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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 51 (2006) 2353–2365 doi:10.1088/0031-9155/51/10/001
A unified approach for inversion problems in
intensity-modulated radiation therapy
Yair Censor
1
, Thomas Bortfeld
2
, Benjamin Martin
2
and Alexei Trofimov
2
1
Department of Mathematics, University of Haifa, Mt Carmel, Haifa 31905, Israel
2
Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical
School, Boston, MA 02114, USA
E-mail: yair@math.haifa.ac.il
Received 20 April 2005, in final form 16 February 2006
Published 27 April 2006
Online at stacks.iop.org/PMB/51/2353
Abstract
We propose and study a unified model for handling dose constraints (physical
dose, equivalent uniform dose (EUD), etc) and radiation source constraints
in a single mathematical framework based on the split feasibility problem.
The model does not impose on the constraints an exogenous objective (merit)
function. The optimization algorithm minimizes a weighted proximity function
that measures the sum of the squares of the distances to the constraint sets.
This guarantees convergence to a feasible solution point if the split feasibility
problem is consistent (i.e., has a solution), or, otherwise, convergence to
a solution that minimally violates the physical dose constraints and EUD
constraints. We present computational results that demonstrate the validity
of the model and the power of the proposed algorithmic scheme.
1. Introduction
In intensity-modulated radiation therapy (IMRT) (see, e.g., Palta and Mackie (2003)), beams
of penetrating radiation are directed at the tumour lesion from external sources. A multileaf
collimator (MLC) is used to split each beam into many beamlets with individually controllable
intensities. There are two principal aspects of radiation teletherapy that call for computational
modelling. The first is the calculation of the radiation dose absorbed in the irradiated tissue
based on a given distribution of beamlet intensities. This dose calculation is a forward
problem. The second aspect is the inverse problem of the first: to find a distribution of
radiation intensities (radiation intensity map) deliverable by all beamlets, which would result in
a clinically-acceptable dose distribution (i.e., such that the dose to each tissue would be within
the desired upper and lower bounds, which are prescribed on the basis of medical diagnosis,
knowledge and experience). To be of practical value, however, this radiation intensity map
must be implementable, in a clinically acceptable form, on the available treatment machine.
Therefore, in addition to the physical and biological parameters of the irradiated object,
0031-9155/06/102353+13$30.00 © 2006 IOP Publishing Ltd Printed in the UK 2353
2354 YCensoret al
the relevant information about the capabilities and specifications of the available treatment
machine (i.e., radiation source) should be taken into account.
The concept of equivalent uniform dose (EUD) was introduced in recent studies to
describe dose distributions with a greater clinical relevance. EUD is defined for tumours
as the biological equivalent dose that, if given uniformly, will lead to the same cell-kill in the
tumour volume as the actual non-uniform dose distribution. EUD can also be defined so as to
be applicable for normal tissues. The use of EUD had been proposed for treatment planning
in Brahme (1984), Kwa et al (1998), Mohan et al (1992), Niemierko (1997). In this paper, we
use the concept of generalized EUD (Niemierko 1999).
EUD-based IMRT optimization was described by Wu et al (2002) and Thieke et al (2003).
The latter proposed to formulate EUD as constraints instead of optimizing the EUD functions.
Motivated by their work we develop and study here a unified theory that enables treatment
of EUD constraints and physical dose constraints in a combined manner. The model can be
extended to include other kinds of dose constraints, such as dose–volume constraints (Bortfeld
et al 1997, Spirou and Chui 1998).
A split feasibility problem was first proposed and analysed by Censor and Elfving (1994),
in the formulation:
find x ∈ C ⊆ R
m
such that Ax ∈ Q ⊆ R
m
(1)
where C and Q are two given closed convex sets in the m-dimensional vector space R
m
, and
A : R
m
→ R
m
is a linear bijection (i.e., a one-to-one and onto mapping) represented by
a full-rank matrix. The algorithm proposed there relied on inversion of the matrix A and
was, therefore, unrealistic for large matrix dimensions. Later, Byrne (2002) developed the
‘CQ-algorithm’ for this problem, which does not call for any matrix inversion.
