Analysis of tumour dose–response data from animal experiments via two TCP models accounting for tumor hypoxia and resensitization

Abstract

To treat animal dose–response data exhibiting inverse dose–response behavior with two tumor control probability (TCP) models accounting for tumor hypoxia and re-oxygenation leading to resensitization of the tumor. One of the tested TCP models uses a modified linear-quadratic (LQ) model of cell survival where both α and β radiosensitivities increase in time during the treatment due to re-oxygenation of the hypoxic tumor sub-population. The other TCP model deals with two types of hypoxia—chronic and acute—and accounts for tumor re-sensitization via oxygenation of the chronically hypoxic and fluctuating oxygenation of the acutely hypoxic sub-populations. The two models are fit using the maximum likelihood method to the data of Fowler et al. on mice mammary tumors irradiated to different doses using different fractionated schedules. These data are chosen since as many as five of the dose–response curves show an inverse dose behavior, which is interpreted as due to re-sensitization. The p-values of the fits of both models to the data render them statistically acceptable. A performed comparison test shows that both models describe the data equally well. It is also demonstrated that the most sensitive (oxic) tumor component has no impact on the treatment outcome. The ability of the tested models to predict and describe the impact of re-sensitization on the treatment outcome is thus proven. It is also demonstrated that prolonged treatment schedules can be more beneficial than shorter ones. However, this may be true only for schedules with small number of fractions, i.e. for hypo-fractionated treatments only.

Appendices

Appendix 1

The Zaider–Minerbo TCP model solved for the case of (n) external fractions is:

$$TCP\left( {T_{n - 1} } \right) = \left[ {1 - \frac{{p_{s} \left( {T_{n - 1} } \right)e^{{\left( {\lambda - \mu } \right)T_{n - 1} }} }}{{1 - \frac{\lambda }{\lambda - \mu }p_{s} \left( {T_{n - 1} } \right)e^{{\left( {\lambda - \mu } \right)T_{n - 1} }} \sum\nolimits_{k = 1}^{n - 1} {p_{s}^{ - 1} \left( {T_{k - 1} } \right)\left[ {e^{{\left( {\mu - \lambda } \right)T_{k} }} - e^{{\left( {\mu - \lambda } \right)T_{k - 1} }} } \right]} }}} \right]^{N}$$

(3)

where N is the initial number of tumor cells, λ and μ are the cell birth rate and the cell natural death rate correspondingly, $T_{n - 1}$ is the total treatment time, $T_{k - 1}$ is the time until after the kth fraction, $p_{s} \left( {T_{k - 1} } \right)$ is the cell survival probability after the kth fraction. If the linear-quadratic (LQ) model with complete repair of the sub-lethal cell damage between fractions is assumed, $p_{s} \left( {T_{k - 1} } \right)$ is:

$$p_{s} \left( {T_{k - 1} } \right) = e^{{ - \left( {\alpha \sum\nolimits_{i = 1}^{k} {d_{i} } + \beta \sum\nolimits_{i = 1}^{k} {d_{i}^{2} } } \right)}} ,$$

(4)

where $d_{i}$ is the dose per the ith fraction delivered homogeneously and α and β are the radiosensitivity parameters of the LQ model. The LQ model parameters are considered to be constants in time in the widely accepted version of the LQ model. However, a version of the model was developed in [11] where the cell radiosensitivity, α, increases in time according to the following function:

$$\alpha \left( t \right) = \alpha_{0} e^{{ - bt^{2} /2}} + \alpha_{m} \left( {1 - e^{{ - bt^{2} /2}} } \right)$$

(5)

Parameter b in Eq. (5) determines the rate of re-oxygenation and hence the rate of re-sensitization of the initially hypoxic region of the tumor. $\alpha_{0}$ is the initial low value of $\alpha$ in the hypoxic region of the tumor and $\alpha_{m}$ is the maximum possible value of α reached asymptotically in time. The cell survival probability is then calculated according to:

$$p_{s} \left( {T_{k - 1} } \right) = e^{{ - \left( {\sum\nolimits_{i = 1}^{k} {\alpha \left( {t_{i} } \right)d_{i} } + \beta \sum\nolimits_{i = 1}^{k} {d_{i}^{2} } } \right)}}$$

(6)

In this study we assume that radio-sensitivity, β, also increases with time simultaneously with α according to the following function:

$$\beta \left( t \right) = \beta_{0} \left( {\frac{\alpha \left( t \right)}{{\alpha_{0} }}} \right)^{2} ,$$

(7)

where $\beta_{0} {\text{ and }}\alpha_{{0}}$ are the initially low values of β and α. The above relationship applied to the maximum (oxygenated) values of $\beta_{ \, } {\text{and }}\alpha$, $\beta_{m} {\text{ and }}\alpha_{m}$, results in the following accepted relation between the oxygen enhancement ratios (OERs) of the radio-sensitivity parameters: $\frac{{\beta_{m} }}{{\beta_{0} }} = OER_{\beta } = \left( {\frac{{\alpha_{m} }}{{\alpha_{0} }}} \right)^{2} = \left( {OER_{\alpha } } \right)^{2}$.

