Int. J. Radiation Oncology Biol. Phys., Vol. 52, No. 1, pp. 224 –235, 2002 Copyright © 2002 Elsevier Science Inc. Printed in the USA. All rights reserved 0360-3016/02/$–see front matter


PII S0360-3016(01)02585-8




*Department of Radiation Oncology, Medical College of Virginia, Virginia Commonwealth University and McGuire Veterans Affairs Hospital, Richmond, VA; †Department of Radiation Oncology, Massachusetts General Hospital and Harvard University, Boston, MA Purpose: The equivalent uniform dose (EUD) for tumors is defined as the biologically equivalent dose that, if given uniformly, will lead to the same cell kill in the tumor volume as the actual nonuniform dose distribution. Recently, a new formulation of EUD was introduced that applies to normal tissues as well. EUD can be a useful end point in evaluating treatment plans with nonuniform dose distributions for three-dimensional conformal radiotherapy and intensity-modulated radiotherapy. In this study, we introduce an objective function based on the EUD and investigate the feasibility and usefulness of using it for intensity-modulated radiotherapy optimization. Methods and Materials: We applied the EUD-based optimization to obtain intensity-modulated radiotherapy plans for prostate and head-and-neck cancer patients and compared them with the corresponding plans optimized with dose–volume-based criteria. Results: We found that, for the same or better target coverage, EUD-based optimization is capable of improving the sparing of critical structures beyond the specified requirements. We also found that, in the absence of constraints on the maximal target dose, the target dose distributions are more inhomogeneous, with significant hot spots within the target volume. This is an obvious consequence of unrestricted maximization target cell kill and, although this may be considered beneficial for some cases, it is generally not desirable. To minimize the magnitude of hot spots, we applied dose inhomogeneity constraints to the target by treating it as a “virtual” normal structure as well. This led to much-improved target dose homogeneity, with a small, but expected, degradation in normal structure sparing. We also found that, in principle, the dose–volume objective function may be able to arrive at similar optimum dose distributions by using multiple dose–volume constraints for each anatomic structure and with considerably greater trial-and-error to adjust a large number of objective function parameters. Conclusion: The general inference drawn from our investigation is that the EUD-based objective function has the advantages that it needs only a small number of parameters and allows exploration of a much larger universe of solutions, making it easier for the optimization system to balance competing requirements in search of a better solution. © 2002 Elsevier Science Inc. Intensity-modulated radiotherapy, IMRT, Optimization, Biologic models, Equivalent uniform dose.

INTRODUCTION The purpose of this paper was to propose an objective function based on the equivalent uniform dose (EUD) for optimization of intensity-modulated radiotherapy (IMRT) plans and to present the results of an investigation based on its application. The concept of EUD for tumors was introduced by Niemierko (1) originally as the biologically equivalent dose that, if given uniformly, would lead to the same cell kill in the tumor volume as the actual nonuniform dose distribution. Later, Niemierko extended the EUD concept to apply to normal tissues as well (2). Presently, most IMRT optimization systems use dose-

and/or dose–volume-based objective functions (3–5). Neither adequately represents the nonlinear response of tumors or normal structures to dose, especially for arbitrary inhomogeneous dose distributions. For instance, if a single voxel or a small number of voxels in a tumor receive a very low dose, it would not have a significant effect on the IMRT plan score. (The numeric value of the objective function is called the “score” of the treatment plan. In this report, the terms “score,” “optimization criteria,” and “objective function” are used interchangeably.) However, the tumor control probability would be greatly diminished as a result of the cold spot. Stated in a different way, for dose- or dose–

Reprint requests to: Qiuwen Wu, Ph.D., Department of Radiation Oncology, Medical College of Virginia, Virginia Commonwealth University, 401 College Street, Richmond, VA 23298. Tel: 904-828-9461; Fax: 804-828-6042; E-mail: [email protected]

Supported by Grants CA 74043 and CA 84430 from the National Cancer Institute. Received Aug 23, 2000, and in revised form Feb 21, 2001. Accepted for publication Feb 26, 2001. 224

