Exploration of tradeoffs in intensity-modulated radiotherapy

Exploration of tradeoffs in intensity-modulated

radiotherapy

David Craft, Tarek Halabi, and Thomas Bortfeld

Department of Radiation Oncology, Massachusetts General Hospital and HarvardMedical School, Boston, MA 02114, USA

E-mail: [email protected]

Abstract. The purpose of this study is to calculate Pareto surfaces in multicriteriaradiation treatment planning, and to analyse the dependency of the Pareto surfaces onthe objective functions used for the volumes of interest. We develop a linear approachthat allows us to calculate truly Pareto-optimal treatment plans, and we apply it toexplore the tradeoff between tumor dose homogeneity and critical structure sparing.We show that for two phantom and two clinical cases, a smooth (as opposed to kinked)Pareto tradeoff curve exists. We find that in the paraspinal cases the Pareto surfaceis invariant to the response function used on the spinal cord: whether the mean corddose or the maximum cord dose is used, the Pareto plan database is similar. This isnot true for the lung studies, where the choice of objective function on the healthylung tissue influences the resulting Pareto surface greatly. We conclude that in thespecial case when the tumor wraps around the organ at risk, e.g. prostate cases andparaspinal cases, the Pareto surface will be largely invariant to the objective functionused to model the organ at risk.

Submitted to: Phys. Med. Biol.

Exploration of tradeoffs in intensity-modulated radiotherapy 2

1. Introduction

Radiotherapy treatment planning almost always requires tradeoffs to be made between

delivering high doses of radiation to the tumor target volume and sparing the

surrounding healthy structures. Recently the concept of multi-objective Pareto

optimization has been introduced into treatment planning, especially for intensity-

modulated radiotherapy, to control the tradeoffs more directly and simultaneously

[5, 9, 24]. Given one or more objective functions for each volume of interest (VOI),

a Pareto optimal treatment plan is one for which there does not exist another plan that

is strictly better in at least one objective function while being no worse in every other

VOI objective. Given a pre-calculated database of Pareto optimal plans, the treatment

planner can interactively navigate through these plans and understand in realtime the

tradeoffs involved in the particular case [9].

With this new capability, it is now of interest to explore the shape of the tradeoff

curves (”Pareto fronts”) for various theoretical and clinical cases. Similar analyses have

been done with single-objective IMRT planning systems [8], which, however, did not

necessarily produce Pareto optimal plans. One question to be asked is about the general

shape of the tradeoff curves: Are they generally smooth such that a gradual change in

one objective (say, the dose in one critical structure) will cause gradual changes in the

other objectives? Or are there ”kinks” in the curves, such that a small change in one

objective will suddenly have a large impact on some other planning objective(s)?

Another question of interest is the sensitivity of the Pareto optimal plans to the

choice of the objective functions for the various VOIs. Specifically, will the tradeoff

curves look completely different if we use different objective functions? The reason

to ask this question is that no general agreement exists with respect to the choice of

the ”right” objective function and both biological and dose-based objectives have been

used [18]. In the case of biological objective functions, the underlying parameters are

uncertain (cf. [14]). To give an example, in the case of the rectum, the Emami tables [6]

do not suggest a volume effect of the dose response, whereas the Burman fit [4] states

a volume exponent [11] of n = 0.12, which corresponds to an EUD model parameter

a = 1/0.12 = 8.3. Indeed, Skwarchuk et al. [21] provide some evidence that the rectum

may be sensitive to large volumes receiving medium to low dose values (see also [10]).

Against this background it is a relief that Romeijn [20] have shown that Pareto

fronts are insensitive to a certain type of change of the objective function: If one function

can be represented as an increasing function of another function, then the two functions

are identical from the point of view of Pareto front computation. Specifically, normal

tissue complication probability (NTCP) can, under certain approximations, be cast as

an increasing function of equivalent uniform dose (EUD), hence NTCP and EUD lead

to the same Pareto fronts.

