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The impact of a rectal spacer and an increased near maximum target dose in VMAT prostate SBRT is studied. For a group of 11 patients (35Gy-in-five-fractions VMAT prostate SBRT) a set of 4 plans were generated, namely two VMAT plans, with D2%⩽37.5Gy (Hom) and with D2%⩽40.2Gy (Het), were created for each of two CT scans taken before (NoSpc) and after (Spc) transperineal spacer insertion. Consequently the methodology for parameter invariant TCP (tumor control probability) plan ranking was applied for comparison of the plans in terms of tumor control. NTCPs (normal tissue complication probabilities) were calculated for rectum and bladder using Lyman's model. For all 11 patients the TCP plan ranking has shown that the Het plans would perform considerably better in TCP terms than the Hom ones. The plans without rectal spacer were ranked worse compared to those with rectal spacer except for one set of Hom plans. The calculated NTCPs for rectum produced by the Het plans were quite similar to the NTCPs of the Hom ones. The rectal NTCPs of the Hom Spc plans were always lower than the NTCPs of the Hom NoSpc plans. The NTCP values for bladder were extremely low in all cases. The use of rectal spacer leads in general to lower risk of rectal complications, as expected, and even to better tumor control. Plans with increased near maximum target dose (D2%⩽40.2Gy) are expected to perform much better in terms of tumor control than those with D2%⩽37.5Gy.

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, OERch, OERah/OERch, 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 (αo/βo(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.

The efficacy of Stereotactic Body Radiation Therapy (SBRT) in early-stage non-small cell lung cancer for severely hypofractionated schedules is clinically proven. Tumour control probability (TCP) modelling might further optimize prescription dose and number of treatment fractions (n). To this end, we will discuss the following controversial questions. Which is the most plausible cell-survival model at doses per fraction (d) as high as 20Gy? Do clinical data support a dose-response relationship with saturation over some threshold-dose? Given the reduced re-oxygenation for severe hypofractionation, is the inclusion of tumour hypoxia in TCP modelling relevant? Can iso-effective schedules be derived by assuming a homogeneous tumour-cell population with α/β≈10Gy, or should distinct cell subpopulations, with different α/β values, be taken into account? Is there scope for patient-specific individualization of n? Despite the difficulty of providing definite answers to the above questions, reasonable suggestions for lung SBRT can be derived from the literature. The LQ model appears to be the best-fitting model of cell-survival even at such large d, and is therefore the preferred choice for TCP modelling. TCP increases with dose, reaching saturation above 90% local control, but there is still uncertainty on the threshold-dose. In silico simulations accounting for variations in tumour oxygenation are consistent with an improved therapeutic ratio at 5-8 fractions instead of the current 3-fraction reference schedules. Tumour hypoxia modelling might also explain how α/β changes with n, identifying the clonogen subpopulation which determines tumour response. Finally, an optimal patient-specific n can be derived from the planned lung dose distribution.

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.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.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).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 n10, thus warning against the adoption of severely hypofractionated schedules.

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.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.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.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.

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.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: [Figure: see text] to describe the population tumor response in case of heterogeneous irradiation is investigated through their fitting to the psuedo-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.

In this work two analytical methods are developed for computing the probability distribution of the number of surviving cells of a repopulating tumor during a fractionated external radio-treatment. Both methods are developed for the case of pure birth processes. They both allow the description of the tumor dynamics in case of cell radiosensitivity changing in time and for treatment schedules with variable dose per fraction and variable time intervals between fractions. The first method is based on a direct solution of the set of differential equations describing the tumor dynamics. The second method is based on the works of Hanin et al. [Hanin LG, Zaider M, Yakovlev AY. Distribution of the number of clonogens surviving fractionated radiotherapy: a long-standing problem revisited. Int J Radiat Biol 2001;77:205-13; Hanin LG. Iterated birth and death process as a model of radiation cell survival. Math Biosci 2001;169:89-107; Hanin LG. A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence. Math Biosci 2004;191:1-17], where probability generating functions are used. In addition a Monte Carlo algorithm for simulating the probability distributions is developed for the same treatment conditions as for the analytical methods. The probability distributions predicted by the three methods are compared graphically for a certain set of values of the model parameters and an excellent agreement is found to exist between all three results, thus proving the correct implementation of the methods. However, numerical difficulties have been encountered with both analytical methods depending on the values of the model parameters. Therefore, the Poisson approximation is also studied and it is compared to the exact methods for several different combinations of the model parameter values. It is concluded that the Poisson approximation works sufficiently well only for slowly repopulating tumors and a low cell survival probability and that it cannot always be used for the description of the tumor dynamics. The two analytical methods and the Monte Carlo method are compared in terms of computational time. The Monte Carlo method is found to be much more time consuming than the analytical methods.