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volume-based objective functions, the penalty imposed for the failure to achieve the prescribed dose is proportional to the dose difference (or the square of the difference), rather than to the loss of tumor control, as would be more appropriate. Dose–volume constraints are simplified surrogates of the underlying biologic effects determining the outcome of treatment. Specifying a single dose–volume constraint is equivalent to stating that, if the volume above the tolerance dose is smaller than the critical volume, no complications will occur. This is a reduced subset and a special case of the critical volume dose–response model (6, 7), in which a functional subunit is destroyed at exactly the tolerance dose and the response occurs when exactly the critical fractional number of the functional subunits is destroyed. (In other words, there is no distribution of responses and the slopes of the dose–response and volume response are infinite.) One could argue that this inadequacy can be overcome by specifying the constraints on the entire dose–volume histogram (DVH) for the anatomic structure. However, there are multiple DVHs (in fact, an infinite number of them) that could lead to an equivalent dose response for a particular organ, but optimization based on each of these DVHs would, in general, lead to different dose responses in other organs and the tumor. Only one of these DVHs will be optimum so far as other organs and the tumor are concerned. Thus, constraining the search to a single DVH for an anatomic structure may miss the overall optimum solution. That multiple DVHs correspond to the same dose response is an important advantage for dose–response-based objective functions. Mathematically speaking, one can state that dose–response functions are highly “degenerate” functions of dose–volume combinations and, therefore, of dose distributions. A dose–response index (e.g., tumor control probability [TCP], normal tissue complication probability [NTCP], EUD, or P⫹, the probability of uncomplicated control) may be considered as a way of summarizing multiple DVHs into a single value similar to the way a DVH summarizes a three-dimensional dose distribution into a single curve. Dose–volume-based objective functions are also degenerate functions of dose distributions (i.e., multiplicity of dose distributions correspond to the same dose– volume constraint), but to a considerably lesser extent. The high degeneracy of dose–response functions makes a large space of biologically equivalent dose distributions for each organ equally acceptable, thus giving greater flexibility to the optimization process to reconcile competing requirements to find a better solution. The limitations of dose- or dose–volume-based criteria have led a number of investigators to propose models for predicting biologic and dose–response indices (6 –10), and consider using them as an alternative or in conjunction with dose- and dose–volume-based criteria. Only a few, rather limited, studies on the use of dose–response-based objective functions for treatment plan optimization have been reported (10, 11). There is considerable controversy about the models for computing dose–response indices and their use

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in optimization. Current dose–response data are sparse and unreliable, and the models for computing dose–response indices are simplistic. It should be remembered, however, that dose and dose–volume indices are even more simplistic measures of the quality of the optimized treatment plan. There is substantial uncertainty in the current dose and dose–volume constraints also. Most of them are specifications based on rather limited clinical experience (12). Beyond the controversies about dose–response data and models, one should be aware that the use of dose–response indices for optimization might also pose some problems. For instance, dose–response-based optimization may lead to very inhomogeneous target dose distributions. Furthermore, it is difficult for clinicians to specify the optimization criteria in terms of certain dose response indices (e.g., TCP, NTCP, and P⫹). This difficulty becomes even more significant when two or more independently optimized plans are to be combined. It is impractical to specify the desired TCP and NTCP of component plans. Such issues regarding dose– response indices may have been obstacles in their clinical use for optimization; however, many of them can be overcome with EUD-based optimization. For example, for all practical purposes, EUD is in dose domain, which makes it easier for the clinician to specify the requirements for plan optimization. Although we have pointed out the limitations of dose– volume-based objectives and the potential advantages of dose–response-based objectives, we must reemphasize that the former are the primary bases for most, if not all, IMRT optimization currently. They have proved quite successful in achieving satisfactory plans. EUD-based optimization is a competing alternative that needs to be considered and that may prove useful as a result of investigations reported in this paper and others to follow. METHODS AND MATERIALS Generalized EUD formalism The original definition of EUD was derived on the basis of mechanistic formulation using a linear– quadratic cell survival model. Recently, Niemierko suggested the phenomenologic form

冉冘 冊

1 EUD ⫽ N



a i

1 a


for both tumors and normal tissues. In this expression, N is the number of voxels in the anatomic structure of interest, Di is the dose in the i’th voxel, and a is the tumor or normal tissue-specific parameter that describes the dose–volume effect. This formulation of EUD is based on the power law dependence of the response of a complex biologic system to a stimulus. This type of relationship has been observed in many biologic phenomena since the mid-19th century and has also been incorporated into the EUD concept. EUD represented by Expression 1 is the “generalized


