In silico simulation of the effect of hypoxia on MCF-7 cell cycle kinetics under fractionated radiotherapy

Abstract

The treatment outcome of a given fractionated radiotherapy scheme is affected by oxygen tension and cell cycle kinetics of the tumor population. Numerous experimental studies have supported the variability of radiosensitivity with cell cycle phase. Oxygen modulates the radiosensitivity through hypoxia-inducible factor (HIF) stabilization and oxygen fixation hypothesis (OFH) mechanism. In this study, an existing mathematical model describing cell cycle kinetics was modified to include the oxygen-dependent G1/S transition rate and radiation inactivation rate. The radiation inactivation rate used was derived from the linear-quadratic (LQ) model with dependence on oxygen enhancement ratio (OER), while the oxygen-dependent correction for the G1/S phase transition was obtained from numerically solving the ODE system of cyclin D-HIF dynamics at different oxygen tensions. The corresponding cell cycle phase fractions of aerated MCF-7 tumor population, and the resulting growth curve obtained from numerically solving the developed mathematical model were found to be comparable to experimental data. Two breast radiotherapy fractionation schemes were investigated using the mathematical model. Results show that hypoxia causes the tumor to be more predominated by the tumor subpopulation in the G1 phase and decrease the fractional contribution of the more radioresistant tumor cells in the S phase. However, the advantage provided by hypoxia in terms of cell cycle phase distribution is largely offset by the radioresistance developed through OFH. The delayed proliferation caused by severe hypoxia slightly improves the radiotherapy efficacy compared to that with mild hypoxia for a high overall treatment duration as demonstrated in the 40-Gy fractionation scheme.

Introduction

Clinical decisions for breast cancer management require comprehensive data coming from multiple laboratory and imaging studies. Digital mammography is a commonly used imaging technique for detection of early-stage breast cancer through observation of characteristic tumor morphology and microcalcifications [1]. Laboratory tests for breast cancer diagnosis and evaluation include blood chemistry, liver function, and ratio of blood urea nitrogen to creatinine. Such tests are essential in characterizing the tumor pathology. Depending on the cancer stage and other pathological information, evidence-based treatment techniques are selected to yield the best tumor control and patient survival [2]. Among the widely used treatments today for breast cancer are radiotherapy, chemotherapy, and surgery [3]. Radiation therapy, in conjunction with hormonal therapy and lumpectomy or mastectomy, is advised to early-stage breast cancer patients [2]. Tumor genetic studies, such as microarray gene expression of biomarkers, also assist in the prognosis of the disease [4].

Important considerations in radiotherapy treatment recommendations are the tumor radiobiological properties such as cell cycle phase distribution, intracellular oxygen tension, radiation-induced damage repair rate, and tumor population kinetics [5]. It was demonstrated in numerous experiments that different stages of the cell cycle exhibit different radiosensitivies. Cells in the S phase manifest peak radioresistance, while the mitosis (M) and the G2 stages are the most radiosensitive phases of the cell cycle [6]. The intracellular oxygen tension affects the radiosensitivty based on the oxygen fixation hypothesis (OFH) mechanism. We refer to intracellular oxygen tension as simply oxygen tension throughout the rest of the article. DNA damage/radicals produced from DNA interaction with radiation-induced hydroxyl radicals are easily reparable. OFH mechanism is described as the binding of a DNA radical with an oxygen molecule, which renders the repair of the DNA damage to be difficult or impossible [7]. Proper DSB repair, either by homologous recombination repair or nonhomologous end-joining, minimizes the incidence of lethal chromosome aberrations and apoptosis activation [5]. Along with cell population kinetics, these radiobiological factors alter the response to different fractionation schemes. A fractionation scheme is defined by a particular combination of dose per fraction and number of fractions with the total absorbed dose given by the product of the two.

Cell cycle progression from the G1 phase to S phase is a prerequisite to DNA replication and subsequently, cell proliferation. Growth factors are necessary to initiate the cell cycle process in the G1 phase, and to ensure that adequate resources and molecular primers are available to carry out the following cell cycle activities. Preparatory events such as licensing of DNA replication origins and DNA damage sensing occur in the G1 phase [8]. Some G1 cell cycle checkpoints such as mammalian target of Rapamycin (mTOR) signaling pathway are in place to monitor the nutritional adequacy [9].

Analyses from various experimental results showed that radiosensitivity changes with cell cycle stage [10, 11]. For a constant absorbed dose, survival fraction of synchronously dividing cells fluctuates depending on the cell cycle phase in which the radiation exposure is delivered [12]. Although the cause of the dependence of radiosensitivity on cell cycle phase during irradiation is still not widely understood, it was suggested that such dependence may be attributed to the presence of error-free homologous recombinant repair (HRR) of DSBs [5]. Prior to DNA replication in S phase, the primary DSB repair mode is through non-homologous end-joining (NHEJ). Incorrect DSB repair through NHEJ may occasionally result in lethal chromosome aberrations such as dicentric, rings, and anaphase bridge. On the other hand, DSB repair in S and G2 cell cycle phases is a combination of NHEJ and homologous recombination repair (HRR). HRR is an error-free DSB repair mechanism that uses homologous chromosome as a template for the repair process. The predominant role of HRR in DSB repair minimizes the overall probability of chromosome aberrations, thus increasing radioresistance [13].

