Effect of cellular quiescence on the success of targeted CML therapy

Abstract

Background: Similar to tissue stem cells, primitive tumor cells in chronic myelogenous leukemia have been observed to undergo quiescence; that is, the cells can temporarily stop dividing. Using mathematical models, we investigate the effect of cellular quiescence on the outcome of therapy with targeted small molecule inhibitors.

Methods and results: According to the models, the initiation of treatment can result in different patterns of tumor cell decline: a biphasic decline, a one-phase decline, and a reverse biphasic decline. A biphasic decline involves a fast initial phase (which roughly corresponds to the eradication of cycling cells by the drug), followed by a second and slower phase of exponential decline (corresponding to awakening and death of quiescent cells), which helps explain clinical data. We define the time when the switch to the second phase occurs, and identify parameters that determine whether therapy can drive the tumor extinct in a reasonable period of time or not. We further ask how cellular quiescence affects the evolution of drug resistance. We find that it has no effect on the probability that resistant mutants exist before therapy if treatment occurs with a single drug, but that quiescence increases the probability of having resistant mutants if patients are treated with a combination of two or more drugs with different targets. Interestingly, while quiescence prolongs the time until therapy reduces the number of cells to low levels or extinction, the therapy phase is irrelevant for the evolution of drug resistant mutants. If treatment fails as a result of resistance, the mutants will have evolved during the tumor growth phase, before the start of therapy. Thus, prevention of resistance is not promoted by reducing the quiescent cell population during therapy (e.g., by a combination of cell activation and drug-mediated killing).

Conclusions: The mathematical models provide insights into the effect of quiescence on the basic kinetics of the response to targeted treatment of CML. They identify determinants of success in the absence of drug resistant mutants, and elucidate how quiescence influences the emergence of drug resistant mutants.

Figure 1. Biphasic decline of the CML cell population as a function of time, for parameters l = 1, d = 1.5, α = 0.01, β = 0.2, I₀ = 10⁸ and J₀ = 10².

The solid line represents log₁₀(x(t)+y(t)), and the dashed lines are log₁₀(g₊exp{λ₊t}) and log₁₀(g_-exp{λ_-t}) (See Text S1, Section 1.1 and 1.2 for details). The time of treatment in this case is T_treat = 72.1 and the switching time is T_switch = 5.1.

Figure 2. The relative amount of CML cells as a fuction of time, in patients treated with Imatinib.

The circles represent experimental data replotted from (a) Michor et al and (b) from Roeder et al ; they show the median values of BCR-ABL transcripts (relative to BCR transcripts in (a) and ABL transcripts in (b)). The vertical bars are the quartiles. The solid lines represent the fitted theoretical curves, formula (7) of Text S1, obtained by a mean-square procedure. The estimated parameter values are: (a) d-l = 0.0502 days⁻¹, β = 0.0065 days⁻¹, α = 10⁻⁵ days⁻¹, J₀ = 0.47; (b) d-l = 0.0278 days⁻¹, β = 0.0067 days⁻¹, α = 0.0004 days⁻¹, J₀ = 0.50. Here J₀ denotes the initial percentage of quiescent CML cells.

Figure 3. Data that document the decline of CML cells during imatinib treatment in two patients, taken from Roeder et al .

Figure 4. The probability of having no fully-resistant mutants at size N for different quiescence parameters.

The numerical simulations are performed according to the theory described in Text S1, Section 2.3. Each figure (a)–(d) shows the probability of no resistant mutants as a function of β (the rate of cell awakening), for 10 different values of α (the rate at which cells become quiescent), α = 0.1, 0.2, … and 1.0. (a) Treatment with m = 1 drugs; all the curves corresponding to different values of α are the same. The parameters are N₀ = 10⁷ and u = 10⁻⁷. (b) Treatment with m = 2 drugs, N = 10¹¹, u = 10⁻⁷. (c) m = 3 drugs, N = 10¹³, u = 10⁻⁶. (d) m = 4 drugs, N = 10¹³, u = 10⁻⁵. In all plots, we took M₀ = 10³, l = 1, d = 0. The reason we used different values of N and u for different values of m is because we chose the parameter regime corresponding to intermediate values of the probability of treatment success. When this probability is nearly 100% or nearly 0, then the dependence on α and β is less apparent and less meaningful.

Figure 5. A schematic demonstrating the number of cell divisions that is needed for a colony of cells to expand from 1 cell to N cells (in the figure, N = 6).

Empty circles represent cycling cells, and gray circles represent quiescent cells. Columns depict states of the colony in consecutive moments of time. The changes are marked by arrows. Two arrows stemming from one cell represent a cell division. A single arrow represents either a cell becoming quiescent or a quiescent cell waking up. (a) A colony without quiescence. (b) A colony with quiescence. In both cases we can see that it takes exactly N-1 = 5 cell divisions to expand to size N; however the process in (b) contains more “events”.

Figure 6. The expected number of one-hit mutants does not depend on the presence of quiescence.

(a) represents a colony with no quiescence, and there is quiescence in (b). The white triangles depict growing colonies of cells (cells with quiescence grow slower). The end size is the same in both cases. Dark triangles represent growing mutant clones inside the colonies. The total number of mutant colonies is the same in both cases (the same number of cell divisions). The mutant colonies in (b) have a longer time to grow, but at the same time they grow slower. Therefore the resulting frequency of mutants is the same in (a) and (b).

Figure 7. A schematic illustrating the argument stating that the probability to produce 2-hit mutants increases with quiescence.

Each rectangle represents a colony of cells. There are three moments of time shown, first we have N = 24, then N = 48 and finally N = 72. Circles represent wild-type cells, and stars–one-hit mutants. Gray shading denotes the state of quiescence for wild-type and mutant cells. In (a) we assume no quiescence (α = 0), whereas in (b) there is a probability to become quiescent (with α = 1/3). The number of cycling 1-hit mutants (empty stars) is the same in (a) and (b ) for the same values of N. The number of quiescent wild-type cells is given by the fraction α of all wild-type cells (e.g. 1/3 in (b)). At each moment of time, one of the cycling cells is picked for reproduction. We can see that the probability to pick a 1-hit mutant is always higher in (b) than in (a), because the fraction of cycling one-hit mutants increases as the tumor grows. Therefore, the probability to create a 2-hit mutant is higher in (b).