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mean” of the nonuniform dose distribution (see Abramowitz and Stegun’s “Handbook of Mathematical Functions” for a definition and the properties of the generalized mean) (13). For a ⫽ ⬁, the EUD is equal to the maximal dose, and for a ⫽ ⫺⬁, the EUD is equal to the minimal dose. For a ⫽ 1, the EUD is equal to the arithmetic mean, and for a ⫽ 0, it is equal to the geometric mean. The EUD mimics dose– response reality more closely than the simple dose–volume parameters. The EUD retains the original meaning for tumors; for normal tissues, it represents the uniform dose, which leads to the same probability of injury as the corresponding inhomogeneous dose distribution. Parameter a may be determined empirically by fitting with published dose–volume data (e.g., those published by Emami et al. (12) or with analogous data used in an individual institutional practice. As an alternative, if no dose–volume data are available, or if the available data are not reliable, a can be treated as a freely adjustable parameter whose value is determined by trial-and-error to achieve the best dose distributions. Figure 1b shows the EUD value as a function of parameter a for an arbitrary DVH for an inhomogeneous dose distribution (Fig. 1a). One notices that for large negative values of a, the EUD is close to the minimal dose, a characteristic consistent with the dose response of highly inhomogeneously irradiated tumors. Thus, parameter a is likely to attain large negative values for tumors. Similarly, for a large positive value of a, the EUD tends to be near the maximal dose, a characteristic of nonuniformly irradiated serial architecture normal tissues such as the spinal cord. There is evidence that, for normal tissues that exhibit a large-volume effect and may be assumed to follow the parallel architecture model (e.g., liver, parotids, and lungs), the dose response may be more closely related to the mean dose (14 –16). For these tissues, the parameter a would be small and positive (⬃1). (Note, that it is not possible to constrain the mean dose when using dose- or dose–volumebased objective functions for plan optimization.) Parameter a can be related analytically to the partial organ dose–volume data. Under partial organ irradiation, the fractional volume V receives a dose D(V) and the remainder receives no dose. For such a case, Expression 1 reduces to EUD ⫽ V 共1/a兲D共V兲


This expression has the same form as the power law model used by Lyman and Wolbarst (17) (and others) to represent the volume effect. The power law model was also used by Kutcher et al. (18, 19) to reduce a DVH into Veff and by Mohan et al. (20) to reduce a nonuniform dose distribution into an effective uniform dose Deff. In fact, the EUD formula (Expression 2) above has exactly the same form as the Deff formula of Mohan et al. Parameter a and the Lyman model parameter n are related by a ⫽ 1/n. Although there is underlying similarity between Veff and EUD in that they

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Fig. 1. (a) Sample DVH used for EUD calculation. (b) EUD for the DVH in (a) as a function of parameter a. Tumors generally have large negative values of a, whereas critical element normal structures (e.g., spinal cord and rectum) have large positive values, and critical volume normal structures that exhibit a large-volume effect (e.g., lungs and parotids) have small positive values.

both use the power law of response, some differences exist. Veff is defined in the volume domain, and EUD is defined in the dose domain. Veff was introduced as a DVH reduction scheme, and EUD represents a generalized mean of nonuniform dose distribution. Most importantly, in contrast with Veff, EUD applies to both tumors and normal structures. EUD may be used to calculate the TCP and NTCP also. Then, in addition to determining the values of parameter a from partial organ irradiation data, they can also be determined directly by comparing the calculated TCPs and NTCPs based on EUD formalism with corresponding ob-

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easily differentiable, attributes that are important for IMRT optimization. Other advantages of the EUD were cited above. These include that the same formalism applies to both tumors and normal structures; EUD mimics the biologic response to dose more closely than do dose– volume relationships; and the higher degeneracy of EUD enables EUD-based optimization to explore a large solution space compared with dose–volume-based objective functions. Although EUD is a hybrid between the physical dose and biologic response, for practical purposes, it is in the dose domain, which allows specification of its desired value in terms of Gray. EUD lends itself to an easy formation of an objective function suitable for optimization. In the work presented here, we used the objective function Fig. 2. Plots of EUD vs. target dose inhomogeneity. The volume of the target is divided into two parts. Each part is irradiated uniformly to different doses. Part 1 receives 65 Gy, part 2 receives doses in the range of 50 – 80 Gy. EUD plots for part 2 ⫽ 0%, 10%, 50%, 90%, and 100% of the target volume are shown. The parameter a was chosen to be ⫺10. Plots indicate that EUD saturates (flattens out) asymptotically when the dose to volume of inhomogeneity (i.e., hot spot, part 2) increases beyond the dose to the rest of the volume (part 1). For instance when 10% of the volume receives 75 Gy, the EUD is approximately 65.5 Gy.