Oxygen consumption of a cell should be sufficient to ensure continuous production of energy required in undertaking cellular activities such as biosynthesis in DNA replication process. Hypoxia-Inducible Factors (HIFs) are responsible for survival and adaptation of cells under hypoxic conditions. Oxygen is used by prolyl hydroxylase enzymes to hydroxylate HIF1$\alpha$ and HIF2$\alpha$ at relevant proline residues [14]. Proline hydroxylation is a requirement in order for the Von Hippel–Lindau (VHL) to bind with HIF. The binding of VHL to HIF, in turn, leads to proteosome degradation of HIF [14, 15]. In the absence of oxygen, HIF stabilizes above its basal concentration due to lack of HIF recognition by VHL. HIF activation during hypoxia predisposes to promotion of angiogenesis through enhanced transcription of vascular endothelial growth factor (VEGF) [16]. Furthermore, HIF inhibits c-Myc, which results in cell cycle arrest through p27 upregulation and reduced cyclin D2 transcription [17, 18]. The diminished cyclin D during hypoxia causes a decrease in active E2F, which is crucial in stabilizing cyclin E expression [9]. As a result, G1/S transition is slowed down or completely inhibited.

Several mathematical models were developed to study the cell cycle regulation under the influence of hypoxia. The effect of hypoxia in the G1/S transition through the cyclin-E2F signaling pathway was mathematically investigated using system of ordinary differential equations (ODEs) in [19]. Zhang et al. also used ODEs to simulate HIF–myc signaling pathway in mesenchymal stromal cells [20]. Results from both studies indicate that decreasing oxygen tension within the hypoxic region can cause a delay in the normal G1/S transition. A delay in G1/S transition will increase the intermitotic time, and reduce the cell proliferation rate. The cell population growth can also be compartmentalized in terms of cell cycle kinetics; the equilibrium fraction of each subpopulation in a particular cell cycle phase is characterized by the cell cycle phase transition rates [21].

Dose-response models are usually developed from mathematical representation of experimental data with an appropriate function [22]. Cell reproductive integrity and cell death are among the biological endpoints used in dose-response models. The linear-quadratic (LQ) function is a widely used model in describing the cumulative effect of different radiation absorbed doses [23]. Modifications of the LQ model were introduced to incorporate the change in dose-response with varying radiobiological factors. For example, oxygen dependence and in-between fraction repair rate were included in the LQ model [24]. A mechanistic investigation of oxygen fixation hypothesis in [25] provided a mathematical relationship between the probability of DNA radical interaction with a single oxygen molecule, and oxygen enhancement ratio (OER). OER is defined as the ratio of the dose required in hypoxic condition to that in an aerated condition to produce the same biological endpoint.

Breast tumors are known to exhibit hypoxic regions that may alter their response to radiotherapy and chemotherapy treatments [26]. Along with oxygen effect in radiosensitivity due to OFH mechanism, impact of prolonged proliferation time as a consequence of hypoxia-induced HIF activation should also be considered in radiotherapy fractionation. The goal of this study is to develop a mathematical model of cell cycle kinetics of hypoxic MCF-7 tumor subjected to fractionated radiotherapy. Both the hypoxia-induced G1/S transition delay and OFH mechanism-induced radioresistance were included in the model. Compartmentalization of tumor cell population to different cell cycle phases was modelled using systems of ordinary differential equations.

Materials and methods

Cell cycle kinetic model with oxygen dependent G1/S transition

Ordinary differential equations (ODEs) have been used to model the time evolution of cell cycle phase distribution of the cell population [27, 28]. An important characteristic of cell cycle kinetics in the cell population level is the transition rates of a particular cell line from one cell cycle phase to another. The applications of mathematical modelling in cell cycle progression can be extended to the more complex protein interaction networks, where the cell cycle regulatory features are depicted [29].

In this study, the mathematical model of Piantadosi et al. in [21] was adopted as the baseline cell cycle kinetics. The baseline cell cycle mathematical model consists of six ODEs that represent the cell population dynamics in the G0, G1, S, G2, and M cell cycle phases. These ODEs are defined as follows:

$$\begin{array}{c}\hfill \frac{d{N}_{G0}(t)}{\mathit{dt}}=k_{G1,G0}{N}_{G1}(t)\frac{N_{\mathit{tot}}(t)}{N_e}-k_{G0,G1}{N}_{G0}(t)-k_d{N}_{G0}(t)\end{array}$$ 1

$$\begin{array}{cc}\hfill \frac{d{N}_{G1}(t)}{\mathit{dt}}=& k_{G0,G1}{N}_{G0}(t)+2k_{M,G1}{N}_M(t)-k_{G1,G0}{N}_{G1}(t)\frac{N_{\mathit{tot}}(t)}{N_e}\hfill \\ \hfill & -k_{G1,S}{N}_{G1}(t)-k_d{N}_{G1}(t)\hfill \end{array}$$ 2

$$\begin{array}{c}\hfill \frac{d{N}_S(t)}{\mathit{dt}}=k_{G1,S}{N}_{G1}(t)-k_{S,G2}{N}_S(t)-k_d{N}_S(t)\end{array}$$ 3

$$\begin{array}{c}\hfill \frac{d{N}_{G2}(t)}{\mathit{dt}}=k_{S,G2}{N}_S(t)-k_{G2,M}{N}_{G2}(t)-k_d{N}_{G2}(t)\end{array}$$ 4

$$\begin{array}{c}\hfill \frac{d{N}_M(t)}{\mathit{dt}}=k_{G2,M}{N}_{G2}(t)-k_{M,G1}{N}_M(t)-k_d{N}_M(t)\end{array}$$ 5

$$\begin{array}{c}\hfill \frac{d{N}_{\mathit{tot}}(t)}{\mathit{dt}}=k_{M,G1}{N}_M(t)-k_d{N}_{\mathit{tot}}(t)\end{array}$$ 6

where $N_j(t)$ is the relative number of tumor cells in cell cycle phase j, and $k_{i,j}$ is the transition rate from cell cycle phase i to j with $i,j\in \{G0,G1,S,G2,M\}$. The relative number of cells in this case is considered as any quantity representing the number of cells. Such quantities can be the number of cell colonies or the number of cells normalized to a fixed initial number. In reality, tumor cells cannot grow indefinitely and is restricted by the limitation or the carrying capacity in tumor size. The parameter $N_e$ is the carrying capacity of the tumor population, which restricts the tumor growth to a certain relative number.

