Internatonal Journal of Cancer Therapy and Oncology www.jcto.org Evaluaton of the generalzed gamma as a tool for treatment plannng optmzaton Emmanoul I Petrou 1,, Ganesh Narayanasamy 3, Eleftheros Lavdas 4, Sotros Stathaks 3, Nkos Papankolaou 3, Bengt K Lnd, Panayots Mavrods, 3 1 Laboratory of Medcal Physcs, Medcal School, Unversty of Patras, Patras, Greece vson of Medcal Radaton Physcs, Karolnska Insttutet and Stockholm Unversty, Stockholm, Sweden 3 vson of Medcal Physcs, epartment of Radaton Oncology, Unversty of Texas Health Scence Center, San Antono, TX, USA 4 epartment of Medcal Radologcal Technologsts, Technologcal Educaton Insttute of Athens, Greece Receved September 9, 014; Revsed November 19, 014; Accepted November 1, 014; Publshed Onlne ecember 01, 014 Orgnal Artcle Abstract Purpose: The am of that work s to study the theoretcal behavor and merts of the Generalzed Gamma (generalzed dose r e- sponse gradent) as well as to nvestgate the usefulness of ths concept n practcal radobologcal treatment plannng. Methods: In ths study, the treatment plannng system RayStaton 1.9 (Raysearch Laboratores AB, Stockholm, Sweden) was used. Fu r- thermore, radobologcal models that provde the tumor control probablty (TCP), normal tssue complcaton probablty (NTCP), complcaton-free tumor control probablty (P+) and the Generalzed Gamma were employed. The Generalzed Gammas of TCP and NTCP, respectvely were calculated for gven heterogeneous dose dstrbutons to dfferent organs n order to verfy the TCP and NTCP computatons of the treatment plannng system. In ths process, a treatment plan was created, where the target and the organs at rsk were ncluded n the same ROI n order to check the valdty of the system regardng the objectve functon P+ and the Generalzed Gamma. Subsequently, sx addtonal treatment plans were created wth the target organ and the organs at rsk placed n the same or dfferent ROIs. In these plans, the mean dose was ncreased n order to nvestgate the behavor of dose change on tssue response and on Generalzed Gamma before and after the change n dose. By theoretcally calculatng these quanttes, the agreement of dfferent theoretcal expressons compared to the values that the treatment plannng system provdes could be evaluated. Fnally, the relatve error between the real and approxmate response values usng the Posson and the Probt models, for the case of havng a target organ consstng of two compartments n a parallel archtecture and wth the same number of clonogens could be nvestgated and quantfed. Results: The computatons of the RayStaton regardng the values of the Generalzed Gamma and the objectve functon (P+) were verfed by usng an ndependent software. Furthermore, t was proved that after a small change n dose, the organ that s beng affected most s the organ wth the hghest Generalzed Gamma. Apart from that, the valdty of the theoretcal expressons that descrbe the change n response and the assocated Generalzed Gamma was verfed but only for the case of small change n dose. Especally for the case of 50% TCP and NTCP, the theoretcal values (ΔPapprox.) and those calculated by the RayStaton show close agreement, whch proves the hgh mportance of the 50 parameter n specfyng clncal response levels. Fnally, the presented fndngs show that the behavor of ΔPapprox. looks sensble because, for both of the models that were used (Posson and Probt), t sgnfcantly approaches the real Δ P around the regon of 37% and 50% response. The present study managed to evaluate the mathematcal expresson of Generalzed Gamma for the case of non-unform dose delvery and the accuracy of the RayStaton to calculate ts values for dfferent organs. Concluson: A very mportant fndng of ths work s the establshment of the usefulness and clncal relevance of Generalzed Gamma. That s because t gves the planner the opportunty to precsely determne whch organ wll be affected most after a small ncrease n dose and as a result an optmal treatment plan regardng tumor control and normal tssue complcatons can be found. Keywords: Prostate Cancer; Radobologcal Treatment Plannng; Generalzed Gamma; TCP; NTCP Introducton Recent technologcal developments have ntroduced dramatc changes n the feld of Radotherapy. 1, Radologcal magng has become more advanced provdng nformaton at functonal level. In ths way, a better assessment of the spread, cell densty and radosenstvty varaton of clonogenc tumor cells can be accomplshed. 3 For normal tssues, nformaton on the locaton and dstrbuton of radaton senstve functonal subunts can be assessed. 4, 5 Furthermore, the possblty of calculatng now the fractonal and through them the composte dose dstrbutons delvered to the pa- Correspondng author: Panayots Mavrods; epartment of Radaton Oncology, Unversty of Texas Health Scence Center, San Antono, TX, USA. Cte ths artcle as: Petrou EI, Narayanasamy G, Lavdas E, Stathaks S, Papankolaou N, Lnd BK, Mavrods P. Evaluaton of the generalzed gamma as a tool for treatment plannng optmzaton. Int J Cancer Ther Oncol 014; (4):00418. OI: 10.14319/jcto.004.18
