American Journal of Oper ations Research, 2011, 1, 160-171
doi:10.4236/ajor.2011.13018 Published Online September 2011 (
Copyright © 2011 SciRes. AJOR
IMRT Optimization with Both Fractionation and
Cumulative Constraints*
Delal Dink1, Mark Langer2, Seza Orcun3, Joseph Pekny1, Ronald Rardin4,
Gintaras Reklaitis1, Behlul Saka4
1School of Chemical Engineering, Purdue University, W. Lafayette, USA
2Department of Radi at i o n Oncology, Indiana University, Indianapol i s , USA
3e-Enterprise Center, Discovery Park, Purdue University, W. Lafayette, USA
4Department of Industrial Engineering, University of Arkansas, Fayetteville, USA
Received June 8, 2011; revised June 30, 2011; accepted July 23, 2011
Radiation therapy plans are optimized as a single treatment plan, but delivered over 30 - 50 treatment ses-
sions (known as fractions). This paper proposes a new mixed-integer linear programming model to simulta-
neously incorporate fractionation and cumulative constraints in Intensity Modulated Radiation Therapy
(IMRT) planning optimization used in cancer treatment. The method is compared against a standard practice
of posing only cumulative limits in the optimization. In a prostate case, incorporating both forms of limits
into planning converted an undeliverable plan obtained by considering only the cumulative limits into a de-
liverable one within 3% of the value obtained by ignoring the fraction size limits. A two-phase boosting
strategy is studied as well, where the first phase aims to radiate primary and secondary targets simultane-
ously, and the second phase aims to escalate the tumor dose. Using of the simultaneous strategy on both
phases, the dose difference between the primary and secondary targets was enhanced, with better sparing of
the rectum and bladder.
Keywords: IMRT, Mixed-Integer Linear Programming, Optimization, Cumulative Dose Constraints,
Fractionation, Two-Phase Planning, Uniform Fractionation
1. Introduction
The success of radiation treatment rests on satisfying
dose limits within critical or healthy structures. These
criteria are determined mostly after years of experience
and increasingly by formal clinical trials. Strictly, the
prescriptions should be stated both in terms of cumula-
tive doses as well as the dose received per treatment ses-
sion known as fraction [1,2]. Both cumulative dose and
per-fraction dose limits appear in the historical reports
and clinical trial protocols that form the experience upon
which modern treatments are based.
In practice, planning optimization is based on cumula-
tive dose limits alone. Numerous methods have been
proposed in the literature to generate radiation therapy
plans. Of these methods, optimization models using
mathematical programming formulations have been de-
veloped to determine the best beamlet intensities [3-11]
and the best aperture intensities [12,13]. Further methods
include randomized approaches, such as simulated an-
nealing [14-17] and genetic algorithms [18-20], and
non-linear gradient techniques [21-24].
This paper investigates a conventional approach of op-
timizing over only cumulative limits as the optimization
constraints alone and then dividing the plan into integral
number of fractions after the optimization in order to
obtain the daily delivery plan. We refer to this approach
as Cumulative First Method. Then we present a new Si-
multaneous Method approach including both cumulative
and fraction size limits simultaneously in the optimiza-
tion which guarantee a plan that can be equally fraction-
ated into a uniform course of the treatment. Both of the
approaches try to achieve the u niform fractionation plan,
where the same doses are delivered in every fraction.
*Supported in part by grants from the National Science Foundation
(ECS 020145, CMMI 0813896) and from the National Institutes o
Health (1R41 CA91688-01) and the Indiana 21st Century Fund grant
The ability of the proposed approach to generate a so-
lution that simultaneously satisfies the cumulative dose
constraints and the fraction size constraints is tested via a
test case, and the effect of not including the fraction sizes
on the obtained objective, the minimum tumor dose and
the normal structures is evaluated. By imposing both the
cumulative and the fraction size limits into the formula-
tion explicitly, the Simultaneous Method guarantees a
solution when any feasible one exists by searching the
feasible solution space with the fractionation criteria in-
In addition to the two uniform fractionation methods,
we also consider a two-phase approach familiar in con-
ventional practice which uses different uniform fractions
in two successive periods of treatment. The primary and
secondary targets are covered as uniformly as possible
during the first phase of the treatment using the fraction
sizes traditionally accepted for disease sterilization. Then,
a second phase is added with the main focus to escalate
the primary target. Relaxing the secondary target mini-
mum fraction size requirements partially in the second
phase of the treatment provides two potential advantages
to the two-phase planning approach over the uniform
fractionation: escalating the tumor dose or dropping the
normal tissue exposure, if not both. Here, these potential
advantages will be explored in Intensity Modulated Ra-
diation Therapy (IMRT) planning where highly non-
uniform beam patterns are possible by dividing the beam
faces into small beamlets or bixels.
