A theoretical approach to the problem of dose-volume constraint estimation and their impact on the dose-volume histogram selection

A theoretical approach to the problem of dose-volume constraint estimation and their impact on the dose-volume histogram selection
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  See discussions, stats, and author profiles for this publication at: A theoretical approach to the problem of dose-volume constraint estimation and their impacton the dose-volume...  Article   in  Medical Physics · October 2006 DOI: 10.1118/1.2237453 · Source: PubMed CITATIONS 6 READS 57 4 authors , including:Pavel Stavrev 94   PUBLICATIONS   533   CITATIONS   SEE PROFILE Nadejda Atila Stavreva 67   PUBLICATIONS   413   CITATIONS   SEE PROFILE All content following this page was uploaded by Pavel Stavrev on 09 January 2017. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.  A theoretical approach to the problem of dose-volume constraintestimation and their impact on the dose-volume histogram selection Colleen Schinkel  Department of Physics, University of Alberta, and Department of Medical Physics,Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta, T6G1Z2, Canada Pavel Stavrev and Nadia Stavreva  Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta, T6G1Z2, Canada B. Gino Fallone a   Department of Physics and Oncology, University of Alberta, and Department of Medical Physics,Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta, T6G1Z2, Canada  Received 23 February 2006; revised 22 June 2006; accepted for publication 29 June 2006;published 30 August 2006  This paper outlines a theoretical approach to the problem of estimating and choosing dose-volumeconstraints. Following this approach, a method of choosing dose-volume constraints based onbiological criteria is proposed. This method is called “reverse normal tissue complication probabil-ity   NTCP   mapping into dose-volume space” and may be used as a general guidance to theproblem of dose-volume constraint estimation. Dose-volume histograms   DVHs   are randomlysimulated, 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 samelevel of NTCP. The points from the averaged DVH are proposed to serve as physical dose-volumeconstraints. The population-based critical volume and Lyman NTCP models with parameter setstaken from literature sources were used for the NTCP estimation. The impact of the prescribedvalue of the maximum dose to the organ,  D max  , on the averaged DVH and the dose-volume con-straint points is investigated. Constraint points for 16 organs are calculated. The impact of thenumber of constraints to be fulfilled based on the likelihood that a DVH satisfying them will resultin an acceptable NTCP is also investigated. It is theoretically proven that the radiation treatmentoptimization based on physical objective functions can sufficiently well restrict the dose to theorgans at risk, resulting in sufficiently low NTCP values through the employment of several appro-priate dose-volume constraints. At the same time, the pure physical approach to optimization isself-restrictive due to the preassignment of acceptable NTCP levels thus excluding possible bettersolutions to the problem. ©  2006 American Association of Physicists in Medicine .  DOI: 10.1118/1.2237453  Key words: dose-volume constraints, NTCP, DVH, inverse planning, physical optimization, bio-logical optimization I. INTRODUCTION The most advanced treatment planning systems to date makeuse of inverse planning software in order to produce plansthat will deliver a high dose to the target while minimizingdose, and thus normal tissue complication probability  NTCP  , to the surrounding structures. This is accomplishedthrough the specification of physical dose-volume objectivesand constraints, and there are often multiple constraints se-lected for a given organ at risk. These constraints are oftenselected based on clinical experience. However, in many in-stitutions, they are chosen based on the dose-response valuespublished by Emami  et al. 1 This work is the first and remainsthe largest compilation of dose-response data to date. It con-tains estimates of doses that lead to 5 and 50% complicationprobability for partial volume irradiation of a variety of or-gans. Tolerance doses are given for relative irradiated vol-umes of   1 3 ,  2 3 , and 1. It is assumed that, in