Petrou et al.: Evaluaton of the generalzed gamma Internatonal Journal of Cancer Therapy and Oncology www.jcto.org tents n a 3-dmensonal mode gves a better vew of the effectveness of the appled treatment confguratons. 6, 7 Ths abundance of nformaton needs to be accurately used n order to acheve a close agreement between treatment plannng and clncal outcome. Modern treatment plannng algorthms try to maxmze the conformaton of the delvered dose dstrbuton to the target volume through three-dmensonal ntensty modulated treatment plannng, whch conforms the hgh dose regon to the shape of the target volume and takes nto account the locaton of the organs at rsk (OAR). 8-11 Commonly, the rradaton protocols apply dose prescrptons and constrants that have been assocated wth certan clncally accepted tumor control and normal tssue complcaton rates. In the current practce, the mean dose n the target volume and addtonal dose-volume ponts n the targets and organs at rsk are manly used n treatment plan optmzaton as objectve functons or constrants, respectvely. However, the clncal outcome of a radotherapy treatment n terms of tumor control and normal tssue complcatons s nearly always lnked to a degree of uncertanty. 1, 13 Ths s partly because two treatment fractons of the same beam confguraton are not exactly the same snce the nature of radaton beams are stochastc at a mcroscopc level. Furthermore, the nter-patent and cellular radosenstvty varatons are generally unknown. For these reasons, the expected outcome of a treatment s expressed as the probablty of havng a certan treatment effect. Radobologcal treatment plannng estmates these probabltes for each target and organ at rsk of a gven clncal case based on the appled dose-dstrbuton and radobologcal data. 14-16 Currently, the standard tools that are used for radotherapy treatment plan evaluaton (e.g. sodose dstrbuton, dose volume hstogram (VH), etc) are based on dose only and they do not take the radobologcal characterstcs of tumors or normal tssues nto account. To cover ths gap, the concepts of Tumor Control Probablty (TCP), Normal Tssue Complcaton Probablty (NTCP) and complcaton -free tumor control (P+) were ntally ntroduced to provde an addtonal plan evaluaton analyss. 8 More recently, the quanttes of the Normalzed ose-response gradent and the Generalzed ose-response gradent were proposed as a supplementary tool n the optmzaton of treatment plans nvolvng non-unform dose delveres, respectvely. 17 The prncple am of ths study s to nvestgate the theoretcal behavor and merts of the Generalzed Gamma (generalzed dose response gradent) concept. The secondary goal s to examne the usefulness and clncal relevance of the Generalzed Gamma n practcal radobologcal treatment plannng through ts mplementaton n the RayStaton treatment plannng system. Methods and Materals efnton of the Generalzed Gamma The normalzed dose-response gradent was ntroduced n the 80s (Brahme 1984) and t has extensvely been used for descrbng the dose-response relatons of both tumors and normal tssues. 18 The normalzed dose-response gradent γ() can be defned at any dose level accordng to the followng expresson: dp( ) ( ) --------------------------Eq. 1 d where, P() s the response of the examned tssue to a gven dose,. The most useful features of γ s that t can be used to predct the change n response from a small change n dose accordng to the followng formula: P( ) ----------------------------Eq. From a clncal and radobologcal pont of vew, t has always been mportant to know the value of the steepest gradent of the dose-response relatonshp. Ths value s denoted as ~ and s defned as follows: ~ ~ P( ~ ), where P( ) P( ~ ) max -----------Eq. 3 In 001, the concept of the normalzed dose-response gradent was generalzed to account for non-unform dose delvery, 17 by explctly extendng the defnton of γ as a functon of a 3-dmensonal dose dstrbuton ( r ) and by replacng the dervatve n Equaton (1) wth a gradent as shown below: ( ) P( ) ----------------------Eq. 4 Ths defnton makes t possble to calculate the normalzed dose-response gradents of dfferent tumors and healthy tssues recevng a gven dose dstrbuton and to relate them to varous clncal endponts. The magntude of the dfferent γ values gves the planner a hnt about the modfcaton of the dose dstrbuton needed n order to most effectvely decrease normal tssue complcatons and maxmze the probablty of complcaton free cure. Mathematcal formulae related to dose-response gradent One of the radobologcal models that have extensvely been used to descrbe the dose-response relaton of tumors and normal tssues s the lnear-quadratc model, whch s gven by the followng mathematcal expresson: e / 50 e ln ln P() exp e -----------Eq. 5 where 50, whch s the dose that produces a gven response to 50% of the patents and γ, whch s the maxmum normalzed gradent of the dose-response curve. 4, 15, 19 In order to