There have been studies where a non-IMRT module is
accompanied by IMRT in the second phase for tumor
boosting [25-27]. Two-phase IMRT planning was also
tested where the coverage volume is changed from whole
pelvic irradiation to prostate only in the second phase
and it was concluded that when the whole pelvic area is
simultaneously irradiated with IMRT higher dose to pri-
mary target and lower dose to the subclinical disease can
be achieved [28]. Simultaneous optimization of the two-
phases for IMRT was addressed in [29] by using a gra-
dient search algorithm with cumulative dose constraints
Here, we show the advantages of adapting our Simul-
taneous Method to two-phase planning. A two-phase
IMRT plan by irradiating the whole pelvic area is ob-
tained with one-step optimization, where the irradiated
volume is kept the same but the dose constraints for the
two-phases are different and the dose difference between
the primary disease and the secondary target is enhanced.
The cumulative minimum dose limits on the secondary
targets are satisfied during the first phase of the treatment.
The goal of the second phase is solely to boost the tumor
dose given the irradiated volume remains as the whole
pelvic. In the second phase, it is no longer the interest of
the planner to satisfy the minimum fraction size re-
quirements of the secondary target since the potential
disease on those sites are already eliminated during the
first phase. Of course, the fraction sizes on the organs at
risk remain to be a concern in the second phase.
This paper is organized as follows. Section 2 develops
all the methods including Cumulative First Method, Si-
multaneous Method, and Two-Phase Planning with Si-
multaneous Method used to generate treatment plans in
this paper. The results from computed plans are pre-
sented in Section 3 on a prostate case. Finally, the con-
clusions and possible future research directions are given
in Section 4.
2. Methods and Materials
In the formulations, each structure is identified as a pri-
mary target, a secondary target, or an organ at risk. The
set of normal tissue indices is denoted by K, and the set
of indices for the secondary target is denoted by V. The
set of tissue points drawn from the primary target is de-
noted by T, that drawn from the k-th secondary target is
denoted as Sk, and that drawn from the k-th organ at risk
is denoted as Hk.
The solution approach in all cases employed a beam-
let-based optimization in which dose to any point could
be expressed as a linear combination of the individual
beamlet intensities. Thus, dose di is the dose received at
tissue point i in the system is defined as
gijg jgi
ax d
 
where xjg is the value of total intensity assigned to beam-
let j of angle g, and ijg are pre-computed dose coeffi-
cients at point i from beamlet j of beam g.
The chosen objective function in all approaches is to
maximize the average tumor dose, a metric which has
been found to be a predictor of tumor persistence after
radiotherapy. It is represented as in (2) where T
represents the total number of tumor points.
Maximize i
In addition, preliminary work showed that the average
tumor dose to be a computationally efficient surrogate
for the minimum tumor dose in the presence of a strong
tumor-dose homogeneity constraint. Both the value of
the objective function and the value of the minimum tu-
mor dose are evaluated in the results.
A homogeneity limit, defined as the ratio of the mini-
mum to the maximum dose received, is enforced in all
models to sustain a near uniform tumor dose profile.
Copyright © 2011 SciRes. AJOR
162 D. DINK ET AL.
min id
 (3) iT
Here dmin represents the minimum tumor dose value
and parameter 0
is the allowed ratio.
An alternative way of enforcing homogeneity in the
tumor instead of specifying a ratio parameter α between
the minimum and maximum tumor doses is to impose
strict lower and upper bounds tumor and tumor
u, in the
tumor, and replace relations (3) with (4) and (5).
i tumor
dl iT
i tumor
du iT
2.1. Cumulative First Method
This conventional approach is based on the assumption
that, if one can find a good plan to satisfy the cumulative
dose limits, an appropriate number of fractions to indi-
rectly satisfy the fraction size limits and to deliver the
plan exists. Constraint sets (6) and (7) ensure that the
maximum cumulative dose constraints for all healthy
tissue points and minimum cumulative dose constraints
for all secondary target points are satisfied. The super-
script “total” represents the cumulative dose limits in the
relations below. The limits tumor and tumor in the con-
straint sets (4) and (5) continue to represent the cumula-
tive lower and upper bounds for the tumor.
l u
du (6) ,
iHkK 
dl (7) ,
iSkV 
Sometimes, dose volume limits are specified to limit
the fractional portion p of a structure volume that can
exceed a threshold dose in a normal tissue. A layering
heuristic employed here to incorporate these limitations
in the above model is as follows: After ordering the op-
timization points of the tissues with partial volume limits
according to their distance to the tumor margin in 3D, the
upper dose limits are assigned to that fraction p of the
points that lie closest to the target boundary, and the
lower dose limits are assigned to (1 – p) fraction of those
points farthest from the target boundary [30].
Once a plan is developed based only on cumulative
dose limits, it still must be divided into a fixed number of
fractions which in today’s practice are taken to be uni-
form. Practical bounds on N can be developed for the
maximum and minimum number of allowed uniform
fractions. An upper bound N comes from the mini-
mum total doses achieved by tumor and secondary tar-
gets and dividing these numbers by the minimum al-
lowed daily dose that should be delivered to each. The
minimum of these ratios results in the upper bound for
number of fractions given below in (8). Similarly, the
lowest number N of allowed fractions is governed by the
need not to exceed the fraction size requirement in any
normal tissue (relation (9)).
min ,
iS kVk tumor
NN ll
iH kKk
NN u
The number of fractions N for a plan should satisfy
. If these fraction size requirements cannot
be satisfied for any integral N, then the plan produced
without consideration of the fraction size requirements is
infeasible. Furthermore, a plan for which no integer
number of fractions will satisfy bounds (8) and (9) on the
fraction size cannot be renormalized to satisfy the frac-
tion size requirements. Violation NN means from
(8) and (9) that
maxmin ,
dailydaily daily
iS kV
iH kKkk
 tumor
 
 
 
Thus, renormalizing the plan by scaling down the
beamlet intensities, xjg, and thus total doses i by factor d
, has only the effect of multiplying both sides of
(10) by the same 0
. The infeasibility would remain.