In this paper, we present a framework that enables a unified approach for handling dose
constraints and radiation source constraints in intensity-modulated radiation therapy. The
main feature of the approach is the rigorous formulation and solution of the feasibility model
in a situation where the constraint sets are split between two Euclidean vector spaces: the
space of beamlet intensity vectors and the space of dose vectors. Our split feasibility problem
is more complicated than that given by (1) and calls for a generalization that will allow finite
families of sets in each of the two relevant spaces. Therefore, we call this framework the
multiple-sets split feasibility approach.
The mathematical analysis of the results presented here is included in our companion
paper (Censor et al 2005).
2. Formulating the unified model
Most clinical constraints are naturally described as constraints on the dose delivered to the
patient. For example, upper and lower dose bounds on each tissue can be described as
minimum and maximum dose constraints on the dose delivered to each voxel. Similarly, EUD
constraints or dose–volume constraints are most naturally described in the dose space. In
contrast, constraints on the deliverable radiation intensities are best described in the intensity
space. The universal example of this is the requirement of non-negative beamlet intensity.
Additional constraints can be described such as limits on the complexity of the intensity map
or limits on the acceptable delivery time.
It is possible to translate constraints formulated in the dose space into constraints in the
intensity space. Unfortunately, the resulting constraints tend to be more complicated than the
original ones. Thus, it is more convenient to leave the constraints in their respective spaces
and develop a framework for combining them. Each constraint is described by a set of either
A unified approach for inversion problems in IMRT 2355
dose vectors or intensity vectors that fulfil that constraint. A satisfactory treatment plan would
be that which is in the intersection of all such sets.
Projection algorithms are frequently used to find a point that belongs to the intersection
of closed convex sets. A projection onto a set is the point within that set that is closest to the
current point. In projection algorithms, a point is repeatedly projected onto each set, according
to some algorithmic scheme, until a point in all of the sets is found.
Unfortunately, for most IMRT inverse problems, there are no possible treatment plans
that satisfy all of the constraints. Thus, we need to find plans that are as close to satisfying
the constraints as possible. To judge this, we introduce proximity functions that measure the
distance of a point to the set. We can still use projection algorithms for this case, with some
modification. We will seek to minimize a weighted sum of the proximity functions.
Let us first define the notation. We divide the entire volume of the patient into I voxels,
enumerated by i = 1, 2,...,I. Assume that T anatomical structures have been outlined,
including planning target volumes (PTVs) and the organs at risk (OAR). We denote the set
of voxels indices in structure t by S
t
. Individual voxels i may belong to several sets S
t
,
i.e., different structures may overlap. We will further assume that the radiation is delivered
independently from each of the J beamlets, which are arranged in a certain geometry and
indexed by j = 1, 2,...,J. The intensities x
j
of the beamlets are arranged in a J -dimensional
vector x = (x
j
)
J
j=1
∈ R
J
,intheJ -dimensional Euclidean space R
J
—the radiation intensity
space.
The quantities d
ij
0, which represent the dose absorbed in voxel i due to radiation of
unit intensity from the jth beamlet (Censor et al 1988a, Bortfeld et al 1993), are calculable
by any forward calculation program. Let h
i
denote the total dose absorbed in voxel i and let
h = (h
i
)
I
i=1
be the vector of doses absorbed in all voxels. We call the space R
I
the dose space.
We can now calculate h
i
as
h
i
=
J
j=1
d
ij
x
j
. (2)
The dose influence matrix D = (d
ij
) is the I ×J matrix whose elements are the d
ij
’s mentioned
above. Thus, (2) can be rewritten as the vector equation
h = Dx. (3)
Finally, let us assume that we have M constraints in the dose space and N constraints in
the intensity space. Let H
m
be the set of dose vectors that fulfil the mth dose constraint, and
let X
n
be the set of beamlet intensity vectors that fulfil the nth intensity constraint. Let us
consider some concrete examples of the sets H
m
and X
n
. Each of the constraint sets H
m
and
X
n
can be one of the specific H and X sets, respectively, described below.
