The segmentation problem in radiation therapy
Céline Engelbeen
Thèse présentée en vue de l’obtention du grade de docteur en sciences Juin 2010
Université Libre de Bruxelles Promoteur: Samuel Fiorini Faculté des Sciences
Université Libre de Bruxelles Département de Mathématique Boulevard du Triomphe - CP 216 1050 Bruxelles
Belgique
Email: cengelbe@ulb.ac.be
Thèse de doctorat présentée en séance publique le 30 juin 2010 à l’Université Libre de Bruxelles.
Jury:
Jean Cardinal, Secrétaire du jury, ULB Jean-Paul Doignon, Président du jury, ULB Konrad Engel, University of Rostock, Germany Samuel Fiorini, Promoteur, ULB
Horst Hamacher, University of Kaiserslautern, Germany Martine Labbé, ULB
Marc Pirlot, Université de Mons (Faculté Polytechnique)
First of all, I would like to warmly thank my advisor, Samuel Fiorini. Thank you for accepting to accompagny me in this adventure. Thank you for your help, your patience, your advice and your availability. Working with you has always been a pleasure. During these four years spent with you, I have not only discov- ered the incredible world of research, but I have also learnt a lot from a human point of view. In every respect, you were a wonderful advisor, and I could not have hoped for a better one.
I also adress special thanks to Martine Labbé, who was my advisor during my master thesis with Samuel and who gave me the opportunity to work on the problem which is the topic of the present thesis.
I also thank the F.R.I.A. for the grants it offered me and the financial support.
During this thesis, I had the chance to meet and work with several people.
Thank you Antje Kiesel for our nice collaboration. Your visit in Brussels and my stays in Rostock were such good experiences. Thank you very much for your friendship. I also thank you and the Department of Radiotherapy and Radio- Oncology at the University Medical Center Hamburg-Eppendorf for providing me with clinical intensity matrices. I also thank Konrad Engel for his collaboration during my stay in Rostock as well as Thomas Kalinowski for his code.
Special thanks to Çiğdem Güler for our collaboration and for welcoming me during my stay in Kaiserslautern. Many thanks to Horst Hamacher for the nu- merous discussions during this stay.
I would like to thank Daniele Catanzaro who helped me in the implementation part of my thesis. With your help and the good atmosphere you installed, I did not only manage to implement my program, but it was also a pleasure to do it.
I thank Maude Gathy and Yvik Swan for their help in the probabilistic aspects of this thesis.
Moreover, I thank Gwenaël Joret and Thibaut Lust for numerous discussions and precious advice all along my thesis.
To me the pratical part of this thesis was really important. I had the chance to have advice of practicians all along my thesis. This is why I want to thank Stefaan Vynckier from the radiation therapy service of Saint-Luc. Thank you for your advice, your answers and your courses. I also thank Pierre Scalliet from the same service and Stephane Simon from the Institut Bordet who helped me in the early stage of that work.
I want to thank Francis Buekenhout, Charlotte Bouckaert and Jacqueline Sen- gier for their support and interest concerning my future. It was really appeasing to talk with you.
I thank all my colleagues from the Department of Mathematics for their hos- pitality, their sympathy and their support.
I take the opportunity to address special thanks to my friends Maude, Sophie, Mélanie, Aurélie, Alessandra, Muriel, Aude, Sébastien, Michaël and Mathieu for being there in good and difficult times.
I would also like to thank my family, in particular my parents who have always supported me and have encouraged me in all my enterprises.
Last but not least, thank you Jacques for your help, your encouragements and your incredible patience. Thank you also for the love and hapiness you bring into my everyday’s life.
Brussels, June 2010.
We study the problem of decomposing a nonnegative integer matrix into a non- negative integer linear combination of certain binary matrices. The properties of these binary matrices can vary as well as the objective function taken into ac- count. This family of problems arises in the elaboration of radiation therapy plans.