2.2. Simultaneous Method
Our proposed Simultaneous Method, unlike the earlier
approach, optimizes over both the cumulative and per
fraction dose limits at the same time, where the total
number of fractions are included explicitly [31]. The
complete problem was specified as a linear or mixed-
integer program, in which doses in each tissue must fall
within an allowed range over each treatment session, and
the cumulative dose distribution across each structure
must meet the specified dose, dose-volume and homo-
geneity limits.
In order to achieve this, objective function (2), homo-
geneity limits (3) or (4)-(5) combined with cumulative
constraints (6) and (7) are retained, and new constraints
(11)-(13) are introduced as follows:
dNu ,
 (11)
dNl (12) ,
iSkV 
i tumor
dNl (13) iT
Here, N represents the total integer number of treat-
ment sessions. Upper and the lower fraction doses and
limits for tumor, secondary targets, and healthy tissues
along with per-fraction doses are denoted by the super-
script “daily” assuming one fraction is delivered per day.
Copyright © 2011 SciRes. AJOR
Copyright © 2011 SciRes. AJOR
The effect of constraints (11)-(13) is to ensure that total
doses i accommodate N times the applicable per-frac-
tion limit on each tissue. Then a feasible fraction plan
can be obtained simply by dividing all doses (equiva-
lently intensities) by N. We choose to treat N as an inte-
ger variable and optimize the number of fractions.
2.3. Two-Phase Planning with Simultaneous
In the two-phase planning approach, there are different
plans and numbers of fractions for the two phases. We
extend our Simultaneous Method by adding subscripts
e = 1, 2 to distinguish doses, intensities, and the numbers
of fractions for the first and second periods respectively.
Table 1 details the mixed-integer program.
In Table 1, constraint (15) puts an upper bound on
cumulative healthy tissue dose, (16) enforces tumor ho-
mogeneity, (17) and (18) limit the maximum healthy tis-
sue dose per each fraction of the two phases, and (19)-
(20) set maximum tumor fraction size in each phase. Con-
straints (21) and (22) applies cumulative and per-fraction
constraints on secondary targets only in phase 1. It should
be noted that the only integer variables of the above
method are N1 and N2, which makes the mixed-integer
linear programming (MILP) formulation a relatively easy
one to handle. The method is tested with the same pros-
tate case and compared to uniform fractionation scenario.
3. Case Studies
CPLEX with branch and bound algorithm was employed
for the following test case. The points for optimization
were distributed throughout the contours, determined
randomly within a structure volume for computation ef-
ficiency for optimization rather than employing a uni-
form point set, yet highly concentrated especially within
the target and the critical structures of interest. [32-36].
For each structure, we presented numbers of points used
in sampling and computed the mean distances to the
nearest neighbor point. Further details are provided be-
low. The influence matrix was calculated by using the
standard radiation therapy software GRATIS [37].
3.1. Effect of Including Fraction Size
Post-Optimally with Cumulative First
The effect of Cumulative First Method, i.e. not including
the fraction size constraints in optimization is examined
for a case of carcinoma of prostate. The data set con-
sisted of 23 CT slices of which 5 are illustrated in Figure
1. The prostate case contained a primary target volume
identified as the prostate gland, a secondary target vol-
ume consisting of the seminal vesicles, and four healthy
tissue structures, namely the bladder, two femoral heads,
rectum and all other surrounding unspecified normal
tissue. Number of sample points used (the mean distance
to the nearest neighbor point) is 400 (0.44 cm) for blad-
der, 1000 (0.38 cm) for both femoral heads together,
1000 (0.28 cm) for rectum, 465 (1.4 cm) for the remain-
ing unspecified tissue, 786 (0.29 cm) for PTV1 (Planning
Target Volume 1), and 252 (0.24 cm) for the seminal
vesicles. The prescriptions for both cumulative and frac-
tion doses are given in Table 2.
Table 1. Two-phase planning with simultaneous model.
Cumulative Conditions
ie ik
 (15)
Phase 1 Conditions Phase 2 Conditions
du N (17),
iHkK 22
du N ,
 (18)
i tumor
dl N
iT (19)2
dl N
dl ,
iSkV (21)
dl , (22)
iSkV 
N1, N2 nonnegative integers (23)
164 D. DINK ET AL.
Table 2. Prostate case study presc r iption.