In the dose space, a typical constraint is that, in a given critical structure S
t
, the dose
should not exceed an upper bound u
t
. The corresponding set H
max,t
is
H
max,t
={h ∈ R
I
| h
i
u
t
, for all i ∈ S
t
}. (4)
Similarly, in the target volumes (TVs), the dose should not fall below a lower bound l
t
.
The set H
min,t
of dose vectors that fulfil this constraint is
H
min,t
={h ∈ R
I
| l
t
h
i
, for all i ∈ S
t
}. (5)
To handle EUD constraints for each volume of interest S
t
, consisting of N
t
voxels, a
real-valued function E
t
: R
I
→ R, called the EUD function, is defined as
E
t
(h) =
1
N
t
i∈S
t
(h
i
)
α
t
1/α
t
. (6)
2356 YCensoret al
The parameter α
t
is a tissue-specific number which is negative for target volumes and positive
for OAR. For α
t
= 1, the EUD function is the mean dose of the organ for which it is calculated.
On the other hand, letting α
t
→∞makes the EUD function approach the maximal value:
max{h
i
| i ∈ S
t
}. The EUD constraint for an upper EUD bound e
t
for a structure S
t
can be
described by the set
H
EUD,t
={h ∈ R
I
| E
t
(h) e
t
}. (7)
Lower EUD bounds can be described similarly.
Choi and Deasy (2002) (theorem 1) showed that, due to non-negativity of the dose, h 0,
the EUD function of (6) is convex for all α
t
1 and concave for all α
t
1. Therefore, the
constraint sets H
EUD,t
are always convex sets in the dose vector space, since they are level sets
(i.e., sets on which the function values are smaller or equal to some fixed real constant) of the
convex functions E
t
(h) for OAR (with α
t
1), or of the convex functions −E
t
(h) for targets
(with α
t
< 0).
In the radiation intensity space, the most prominent constraint is the non-negativity of the
intensities, described by the set
X
+
={x ∈ R
J
| x
j
0, for all j = 1, 2,...,J}. (8)
Thus, we have a multiple-sets split feasibility problem, where some constraints (the non-
negativity of radiation intensities) are defined in the radiation intensity space R
J
and other
constraints (the upper and lower bounds on dose and the EUD constraints) are defined in the
dose space R
I
, and the two spaces are related by a linear transformation D (according to (3)).
The unified problem can be formulated as follows:
find x
∗
∈ X
+
∩
∩
N
n=1
X
n
such that h
∗
= Dx
∗
and h
∗
∈
∩
M
m=1
H
m
. (9)
3. Solving the split feasibility problem
3.1. Projections
Next, we describe the operators that are used to project a point to the nearest point that fulfils
the constraint, i.e., belongs to the set H
m
or X
n
. We refer to projection operators in the dose and
intensity spaces as P
H
m
and P
X
n
, respectively. (Projection algorithms and general projection
operators are discussed in detail in the appendix.)
Let us look at some examples of projection operators in the dose space first. In critical
structures, it is often required that no voxel should receive a dose above a certain tolerance
bound, i.e., a constraint H
t
of the form (4) is satisfied. In this case the corresponding projection
operator P
H
max,t
(h) simply cuts off doses beyond u
t
in a given structure S
t
.Theith component
of the projection, i.e., the dose at the ith voxel, is given for all i = 1,...,I by
P
H
max,t
(h)
i
=
u
t
if i ∈ S
t
and u
t
<h
i
h
i
otherwise.
(10)
In a target volume, on the other hand, it is requested that the dose should be above the
prescription dose l
t
, i.e., for all i:
P
H
min,t
(h)
i
=
l
t
if i ∈ S
t
and l
t
>h
i
h
i
otherwise.
(11)
Approximate EUD projectors P
H
EUD,t
(h) are derived as in Thieke et al (2003), for the convex
constraints H
EUD,t
(given by equation (7)).
