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Year : 2021  |  Volume : 17  |  Issue : 4  |  Page : 1125-1131

A mathematical formulation for volume expansions in contouring for radiotherapy planning

1 Department of Radiation Oncology, Manipal Hospitals, New Delhi, India
2 Department of Radiation Oncology, Fortis Memorial Research Institute, Guragon, Haryana, India

Date of Submission16-Aug-2019
Date of Decision09-Oct-2019
Date of Acceptance30-Dec-2019
Date of Web Publication10-May-2021

Correspondence Address:
Biplab Sarkar
Department of Radiation Oncology, Manipal Hospitals, Dwarka, New Delhi - 110 075
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/jcrt.JCRT_614_19

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 > Abstract 

Context: This research describe the characteristic volume expansion of a moving target as a function of differential margins.
Aim: We aimed to ascertain the volume change after giving margin for clinical and set up uncertainties including generating internal target volume (ITV) for moving target.
Materials and Methods: Settings and Design – Spheres of diameter (0.5–10 cm) with differential expansion of 1–15 mm were generated using a mathematical formula. Moving targets of radius 1–5 cm were generated, and the resultant volume envelopes with incremental motion from 1 to 20 mm were obtained. All relative volume change results were fitted with mathematical functions to obtain a generalized mathematical formula.
Statistical Analysis Used: None.
Results: The percentage increase in volume (%ΔVp) was much more pronounced for smaller radius target. For moving target with relatively smaller radius, %ΔVp is predominant over the absolute volume change and vice versa in case of larger radius. Mathematical formulae were obtained for %ΔVp as a function of radius and expansion and for %ΔVp in ITV volume as a function of radius and tumor movement.
Conclusions: This study provides an idea of volume change for various expansions for various size targets and/or moving target for different range of movements. It establishes a correlation of these volume changes with the changing target size and range of movements. Finally, a clinically useful mathematical formulation on volume expansion has been developed for rapid understanding of the consequence of volume expansion.

Keywords: Margins, radiotherapy, target, volume

How to cite this article:
Munshi A, Sarkar B, Paul S, Chaudhari BB, Chauhan RS, Ganesh T, Mohanti BK. A mathematical formulation for volume expansions in contouring for radiotherapy planning. J Can Res Ther 2021;17:1125-31

How to cite this URL:
Munshi A, Sarkar B, Paul S, Chaudhari BB, Chauhan RS, Ganesh T, Mohanti BK. A mathematical formulation for volume expansions in contouring for radiotherapy planning. J Can Res Ther [serial online] 2021 [cited 2022 Nov 26];17:1125-31. Available from: https://www.cancerjournal.net/text.asp?2021/17/4/1125/315672

 > Introduction Top

The past 2–3 decades have seen the evolution of radiotherapy from two-dimensional (2D) to 3D and volumetric planning. Image-based radiotherapy is the norm in most radiotherapy centers. Various ICRU recommendations serve as guidelines for the concepts of gross tumor volume (GTV), clinical target volume (CTV), internal target volume (ITV), and planning target volume (PTV)[1],[2],[3] ICRU reports (ICRU50, ICRU62, and ICRU83) have detailed the issues of margins to the gross tumor for microscopic diseases, motion, and setup uncertainty.

Aided with modern imaging facilities such as computed tomography (CT), magnetic resonance imaging, and biological imaging like positron emission tomography, radiation oncologists and physicists have become more fine and precise in delineation of target volume and organ at risk (OAR). Furthermore, different imaging modalities on couch, for example, cone beam CT, Portal Imaging, and ExacTrac® (Brainlab AG, Germany) have reduced setup-related uncertainties to a significant extent.[4],[5],[6],[7]

CTV relates to the extent of microscopic extension beyond the gross disease. The margin that deals with setup uncertainties is the PTV whereas ITV relates to tumor motion.[6],[8]