Site Volume
Cumulative Dose
Objective (Gy)
Objective (Gy)
Prostate Tumor PTV1 79
2 100%
Seminal Vesicles PTV2 56
2 100%
85 100%
Bladder Normal
72 100%
Femoral Heads (2) Normal 50
85 100%
Rectum Normal
Other unspecified normal tissue Normal 85 2 100%
Figure 1. Outlines of 5 of the 23 transverse slices of the
prostate case taken through the inferior portion of the
treatment volume, with optimization sample points shown
within structures of interest. The densest collection of
points is taken within and on the boundary of the prostate
target. Also shown are samplings from the two femurs, the
rectum, and the unspecified normal tissue within the exter-
nal surface.
The prescription includes a set of limits on the maxi-
mum doses that can be received in a structure as well as
partial volume limits placed on the bladder, rectum and
femoral heads. Those constraints are reduced to point-
by-point upper limits on those tissues by protecting the
points farthest from the tumor as explained in Subsection
2.1. Table 2 also includes upper or lower bounds on the
doses that can be given to structures of interest each frac-
tion. A lower bound on the dose per fraction to PTV1
and PTV2 (Planning Target Volumes 1 & 2), and an up-
per bound on the dose per fraction to any normal tissue
structure are set at 2 Gy. Tumor dose homogeneity is
required as in constraints (4) and (5) by a minimum dose
limit of 79 Gy and maximum dose limit of 84 Gy to
PTV1. In addition to the minimum cumulative dose limit
on secondary target, a maximum dose limit of 84 Gy is
also assigned to ensure that the surrounding structures of
the tumor remain below the maximum tumor dose. The
number of fields (beam angels) used in the plan is 9, with
the angles pre-specified.
A treatment plan was prepared first with Cumulative
First Method without regarding to the fraction size limits,
but only regarding to the cumulative dose limits, using
the optimization model described in Subsection 2.1. The
resulting dose-volume histograms (DVH) for the target,
bladder and rectum are shown in Figure 2. The DVHs
show the dose received by the volume percentage of the
critical structures. All of the extreme dose and partial
dose volume conditions expressed in Table 2 were satis-
fied over the points employed in the optimization rou-
Table 3 shows the values obtained for the extreme
doses delivered to target or normal tissues alongside the
fraction size and cumulative dose limits which apply to
them. The final column shows the number of fractions
(treatment sessions) that can be prescribed under each
separate condition for a normal tissue or target structure,
found by applying relations (8) and (9), and substituting
D05 and D95 values for the maximum and minimum
doses. Here, D05 and D95 denote the minimum dose
delivered to the hottest 5% and 95% of the structure,
Scanning Ta ble 3 shows that the produced plan meets
the required cumulative dose conditions, but a number of
fractions cannot be determined such that all the fraction
Copyright © 2011 SciRes. AJOR
size requirements are met. The unspecified tissue, blad-
der and rectum conditions require more fractions to be
delivered than the maximum allowed by the condition on
the secondary target (PTV2), and bladder and rectum
require more fractions than allowed by the primary target.
For example, the bladder requires a minimum of 42 frac-
tions to satisfy its fraction size goals with this plan, while
the PTV2 requires a maximum of 28 fractions. Renor-
malizing this plan to satisfy the fraction size require-
ments is not possible for any number of treatment ses-
sions as explained above.
Instead of trying to fractionate the plan after the opti-
mization, the fractionation was imposed as part of the
optimization problem for the same case using the Simul-
taneous Method of Subsection 2.2. The results are given
in the form of dose-volume histograms for target, bladder
and rectum in Figure 3. The total number of fractions, N,
is optimized and found that the plan obtained with both
fraction size and cumulative dose limits can be delivered
in 41 equal fractions.
Table 4 compares the relevant statistics on the ex-
treme doses received in targets and healthy tissues under
with and without fraction size limits. With either model,
both the extreme dose values and the partial volume
doses satisfied their required limits. As can be seen, the
minimum primary target dose in the model that satisfied
the fraction size limits was only 1.2% less (83.88 vs
82.91 Gy) than the infeasible solution found by removing
the fraction size requirement.
Comparing Figure 2 and Figure 3, there are no sig-
nificant differences in the rectum and target dose distri-
bution; however the bladder dose in the 60% - 20% vol-
ume region increases when fraction size limits are in-
cluded to the planning. Still, when the extreme doses are
considered, the drop in the bladder and rectal doses are
greater than the 0.26% drop in the primary target dose,
Table 3. Dose statistics of the prostate case obtained by optimizing with cumulative first method (Disregarding fraction con-
Structure Dose Statistics
Required Cumulative
Dose (Gy)
Achieved Cumulative
Dose (Gy)
Dose per Fraction
Limit (Gy)
Resulting Bound on
Number of Fractions
Target Min (D95)
83.88 2.0 41
PTV2 Min (D95)
56.09 2.0 28
Bladder Max (D05)
85.0 83.74 2.0 42
FemHead1 Max (D05)
72.0 48.50 2.0 25
FemHead2 Max (D05)
72.0 47.12 2.0 24
Rectum Max (D05)
85.0 82.99 2.0 42
Otherwise unspecified tissue Max (D05) 85.0 63.74 2.0 32
Table 4. Comparison of cumulative dose statistics of the prostate case given by cumulative first and simultaneous methods.