写f, j



where the component subscore fj may be either fT ⫽

1 EUD 0 1⫹ EUD

冉 冊



for tumors, or served values, or, alternatively, with those calculated with other models. A property of EUD of importance in its use in the optimization of plans is illustrated in Fig. 2. Figure 2 shows the variation of EUD as a function of target dose inhomogeneity. In the simple example shown, the target volume is divided into two subvolumes V1 and V2. V1 receives 65 Gy uniformly and V2 receives differing doses. Plots indicate that EUD saturates asymptotically when the dose to V2 is increased ⬎65 Gy. For instance, when 10% of the volume receives 75 Gy, the equivalent uniform dose is approximately 65.5 Gy. The interpretation of this saturation effect for tumors is obvious if we remember that EUD is a proxy for the corresponding TCP. That is, a boost to a relatively small subvolume of the tumor is ineffective regardless of the magnitude of the boost dose. Note that the plot is not symmetrical. The EUD drops quickly even if only a small part of the tumor is significantly underdosed. For example, when 10% of the volume receive 55 Gy, the EUD is 62.5 Gy. These characteristics mean that, when EUD is used as a measure of quality of the target dose distribution, (1) the presence of a hot spot will have an insignificant advantage, but (2) underdosing will reduce the quality of a plan significantly. As discussed in the “Results” section, these characteristics are able to explain some of the observed consequences of EUD-based optimization. Objective function based on EUD An advantage of the EUD defined in this manner is its simplicity. It has only one adjustable parameter and is

f OAR ⫽

1 EUD 1⫹ EUD 0

冉 冊



for normal tissues (organs at risk). Formulas 4 and 5 are a simple way of parameterization of the assumed sigmoidal shape of the f function in terms of EUD using a logistic function. A different function (e.g., the error function) that exhibits similar sigmoid behavior could also have been used. We chose the logistic form for its suitability for differentiation, its simplicity, and because it is a popular function among investigators working with dose– response data. For target volumes, EUD0 is the desired dose parameter. For normal structures, it is the maximal tolerable uniform dose that may be D5, D50 (the uniform doses that lead to 5% or 50% complication probability, respectively), or another value, depending on the organ at risk. Parameter n is akin to the weight or penalty that indicates the importance of the structure-specific end point. Figure 3 illustrate the variation of subscores for tumors and normal tissues, respectively, as a function of the EUD and penalty values. A value of 50 Gy for EUD0 was used for these plots. The values of subscores fj vary from 0 to 1. It is evident that for higher penalties, the rise (or drop) in the value of the subscore is steeper when the EUD deviates from EUD0. For a tumor, the subscore attains a small value when the equivalent uniform dose falls significantly below EUD0. Similarly, for a normal structure


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Optimization based on F can be shown to lead to the same results as those based on F⬘, because ⭸F⬘ ⭸F ⫽F 䡠 ⭸w j ⭸w j


The steepest descent optimization method requires the first derivatives, which can easily be calculated as follows: ⭸F⬘ ⫽ ⭸w j

⭸F⬘ ⭸EUD ⭸D 䡠 䡠 冘 ⭸EUD ⭸D ⭸w i





where D a⫺1 ⭸EUD i ⫽ EUD 䡠 ⭸D i D ai



and ⭸F⬘ ⭸EUD ⭸F⬘ ⭸EUD

⫽ T

n 䡠 fT 䡠 EUD

⫽ ⫺ OAR

冉 冊 冉 冊 EUD 0 EUD

n 䡠 f OAR 䡠 EUD



(10) n


Since D i ⫽ 冘j K ij 䡠 w j , where Di is the dose value at calculation point i, wj is the weight of ray j, and Kij is the dose contribution of ray j to point i per unit weight, ⭸D i ⫽ K ij ⭸w j

Fig. 3. Typical objective function components for (a) target and (b) organs at risk for various penalties.

the subscore becomes small when the equivalent uniform dose exceeds the tolerance limit. In contrast with the quadratic dose-difference form typically employed in dose- and dose–volume-based objective functions, where the goal of optimization is to minimize the objective function (5), for objective function represented by Equations 3, 4, and 5, the goal of optimization is to maximize the objective function. We used a gradient (steepest descent) method for optimization as described below: Because 0 ⬍ F ⱕ 1, we can optimize F directly or its logarithm