Knowledge of geometrical uncertainties in radiotherapy is steadily increasing.[9] Several publications have presented data on organ motion and setup accuracy.[6] Various PTV generation formulae are available including those by Van Hark and Stroom et al.[10],[11],[12],[13] While CTV is a margin that is essentially obtained from literature, individual centers are supposed to obtain their own PTVs through departmental setup variability data analysis.[6],[14],[15]

As a result of varying biological data regarding GTV to CTV expansion as well as various ways of interpreting the data used to generate PTV, the treating clinician often does not have a sacrosanct figure to be used for expansion for generating CTV and PTV.[16]

Increase in CTV margin (or PTV margins) may decrease the locoregional failure However, there is a normal tissue complication probability cost function attached to increment in margin.[17]

Knowledge of the correct CTV and PTV margin is still evolving. Therefore, it is critical that the treating radiation oncologist is cognizant of the effect of prescribing varied volumetric expansions.

In this study, we have tried to establish a mathematical relationship between expansion in PTV/CTV volume as a function of radius vector of GTV. As a subcomponent of the study, we studied the relationship between expansion in volume and the generalized radius vector of GTV for a hypothetical moving target.

 > Materials and Methods Top

Static gross tumour volume/clinical target volume to planning target volume expansion

Serial volume of spheres was generated using an in-house made Microsoft excel® program using formula:

Where r is the radius vector of the spherical PTV

This study demonstrate eleven hypothetical spheres of radius 0.5 cm, 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, 7 cm, 8 cm, 9 cm, 10 cm. These spheres present the baseline volume in this study. Further PTV margin was given from 0 mm to 15 mm with increments of 1 mm, i.e., the serial expansions that were given to each baseline volume were 1 mm, 2 mm, 3 mm, 4 mm, and so on up to 15 mm. The absolute change (ΔAV) and relative change in (%ΔVp) in PTV volume against GTV volume was calculated as a function of radius and expansion, given in [Table 1] and [Table 2], respectively. Further, a mathematical relationship was formulated for (%ΔVp) as a function of radius (%ΔVp®) as well as the differential expansion (%ΔVp[x]). The results obtained were cross verified in the following manner. Serial spheres of same radii as hypothetical GTVs were made using sphere generation tool in iPlan planning system (V4.5.1) (Brainlab AG, Germany). These were given serial expansions to cross-check the values provided in the excel sheets.
Table 1: Change in absolute volume for spheres of different radius for different expansions

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Table 2: Change in percentage volume (%ΔVp) for spheres of different radius for different expansions

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Hypothetical four-dimensional internal target volume generation

To incorporate the tumor motion and generate the ITV (ICRU-83), the GTV center was moved in one direction from 1 to 20 mm (1 mm increment). ITV volume was calculated using the following formula:

Where h is twice the tumor movement in one direction (i.e., total shift). The calculation geometry for equation 2 is schematically shown in [Figure 1]. Absolute (ΔAITV) and relative (Δ%ITV) change in ITV volume for spheres of radius from 1 to 5 cm were calculated for a 1-cm increment and noted in [Table 3]. The ΔAITV and Δ%ITV were calculated up to 20 mm movement only as this is the clinically relevant movement in most of the cases [Figure 1]. Further, a simple mathematical relationship was formulated between the Δ%ITV radius and the total shift.
Figure 1: Internal target volume for a moving target (gross tumour volume) having Radius r unit. The Internal target volume is the combination of two hemisphere and one cylinder all having radius equal to the gross tumor volume radius and cylinder height is twice the isocenter shift

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Table 3: Change in absolute volume (in cubic centimeter) and percentage volume change of hypothetical moving target of different radius for different range of movement

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 > Results Top

Static gross tumor volume/clinical target volume to planning target volume expansion

The values in the spread sheet using the mathematical formula and the ones generated by iplan were completely matching with each other. Absolute increase in volume as a function of radius expansions has been shown in [Table 1].