A transition rate is equal to the fraction of the relative number of tumor cells that leaves or enters a particular cell cycle phase per unit time. For example, $k_{G1,S}{N}_{G1}(t)$ in Eq. 3 describes the relative number of cells in the G1 phase that are entering the S phase, while $-k_{S,G2}{N}_S(t)$ refers to the relative number of cells leaving the S phase. Parameter values for Eqs. 1 to 6 were chosen such that the proportion of each subpopulation in a cell cycle phase to the total population matches those from experimentally derived fractions in [30]. To ensure that the solutions approach the positive equilibrium, the constraint in G1/G0 transition rate in [21] was also imposed:

$$\begin{array}{c}\hfill k_{G1,G0}=\frac{(k_d+k_{G0,G1})\{2k_{G1,S}{k}_{S,G2}{k}_{G2,M}{k}_{M,G1}-(k_d+k_{G1,S})(k_d+k_{S,G2})(k_d+k_{G2,M})(k_d+k_{M,G1})\}}{k_d(k_d+k_{S,G2})(k_d+k_{G2,M})(k_d+k_{M,G1})}\end{array}$$ 7

A summary of the parameter description and values is given in Table 1.

Table 1.

Description and values of parameters in Eqs. 1, 2, 3, 4, 5 and 6

Parameter	Description	Value
$k_d$	Cell loss rate	0.580 day-1
$k_{G0,G1}$	G0-G1 transition rate	0.150 day-1
$k_{G1,S}$	Aerated G1-S transition rate	3.491 day-1
$k_{S,G2}$	S-G2 transition rate	3.254 day-1
$k_{G2,M}$	G2-M transition rate	8.451 day-1
$k_{M,G1}$	M-G1 transition rate	24.587 day-1
$k_{G1,G0}$	G1-G0 transition rate	Calculated (Eq. 7)

The parameter $k_d$ is the cell loss rate, which takes into account naturally occurring cell death from apoptosis and necrosis, and cell migration. $k_d$ was estimated using the relationship between the cell loss factor (CLF) and the potential doubling time $T_{\mathit{pot}}$ [1, 21]:

$$\begin{array}{c}\hfill k_d\approx \frac{\mathit{CLF}}{1-CLF}\frac{ln2}{T_d}=\frac{CLF(ln2)}{T_{\mathit{pot}}}\end{array}$$ 8

where $T_d$ is the actual doubling time. Using the parameter values $CLF=0.71$ for adenocarcinomas [31], and $T_{\mathit{pot}}=20.36\text{hrs}$ [32], we have arrived with the value $k_d=0.58{\text{day}}^{-1}$.

Activation and coordination of cell cycle events are controlled by the corresponding cyclins and cyclin-dependent kinases (Cdks) [33]. Cyclin D is involved in the G1 phase cell cycle progression while cyclin E activity is responsible for the G1/S transition. Complex formation between cyclin D and cdk4/cdk6 leads to inactivation of Rb by phosphorylation. The process permits E2F to escape sequestration by unphosphorylated Rb [34]. E2F acts as a transcription factor of genes crucial in the G1 phase progression, as well as cyclin E production. The interplay between cyclin D and E2F is responsible for the shortened duration of G1 phase for cells with overexpressed cyclin D [35].

A mathematical simulation of the cyclin D - HIF dynamics was presented in the study of Bedessem and Stephanou in [19]. Their results showed that a decrease in oxygen tension causes a delay in the G1/S transition, which extends the G1 phase duration. The variables in the system of ODEs in [19] are cyclin D (CYCD(t)), cyclin E (CYCE(t)), SCF (SCF(t)), unphosphorylated Rb (RB(t)), cell mass ($M_c(t)$), and unphosphorylated E2F (E2F(t)); the ODE system is defined as:

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}CYCD(t)=a_1-a_{3}HIF(P)-a_{2}CYCD(t)\end{array}$$ 9

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}CYCE(t)=b_1{M}_c(t)E2F_A(t)-b_{2}CYCE(t)SCF(t)\end{array}$$ 10

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}SCF(t)=c_1\frac{1-SCF(t)}{J_1+1-SCF(t)}-c_2\frac{SCF(t)CYCE(t)}{J_2+SCF(t)}\end{array}$$ 11

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}RB(t)=d_2-(d_3+d_{1}CYCD(t))RB(t)\end{array}$$ 12

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}{M}_c(t)=e_1{M}_c(t)(1-\frac{M_c(t)}{M_e})\end{array}$$ 13

$$\begin{array}{c}\hfill \frac{d}{\mathit{dt}}E2F(t)=f_1(E2F_{\mathit{tot}}-E2F(t))\end{array}$$ 14

The following algebraic expression of HIF mediates the dependence of cyclin D dynamics to oxygen tension $P$:

$$\begin{array}{c}\hfill HIF(t)=H_0{e}^{a_4(1-P)}\end{array}$$ 15

$E2F_A(t)$ is the concentration of E2F that is not in complex formation with Rb, representing the unphosphorylated and active form of E2F [19, 36]:

$$\begin{array}{c}\hfill E2F_A(t)=(1-\frac{E2F_{\mathit{Rb}}(t)}{E2F_{\mathit{tot}}})E2F(t)\end{array}$$

$$\begin{array}{c}\hfill E2F_{\mathit{Rb}}(t)=E2F(t)-\frac{RB(t)E2F(t)}{E2F_{\mathit{tot}}+RB(t)+\sqrt{{(E2F_{\mathit{tot}}+RB(t))}^2-4E2F_{\mathit{tot}}RB(t)}}\end{array}$$ 16

The associated parameter values of Eqs. 9 to 16 are listed in Table 2.