Purpose To determine from the number of trials, n , and the number of observed successes, k the most probable value, the variance and the confidence limits of the probability of success, p , in animal experiments and clinical studies subject to binomial statistics. Method In such experiments the probability of success is an unknown parameter. The Bayesian approach to the problem is advocated, based on constructed distribution of the probability of success. Results A simple Matlab code for the calculation of the confidence limits according to the proposed method is provided. The most probable, the mean, the variance and the confidence limits are calculated applying the usual definitions of these characteristics. Conclusion The proposed method works for any number of trials – large and small and all possible values of the number of successes, including k = 0 and k = n , providing exact formulae for the calculation of the confidence limits in all cases.

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.

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: D{sub 50} and {gamma}{sub 50}. In the general case, the TCP model can be written in terms of D{sub 50}, {gamma}{sub 50} 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 themore » fundamental form of the Poisson-based population TCP model can be specified completely in terms of D{sub 50} and {gamma}{sub 50}: TCP=(1/2) erfc[{radical}({pi}){gamma}{sub 50}(D{sub 50}/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.« less

The variation between individuals in their dose-response characteristics complicates attempts to extract estimates of radiobiological parameters (e.g. alpha, beta, etc) from fits to clinical dose-response data. The use of 'population' dose-response models that explicitly account for this variability is necessary to avoid obtaining skewed parameter estimates. In this work, we evaluated an example of a 'population' tumour control probability (TCP) model in terms of its ability to provide reliable parameter estimates. This was accomplished by performing fits of this population model to 'pseudo' data sets, which were generated with Monte Carlo techniques and based on preset values for the various radiobiological parameters. The fitting exercises illustrated considerable correlations between the model parameters. Especially significant was the large correlation observed between the parameter mu=alpha/sigmaalpha used to characterize the level of population heterogeneity in radiosensitivity and the alpha/beta parameter typically used to describe the response to fractionation. The results imply that fits to clinical data may not be able to distinguish between tumours exhibiting a high degree of heterogeneity and a strong beta-mechanism and those containing little heterogeneity and having a weak beta-mechanism. One implication is that basing the design of optimal fractionation regimes on such fitting results may be error-prone. If in vitro assays are to be used to independently determine biologically reasonable ranges for parameter values, an accurate knowledge of the relationship between in vitro and in vivo dose-response characteristics is required.

In this work we further develop the modeling of tumor dynamics by proposing a mechanism of tumor resensitization that is based on the process of reoxygenation. Reoxygenation is modeled using the concept of nonstationary diffusion of oxygen. This leads to the derivation of an explicit expression for the radiosensitivity parameter that predicts a radiosensitivity that increases with time. To account for the resensitization mechanism, the time-dependent expression for the radiosensitivity is then incorporated within a tumor control probability (TCP) model that already includes tumor cell repopulation and repair. We fit a set of experimental animal TCP curves corresponding to several different fractionation regimes using both the modified (with resensitization) and unmodified (without resensitization) versions of the TCP model. In comparison to the unmodified model, the modified model produces statistically superior fits, and is able to describe an "inverse" dose-fractionation behavior present in the data.

Objective To develop a model describing the relationship between the parotid gland radiation dose and salivary flow reduction. Salivary function was described by the "relative flow reduction" (RFR)—a continuous variable in contrast to the traditional binary response used in normal tissue complication probability estimations. Methods and materials Twenty-three patients with squamous cell carcinoma of the head and neck who were treated with intensity-modulated radiotherapy (RT) were the subject of this study. Of these patients, 19 had sufficiently long follow-up to be eligible for analysis. All were treated with curative intent, most (14 of 19) in the postoperative setting. The planning objectives were to deliver a mean dose of 50, 60, or 70 Gy, respectively, to low-risk microscopic, high-risk microscopic, and gross disease areas, while maintaining a mean dose of ≤20 Gy to the spared portion of one or both parotid glands. The mean dose to all parotid glands (right and left) was 30.2 Gy. All submandibular glands received >50 Gy when not surgically removed. Whole-mouth saliva collections, including both stimulated and unstimulated saliva flow, were obtained before treatment and at regular intervals after RT. These measurements were converted to the RFR by comparing the posttreatment and pretreatment flow rates. Any follow-up flow rates greater than baseline were scored as 0 relative reduction. We used Lyman's model to relate the equivalent uniform dose to RFR at various points for each patient. The equivalent uniform dose was calculated using the linear quadratic model, with an assumed α/β ratio of 3 Gy for the parotid gland. Measurements were modeled 1–3 months after RT (early) and >6 months after RT (late), and using the best and worst measurements, regardless of when measured. Results Fitting the Lyman model to RFR data of unstimulated flow revealed a statistically significant dose–complication relationship. We observed a stepwise reduction in flow, with the threshold dose D 50 at 2 Gy per fraction (D 50 ) increasing from 12.4 Gy (early) to 43.9 Gy (late). For the worst and best flow measurements, the corresponding D 50 (2 Gy/fr) was 13.0 Gy and 40.1 Gy, respectively. For most stimulated flow measurements, a weak relationship was found between the RFR and equivalent uniform dose. In those cases, the model did not yield a statistically significant description of the data. However, in the case of late measurements, the relationship was statistically significant and similar to that seen in the unstimulated cases, with a D 50 (2 Gy/fr) of 47.5 Gy. Conclusion We observed a strong relationship between the generalized mean parotid gland dose and RFR. The threshold dose increased markedly between the early and late measurements, indicating a statistically significant recovery effect in this tissue. Compared with unstimulated flow, the RFR for stimulated flow was not described as well by the model, because the effect of the stimulant was not included in the model.

