Modeling treatment-dependent glioma growth including a dormant tumor cell subpopulation

We analyze the proportions of two different GBM cell phenotypes, dormant (D, please refer to Table 1 for symbols used in the equations) and rapidly proliferating (P) cells, in a mathematical model including the influence of different treatment conditions. In the following, we characterize the cells in terms of their fitness, which we define as the growth rate in comparison to cells of the other phenotype. Dormant cells always have a very low or even zero growth rate ε, which we assume to be independent of the exact composition of the population and the treatment condition. Rapidly proliferating P cells, on the other hand, have a very large fitness advantage compared to dormant cells, which means they proliferate much faster, but they also compete with each other for space and resources. Facing another P cell, a focal P cell has an intermediate fitness, which we assume to be still much larger than the growth rate of D cells ε. Their fitness therefore depends on the relative fraction of D vs. P cells. Due to the very slow growth of D cells, P cells will represent the vast majority of glioblastoma cells in the absence of treatment.

Under treatment conditions, however, the population composition changes. Even though D cells still have the same very low (or zero) growth rate ε, P cells experience a fitness cost λ due to treatment. The reduction of the fitness due to treatment only applies to P cells, because cytotoxic drugs mostly affect rapidly dividing cells. The fitness cost parameter λ can be adjusted to account for the strength of the applied treatment. In principle we can continuously vary this parameter. However, for simplicity we focus on two different treatment strategies: In high dosage (HD) chemotherapy the treatment strength parameter λ is large compared to the growth rate of the P type. Since high dosage chemotherapy has strong side effects for the whole organism (for GBM: [3]), in reality this treatment strategy cannot be maintained for extended time periods. Therefore, strong treatment needs to be applied in turns with weaker or no treatment. For low dosage (LD) chemotherapy, λ means only a small reduction of the growth rate of the P cells. As the side-effect stress to the organism should also be lower, this treatment regime could be applied for longer time spans.

Dormant (D) and rapidly proliferating (P) phenotypes in glioblastoma and their aforementioned interactions can be described by the following payoff matrix [18]:

This matrix gives the fitness for each type if confronted with any of the two other types. Here, we find for example that the fitness of a focal P cell interacting with a D cell is 1 − ε − λ, which includes both the small or zero growth rate of D cells ε and the fitness cost for P cells under treatment λ.

In the following, we include these fitness effects and phenotypic conversion into a set of ordinary differential equations. In general, the growth of a whole cell population can be explained in terms of a differential equation that describes the change in the number of individuals over time

Here n is the number of individuals, t is the time and r(n, t) is the growth rate, which can itself depend on the number of cells and the time.

At first, we focus on the number of D cells, n _D , in the population over time, which have a very small but constant growth rate ε

For P cells on the other hand, the growth rate of n _P, given by the average fitness from the payoff matrix (weighted to the cell fractions), changes with the composition of the population

Since the system under consideration is constrained, both in terms of nutrients and space, in reality the cell population only grows exponentially as indicated by the growth equations in the very beginning of the process where the constraints regarding space or nutrients are negligible. However, we are mainly interested in the fraction of D cells \( {x}_D=\frac{n_D}{n_D+{n}_{\mathrm{P}}} \) in the population and vice versa the fraction of P cells \( {x}_{\mathrm{P}}=1-{x}_D=\frac{n_{\mathrm{P}}}{n_D+{n}_{\mathrm{P}}} \). To obtain the change in fractions for both types, we subtract the average growth rate \( \overline{f} \) of the population from both individual growth rates,

From this we obtain two differential equations for the fractions of D and P cells,

Next, we include the spontaneous conversion between phenotypes with a constant rate σ, which is independent of the cellular growth. This leads to an additional term to the differential equation of both phenotypes

These equations have the important difference to the usual replicator-mutator equation [15] that phenotype conversion is a spontaneous process with a constant rate and is independent of the growth in the population. This allows conversion from D to P even if D cells do not grow at all.

Using these equations, we model different therapy schedules combining different treatment strengths in different cycling time plans. Since the equations are nonlinear, we use numerical integration with Odeint of the Python library Scipy¹ to examine the temporal dynamics of the system under different treatment regimes. Additionally we analytically determine the fixed points of the system and their stability.

The temporal dynamics of the proportion of D against P cells in GBM strongly depends on the treatment conditions. Therefore, we first analyze the fixed points of the dynamical system and how they change for different treatment strengths λ, without considering possible conversions of phenotypes. The fixed points mark a stable equilibrium between the portions of P and D cells under certain, predefined conditions and are found by setting Eq. 1 to zero. Of particular interest are stable fixed points, as the system returns into this state again after a small perturbation [26].