Table 2.

Parameter values used in the model of Bedessem and Stephanou in [19]. The Michaelis-Menten coefficients are abbreviated as MM

Parameter	Description	Value
$a_1$	Cyclin D synthesis rate	0.51
$a_2$	Cyclin D spontaneous degradation rate	1.00
$a_3{H}_0$	HIF-mediated degradation rate	$8.5\times {10}^{-3}$
$a_4$	Fitting parameter from HIF measurements	2.50
$b_1$	active E2F-induced and cell growth-dependent synthesis rate of cyclin E	$1.8\times {10}^{-2}$
$b_2$	Cyclin E spontaneous degradation rate	0.50
$c_1$	SCF activation rate	1.00
$c_2$	SCF inactivation rate	14.0
$J_1$	MM coefficient for SCF activation	0.04
$J_2$	MM coefficient for SCF activation	0.04
$d_1$	Unphosphorylated Rb synthesis rate	0.20
$d_2$	Cyclin D-mediated Rb phosphorylation rate	0.10
$d_3$	Unphosphorylated Rb spontaneous degradation rate	0.10
$e_1$	Cell growth rate	$5.0\times {10}^{-3}$
$M_e$	Maximal size of cell growth	10
$f_1$	E2F dephosphorylation rate	$1.6\times {10}^{-2}$
$E2F_{\mathit{tot}}$	Scaled sum of total E2F (both phosporylated and unphosphorylated)	1
$CYC{E}_{\mathit{thresh}}$	Cyclin E amount required for progression to S phase	0.15

Equations 9 to 14 were numerically solved with the same initial conditions used in [19] at varying oxygen tensions. The G1 phase duration for a particular oxygen tension was obtained from the numerical solution of cyclin E variable. A significant increase in the amount of cyclin E above a particular threshold, denoted as $CYC{E}_{\mathit{thresh}}$, signals the onset of transition from G1 to S phase. For a particular oxygen tension, the time interval ($\mathrm{\Delta }{\tau }_{G1}$) between $t=0$ and the time it takes for CYCE(t) to reach $CYC{E}_{\mathit{thresh}}$ is considered as the G1 phase duration. In the aerated condition where P is sufficiently high, $\mathrm{\Delta }{\tau }_{G1}$ becomes relatively constant. We define the normalized G1 phase duration ($\overline{\mathrm{\Delta }{\tau }_{G1}}(P)$) as:

$$\begin{array}{c}\hfill \overline{\mathrm{\Delta }{\tau }_{G1}}(P)=\frac{\mathrm{\Delta }{\tau }_{G1}(P)}{\mathrm{\Delta }{\tau }_{G1}(P=100\text{mmHg})}\end{array}$$ 17

where $\mathrm{\Delta }{\tau }_{G1}(P=100\text{mmHg})$ is the G1 phase duration for an oxygen tension of 100 mmHg. $\mathrm{\Delta }{\tau }_{G1}(P=100\text{mmHg})$ was regarded as the normalization factor since its deviation from the asymptote $\mathrm{\Delta }{\tau }_{G1}(P\to \infty )$ is relatively negligible. The following function:

$$\begin{array}{c}\hfill G(P)=\frac{1}{\lambda P+\rho }+1\end{array}$$ 18

was fitted to $\overline{\mathrm{\Delta }{\tau }_{G1}}(P)$ using the least squares optimization method. Equation 18 establishes a simplified mapping between oxygen tension and G1 phase relative duration. An illustrative overview of how the tumor cell cycle kinetics is affected by oxygen tension is shown in Fig 1.

Fig. 1.

The cell cycle compartments and the influence of oxygen tension on G1/S phase transition rate via the cyclin-E2F signaling pathway (Eqs. 9 to 16)

An increase in G1 phase duration is associated to a decrease in G1/S phase transition. Based on the adopted cell cycle model in [21], the oxygen tension-dependent duration of the G1 phase is:

$$\begin{array}{c}\hfill {\tau }_{G1}=\frac{1}{k_{G1,S}+k_d}\Rightarrow \frac{1}{\mu (P)k_{G1,S}+k_d}\end{array}$$ 19

where $\mu (P)$ is a oxygen tension-dependent multiplicative factor that modifies the G1/S transition rate. Using the definition of normalized G1 phase duration $\overline{\mathrm{\Delta }{\tau }_{G1}}(P)\to G(P)$, the relation:

$$\begin{array}{c}\hfill \frac{1}{\mu (P)k_{G1,S}+k_d}=\frac{1}{k_{G1,S}+k_d}G(P)\end{array}$$

$$\begin{array}{c}\hfill \mu (P)=\frac{k_{G1,S}+(1-G(P))k_d}{G(p)k_{G1,S}}\end{array}$$ 20

was obtained. If P is in the aerated range such that $G(P)\approx 1$, $\mu (P)$ reduces to 1 and the G1 phase duration is equal to the unperturbed G1 phase duration in [21]. Equation 20 is valid under the condition:

$$\begin{array}{c}\hfill P\ge (\frac{k_d}{k_{G1,S}}-\rho )\frac{1}{\lambda }\end{array}$$ 21

$\mu (P)$ was multiplied to the $k_{G1,S}{N}_{G1}(t)$ term in both Eqs. 2 and 3 as the oxygen-dependent correction to the G1/S phase transition rate.