Volume Number 4 014 Internatonal Journal of Cancer Therapy and Oncology 3 www.jcto.org calculate the normal tssue complcaton probablty (NTCP) of an OAR to a gven heterogeneous dose dstrbuton, the relatve seralty model was used: 4, 15, 0, 1 1/ s M s V PI (,) V 1(1()) P ----------------Eq. 6 1 where, M s the total number of voxels or subvolumes n the tssue, ΔV s the fractonal rradated subvolume of an organ compared to the reference volume, Vref for whch the values of 50 and γ were calculated and s s the relatve seralty parameter that characterzes the nternal organzaton of the organ. P() s the probablty of response of the organ havng reference volume and beng rradated to dose as descrbed by Equaton (5). Regardng the calculaton of the TCP t s assumed that the tumor has a parallel structural organzaton snce the eradcaton of all the clonogenc cells s requred. Ths prerequste s satsfed by the followng relatonshp: 15 M V P(,)() V P ---------------Eq. 7 1 Fnally, regardng the calculaton of the Generalzed Gamma, the followng relatonshp was used: n n P()() P P ()() P ---- Eq. 8 P P 1 1 where, s the quas-unform dose n voxel and γ s the local contrbuton of voxel to the Generalzed Gamma and t s gven by the followng equaton: P () --------------------Eq. 9 Treatment plannng specfcaton and optmzaton by usng RayStaton The platform that was used for conductng the present study s RayStaton 1.9 (Raysearch Laboratores AB, Sweden). RayStaton s among the few treatment plannng systems that can produce treatment plans combnng dfferent radaton modaltes and optmze them usng radobologcal measures such as the Generalzed Gamma. In ths study, data from prostate cancer cases were used. So, the target of the examned treatment plans was the prostate regon and the optmzaton algorthm calculated the value of the Generalzed Gamma for ths target whle optmzng the plan. urng ths process, the number of beams, ther energy and portals as well as the overall geometry of the treatment confguraton were specfed. More specfcally, the MLC Step and Shoot IMRT radaton modalty was used n all the plans. Three 6MV beams were used (gantry angles: 7, 180 and 88 degrees) to acheve an acceptable treatment plan. The dose prescrpton and number of fractons n every plan were determned by the optmal P+ value, whereas the predetermned dose per fracton was.0 Gy. The prmary organs at rsk (OAR) nvolved n ths case are bladder and rectum. Also, for the dfferent goals of the study, dfferent regons of nterest (ROI) were defned. A number of measures were used as evaluators of the qualty of the treatment plan such as the mnmum dose to the target ( mn), maxmum doses to the OARs (max, whch corresponds to 1cc) as well as dose volume hstogram (VH) constrants. Furthermore, radob o- logcal measures such as the Tumor Control Probablty (TCP), the Normal Tssue Compatblty Probablty (NTCP) and the complcaton-free tumor control probablty ( P+) were employed. The TCP functon corresponds to the probablty of tumor control, whereas the NTCP functon corresponds to the probablty of havng normal tssue complcatons, due to radotherapy. Fnally, the objectve functon P+ corresponds to the probablty of achevng tumor control wthout havng any normal tssue complcatons and t was used as the prmary objectve functon n the optmzaton of the dfferent plans. After defnng all the above parameters and functons, seven treatment plans were produced n order to examne the mpact of dfferent factors on the values of the examned radobologcal measures. For ths purpose, the number and type of OARs nvolved, the radobologcal parameters of the nvolved tssues and the tssues ncluded n the ROI were vared. The values of the radobologcal measures (TCP, NTCP, P+ and Generalzed Gamma) were calculated both by the treatment plannng system as well as by an ndependent software. Ths software uses the VHs fles of the target and OARs (that are exported by RayStaton) as nput for the calculatons of the TCP, NTCP, P+ and Generalzed Gamma values. It has also an nternal lbrary wth values for the radobologcal parameters of the dfferent tumors and OARs for dfferent models. These values are exactly the same as those used by the RayStaton treatment plannng system. Treatment plannng specfcaton and optmzaton usng RayStaton In order to llustrate the characterstcs of the Generalzed Gamma concept and to dentfy the accuracy of the RayStaton treatment plannng system n calculatng the values of the dfferent related radobologcal measures, a number of comparsons were performed. Frstly, the values of Generalzed Gamma of the TCP and NTCP measures were calculated for the case of a heterogeneous dose dstrbuton delvered to dfferent tssues. These values were calculated both by the RayStaton treatment plannng system as well as by the ndependent software. Ths comparson was conducted n order to verfy the calculated values of the Generalzed Gamma, TCP and NTCP from the TPS. For the prostate cancer case that was used, the frst treatment plan that was produced ncluded all the nvolved tssues (namely target and OARs) n the same ROI, whch means that