F⬘ ⫽ ln F ⫽

冘ln f j




The EUD formalism developed in this section was applied to optimize prostate and head-and-neck IMRT plans and evaluate the merits of EUD-based objective functions relative to dose–volume-based objective functions. RESULTS EUD-based optimization for prostate IMRT plans We first applied EUD-based IMRT optimization to a prostate case and compared the results with dose–volumebased optimization. Details of the dose–volume objective function used are described by Wu and Mohan (5). The number of dose–volume constraints and their parameters were adjusted by trial-and-error to obtain an optimal plan. Figure 4 shows the dose distributions for IMRT plans optimized using dose–volume-based and EUD-based objective functions. The organs at risk were the rectum and bladder. All plans used identical configurations of five coplanar 18-MV photon beams placed at equi-spaced gantry angles. The plans were normalized to deliver the prescription dose of 70 Gy to 99% of the target volume. Figure 4a is the

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Fig. 4. Sagittal isodose distributions for prostate IMRT plans designed using (a) dose–volume-based criteria, (b) EUD-based criteria, and (c) EUD-based criteria with target inhomogeneity constraints.

IMRT plan based on dose–volume objectives, and Fig. 4b is the plan obtained using EUD-based objectives. The corresponding DVHs are shown in Fig. 5. The EUD objective function parameters used are listed in Table 1. It is clear that, for the same minimal target dose, sparing of organs at risk is greatly improved in the EUD-based plan. Furthermore, a sharp dose gradient at the interface between the target and organs at risk is produced. This is evidenced by the bunching of isodose lines in these regions. The maximal target dose also increased significantly. Such a result is expected for reasons explained in “Methods and Materials” and is evident from the form of the EUD vs. dose inhomogeneity curve, which flattens out asymptotically when dose to a fractional volume of the target is increased above the prescription dose (Fig. 2). This feature of inhomogeneously irradiated target volumes makes the objective function insensitive to hot spots within the target volume

and, as a consequence, unable to effectively distinguish between high and moderate degrees of target dose inhomogeneities. Although hot spots in the target may be considered beneficial in some clinical situations, they are generally not desirable. We used the following approach to limit the target dose inhomogeneity. We represented the target both as a target (with EUD0 value equal to the prescription dose) and as an embedded “virtual” normal tissue to be protected from high doses (with an EUD0 value greater than prescription dose specified as the normal tissue constraint). This is a logical approach, because the primary reason we need to impose an upper limit to the dose within the target volume is to limit the damage to the embedded normal tissues. Presently, no data are available that can be used to determine the EUD0 for embedded normal tissues. At the same time, it should be noted that the acceptable or desired level of target volume inhomogeneity is subjective and arbitrary. Therefore, in the work presented here, we used EUD0 for embedded normal tissues as an adjustable parameter whose value is determined by trial-and-error to achieve the desired level of dose homogeneity. Figure 6 shows the typical form of the objective function when the target is treated both as a target and as a virtual normal structure. Figure 4c shows the dose distribution, and Fig. 7 shows the DVH for the plan with this form of dose Table 1. EUD-based optimization parameters for prostate target, bladder, and rectum

a† EUD0 (Gy) n† Fig. 5. Dose–volume histograms for the prostate plans of Fig. 4. Target, rectum, and bladder are shown. Solid lines indicate dose– volume-based criteria and dashed lines EUD-based criteria with no constraint on target dose inhomogeneity.





⫺10.0 72 20

10.0 76 20

6.0 35 6

6.0 35 6

Abbreviation: EUD ⫽ equivalent uniform dose. * Contains parameters for the target treated as virtual normal tissue to limit dose inhomogeneity. † Unitless.


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Fig. 6. Illustrative objective function for the target when it is also assumed to be an embedded “virtual” normal tissue to constrain hot spots. The objective function parameters for the illustrative example shown are EUD0 ⫽ 50 Gy, n ⫽ 20 for the subscore component representing the volume as a tumor, and EUD0 ⫽ 70 Gy, n ⫽ 10 for the subscore component representing the volume as an embedded normal tissue.

homogeneity constraints. This experiment resulted in a significant reduction in the target dose inhomogeneity. The rectum and bladder dose distributions were degraded somewhat but were still significantly better compared with the dose–volume-based plan. EUD-based optimization of head-and-neck IMRT plans Figure 8 compares the dose distributions for a head-andneck IMRT plan optimized using dose–volume-based and EUD-based objective functions. The organs at risk were the parotids, spinal cord, and brainstem. All plans used identical configurations of nine coplanar 6-MV photon beams placed at equi-spaced gantry angles. Plans were normalized to deliver the prescription dose of 68 Gy to 98% of the target

Fig. 7. Comparison of DVHs of prostate plans with EUD basedoptimization and with (solid lines) and without (dashed lines) dose inhomogeneity constraints on the target.