Radiation inactivation rate with OER dependence

The eradication of tumor population as a result of radiation exposure can be defined using dose-response functions. Linear-quadratic (LQ) model relates the cell survival S to the radiation absorbed dose D using the following function:

$$\begin{array}{c}\hfill S(D)=exp(-\alpha D-\beta D^2)\end{array}$$ 22

where $\alpha$ and $\beta$ are the fitting parameters [1]. For MCF-7 cell line irradiated with 6 MV beam quality, the parameter values are $\alpha =0.2{\text{Gy}}^{-1}$ and $\beta =0.09{\text{Gy}}^{-2}$ [37]. Dose-response models should be converted to their equivalent cell death rates to enable merging with the appropriate DEs. This conversion was achieved by initially considering the simple DE:

$$\begin{array}{c}\hfill \frac{\mathit{dN}}{\mathit{dt}}=-\gamma (D)N(t)\end{array}$$ 23

where N(t) is the number of cells at time t, and $\gamma (D)$ is the rate of radiation-induced cell death for an absorbed dose D. The solution to Eq. 23 for a constant D is:

$$\begin{array}{c}\hfill N(t)=N_{0}exp(-\gamma (D)t)\end{array}$$ 24

Equations 23 and 24 describe a system of cell population where cell death due to irradiation is the only factor that changes the fraction of cells.

For an absorbed dose D delivered at $t=0$ with an irradiation time of $\tau$, $N(\tau )/N_0$ should be equal to the LQ survival fraction:

$$\begin{array}{c}\hfill \frac{N(\tau )}{N_0}=exp(-\alpha D-\beta D^2)=exp(-\gamma (D)\tau )\end{array}$$ 25

The boundary condition previously employed entails that the initiation of radiation-induced cell death is instantaneous. Although cell death is an extended process, cells marked for apoptosis and mitotic catastrophe are irreversible [5], which justifies the assumption implied by the boundary condition. From Eq. 25, the dose-dependent cell death rate is given by:

$$\begin{array}{c}\hfill \gamma (D)=\frac{\alpha D+\beta D^2}{\tau }\end{array}$$ 26

The oxygen-dependence of parameters $\alpha$ and $\beta$ in the LQ dose-response model due to OFH mechanism should be identified. A study in [38] modified the LQ cell survival model into the following equation:

$$\begin{array}{c}\hfill S(D,P)=exp(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)\end{array}$$ 27

In this study, the following OER expression was used [39]:

$$\begin{array}{c}\hfill OER=\frac{m{K}_o+P}{K_o+P}\end{array}$$ 28

where m is the maximum achievable OER, and $K_0$ is the value of oxygen tension in which OER results in half of its maximum interval. Since many mammalian cells have $m=3.0$ and $K_o=0.5\%{\text{O}}_2\text{or}3\text{mmHg}$ in their characteristic OER for low linear energy transfer radiations [5, 40], these values were adopted to represent OER function of MCF-7 cells in Eq. 28. In relation to Eqs. 25, 26 and 27, the radiation-induced cell death rate becomes:

$$\begin{array}{c}\hfill \gamma (D,P)=\frac{1}{\tau }(\frac{\alpha }{OER(P)}D+\frac{\beta }{OE{R}^2(P)}{D}^2)\end{array}$$ 29

which considers the modulation of radiosensitivity by varying oxygen tension.

Since the cell death rate in Eq. 29 is applicable to a tumor population that are asynchronous in cell cycle phase, cell cycle-specific radiosensitivities were factored in by multiplying a calibration constant to Eq. 29. The calibration constant ${ϵ}_i$ for a cell cycle phase i is given as:

$$\begin{array}{c}\hfill {ϵ}_i=\frac{q_i}{\overline{q}}=\frac{q_i}{{\sum }_j{q}_j{f}_j}\end{array}$$ 30

where $j\in$ (G1, S, M), $q_i$ is the cell cycle phase-dependent death rate, and $f_j$ is the fraction of tumor cell population in the cell cycle phase j. Since cells in G2 and M phases have approximately equal radiosensitivity, their tumor subpopulation were treated as one in the normalization of cell cycle-dependent death rate. The parameter values $q_{G1}=0.06{\text{hr}}^{-1}$, $q_S=0.02{\text{hr}}^{-1}$, and $q_M=0.1{\text{hr}}^{-1}$ in [41] were used. The associated fractions $f_j$ for the cell cycle phases were obtained by solving Eqs. 1 to 6, and subsequently normalizing the equilibrium subpopulation in a particular cell cycle phase to the total tumor equilibrium population $N_e$. Equation 30 serves as the calibration constant for the cell cycle-specific radiosensitivity, which is the ratio of the cell death rate for tumor cells in cell cycle phase i to the mean cell death rate of the entire tumor population in the G1, S, and G2/M phases.