This version of the model retains the number of free parameters as in the version of the model where β is constant.

For the case of both α and β changing in time Eq. (4) transforms into:

$$p_{s} \left( {T_{k - 1} } \right) = e^{{ - \left( {\sum\nolimits_{i = 1}^{k} {\alpha \left( {t_{i} } \right)d_{i} } + \sum\nolimits_{i = 1}^{k} {\beta \left( {t_{i} } \right)d_{i}^{2} } } \right)}}$$

(8)

where $\alpha \left( {t_{i} } \right)$ and $\beta \left( {t_{i} } \right)$ are calculated according to Eqs. (5) and (7) for the moment of the ith fraction delivering dose $d_{i}$ correspondingly.

Appendix 2

The TCP is calculated according to a Poisson based expression:

$$TCP = e^{{ - N_{s} }} = e^{{ - \left( {N_{o}^{n} + N_{ah}^{n} + N_{ch}^{n} + \tilde{N}_{o}^{n} } \right)}}$$

(9)

where $N_{s}$ is the sum of the mean number of surviving oxic,$N_{o}^{n}$, acutely hypoxic, $N_{ah}^{n}$, chronically hypoxic, $N_{ch}^{n}$, and chronically hypoxic turned oxic cells, $\tilde{N}_{o}^{n}$, after n fractions of irradiation. The mean number of oxic cells surviving n fractions of irradiation to dose d per fraction is:

$$N_{o}^{n} = N_{o}^{o} \left[ {S_{o} } \right]^{n} e^{\lambda T} ,$$

(10)

where $N_{o}^{o}$ is the initial number of oxic cells and T is the total treatment time, λ is the cell repopulation rate, $S_{o} = e^{{ - \alpha_{o} d - \beta_{o} d^{2} }}$ is the cell survival probability according to the LQ model of the oxic cells after irradiation to dose d.

The mean number of acutely hypoxic cells after n fractions of irradiation to dose d per fraction is:

$$N_{ah}^{n} = N_{ah}^{o} \left[ {\left( {1 - C} \right)S_{o} + C \cdot S_{ah} } \right]^{n} e^{\lambda T} ,$$

(11)

where $N_{ah}^{o}$ is the initial number of acutely hypoxic cells and $S_{ah} = e^{{ - \alpha_{ah} d - \beta_{ah} d^{2} }}$ is the cell survival probability of the acutely hypoxic cells when irradiated to dose d.

The mean number of chronically hypoxic cells surviving n fractions of irradiation to dose d per fraction is:

$$N_{ch}^{n} = N_{ch}^{o} \left( {S_{ch} } \right)^{n} \prod\limits_{i = 1}^{n - 1} {\left( {1 - B\left( {\Delta t_{i} } \right)} \right)} ,$$

(12)

where $N_{ch}^{o}$ is the initial number of chronically hypoxic cells, $1 - B(\Delta t_{i} ) = e^{{ - a\Delta t_{i} }}$ is the fraction of chronically hypoxic cells, which remain hypoxic between the $i{\text{th}}$ and the $(i + 1){\text{th}}$ irradiation and $S_{ch} = e^{{ - \alpha_{ch} d - \beta_{ch} d^{2} }}$ is the cell survival probability of the chronically hypoxic cells when irradiated to dose d.

The mean number of chronically hypoxic turned oxic cells surviving n fractions of irradiation to dose d per fraction is:

$$\tilde{N}_{o}^{n} = N_{ch}^{o} \sum\limits_{k = 1}^{n - 1} {B\left( {\Delta t_{k} } \right)} \prod\limits_{j = 1}^{k - 1} {\left( {1 - B\left( {\Delta t_{j} } \right)} \right)} \left( {S_{ch} } \right)^{k} \left( {S_{o} } \right)^{n - k} e^{{\lambda \sum\nolimits_{m = k + 1}^{n - 1} {\Delta t_{m} } }} ,$$

(13)

and $B(\Delta t_{i} ) = 1 - e^{{ - a\Delta t_{i} }}$ is the fraction of chronically hypoxic cells, which re-oxygenate between the $i{\text{th}}$ and the $(i + 1){\text{th}}$ irradiation.