each case, ahomogeneous dose is delivered to the given relative volumewhile the rest of the organ receives no dose. Thus, any dose-response data from Emami  et al.  are equivalent to single-stepdose-volume histograms   DVHs  . During the majority of treatments, the organs at risk are irradiated heterogeneouslyas opposed to homogeneously. Therefore, using any of theEmami 5% complication rate dose-volume points, or combi-nations of them, as constraints would likely fail to produce atreatment plan that would yield the desired NTCP of 5% orless.To avoid the difficulties that could result from using rawclinical dose-response data   such as the Emami  et al.  esti-mates   directly as constraints, one might consider using bio-logical, rather than physical, inverse planningoptimization. 2–6 That is, specify a constraint NTCP value foreach organ at risk instead of a physical dose-volume point.Then the dose to the organ would be limited based on NTCPmodels and parameters reflecting clinical dose-volume char-acteristics of different tissues. Inverse planning can, in prin-ciple, use NTCP constraints directly. For example, an inten-sity modulated radiation therapy   IMRT   plan can vary the 3444 3444Med. Phys. 33  „ 9 … , September 2006 0094-2405/2006/33 „ 9 …  /3444/16/$23.00 © 2006 Am. Assoc. Phys. Med.  beamlet weights to satisfy both the physical and radiobio-logical constraints simultaneously. 3–6 Although biologicaloptimization is not a new concept, 7,8 it is not currently avail-able as an option for inverse planning on commercially avail-able treatment planning systems. The main reason why bio-logical constraints are not routinely used for inverse planningis the lack of a sufficient amount of clinical dose-responsedata on which to base NTCP model parameter estimates. 9–11 Misinterpretation of model formalism and assumptions alsocontribute to this problem. Due to the incompleteness of theclinical dose-response data available currently, biological op-timization for inverse planning is generally discouraged. 9,11 This is a puzzling fact, considering that almost three decadeshave passed since the introduction of the concept. The work of Emami  et al.  unfortunately did not provoke an appropriatedata gathering “rush,” which would have lead to the creationof sufficiently large data sets. Different researchers havestarted analyzing small data sets of real clinical data, andalternative sets of parameter estimates for different NTCPmodels have been reported. 12–27 Some of the reports, though,use data obtained under different conditions   tumorradiosensitizing, 14,24,27 surgical or nonsurgical intrusion, 28 dose-volume versus dose-wall histograms, 29–31 etc.  . There-fore, care should be taken that the application of these pa-rameter estimates be consistent with the conditions underwhich they were derived. 32 The purpose of applying physical dose-volume constraintsis to produce a plan that results in a low complication prob-ability, and the problem remains that the Emami dose-volume points are sometimes used as constraint points. Inthis paper, we present a method that enables the calculationof physical dose-volume constraints that are based on NTCPmodels for the purpose of inverse planning optimization.Specifically, we apply a Monte Carlo method of reverseNTCP mapping 33 to calculate dose-volume constraints for 16organs for which parameter value information isavailable. 34,35 The method makes use of the random DVHgenerator introduced in our companion work. 36 The NTCPfor each randomly generated DVH is estimated by applica-tion of the Lyman 37,38 and the critical volume NTCPmodels. 39–43 The investigation of the impact of these twowell-known NTCP models on the dose-volume constraintsestimation is the second purpose of this study. Dose-volumeconstraint points are calculated by interpolating from the av-erage of all DVHs with NTCP=5±0.5%. It is shown thatthese points have the potential to increase the probability thatthe inversely planned treatment will lead to an acceptablylow NTCP for the organs at risk. II. BACKGROUND We give some definitions and a short discussion of themodels and parameters necessary for the understanding of our present study. A. Some definitions  Integral dose-volume histogram : Defines the volume  V  int  ,which is irradiated to at least a dose  D : V  int   D   =  StructureOfInterest     D  r        −  D  d  3 r      where    is the Heaviside step function and  D  r        is the dosedistribution in the structure of interest.This definition of the dose-volume histogram was initiallyused implicitly by Hristov  et al. 44 From the definition of anintegral DVH it is clear that any monotonically decreasingfunction in the region   0,1    0,1   could represent a nor-malized integral DVH.  