A unified approach for inversion problems in IMRT 2357
In the intensity space, for the non-negativity constraint X
+
(equation (8)), the
corresponding projection operator cuts off negative intensities as follows:
P
X
+
(x)
j
=
0ifx
j
< 0
x
j
otherwise.
(12)
Another example of a projector in the intensity space is the intensity gradient projector,
which is relevant, e.g., in the optimization of particle therapy delivery with continuous beam
scanning (Trofimov and Bortfeld 2003). Assuming that the beamlets are numbered in the
order of scanning, from the first to the J th, if the change in intensity between two consecutive
beamlets, (j −1) and j , is constrained to a maximum value , then the corresponding gradient
projector is
P
X
∇
(x)
j
=
x
j−1
+ if x
j
>x
j−1
+
x
j−1
− if x
j
<x
j−1
−
x
j
otherwise.
(13)
Multileaf collimator constraints for IMRT delivery can be enforced in a similar manner.
In practice, it is often impossible to fulfil all constraints simultaneously. In this infeasible
case, it makes sense to find a solution that is as close as possible to a feasible solution.
Using the square Euclidean norm as a closeness measure, a suitable objective function, called
proximity function, is
F(x) =
1
2
M
m=1
w
H
m
P
H
m
(Dx) − Dx
2
+
1
2
N
n=1
w
X
n
P
X
n
(x) − x
2
(14)
where each of the projectors P can be one of the sample projection operators mentioned
above, and
w
H
m
M
m=1
,
w
X
n
N
n=1
are their corresponding weight factors. Note that there
may be multiple constraints, i.e., multiple projection operators, per anatomical structure. For
example, in the target volumes one usually aims to keep the dose above a lower bound l
t
,but
at the same time below an upper bound u
t
, often with various degrees of priority (expressed
by weights w
H
m
, i.e., penalties for under- and overdose).
3.2. Algorithm
We seek an x
∗
that minimizes the proximity function (equation (14)). We do that by applying
the gradient ∇
x
to F(x)and using a result from Aubin and Cellina (1984), which states that
∇
x
P
X
n
(x) − x
2
= 2
P
X
n
(x) − x
. (15)
Using the chain rule we calculate the gradient with respect to the dose projectors:
∇
x
P
H
m
(Dx) − Dx
2
= 2D
T
P
H
m
(Dx) − Dx
(16)
where D
T
is the transposed matrix.
The general iterative gradient projection scheme, with the stepsize s, designed to find a
minimum of F(x)subject to x ∈ , where ⊆ R
J
is some constraint set, whose projector is
P
,is
x
(k+1)
= P
(x
(k)
− s∇
x
F(x
(k)
)). (17)
Inserting equations (15) and (16), and using P
= P
X
+
, yields the algorithm:
x
(k+1)
= P
X
+
x
(k)
− s
D
T
M
m=1
w
H
m
P
H
m
(Dx
(k)
) − Dx
(k)
+
N
n=1
w
X
n
P
X
n
(x
(k)
) − x
(k)
.
(18)
2358 YCensoret al
Figure 1. The optimized dose distribution on a transversal CT slice.
(This figure is in colour only in the electronic version)
The projection P
X
+
must be applied at each iteration step, after all other projections, to
eliminate the unphysical negative intensities. As shown in Censor et al (2005), the algorithm
of equation (18) generates a sequence {x
(k)
} of intensity vectors that will converge towards x
∗
if the stepsize s is chosen appropriately.
The algorithm (18) is similar to that of Thieke et al (2003). However it does not involve
scaling of the gradient with second derivatives, and is formulated for a more general case.
3.3. Test case
To demonstrate the practical usefulness of our algorithm, we applied it to a clinical case of a
tumour in the thorax. Figure 1 shows the geometry of the gross tumour volume (GTV), the
planning target volume and the surrounding critical structures in a transversal CT slice. A
thorax case was chosen because it involves serially organized organs such as the spinal cord
and oesophagus, and parallel organs such as the lung. Dose prescriptions and constraints were
given as lower and/or upper dose bounds for the target volume and the serially organized
critical structures. These upper and lower dose bounds, as well as the weights of importance
w
H
, are summarized in table 1. The weights of importance were normalized by the number of
voxels in the respective organ. Clearly, because the target lower dose bound was set equal to
the upper dose bound, there is no feasible solution in this case. For the lung, an EUD bound
was used. The lung constraints are displayed in table 2.
