[en] Purpose: To compare the partially wide tangent (PWT) technique of breast and internal mammary chain irradiation with photon/electron (P/E) and standard tangent (ST) techniques in terms of dose homogeneity within breast and the dose to critical structures such as the heart and lung. Methods and Materials: Sixteen left breast cancer patients underwent CT simulation. The breasts, lungs, heart, and internal mammary chain were contoured and treatment plans generated on a three-dimensional planning system (Helax-TMS). Results: The mean dose to the left breast volume with the ST, P/E, and PWT techniques was 94.7%, 98.4%, and 96.5%, respectively (p=0.029). The left lung received the lowest mean dose with the ST technique (13.9%) compared with PWT (22.8%) and P/E (24.3%). The internal mammary chain volume was most consistently treated with the PWT (mean dose 99%) vs. P/E (86%) and ST (38.4%) techniques. The heart received the least dose with ST (mean dose 6.7%) vs. PWT (10.3%) and P/E (19%). The PWT treated the greatest amount of contralateral breast (mean dose 5.8%) vs. ST (3.2%) vs. P/E (2.8%). Conclusion: The PWT technique treats the internal mammary chain with acceptable toxicity to major organs, especially the heart, and with reasonable dose homogeneity in patients with mastectomy or intact breasts

[en] We have compared two methods of estimating the cellular radiosensitivity of a heterogeneous tumour, namely, via cell-survival and via tumour control probability (TCP) pseudo-experiments. It is assumed that there exists intra-tumour variability in radiosensitivity and that the tumour consists predominantly of radiosensitive cells and a small number of radio-resistant cells.Using a multi-component, linear-quadratic (LQ) model of cell kill, a pseudo-experimental cell-survival versus dose curve is derived. This curve is then fitted with a mono-component LQ model describing the response of a homogeneous cell population. For the assumed variation in radiosensitivity it is shown that the composite pseudo-experimental survival curve is well approximated by the survival curve of cells with uniform radiosensitivity.For the same initial cell radiosensitivity distribution several pseudo-experimental TCP curves are simulated corresponding to different fractionation regimes. The TCP model used accounts for clonogen proliferation during a fractionated treatment. The set of simulated TCP curves is then fitted with a mono-component TCP model. As in the cell survival experiment the fit with a mono-component model assuming uniform radiosensitivity is shown to be highly acceptable.However, the best-fit values of cellular radiosensitivity produced via the two methods are very different. The cell-survival pseudo-experiment yields a high radiosensitivity value, while the TCP pseudo-experiment shows that the dose-response is dominated by the most resistant sub-population in the tumour, even when this is just a small fraction of the total. (note)

[en] Purpose. To analytically investigate the possibility of a parameter invariant ranking of radiotherapy (RT) plans based on comparing the tumor control probabilities (TCPs) produced by the competing plans for different values of the radiobiological model parameters determining the radiation response. Method. Individual TCP models based on the Single hit model of cell kill and on the linear-quadratic (LQ) model of cell damage, with and without repopulation, are considered. The tumor dose distributions in case of heterogeneous dose irradiation are described by a Gaussian distribution function on the basis of which a TCP expression is derived depending only on the mean dose to the tumor and its standard deviation and the TCP model parameters. Results. It is shown that in case of homogeneous dose to the tumor the plan ranking in terms of TCP is parameter invariant. In case of heterogeneous dose to the tumor there are cases when the plan ranking is parameter invariant and cases when the parameter invariance is violated. An interesting dependence of the extent of the parameter invariance violation on the model of cell kill as well as on the size and repopulation rate of the tumor is noted. Conclusion. We conclude that in many cases RT plan ranking in terms of TCP is parameter invariant. However, since there exist cases where the parameter invariance is lost an investigation of the specific plans to be ranked should be performed applying the proposed approach

[en] Purpose. To investigate the capacity of two phenomenological expressions to describe the population tumor response in case of a heterogeneous irradiation of the tumor. The generalization of the individual tumor control probability (TCP) models to include the case of a heterogeneous irradiation is a trivial problem. However, an analytical solution that results in a closed form population TCP formula for the heterogeneous case is, unfortunately, a very complex mathematical problem. Therefore we applied a numerical approach to the problem. Method. Pseudo-experimental data sets are constructed through the generation of dose distributions and population TCP data obtained by a numerical solution of a multi-dimensional integral over an individual TCP model. The capacity of the following two phenomenological - Poisson and equivalent uniform dose (EUD) based - TCP expressions: to describe the population tumor response in case of heterogeneous irradiation is investigated through their fitting to the pseudo-experimental data sets. Results and conclusions. While both expressions produce statistically acceptable fits to the pseudo-experimental data within 2% TCP error band, the use of the second expression is preferable since it produces considerably better fits to the data sets