In this paper we take this analysis one step further. We explore the sensitivity of

Pareto fronts to more substantial changes of the objectives. We focus our exploration

on linear objectives because of the simplicity of linear models and the fact that they

Exploration of tradeoffs in intensity-modulated radiotherapy 3

guarantee optimality of solutions. Specifically, we explore the sensitivity to switching

objectives between two extreme dose response models of normal tissues: maximum dose

and mean dose. The former is relevant for serially organized normal tissues, and the

latter for parallel structures such as lung [22]. We also include an intermediate dose

response model with a combination of maximum dose and mean dose response. If

Pareto fronts are similar for these extremes, we argue that they should be similar for

other cases as well.

2. Materials and methods

The equivalent uniform dose (EUD) [13] function translates a heterogeneous dose

distribution on a strucure into a single value representing the uniform dose that would

have the same biological effect. In its generalized form [15] this nonlinear function is

given by EUD=(

1|V |∑

i∈V dai

)1/a, where |V | is the number of voxels in the structure, di

is dose delivered to voxel i, and a is an organ-specific parameter related to the serial

(a → ∞) versus parallel (a = 1) structure of the organ being modeled. Thieke et

al. [23] presented an alternate version of Niemerko’s nonlinear EUD, which we refer

to as the linear EUD or the αEUD, which covers serial to parallel organs using the

equation: αEUD=α ·maxi∈V di + (1− α) · 1|V |∑

i∈V di. With the αEUD as our indicator

for organ response to a non-uniform dose, we use the following linear formulation for

the multi-criteria optimization (MCO) problem:

minimize{EUDj}, j = 1..N

d = Dx,

di <= Mj, ∀i ∈ Vj, j = 1..N

di >= mj, ∀i ∈ Vj, j = 1..N

EUDj = αj ·maxi∈V (j)di + (1− αj) ·1

|Vj|∑i∈Vj

di, j = 1..N

EUDj ≤ Uj

x >= 0. (1)

d is a vector of doses di in each voxel i, D is the dose deposit matrix, and x is the

vector of beamlet intensities, to be determined. Mj and mj are maximum and minimum

voxel dose values for structure j (hard constraints), respectively, and Vj are the voxels of

structure j. N is the number of objective functions. The ’max’ term in the computation

of the EUDs, while not linear as is, can be rewritten with auxilliary variables and

constraints using standard techniques, see for example [1]. The EUD values may also

have hard upper bounds Uj, thus restricting the domain of Pareto optimal solutions,

and keeping all solutions clinically relevant. The min/mean version of the αEUD for

tumors can be included in Program 1 (with maximization in the objective), or tumors

can be included in the formulation as is using a hard minimum on the voxels and α = 1

to reduce the maximum voxel dose, thus achieving tumor dose homogeneity, as we do

Exploration of tradeoffs in intensity-modulated radiotherapy 4

in this paper.

The objective, minimize {EUDj}, j = 1..N is shorthand for ’find all Pareto optimal

solutions using EUDj, j = 1..N as the objective functions’. We calculate a discrete set

of Pareto optimal solutions to Problem 1 using both the standard approach of altering

weights in a weighted sum objective scalarization approach, and the normalized normal

constraint (NC) method [12]. The NC method is graphically depicted in Figure 1 for a

two dimensional Pareto surface (this method generalizes to higher dimensions as well as

nonlinear formulations, see [12]). The method begins by individually optimizing the two

objective functions F1(x) and F2(x). In objective space, these two solutions are termed

anchor points, and denoted by Y1 and Y2. The line connecting Y1 and Y2 is populated

by an even spread of points. At each of these points Y , an optimization minimizing

F2 is performed with the constraint which makes the shaded region in Figure 1 feasible

and the rest infeasible. This constraint is given by N ′V ≤ 0 (scalar product) where V

is the vector from Y to the trial point (F1, F2), and N is a vector from Y1 to Y2. The

guarantee of an even spread of points comes at a cost of increased computuational time,

which in practice has turned out to be significant, and may decrease the appeal of the

algorithm for very large scale IMRT instances.