The irradiation geometry was a coplanar beam arrangement with gantry angles of 110
◦
,
180
◦
, 220
◦
, 260
◦
and 320
◦
(clockwise). The angles were taken from the clinical treatment
plan with which the patient was actually treated. The beam energy was 6 MV. There was
a total of 125 × 75 × 82 = 768, 750 voxels, measuring 2.92 × 2.5 × 2.92 mm
3
each. The
number of beamlets was between 225 and 336 per beam (1460 in total) and their size was 5 ×
5mm
2
. The dose operator D was pre-calculated with the KonRad program (Preiser et al 1997).
Specifically, the d
ij
matrix elements were calculated with a pencil beam algorithm (Bortfeld
A unified approach for inversion problems in IMRT 2359
Table 1. Physical dose prescriptions (lower and upper dose bounds l
t
and u
t
) and their
corresponding weights of importance (w
H
) used for the example case.
Volume Lower (Gy) Weight Upper (Gy) Weight
PTV 72 1000 72 200
Oesophagus – – 60 30
Spinal cord – – 50 1000
Heart – – 60 30
Table 2. Equivalent uniform dose bounds with their α
t
parameters (see the EUD definition in (6)),
and the corresponding weights of importance (w
H
).
Vo l u m e E U D ( G y ) α Weight
Right lung 15 1 50
Left lung 15 1 50
et al 1993, Thieke et al 2003) and stored in a file. Our algorithm, given by equation (18), was
implemented in a dedicated temporo-spatial radiotherapy in-house optimization tool, called
‘opt4D’ (Trofimov et al 2005). For the present application, ‘opt4D’ was used in the static
(three-dimensional) mode. The projection operator P
H
EUD
onto each EUD constraint set was
implemented using the method of Thieke et al (2003).
The stepsize s was calculated as follows: the iteration was initialized with x
(0)
= 0 and
the first iteration was performed with a stepsize s = 1. This resulted in a vector of beamlet
intensities x
(1)
and its corresponding dose vector Dx
(1)
. Then a normalization factor κ was
defined such that the average target dose of κDx
(1)
was identical to the prescribed target dose.
The purpose of this exercise was to put the dose range in the right order of magnitude. The
vector of intensities x
(1)
was then multiplied by κ and all subsequent iterations were performed
with a stepsize of s = κ. To test the convergence behaviour with different stepsizes s,we
performed iterative algorithmic runs with s = 2κ, s = 4κ and so forth.
The heuristic stopping criterion was that the relative difference between the proximity
functions of two consecutive iteration steps should be below 0.2%. The convergence behaviour
of the proximity function (14), for different stepsizes s, is plotted in figure 2.Fors = κ it
took 85 iteration steps to reach the stopping criterion. In this specific case, it turned out that
s = 2κ resulted in the fastest convergence, in 65 steps, without leading to oscillations. The
calculation time per iteration step was 7.2 s on a 2.4 GHz Pentium 4 PC running Linux.
The resulting dose distribution is shown superimposed on a transversal CT slice in figure 1,
and in the form of a dose–volume histogram in figure 3.
4. Discussion
We used physical dose constraints, non-negativity of beamlets’ intensities and equivalent
uniform dose constraints to describe the proposed framework and demonstrate its
computational viability. These constraints are formulated in two different Euclidean vector
spaces. Some constraints, such as the physical dose constraints, can be formulated in either
the space of radiation intensity vectors or of dose vectors (as in equations (4), (5) and
(A.7)). However, the EUD constraints are formulated exclusively in the space of dose vectors.
