An off-line strategy for constructing a patient-specific planning target volume in adaptive...

PII S0360-3016(00)00608-8

PHYSICS CONTRIBUTION

AN OFF-LINE STRATEGY FOR CONSTRUCTING A PATIENT-SPECIFICPLANNING TARGET VOLUME IN ADAPTIVE TREATMENT PROCESS FOR

PROSTATE CANCER

DI YAN, D.SC., DAVID LOCKMAN, D.SC., DONALD BRABBINS, M.D., LAURA TYBURSKI, B.S.,AND

ALVARO MARTINEZ, M.D., F.A.C.R.

Radiation Oncology, William Beaumont Hospital, Royal Oak, MI

Purpose: To improve the efficacy of dose delivery and dose escalation for external beam radiotherapy of prostatecancer, an off-line strategy for constructing a patient-specific planning target volume is developed in the adaptiveradiotherapy process using image feedback of target location and patient setup position.Materials and Methods: We hypothesize that a patient-specific confidence-limited planning target volume(cl-PTV), constructed using an initial sequence of daily measurements of internal target motion and patient setuperror, exists and ensures that the clinical target volume (CTV) in the prostate cancer patient receives theprescribed dose within a predefined dose tolerance. A patient-specific bounding volume to correct for targetlocation and compensate for target random motion was first constructed using the convex hull of the firstk daysof CT measurements. The bounding volume and the initial days of CT measurements were minimized based ona predefined dosimetric criterion. The hypothesis was tested using multiple daily CT images by mimicking theactual treatment of both conventional 4-field-box and intensity-modulated radiotherapy (IMRT) on each of30 patients with prostate cancer. For each patient, a patient-specific setup margin was also applied to thebounding volume to form the final cl-PTV. This margin was determined using the random setup errorpredicted from the initial days of portal imaging measurements and the residuals after correcting for thesystematic setup error.Results: The bounding volume constructed using daily CT measurements in the first week of treatment areadequate for the conventional beam delivery to achieve maximum dose reduction in the CTV of 2% or less of theprescription dose, for at least 80% of patients (p 5 0.08), and 4.5% or less for 95% of patients (p 5 0.1). However,for IMRT delivery, 2 weeks of daily CT measurements are required to achieve a similar level of the dosimetriccriterion, otherwise the maximum dose reduction of 7%, on average, in the CTV is expected. Furthermore, thepatient-specific setup margin required for the IMRT treatment is at least twice larger than that for theconventional treatment, to maintain the same dosimetric criterion. As compared to the conventional PTV, thevolume of cl-PTV is significantly reduced, while maintaining the same dosimetric criterion.Conclusion: The cl-PTV for prostate treatment can be constructed within the first week of treatment using thefeedback of imaging measurements. The cl-PTV has the capability to exclude the systematic variation andcompensate for the patient-specific random variation on target location and patient setup position. This impliesthat in the current off-line image feedback adaptive treatment process, a single plan modification can beperformed within the second week of treatment to improve the efficacy of dose delivery and dose escalation forexternal beam therapy of prostate cancer. © 2000 Elsevier Science Inc.

Prostate cancer treatment, Off-line image feedback, Patient-specific planning target volume, Conventional andIMRT dose distributions, Adaptive radiation therapy process.

INTRODUCTION

Expectations of improvement on local tumor control havestimulated a rapid development of advanced technology inradiotherapy planning and dose delivery. To implementthese technologies in the radiotherapy clinic effectively, theconventional treatment needs to be renovated to include also

an optimal and adaptive control process (1). One goal forapplying the feedback control strategy to current treatmentprocess is to improve the efficacy of the dose delivery byincorporating patient image feedback obtained during thecourse of his radiation treatment.

Many investigators have expressed concerns for geomet-ric miss of target volume during the daily radiation dose

Reprint request to: Dr. Di Yan, Department of Radiation On-cology, William Beaumont Hospital, 3601 West Thirteen MileRoad, Royal Oak, MI 48073-6769. E-mail: [email protected]—We would like to acknowledge Dr. DavidJaffray, Dr. Michael Sharpe, and Dr. John Wong for valuablediscussion and support. Special thanks to Kamal Kota and Peter

Girimonte for their hard work on patient image management andtreatment planning. We also thank our physician staff and thera-pists who made this study possible. This work is supported in partby NCI Grant-CA71785.

Accepted for publication 5 October 1999.

Int. J. Radiation Oncology Biol. Phys., Vol. 48, No. 1, pp. 289–302, 2000Copyright © 2000 Elsevier Science Inc.Printed in the USA. All rights reserved

0360-3016/00/$–see front matter

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delivery. Results from numerous imaging studies show con-clusive evidence that large variation on patient internalorgan location (2–10) and daily setup position (11–18)occurs during external beam treatment. Although patientdata have been analyzed differently, some generalizationscan be made. The variation is significant, and the conven-tional margin in the planning target volume (PTV) (19) isnot always adequate to ensure that the prescribed dose isactually absorbed in the clinical target volume (CTV). As aconsequence, highly conformal therapy may increase therisk of marginal tumor recurrence.