Structure Statistics Cumulative First Method (Gy) Simultaneous Method (Gy)
Max Dose (D05) 84.00 84.00
Min Dose (D95) 83.88 82.91
Mean Dose 83.97 83.75
PTV2 Min Dose (D95) 56.09 82.00
Bladder Max Dose (D05) 83.74 81.97
FemHead1 Max Dose (D05) 48.50 48.66
FemHead2 Max Dose (D05) 47.12 50.68
Rectum Max Dose (D05) 82.99 80.74
Otherwise unspecified tissue Max Dose (D05) 63.74 63.06
Copyright © 2011 SciRes. AJOR
166 D. DINK ET AL.
Figure 2. DVH obtained with optimization of the prostate case with cumulative first method
(Disregarding fraction constraints).
Figure 3. DVH obtained with optimization of the prostate case with fraction size and cumulative
dose limits together (Simultaneous method).
drops of 1.77 Gy and 2.25 Gy respectively.
Noteworthy is the change in the dose delivered to the
secondary target (PTV2). The lower dose bound for each
treatment session of 2.0 Gy placed on this structure could
not be satisfied simultaneously with an upper dose bound
for each session of 2.0 Gy placed on normal tissues if the
model that eliminated the fractionation terms were used.
Once the fractionation terms were added, the minimum
dose in the PTV1 fell by 0.26% but the minimum dose in
the PTV2 rose from 56 Gy to 82 Gy.
A comparison of isodose plots for the plans constructed
with and without a limit on fraction sizes delivered to the
critical tissues is shown in Figures 4(a) and (b). The
isodose plots are very similar, showing that a deliverable
plan can be obtained reaching the same cumulative doses
when the fractionation considerations are included in the
3.2. Two-Phase Planning versus Uniform
Fractionation with Simultaneous Method
The prostate case studied above with uniform fractiona-
tion with simultaneous method is further analyzed with
the two-phase formulation given in Subsection 2.3. Both
cumulative and fraction size limits are still enforced,
however the minimum fraction size limits on PTV2 is
removed after the minimum cumulative dose limit of 56
Gy is reached in each and every point of this structure as
in the model in Table 1.
The results are given in Table 5. The optimized num-
bers of fractions for the first and second phases are 28
and 13 respectively, matching the total of 41 equal frac-
tions with the previous plan. Keeping in mind that there
is a minimum and maximum dose limit on the primary
target PTV1, insignificant difference is observed on the
Copyright © 2011 SciRes. AJOR
(a) (b)
Figure 4. (a) Isodose surfaces for the prostate plan obtained with cumulative first method; (b) Isodose surfaces for the pros-
tate plan obtained with simultaneous method.
Table 5. Comparative results of the prostate case with the two-phase model.
Structure Statistics
Uniform Fractionation with
Simultaneous Method (Gy) Two-Phase Model (Gy)
Max Dose (D05) 84.00 84.00
Min Dose (D95) 82.91 82.92 Target
Mean Dose 83.75 83.76
Max Dose (D05) 84.00 83.46
Min Dose (D95) 82.00 64.90 PTV2
Mean Dose 82.98 75.70
Bladder Max Dose (D05) 81.97 81.73
FemHead1 Max Dose (D05) 48.66 47.38
FemHead2 Max Dose (D05) 50.68 49.09
Rectum Max Dose (D05) 80.74 80.37
Otherwise unspecified tissue Max Dose (D05) 63.06 64.59
Number of fractions 41 28 + 13
PTV1 dose distribution between the two methods. How-
ever, the decrease in the PTV2 seminal vesicles is quite
noteworthy. It can be observed further in detail in the
dose-volume histograms given in Figure 5 that deliver-
ing the plan in two different phases helps to spare the
vesicles from significant amount of radiation. Unneces-
sary exposure to radiation in those irradiated areas in-
creases the potential of complications at the end of the
treatment, although there is a minimum cumulative limit
on the secondary targets; it is not the intention of the
planner to radiate them above 56 Gy. Furthermore, al-
though the extreme values do not show much difference
between the two approaches, the dose volume histograms
given in Figure 6 demonstrate that removing fraction
size requirement from seminal vesicles during the second
phase of the treatment helps to reduce the bladder dose
on average significantly and rectum exposure slightly as
well. Although the extreme dose limits on these two or-
gans at risk are almost the same with the two methods,
the doses at different volume percentages are quite far
The isodose surfaces given in Figure 7 for the two ap-
proaches provide another tool to observe the improve-
ent with the two-phase model. The plots on the left m
Copyright © 2011 SciRes. AJOR
168 D. DINK ET AL.
Figure 5. Comparison of DVH of vesicles for uniform fractionation and two-phase simultane ous models.
Figure 6. Comparison of DVH of bladder and rectum for uniform fractionation and two-phase simultaneous models.
correspond to the uniformly fractionated plan with the
simultaneous method, while the ones on the right are
obtained with two phase planning. The color scales on all
four plots are kept the same for fair comparison. Al-
though there is no apparent difference on the transverse
slice at z = 0, the second slice at z = 4.4 which includes
the bladder, rectum and the seminal vesicles demonstrate
the decrease in radiation exposure in all parts of the irra-
diated volume.