Similarly, the non-negativity constraints on the radiation intensities are formulated exclusively
in the radiation intensity space. Thus, there is no escape from addressing constraints that reside
in both of these different spaces. Additionally, these constraints have a different mathematical
2360 YCensoret al
10 20 30 40 50
10
2
10
3
10
4
s = 0.25κ
s = 0.5κ
s = 1κ
s = 2κ
s = 4κ
s = 8κ
Number of iterations
Proximity Function
Convergence for different stepsizes s
Figure 2. The value of the proximity function (14) versus the iteration number for different
stepsizes s. The definition of κ is given in subsection 3.3.Fors = κ it took 85 iterations to reach
the 0.2% relative difference stopping criterion.
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
Dose (Gy)
Fractional volume (%)
Rt Lung EUD = 16.8 Gy
Lt Lung EUD = 9.6 Gy
Spinal Cord
Oesophagus
Heart
GTV
PTV
Figure 3. The dose–volume histograms for the test case described in subsection 3.3.
nature, namely, physical dose constraints are linear while EUD constraints are convex but
nonlinear. To accommodate such constraints we developed a logical framework for performing
a search that iterates in each of the two spaces and correctly passes back and forth between
the spaces, without resorting to the inverse matrix of D.
More generally, our framework fits any situation in which an inversion problem in IMRT
can be presented by constraint sets that are split between the beamlet intensity space and
the dose space. For example, the multileaf collimator (MLC) hardware constraints that are
integrated into the IMRT inversion problem by Cho and Marks (2000) could be included. The
unified new model relies on the multiple-sets split feasibility problem formulation, that we
proposed in Censor et al (2005), which is further developed here to accommodate the specific
A unified approach for inversion problems in IMRT 2361
IMRT situation and to establish a valid framework for handling both physical dose constraints
and EUD constraints in a unified manner.
Our unified framework and algorithm also cover the situation in which the constraints
are inconsistent, i.e., it is impossible to satisfy all of them. In such a case, our simultaneous
iterative projections algorithm minimizes a proximity function that measures how close the
constraints are being satisfied. The original split feasibility problem of Censor and Elfving
(1994) addressed only the consistent case and the iterative algorithm proposed there employed
the inverse D
−1
of the dose matrix D that maps the radiation intensity space to the dose space.
This is a practical disadvantage since the inversion of a large matrix is a computationally
difficult task. However, recently Byrne (2002) refined the algorithms of Censor and Elfving
(1994) so that only the transposed matrix D
T
is needed, and not D
−1
. The existing algorithms
for the split feasibility problem, which currently handle only a single constraint set in each of
the spaces, had to be properly modified to handle multiple constraints.
A further useful modification can be implemented here by replacing, in (14) and (18),
the orthogonal projections onto the EUD constraint sets by subgradient projections (see, e.g.,
Censor and Lent (1982) or Censor and Zenios (1997) (subsection 5.3)). These projections
do not require the (iterative) minimization of distance between the point and the set but are
rather given by closed-form analytical expressions. In the vector space of radiation intensities,
this would make no difference because the physical dose constraints there are linear, but in
the dose vector space the EUD constraints are nonlinear and, thus, the algorithm benefits
from replacing orthogonal projections by subgradient projections because the latter are easier
to compute. Subgradient projections are discussed in more detail in the appendix, where
examples of relevant operators are also given.
Treating physical dose and EUD as constraints, instead of optimizing respective functions,
has been previously proposed, thus creating what we now call a ‘split feasibility problem’. In
our approach, this idea is cast into a unified framework based on a rigorous foundation. The
multiple-sets split feasibility approach applies to any inversion problem, in its fully-discretized
formulation, where the constraints are split between two different vector spaces that are related
by a linear transformation.
Acknowledgments
We thank Dr John Wolfgang and Dr Noah Choi of Massachusetts General Hospital for
providing the clinical example case. This research is supported by grant no 2003275 from
the United States–Israel Binational Science Foundation (BSF). The work of Y Censor on
this research was also supported by a National Institutes of Health (NIH) grant no HL70472.