Indeed, geometric variation in patient daily treatmentstrikes at the very foundation of conformal radiation ther-apy, where small PTV margins and sharp dose gradient atthe edge of target volume are expected to escalate tumordose while avoiding deleterious sequelae. With the currentstandard of clinical practice, the reduction of the PTVmargin to achieve a high level of dose escalation could beextremely detrimental to the success of conformal radiationtherapy. Therefore, it is critical to develop a new practicaltreatment process to maximize the effect of conformal ther-apy. In this study, an off-line image feedback strategy isproposed to generate a patient-specific confidence-limitedPTV (cl-PTV) in an adaptive treatment process for prostatecancer. The utility of cl-PTV was evaluated and analyzedprospectively and retrospectively using daily treatment im-aging data.

MATERIALS AND METHODS

From October 1997 to March 1999, 30 patients withorgan-confined prostate cancer were prospectively enrolledand consented to an adaptive radiation treatment process inthe Department of Radiation Oncology, William BeaumontHospital. Clinical implementation of the adaptive treatmentprocess has been discussed and reported (20). Briefly, eachpatient had a pretreatment planning CT scan of 3-mm sliceswith urethral contrast, followed by three-dimensional (3D)treatment planning. The initial PTV included the prostateand/or seminal vesicles (CTV), plus a 1-cm uniform margin.Treatment is planned to deliver the prescribed dose to thePTV using a 4-beam-box technique. The PTV-to-field-edgemargin is designed as 7 mm elsewhere, but 11 mm at thesuperior and inferior borders of the PTV by considering thedosimetric penumbra of the multileaf collimator (21). Oneach of the first 4 days of treatment, a CT scan encompass-ing the entire bladder to below the ischium was obtained,and the daily CTV was contoured to assess treatment targetmotion. In addition, a daily portal image for each treatmentfield was taken using an electronic portal imaging device(EPID), and daily setup error was measured. The patient-specific data of daily target motion and setup error wereexported to a prediction model to form the cl-PTV. Then,the individual treatment plan was modified to deliver theprescribed dose to the cl-PTV by either continuing the4-field treatment or using beam intensity-modulated radio-therapy (IMRT) for the remaining treatment. To gain con-

fidence in the cl-PTV, each patient was also monitored withadditional biweekly CT scanning and daily portal imagingfollowing the plan modification. Therefore, throughout theentire treatment course each patient had an average of 18daily CT scans and 120 daily portal images.

One of the most important procedures in the adaptivetreatment process is utilization of the patient image feed-back of internal target motion and setup error for the cl-PTVconstruction. Unlike a conventional PTV, the purpose of thecl-PTV is to eliminate the effect of the systematic compo-nents and compensate for the random components of pa-tient-specific geometric variation due to internal target mo-tion and daily setup error. The following sections in thispaper will focus on the steps of cl-PTV construction andevaluation. To simplify the problem, internal target motionand patient setup error are considered separately in thestudy, although the proposed method does not require theassumption of independence between these two motions.

Constructing a patient-specific bounding volume forinternal target motion

We first hypothesized that a patient-specific boundingvolume, constructed using an initial sequence of daily mea-surements of internal organ motion, exists and ensures thatthe CTV in the patient treatment for prostate cancer receivesthe prescribed dose within a predefined dose tolerance. Totest the hypothesis, daily CT images of 30 prostate cancerpatients were used to mimic the patient target motion in theactual treatment process. To separate internal target motionfrom entire geometric variation occurring in external beamtreatment, the daily CT images were registered using 3Dtreatment planning software (Pinnacle, ADAC Laboratories,Milpitas, CA) to the initial planning CT image with respectto the same pelvic bony structure, and the daily CTVs weredelineated by a well-trained person. A bounding volume,defined as the volume that represents the maximum excur-sions of the CTV as measured on a number of daily deter-minations of CTV, was then formed as the convex hull ofthe firstk days of the measured CTV on each transverse CTslice (see Fig. 1). The major advantage in forming thebounding volume by applying a few days of CTV is toeliminate the effect of the systematic variation in internaltarget location from the cl-PTV definition. This importantproperty can not be achieved in conventional treatmentplanning in which a single static CTV is employed for PTVdefinition.

To minimize the bounding volume, the minimal numberof CT measurements,k*, in the initial treatment days wasdetermined using the following dosimetric criterion. Themaximum dose reduction in the CTV due to internal targetmotion is no more thanD% of the prescribed dose when thebounding volume is applied for the remainder of treatment.The search method fork* is described in Appendix A.Basically, the bounding volume constructed using the firstkdays of CTV was mimicked as a planning target volume,denoted PTVo(k). A planning dose distribution was thencreated to cover the PTVo(k).

290 I. J. Radiation Oncology● Biology ● Physics Volume 48, Number 1, 2000

The maximum dose reduction in the CTV due to internaltarget motion of the individual was calculated by examiningthe daily determinations of CTV in the dose distribution andusing a backward convolution method (Appendix 1). In thisstudy, we did not assume that internal target motion wasrigid-body motion and had a Gaussian distribution. There-fore, daily CTV motion was not modeled as a linear trans-formation of the initial planning CTV. Instead, the maxi-mum dose reduction was evaluated based on the actualshape, size, and location of the prostate and/or seminalvesicles through the treatment course.