4. Discussions and Conclusions
In this paper, the problem of producing an IMRT plan
that meets fraction size and cumulative dose limits on
tumor and normal tissues simultaneously is addressed.
The common approach of employing the cumulative dose
limits alone and considering fractionation as a post-op-
timization stage, i.e. The Cumulative First approach, was
demonstrated on a prostate case for which the dose dis-
tribution planned without considering fraction size re-
quirements could not subsequently be partitioned into
uniform sessions so as to satisfy the limits. It was ob-
served that a plan that is feasible with respect to the ac-
cumulated doses may no longer remain feasible with
uniform fractions once upper and lower fraction size
limits are specified. The objective obtained upon impos-
ing fraction size and cumulative limits was only 0.26%
less than that which could be found even if no fraction
size requirements were posed, a difference smaller than
the uncertainty in the delivered dose or in the minimum
dose that can be delivered in a treatment session. Hence,
both cumulative and fraction size limits could be used in
the planning and the fraction size requirements could be
accommodated without mateially losing the level of dose r
Copyright © 2011 SciRes. AJOR
(a) (b)
Figure 7. Isodose surfaces for transverse slices of z = 0 and z = 4.4 prostate case with (a) Simultaneous uniform fractionation
(b) Simultaneous two-phase planning.
escalation which can be achieved with IMRT.
In operations research, it is common practice to repre-
sent a problem with the least constraints possible by
leaving dependent equations out to reduce solution time.
However, as the experience of this research showed, the
cumulative dose and fraction size limits are neither de-
pendent nor redundant, and none of them should be
eliminated from the formulation. Further, an analysis of
bounds proved that no renormalization of the prescrip-
tion would improve the plans developed based on the
cumulative limits alone.
In the paper, it was also questioned if Simultaneous
IMRT approach would benefit from combining with a
conventional planning method: two-phase planning. An
MILP formulation was proposed incorporating all cumu-
lative and fraction size dose limits and optimizing the
number of fractions for each phase. A two-phase solution
to the prostate case without changing the irradiated mar-
gins was found in which the same tumor dose exposure
was achieved while the dose to healthy structures like
bladder and rectum were lessened and unwanted further
exposure on the secondary target was avoided in com-
parison to the uniformly fractionated plan. Moreover, it
has been our experience that in some cases in addition to
sparing the healthy organs, the two-phase planning also
helps to escalate the tumor dose.
One might assume that the detailed intensity modula-
tion achieved with multi-leaf collimators or other modu-
lation techniques might not require partitioning the
problem into two phases in order to improve the quality
of the treatment. Yet, it is demonstrated that, IMRT can
benefit from the conventional tumor boosting approach.
Results building on the simultaneous approach establish
that releasing the fraction size minimum on the secon-
Copyright © 2011 SciRes. AJOR
170 D. DINK ET AL.
dary target in a second phase of the therapy can favor
tumor dose escalation or reduce the normal tissue expo-
sure. Hence, the two-phase planning is still a method to
explore in the era of detailed intensity maps.
The solution times are also compared for the prostate
case. It increases from order of minutes to hours for the
uniform fractionation versus the two-phase model. The
main contributor to this change is the increase in the size
of the problem, as the number of variables doubles and
the number of constraints for the two epoch model in-
creases as many as the total number of tissue points,
which is close to four thousand in this specific case. It
requires further research to employ efficient heuristics
and approximations in order to benefit from the simulta-
neous and two-phase approaches in the clinics in practice.
It should be noted that, although, the optimization method
used here is linear programming, there is no reason not to
think that the same conclusions could be reached when
this analysis of simultaneous and two-phase methods was
repeated with other optimization tools.
Finally, although the two-phase approach shows prom-
ise as it is, it would be of interest to the therapy planner
to investigate the possibility of allowing different beam
orientations during the two phases of the therapy.
5. Acknowledgements
We would like to thank Seda Gumrukcu for her contri-
bution in managing submissions of this work.
6. References
[1] Q. Wu, M. Manning, R. Schmidt-Ulrich and R. Mohan,
“The Potential for Sparing of Parotids and Escalation of
Biologically Effective Dose with Intensity-Modulated
Radiation Treatments of Head and Neck Cancers: A
Treatment Design Study,” International Journal of Ra-
diation Oncology, Biology, Physics, Vol. 46, No. 1, 2000,
pp. 195-205. doi:10.1016/S0360-3016(99)00304-1
[2] A. I. Blanco and K. S. C. Chao, “Intensity-Modulated
Radiation Therapy and Protection of Normal Tissue Fun-
ction in Head and Neck Cancer,” In: C. A. Perez and L.
W. Brady, Eds., Principles and Practice of Radiation
Oncology Updates, Lippincott Williams & Wilkins Heal-
thcare, New York, 2002, pp. 2-12.
[3] M. Langer, R. Brown, M. Urie, J. Leong, M. Stracher and
J. Shapiro, “Large Scale Optimization of Beam Weights
under Dose-Volume Restrictions,” International Journal
of Radiation Oncology, Biology, Physics, Vol. 18, No. 4,
1990, pp. 887-893. doi:10.1016/0360-3016(90)90413-E
[4] M. Langer, P. Kijewski, R. Brown and C. Ha, “The Effect
on Minimum Tumor Dose of Restricting Target-Dose
Inhomogeneity in Optimized Three-Dimensional Treat-
ment of Lung Cancer,” Radiotherapy and Oncology, Vol.