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volume. The detailed plan design strategy has been described elsewhere (21). Figure 8a is the IMRT plan based on the dose–volume objectives, and Fig. 8b is the plan obtained using the EUDbased objectives with constraint on target dose inhomogeneity. Figure 8c shows the EUD-based plan with no constraint on target dose inhomogeneity. The corresponding DVHs are shown in Fig. 9. The objective function parameters used for the EUD-based optimization are listed in Table 2. Figure 10 compares the DVHs for the EUD-based plans with and without the inhomogeneity constraints on the target. The computed target EUD values (1) for the dose distributions shown in Fig. 8 were 71.8 Gy for the dose– volume-based plan, 73.2 Gy for the EUD-based plan with target dose inhomogeneity constraint, and 77.7 Gy for the EUD-based plan with no target dose inhomogeneity constraint. As in the case of the prostate plans, the use of the EUD-based objective function resulted in a significant reduction in the normal tissue dose and a higher dose to the target and about the same dose to the nodes. As before, the use of the EUD-based objective function without a constraint on the high dose in the target resulted in target dose distributions with a high degree of inhomogeneity. When the target volume was assumed to be a virtual normal structure as well, the target dose inhomogeneity was reduced considerably with a modest degradation in normal tissue sparing. It should be pointed out that similar dose inhomogeneity was observed when TCP and NTCP-based objective functions were used, as has been reported by Wang et al. (11). Thus, it would seem that one-component dose–responsebased objective functions may not, by themselves, be able to control dose inhomogeneity in the target volume, and that the two-component dose–response objective functions are effective in producing results consistent with clinical experience and practice. To test if our results and conclusions were valid generally, we applied similar techniques to a group of other head-and-neck IMRT patients with diverse anatomic geometries and compared the plans based on dose–volume-based objective functions with those based on EUD. The EUD plans showed a significant, although variable, degree of improvement in the sparing of the spinal cord, larynx, and parotids. Figure 11 summarize the dose–volume information for these patients. A question that can be raised is whether it is possible to achieve the same IMRT solution with dose–volume-based optimization as with EUD-based optimization. The answer is yes, in principle. However, it will require a priori knowledge of the solution achievable with EUD-based optimization. One could then specify it in terms of a number of dose–volume constraints for each anatomic structure and use the optimization process to determine the intensity distributions required to deliver the same dose distributions as would be possible with EUD-based optimization. The difficulty obviously is that it is not possible to know a priori

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Fig. 8. Dose distributions in a transverse image section through a head-and-neck tumor for (a) dose–volume-based objectives, (b) EUD-based objectives with dose homogeneity constraint, and (c) EUD-based objectives without dose homogeneity constraint.

the optimal solution achievable with EUD-based optimization. Without such knowledge, it would be necessary to resort to considerable trial-and-error effort to obtain a dose– volume-based solution that is the same as or similar to the EUD-based solution. On the other hand, if EUD-based optimization consistently achieves plans with similar DVHs for the same class of problems, one could use the general features of such DVHs to modify requirements for dose–volume-based optimization. Although the results may not be exactly the same, they are likely to be a good approximation of those achievable with EUD-based optimization. As an illustration, we applied this strategy for the head-and-neck example presented above. The use of EUD-based optimization led to a reduction in the spinal cord dose significantly. Using this as a clue, we repeated the dose–volume-based optimization with a more stringent constraint on the cord while maintaining the same dose–volume limits to all other structures. The results are shown in Fig. 12. The cord dose was reduced with practically no change in the doses to other structures. DISCUSSION One of the reasons that the EUD-based optimization leads to greater normal tissue sparing is the following. In dose- or dose–volume-based optimization, it is typical to require the target dose to be within a specified range of values. The optimization system assesses a penalty to the plan score for each voxel in which the dose is outside this range. Similarly, for an organ at risk, if the dose–volume values are within the tolerance limits, no penalty is assessed. Therefore, the optimization process will not attempt to improve the plan once the specified criteria have been met. In contrast, EUD-based optimization will continue to improve the plan even beyond