Different treatment modality causes different dose distri-bution around treatment volume, which should be consid-ered in the PTV design. To study this feature, we alsoexamined the sensitivity of PTVo(k) to different planningdose distributions. The dose distribution was created usingeither conventional 3D treatment planning for the 4-beam-box delivery or inverse planning for the IMRT delivery. Forthe IMRT delivery, a multi-static-field delivery techniquewas applied (22, 23), where 5 coplanar beams were ar-ranged with equal angles starting at the patient anteriordirection. The resolution of beam intensity in the IMRTdelivery was 1 cm3 1 cm. An inverse planning software(KonRad, MRC, Heidelberg, Germany) was used to opti-mize the treatment plan with respect to following objec-tives: (a) maximizing the minimum dose and maintaining15% of dose heterogeneity in PTV; (b) no more than 5% ofrectal wall volume has a dose of 75.6 Gy or more; (c) nomore than 30% of rectal wall volume has a dose of 72 Gyor greater; (d) no more than 40% of rectal wall volume hasa dose of 65 Gy or more; and (e) no more than 50% ofbladder volume has a dose of 75.6 Gy or greater. In thisimplementation, average dose gradient within a range of 1cm around the treatment volume generated by the inverseplanning was always higher than the one generated by theconventional planning.

After calculating the maximum dose reduction for eachgiven k, a standard statistical analysis (The null hypothesistest for proportions with an adequate statistical significance,p value.) was performed, to determine the minimal numberk* required to achieve the specified dosimetric criterion fora proportion of patients. This study was also repeated with-

out including the initial planning CTV in PTVo(k) construc-tion to examine if there is extra bias introduced in thepretreatment CT simulation procedure.

Determining patient-specific margins for patient setuperror

Different from internal target motion, patient setup errorwas evaluated as rigid-body motion of the pelvic bonystructure of the patient with respect to treatment machinecoordinate. Therefore, patient-specific 3D setup error wasmeasured from daily portal images and quantified using alinear transformation with respect to the digitally recon-structed radiograph (DRR) image created in the treatmentplanning. Because the bounding volume PTVo(k*) was arigid object fixed during the treatment course with respect tothe patient’s bony structure, the resulting linear transforma-tion was applied to represent the daily setup position ofPTVo(k*) with respect to the treatment beam. As has beendiscussed and tested clinically (24, 25), the patient-specificsetup error was characterized using the mean, representingthe systematic setup error, and the standard deviation, rep-resenting the random setup error, and the systematic andrandom setup errors were predicted from the portal imagingmeasurements in the early days of treatment.

After 4–6 initial days of portal imaging, the predictedsystematic and random setup errors were calculated usingthe method discussed previously (24). A correspondingresidual from the systematic error prediction was also de-termined using the confidence interval estimate (Appendix2). The predicted systematic setup error was corrected in ourclinic by translating the initial planning CT image withrespect to the initially planned beams’ position, instead ofphysically moving the patient or treatment couch. NewDRRs were regenerated for the new reference images aftercorrecting the systematic setup error. In the current clinicalimplementation, we compensate for rotational errors of pa-tient setup by considering them as extra residuals (Appendix2). The reason for handling rotational errors different fromtranslational errors is mainly because in the current imple-mentation, the 3D setup error is measured by averagingmultiple two-dimensional beams’ alignment. However,there is no technical difficulty with correcting the systematic

Fig. 1. Illustration of the convex hull of multiple clinical target volumes (CTVs) and planning target volume (PTV).

291Patient-specific PTV for prostate cancer treatment● D. YAN et al.

rotational error by also rotating the initial planning CTimage, when a full 3D setup error alignment tool is imple-mented in the clinic. The predicted random setup error andthe corresponding residuals were then used to design apatient-specific setup margin for the remaining treatmentbased on the following dosimetric criterion. The maximumdose reduction in PTVo(k*) due to patient daily setup erroris no more thand% of the prescribed dose, when thepatient-specific setup margin,p 5 (px, py, pz) in 3D, isapplied to PTVo(k*) for the remainder of treatment. Optimaldesign of the patient-specific setup margin is detailed inAppendix 2. Briefly, for a given margin of setup error, themaximum dose reduction due to daily setup error inPTVo(k*) was calculated. The calculation was performed byconvolving the distribution of daily setup error with theplanned dose distribution calculated by applying thePTVo(k*) plus the margin of setup error as the planningtarget volume. The optimal patient-specific margin was thenselected as the minimal margin,p*, that had the maximumdose reduction in PTVo(k*) less than or equal tod%. Asdescribed in the previous section, two distinct planning dosedistributions, corresponding to the 4-beam-box treatmentand the IMRT, were separately used in the setup margindetermination.

Finally, the cl-PTV is constructed as PTVo(k*) plus thepatient-specific setup marginp* after selecting of the dosereduction toleranceD% for internal target motion andd%for patient setup error.