21, No. 4, 1991, pp. 245-256.
[5] M. Langer, E. K. Lee, J. O. Deasy, R. L. Rardin and J. A.
Deye, “Operations Research Applied to Radiotherapy, An
NCI-NSF-Sponsored Workshop,” International Journal
of Radiation Oncology, Biology, Physics, Vol. 57, No. 3,
2003, pp. 762-768. doi:10.1016/S0360-3016(03)00720-X
[6] E. K. Lee, T. Fox and I. Crocker, “Integer Programming
Applied to Intensity-Modulated Radiation Treatment
Planning,” Annals of Operations Research, Vol. 119, No.
1-4, 2003, pp. 165-181. doi:10.1023/A:1022938707934
[7] E. K. Lee, T. Fox and I. Crocker, “Simultaneous Beam
Geometry and Intensity Map Optimization in Inten-
sity-Modulated Radiation Therapy,” International Jour-
nal of Radiation Oncology, Biology, Physics, Vol. 64, No.
1, 2006, pp. 301-320. doi:10.1016/j.ijrobp.2005.08.023
[8] H. E. Romeijn, R. K. Ahuja, J. F. Dempsey, A. Kumar
and J. G. Li, “A Novel Linear Programming Approach to
Fluence Map Optimization for Intensity Modulated Ra-
diation Treatment Planning,” Physics in Medicine and
Biology, Vol. 48, No. 21, 2003, pp. 3521-3542.
[9] H. E. Romeijn, R. K. Ahuja and J. F. Dempsey, “A New
Linear Programming Approach to Radiation Therapy
Treatment Planning Problems,” Operations Research,
Vol. 54, No. 2, 2006, pp. 201-216.
[10] F. Preciado-Walters, R. L. Rardin, M. P. Langer and V.
Thai, “A Coupled Column Generation, Mixed Integer
Approach to Optimal Planning of Intensity Modulated
Radiation Therapy for Cancer,” Mathematical Program-
ming, Vol. 101, No. 2, 2004, pp. 319-338.
[11] A. Tuncel, “Methods and Algorithms for Radiation
Therapy Optimization under Dose-Volume Restrictions,”
Ph.D. Dissertation, Purdue University, West Lafayette,
[12] F. Preciado-Walters, M. P. Langer, R. L. Rardin and V.
Thai, “Column Generation for IMRT Cancer Therapy
Optimization with Implementable Segments,” Annals of
Operations Research, Vol. 148, No. 1, 2006, pp. 65-79.
[13] H. E. Romeijn, R. K. Ahuja, J. F. Dempsey and A.
Kumar, “A Column Generation Approach to Radiation
Therapy Treatment Planning Using Aperture Modula-
tion,” SIAM Journal on Optimization, Vol. 15, No. 3,
2005, pp. 838-862. doi:10.1137/040606612
[14] S. Webb, “Optimization by Simulated Annealing of Three-
Dimensional Conformal Treatment Planning for Radia-
tion Fields Defined by a Multileaf Collimator,” Physics in
Medicine and Biology, Vol. 36, No. 9, 1991, pp. 1201-
1226. doi:10.1088/0031-9155/36/9/004
[15] S. M. Morril, R. G. Lane and I. I. Rosen, “Constrained
Simulated Annealing for Optimized Radiation Therapy,”
Computer Methods and Programs in Biomedicine, Vol.
33, No. 3, 1990, pp. 135-144.
[16] G. S. Mageras and R. Mohan, “Application of Fast
Simulated Annealing to Optimization of Conformal Ra-
Copyright © 2011 SciRes. AJOR
Copyright © 2011 SciRes. AJOR
diation Treatments,” Medical Physics, Vol. 20, No. 3,
1993, pp. 639-647. doi:10.1118/1.597012
[17] M. Langer, S. Morrill, R. Brown, O. Lee and R. Lane, “A
Comparison of Mixed-Integer Programming and Fast
Simulated Annealing for Optimizing Beam Weights in
Radiation Therapy,” Medical Physics, Vol. 23, No. 6,
1996, pp. 957-964. doi:10.1118/1.597857
[18] M. Langer, S. Morrill, R. Brown, O. Lee and R. Lane, “A
Genetic Algorithm for Generating Beam Weights,” Medi-
cal Physics, Vol. 23, No. 6, 1996, pp. 965-971.
[19] G. A. Ezzel, “Genetic and Geometric Optimization of
Three-Dimensional Radiation Therapy Treatment Plan-
ning,” Medical Physics, Vol. 23, No. 3, 1996, pp. 293-
305. doi:10.1118/1.597660
[20] X. Wu, Y. Zhu, J. Dai and Z. Wang, “Selection and De-
termination of Beam Weights based on Genetic Algo-
rithms for Conformal Radiotherapy Treatment Planning,”
Physics in Medicine and Biology, Vol. 45, No. 9, 2000,
pp. 2547-2558. doi:10.1088/0031-9155/45/9/308
[21] P. S. Cho, S. Lee, R. J. Marks, S. Oh, S. G. Sutlief and M.
