming that pre-existing resistant population of cells is present at beginning of treatment. The sensitive cells have the same growth rate as in Figure 2, and the resistant cell have the same growth rates as in Figure 4.
Figure 7.
The effect of the dedifferentiation rate on differentiated cancer cells during pulsatile therapy.
The figure shows the dynamics of differentiated cancer cells in response to a treatment strategy in which the drug is administered for 30 days, followed by a treatment holiday of 30 days during each pulse. Panel a shows the effects of a drug which inhibits cancer stem cell proliferation and their differentiation to progenitors, while panel b demonstrates the effects of a drug which additionally inhibits the production of differentiated cancer cells from progenitors. Parameters are for i = 0,1,2,
,
, and
in (a) and
in (b). For both panels, the off-treatment parameters are identical to those in Figure 2.