4 Petrou et al.: Evaluaton of the generalzed gamma Internatonal Journal of Cancer Therapy and Oncology www.jcto.org all the organs were meant to receve the same radaton doses. Furthermore, addtonal plans were developed, where ndependent ROIs were used and the dfferent organs were rradated wth non-unform dose dstrbutons. In these cases, the target receved the hghest dose and the OARs receved a smaller percentage of the target dose. After the physcal optmzaton of the dfferent treatment plans, the radobologcal modalty of RayStaton was used to calculate the values of TCP, NTCP, P+ as well as the Generalzed Gamma values for the target and the OARs. Those values were subsequently assocated wth the theoretcal formulae that gve the Generalzed Gamma value and the change n response, whch follows a small change n the mean dose ( ). In ths study, the mean dose was ncreased by 5% n the frst treatment plan (all the organs ncluded n the same ROI) and 5% n the second plan (organs treated as dfferent ROIs). Equaton ( ) s vald only for unform dose dstrbutons and t gves an approxmate soluton to the change n response after a small change n dose. In order to obtan a more accurate quantfcaton of the change n response, one should preferably calculate the γ-value as a functon of dose or P(). Assumng Posson statstcs, the response probablty for unform dose s: P() exp() N e a ---------------------------Eq. 10 0 so that the normalzed dose response gradent becomes: dp() ()() ln() P a P ----------Eq. 11 d Snce Equaton (10) gves a ln N 0 ln( ln P( ) and ln N o /e, the normalzed dose-response gradent for non-unform doses s thus approxmately gven as: ()() ln()( P ln( Pln())) e P -----Eq. 1 The relatve change n response as a functon of the relatve change n dose can thus be approxmated as: ---------Eq. 13 P()() ln()( Pln( ln())) P e P or alternatvely, f the mean dose s known -------Eq. 14 P()() ln()( Pln(ln )) P e In many crcumstances, Equaton (1) can be further approxmated as: ()() ln() P P e -------------------Eq. 15 where, exact equalty prevals when P()=e -1. Hence, the approxmate relatve change n response as a functon of the relatve change n dose (Equaton (13)) becomes P()()*ln() P P e ------------------Eq. 16 50 After the accuracy valdaton of the Generalzed Gamma value calculatons wth RayStaton, an evaluaton of practcal merts of Generalzed Gamma was performed. For ths purpose, the functonalty of the concept was studed for the case of a target volume consstng of two compartments. Frstly, the target was segregated nto compartments recevng doses, (-ε) and (+ε), where ε s the dose varance, and the correspondng responses are denoted as P1 and P, respectvely. So, accordng to Equaton (16) and assumng that the responses are governed by Posson statstcs, the Generalzed Gamma wll be: v1 v P()()() P P ---------------------Eq. 17 1 1 a* *(1 ) N where, 1 = - ε and = + ε, 0 * e P e, a* *(1 ) 0e P e N, e 50Gy and N0 e. P P * P, a = (lnn0 lnln)/50, 50 = 1 Based on those formulas, generalzed gamma s gven by the followng expresson: (1)(1) *{(1)( / )(1)( / ) a a P a N e } a N e ---Eq.18 0 0 The gradent of the dose-response functon s also gven by: P (1)(1) ()()*{( P / )( a N a a / 0) e } a N e ----Eq. 19 0 In order to calculate the response of both compartments after a small ncrease n dose, (Δ or δ) to get the theoretcally estmated value for the change n response ΔP=P(+Δ)-P() the followng formulatons were used n Equaton (17): * *(1 )*(1 ) 0 * a N e N 0 P e, * e P e, 1 1 * 1 a* *(1 )*(1 ) P( ) P P ----------Eq. 0 In ths study, we used ΔPreal=P(+δ)-P() to represent the real value of the change n response. An approxmate value for ΔP was calculated usng the Equaton ( and 19). Havng calculated both ΔPreal and ΔPapprox., the quantty (ΔPapprox.- ΔP real)/δpreal was plotted as a functon of the prescrbed dose. Fnally, the above procedure was repeated assumng that the responses of the tssues are governed by normal dstrbuton statstcs. For ths nvestgaton, the normal cumulatve dstrbuton was employed for specfed mean value μ and standard devaton σ values. In ths study, the mean and standard devaton values were set equal to 50 = 50 Gy and σ = 50/γ(π) 1/. The cumulatve dstrbuton returns a sgmodal functon for TCP and NTCP. Results and scussons Takng nto account the above basc defntons and formulas, as well as the calculatons and the data from the RayStaton treatment plannng system, the theoretcal behavor and merts of the generalzed dose-response gradent were nvestgated. Addtonally, the nfluence of the organ archtecture (parallel-seral) and the usefulness of the generalzed