H. Phillips, “Optimization of Intensity Modulated Beams
with Volume Constraints Using Two Methods: Cost
Function Minimization and Projections onto Convex
Sets,” Medical Physics, Vol. 25, No. 4, 1998, pp. 435-
443. doi:10.1118/1.598218
[22] D. H. Hristov and B. G. Fallone, “A Continuous Penalty
Function Method for Inverse Treatment Planning,” Medi-
cal Physics, Vol. 25, No. 2, 1998, pp. 208-223.
[23] S. V. Spirou and C. S. Chui, “A Gradient Inverse Plan-
ning Algorithm with Dose-Volume Constraints,” Medical
Physics, Vol. 25, No. 3, 1998, pp. 321-333.
[24] Q. Wu and R. Mohan, “Algorithms and Functionality of
an Intensity Modulated Radiotherapy Optimization Sys-
tem,” Medical Physics, Vol. 27, No. 4, 2000, pp. 701-
711. doi:10.1118/1.598932
[25] P. Cheung, K. Sixel, G. Morton, D. A. Loblaw, R. Tirona,
G. Pang, R. Choo, E. Szumacher, G. Deboer and J. P.
Pignol, “Individualized Planning Target Volumes for In-
trafraction Motion during Hypofractionated Intensity-
Modulated Radiotherapy Boost for Prostate Cancer,” In-
ternational Journal of Radiation Oncology, Biology, Phy-
sics, Vol. 62, No. 2, 2005, pp. 418-425.
[26] M. Guerrero, X. A. Li, L. Ma, J. Linder, C. Deyoung and
B. Erickson, “Simultaneous Integrated Intensity-Modu-
lated Radiotherapy Boost for Locally Advanced Gyneco-
logical Cancer: Radiobiological and Dosimetric Consid-
erations,” International Journal of Radiation Oncology,
Biology, Physi cs, Vol. 62, No. 3, 2005, pp. 933-939.
[27] M. L. Cavey, J. E. Bayouth, M. Colman, E. J. Endres and
G. Sanguineti, “IMRT to Escalate the Dose to the Pros-
tate While Treating the Pelvic Nodes,” Strahlenther On-
kol, Vol. 181, No. 7, 2005, pp. 431-441.
[28] X. A. Li, J. Z. Wang, P. A. Jursinic, C. A. Lawton and D.
Wang, “Dosimetric Advantages of IMRT Simultaneous
Integrated Boost for High-Risk Prostate Cancer,” Inter-
national Journal of Radiation Oncology, Biology, Physics,
Vol. 61, No. 4, 2005, pp. 1251-1257.
[29] R. A. Popple, P. B. Prellop, S. A. Spencer, J. F. De Los
Santos, J. Duan, J. B. Fiveash and I. A. Brezovich, “Si-
multaneous Optimization of Sequential IMRT Plans,”
Medical Physics, Vol. 32, No. 11, 2005, pp. 3257-3266.
[30] S. M. Morrill, R. G. Lane, J. A. Wong and I. I. Rosen,
“Dose-Volume Considerations with Linear Programming
Optimization,” Medical Physics, Vol. 18, No. 6, 1991, pp.
1201-1210. doi:10.1118/1.596592
[31] D. Dink, “Approaches to 4-D Intensity Modulated Radia-
tion Therapy Planning with Fraction Constraints,” Ph.D.
Dissertation, Purdue University, West Lafayette, 2005.
[32] S. M. Morrill, I. I. Rosen, R. G. Lane and J. A. Belli,
“The Influence of Dose Constraint Point Placement on
Optimized Radiation Therapy Treatment Planning,” In-
ternational Journal of Radiation Oncology, Biology, Phy-
sics, Vol. 19, No. 1, 1990, pp. 129-1241.
[33] A. Niemierko and M. Goitein, “Random Sampling for
Evaluating Treatment Plans,” Medical Physics, Vol. 17,
No. 5, 1990, pp. 753-762. doi:10.1118/1.596473
[34] A. Niemierko and M. Goitein, “Letter to the Editor,
Comments on ‘Sampling Techniques for the Evaluation
of Treatment Plans’,” Medical Physics, Vol. 20, No. 1,
1993, pp. 1377-1380. doi:10.1118/1.597103
[35] X.-Q. Lu and L. M. Chin, “Sampling Techniques for the
Evaluation of Treatment Plans,” Medical Physics, Vol. 20,
No. 1, 1993, pp. 151-161. doi:10.1118/1.597096
[36] R. Acosta, W. Brick, A. Hanna, A. Holder, D. Lara, G.
McQuilen, D. Nevin, P. Uhlig and B. Salter, “Radiother-
apy Optimal Design: An Academic Radiotherapy Treat-
ment Design System,” In: J. W. Chinneck, B. Kristjans-
son, and M. J. Saltman, Eds., Operations Research and
Cyber-Infrastructure, Springer, New York, 2009, pp. 401-
425. doi:10.1007/978-0-387-88843-9_21
[37] Sherouse Systems Inc., 2011.