In the current state-of-the art of clinical inverse planning, the design of clinically acceptable IMRT plans is predominantly based on the optimization of physical rather than biological objective functions. A major impetus for this trend is the unproven predictive power of radiobiological models, which is largely due to the scarcity of data sets for an accurate evaluation of the model parameters. On the other hand, these models do capture the currently known dose-volume effects in tissue dose-response, which should be accounted for in the process of optimization. In order to incorporate radiobiological information in clinical treatment planning optimization, we propose a hybrid physico-biological approach to inverse treatment planning based on the application of a continuous penalty function method to the constrained minimization of a biological objective. The objective is defined as the weighted sum of normal tissue complication probabilities evaluated with the Lyman normal-tissue complication probability model. Physical constraints specify the admissible minimum and maximum target dose. The continuous penalty function method is then used to find an approximate solution of the resulting large-scale constrained minimization problem. Plans generated by our approach are compared to ones produced by a commercial planning system incorporating physical optimization. The comparisons show clinically negligible differences, with the advantage that the hybrid technique does not require specifications of any dose-volume constraints to the normal tissues. This indicates that the proposed hybrid physico-biological method can be used for the generation of clinically acceptable plans.

In this work we study the descriptive power of the main tumor control probability (TCP) models based on the linear quadratic (LQ) mechanism of cell damage with cell recovery. The Poisson, binomial, and a dynamic TCP model, developed recently by Zaider and Minerbo are considered. The Zaider-Minerbo model takes cell repopulation into account. It is shown that the Poisson approximation incorporating cell repopulation is conceptually incorrect. Based on the Zaider-Minerbo model, an expression for the TCP for fractionated treatments with varying intervals between two consecutive fractions and with cell survival probability that changes from fraction to fraction is derived. The models are fitted to an experimental data set consisting of dose response curves that correspond to different fractionation regimes. The binomial TCP model based on the LQ mechanism of cell damage solely was unable to fit the fractionated response data. It was found that the Zaider-Minerbo model, which takes tumor cell repopulation into account, best fits the data.

A method for determining the optimum dose distribution in the planning target volume is proposed when target volumes are deliberately enlarged to account for tumour mobility in external beam radiotherapy. The optimum dose distribution is a dose distribution that will result in an acceptable level of tumour control probability (TCP) in most of the arising cases of tumour dislocation. An assumption is made that the possible shifts of the tumour are subject to a Gaussian distribution with mean zero and known variance. The idea of a reduced (mean in ensemble) tumour cell density is introduced. On this basis, the target volume and dose distribution in it are determined. The tumour control probability as a function of the shift of the tumour has been calculated. The Monte Carlo method has been used to simulate TCP distributions corresponding to tumour mobility characterized by different variances. The obtained TCP distributions are independent of the variance of the mobility because the dose distribution in the planning target volume is prescribed so that the mobility variance is taken into account. For simplicity a one-dimensional model is used but three-dimensional generalization can be done.

This paper offers a general mathematical approach to dose distribution optimization which allows tumours with different degrees of complexity to be considered. Two different biological criteria - A) keeping the control probability of the different parts of the tumour (local tumour control probability) uniform throughout the tumour and B) minimizing the mean dose delivered to the tumour are studied. For both criteria we impose the requirement that the whole tumour control probability be kept on a certain desired level. It is proved that the adoption of the first criterion requires a dose distribution logarithmic with the cell density and proportional to the inverse of the cell radiosensitivity while the adoption of the second criterion necessitates a homogeneous dose distribution when the cell radiosensitivity is constant. The corresponding formula for the dose distribution in case of heterogeneous cell radiosensitivity is also given. The two criteria are compared in terms of local tumour control probability and mean dose delivered to the tumour. It is concluded that maintaining constant local tumour control probability (criterion A) may be of greater clinical importance then minimizing the mean dose (criterion B).