The change in absolute volume (AV) was most for 15 mm expansion and least for 1 mm expansion [Table 1]. However, the percentage increase for sphere size 2.5 cm or less was remarkable. For 1 cm sphere with a 15 mm expansion, the volume change was 14.6%; but, for a 10 cm sphere, the change in volume with a similar expansion was only 52%. For a sphere of radius 0.5 cm, the change in percentage volume was 63.0% as noted in [Figure 2] and [Table 2]. From [Figure 2], it is readily appreciable that percentage volume increase curve as a function of radius and as a function of expansion bends very sharply for spheres below 2.5 cm radius and distinctly different at radius 1 cm or below. %ΔVp curve as a function of expansion starts rising steeply for 1 cm and 2 cm radius sphere as compared to sphere of larger size. For an expansion of 15 mm, change in the AV for sphere of radius 0.5 cm is nearly doubled as compared to 1 cm radius. However, the percentage change is more than quadrupled. In the intermediate range, i.e., 5 cm radius, the change in the absolute and percentage volume, for an expansion of 1 mm to 15 is 555.9–1150.8 cc and 6%–120%, respectively. This indicates a doubling of AV and an increase in percent volume by a factor of 20. For sphere of 10 cm radius, the change in the percentage volume is 17 times with a corresponding increase in AV of 1.5 times as indicated in [Table 1] and [Table 2] and [Figure 2]. As the size of sphere increases, the differential change in AV reduces as a function of the increasing radius. [Figure 2] represent the relative variation of the % volume as a function of the radius vector; which gets saturated at near 2.5 cm. The curves indicated in [Figure 2] can be fitted with a mathematical formulation. If r is the generalized radius vector expressed in cm, %ΔVp is expressed as a function of expansion (%ΔVp = f[x]) in mm, and a function of radius (%ΔVp = f[x]) is expressed, respectively, as:
Figure 2: Percentage volume change (a) as a function of expansion in millimeter and (b) as a function of radius in cm

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It may be difficult to use Equation (3) and (4) readily in clinical work, and hence, we specify a simplified formula in appendix 1 and 2 for %ΔVp (x) and (%ΔVp®), respectively. The exact fitting functions for all the curves are given in appendix eI and eII.

Hypothetical 4D ITV result: For moving sphere of different radius, the increase in volume for various increments of movement has been shown in [Table 3] and [Figure 3]. For the moving target of the 1 cm radius, the change of percentage volume shows a high gradient against the displacement. However, the gradient saturates on increase of the radius. Similar effect can be observed in the absolute change of volume also [Figure 3]b; however, the manifestation is much less pronounced form than that of the percentage volume. For example, AV change of ITV for 2 cm movement for spheres of 1, 4, and 5 cm was 16.8 cc, 469.3 cc, and 838.1 cc, respectively. However, relative change is 300%, 100%, and 60%, respectively. This indicates the change in the absolute and relative volume is predominant in case of larger and smaller radius, respectively. The relative change in ITV can be expressed as a simple formula, if r is the radius of the sphere in cm and h is the total shift in cm then % change in ITV volume (%ΔVITV) is expressed as:
Figure 3: Percentage change (a) and absolute change (b) in volume of a hypothetical moving target of different radius for different range of movement

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As there is no fitting error for these curves due to its straight line nature, exact fitting parameters are not required in this case. For simplicity, a further set of equations is given in appendix eI.

 > Discussion Top

[Figure 2]a specifies the %ΔVp (x) as a function of expansion. The nature of the curves is parabolic and therefore can be fitted with a polynomial of degree two as given in equation B1-B11 in appendix. However, the generalized equation (Equation 3) is having a relatively complicated form. It can be noted that the radius of 0.5 cm could not be fitted in the general form of %ΔVp (x). In case of %ΔVp® [Figure 2]b, the curves do not fit to any common mathematical form. Therefore, the y axis was converted into a Log10 scale for plotting. There is a mathematical approximation (fitting error) in fitting these curves shown in [Figure 2]b. Mathematically, more accurate method would be using a Taylor series; however, in a clinical setting, this may give an unnecessary accuracy as well as nonusability of the mathematical formulations. In case of ITV, both relative and AV change is a straight line function [Figure 3]a and [Figure 3]b of shift and therefore can be represented by a simple mathematical formula as given Equation 5.