In this paper we consider mainly two dimensional tradeoffs: tumor dose

homogeneity versus OAR sparing. We consider sensitivity of the Pareto surfaces to

the OAR objective function by computing Pareto surfaces for different values of α.

When comparing the Pareto surface produced with the ’maximum dose’ (α = 1) OAR

objective with that produced by the ’mean dose’ (α = 0) objective, we compute the mean

dose of the ’maximum dose’ plans, and plot both surfaces in the mean dose objective

space. We also investigate the use of a max/mean objective with α = 0.5, and plot it

in the mean dose space as well.

We apply the optimization formulation to two phantom geometries and two actual

clinical cases, a paraspinal tumor (spinal cord is the OAR) and a lung tumor (the healthy

lung tissue surrounding the tumor is the OAR). The phantoms are based on the RTOG

IMRT benchmark phantom (www.rtog.org), which consists of a horseshoe tumor and

circular OAR. To mimic a paraspinal case, we use the RTOG geometry as specified. For

the lung phantom, we swap the OAR and the tumor of the RTOG phantom and resize

the geometry to make it more fitting for a lung case. In each phantom case, a minimum

dose of 0.95 Gy is enforced for the tumor, and the maximum dose is kept below 1.2

Gy. See Figure 2 for beam configurations, voxel sizes, and organ geometry. For the

dose-deposit pencil beams for the phantom cases we use a double Gaussian fit equation

based on [7]:

d(z, x) = e−µz 1

2

[w1 erf

(x + W/2

b1

)+ w2 erf

(x + W/2

b2

)

+ w1 erf

(−x + W/2

b1

)+ w2 erf

(−x + W/2

b2

)], (2)

where W = 0.5 cm is the width of the beamlet, z is the axial distance, x is the transversal

distance, µ = 0.04 cm−1, w1 = 0.8, w2 = 0.2, b1 = 0.35 cm, and b2 = 1.7 cm. For

Exploration of tradeoffs in intensity-modulated radiotherapy 5

the clinical cases, we use the IMRT planning package KonRad [3, 19, 17, 16, 2] that

was originally developed at the German Cancer Research Center (DKFZ) for the dose

calculation.

The entire source code (MCLP-IMRT, multicriteria linear programming for

IMRT) is available on request of the authors as open source software, including

the underlying linear programming solver GNU Linear Programming Kit (GLPK,

www.gnu.org/software/glpk/glpk.html). The linear programming model, written in

the AMPL (www.ampl.com) modeling language, is compatible with GLPK as well as

commercial solvers such as MOSEK (www.mosek.com) and CPLEX (www.cplex.com),

which are needed for large cases.

3. Results

The results for the paraspinal phantom and the lung phantom are shown in Figures 3

and 4 respectively. The paraspinal phantom, with the OAR surrounded by the horseshoe

tumor, displays a large invariance to the choice of α (the organ parameter of the OAR)

whereas the lung phantom, with the large OAR and a small embedded tumor is very

sensitive to α. The leftmost portion of the curves in each figure represent points of

high tumor dose homogeneity, and consequently the plans that put the most dose to

the OAR. For the extreme leftmost point, the tumor anchor point, the OAR objective

is ignored in the optimization, which is why the Pareto curves are consistently closer to

each other on the left portions of the graphs.