An analytic solution to the problem of finding the value of the dose DR delivered to the edge of the tumour in the case of the tumour model studied by Webb and Nahum is suggested.

The dose heterogeneity in the tumor is often described as being normally distributed. Besides the normal distribution we propose the Fermi function describing Fermi statistics as a possible dose heterogeneity descriptor. In order to demonstrate the adequacy of the proposed functions as dose distribution descriptors 30 clinical gross tumor volume (GTV) dose-volume histograms (DVHs) are gathered and fit with the examined functions.

The aim of this study is to develop a magnetic resonance imaging (MRI)-based treatment planning procedure for intracranial lesions. The method relies on (a) distortion correction of raw magnetic resonance (MR) images by using an adaptive thresholding and iterative technique, (b) autosegmentation of head structures relevant to dosimetric calculations (scalp, bone and brain) using an atlas-based software and (c) conversion of MR images into computed tomography (CT)-like images by assigning bulk CT values to organ contours and dose calculations performed in Eclipse (Philips Medical Systems). Standard CT + MRI-based and MRI-only plans were compared by means of isodose distributions, dose volume histograms and several dosimetric parameters. The plans were also ranked by using a tumor control probability (TCP)-based technique for heterogeneous irradiation, which is independent of radiobiological parameters. For our 3 T Intera MRI scanner (Philips Medical Systems), we determined that the total maximum image distortion corresponding to a typical brain study was about 4 mm. The CT + MRI and MRI-only plans were found to be in good agreement for all patients investigated. Following our clinical criteria, the TCP-based ranking tool shows no significant difference between the two types of plans. This indicates that the proposed MRI-based treatment planning procedure is suitable for the radiotherapy of intracranial lesions.

Background. A property of the integral dose-volume histogram (DVH) space is analytically investigated in this work. A curve called an alpha-iso-NTCP (normal tissue complication probability) envelope is constructed by connectingpoints belonging to step-like integral DVHs, each corresponding to homogeneous partial organ irradiation of a relative volume vk to dose Dk such that the resulting NTCP has, in all cases, a particular value alpha. The two subspaces into which the envelope divides the DVH space are analytically explored in terms of the equivalent uniform doses (EUDs) corresponding to the different DVHs. It is theoretically proven that any DVH, other than the step-like DVH, passing through a point (Dk1 vk) from the alpha-iso-NTCP envelope, will result in an NTCP > alpha. Conclusions. Thus, it is proven thatany DVH that at least partially lies above the envelope results in an NTCP> alpha. For some of the DVHs lying under the envelope, e.g. those that are tangential to the envelope, it is also true that the resulting NTCP > alpha. However, it was numerically demonstrated elsewhere that there east DVHslying entirely in the lower subspace that result in an NTCP < alpha. Therefore, one can conclude that since there is a chance that a DVH lying under the alpha-iso-NTCP envelope will result in NTCP < alpha, it would be preferable in the treatment optimization process to seek solutions for DVHs lying entirely under an iso-NTCP envelope and avoid solutions that have DVHs above an iso-NTCP envelope. Izhodišča. V članku smo raziskali lastnost prostora integralnega dozno-volumskega histograma (DVH). Krivuljo enake verjetnosti poškodbe zdravega tkiva (alfa-izo-NTCP (normal tissue complication probability)) smo oblikovali s povezovanjem točk, ki pripadajo intervalom vk stopničastega integralnega DVH s pripadajočo dozo Dk in posledičnim NTCP z vrednostjo alfa. Podprostora, na katera krivulja razdeli DVH prostor, smo raziskali v smislu ekvivalentnih uniformnih doz (EUD), ki pripadajo različnim DVH-jem. Zaključki.Teoretično je dokazano, da ima vsak DVH, ki je različen od stopničastega DVH in ki gre skozi točko (Dk1 vk) alfa-izo-NTCP ovojnice, NTCP > alfa. Iz česar sledi, da vsakemu DVH, ki vsaj delno leži nad ovojnico, pripada NTCP > alfa. Nekaterim DVH, ki ležijo pod ovojnico, npr. se je dotikajo, prav tako pripada NTCP > alfa. Vendar je bilo numerično dokazano, daima DVH, ki v celoti leži v spodnjem podprostoru, NTCP < alfa. Glede na slednje bi bilo v postopku optimizacije zdravljenja zaželeno iskati rešitve z DVH povsem pod izo-NTCP ovojnico in se izogibati tistim, ki ležijo nad njo.