Selecting the dose reduction tolerance for cl-PTVconstruction

The cl-PTV is constructed to ensure that the maximumdose reduction in the CTV due to internal target motion andpatient setup error is no more than a predefined tolerance.Selection of the dose-reduction tolerance is a nontrivialissue. Intuitively, selecting a larger dose-reduction tolerancefor the CTV produces a smaller cl-PTV resulting in lessnormal tissue irradiated. Consequently, a higher dose can beprescribed to the treatment. Therefore, optimal solutionshould be determined by alternating the dose reductiontolerance and the prescription dose escalation. However,this solution can not be obtained with confidence using thecurrent knowledge of clinical dose response for the treat-ment target and critical normal structures.

The dose reduction tolerance can also be selected basedon the current standard recommendation for dose variationin treatment planning and delivery, ICRU24 (26), where thedose variation due to patient geometric variation is expectedto be no more than 3% of the prescription dose. In ourclinical implementation, the cl-PTV was conservativelyconstructed with this consideration. The bounding volumePTVo(k*) was constructed withD% 5 2% to ensure that themaximum dose reduction in the CTV due to internal targetmotion was no more than 2% of the prescription dose.Meanwhile, the setup marginp* was selected withd% 51% to ensure that the maximum dose reduction in thePTVo(k*) due to patient setup error was no more than 1% of

the prescription dose. Accordingly, the direct summation ofthe dose reduction tolerances,D% 1 d%, ensures that nopoint in the CTV has dose reduction due to internal targetmotion and patient setup error beyond 3% of the prescrip-tion dose. With knowledge of the probabilistic nature of thedose reductions due to internal target motion and patientsetup, the tolerances ofD% andd% can also be added in thequadrature, such as=(D%)2 1 (d%)2 # 3%.

RESULTS

Figures 2a and 2b show the error bar plot of the maxi-mum dose reduction in the CTV due to internal targetmotion, when the bounding volume PTVo(k) was appliedfor the 4-beam-box treatment with and without inclusion ofthe initial planning CTV delineated from pretreatment CTsimulation image. In these results, the CTV is defined as theprostate and seminal vesicles. The two plots are similar,indicating that the dosimetric criterion can be achieved inboth cases with similar number of CT measurements. Inthese figures, we also report the results as dose reductiontolerance achieved by 80% of patients (p 5 0.08), as well as95% of patients (p 5 0.1), for different days of CT mea-surement. In our current clinical implementation,k* 5 5days of CT measurement (initial planning CT plus 4 dailytreatment CT) was selected to achieve the maximum dosereduction due to internal target motion of 2% or less of theprescription dose for 80% of patients (p 5 0.08), and 4.5%or less of the prescription dose for 95% of patients (p 50.1). In this case, 3 of the 30 patients have the maximumdose reduction in the CTV larger than 2%, and they areequal to 2.3%, 3.6% and 4.5%, respectively.

The same study was also applied with the CTV defined asprostate alone. In this case, the dose reduction toleranceachieved by the same number of CT measurements wassimilar to the CTV defined as the prostate and seminalvesicles. The results are shown in Fig. 3.

Figure 4 shows similar data as in Figs. 2 and 3, but for theIMRT treatment. The results demonstrate that PTVo(k) con-structed using the same number of CT measurements is notadequate to achieve the same level of the dosimetric crite-rion as the 4-beam-box treatment. On average, 7% dosereduction in the CTV is expected by applying the PTVo(5).Therefore, if maximal 2% dose reduction due to internaltarget motion should be maintained, one must either in-crease the number of CT measurements, or add extra marginto PTVo(5) for the IMRT.

Figure 5 shows the size of PTVo(5) with and without theinitial planning CTV for all patients. The PTVo(5), whichincludes the initial planning CTV, has a volume 5.3%larger, on average, than the one without including the initialplanning CTV. As discussed in the next section as well as inAppendix 3, this volume difference indicates an extra biason the CTV localization introduced in the initial CT simu-lation procedure.

Figure 6 shows the relationship between the randomsetup error and the setup margin that is required to meet the


dosimetric criterion (Maximum dose reduction in thePTVo due to daily random setup error# 1% of theprescribed dose). As expected, the margin required forthe IMRT is larger than the one for the conventionaltreatment. For example, to compensate for 3-mm randomsetup error, 1-mm setup margin will be adequate for the4-beam-box treatment, but 3-mm setup margin will beneeded for the IMRT to maintain the same dosimetriccriterion.