Part of this work was done at the Center for Computational Mathematics and Scientific
Computation (CCMSC) at the University of Haifa and supported by research grant no 522/04
from the Israel Science Foundation (ISF). T Bortfeld and A Trofimov acknowledge support
from the National Cancer Institute programme project grant no 5-P01-CA21239-25 and grant
no 1-R01-CA103904-01. This work was supported in part by CenSSIS, the Center for
Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program
of the National Science Foundation (Award Number EEC-9986821).
Appendix. Projection algorithms
Projection algorithms employ projections onto convex sets in various ways. They may use
different kinds of projections and, sometimes, even use different projections within the same
2362 YCensoret al
algorithm. They serve to solve a variety of problems which are either of the feasibility or the
optimization types. They have different algorithmic structures, of which some are particularly
suitable for parallel computing, and they demonstrate nice convergence properties and/or
good initial behaviour patterns. This class of algorithms has witnessed great progress in recent
years and its member algorithms have been applied with success to fully-discretized models of
problems in image reconstruction and image processing (Bauschke and Borwein 1996, Censor
and Zenios 1997, Stark and Yang 1998).
The convex feasibility problem is a fundamental problem in many areas of mathematics
and the physical sciences (see, e.g., Combettes (1994) and references therein). It has been used
to model significant real-world problems in image reconstruction from projections (Herman
1980), in radiation therapy treatment planning (Censor et al 1988b, Censor 2003), and in
crystallography (Marks et al 1999), to name but a few, and has been used under additional
names such as set theoretic estimation or the feasible set approach. A common approach
to such problems is to use projection algorithms (see, e.g., Bauschke and Borwein (1996)),
which employ orthogonal projections (i.e., nearest point mappings) onto the individual closed
convex constraint sets C
i
in an n-dimensional vector space R
n
. The orthogonal projection
P
(
ˆ
x) of a point
ˆ
x ∈ R
n
onto a closed convex set ⊆ R
n
is defined by
P
(
ˆ
x) = argmin{
ˆ
x − x|x ∈ } (A.1)
where · is the Euclidean norm in R
n
. Frequently a relaxation parameter is introduced so
that
P
(
ˆ
x,λ) = (1 − λ)
ˆ
x + λP
(
ˆ
x) (A.2)
is the relaxed projection of
ˆ
x onto with relaxation λ. Orthogonal projections are easy to
calculate for linear sets such as hyperplanes, half-spaces, hyperslabs, or for balls, etc. But for
a general (nonlinear) convex set, finding the orthogonal projection of a given point requires
the solution of a subsidiary constrained optimization problem of equation (A.1) to minimize
the distance of the given point to the set over all points in the set. Therefore, it is useful to
employ subgradient projections instead. Let the convex set be given as a level-set of a convex
function f : R
n
→ R, i.e.,
={x ∈ R
n
| f(x) 0} (A.3)
and let
ˆ
x ∈ R
n
be a given point. Then the subgradient projection of
ˆ
x onto the set of (A.3)
is the point
(
ˆ
x) given by
(
ˆ
x) =
ˆ
x −
f(
ˆ
x)
g(
ˆ
x)
2
· g(
ˆ
x) if
ˆ
x/∈
ˆ
x if
ˆ
x ∈
(A.4)
where g(
ˆ
x) is a subgradient vector of the function f calculated at
ˆ
x. Note that, according to the
traditional definition, this is not a projection, because
(
ˆ
x)does not even need to belong to .
Nevertheless, this is a very useful type of ‘projection’, because it has a closed-form formula
and does not require one to solve an optimization problem as required in equation (A.1).
For the notion of subgradients of convex function see, e.g., Hiriart-Urruty and Lemar
´
echal
(2001), Boyd and Vandenberghe (2003). For subgradient projection algorithms for feasibility
problems see, e.g., Censor and Zenios (1997) (section 5.3) or Bauschke and Borwein (1996).