In this work, the functional form similarity between the individual and fundamental population TCP models is investigated. Using the fact that both models can be expressed in terms of the geometric parameters gamma50 and D50, we show that they have almost identical functional form for values of gamma50 >= 1. The conceptual inadequacy of applying an individual model to clinical data is also discussed. A general individual response TCP expression is given,parameterized by Df and gammaf - the dose corresponding to a control level of f, and the normalized slope at that point. It is shown that the dose-response may be interpreted as an individual response only if gamma50 is sufficiently high. Based on the functional form equivalency between the individual and the population TCP models, we discuss the possibility of applying the individual TCP model for the case of heterogeneous irradiations. Due to the fact that the fundamental population TCP model is derived for homogeneous irradiations only, we propose the use of the EUD, given by the generalized mean dose, when the fundamental population TCP model is used to fit clinical data. Raziskali smo podobnost TCP modelov za posameznika in populacijo. Z ozirom na to, da lahko oba modela opišemo z geometrijskima parametroma gama50 in D50, smo pokazali, da dobita skoraj enako obliko pri vrednostih gama50 >= 1. Obravnavali smo neprimernost uporabe modela za posameznika na kliničnih podatkih. Podali smo splošen izraz za TCP in ga parametrizirali z Df in gamaf - z dozo, kjer TCP zavzame vrednost f in z normaliziranim naklonom v tej točki. Izkazalo se je, da lahko krivuljo interpretiramo kot odziv posameznika le ob dovolj velikem gama50. Na osnovi podobnosti TCP modelov za posameznika in populacijo smo obravnavali možnost uporabe slednjega v primeru neenakomernega obsevanja. Ker osnovni TCP model za populacijo predpostavlja enakomerno obsevanje, smo predlagali, da se pri uporabi na kliničnih podatkih vrednost parametra EUD izenači s povprečno dozo.

In volumetric-modulated arc therapy (VMAT) prostate stereotactic body radiotherapy (SBRT), dose coverage of the planning target volume (PTV) becomes challenging when the sparing of rectum, bladder and urethra is strictly pursued. Our current 35-Gy-in-five-fraction plans only assure 33.2 Gy to ≥95% PTV ([Formula: see text] ≥ 95%). Looking for an improved [Formula: see text], increased near-maximum target dose (D2%) and prostate-rectum spacer insertion were tested.For 11 patients, two VMAT plans, with D2% ≤ 37.5 Gy (Hom) or D2% ≤ 40.2 Gy (Het), on each of two CT studies, before or after spacer insertion, were computed. All plans assured [Formula: see text] ≥95%, and1 cm(3) of rectum, bladder and urethra receiving ≥35 Gy. By hypothesis testing, several dose-volume metrics for target coverage and rectal sparing were compared across the four groups of plans. The impact of spacer insertion on the fractions of rectum receiving more than 18, 28 and 32 Gy ([Formula: see text]) was further tested by linear correlation analysis.By hypothesis testing, the increased D2% was associated with improvements in target coverage, whereas spacer insertion was associated with improvements in both target coverage and rectal [Formula: see text]. By linear correlation analysis, spacer insertion was related to the reductions in rectal [Formula: see text] for X ≥ 28 Gy.A slightly increased D2% or the use of spacer insertion was each able to improve [Formula: see text]. Their combined use assured [Formula: see text] ≥ 98% to all our patients. Spacer insertion was further causative for improvements in rectal sparing.For VMAT plans in prostate SBRT, the distinct dosimetric usefulness of increased D2% and of the use of spacer insertion were validated in terms of target coverage and rectal sparing.

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.

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.

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 gamma50. In the general case, the TCP model can be written in terms of D50, gamma50 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 gamma50: TCP= 1/2 erfc[square root of pi(gamma50)(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.

Radiotherapy treatment planning is becoming ever more complex, driven largely by the growing use of intensity modulated radiotherapy (IMRT). There is a corresponding need to improve the computer-based programs being designed to optimize the process of planning the spatial and temporal distributions of dose for such treatments. This article reviews the general objectives of treatment planning, and also the various physical and biological/statistical criteria that have been explored to serve as the bases for optimization objective functions. Keywords: radiotherapy treatment planning; optimization of treatment planning; radiotherapy optimization; radiotherapy treatment planning dose distribution; radiotherapy dose-response model biological dose-response modeling

The effects of radiation damage, tumour repopulation and cell sublethal damage repair and the possibility of extracting information about the model parameters describing them are investigated in this work. Previously published data on two different cultured cell lines were analysed with the help of a tumour control probability (TCP) model that describes tumour cell dynamics properly. Different versions of a TCP model representing the cases of full or partial cell recovery between fractions of radiation, accompanied by repopulation or no repopulation were used to fit the data and were ranked according to statistical criteria. The data analysis shows the importance of the linear–quadratic mechanism of cell damage for the description of the in vitro cell dynamics. In a previous work where in vivo data were analysed, the employment of the single hit model of cell kill and cell repopulation produced the best fit, while ignoring the quadratic term of cell damage in the current analysis leads to poor fits. It is also concluded that more experiments using different fractionation regimes producing diverse data are needed to help model analysis and better ranking of the models.