Simulation of the model for different oxygen tension and fractionation scheme

Integrating all mathematical formulations in Sects. 2.1 and 2.2, we have arrived with a cell cycle kinetics model that includes hypoxic and irradiation effects:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{N}_{G0}(t)}{\mathit{dt}}=& k_{G1,G0}{N}_{G1}(t)\frac{N_{\mathit{tot}}(t)}{N_e}-k_{G0,G1}{N}_{G0}(t)-k_d{N}_{G0}(t)\hfill \\ \hfill & -\frac{{ϵ}_{G1}}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_{G0}(t)\hfill \end{array}\end{array}$$ 31

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{N}_{G1}(t)}{\mathit{dt}}=& k_{G0,G1}{N}_{G0}(t)+2k_{M,G1}{N}_M(t)-k_{G1,G0}{N}_{G1}(t)\frac{N_{\mathit{tot}}(t)}{N_e}\hfill \\ \hfill & -\mu (P)k_{G1,S}{N}_{G1}(t)-k_d{N}_{G1}(t)-\frac{{ϵ}_{G1}}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_{G1}(t)\hfill \end{array}\end{array}$$ 32

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{N}_S(t)}{\mathit{dt}}=& \mu (P)k_{G1,S}{N}_{G1}(t)-k_{S,G2}{N}_S(t)-k_d{N}_S(t)\hfill \\ \hfill & -\frac{{ϵ}_S}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_S(t)\hfill \end{array}\end{array}$$ 33

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{N}_{G2}(t)}{\mathit{dt}}=& k_{S,G2}{N}_S(t)-k_{G2,M}{N}_{G2}(t)-k_d{N}_{G2}(t)\hfill \\ \hfill & -\frac{{ϵ}_M}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_{G2}(t)\hfill \end{array}\end{array}$$ 34

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{N}_M(t)}{\mathit{dt}}=& k_{G2,M}{N}_{G2}(t)-k_{M,G1}{N}_M(t)-k_d{N}_M(t)\hfill \\ \hfill & -\frac{{ϵ}_M}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_M(t)\hfill \end{array}\end{array}$$ 35

$$\begin{array}{c}\hfill \frac{d{N}_{\mathit{tot}}(t)}{\mathit{dt}}=k_{M,G1}{N}_M(t)-k_d{N}_{\mathit{tot}}(t)-\frac{1}{\tau }(\frac{\alpha }{\mathit{OER}}D+\frac{\beta }{OE{R}^2}{D}^2)N_M(t)\end{array}$$ 36

No cell cycle-dependent calibration constant was used in $N_{\mathit{tot}}(t)$ as it represents the conglomerate radiosensitivity of the entire tumor population similar to the derivation Eq. 29. Numerical computation of the ODEs using adaptive Runge–Kutta of order 4 (RK4 solver) was carried out in Scilab 6.0.1. The following are the different cases considered in the numerical simulation of Eqs. 31 to 36:

1. Varying oxygen tension: The system of ODEs was simulated at different oxygen tensions without radiotherapy treatment, $D=0$. The assumption that oxygen tension is time-invariant was employed in all simulations.

2. Varying oxygen tension and radiotherapy fractionation pattern: Two different breast radiotherapy fractionation schemes from [42] were investigated. The treatment parameters for the two fractionation schemes are given in Table 3. During the dose fraction delivery, D is equal to the dose per fraction with a constant dose rate of 1 Gy/min. In between the fractions, the absorbed dose rate is zero. All fractionation types were delivered starting at $t=0$ with one dose fraction delivered per day. Different oxygenation conditions (P) for each fractionation scheme were simulated.

Table 3.

Radiotherapy fractionation schemes simulated in the developed mathematical model

Total absorbed dose	Dose per fraction	Number of fractions	Irradiation time per fraction ($\tau$)
30 Gy	6 Gy	5	6 mins
40 Gy	2.67 Gy	15	2.67 mins

Model assumptions

1. The average oxygen tension is spatially and temporally constant. This assumption requires to compartmentalize the tumor region according to their oxygen tension such that each compartment has relatively small deviations from its average oxygen tension. In addition, the weighted average oxygen tension can be used to represent a given oxygen distribution in the same way the effective energy represent the photon and charged particle beam spectrum. In this case, the polyenergetic beam is specified in terms of the effective energy, which is defined as the equivalent monoenergetic beam that will result in the same transmission characteristics as the polyenergetic beam [43]. The similar perspective can be used in the developed cell cycle kinetics model in that we use an effective oxygen tension that will result in the same overall G1-S transition rate and OFH response.

2. Radiation-induced damage and hypoxic effects are instantaneous. An alternative interpretation is that the fraction of damaged cells that will be repaired is also instantaneous.

3. Prior to the delivery of the first fraction, the MCF-7 tumor population is in the equilibrium state. The equilibrium point of each cell cycle phase for a particular oxygen tension was obtained from the simulation of case 1 in Sect. 2.3.

4. The cyclin-E2F signaling pathway represents the average cyclin dynamics of the tumor population or tumor area in question.

Results and discussion

Mapping of oxygen tension to relative G1 phase duration

The ODEs of the cyclin-E2F signaling pathway in Eqs. 9 to 15 were numerically solved for different values of oxygen tension P. By referring to Eq. 15, a decrease in $P$ causes an increase in the amount of HIF equilibrium. The change in HIF equilibrium also alters the dynamics of cyclin D. This cascades into disparities in dynamical behavior of the system for different P values, which eventually leads to changing G1 phase duration as demonstrated in [19]. The resulting G1 phase duration as a function of oxygen tension, and the corresponding fitted function using Eq. 18 are illustrated in Fig. 2. Relatively small deviation of the fitted function from the numerically simulated G1 phase duration was attained, making Eq. 18 with parameter values $\lambda =0.457$ and $\rho =-0.019$ an acceptable representation of the oxygen-dependent G1 phase duration.

Fig. 2.