For the paraspinal phantom case, Figure 3, each of the three OAR anchor point

solutions push the tumor maximum dose to its absolute maximum level of 1.2, but for

the lung phantom case, Figure 4, the only OAR anchor solution that pushes the tumor to

its most heterogeneous dose distribution is the max optimized OAR case: pushing down

the OAR maximum dose is a harder problem than pushing down the mean dose. The

maximum dose occurs near the tumor boundary and is highly correlated with the tumor

minimum dose. The only way to bring down the OAR maximum dose while keeping

the same tumor minimum dose is to increase the local dose gradient, which, in turn,

increases the maximum tumor dose. The fact that reducing the maximum OAR dose in

the lung phantom is a ’more difficult’ problem than minimizing the mean dose is further

evidenced by the two iso-dose plots in Figure 4: iso-dose number 1 plan effectively uses

only two out of five beams to cover the OAR while iso-dose number 2, with maximum

dose as the OAR objective function, uses all 5 beams in a more complicated manner,

in order to spread out the dose along the OAR boundary. The Pareto plan databases

are highly dependent on α for the lung phantom case and widely different strategies are

employed for the mean and the max objectives.

These results (insensitivity to α in the paraspinal phantom but not in the lung

phantom) are paralleled in the clinical cases, shown in Figures 6–9. We do not expect

the results to be as clean in the clinical cases since the clinical cases are not as idealized

as the phantoms. Nevertheless, we see the same trend: for the paraspinal case in Figure

Exploration of tradeoffs in intensity-modulated radiotherapy 6

6 the mean OAR dose for the ’max optimized’ plans is typically 50% higher compared

to the mean OAR dose for the ’mean optimized’ plans. However, for the lung case the

mean doses for the two Pareto sets differ by a factor of 3. Thus we have support for

our main result: when the tumor wraps around a smaller OAR we expect invariance of

the Pareto surface to the choice of OAR organ response parameter α. Iso-doses for the

paraspinal and lung case iso-center slices are shown in Figures 8 and 10 respectively.

Although in the paraspinal case, the invariance appears to be much less than the

phantom paraspinal case, this is only true across the full range of α ∈ [0, 1]. Across

a restricted range, approximately α ∈ [0.2, 0.8], there is a much larger degree of

invariance, as suggested by the proximity of the α = 0.5 curve to the mean Pareto

curve. This is explained by the three dimensional Pareto surface, Figure 7, where the

OAR objective function is broken up into its separate mean and max components, i.e.

mean and max are considered as two separate objectives. This figure shows a kink in the

tradeoff between the maximum and the mean OAR dose at each tumor homogeneity

level. Due to the sharpness of the kink, the kink solution has a mean value close to

the optimal mean value and also a maximum value close to the optimal maximum

value, and this is clearly a smart choice for planning. Such a sharp kink is also helpful

since any α near 0.5 will yield this kink solution, as shown by the red line in Figure

7. To see this, note that the two dimensional Pareto surface corresponding to the

combined objective α(max dose) + (1 − α)(mean dose) for a particular value of α can

be reconstructed from the three dimensional Pareto surface in Figure 7. The Pareto

optimal pairs (α(max dose) + (1− α)(mean dose), tumor max) in the two dimensional

problem correspond to points (max dose, mean dose, tumor max) on the Pareto surface

of the three dimensional problem that satisfy

∂max dose

∂mean dose= −1− α

α, (3)

i.e. points where the three dimensional Pareto surface has slope of −(1 − α)/α in the

OAR max/mean plane . Therefore, for a given α, to find the Pareto optimal plan for

the two dimensional tradeoff from the three dimensional surface, one draws a line with

slope −(1 − α)/α tangent to the tumor iso-level of interest. The intersection is at the

two dimensional Pareto optimal plan. Using this visualization of the optimization, one

can see that a wide range of α values will all pick out a kink if one exists.

In the lung cases, no such kink exists. Figure 5 shows the OAR max/mean tradeoff

for the lung phantom. The large tradeoff that exists between these two objectives,

apparent by the gap in the Pareto curves in Figure 4, is seen here to be smooth. This

figure reveals that values of α near 0.5 will yield a relatively flat two dimensional tradeoff

between OAR EUD and tumor maximum dose, i.e. decreasing the tumor maximum dose

level will not increase the OAR EUD greatly, since the tumor iso-dose levels are close

together for α ≈ 0.5.