For our system, there is only one stable fixed point for each treatment condition (Fig. 1a). If we consider the case of no phenotype conversion, σ = 0, we can give the exact position of this point for each treatment condition λ. The fraction of D cells at the fixed point is then given by

For small treatment strengths λ the fraction of D cells in the population at the stable fixed point is zero, but after reaching a threshold, the fraction of dormant cells increases linearly with λ until the whole population consists of dormant cells at very high treatment strengths.

With conversion between the two phenotypes (σ > 0), the analytical calculation for the stable fixed points is more difficult and the result is not instructive. By contrast to the previous results without phenotype conversion, there is always a small proportion of dormant cells in the population, even at very low treatment strengths. The proportion of dormant cells at the fixed points increases immediately with increasing treatment strength until it approaches a maximum at high treatment strengths. For both D cell growth rates ε (ε = 0, orange lines; and ε = 0.1, blue lines) the population composition is very similar or even the same without phenotype conversion at very small or large treatment strengths λ. In contrast, the largest effect of ε on the population composition is at intermediate values of treatment strength.

The average fitness \( \overline{f} \) for the whole tumor cell population including P and D cells decreases linearly from the maximum at treatment strength λ = 0 until it reaches the minimum of \( \overline{f}=\varepsilon \) at the point where the fraction of dormant cells in the population starts to increase (Fig. 1b). Interestingly, with spontaneous conversion σ > 0, the average fitness at the fixed point can become smaller than ε and even negative for high treatment strengths, potentially leading to a shrinking tumor. This is caused by conversion of D cells into P cells which are then susceptible to treatment.

To test our mathematical model of phenotype composition upon treatment, we used LN229 cells as an experimental in vitro model. We treated these cells for 10 days with temozolomide (TMZ), the most commonly used cytotoxic drug in glioma therapy. In a first step, we focused on different treatment strength and analysed the portions of surviving cells in comparison to control cultures and the percentage of cells expressing H2BK (histone cluster 1), a marker of glioma dormancy [24, 25], alongside with incorporation of BrdU in the late treatment phase (day 6–10). In general, after 10 days of treatment, samples stimulated with 5, 50 and 100 μg/ml TMZ had significantly less total cell numbers than control treated cells (Fig. 2a). By immunocytochemistry of H2BK, we could detect and quantify the fraction of dormant cells within the cultures, and by adding BrdU to the cells from day 6 of treatment and immunocytochemical staining of BrdU, we could in parallel mark cells that incorporate BrdU in the DNA (examples of microscopic pictures in Fig. 2b). While DMSO-treated control cells showed a low fraction of H2BK-positive cells (mean: 9.7% ± 3.5), TMZ treatment yielded increased numbers of dormant cells reaching a plateau at high concentrations (5 μg/ml: mean 26.8% ± 9.0, 50 μg/ml: mean: 82.8% ± 5.3, 100 μg/ml: 87.7% ± 8.0, compare Fig. 2b, grey graph portions). In parallel, we investigated the incorporation of BrdU in the DNA and determined a high portion (66.0% ± 7.8) of BrdU positive cells in the control cultures and lower portions upon TMZ treatment (5 μg/ml: 53.6% ± 14.5; 50 μg/ml: 33.4% ± 5.3; 100 μg/ml: 33.7% ± 10.1, compare Fig. 2b, hatched graph portions). Interestingly, BrdU incorporation also took place in TMZ treated cultures, so that staining for the dormancy marker H2BK and for BrdU could be observed in the very same cells (compare examples of microscopic photographs in Fig. 2b) indicating that cell cycle arrest may occur after the S phase of the cell cycle. Together with our experiments described in the following section and Fig. 2c, showing that dormant cells hardly divide within our experimental time frame, these observations suggest that dormant glioma cells are not or only very slowly cycling. Furthermore, taking into account that we use a clonal cell line, the occurrence of dormant cells needs to be a phenotypic adaption to the environmental conditions as all cells are genetically homogenous (as proven by routinely STR profiling, compare Materials and Methods section).

To investigate if the conversion to a dormant phenotype depended on the cell density, initially, we determined in a DiO retention assay that nearly all cells (98.3% ± 1.2) retain the green fluorescent dye when treated with 100 μg/ml (“high dose”) TMZ for 10–12 days, while in control cultures (treated with equal volumes of the solvent DMSO) only 2.9% ± 1.7 retained the dye (Fig. 2c, left part). The vital dye is included in every cell at the moment of staining, and is transferred to every daughter cell upon cell division. However, this means the staining is diminishing after several divisions of rapidly proliferating cells, but retained in non-proliferative or very slowly cycling dormant cells. Thus, assuming that nearly all cells that survive treatment with 100 μg/ml TMZ are dormant in our particular setting, we determined the relative incidence of phenotype conversion and the influence of the cell density on this conversion factor by determination of TMZ surviving cells in relation to (extrapolated) total cell numbers of control (DMSO treated) cultures. In our experimental setting, a portion of 0.68% ± 0.13 cells of initially seeded 50,000 LN229 glioblastoma cells survived this high dose treatment, while in cultures of initially seeded 150,000 cells, the portion of surviving cells was nearly similar (0.66% ± 0.13; Fig. 2c, right part) underlining the assumptions of our theoretical model.

