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This present paper presents an analytical description and numerical simulations of the influence of macroscopic intercell dose variations and intercell sensitivity variations on the probability of controlling the tumour. Computer simulations of tumour control probability accounting for heterogeneity in dose and radiation sensitivity were performed. An analytical expression for tumor control probability accounting for heterogeneity in sensitivity was also proposed and validated against simulations. The results show good agreement between numerical simulations and the calculated TCP using the proposed analytical expression for the case of a heterogeneous dose and sensitivity distributions. When the intratumour variations of dose and sensitivity are taken into account, the total dose required for achieving the same level of control as for the case of homogeneous distribution is only slightly higher, the influence of the variations in the two factors taken into account being additive. The results of this study show that the interplay between cell or tumour variation in the sensitivity to radiation and the inherent heterogeneity in dose distribution is highly complex and therefore should be taken into account when predicting the outcome of a given treatment in terms of tumor control probability.

Modern advanced radiation therapy makes use of radiobiological models for tumor control probability (TCP) for predicting the outcome and optimizing the treatment as well as for treatment evaluation. The most widely used model for the calculation of the probability of eradicating the tumor is based on the linear-quadratic (LQ) model for cell survival in conjunction with Poisson statistics describing the distribution of the clonogenic cells surviving at the end of the treatment. The standard expression for TCP assumes that all the cells in the irradiated volume have the same intrinsic sensitivity to radiation and the same sensitivity to changes in the fractionation. Furthermore, the model also assumes that all the cells receive the same dose. Refined versions of the model proposed solutions for accounting for heterogeneities in the dose distribution by dividing the tumor volume into subvolumes down to voxel size in which the dose and the radiation sensitivity were assumed to be constant [

However, to the best of our knowledge, the combined influence on TCP from both the deterministic and the intrinsic stochastic heterogeneity in dose delivery and the variation of the sensitivity of the cells to radiation on TCP has not yet been fully explored.

It is therefore the aim of the current paper to present a comprehensive analysis of the TCP, after irradiation with heterogeneous dose distributions in the presence of differences between cells with respect to their radiation sensitivity, in fractionated radiation therapy. The analysis is based on the LQ description of the cell survival and includes the effect of cell proliferation. The study explores the effect on TCP when exposing cell populations with various irradiation patterns under different assumptions of the cells sensitivity or with intertumor variation in sensitivity.

Closed form expressions for the tumor control probability that take into account sensitivity variation within a tumor has, to the best of our knowledge, not been published. Therefore, an analytical formalism for calculations of TCP has been proposed, in order to determine and explain the dose response in presence of heterogeneity either in dose to the cells or sensitivity of the cells within a tumor.

The probability of controlling the tumors under the assumption that the LQ model describes the response of individual cells to radiation and that a Poisson distribution describes the number of clonogens

In fractionated radiation therapy the cells that survive irradiation will get the chance to proliferate. In order to account for proliferation one could introduce a proliferation factor depending on the clonogen doubling time,

According to Deasy [

In the present paper the probability of controlling the tumor was calculated as the average TCP, resulting from the simulation of the irradiation of a large number of tumors having the same average growth kinetics and radiation sensitivity and containing the same number of initial clonogens. The simulations are based on the assumption that all clonogens have to be eradicated in order to control the tumor. The cell survival is sampled from a Bernoulli distribution where the probability for a given cell to survive is described by the LQ model:

If the cells potential doubling time is

In order to account for variations in the sensitivity of the cells to radiation, one could assume that the individual radiation sensitivity of the cells in a population follows a given log-normal distribution [

the

the

An analytical expression for TCP

If the sensitivity to radiation is heterogeneous, a similar formalism for determining the tumor control probability could be applied by performing a Taylor expansion of the expression for the tumor control probability around the mean

Several approaches for determining the probability of tumor control have been presented in the previous section: the Poisson-LQ model including the effect of proliferation, (

Comparison between dose response calculated with the Poisson expression where the repopulation correction is made by the multiplicative factor

The results in Figure

Figure

Intratumor variation in dose and sensitivity for a tissue with

In order to quantify the effects on TCP from two or more heterogeneities, one can define a coefficient of synergy:

For a given type of tumors with respect to histopathology and stage, the shape, position, and slope of clinically derived dose response curves depend on the actual heterogeneity of the parameters describing the response of the individual tumors. Figure

Interpatient variation in sensitivity, intratumor variation in dose, and a combination of those two for a tissue with

The analytical approach used for describing the influence of the heterogeneity in sensitivity on the control probability was validated by comparing the TCP calculated using (

Comparison of the analytical expression for sensitivity variations given by (

The influence of the heterogeneity in dose on the control probability described by (

Modern advanced radiation therapy implies the use of functional and molecular imaging for mapping the tumor in terms in radiation sensitivity and nonhomogeneous dose distributions as in dose-painting approaches. The findings of this study have the potential to contribute to the better understanding of the interplay between variations in radiation sensitivity between cells within the tumor as determined based on functional and molecular imaging and the delivered dose distribution, which is currently one of the most exciting challenges in the clinical practice.

The results of the modeling and the simulations performed in this study show that the interplay between cell or tumor variation in the sensitivity to radiation and the inherent heterogeneity in dose distribution should be taken into account when predicting the outcome of a given treatment in terms of control probability. Furthermore, the findings of this paper should also be taken into account when attempting to determine the parameters describing the radiation sensitivity based on clinically derived dose response curves.

The authors declare that there is no conflict of interests regarding the publication of this paper.