Volume Number 4 014 Internatonal Journal of Cancer Therapy and Oncology 5 www.jcto.org dose-response gradent n practcal radobologcal treatment plannng by usng the RayStaton platform were also studed. Verfcaton of RayStaton computatons Frst, a verfcaton of the calculatons of RayStaton regardng the values of the TCP and NTCP quanttes was performed usng an ndependent homemade software. At frst, both the target and the normal tssue were assumed to belong to the same ROI, whose volume was characterzed by the same radobologcal parameters (Table 1) and for ths reason they both receved the same unform dose. Based on the calculatons shown n the Appendx, the results for the TCP, NTCP and Generalzed Gammas are presented n Table. As t can be notced n Table, the values of TCP and NTCP are almost dentcal, whch means that the calculatons of RayStaton were verfed properly. The mnor dfferences that appear between TCP and NTCP as well as the respectve Generalzed Gamma values stem from the fact that RayStaton calculates the value of the objectve functon P+ usng the expresson P+=PB(1-PI) nstead of P+=PB-PI that was used by the ndependent software. The determnaton of the optmal soluton corresponds to the determnaton of the maxmum P+ value. At ths pont, the value of PB s not equal to that of PI but to the dervatve of P+. TABLE 1: Radobologcal and physcal parameters used n the calculaton of the TCP and NTCP measures. 50 (Gy) γ s N0 n (fractons) α/β (Gy) α (Gy -1 ) β (Gy - ) 50.0 6.0 0.7 1.1x10 5 30 3.0 0.0 0.0667 50: ose n whch the response probablty s 50% γ: maxmum normalzed value of dose-response gradent α, β: fractonaton parameters of LQ-Posson model s: relatve seralty parameter that characterze the nternal organzaton of the organ N0: ntal number of clonogenc cells for tumors TABLE : Generalzed Gamma values for TCP and NTCP. TCP (%) NTCP (%) Generalzed Gamma for TCP Generalzed Gamma for NTCP 19.07 19.53 6.75 6.79 TABLE 3: Summary of the results of Plan 1, whch was optmzed usng a sngle ROI and 50 value for the target and bladder. However, the rest of the radobologcal parameters of two organs were dfferent. Organs 50 (Gy) γ α/β (Gy) s TCP (%) NTCP (%) GenGamma Prostate 60.0 4. 10.0-53.8-4.6 Bladder 60.0 3.0 3.0 0.18-54.3 4.0 P+ = 4.6% Subsequently, further expermental treatment plans were produced n order to examne the behavor of the objectve functon P+, as well as the value of Generalzed Gamma and n ths way the effectveness of the system. Based on the fact that both organs belong to the same ROI, we expect a small dfference n the Generalzed Gamma values due to the dfferences n γ and α/β values of the target and bladder. Also, the Generalzed Gamma values are a lttle dfferent due to the dfferent response probabltes and parameter values. Accordng to our results (Table 3), P+ s not zero, whch ndcates that the expresson used n RayStaton for P+ calculaton s P+ = PB (1-PI). Theoretcal and expermental approach of Generalzed Gamma In Table 4, the results of another case n whch the organs at rsk belong to the same ROI as the target, are shown. In ths case, however, there are two OARs and dfferent radobologcal parameters characterze the target and the two OARs. Furthermore, dfferent physcal constrants were mposed durng the development of the two plans (e.g. the mnmum dose to the target was set to 60 Gy n Plan 3). In Table 4 t s notced that the NTCP and the generalzed gamma have the same value for both OARs, somethng that was expected because they receve exactly the same dose and they have the same radobologcal parameters. In Table 5, the RayStaton results for the case that the OARs belong to a ROI that s dfferent than that of the target are shown. Those results are derved from three treatment plans, whch were optmzed usng dfferent combnatons or organs nvolved (prostate, bladder and rectum), dfferent radob o- logcal parameters and dfferent physcal constrants (e.g. the mnmum dose to the target was set to 60 Gy n Plan 3). It can be seen that when the prescrpton dose constrant changed, the response of rectum was affected much more than those of the target and bladder and ths s also reflected on the Generalzed Gamma values.