[en] A population tumor control probability (TCP) model for fractionated external beam radiotherapy, based on Poisson statistics and in the limit of large parameter heterogeneity, is studied. A reduction of a general eight-parameter TCP equation, which incorporates heterogeneity in parameters characterizing linear-quadratic radiosensitivity, repopulation, and clonogen number, to an equation with four parameters is obtained. The four parameters represent the mean and standard deviation for both clonogen number and a generalized radiosensitivity that includes linear-quadratic and repopulation descriptors. Further, owing to parameter inter-relationship, it is possible to express these four parameters as three ratios of parameters in the large heterogeneity limit. These ratios can be directly linked to two defining features of the TCP dose response: D₅₀ and γ₅₀. In the general case, the TCP model can be written in terms of D₅₀, γ₅₀ and a third parameter indicating the ratio of the levels of heterogeneity in clonogen number and generalized radiosensitivity; however, the third parameter is unnecessary when either of these two sources of heterogeneity is dominant. It is shown that heterogeneity in clonogen number will have little impact on the TCP formula for clinical scenarios, and thus it will generally be the case that the fundamental form of the Poisson-based population TCP model can be specified completely in terms of D₅₀ and γ₅₀: TCP=(1/2) erfc[√(π)γ₅₀(D₅₀/D-1)]. This implies that limited radiobiological information can be determined by the analysis of dose response data: information about parameter ratios can be ascertained, but knowledge of absolute values for the fundamental radiobiological parameters will require independent auxiliary measurements

[en] In small NSCLC, 88% local control at three years from SBRT was reported both for schedule (20–22 Gy ×3) (Fakiris et al 2009 Int. J. Radiat. Oncol. Biol. Phys. 75 677–82), actually close to (18–20 Gy ×3) if density correction is properly applied, and for schedules (18 Gy ×3) and (11 Gy ×5) (Palma et al 2012 Int. J. Radiat. Oncol. Biol. Phys. 82 1149–56). Here, we compare our computed iso-TCP = 88% dose per fraction (d₈₈) for three and five fractions (n) with such clinically adopted ones. Our TCP model accounts for tumour repopulation, at rate λ (d⁻¹), reoxygenation of chronic hypoxia (ch-), at rate a (d⁻¹) and fluctuating oxygenation of acute hypoxia (ah-), with hypoxic fraction (C) of the acutely hypoxic fractional volume (AHF). Out of the eight free parameters whose values we had fitted to in vivo animal data (Ruggieri et al 2012 Int. J. Radiat. Oncol. Biol. Phys. 83 1603–8), we here maintained (a(d⁻¹), C, OER_ch, OER_ah/OER_ch, AHF, CHF) = (0.026, 0.17, 1.9, 2.2, 0.033, 0.145) while rescaling the initial total number of clonogens (N^o) according to the ratio of NSCLC on animal median tumour volumes. From the clinical literature, the usually assumed (α_o/β_o(Gy), λ(d⁻¹)) = (10, 0.217) for the well-oxygenated (o-)cells were taken. By normal (lognormal) random sampling of all parameter values over their 95% C.I., the uncertainty on present d₈₈(n) computations was estimated. Finally, SBRT intra-tumour dose heterogeneity was simulated by a 1.3 dose boost ratio on 50% of tumour volume. Computed d₈₈(±1σ) were 19.0 (16.3; 21.7) Gy, for n = 3; 10.4 (8.7; 12.1) Gy, for n = 5; 5.8 (5.2; 6.4) Gy, for n = 8; 4.0 (3.6; 4.3) Gy, for n = 12. Furthermore, the iso-TCP = 88% total dose, D₈₈(n) = d₈₈(n)*n, exhibited a relative minimum around n = 8. Computed d₈₈(n = 3, 5) are strictly consistent with the clinically adopted ones, which confirms the validity of LQ-model-based TCP predictions at the doses used in SBRT if a highly radioresistant cell subpopulation is properly modelled. The computed minimum D₈₈(n) around n = 8 suggests the adoption of 6 ≤ n ≤ 10 instead of n = 3 in SBRT of small NSCLC tumours. (paper)

[en] This paper outlines a theoretical approach to the problem of estimating and choosing dose-volume constraints. Following this approach, a method of choosing dose-volume constraints based on biological criteria is proposed. This method is called ''reverse normal tissue complication probability (NTCP) mapping into dose-volume space'' and may be used as a general guidance to the problem of dose-volume constraint estimation. Dose-volume histograms (DVHs) are randomly simulated, and those resulting in clinically acceptable levels of complication, such as NTCP of 5±0.5%, are selected and averaged producing a mean DVH that is proven to result in the same level of NTCP. The points from the averaged DVH are proposed to serve as physical dose-volume constraints. The population-based critical volume and Lyman NTCP models with parameter sets taken from literature sources were used for the NTCP estimation. The impact of the prescribed value of the maximum dose to the organ, D_max, on the averaged DVH and the dose-volume constraint points is investigated. Constraint points for 16 organs are calculated. The impact of the number of constraints to be fulfilled based on the likelihood that a DVH satisfying them will result in an acceptable NTCP is also investigated. It is theoretically proven that the radiation treatment optimization based on physical objective functions can sufficiently well restrict the dose to the organs at risk, resulting in sufficiently low NTCP values through the employment of several appropriate dose-volume constraints. At the same time, the pure physical approach to optimization is self-restrictive due to the preassignment of acceptable NTCP levels thus excluding possible better solutions to the problem