Normalized G1 phase duration obtained from numerical simulation of Eqs. 9 to 16 at different oxygen tension values

Equation 19 defines the relationship between the G1 phase duration and the G1/S transition rate. The corrected G1/S transition rate in Eq. 19 is equal to the product of $\mu (P)$ and $k_{G1,S}$. As shown in Fig. 3, G1/S transition ratio $\mu (P)$ is asymptotic to 1 for a sufficiently high oxygen tension, and decreases at lower oxygen tension values. The obtained G1/S transition rate as a function of oxygen tension agrees with the qualitative description of hypoxic effects on cell cycle kinetics [44] and with the G1 phase duration curve in [19].

Fig. 3.

The plot of the correction factor $\mu (P)$ from Eq. 20

Cell cycle kinetics with varying oxygen tension values

Repopulation is also a crucial consideration in radiotherapy treatment prescription. As repopulation rate increases, the capability of the tumor population to recover in number after each radiotherapy fraction also improves. The extension of the G1 phase duration caused by hypoxia results in the increase of cell proliferation time. This retardation of tumor growth by hypoxia can affect treatment outcomes of modalities such as radiotherapy and chemotherapy [45]. For example, accelerated repopulation in some tumors actuates the use of hypofractionated radiotherapy to compensate for the rapid tumor growth [46].

The cell cycle kinetics for $p=100\text{mmHg}$, $p=5\text{mmHg}$, and $p=2\text{mmHg}$ are shown in Fig. 4a, b, c, respectively. Small oxygen tension values result in slower growth rate, which translates to larger growth period required before the tumor population reaches the carrying capacity for the same initial conditions. An experimental work of Sutherland et al. in [47] demonstrated the population growth of $3\times {10}^5$ cultured MCF-7 cells. In their experimental result, the tumor cell population reaches the plateau of approximately $9\times {10}^7$ cells at around $t=13$ days. For an initial tumor population of 3.33 and $N_e=1000$ in Fig. 4a, the ratio of the initial to final relative number of tumor cells is $3.33/1000=3.33\times {10}^{-3}$. Similarly, the ratio of the initial to final number of cells in the experimental work in [47] is also close to $(3\times {10}^5)/(9\times {10}^7)=3.33\times {10}^{-3}$. The experimental and the simulated growth curves manifested a start of the plateau region at $t=13\text{days}$ and $t=18\text{days}$, respectively. Thus, the mathematical model of MCF-7 cell cycle kinetics provided in the study is a good approximate to the actual MCF-7 cell population growth.

Fig. 4.

MCF-7 cell cycle kinetics at oxygen tension (a) $P=100\text{mmHg}$, (b) $P=5\text{mmHg}$, and (c) $P=2\text{mmHg}$

Another observation extracted from Fig. 4a, b, c, is the changing proportion of tumor subpopulations in the G0 and G1 phases. As the oxygen tension decreases, the equilibrium of G0 tumor subpopulation decreases while that of the G1 tumor subpopulation increases. A comparison of the simulated tumor subpopulation fraction $f_j$ in the cell cycle phase j with the experimental fractions of cell cycle phase subpopulations in [30] is provided in Table 4. The relatively good agreement between the simulated and experimental growth curves, including the corresponding fractional cell cycle phase subpopulations, justifies the validity of the selected parameters in Eqs. 1 to 6 for MCF-7 cell cycle kinetics. The invariance of the equilibrium of S, G2, and M phase subpopulations with changing oxygen tension is attributed to the independence of the corresponding equilibrium points to $\mu (P)k_{G1,S}$ and $k_{G1,G0}$ [21]. Based on the definition of G0 phase, the mathematical growth fraction should increase with decreasing G0 subpopulation in hypoxic conditions [21].

Table 4.

Corresponding fractions of tumor subpopulation in different cell cycle phases to the total population. Experimental results from [30] are also provided

Tumor subpopulation fraction	Model	Experimental results in [30]
$f_{G1}$	0.707	0.717
$f_S$	0.197	0.191
$f_M$	0.099	0.092

Cell cycle kinetics with varying oxygen tension and radiotherapy fractionation pattern

Although the delayed repopulation induced by hypoxia may be beneficial in elevating tumor control, this is greatly counteracted by the reduction of overall radiosensitivity due to OFH mechanism. A numerical simulation of the developed mathematical model allows the assessment of the two-fold hypoxic effects on different fractionation schemes. The MCF-7 cell cycle kinetics with different oxygen condition and under different radiotherapy fractionation schemes listed in Table 3 are shown in Figs. 5 and 6.

Fig. 5.

MCF-7 cell cycle kinetics under 30-Gy fractionated radiotherapy schedule in Table 3 and at oxygen tension (a) $P=100\text{mmHg}$, (b) $P=5\text{mmHg}$, and (c) $P=2\text{mmHg}$

Fig. 6.

MCF-7 cell cycle kinetics under 40-Gy fractionated radiotherapy schedule in Table 3 and at oxygen tension (a) $P=100\text{mmHg}$, (b) $P=5\text{mmHg}$, and (c) $P=2\text{mmHg}$

Figures 5a and 6a illustrate the effect of fractionated radiotherapy on cell cycle kinetics in an aerated condition. The tumor population was initially in the equilibrium as outlined in Assumption 3 in Sect. 2.4 before irradiation. Cell death rate due to radiation reduces a large fraction of the tumor population during the delivery of the first fraction. The tumor subpopulation in the G2/M phase experienced the largest degree of cell death as they have the highest radiosensitivity among the cell cycle phase compartments. On the other hand, the S phase tumor subpopulation is the least affected by the irradiation. These events occur such that the tumor population is predominated by the radioresistant subpopulation in the cell cycle phase at the end of the delivery of dose fraction.