6 Petrou et al.: Evaluaton of the generalzed gamma Internatonal Journal of Cancer Therapy and Oncology www.jcto.org TABLE 4: Summary of the results of two treatment plans, whch were optmzed usng a sngle ROI for the target and OARs but dfferent combnatons of radobologcal and physcal constrants (e.g. mnmum dose to the target). Organs Prostate Bladder Rectum 50 (Gy) 60.0 80.0 80.0 γ.0.0.0 s - 0.00001 0.00001 α/β (Gy) 10.0 10.0 10.0 PLAN TCP (%) 61.9 - - NTCP (%) - 11.3 11.3 GenGamma.14 1.34 1.34 P+ (%) 48.7 PLAN 3 TCP (%) 7.4 - - NTCP (%) - 0.0 0.0 GenGamma 1.80 1.89 1.89 P+ (%) 46.3 TABLE 5: Summary of the results of three treatment plans, whch were optmzed usng dfferent combnatons or organs nvolved (prostate, bladder and rectum) and dfferent physcal parameters (e.g. mnmum dose to the target). Organs Prostate Bladder Rectum 50 (Gy) 5.7 60.0 60.0 γ 4. 3.0. s - 0.18 1.0 α/β (Gy) 10.0 3.0 3.0 PLAN 1 TCP (%) 95.3 - - NTCP (%) - - 4.1 GenGamma 0.68-0.65 P+ (%) 91.4 PLAN TCP (%) 93.4 - - NTCP (%) - 1.3 4. GenGamma 0.94 0.31 0.66 P+ (%) 88.5 PLAN 3 TCP (%) 95.3 - - NTCP (%) - 7.3 3.7 GenGamma 0.77 1.7.49 P+ (%) 59.6 After specfyng the expermental treatment plans at RayStaton and extractng the respectve theoretcal Generalzed Gamma values, we tred to assocate the expermental results for change n response, ΔP, wth the theoretcal formulae gven n Equaton (, 14 and 16). In Tables 6-10, the results for theoretcal and expermental changes n response probablty (ΔPtheor. and ΔPexp.) for each case and correspondng treatment plan, are presented. For the scenaro of the organs wthn the same ROI (Plan of Table 4), a set of values for a 5% ncrease n mean dose was acqured. Based on the results shown n Table 6, the expermental value of ΔP s closer to that of Equaton (). As a result, the expermental value can be used n Equaton ( ) to calculate the generalzed gamma after the ncrease n dose. Table 6 shows that the values of the calculated generalzed gamma of the dfferent organs dffer, whch stems from the fact that the ΔP values are not dentcal. TABLE 6: Comparson of the ΔPexp. and ΔPtheor. values for a 5% ncrease n dose. The Generalzed Gamma values before and after the change n dose are also shown. These results stem from Plan of Table 4. Organs Prostate Bladder Rectum Pexp. (%) 9.75 7.80 7.80 Peq. (%) 10.69 6.7 6.7 Peq.14 (%) 9.10 5.65 6.65 Peq.16 (%) 8.74 6.68 6.68 GenGammabefore.14 1.34 1.34 GenGammaafter 1.83 1.84 1.84 The same procedure was repeated for the case that the target organ and the OARs belong to dfferent ROIs (Plan 1 of Table 5) and the results are presented n Tables 7. Accordng to those results t can be concluded that a 5% change n dose results n a change n response, whch s close to that derved by Equaton (16). Ths means that the γ() can be calculated usng the expresson ( ) P( )*ln P( )* e*. TABLE 7: Comparson of the ΔPexp. and ΔPtheor. values for a 5% ncrease n dose. The Generalzed Gamma values before and after the change n dose are also shown. These results stem from Plan 1 of Table 5. Organs Prostate Rectum Pexp. (%).0 4.19 Peq. (%) 3.38 3.5 Peq.14 (%) 3.55.86 Peq.16 (%).61 3.93 GenGammabefore 0.68 0.65 GenGammaafter 0.5 0.79 Calculatng the ΔPtheor. for the case that the target organ and the OARs belong to dfferent ROIs, and applyng physcs dose constrants (Plan of Table 5), t s notced that for 5% change n dose the change n response, ΔP s close to that gven by Equaton (16). The respectve results for ΔP and Generalzed Gamma are presented n Table 8. TABLE 8: Comparson of the ΔPexp. and ΔPtheor. values for a 5% ncrease n dose. The Generalzed Gamma values before and after the change n dose are also shown. These results stem from Plan of Table 5. Organs Prostate Bladder Rectum Pexp. (%) 3.36.41 4.3 Peq. (%) 4.68 1.54 3.8 Peq.14 (%) 4.7 1.41.84 Peq.16 (%) 3.56.31 3.94 GenGammabefore 0.94 0.31 0.66 GenGammaafter 0.7 0.46 0.79
Volume Number 4 014 Internatonal Journal of Cancer Therapy and Oncology 7 www.jcto.org Accordng to the results shown n Tables 4-8 the theoretcal value of response change, ΔP s n good agreement wth the value returned by RayStaton provdng a verfcaton of the valdty of Equaton (16). The last part of ths study concerns the assocaton of the values of dfferent quanttes between RayStaton and ther theoretcal calculaton regardng the behavor of the change n response, ΔP and the Generalzed Gamma for the 50 pont of the dose-response relaton. For ths purpose, two treatment plans were developed by optmzng them so that the correspondng TCP and a NTCP values are almost 50%. In these cases, the behavor of ΔP and that of Generalzed Gamma are shown n Tables 9 and 10, respectvely. Ths was done by performng the prevously descrbed process, whch nvolves the calculaton of ΔPtheor. and the comparson wth the respectve ΔPexp. value. As t can be notced from Tables 9 and 10, the ΔPexp. value s n qute good agreement wth the ΔPtheor. gven by Equaton () n both cases. Ths has resulted n the respectve Generalzed Gamma values to be almost the same as the ntal values before the ncrease. In Table 10, the Generalzed Gamma values before and after the change n dose for both plans are presented both as