[en] The aim of the work is to investigate the impact of radiation-independent (natural or spontaneous) tumor cell death on tumor control probability (TCP) during and following fractionated external-beam radiotherapy employing both analytical and numerical methods. The analytical method solves a TCP model accounting for tumor repopulation and non-radiation tumor cell death during fractionated external-beam radiotherapy. The numerical method is based on a Monte Carlo simulation of the processes of radiation-induced cell kill, as well as cell division and natural cell death randomly taking place in the time interval between fractions. Distributions of the number of surviving cells are constructed using the Monte Carlo method for cases with and without natural cell death. The analytically and numerically calculated values of TCP were found to be in excellent agreement (as shown in the Method and materials section), thereby validating both methods. The TCP model is then fitted to two different experimental data sets with the aim of determining the model parameter values, primarily the natural death rate. Two versions of the linear-quadratic model of cell damage—with and without assumed re-sensitization of the tumor cells during treatment—are used. In two of the fits a strong correlation between the repopulation and spontaneous cell death rates is observed. It was possible to determine separately the values of the two rates only in the fit of the model with resensitization to the most diversified data set consisting of seven different fractionation regimes. The observed correlation together with a theoretical consideration leads to the conclusion that in most cases it is the net effect of the two processes of birth and death rather than the processes separately that determines treatment outcome. However, depending on the values of the rates of the two processes and the duration of the treatment, the treatment outcome may be more accurately determined by the absolute values of the two rates rather than just by their difference. (paper)

[en] Purpose: To verify whether a tumor control probability (TCP) model which mechanistically incorporates acute and chronic hypoxia is able to describe animal in vivo dose–response data, exhibiting tumor reoxygenation. Methods and Materials: The investigated TCP model accounts for tumor repopulation, reoxygenation of chronic hypoxia, and fluctuating oxygenation of acute hypoxia. Using the maximum likelihood method, the model is fitted to Fischer-Moulder data on Wag/Rij rats, inoculated with rat rhabdomyosarcoma BA1112, and irradiated in vivo using different fractionation schemes. This data set is chosen because two of the experimental dose–response curves exhibit an inverse dose behavior, which is interpreted as due to reoxygenation. The tested TCP model is complex, and therefore, in vivo cell survival data on the same BA1112 cell line from Reinhold were added to Fischer-Moulder data and fitted simultaneously with a corresponding cell survival function. Results: The obtained fit to the combined Fischer-Moulder-Reinhold data was statistically acceptable. The best-fit values of the model parameters for which information exists were in the range of published values. The cell survival curves of well-oxygenated and hypoxic cells, computed using the best-fit values of the radiosensitivities and the initial number of clonogens, were in good agreement with the corresponding in vitro and in situ experiments of Reinhold. The best-fit values of most of the hypoxia-related parameters were used to recompute the TCP for non–small cell lung cancer patients as a function of the number of fractions, TCP(n). Conclusions: The investigated TCP model adequately describes animal in vivo data exhibiting tumor reoxygenation. The TCP(n) curve computed for non–small cell lung cancer patients with the best-fit values of most of the hypoxia-related parameters confirms previously obtained abrupt reduction in TCP for n < 10, thus warning against the adoption of severely hypofractionated schedules.

[en] A very important issue in contemporary inverse treatment radiotherapy planning is the specification of proper dose-volume constraints limiting the treatment planning algorithm from delivering high doses to the normal tissue surrounding the tumor. Recently we have proposed a method called reverse mapping of normal tissue complication probabilities (NTCP) onto dose-volume histogram (DVH) space, which allows the calculation of appropriate biologically based dose-volume constraints to be used in the inverse treatment planning. The method of reverse mapping requires random sampling from the functional space of all monotonically decreasing functions in the unit square. We develop, in this paper, a random function generator for the purpose of the reverse mapping. Since the proposed generator is based on the theory of random walk, it is therefore designated in this work, as a random walk DVH generator. It is theoretically determined that the distribution of the number of monotonically decreasing functions passing through a point in the dose volume histogram space follows the hypergeometric distribution. The proposed random walk DVH generator thus simulates, in a random fashion, trajectories of monotonically decreasing functions (finite series) that are situated in the unit square [0,1]x[1,0] using the hypergeometric distribution. The DVH generator is an important tool in the study of reverse NTCP mapping for the calculation of biologically based dose-volume constraints for inverse treatment planning