Exploration of tradeoffs in intensity-modulated radiotherapy 7

4. Discussion

Multi-criteria IMRT optimization facilitates understanding of the tradeoffs in tumor

coverage and OAR sparing in radiation treatment planning. Access to a Pareto optimal

surface allows treatment planners and physicians to evaluate in real-time how changes

in the dose to some structure impacts the doses to other structures. In this paper we

present a method to calculate Pareto surfaces for IMRT planning and investigate the

sensitivity of the resulting database of Pareto plans to the choice of OAR response

function.

A persistent difficulty which affects both scalar and multi-objective optimization is

the lack of scientific agreement over the appropriate objective functions to use, further

complicated by the fact that it is unlikely that a single value can capture the effects of a

heterogeneous dose distribution on a single OAR or target volume. MCO partly resolves

this issue by allowing multiple objective functions on a single organ without having to

combine them with artifical weights, but the choice of the objective functions is still

problematic. However, MCO offers another advantage regarding the uncertainties over

the best objective functions to select.

Any two functions which are positively correlated are the same in terms of the

generation of the Pareto surface‡. In a previous work, [20], such correlation of two

functions was discussed in the context of mathematical correlation, and included as the

main examples the correlation between EUD and NTCP and TCP. In this current work,

we investigate IMRT patient/beam geometries which produce correlations in objective

functions that are not in general mathematically correlated. The particular example

we use is the linear EUD, α · (max dose) + (1 − α) · (mean dose), for different values

of α ∈ [0, 1], which we employ due to its simplicity and because linear programs are

solvable to optimality, thus ensuring our plans are exactly Pareto optimal. It may be

more appropriate, especially in a multi-criteria setting, to have maximum dose and mean

dose as separate objective functions, but combining them into a single EUD reduces

the dimensionality of the Pareto surface, which is an important consideration for large

dimensional Pareto surfaces.

If the maximum and mean doses of a structure are correlated, then the choice of

α is irrelevant from the point of view of MCO. In this paper, we show that when the

tumor surrounds a relatively small OAR, as in the paraspinal phantom and the actual

paraspinal case, minimizing the OAR maximum is similiar to the problem of minimizing

the OAR mean. In order to reduce the mean of the OAR, one must bring the dose in

the voxels adjacent to the tumor down as much as possible, i.e. one must reduce the

maximum dose. Due to its size and location in the tumor, one does not have much

separate control over the mean and max doses in the OAR (in the limiting case of a

structure consisting of a single voxel, maximum and mean dose are of course equal).

‡ This is not true in the case of single objective optimization when multiple objectives are scalarizedby a weighted sum. In this case, substituting one objective function for another positively correlatedone is the same as altering the weight of that function, which obviously changes the final result.

Exploration of tradeoffs in intensity-modulated radiotherapy 8

Thus, a large common boundary between the tumor and the OAR, large relative to the

size of the OAR, is sufficient for a strong correlation between the maximum and the

mean doses of the OAR. With a large OAR, a coupling between the maximum dose

and the mean dose need not exist, as shown in the lung studies, which reveal a large

difference in plans when the max dose is minimized versus the mean dose. Fortunately

however, large organs like the lung are often treated as parallel structures, meaning

that a single voxel representing the mean dose to the lung can be used in a linear

programming setting.

In the clinical paraspinal case, we do not find as strong a correlation between the

mean and the maximum dose when we optimize, as indicated by the larger gap in the

Pareto curves of Figure 6. This is most likely due to the fact that there is a larger gap

between the tumor and the OAR as compared with the phantom study which means the

maximum dose to the OAR is not occuring at a ring of voxels on the common interface,

but is more localized. The small tradeoff that does exist between the maximum and

the mean dose is explored in Figure 7, and displays a kink in this tradeoff. In words,

such a kink means that, for a fixed tumor homogeneity level, one can reduce the max

dose to the OAR without impacting the mean dose greatly, until a threshold where

further reduction in the OAR max causes the OAR to be flooded with radiation. More

importantly, the existence of a kink in this curve means that the original 2D Pareto

surface, while not insensitive to the full range of α, is insensitive to α values that find

the kink of the curve, approximately α ∈ [0.2, 0.8].