Our exhaustive literature search revealed that no article has been published till date to address this issue. It has been a constant effort of radiation oncology community to reduce the irradiation volume for obvious reason of minimizing the toxicity and maximizing the dose delivery. From era of 2D radiotherapy, we have moved to intensity-modulated radiotherapy (IMRT), stereotactic body radiotherapy (SBRT), stereotactic radiosurgery (SRS) tomotherapy, and volumetric-modulated arc therapy (VMAT).[18],[19],[20],[21],[22],[23],[24],[25],[26]

Constant technological and procedural updates are coming to reduce the treatment volume, and the margins we are using. There are evidence-based guidelines for defining different target volumes that can be treated by precise techniques like SBRT/SRT.[27],[28],[29]

Knowledge of radiobiology from QUANTEC data emphasized the importance of irradiation volume and volume-dependent toxicities.[30]

Pronounced dose–volume effect for parallel structures makes it more important to evaluate the volume we radiated when they are the OAR. In the post-QUANTEC era of volumetric dosimetry, we found it appropriate to generate a simple relationship of volume change with target size for static as well as moving target. This simple numerical data presented in our study shall be a helpful guide to the radiation oncologist and medical physicist during the critical process of treatment planning. The obvious side effect of the volume expansion is the radiotherapy-related toxicity, for both early and late effects. The radiation related toxicities manifested in different form for different sites; such as pneumonitis in lung; nausea, vomiting, diarrhea in the pelvic; and abdominal radiotherapy. Nevertheless, the detail discussion of the volume expansion and its clinical/side effect correlation is beyond the scope of the present article.

The second part of our study deals with moving targets, and the resultant PTV “enveloped” SBRT has become a common practice in most of the modern radiotherapy centers. Planning for moving target is based on 4D imaging data or non-4D imaging dataset. In both cases, treatment volume always remains critical. This experimental data of our study can actually provide an idea of treatment volume radiation oncologist is planning while giving margins. Be it 4D data for internal GTV (iGTV)/ITV generation or simply generation of CTV and PTV, this simple correlation of volume increment with target size and target movement can play a very important role during planning. Radiation oncologist and physicists can do good amount of preplanning before the actual treatment planning is done. From the target size and target movement for moving target, the final treatment volume can be estimated to a certain extent with close approximation. These findings can be especially useful where expansions are critical like in SRS, Para spinal SBRT, and any clinical situation where tumor lies in close proximity to critical OAR. In 4D scan-based planning, these data will provide an idea of expanded volume priory, and radiation oncologist can estimate its potential implications and plan accordingly.

These experimental data can be used for clinical decision-making. For example, in lung SBRT, usually tumor up to 5 cm is treated and various motion management techniques are used for different size targets. Different expanded tumor volumes like internal target volume (ITV) of internal gross tumour volume (iGTV) were formed from a time averaged (4D) CT Scan. This study seamlessly emphasizes treating smaller size targets avoiding gating or tumor tracking, as the volume of irradiation in ITV-based planning will be much less than that of a larger size target. We emphasize the importance of volume expansion by demonstrating how to apply this method uniformly for both GTV to CTV and CTV to PTV expansions. Our study is not without some criticisms. There may also be some caveats to the findings of our study. In real life, tumors are rarely completely spherical. An irregularly shaped volume (having the same AV as a sphere) may not have the same resultant volume after 5 mm expansion as a sphere with 5 mm expansion. However, the gross serial trends after expansion even of this irregular volume would follow the trends demonstrated in our study. In this study, we have not formulated a mathematical equation for absolute value change for expansion as well as for moving target; this is because the AV change is readily quantifiable from the DVH parameters.