The critical volume (CV) normal tissue complication probability (NTCP) model was used to fit experimental data on radiation pneumonitis in mice to test the model and determine the values of the model parameters characterizing the lung structure: relative critical volume and cell radiosensitivity. The entire lungs of mice from ten different strains were irradiated acutely and homogeneously to different doses. The experimental animals from the different strains expressed different radiation sensitivities, forming ten well-defined dose-response curves. The most widely accepted biological NTCP model (the individual CV NTCP) readily applicable to cases of acute uniform irradiation was used to fit all the individual dose-response curves simultaneously. To account for the apparent difference in the response of the different strains, it was assumed that the strains differed in their (cell) radiosensitivity. The maximum likelihood method of fitting was used. The obtained fit was statistically highly acceptable. The best-fit value of the relative critical volume, mu, was 78%, which is extremely close to the histologically observed value of around 72%. The values of radiosensitivity, alpha, ranged between 0.26 and 0.37 Gy(-1) for the different strains. The best-fit numbers of functional subunits (FSU) constituting the lung, N, and the number of cells in an FSU, N(o), were implausibly low: N = 9 and N(o) = 23, respectively. The best-fit value of N(o)N was a very small number that was unlikely to correspond to the total number of cells comprising the lung, suggesting that a different interpretation of N and N(o) was required. The individual CV model provided a simultaneous description of the individual responses of different mouse strains through assumed interindividual variability in alpha only. A new interpretation is given to the entities corresponding to N(o) and N. N(o)N is interpreted as the number of certain elementary structures. These structures are considered to be equivalent to the classical functional subunit, which is much larger than a cell and plays a fundamental role in determining the radiation response of the organ. N is identified as the number of the few large subdivisions of the lungs, M = microN is the number that have to be damaged for the lung to fail. N(o) is interpreted as the mean number of elementary structures (FSU) per large subdivision. This imposes a picture of damage to large, contiguous subdivisions containing many FSU, which is consistent with the histological appearance of the lungs of mice in respiratory distress. This picture is in marked contrast to the random distribution of small areas of damage expected for the small size of an FSU. This random distribution is characteristic of earlier stages of the development of radiation pneumonitis, suggesting that some additional process spreads injury from damaged FSU to adjacent, undamaged FSU during the terminal phase.

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. © 2003 Elsevier Science Inc.

This paper presents a complete set of formulae for the position (D50) and the normalized slope (gamma50) of the dose-response relationship based on the most commonly used radiobiological models for tumours as well as for normal tissues. The functional subunit response models (critical element and critical volume) are used in the derivation of the formulae for the normal tissue. Binomial statistics are used to describe the tumour control probability, the functional subunit response as well as the normal tissue complication probability. The formulae are derived for the single hit and linear quadratic models of cell kill in terms of the number of fractions and dose per fraction. It is shown that the functional subunit models predict very steep, almost step-like, normal tissue individual dose-response relationships. Furthermore, the formulae for the normalized gradient depend on the cellular parameters alpha and beta when written in terms of number of fractions, but not when written in terms of dose per fraction.

A method that allows a straightforward implementation of dose-volume constraints in gradient algorithms for inverse treatment planning is presented. The method is consistent with the penalty function approach, which requires the formulation of an objective function with penalty terms proportional to the magnitudes of constraint violations. Dose constraints with respect to minimum and maximum target dose levels are incorporated in quadratic, dose-penalty terms. Analogously, quadratic volume-penalty terms in the objective function reflect the violation of dose-volume constraints imposing limits on the fractions of healthy organ volumes that can be irradiated above specified dose levels. It has been demonstrated that within the framework of this formulation neither modified objective functions nor finite difference gradient calculations are necessary for the incorporation of gradient minimization algorithms. As an example, a simple steepest descent algorithm is presented along with its application to illustrate prostate and lung cases.

This work investigates the existing biological models describing the response of tumours and normal tissues to radiation, with the purpose of developing a general biological model of the response of tissue to radiation. Two different types of normal tissue behaviour have been postulated with respect to its response to radiation, namely critical element and critical volume behaviour. Based on the idea that an organ is composed of functional subunits, models have been developed describing these behaviours. However, these models describe the response of an individual, a particular patient or experimental animal, while the clinically or experimentally observed quantity is the population response. There is a need to extend the models to address the population response, based on the ideas we have about the individual response. We have attempted here to summarize and unify the existing individual models. Finally, the population models are investigated by fitting to pseudoexperimental sets of data and comparing them with each other in terms of goodness-of-fit and in terms of their power to recover the values of the population parameters.