Finally, Fig. 7 shows the equivalent uniform margin ofcl-PTV 5 PTVo(5) 1 patient-specific setup margin (withrespect to the confidence 12 a 5 0.9 and the dose toleranced% 5 1%), for the 4-beam-box treatment with the CTVdefined as the prostate plus seminal vesicles as well asprostate alone. The equivalent uniform margin for eachcl-PTV was calculated as the uniform margin for a conven-tional PTV (the initial CTV1 the uniform margin) that hasvolume equal to that of cl-PTV. Compared to the conven-

Fig. 2. The maximum dose reduction (% of prescribed dose, Rx) in the clinical target volume (CTV) (prostate1 seminalvesicles) due to internal target motion, when the bounding volume PTVo(k) was applied for the 4-beam-box treatment(a) with and (b) without including the initial planning CTV. Error bar indicates the mean and one standard deviation ofmaximum dose reduction for a given number of CT measurement days, and a dash line represents the dose-reductiontolerance achieved by 80% of patients (p 5 0.08). A solid line indicates the maximum dose reduction among the 30patients, which also represents the dose-reduction tolerance achieved by 95% of patients (p 5 0.1).


tional PTV, the cl-PTV margin can be reduced, on average,by a factor of two, while maintaining the same dosimetriccriterion.

DISCUSSION

In this paper, an off-line adaptive strategy of PTVreconstruction is introduced by utilizing feedback of im-

aging measurements on target location and patient setupposition. In principle, intertreatment target motion anddaily setup error can be modeled as two stochastic pro-cesses, where target location and patient setup position inthe pretreatment simulation represent the initial randomsamples in the processes. Target location and setup po-sition in pretreatment simulation have been convention-ally used as the reference in treatment planning for the

Fig. 3. The maximum dose reduction (% of prescribed dose, Rx) in the clinical target volume (CTV) (prostate) due tointernal target motion, when the bounding volume PTVo(k) was applied for the 4-beam-box treatment (a) with and (b)without including the initial planning CTV. Error bar indicates the mean and one standard deviation of maximum dosereduction for a given number of CT measurement days, and a dash line represents the dose reduction tolerance achievedby 80% of patients (p 5 0.08). A solid line indicates the maximum dose reduction among the 30 patients, which alsorepresents the dose reduction tolerance achieved by 95% of patients (p 5 0.1).


entire treatment. Therefore, any discrepancy between thereference and the mean target location or mean setupposition in the actual treatment causes a systematic vari-ation on the patient treatment geometry. In theory (seeAppendix 3), the systematic variation among patients hasa similar magnitude as the random variation if similar

procedures on patient setup and organ location are ap-plied for both simulation and treatment. This is becausethe systematic variation is dependent on the patient setupposition and target location at treatment simulation whichstatistically have similar probability distribution as one intreatment. This phenomenon has been observed in a

Fig. 4. The maximum dose reduction (% of prescribed dose, Rx) in the clinical target volume (CTV) due to internaltarget motion, when the bounding volume PTVo(k) was applied for the intensity-modulated radiotherapy (IMRT)treatment with including the initial planning CTV. (a) CTV5 prostate1 seminal vesicles; (b) CTV5 prostate. Errorbar indicates the mean and one standard deviation of maximum dose reduction for a given number of CT measurementdays, and a dash line represents the dose reduction tolerance achieved by 80% of patients (p 5 0.08). A solid lineindicates the maximum dose reduction among the 30 patients, which also represents the dose reduction toleranceachieved by 95% of patients (p 5 0.1).


previous study of patient setup error (27), and it is alsotrue for internal target motion. Furthermore, a largersystematic variation could be expected if extra clinicalprocedures were introduced into the pretreatment simu-lation, i.e., urethral contrast, rectum laxation used for thesimulation of prostate treatment, and/or position verifi-cation on the simulator following by the CT simulationand 3D planning. The association between the systematicvariation and random variation also indicates that the two

variations are not independent of each other. If we clas-sify patients into specific groups based on the randomvariation of the individual, a group of patients who havea larger random variation in their target location or setupposition will potentially have a larger systematic varia-tion, as well (see Appendix 3). Unfortunately, this im-portant feature has been consistently neglected in previ-ous studies of the effect of patient geometric variationson the planning dose distribution and PTV construction.

Fig. 5. The volume (cc) of PTVo(5) with and without the initial planning CTV (prostate1 seminal vesicles). The meansand standard deviations are 1176 35 and 1116 34 (cc). CTV5 clinical target volume.

Fig. 6. The relationship between the random setup error,s (mm), and the minimum setup margin, cz s (mm), requiredto keep the dosimetric criterion (maximum dose-reduction tolerance in the PTVo due to daily setup error# 1% of theprescribed dose). IMRT5 intensity-modulated radiotherapy.


In contrast, the cl-PTV constructed in the current studyaccounts for this.

In most of the conventional and IMRT treatment pro-cesses, the systematic variation and random variation oftarget location and patient setup position have not beendistinguished. Therefore, the effect of both variations ontreatment dose delivery to the CTV should be consideredin the PTV design. As a consequence, the PTV marginhas to be large enough to compensate for both variations.Recent study (28) has demonstrated that the ratio of theconventional PTV margin used to compensate for thesystematic variation vs. the random variation is approx-imately 3:1. This implies that a large portion of PTVmargin in current treatment is intended to compensate forthe systematic variation. Therefore, if the systematicvariation in the individuals can be identified and elimi-nated in a revised treatment process, the PTV margin orvolume can be significantly reduced. Consequently, moreeffective dose delivery and dose escalation can be per-formed. The cl-PTV constructed in our off-line imagefeedback adaptive treatment process is designed toachieve this aim within a minimal number of treatmentdays. However, in principle we can not expect to com-pletely eliminate the systematic variation by applyingfew days of imaging measurements and single PTV mod-ification for the individual treatment. The statistical re-sidual from the estimation of the systematic variation willalways remain, and is inversely proportional to the squareroot of the number of measurements. Therefore, thechoice of minimum days of measurements in our currentadaptive treatment process for the cl-PTV construction isa compromise of clinical practice, instead of the global