the required limits. This is also evident from a mathematical perspective from the examination of Eqs. 8 through 11. Partial derivatives are always positive for the target (Eq. 10) and negative for the organ at risk (Eq. 11), irrespective of the constraint value specified. This means that the optimization process continues to increase the target subscore and decrease the normal tissue subscore until they balance each other out. In contrast, for dose- and dose–volume-based objectives, the derivatives of subscores for each structure may change sign during optimization or become 0 when the specified criteria are met (5). This means that the dose distribution optimization for that structure will stop, even though a better solution may exist. In principle, optimization based on dose or dose–volume criteria may also be allowed to continue to improve the plan after the required constraint is met. For instance, an additional lower dose- or dose–volume constraint term may be incorporated into the objective function with a smaller penalty factor. However, it would require considerable trialand-error to achieve equivalent results. Another reason for the greater normal tissue sparing may lie in the higher degeneracy of EUD-based objective functions compared with dose–volume-based objective functions. The availability of a larger space of solutions allows greater flexibility in finding a solution with lower score. Although EUD-based optimization may lead to lower than required normal tissue sparing, one could question the significance of such an achievement. For some organs, the spinal cord for example, the reduction in dose to levels well below the tolerance dose may not be considered necessary. However, if it can be accomplished without compromising the target dose, there may be advantages in doing so. For example, such a reduction may provide a “tolerance reserve,” which may be important for dose escalation and in


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Fig. 9. DVHs for the target volume, electively treated nodes, and various critical structures corresponding to dose distributions shown in Fig. 8a and b. Solid and dashed lines indicate dose– volume-based criteria and EUD-based criteria with dose inhomogeneity constraint, respectively.

the event of repeated treatment. In addition, for some other organs, such as the parotids, there is evidence of damage (14) even at doses lower than previously considered acceptable (12). The probability and the rate of recovery from

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Fig. 10. DVHs for various structures corresponding to dose distributions shown in Fig. 8b and c. Solid lines and dashed lines indicate EUD-based optimization without target dose inhomogeneity constraint and with constraint, respectively.

xerostomia are functions of volumes receiving high doses, as well as the magnitude of the dose. In general, it may be desirable to reduce the dose to nontarget tissues to as low a level as feasible. We should point out that although in this paper we examined the potential of EUD-based optimization relative

Table 2. EUD-based optimization parameters for head-and-neck target, and normal structures brainstem, cord, parotids, and larynx

a† EUD0 (Gy) n†







⫺8.0 71 10

4.6 80 6

4.6 49 6

7.4 43 5

5.0 30 5

7.4 45 5

Abbreviation: EUD ⫽ equivalent uniform dose. * Contains parameters for the target treated as virtual normal tissue to limit dose inhomogeneity. † Unitless.

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Fig. 11. Comparison of DVH for three additional head-and-neck patients. Solid lines indicate IMRT plans based on dose–volume criteria; dashed lines indicate plans based on EUD-based criteria with dose inhomogeneity constraints. (a, b) Patient 2, (c, d) Patient 3, and (e, f) Patient 4. ETV ⫽ electively treated uninvolved nodes; CTV ⫽ clinical target volume, which includes microscopic disease surrounding the gross target volume (GTV).

to dose–volume-based objectives, we judged the quality of the resulting plans using the traditional parameters (e.g., DVHs). One might be inclined to think that there is an inherent conflict in this process. We do not believe this is the case. A dose–response function is simply a superset of dose–volume functions. The latter is a more restrictive criteria for the evaluation of plans than the former. A plan judged to be acceptable by dose–volume criteria must also be acceptable according to dose–response criteria. The opposite may not necessarily be true. Unfortunately, there is very little experience with the use of EUD and other dose–response indices for plan evaluation. Thus, it is safe to assume that, for the foreseeable future, we will need to continue to use dose and dose–volume-based quantities for plan evaluation. Theoretically, a plan optimized using a particular objective function would be the best plan when evaluated using the same criteria. For example, a plan developed using dose–volume criteria would be the best plan if evaluated using the same dose–volume criteria. Similarly, a plan developed using the EUD-based criteria would the best plan if evaluated using EUD-based criteria. Because we have accepted the use of the DVH for plan evaluation, one could question the value of using EUD-based optimization and ask “Why should EUD-based optimization lead to plans that may be superior compared with plans designed using the dose–volume based objective functions, when DVHs are