5. Conclusion

We have presented a method to compute the Pareto surface exactly for the general IMRT

inverse planning problem using a linear formulation. Pareto surfaces allow treatment

planners and physicians to understand the tradeoffs for individual patient plans. We

apply the model to four cases, two phantoms and two real data sets, and for each analysis

we produce three pareto curves by varying the objective function on the OAR: mean

dose, maximum dose, and the average of the mean and maximum dose.

For the lung phantom and the real lung case, we find that the choice of the

objective function on the OAR is very influential on the resulting Pareto database of

plans: minimizing the mean lung dose is a much different problem than minimizing the

maximum lung dose. In contrast, for the paraspinal cases, the choice of the OAR

parameter α is less important: whether you minimize the mean OAR dose or the

maximum OAR dose, it is necessary to reduce the dose to the the ring of voxels of the

OAR adjacent to the tumor which are at the maximum dose level. The smallness of the

OAR, the large common boundary it has with the tumor, and the physical limitations

of the dose deposit mechanism (i.e. the inability to create arbitrary dose distributions)

create an invariance to the choice of α in the paraspinal cases.

Exploration of tradeoffs in intensity-modulated radiotherapy 9

Acknowledgments

We are grateful to Drs. Karl-Heinz Kufer, Michael Monz, and Alexander Scherrer from

Fraunhofer ITWM, Kaiserslautern, and Jan Unkelbach and Monika Uhrig from DKFZ,

Heidelberg, for helpful discussions of this work, and for their hospitality during the

optimization workshop in Kaiserslautern, Germany. This work was supported by NCI

Grant 1 R01 CA103904-01A1: Multi-criteria IMRT Optimization.

[1] D. Bertsimas and J. N. Tsitsiklis. Introduction to linear optimization. Athena Scientific, 1997.[2] T. Bortfeld, J. Burkelbach, R. Boesecke, and W. Schlegel. Methods of image reconstruction from

projections applied to conformation radiotherapy. Physics in Medicine and Biology, 35:1423–1434, 1990.

[3] T. Bortfeld, J. Stein, and K. Preiser. Clinically relevant intensity modulation optimization usingphysical criteria. In D.D. Leavitt and G. Starkschall, editors, XIIth International Conferenceon the Use of Computers in Radiation Therapy, pages 1–4, Madison, 1997. Medical PhysicsPublishing.

[4] C. Burman, G. J. Kutcher, B. Emami, and M. Goitein. Fitting of normal tissue tolerance data toan analytic function. Int J Radiat Oncol Biol Phys, 21(1):123–35, 1991. Journal Article.

[5] C. Cotrutz, M. Lahanas, C. Kappas, and D. Baltas. A multiobjective gradient-based doseoptimization algorithm for external beam conformal radiotherapy. Phys Med Biol, 46(8):2161–75, 2001. Journal Article.

[6] B. Emami, J. Lyman, A. Brown, L. Coia, M. Goitein, J. E. Munzenrider, B. Shank, L. J. Solin,and M. Wesson. Tolerance of normal tissue to therapeutic irradiation. Int J Radiat Oncol BiolPhys, 21(1):109–22, 1991. Journal Article Review Review, Tutorial.

[7] M. Engelsman. How much margin reduction is possible through gating or breath hold? Physicsin Medicine and Biology, 50:477–490, 2005.

[8] M.A. Hunt, C.Y. Hsiung, S.V. Spirou, C.S. Chui, H.I. Amols, and C.C. Ling. Evaluation of concavedose distributions created using an inverse planning system. Int. J. Radiation Oncology Biol.Phys., 54(3):953–62, 2002.