optimal of feedback control. If the frequency of dailymeasurements could be increased without excessively tax-ing the clinic, a different feedback and control strategyshould be applied to achieve the maximal treatment im-provement. The other advantage of increasing the frequencyof image measurement is to monitor a possible time-trend ininternal target motion and/or patient setup error. In ourcurrent clinical implementation, a weekly CT scan is con-tinuously acquired for the quality insurance. Extra modifi-cation could be possible if unexpected variation occurred.

The dependence of dose distributions on treatmentmodality must be considered in the PTV design. Asdemonstrated in this study, treatment with a sharpergradient of the dose distribution at the target edge, suchas the IMRT treatment, requires a larger planning targetvolume or PTV margin to achieve the same level ofdosimetric criteria. Therefore, applying the same PTVmargin as conventional treatment or even reducing PTVmargin for IMRT treatment will be potentially risky tothe treatment outcome unless the treatment process isrevised to either reduce the geometric variation or adaptthe treatment to the variation by applying image feedbackstrategy. Fundamentally, selection of CTV-to-PTV mar-gin and/or dose gradient at the edge of PTV is an opti-mization problem, which also requires a consideration ofthe volume effects on critical normal tissues, and thepotential for dose escalation. The major challenge insolving this problem is to quantify the cumulative dose aswell as the fractional dose in the moving organs, espe-cially for non-rigid body motion (29). With this quanti-fication, treatment process can also be further refined sothat on-line imaging feedback of patient geometry and

Fig. 7. The equivalent uniform margin (mm) of cl-PTV for the 4-beam-box treatment. The cl-PTV was constructed usingthe dosimetric criterion: the maximum dose reduction tolerance in the CTV (prostate1 seminal vesicles as well asprostate alone) due to internal organ motion and daily setup error is less than or equal to 3% of the prescribed dose. Themean and the standard deviation are 7.26 2 mm and 6.16 1.7 mm, respectively.


optimal control of dose delivery can be applied on dailybasis. In this case, treatment efficacy of hypo-fraction-ation or even variable fractionation regimes can be rea-sonably explored and optimized in an on-line imagefeedback adaptive treatment process, by quantifying thedaily dose distribution, and estimating the cumulativedose distribution as well as the corresponding radiobio-logical effect in critical normal structures.

In summary, the cl-PTV for prostate treatment can be

constructed within the first week of treatment using treat-ment images as feedback. The cl-PTV has the capabilityto exclude the systematic variation and compensate forthe patient-specific random variation in both target loca-tion and patient setup position. This implies that in thecurrent off-line image feedback adaptive treatment pro-cess, a single plan modification can be performed withinthe second week of treatment to improve the efficacy ofexternal beam therapy for prostate cancer.

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APPENDIX 1

Let CTVi, i 5 0, 1,...,M represent theM 1 1 determi-nations of the individual CTV obtained from the pretreat-ment CT scan (i 5 0) and theM days of treatment CT scans.All these CTVs are defined with respect to the referencecoordinate of rigid bony structures, following bony regis-tration to eliminate the effect of patient setup error. The

bounding volume, PVTo(k) 5 ConvexHull Høi50

k

CTViJ, is

then constructed as the convex hull of the union of CTVi,i 5 0, 1, . . . ,k , M, on each of transverse CT slices (Fig. 1).

For a givenk, the maximum dose reduction in the CTVdue to internal target motion is evaluated by mimicking thePTVo(k) as a planning target volume for planning dosecalculation. To simplify the evaluation, a functionD( z ) wasgenerated to represent the dose decrement outside ofPTVo(k). The functionD(x) 5 [Dp 2 D(x)] / Dp, shown inFig. 8, is defined as the maximum dose decrement amongthe directions which are normal to the edge of the PTVo(k),where D(x) is the dose at pointx in the dose penumbraregion calculated for a given beam delivery technique, andDp is the prescribed dose level.

To calculate dose reduction outside of PTVo(k), a fre-quency function of the target occupancy during the entirecourse of treatment is constructed as follows. First, denoteCTV 5 { vp u p 5 1, . . . , P} as the set ofP volumeelements. We also define CTVi 5 { xi [ N3 u xi 5 argvp,@vp [ CTV}, for each i 5 0, 1,..., M, to be the set ofpositions of the volume elements at each treatment delivery,andS(vp) 5 { xi u xi 5 argvp, @i 5 0, 1, . . . ,M} as theset of positions at all treatment deliveries for a given volume

elementvp [ CTV. A function, f(z) onøi51

M

CTVi , R3 is

then defined as the frequency measure of the CTV occu-pancy as follows,

f~ x! 5 m/M, x [ øi50

M

CTVi,

if and only if ?I 5 { i1, . . . , im} # {0, 1, . . . , M}, suchthat@i j [ I , x [ CTVi, and@i [ I c # {0, 1, . . . , M},x [/ CTVi.