Iso-NTCP envelope : The curve  v   D   defined by the rela-tionship NTCP   D , v  =   %, where  D  is the dose of partialhomogeneous irradiation of the relative volume  v  will becalled an    % iso-NTCP envelope.The    % iso-NTCP envelope has a very interesting prop-erty: If a DVH is tangential to or crosses the envelope, sothat a part of the DVH curve happens to be above it, theNTCP in which this DVH results is higher than    %.   dose-volume constraint vicinity : Consider an integraldose-volume histogram,  DVH  k  , with dose-volume points   D , v   and a maximum dose of   D max, k   at  v =0. If, for a par-ticular dose-volume constraint point   v i ,  D i   i =1,..., n  , thefollowing condition is met:min    v  −  v i  2 +    D  −  D i  D max, k   2  ,then this DVH belongs to the   -vicinity of the given con-straint and is said to satisfy the   - criterion . This definitionselects functions that are crossing a circle of radius   arounda constraint point. B. NTCP models 1. The Lyman   „  Sigmoidal dose response   …   NTCP model  The Sigmoidal dose response   SDR   model, first intro-duced by Lyman, 37 describes the dose-response of normaltissues as follows:  NTCP =    EUD −  D 50 mD 50  ,   1  where    is the probit function    x    =1   2    −   x  exp  −  t  2 2   dt   =12  1 +  erf    x    2   .   2  The equivalent uniform dose   EUD  45 is defined as theuniform organ dose that would produce the same effect asthe given heterogeneous dose distribution, as specified by adifferential dose-volume histogram   dDVH   defined by thepoints    D  j , v  j  . The EUD or generalized mean dose   GMD  ,which in this case is chosen to represent the EUD, is calcu-lated from the dDVH as follows: 46–48  EUD =  GMD =    j v  j  D  j 1/  n  n .   3  There are three parameters that determine the response of normal tissues to radiation according to the Lyman model:  m , 3445 Schinkel  et al. : Theoretical approach to dose-volume constraint estimation 3445Medical Physics, Vol. 33, No. 9, September 2006  n , and  D 50 . The dose-volume dependence of a tissue is de-termined by the parameter  n ,  m  gives the slope of the dose-response curve, and  D 50  is the dose that gives a 50% com-plication rate and thus determines the position of theresponse curve. 2. Critical volume population model  The critical volume   CV   model 40,41,43 is based on theidea that organs are composed of functional subunits   FSUs  and that a complication occurs when a certain number of these FSUs are destroyed. The response of different tissues isdetermined by the application of binomial statistics. Here weuse the population CV NTCP model, 49,35,42 which takes intoaccount interpatient variability in normal tissue response anddescribes dose-volume response averaged over a populationof individuals:  NTCP  pop     − ln  − ln   ¯  d    + ln  − ln   cr   −     cr   /    cr   ln   cr   ,   4  where   ¯  d   =    j v  j     2     50 FSU  ln   D  j  /   D 50 FSU   .   5  For the CV population model, it is assumed that the interpa-tient variability is limited to the parameter    cr    the meancritical relative volume  . The parameters for this model in-clude the mean critical volume    cr  , the population variationin this parameter      cr  , the position of the FSU dose response  D 50 FSU  , and the slope of the FSU dose response    50 FSU  . C. Model parameters For the calculations in this work, we use the CV popula-tion model parameters from Stavrev  et al. 35 These authorsestimated parameters that are based on the dose-response es-timates of Emami  et al. 1 for each of 16 types of normaltissue. For the SDR model proposed by Lyman, we use pa-rameters derived by Burman  et al. 34 that are also based onthe Emami  et al. 1 data. Burman  et al. 34 provided SDR pa-rameter estimates for 27 organs in total. For this work, ourdatabase consists of 16 organs for which both SDR and CVpopulation NTCP model parameters exist. III. METHOD A. Reverse mapping of NTCP onto DVH space—Atheoretical approach for dose-volumeconstraint estimation In general, the reverse mapping method is defined as fol-lows:  i   Generate monotonically decreasing dose-volume his-togram functions.  