calculated by RayStaton as well as theoretcally. TABLE 9: Summary of the results of three treatment plans, whch were optmzed usng dfferent combnatons or organs nvolved (prostate, bladder and anteror rectum) and dfferent physcal parameters. Comparson of ΔPtheor. to ΔPexp. for 5% ncrease n dose. Upper panel: TCP 50%, Lower panel: NTCP 50%. Organs Prostate Bladder 50 (Gy) 60.0 18.4 γ.0.0 s - 0.00001 α/β (Gy) 10.0 10.0 TCP (%) 49.9 - NTCP (%) - 35.9 GenGamma 1.64.18 P+ (%) 31.4 50 (Gy) 60.0 15.5 γ.0.0 s - 0.00001 α/β (Gy) 10.0 10.0 TCP (%) 36.7 - NTCP (%) - 48.8 GenGamma 1.61.19 P+ (%) 18.9 After verfyng the expermental wth the theoretcal value of Generalzed Gamma for the scenaro of ncreasng the mean dose by 5%, t was notced that the expermental value of Generalzed Gamma after the dose ncrease remans almost the same as the ntal one. Ths s qute mportant for the verfcaton of the accuracy of the TPS and ts calculatons because by takng nto account the 50 and ts propertes, the behavor of the Generalzed Gamma was shown to be the expected one. TABLE 10: Comparson of ΔPtheor. to ΔPexp. for 5% ncrease n dose. Upper panel: TCP 50%, Lower panel: NTCP 50%. Organs Prostate Bladder PLAN 1 Pexper (%) 7.8 10.76 Peq. (%) 8.0 10.90 Peq.14 (%) 13.14 10.81 Peq.16 (%) 9.43 9.99 GenGammabefore (theor.) 1.64.18 GenGammaafter (theor.) 1.55.3 GenGammabefore (exp.) 1.64.18 GenGammaafter (exp.) 1.56.15 PLAN Pexper (%) 7.91 10.99 Peq. (%) 8.05 10.95 Peq.14 (%) 13.16 11.06 Peq.16 (%) 10.00 9.5 GenGammabefore (theor.) 1.61.19 GenGammaafter (theor.) 1.61.0 GenGammabefore (exper.) 1.61.19 GenGammaafter (exper.) 1.6.05 Relatve error of ΔPapprox. for target volume consstng of two compartments Before examnng the case of bfurcaton of target volume nto two compartments and the dependence of ΔPapprox. on the receved dose, the behavor of Generalzed Gamma was studed as a functon of ε. As shown n Fgure 1, we can notce that the generalzed gamma decreases as the value of ε ncreases. FIG. 1: The Generalzed Gamma as a functon of the dose change, ε. Fgure 1 could be qute comparable wth the fndngs of the study of Lnd et al. 17, where γ had been computed for the case of havng dfferent number of clonogens for the two compartments as shown n Fgure.
8 Petrou et al.: Evaluaton of the generalzed gamma Internatonal Journal of Cancer Therapy and Oncology www.jcto.org value and standard devaton. In ths analyss, the mean value μ = 50Gy and the standard devaton s σ=50/(γ*(π) 1/ ). The expectaton from ths calculaton s the sgmodal curve that represents the response functon. For the computaton of ΔPapprox, the calculaton of generalzed gamma was not needed, due to the fact that the gradent of P s a cumulatve densty functon (CF) and tends towards a probablty densty functon (PF). Fgure 4 shows the behavor of the relatve devaton ΔPapprox as a functon of dose, usng the Probt (error) functon. FIG. : γ() for the two compartments of the target recevng doses 1 and and havng N01 and N0 number of clonogens. Furthermore, the relatve error ΔPapprox as a functon of dose has been studed and plotted ( Fgure 3) for the case of the target volume consstng of two compartments wth N1 and N (=N0) number of clonogens recevng doses 1 and, respectvely. More specfcally, the Posson model was used to calculate the response P() for the two compartments wth doses ( +ε) and ( -ε) and the response P(+Δ) for ncreasng the dose at by Δ. In addton, the values of ΔPreal=P(+Δ)-P() and ΔPapprox.=Δ/*gamma were calculated and ther relatve devaton as a functon of the prescrbed dose was plotted. The result s presented n Fgure 3, n whch ε = 0 and δ = 1%. FIG. 4: ΔPapprox as a result of the probt functon. Even wth the Gaussan-shaped curve, t can be notced that the maxmum value of the curve occurs at a dose slghtly hgher than 50Gy. espte that fact, we notce from the followng plot ( Fgure 5) that the best approxmaton between ΔPreal and ΔPapprox. appears at 50, whch s 4.34*10-4. From Fgure 5 t can be easly concluded that the probt functon gves a smoother curve than the Posson model and provdes a much better agreement between the ΔPreal and ΔPapprox. quanttes. FIG. 3: Relatve error of ΔPapprox as functon of dose, for ε=0 and δ= 1% In Fgure 3, t can be notced that the pont at whch the best agreement between ΔPreal and ΔPapprox. occurs s the 37 pont. Ths pont corresponds to a 37% response and the relatve dfference s 9.76*10-4. That makes sense consderng the behavor of the Posson model at 37. Another model was also used for calculatng the response, P n order to plot the relatve error of ΔPapprox, expectng more precse results and smoother curve. In ths case, the response P s represented by the normal cumulatve dstrbuton functon, whch s computed n terms of a specal functon called Probt functon (error functon), for specfc mean FIG. 5: The relatve devaton of ΔP usng probt functon, for ε = 0 and δ = 1%. The results of ths study are strongly dependent on the accuracy of the radobologcal models and the parameters descrbng the dose-response relaton of the dfferent tumors and normal tssues. Ths accuracy depends on many factors such as the mathematcal formalsm of each model, the assumptons on whch t s based, the bologcal mechansms