Therefore, the measure functionw(z) could have its values

asf0 5 0, f1 5 1/M, . . . , fm 5 m/M, . . . , fM 5 1. Now, letE(vp) be a subset inø

i51

M

CTVi, which contains all possible

treatment positions for a given volume elementvp [ CTV,i.e., S(vp) # E(vp). The ideal construction ofE(vp) is, ofcourse,E(vp) 5 S(vp), if we can label the volume element inthe treatment course (29), otherwise a possible minimumsubset, which contains theS(vp) in a large probability, needsto be constructed based on the knowledge of internal targetmotion. TheE(vp) was constructed, in this calculation, byconsidering the distribution of CTV motion along each axisof the patient coordinate, as a cylinder normal to the surfaceof the PTVo(k). Then, the point inE(vp), which has aconstant frequencyf and the maximum dose reduction, willbe

j~E~vp!, f ! 5 arg maxj[E~vp!

$D~j!uf~j! 5 f %.

With these definitions and constructions, which are alsoillustrated in the Fig. 9, the upper bound of the dose reduc-

Fig. 8. Dose reduction (% of prescribed dose, Rx) outside of the planning target volume (PTV) for the 4-beam-box andthe intensity-modulated radiotherapy (IMRT) treatment.


tion for the volume elementvp [ CTV in the entiretreatment course can be calculated as,

dD~vp! 5 Om51

M2k

D~j~E~vp!, fm!! 3 ~ fm 2 fm21!.

Accordingly, the upper bound of the maximum dose reduc-tion in the CTV can be calculated as,

dDmax 5 Maxvp[CTV

dD~vp! 5 Om51

M2k

D~j~E~vp!, fm!! 3 ~ fm 2 fm21!.

Above models the estimation of the maximum dose reduc-tion in CTV due to internal target motion. The numericalimplementation and mathematical discussion regarding theproper sampling and convergence are quite long, and havebeen included in the other paper.

Finally, using the dosimetric criterion, “the maximumdose reduction in the CTV due to internal target motion isno more thanD% of the prescribed dose when the boundingvolume is applied for the remainder of treatment,” theminimal number of the daily CT measurements,k*, isdetermined as the firstk such thatdDmax # D%.

APPENDIX 2

As has been early investigated (30), the dose reduction inPTVo(k*) for a predefined setup margin can be directlyderived by convolving the dose distribution with the distri-bution of patient setup error, in each primary direction of thepatient.

Let N(S, m, s2) be the normal distribution for the 3Dsetup error« [ N3 of the individual with the probability

densityr~«, m, s! 51

Î2p z sz expS2

~« 2 m!2

2s2 D, where the

mean,m, and the standard deviation,s, represent the cor-responding systematic and random setup errors. The sys-tematic and random setup errors are predicted asm and s,respectively, from the initialk days of portal imaging mea-surement.

The following derivation will focus on the design of setupmargin to compensate for the random setup error,s, andresiduals,g 5 g1 1 g2. The residual,g1, which comes fromthe prediction of the systematic error, is calculated based onthe 1 2 a confidence interval estimate of them, as g1

5 ta/ 2,k21 zs

Îkandim 2 mi # g1, where the factorta/2 2 1,

has thet distribution withk 2 1 degrees of freedom. Theresidual,g2, which results from 3D out-of-planar rotations,is determined using the largest deviation between single-beam alignment result (two-dimensional) and the average ofmultibeam alignment results. To simplify notation, we limitour derivation to the one-dimensional case. However, theconclusion keeps no change for the 3D case, provided thatthe setup errors in any two dimensions are independent eachother, and the setup margin is constructed using the full 3Dtechnique.

First, we define the patient-specific setup margin,p 5

g 1 c z s. The factor,s 5 Î k 2 1

x212a,k21

z s, is the upper

bound of the random setup errors based on the 12 aconfidence interval estimate of thes, wherex2

12a,k21 hasthe x2 distribution with k 2 1 degrees of freedom. Thefactorc is determined from the predefined dosimetric crite-

Fig. 9. Schematic plot of the dose-reduction calculation for internal target motion. The dashed lines represent isodosesoutside the PTVo. The shaded region represents the clinical target volume occupancy distribution,f(x), with the lightestregionf(x) 5 f1, the medium regionf(x) 5 f2 , and the darkest regionf(x) 5 f3.


rion, recalling as the maximum dose reduction in PTVo(k*)due to daily setup error# d% of the prescribed dose, whenthe marginp 5 g 1 c z s is applied to the PTVo(k*) for thetreatment. Therefore, the factorc becomes a function of theupper bound of the random setup error,s and the prese-lected dose tolerance,d%, for a given dose distribution,D( z ), defined in Appendix 1 (Fig. 8). This function,c(s,d%), can be obtained by convolvingD( z ) andr(e, g, s) asfollows (Fig. 10),

dD~p! 5 Ep

`

D~« 2 p! z r~«, g, s! z d« # d%.