ii   Calculate the NTCPs corresponding to these dose-volume histogram functions.  iii   Identify DVH functions resulting in a user-specifiedNTCP interval. Plot all these DVHs.  iv   From   iii  , calculate the probability   frequency   of aDVH, with a user-specified NTCP interval, to passthrough a given point in the dose-volume histogramspace.  v   From   iv  , calculate the averaged and the most prob-able DVHs. These two curves may each serve as asource of dose-volume constraint points, for the pro-cess of inverse treatment planning by physical objec-tive functions. 1. Generation of random DVHs  The first step of the reverse mapping process involves thegeneration of   N   random integral DVH curves that decreasemonotonically from a relative volume of 1 and a relativedose of 0 to   0,1  . The proper theory of DVH generation ispresented in our companion work. 36 In this paper, it wastheoretically determined that the distribution of the numberof monotonically decreasing functions passing through apoint in the dose-volume histogram space follows the hyper-geometric distribution. The generator that we use in thissimulation is based on the random walk theory and simulatesin a random fashion trajectories corresponding to monotoni-cally decreasing functions   finite series   situated in the unitsquare   0,1    1,0   subject to the hypergeometric distribu-tion. 2. Scaling the random DVHs  To calculate NTCP, the relative dose values of the integralDVHs must first be scaled to absolute doses. That is, wemust multiply the relative dose points of each randomly gen-erated integral dose-volume histogram by a maximum dosevalue appropriate to each organ of interest. The maximumdose of the  k  th randomly generated DVH,  DVH  k    where  k  =1,...,  N    is designated as  D max, k   and is calculated using theexpression  D max, k   =  D 5  +  n k    D 99 −  D 5  ,   6  where  n k   is a uniform randomly generated number   between0 and 1  . We have chosen the uniform distribution for  n k  because there is no reason to believe that the possible maxi-mum doses should have any other distribution. The dose val-ues  D 5  and  D 99  are those that give a NTCP of 5 and 99%,respectively, assuming uniform whole-organ irradiation andare thus different for each of the 16 organs. Two sets of   D 5 and  D 99  values were calculated: one based on the CV popu-lation model and one based on the Lyman model. Table Ishows a list of all 16 organs along with the calculated  D 5  and  D 99  values. Examples of typical clinical  D 5  and  D 99 , alongwith the treatments associated with those values, are shownfor comparison. It should be emphasized that the clinicalvalues for the minimal and maximal doses do not necessarilycorrespond to NTCPs of 5 and 99%. Instead, the clinicalvalues for these parameters indicate the observed range of maximum organ dose during actual treatments. To avoid con-fusion, we will hereafter refer to these clinical values as  D low and  D high . Note that the upper and lower limits of clinicalorgan dose   Table I   are sometimes extreme in comparison to 3446 Schinkel  et al. : Theoretical approach to dose-volume constraint estimation 3446Medical Physics, Vol. 33, No. 9, September 2006  the calculated limits   for example, an organ may have a typi-cal clinical  D low  of 0  . Clinical maximum organ dose for agiven treatment depends on factors such as where the plan-ning target volume   PTV   is located in relation to the normaltissue of interest and the maximum dose prescribed to thePTV. Organs that have a clinical  D low  value of zero   Table I  reflect the fact that for the type of treatment listed, they maynot be within the radiation field at all   the esophagus, forexample  . On the other hand, there are some normal tissuesthat have a good chance of receiving a significant dose dur-ing treatment of a tumor in its vicinity   the lung, for ex-ample, will always receive a relatively high maximum doseduring lung tumor treatments  .For the purpose of customization and to avoid biasingbetween the results for different organs, we have chosen touse the calculated  D 5  and  D 99  to define the range of   D max  instead of the