This study gives an idea of ITV volume for various movements for different size spheres in one direction only. Although target movement is not unidirectional but a complex 3D in nature, in actual clinical scenario, these observations can predict the final ITV volume with close approximation. These results can be interpolated to a good extent to appreciate the volume change for a target of similar volume for a particular range of movement. We have also not done the dosimetric analysis of the consequences of such expansions but such analyses would be subject to two major pitfalls: (1) A complete analysis of all possible sphere sizes would again be limited to regular shapes only and (2) dosimetry would be subject to techniques employed for plan generation (3D conformal radiotherapy, IMRT, and VMAT) and each technique would yield its own result. This issue would be the subject of a subsequent study.

This study provides an idea of volume change for various expansions for various size targets and the volume changes for moving target of various radius for different range of movements. Finally, it finds out a correlation of these volume changes with the changing target size and range of movements.

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Conflicts of interest

There are no conflicts of interest.

 > References Top

ICRU50. Prescribing, Recording and Reporting Photon Beam Therapy. Bethesda, MD: ICRU; 1993.  Back to cited text no. 1
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Jassal K, Munshi A, Sarkar B, Paul S, Sharma A, Mohanti BK, et al. Validation of an integrated patient positioning system: Exactrac and iViewGT on synergy platform. Int J Cancer Ther Oncol 2014;2:020212.  Back to cited text no. 5
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Paul S, Sarkar B, Ganesh T, Munshi A, Kumar R, Mohanti BK. EP-1641: PTV margin calculation and time dependency monitoring of intrafraction isocenter movement in lung SBRT by exactrac. Radiotherapy and Oncology 2014;111:S221.  Back to cited text no. 8
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Munshi A, Sarkar B, Roy S, Ganesh T, Mohanti BK. EP-1667: Dose fall off patterns and the OAR effect-experience of Linac based frameless radiosurgery. Radiother Oncol 2016;119:S778-9.  Back to cited text no. 23
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Goswami J, Patra NB, Sarkar B, Basu A, Pal S. Dosimetric comparison between conventional and conformal radiotherapy for carcinoma cervix: Are we treating the right volumes? South Asian J Cancer 2013;2:128-31.  Back to cited text no. 25
[PUBMED]  [Full text]  
Sarkar B, Munshi A, Manikandan A, Roy S, Ganesh T, Mohanti BK, et al. A low gradient junction technique of craniospinal irradiation using volumetric-modulated arc therapy and its advantages over the conventional therapy. Cancer Radiother 2018;22:62-72.  Back to cited text no. 26
Sarkar B, Goswami J, Basu A, Sriramprasath S. Dosimetric comparison of three dimensional conformal radiotherapy and intensity modulated radiotherapy in high grade gliomas. Polish J Med Phys Eng 2011;17:75-84.  Back to cited text no. 27
Sarkar B, Pradhan A, Munshi A. Do technological advances in linear accelerators improve dosimetric outcomes in stereotaxy? A head-on comparison of seven linear accelerators using volumetric modulated arc therapy-based stereotactic planning. Ind J Cancer 2016;53:166.  Back to cited text no. 28
Munshi A, Sarkar B, Anbazhagan S, Giri UK, Kaur H, Jassal K, et al. Short tangential arcs in VMAT based breast and chest wall radiotherapy lead to conformity of the breast dose with lesser cardiac and lung doses: a prospective study of breast conservation and mastectomy patients. Australasian physical & engineering sciences in medicine. 2017 Sep 1;40(3):729-36.  Back to cited text no. 29
Marks LB, Yorke ED, Jackson A, Ten Haken RK, Constine LS, Eisbruch A, et al. Use of normal tissue complication probability models in the clinic. Int J Radiat Oncol Biol Phys 2010;76:S10-9.  Back to cited text no. 30


  [Figure 1], [Figure 2], [Figure 3]

  [Table 1], [Table 2], [Table 3]


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