It is well known that if the convex function f is differentiable at the point
ˆ
x then it has
a unique subgradient there which is equal to the gradient ∇f(
ˆ
x). The relaxed subgradient
projection with relaxation parameter λ may also be introduced by
(
ˆ
x,λ) = (1 − λ)
ˆ
x + λ
(
ˆ
x). (A.5)
A unified approach for inversion problems in IMRT 2363
Projection algorithmic schemes for the convex feasibility problem are, in general, either
sequential or simultaneous or block-iterative (see, e.g., Censor and Zenios (1997)fora
classification of projection algorithms into such classes, and the review paper of Bauschke and
Borwein (1996) for a variety of specific algorithms of these kinds). Recently, Censor and Tom
(2003) have investigated yet another class of projection algorithms, called string-averaging
algorithms.
A.1. Projections onto a generalized box
The projection P
H
(h), in our algorithm, is easy to perform since H is a generalized box (i.e.,
is the product of all individual H
m
). For all i = 1, 2,...,I,
P
H
(h)
i
=
l
i
if h
i
<l
i
h
i
if l
i
h
i
u
i
u
i
if u
i
<h
i
.
(A.6)
Note that if the physical dose constraints H
m
(e.g., equations (4) and (5)) were defined in
the intensity space R
J
by the sets
X
i
={x ∈ R
J
| l
i
(Dx)
i
u
i
} (A.7)
then the projections P
X
i
(x) would have to be calculated as
P
X
i
(x) =
x +
u
i
− (Dx)
i
d
i
2
· d
i
if u
i
<(Dx)
i
x if l
i
(Dx)
i
u
i
x +
l
i
− (Dx)
i
d
i
2
· d
i
if (Dx)
i
<l
i
(A.8)
where d
i
= (d
ij
)
J
j=1
is the J -dimensional vector that forms the ith column of the dose
matrix D.
A.2. Subgradient projections for the EUD constraints
The subgradient projections
H
(h) for the EUD constraints are calculated from (6), (7) and
(A.4) as follows. The EUD function of (6) is differentiable and the ith component of its
gradient ∇E
t
(h) is calculated, for all i ∈ S
t
, by
(∇E
t
(h))
i
=
1
α
t
1
N
t
i∈S
t
(h
i
)
α
t
(1/α
t
)−1
·
1
N
t
α
t
(h
i
)
α
t
−1
=
1
N
t
1/α
t
i∈S
t
(h
i
)
α
t
(1/α
t
)−1
· (h
i
)
α
t
−1
. (A.9)
To obtain
H
(h) for a target volume (TV), with α
t
< 0, we rewrite (7)as
H
(TV)
EUD,t
={h ∈ R
I
|−E
t
(h) −e
t
} (A.10)
which makes it a level set of the convex function −E
t
(h). Now we calculate
(TV)
H
EUD,t
(h
(t)
) for
individual target(s), and this is done by using (A.4) with x = h
(t)
,f(h
(t)
) =−E
t
(h
(t)
) + e
t
,
and with g(h
(t)
) =∇(−E
t
(h
(t)
)), so that
(TV)
H
EUD,t
(h
(t)
) =
h
(t)
+
E
t
(h
(t)
) − e
t
∇(−E
t
(h
(t)
))
2
·∇(−E
t
(h
(t)
)) if h
(t)
/∈ H
(TV)
EUD,t
h
(t)
if h
(t)
∈ H
(TV)
EUD,t
.
(A.11)
2364 YCensoret al
Similarly, for an organ at risk (OAR, with α
t
1):
H
(OAR)
EUD,t
={h ∈ R
I
| E
t
(h) e
t
} (A.12)
and
(OAR)
H
EUD,t
(h
(t)
) is obtained by plugging x = h
(t)
,f(h
(t)
) = E
t
(h
(t)
) − e
t
, and g(h
(t)
) =
∇E
t
(h
(t)
), into (A.4):
(OAR)
H
EUD,t
(h
(t)
) =
h
(t)
−
E
t
(h
(t)
) − e
t
∇E
t
(h
(t)
)
2
·∇E
t
(h
(t)
) if h
(t)
/∈ H
(OAR)
EUD,t
h
(t)
if h
(t)
∈ H
(OAR)
EUD,t
(A.13)
which turns out equivalent to (A.11).
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