Following irradiation, the tumor begins to repopulate and the G0+G1 phase tumor subpopulation takes over the S phase tumor subpopulation. This can be attributed to the higher equilibrium points associated to both G0 and G1 phase subpopulations. As a consequence, the tumor population becomes predominated by the more radiosensitive subpopulation in the G0 and G1 phases, rendering the tumor to become more susceptible to radiation damage in the next dose fraction. These results demonstrate the principle of redistribution of cell cycle phase, which prompts the utilization of fractionated radiotherapy in cancer treatment [48].

The influence of oxygen on radiosensitivity through the OFH mechanism is indicated by the changes in survival fraction after each dose fraction delivery. Each tumor subpopulation in Fig. 5a has smaller survival fraction after the first dose fraction compared to that with corresponding tumor subpopulations in Fig. 5b, c. The Numerical simulation results also describe the modification of MCF-7 cell cycle kinetics by hypoxia in a fractionated radiotherapy. Figure 4a, to c illustrated that the equilibrium points of G0 and G1 phase tumor subpopulations are modulated by the oxygen tension while having a constant sum. However, the delayed G1/S phase transition rate for a small oxygen tension causes a large fraction of the tumor population to initially accumulate in the G1 cell cycle phase. As a result, cell cycle kinetics in Fig. 5c have large differences in relative number of cells between G0+G1 and S tumor subpopulation after each fraction. Thus, cell cycle phase redistribution becomes nonexistent for severe hypoxia during fractionated radiotherapy.

The longer fractionated radiotherapy treatment with a total absorbed dose of 40 Gy in 15 fractions was also considered in the study. As seen in the aerated condition in Fig. 6a, redistribution of cell cycle phase occurs such that the tumor population becomes predominated by G0+G1 subpopulation following each fraction delivery. The G0+G1 tumor subpopulation is greater at all times during the whole radiotherapy treatment period for $P=5\text{mmHg}$ (Fig. 6b). A decrease in oxygen tension further enhances the discrepancy between the tumor subpopulations in different cell cycle phases (Fig. 6c) with the G0+G1 tumor subpopulation occupying the largest survival fraction. This may present advantages in terms of retaining a large portion of the tumor population in the more radiosensitive phase. However, the increase in OER at low oxygen tension significantly reduces the radiation inactivation rate, which raises the survival fraction beyond that with aerated.

The time evolution plots of the total MCF-7 tumor population for the two fractionation types with different oxygen tensions are provided in Fig. 7a, b. Table 5 shows the summary of survival fraction immediately after the radiotherapy treatment. In both fractionation schemes, the net effect of hypoxia is to increase population radioresistance. For 40 Gy fractionation scheme, a higher survival fraction after the treatment duration was obtained for $p=5\text{mmHg}$ compared to the survival fraction of tumor population with $p=2\text{mmHg}$. As shown in Fig. 7b, the tumor population corresponding to $p=5\text{mmHg}$ becomes higher in number compared to that with $p=2\text{mmHg}$ after $t=9\text{days}$. The phenomenon is a consequence of the increased relative contribution of slowing down the proliferation rate by hypoxia for a prolonged treatment duration. The mathematical model provided a quantitative estimate of the contrasting effect of the delayed proliferation to radioresistance acquired from OFH mechanism in radiotherapy.

Fig. 7.

MCF-7 total tumor population with different oxygen tension for a (a) 30-Gy and (b) 40-Gy fractionation scheme

Table 5.

Relative number of cells of the tumor population subsequently after completion of fractionated radiotherapy treatment

Oxygen Tension (mmHg)	30 Gy scheme	40 Gy scheme
100	$6.263\times {10}^{-3}$	0.081
5	1.605	11.89
2	4.874	8.947

Prescribing a radiotherapy fractionation should take into account pathologic and radiobiologic characteristics of the tumor [3, 49]. The developed mathematical model can be used in planning and optimization of radiotherapy treatment. Simultaneously taking into account both the hypoxia-induced inhibition of proliferation, and OFH mechanism allows a more accurate evaluation of the behavior of cell cycle kinetics under radiotherapy treatment. In the clinical scenario, oxygenation measurements from imaging studies can be utilized as inputs in the mathematical model to predict treatment outcome of a given fractionation scheme. Optimization methods may be employed to adjust the dose required to tumor cells in the hypoxic region. The mathematical model can also be extended by using system of partial differential equations to include absorbed dose and oxygen inhomogeneities.

Conclusion

Many radiobiologic factors influence the radiation response of the tumor population. In the developed mathematical model, hypoxic effects, cell cycle phase redistribution, and repopulation of MCF-7 cells were demonstrated to occur. Hypoxia effects include the extension of G1 phase duration and increased radioresistance due to OFH mechanism. Validation of the growth curve and proportion of subpopulation of each cell cycle phase partially reinforces the applicability of the mathematical model to clinical applications. Results from the 30 Gy fractionation scheme showed that surviving relative number after treatment is negatively correlated to oxygen tension. For a longer treatment duration, as in the 40 Gy fractionation scheme, the contribution of the delayed proliferation becomes appreciable such that surviving relative number after treatment slightly increases compared to that with higher oxygen tension. A more detailed assessment of the ODE system using stability analysis should be carried out.

Funding

The author has not received specific funding for this work.

Data availability

Data available on request from the author.

Declarations

Ethical approval

This article does not contain any studies with human or animal subjects performed by the author.

Conflict of interest

The author declares that they have no conflict of interest.