Purpose: To propose an objective function for TCP/NTCP curve fitting.Method and Materials: One of the widely used objective functions by physicists, when fitting theoretical models to experimental data, is the χ 2 one, where: χ 2 = Σ i (( Y theoretical i ‐ Y experimental i )/ σ experimental i ) 2 which is the maximum likelihood function in case of Gaussian random variables. In the case of a binary outcome, as is the nature of clinical or experimental animal radiation outcome data, the random variable has a binomial distribution and the maximum likelihood function for ungrouped data becomes: L =Σ responders ln ( P theor ( par,DVH )) +Σ non‐responders ln (1‐( P theor ( par,DVH ))) , where Ptheor stands for TCPtheor or NTCPtheor. The maximization of L is often used by different authors for the estimation of the TCP/NTCP model parameters. Results: However, sometimes χ 2 is used to fit TCP/NTCP functions for different purposes like theoretical comparison between different models. In this case the application of χ 2 becomes inaccurate and inadequate especially when the function values are close to the ends of the interval in which they are defined, namely 0 or 1. This is why we propose t he application of the double logarithmic transformation, presuming that the random variable is not normally but log‐log‐normally distributed. Conclusion: Thus the objective function in case of model comparison would become: χ 2 = Σ i ((− ln (− ln ( P theoretical i )+ ln (− ln ( P ‘experimental’ i ))/σ log_‘experimental’ i ) 2 where σ log_‘experimental’ i = − σ ‘experimental’ i / ( P ‘experimental’ i ln ( P ‘experimental’ i )) . Here P theoretical i stands for the values of P predicted by one of the models that are being compared and P ‘experimental’ i stands for the values of P predicted by the other model.

Purpose: To demonstrate that the functional form of the population tumor control probability (TCP) model is well approximated by the individual TCP model. Method and Materials: Due to inter‐parameter relations, the parameters of the population based Poisson TCP model in the limit of large heterogeneity reduce to the geometric parameters D 50 and γ50:1 TCP pop =0.5erfc[π 0.5 γ 50 (D 50 /D − 1)] . On the other hand the individual TCP model can also be written in the terms of the geometrical characteristics of the dose‐response relationship — D 50 and γ50: TCP ind =0.5 exp [2γ 50 /( ln 2)(1‐D/D 50 )] = exp [− N o exp (−α D )] . For typical clinical values2 of the parameters D 50 and γ50, we evaluate and compare the individual and population‐based TCP models. Results: The two models are approximately equivalent for different values of D 50 and γ50. When plotted, the individual and population‐based TCP models almost overlap over a wide range of doses for all D 50 values and γ50 higher than 1. When γ50 is less than 1 both functions start to differ from one another. Conclusion: The equivalency between the individual and population TCP models is demonstrated. The individual TCP model has been observed to fit clinical datasets reasonably well, and this phenomenon may be attributed to the similarity of the models. When fitted to clinical data, the individual TCP model will produce parameter estimates completely emptied from their biological meaning and become purely phenomenological.

Purpose: To further develop the modeling of tumor dynamics, we propose a mechanism of tumor resensitization based on reoxygenation, leading to the derivation of a time-dependent expression for the radiosensitivity, α(t), that can be incorporated into tumor control probability (TCP) models used to fit fractionated dose response data. Method and Materials: To model the process of reoxygenation, we first assume that a tumor's radiosensitivity is dictated by the oxygenation level of an inner core of tumor cells, which is initially hypoxic, and thus radioresistant. Using an equation to describe the diffusion of oxygen to this inner core from outer tumor layers, and by presuming a relationship between α and the oxygen concentration, an expression for α(t) can be derived. This expression is then incorporated within a TCP model that already includes tumor cell repopulation and repair. We fit a published set of experimental animal TCP curves corresponding to several different fractionation regimes using both the modified (with resensitization) and unmodified (without resensitization) versions of the TCP model. To investigate the importance of the β mechanism, we also fit a version of the model with resensitization in which the linear-quadratic (LQ) parameter β was set to zero. Results: The modified model with resensitization, and with a non-zero β component, produced statistically superior fits. Specifically, only this model was able to describe an “inverse” dose-fractionation behavior present in the data, which indicated that less dose was required to achieve a given tumor control with five fractions than with three. Conclusion: Resensitization and the β mechanism may be important in the description of the dose-response for a small number of fractions. Since it may in the future be possible to alter radiotherapy schemes after monitoring the tumor response in the first few treatments, consideration of such factors may prove clinically useful.