clinical values. The minimum value,  D 5 , waschosen arbitrarily to eliminate the generation of DVHs withunrealistically low NTCP. Also, in half of the organs   TableI  , the clinical  D low  values are relatively close to  D 5 . Ideally,we would like to generate a decent number of low-NTCPDVHs to choose the constraints from, because our ultimategoal is to generate constraint points that will have a goodchance of producing a clinically acceptable NTCP. We real-ize that, for some sites, the range of maximum delivereddose may be significantly different than  D 5 –  D 99 . This case isalso investigated in our work, namely the impact of   D max  range on the dose-volume constraint estimation.In addition, part of this work involves grouping the ran-domly generated DVHs into intervals according to the result-ing NTCP values from each of the two different models, andthen calculating the averaged DVH for each NTCP interval  intervals are 0–10,  ¼ , 90–100%  . In order to explore theDVH space properly and produce the averages correspond-ing to different NTCP ranges, we require a sufficient numberof those DVHs to yield the corresponding NTCPs. While theclinical  D low  and  D high  are encountered more readily duringradiation treatments than the calculated values of   D 5  and  D 99 ,they may result in a bias of the distribution of all possibleNTCPs for a critical structure. That is, some NTCP rangesmay contain only a small sample of DVHs. To avoid thispotential problem and to ensure that there will be a largeenough number of DVHs in each NTCP interval, we chose touse the calculated values of   D 5  and  D 99  to define the range of   D max  . 3. Probability that a DVH, with a user-specified NTCP, passes through a given point in the dose- volume histogram space  Following the method outlined in Secs. III A   i   andIII A   ii  ,  N   integral DVH curves have now been generatedfor a given organ at risk. We proceed by evaluating theNTCP of each integral DVH. To do this, a differential DVHis calculated from each of the  N   integral ones and then used,along with the SDR and CV population models with appro-priate parameter values, to evaluate the NTCPs.The ratio of the number of DVHs resulting in a user-specified NTCP range   a , b   that pass through a given pointin the dose-volume space    D , v   to the number of all gener-ated DVHs passing through this point: T ABLE  I. Estimates for the clinical maximum dose range to 16 critical structures    D low ,  D high   that typicallyoccur during the listed treatments   values based on treatments given at the Cross Cancer Institute  . Also shownare values for the maximum dose range    D 5  and  D 99   calculated according to both the Lyman and CV popula-tion models. The parameters  D 5  and  D 99  are used to scale randomly generated DVHs appropriately to calculateconstraint points using the reverse mapping method. The following abbreviations are used: CNS—centralnervous system; PTV—planning target volume; H&N—head and neck.Organ Treatment TypeClinical Lyman CV Pop.  D low  Gy   D high  Gy   D 5  Gy   D 99  Gy   D 5  Gy   D 99  Gy  Lung Radical lung treatment   60 Gy to PTV   60 65 17.3 34.8 18.7 46.5Liver Abdomen, e.g., stomach cancer   45 Gy to PTV   30 50 30.1 54.0 30.2 52.0Brain CNS, e.g., Glioblastoma   60 Gy to PTV   60 65 45.2 80.9 45.8 85.1Heart Radical lung treatment   60 Gy to PTV   0 60 40.1 59.2 40.6 61.0Kidney — — — 23.4 34.5 22.8 40.0Esophagus Radical lung treatment   60 Gy to PTV   0 65 55.7 85.4 56.2 82.4Stomach Abdomen, e.g., stomach cancer   45 Gy to PTV   40 55 50.0 86.2 53.0 80.7Brachial plexus Radical lung treatment   60 Gy to PTV   0 50 60.2 95.9 60.6 87.4Bladder Prostate   74 Gy to PTV   70 76 65.5 100.5 65.8 98.3Mandible H&N   70 Gy to PTV   50 75 60.2 88.7 60.8 83.5Brain stem CNS, e.g., Glioblastoma   60 Gy to PTV   0 55 50.0 86.2 49.6 87.1Larynx H&N   70 Gy to PTV   55 75 70.1 94.0 70.6 91.1Small intestine Abdomen, e.g., stomach cancer   45 Gy to PTV   40 55 40.5 75.5 41.2 71.7Colon Abdomen, e.g., stomach cancer   45 Gy to PTV   40 55 45.1 69.1 45.6 66.6Spinal cord H&N   70 Gy to PTV   40 50 47.4 93.6 46.7 86.1Skin Breast   50 Gy to PTV   30 60 56.2 89.5 54.2 88.9 3447 Schinkel  et al. : Theoretical approach to dose-volume constraint estimation 3447Medical Physics, Vol. 33, No. 9, September 2006
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