[9] K-H. Kufer, H.W. Hamacher, and T.R. Bortfeld. A multicriteria optimization approach for inverseradiotherapy planning. In T.R. Bortfeld and W. Schlegel, editors, Proceedings of the XIIIthICCR, Heidelberg 2000, pages 26–29. Springer, 2000.

[10] A.J. Lomax, M. Goitein, and J. Adams. Intensity modulation in radiotherapy: photons versusprotons in the paranasal sinus. Radiotherapy and Oncology, 66:11–18, 2003.

[11] J.T Lyman. Complication probability as assessed from dose-volume histograms. RadiationResearch, 104:S13–S19, 1985.

[12] A. Messac, A. Ismail-Yahaya, and C. A. Mattson. The normalized normal constraint methodfor generating the pareto frontier. Journal of the International Society of Structural andMultidisciplinary Optimization (ISSMO), 25:86–98, 2003.

[13] A. Niemierko. Reporting and analyzing dose distributions: a concept of equivalent uniform dose.Med Phys, 24(1):103–10, 1997. Journal Article.

[14] A. Niemierko. Comparing rival dose distribution using TCP predictions: sensitivity analysis.Proceedings of the 40th AAPM Annual Meeting. Medical Physics, San Antonio, TX, 1998.

[15] A. Niemierko. A generalized concept of equivalent uniform dose. Med. Phys., 26:1100, 1999.[16] S. Nill. Development and application of a multi-modality inverse treatment planning system. PhD

thesis, University Heidelberg, 2001.[17] U. Oelfke and T. Bortfeld. Inverse planning for photon and proton beams. Med. Dosim., 26:113–

124, 2001.[18] J.R. Palta and T.R. Mackie, editors. Intensity-modulated radiation therapy: The state of the art.

Medical Physics Publishing, Madison, WI, 2003.[19] K. Preiser, T. Bortfeld, K. Hartwig, W. Schlegel, and J. Stein. A new program for inverse

Exploration of tradeoffs in intensity-modulated radiotherapy 10

radiotherapy planning. In D.D. Leavitt and G. Starkschall, editors, XIIth InternationalConference on the Use of Computers in Radiation Therapy, pages 425–428, Madison, 1997.Medical Physics Publishing.

[20] H.E. Romeijn, J.F. Dempsey, and J.G. Li. A unifying framework for multi-criteria fluence mapoptimization models. Physics in Medicine and Biology, 49:1991–2013, 2004.

[21] M.W. Skwarchuk, A. Jackson, M.J. Zelefsky, and E.S. Venkatraman. Late rectal toxicity afterconformal radiotherapy of prostate cancer. Int. J. Radiation Oncology Biol. Phys., 47:103–113,2000.

[22] L. S. Stefan, J.C. Kwa, M. Theuws, A. Wagenaar, M.F. Eugene, L.J. Damen, P.B. Boersma, S.H.Muller, and J.V. Lebesque. Evaluation of two dosevolume histogram reduction models for theprediction of radiation pneumonitis. Radiotherapy and Oncology, 48:61–69, 1998.

[23] C. Thieke, T.R. Bortfeld, and K-H. Kufer. Characterization of dose distributions through the maxand mean dose concept. Acta Oncologica, 41:158–161, 2002.

[24] Y. Yu. Multiobjective decision theory for computational optimization in radiation therapy. MedPhys, 24(9):1445–54, 1997. Journal Article.

Exploration of tradeoffs in intensity-modulated radiotherapy 11

FIGURES

F1

YN

Z*

V

F2

Y2

Y1

Figure 1. Normal constraint method. Anchor points Y1 and Y2 are calculated byoptimizing F1 and F2 individually. The line connecting Y1 and Y2 is populated byan even spread of points. At each of these points Y , an optimization minimizing F2

is performed with the constraint which makes the shaded region feasible and the restinfeasible. This constraint is given by N ′V ≤ 0 where V is the vector from Y to thetrial point (F1, F2), and N is a vector from Y1 to Y2. The optimal solution, Z∗, is apoint on the Pareto surface.