Purpose: A model is developed that describes the tumor dynamics of a repopulating tumor during a fractionated external radio treatment.Method and Materials: An iterative algorithm is constructed for the calculation of the probability distribution of the surviving number of tumor cells for any point in time during the treatment and for post treatment times. This is accomplished through the assumption, that cell repopulation takes place only during the intervals between the fractions of irradiation. Thus, the processes of cell repopulation and of radiation‐induced cell kill are considered decoupled and are therefore modeled separately. Binomial statistics is applied to describe the radiation induced cell damage. Natural death is ignored. Tumor repopulation is modeled through the construction of differential equations for the probability that a certain number of cells exist at a certain point in time. The differential equations are solved in a straightforward manner. Results: The probability of zero survivals is compared numerically to the Zaider‐Minerbo tumor control probability (TCP) model for an arbitrarily chosen regime of treatment. An excellent agreement is found to exist between the two expressions. The Poisson approximation of the probability distribution of the number of cells is also studied and it is compared to the proposed method for several different combinations of the model parameter values. Conclusion: It is concluded that the Poisson approximation works well only for slowly repopulating tumors and for rather high cell radiosensitivity (i.e. low cell survival probability after a fraction of radiation). Therefore, the use of the exact method for the description of the radiation response of repopulating tumors is recommended.

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 inmore » the study of reverse NTCP mapping for the calculation of biologically based dose-volume constraints for inverse treatment planning.« less

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.

Dose‐volume constraints are essential for treatment planning optimization using physical objective functions. A theoretical approach to the problem of choosing dose‐volume constraints based on the reverse normal tissue complication probability (NTCP) mapping into dose‐volume space is developed. Dose‐volume histograms (DVHs) are randomly simulated and those resulting in clinically acceptable levels of complication, e.g. NTCP of 5±0.5%, are selected and averaged producing a mean DVH which is proven to result in the same level of NTCP. The points from the averaged DVH are proposed to serve as dose‐volume constraints for treatment planning optimization using physical objective functions. The population based Critical Volume and the Lyman NTCP models with parameter sets taken from literature were used for the NTCP estimation. Constraint points for 16 organs are calculated. These dose‐volume constraints are not unique and depend on the range in which the maximum dose to the organ at risk, D max, is allowed to vary. 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 pre‐assignment of acceptable NTCP levels thus excluding possible better solutions to the problem.

Purpose: To analytically investigate a property of the integral dose‐volume histogram (DVH) space. Method and Materials: A curve called an α‐iso‐NTCP envelope is constructed by connecting points belonging to step‐like integral DVHs each corresponding to partial organ homogeneous irradiation of relative volumes να to dose levels D α such that the resulting NTCP is in all cases α %. The two subspaces into which the envelope divides the DVH space are analytically explored through comparing the values of the equivalent uniform doses (EUDs) corresponding to the different DVHs and using the fact that NTCP is a monotonic function of EUD as well as the monotonic nature of the integral DVH itself. Results: It is theoretically proven that any DVH passing through a point (D α, να) from the α‐iso‐NTCP envelope, i.e. any DVH that crosses or is tangential to the envelope, will result in an NTCP⩾α%, the equality being valid only for the step‐like DVH corresponding to the partial organ homogeneous irradiation of να to D α,. Thus, it is proven that any DVH that at least partially lies above the envelope result in NTCP>α%. For some of the DVHs lying under the envelope, e.g. those that are tangential to the envelope, it is also true that the resulting NTCP>α%. However, it was numerically demonstrated elsewhere that there exist DVHs lying entirely in the lower subspace that result in NTCP

Purpose: The purpose of this study is to develop a complete treatment planning procedure for radiotherapy of brain lesions based solely on magnetic resonance imaging.Method and Materials: The MR‐based treatment planning procedure relies on converting the MRimages into CT‐like images by assigning electron density information to organ structures (i.e. brain, bone and scalp). First step in the process is to correct the MRimages for 3D geometrical distortions by applying a novel distortion correction procedure. The next stage is to segment the datasets into anatomical structures by using an automatic segmentation tool suited for MRbrainimages. Once the MRimages contain both the target volume and the electron density information, they are ready to be used for dose calculations. The resulting CT+MR and MR‐based plans were compared in terms of isodose distributions, DVHs and NTCP/TCPs. A plan ranking TCP‐based method for heterogeneous irradiation, which does not require the knowledge of radiobiological parameters, is used. Results: For all patients investigated, we found that MR‐based plans (1.5T and 3T) are in good agreement (within 1%) with their corresponding CT+MR‐based plans in terms of PTV dose coverage, DVHs and NTCP/TCPs. We compared tumor contours drawn on both 1.5T and 3T MRimages in terms of shape, volumes and their impact on CRT plans. For all patients the delineation of the tumor was simpler for 3T images due to higher contrast. For some patients the tumor volumes drawn on the 3T images were up to 60% higher than on 1.5T images. RT plan ranking shows that the 3T plans are significantly better than 1.5T ones. Conclusion: The proposed MR‐based treatment planning procedure was found to perform as good as the current clinical procedure based on CT+MR. Due to a higher contrast the tumor may be significantly better delineated on 3T images.