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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.