Thus, treatment with TMZ significantly reduced total cell numbers of LN229 cells, while the share of dormant cells within the culture, as detected by the dormancy marker H2BK, was drastically elevated. The incidence of conversion to dormancy did not depend on cell densities in our particular experimental setting.

Next, we use our model to analyze the dynamics of the population composition for periodically changing treatment conditions. One example trajectory for a growth rate for D cells ε = 0 and a conversion rate between phenotypes σ =0.1 is depicted in Fig. 3. The fraction of D and P cells in the population alternates between the fixed points corresponding to the momentary treatment condition. The trajectory starts with a phase of no treatment, which is characterized by a high average growth rate and a cell population composition of mostly P cells and only very few D cells. After the first phase of unconstrained growth large parts of the tumor are removed (e.g. by surgery), leaving only a small number of cancer cells. Under the following high dosage treatment conditions, the dormant phenotype has the highest fitness and takes over the population. The relative fraction of D cells will increase until the steady state under high treatment conditions is reached. The impact of treatment on P cells leads to a strong initial decline in average growth rate, until the population has a significant proportion of dormant cells and the growth rate starts to recover slightly.

Under the following low dosage treatment conditions, P cells (making up a small fraction of the whole population at the end of the high dosage treatment) are less affected by the treatment and now grow faster. The average growth rate will have a maximum when then relative fraction of P cells in the population is still low, since they have a competitive advantage over D cells, and then declines afterwards towards an equilibrium well above the high dosage growth rate. Accordingly, the total number of cells increases strongly again in this regime.

Switching the order of high dosage and low dosage treatment only has a small effect on total number of cells: If treatment starts with low dosage, the system will go into a state with a slightly higher fraction of dormant cells, which makes it less susceptible to the following high dosage treatment. Starting with low dosage therefore does not help to reduce the tumor size.

In Fig. 4 we compare three different treatment schedules: just one switch from initial high (H) dose to low (L) dose treatment (HHHHHHLLLLLL, Fig. 4a, each instance of the letter H or L corresponds to the same time interval), slow cyclic switching (HHHLLL, Fig. 4b), and fast cyclic switching (HLHL, Fig. 4c) for two different growth rates of D cells (left panels ε = 0 and right panels ε = 0.1). In case of only switching once, the fixed points for each treatment are quickly reached. At high dosage treatment the number of cells increases very slowly or even decreases. In the following low treatment phase, however, P cells take over growing particularly fast and jeopardizing any positive effect from the previous strong treatment. This is true for both treatment strengths of the high dosage phase.

For the fast switching treatment schedule (HLHL, Fig. 4c), the fixed point of the population proportion is not reached before the treatment changes again. Therefore the population dynamics stays between the two stable fixed points for the two treatment regimes, but does not reach them. By contrast, in the slow treatment switching regime (HHHLLL, Fig. 4b) the fixed points for both high and low dosage treatment are reached such that the composition of the cell population essentially resembles the case of just one switching event (Fig. 4a). However the time spent at these fixed points is still significantly reduced compared to only a single switch.

The bottom panels of Fig. 4a, b and c show the total number of cells based on the average fitness of the population under the assumption of exponential growth. When the growth rate is positive the cell population grows, otherwise it shrinks. Interestingly, the average growth rate of the population is well below zero only for a short period during the high dosage treatment and only if the share of P cells is still very high and the fraction of D cells in the population is small. However, in this regime the fitness recovers fast and approaches equilibrium with an average fitness close to zero, such that the total number of cells does not change anymore. The strongly negative growth rate directly after switching to the high dosage treatment is therefore the reason why the number of cells for quickly changing treatment regimes is significantly smaller than for slowly changing treatment cases.

To systematically examine the effect of switching treatment cycles on the growth rate of the population, we analyze the temporal dynamics of the population size for varying treatment cycle durations and two different growth rates of D cells ε (Fig. 5). Unsurprisingly, a lower growth rate of D cells has a diminishing effect on the overall growth of the cells. For increasing treatment cycle length cancer cells have an increasing overall growth, while maintaining the same total high and low dosage treatment durations. The overall growth rate approaches a maximum with increasing treatment cycle length when the dynamics reaches equilibrium in each cycle.

Taken together, our mathematical model allows us to theoretically predict the fitness and proportions of rapidly proliferating and dormant tumor cells under different treatment conditions. Strengthening our theoretical assumptions we could exemplarily show the effect of high and low chemotherapy doses on the cell numbers and the proportion of a dormant cell phenotype in cultured GBM cells in vitro. Simulating different therapy schedules, we observed that fast switching of low and high treatment doses yields a lower total tumor cell number at equal total drug dose in comparison to low switching schedules.