used for the evaluation of the plans?” The main reason, as mentioned above, is the higher degeneracy, which makes it easier for the optimization system to reconcile competing requirements in search of the optimal solution. CONCLUSION We have proposed and implemented an IMRT plan optimization objective function based on the concept of EUD and demonstrated its feasibility for clinical use. The formalisms involved are elegant and simple compared with those for objective functions using other dose–response quantities such as TCP and NTCP. The simplicity of EUD is due in part to the fact that the same formalism is used for both tumors and normal tissues and in part that the desired EUD values can be easily related to conventionally required dose or dose–volume limits. The characteristics of EUD also make it easy to formulate objective functions for IMRT optimization and allow relatively easy representation of clinical goals by the user. EUD-based optimization requires only a limited number of parameters (a, EUD0, and relative weighting or penalty factor). Because a and EUD0 are organ specific and can be derived from observed dose–response data, this leaves the penalty factor (n) as the only objective function parameter to be adjusted for optimization. In contrast, dose–volume-based objectives may require specification of multiple constraints for


I. J. Radiation Oncology

● Biology ● Physics

Fig. 12. DVHs for the head-and-neck case reoptimized with more stringent dose–volume-based criteria. Solid lines indicate the original plan; dashed lines, the plan with a more stringent constraint on the cord.

each anatomic structure and may still not be able to represent its dose response adequately. We have shown that EUD-based optimization provides improved sparing of normal tissues for the same or higher target dose. We also found that EUD-based optimization, without constraints, is not able to limit dose homogeneity within the target volume. This is because EUD (and TCP) are insensitive to hot spots within the target volume. Therefore, to limit target dose inhomogeneity, the objective function for the target volumes must use a second term representing embedded normal tissue with an empirically determined value of desired EUD. It is evident from our studies that EUD-based optimization has the flexibility to explore a much wider space of solutions and is able to find solutions that may escape detection if doseor dose–volume-based optimization is used. Although the

Volume 52, Number 1, 2002

EUD-based objective function has an advantage in that it needs only a small number of parameters and allows the exploration of a large solution space, the very small number of parameters might also be a drawback. It means that the EUD-based objective function has less control over fine details of dose distributions. It is then conceivable that on occasion EUDbased optimized dose distribution in some of the anatomic structures may not be compatible with conventionally acceptable DVHs. Although this did not occur in the cases we have studied so far, if it were to happen, an effective alternative would be to use EUD-based optimized intensity distributions as a starting point and make fine adjustments based on dose– volume optimization. For such fine tuning, we would not use the same dose–volume constraints as we would have had we used dose–volume objectives from the beginning. Multiple dose–volume combinations, derived from the EUD-optimized DVHs, would be specified as constraints. For those structures for which further improvement is desired, the dose–volume constraints would be adjusted accordingly. In principle, the solution found with the EUD-based objective function can also be reached using the dose–volume-based objective function from the beginning. Unfortunately, we do not know a priori what that solution is and, as a consequence, we do not know what to require of the optimization system. Furthermore, achievement of the same or similar solution as the EUD-based solution will require specification of a large number of dose–volume constraints for each anatomic structure and considerably greater trial-and-error to adjust dose– volume parameters of each constraint. Our general conclusion based on the results of this study is that the main power of EUD-based optimization is its ability to find suitable solutions that may not otherwise be apparent. The results of our study are consistent with previous studies using biology-based objectives that indicated that dose–response objective functions have the potential for finding improved treatment plans. We plan to test the EUD-based optimization for other treatment sites and are in the process of implementing it for clinical use for prostate and head-and-neck cancer. Finally, we should mention that by using a gradient technique to optimize the IMRT plans, we have implicitly assumed that there are no multiple minima in the objective function. The potential impact of multiple local minima traps on the solution found is an unknown when using dose–response-based objectives. Local minima are known to exist in dose–volume-based objective functions (22). There is no doubt that they may also exist in dose–responsebased objective functions. It is also reasonable to assume that because of the higher degeneracy of dose–response functions, the number of local minima is also greatly increased. What is not clear is whether these multiple minima, presumably larger in number, are a hindrance or an advantage (relative to dose–volume-based objectives) in finding a satisfactory solution. Our preliminary tests did not reveal any evidence of the optimization process based on EUD getting trapped in a local optimum. This issue also needs further examination.

Optimization of IMRT plans

● Q. WU et al.


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deviates from EUD0. For a tumor, the subscore attains a. small value when the equivalent uniform dose falls sig- nificantly below EUD0. Similarly, for a normal ...

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