Volume Number 4 014 Internatonal Journal of Cancer Therapy and Oncology 9 www.jcto.org that t accounts for etc. In ths way, certan models descrbe better the data from certan clncal condtons (e.g. cancer type, treatment protocol) than other models and the opposte happens n other cases. Furthermore, the determnaton of the model parameters expressng the effectve radosenstvty of the tssues s subject to uncertantes mposed by the naccuraces n the patent setup durng radotherapy, lack of knowledge of the nter-patent and ntra-patent radosenstvty and nconsstences n treatment methodology. Consequently, the determned model parameters (such as the 50, γ and s) and the correspondng dose-response curves are characterzed by confdence ntervals. In the present study, most of the tssue response parameters have been taken from recently publshed clncal studes, where these parameter confdence ntervals have been reduced sgnfcantly. In ths study, due to the fact that many parameters affect the calculaton of the TCP/NTCP values (such as the clncal endponts of the dfferent tssues, the treatment protocol etc.) t would not be possble to provde any global error bars to those values. Ths could be possble only f the clncal problem examned was reduced to a very specfc clncal stuaton where a TCP uncertanty of about 5% s usually expected. However, due to the comparatve nature of the study, ths uncertanty n the absolute knowledge of the expected responses does not affect the general conclusons of ths nvestgaton. Concluson The present analyss ndcates that the concept of Generalzed Gamma s very mportant for the optmzaton of a treatment plan because t s a factor that can predct the organ whch wll most lkely be affected after a change n dose. That s because the hghest change n response s observed for the organ wth the hghest Generalzed Gamma. Irrespectve of whch organ has the hghest response, n a gven clncal case the hghest change n response after a small change n dose cannot be predcted wth certanty because usually the organ wth the hghest TCP or NTCP tend to have the hghest change n response after a small ncrease n dose. In the scenaro of a small change n dose, the organ wth the hghest Generalzed Gamma factor s most lkely the organ that s affected the most. Another sgnfcant fndng to be hghlghted s the agreement between the theoretcally calculated values of TCP, NTCP, P+ and Generalzed Gamma wth those from the RayStaton treatment plannng system. Conflct of nterest The authors declare that they have no conflcts of nterest. The authors alone are responsble for the content and wrtng of the paper. Appendx Generalzed Gamma calculatons for TCP and NTCP Usng Equaton (5), the followng formula for voxel response can be derved: * P exp( N0 *exp( a *)) ----------------Eq. A1 n where, -------------Eq. A e ln(ln ) a e ln(ln ) d a 50 1 / 50 d d =/n s the dose per fracton and n s the number of fractons. α and β are the fractonaton parameters of the lnear-quadratc (LQ) model and account for the early and late effects expected. -4 N0 s the ntal number of the clonogenc cells for tumors or the ntal number of functonal subunts for healthy tssues. Parameters α and β are specfc for every organ and specfc for the knd of njury (endpont) consdered and can be calculated only from clncal data. Based on Equaton (A1, A and 7), the TCP, NTCP and correspondng generalzed gamma values for the fve bn VH shown n Table A1 are calculated by the followng equatons: TCP P 5 V ---------------------------------Eq. A3 1 N a a -----Eq. A4 V P * * * * 0 *exp( *)*( *)* n n 5 GenGammaTCP = P * *TCP 1 5 V 1/ 1 P * * * 0 *exp( *)*( *)* n n NTCP(1(1)) P -----------------Eq. A5 ----------------Eq. A6 P N a a P ----Eq. A7 GenGammaNTCP= (1) s 5 * NTCP s V s V 1( 1) s P ---Eq. A8 *(1) * P*(1) V* * P * s* P P s 1 TABLE A1. Radobologcal parameters and calculatons of the generalzed gamma for a tumor and a healthy tssue, respectvely. 50 (Gy) γ s α/β (Gy) N0 n (fractons) α (Gy -1 ) β (Gy - ) 50 6.0 0.7 3.0 1.1*10 5 30 0.0 0.0667 ose V P P V θp/θ (1-P s ) V θp/θ θp/θp (Vol) * * 50. 0.50 0.144 0.6163 10.86 0.981 0.118 1.4918 50.4 0.15 0.1556 0.795 4.7540 0.9611 0.1174 0.7411 50.6 0.50 0.1951 0.6646 8.7868 0.9085 0.1355 1.7149 50.8 0.15 0.30 0.889 4.0593 0.9476 0.145 0.9114 51.0 0.50 0.51 0.7085 7.4999 0.8869 0.1483 1.979 TCP (%) Generalzed Gamma for TCP NTCP (%) Generalzed Gamma for NTCP 19.07 6.75 19.53 6.79
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