Exploration of tradeoffs in intensity-modulated radiotherapy 12

−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

x (cm)

y (c

m)

Tumor

OAR R

1

R3 = 3.8 cm

R2

R1 = 1 cm

R2 = 1.3 cm

IMRT information: 5 equispaced gantry angles,symmetric, beamlet size = 0.5 cm voxel size = 0.2 cm x 0.2 cm

Phantom radius = 8 cm

Paraspinal phantom

Figure 2. RTOG benchmark geometry used as our paraspinal phantom, values asshown. When used as the lung tumor phantom, we switch the OAR and the tumor,reduce the field size of the beams to cover only the tumor, and use R1 = 1.5, R2 = 1.95and R3 = 5.7 cm.

0.95 1 1.05 1.1 1.15 1.2 1.250.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tumor max dose (relative dose)

oar m

ean

dose

(rel

ativ

e do

se)

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

x (cm)

y (c

m)

0.2

0.2

0.2

0.4

0.4 0.4

0.6

0.6

0.6 0.60.6

0.8 0.8

0.95

0.95

1.051.05

1.05

1.05

1.05 1.

05

1.05 1.05

1.05

mean optimizedα=0.5 optimizedmax optimized

Figure 3. Pareto surfaces for paraspinal RTOG phantom geometry. Tradeoffis between tumor homogeneity, measured as the maximum tumor dose, and OARmaximum dose, mean dose, or the average between them. The minimum tumor doseis fixed at 95% prescription level. The red X indicates the plan for which the iso-doselevels are shown.

Exploration of tradeoffs in intensity-modulated radiotherapy 13

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Figure 4. Pareto surfaces for lung phantom geometry. Clearly the choice of the OARresponse model influences the resulting Pareto plan database (left). Iso-dose number1 (middle) is a mean OAR dose optimized plan (corresponding to the red X in the leftfigure), and here one can see clearly that the tumor is covered by the two lower beams.In contrast, a much different plan is used when the maximum dose is used as the OARindicator function (right). In this case, iso-dose 2 shows, as expected, that all 5 beamsare used in order to reduce the maximum dose to the lung. This plan however resultsin a high mean dose to the lung, as shown by the Pareto curves.

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tumor max=1.025tumor max=1.05tumor max=1.2

Figure 5. Three dimensional Pareto surface for the lung phantom case (Figure 4),obtained by splitting the OAR EUD objective into its separate components, mean doseand maximum dose. At each tumor maximum level, a smooth tradeoff exists betweenthe OAR maximum dose and the OAR mean dose.

Exploration of tradeoffs in intensity-modulated radiotherapy 14

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ctv+gtv boostcordR.KidneyL.Kidney

Figure 6. Paraspinal case. The red X indicates the plan for which the DVH is shown.A minimum dose of 45 Gy is used for the tumor.

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tumor max=55tumor max=60tumor max=70tumor max=80

Figure 7. Paraspinal case. Viewing the mean and the maximum OAR doses asseparate objectives, we construct a three dimensional Pareto surface. The max meantradeoff of the OAR displays a kink, which implies that a range of α values near 0.5will yield this kink solution in the original two dimensional problem. The red line plotsconstant values of the max/mean EUD for α = 0.5 (the slope is −(1− α)/α = −1, asdiscussed in the text).

Exploration of tradeoffs in intensity-modulated radiotherapy 15

Figure 8. Paraspinal case, iso-dose plot for plan depicted by red X in Figure 6.

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tumorR.LungL.Lungcord

mean optimizedα=0.5 optimizedmax optimized

Figure 9. Lung case. The red X indicates the plan for which the DVH is shown. Aminimum dose of 97% is used for the tumor.

Exploration of tradeoffs in intensity-modulated radiotherapy 16

Figure 10. Lung case, iso-dose plot for plan depicted by red X in Figure 9. Theclinical target volume is outlined by the green contour and the planning target volumeby the yellow contour. The CTV homogeneity was used for the tumor objective.