In our implementation, the factorc(s, d%) was determinedanalytically by first fitting the dose-reduction curve as,

D~ x! 5 1 2 a1 2 b1 z exp~l1 z x! for 0 # x # t

D~ x! 5 1 2 a2 2 b2 z exp~l2 z x! for t # x,

and then solving the integral as,

dD~p! 5 H ~1 2 a1! z @1 2 N~p, 0, s2!#

2 b1 z expS1

2s2l1

2 2 l1pD z [1 2 N~p, s2l1, s2#J2 H ~1 2 a1! z [1 2 N~p 1 t, 0, s2#

2 b1 z expS1

2s2l1

2 2 l1pD z [1 2 N~p 1 t, s2l1, s2#J1 H ~1 2 a2) z @1 2 N~p 1 t, 0, s2!#

2 b2 z expS1

2s2l2

2 2 l2pD [1 2 N~p 1 t, s2l2, s2]J ,

whereN~x, v, s2! 51

2F1 1 erfSsign~x 2 v! z~x 2 v!

Î2sDG,

andsign(x 2 v) 5 1 1, if x $ v and2 1, if x , v.The factorc is illustrated as a function ofs in Fig. 6 for

the given toleranced% 5 1%.

APPENDIX 3

In this study, we focus on the treatment position of CTV.However, the following discussion can be also extended toconsider the position of any volume of interest (target ornormal organ) in the process of fractional radiotherapy. Tosimplify the mathematical notation, let the symbolX repre-sent the treatment position of a volume element in a volumeof interest. Motion of the volume can be either quantified byapplying a linear transformation matrix to all volume ele-ments in the volume of interest, if the motion is rigid bodymotion (i.e., specific bony structure), or calculating organdeformation for non-rigid body motion (i.e., patient softtissue) (29). The position of the volume element at thetreatment simulation and in the entire course of treatmentcan be mathematically represented as a discrete stochasticprocess (or random sequence), {X0, X1, . . ., XN}, where X0

represents the position in pretreatment simulation (or aplanning position) andX1, . . ., XN represent the actualtreatment positions of the element at treatment fraction 1 toN. Accordingly, a sequence of element positions {x0, x1, . . .,

xN} for a completed treatment of the individual is simply aobserved sample of the random sequence.

Without loss generality, we first classify patients withsame cancer treatment into separated groups based on theirrandom variations, and assume that they receive similarsetup and target localization procedure in the pretreatmentsimulation and treatment deliveries. Hence, for each groupof patient, the random variablesXi, i 5 0, 1, . . .,N, areindependent each other and have the same mean and vari-ance.

In the radiation treatment, the pretreatment simulationposition, X0, has been used as the reference position oftreatment planning and for treatment delivery. Therefore,the deviation between the actual treatment positionXi . 0

and the reference positionX0 represents an unfavorabledisplacement due to either internal organ motion or patientsetup error at theith treatment. Conventionally, the standarddeviation of the position displacement between treatmentand reference under the condition of a given reference

Fig. 10. Schematic plot of the dose variation and setup marginevaluation for patient setup error (PTV, planning target volume).


position x0, mathematically symbolized asSD[Xi.0 2X0 u X0 5 x0], is used to represent the random positiondisplacement of the individual. It can be easily derived thatSD[Xi.0 2 X0 u X0 5 x0] 5 SD[Xi] 5 s. Similarly, theconditional expectation of the difference between the ithtreatment position and the reference position,E[Xi 2X0 u X0 5 x0] 5 E[Xi] 2 x0 5 m i( x0), represents thesystematic position displacement of the individual at thetreatmenti. EqualE[Xi] or m i( x0) 5 m( x0) for all i alsoimplies that no time-trend on patient position appears duringthe treatment course. In this case, treatment position of theindividual in the given patient group has a common randomposition displacement,s, but different systematic positiondisplacement,m(x0), in terms of the simulation positionx0

of the individual. Finally, it is easy to calculate the standarddeviation of the systematic position displacement amongpatients in the group,SD[m(X0)] 5 SD[E[Xi] 2 X0] 5SD[X0] 5 s. This indicates that the standard deviation of

the systematic displacements among the patients has thesame magnitude as the random position displacement.Therefore, if we name the systematic position variation tobe the standard deviation of the systematic position dis-placements among patients, and the random position varia-tion to be the mean of the random position displacementsamong patients, the systematic variation will have the samemagnitude as the random variation on patient geometry.

On the other hand, if an extra procedure is included in thepretreatment simulation, for example contrast for CT imag-ing or extra verification simulation after the CT planning,there will be an extra random event,j, with the standarddeviation,sj, including in the pretreatment simulation po-sition X0. Therefore, the standard deviation of the system-atic displacements will beSD[m(X0 1 j)] 5 SD[E[Xi] 2(X0 1 j)] 5 (s2 1 sj

2)1/ 2 . s. This implies that thesystematic variation will be larger than the random varia-tion.


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