First we have considered the problem of finding a decomposition of the input matrix that minimizes the sum of the coefficients using constrained binary matri- ces, the most important constraint being the consecutive ones property. For this first type of problem, our main contributions are:
• the determination of the complexity of the decomposition problem, where the binary matrices have to respect the interleaf distance constraint, as well as a polynomial time algorithm to solve it;
• a new polynomial time algorithm to solve the decomposition problem, where the binary matrices have to respect the interleaf motion constraint;
• the determination of the complexity of the decomposition problem where the binary matrices have to respect the tongue-and-groove constraint and where the input matrix is binary.
Second we have dealt with the problem of finding a decomposition of the input matrix that minimizes the number of binary matrices used in the decomposition.
For this problem our main contributions are:
• an exact formulation as well as an exact polynomial time algorithm to solve this problem for the case where the highest coefficient in the input matrix is smaller than a given constant;
• new lower bounds on the minimum number of binary matrices;
• a new heuristic to tackle the problem.
We have also generalized these results to the case where we want first to minimize the sum of the coefficients and afterwards the number of binary matrices used in the decomposition.
Finally we have studied the decomposition problem in case the set of binary matrices at our disposal do not allow to exactly decompose the input matrix (we are given an explicit set of binary matrices that can be used in the decomposition).
We have considered the problem of finding a decomposable nonnegative integer matrix whose`1-distance to the input matrix is minimized and whose`∞-distance to the input matrix is bounded by a given constant. For this version of the decomposition problem our main contributions are:
• a proof that in the particular case where the input matrix has only one row the problem can be solved in polynomial time;
• a proof that the general problem is not only NP-hard but is also hard to approximate within an additive error of O(mn) where m and n define the size of the input matrix;
• a proof that for the particular case where the input matrix has only two rows the problem is hard to approximate within an additive error ofO(n).
We give an overview of the dissertation at the end of Chapter 1 and give some outlook for future research in the Conclusion and outlook. This thesis is partly based on the following publications:
i. Constrained Decompositions of Integer Matrices and their Applications to Intensity Modulated Radiation Therapy with S. Fiorini [20]
ii. A closest vector problem arising in radiation therapy planning with S. Fiorini and A. Kiesel (University of Rostock) [21]
iii. Binary matrix decompositions without tongue-and-groove underdosage for radiation therapy planning with A. Kiesel (University of Rostock) [22].
Acknowledgments 3
Introduction 5
1 The segmentation problem: motivations and description 9
1.1 The elaboration of radiation therapy plans . . . 9
1.2 The segmentation problem . . . 14
1.3 Organization of the thesis . . . 20
2 The beam-on time problem 23 2.1 The standard decomposition algorithm . . . 25
2.2 The interleaf distance constraint . . . 29
2.3 The interleaf motion constraint . . . 38
2.4 The tongue-and-groove constraint . . . 44
2.4.1 Decompositions of binary matrices . . . 47
3 The cardinality of the decomposition 61 3.1 The one column case . . . 62
3.2 The one row case . . . 69
3.3 General case: an exact formulation . . . 73
3.4 Lower bounds via column generation . . . 80
3.4.1 The column generation algorithm . . . 80
3.4.2 Experimentations . . . 84
3.4.3 Heuristics . . . 86
4 The approximation problem 89
4.1 The closest vector problem . . . 91
4.2 Polynomial cases . . . 92
4.2.1 Minimum cost flow problem . . . 93
4.2.2 The minimum separation constraint . . . 95
4.3 General case . . . 96
4.3.1 Hardness . . . 97
4.3.2 Approximation algorithm . . . 100
4.4 Further hardness result . . . 104
4.5 Incorporating the beam-on time into the objective function. . . . 108
Conclusion and outlook 111
Notations 113
Index 115
Bibliography 117
The segmentation problem: motivations and description
The segmentation problem depicted in the present thesis is motivated by medicine, and more particularly by radiation therapy. Indeed, the different versions of this problem that we consider here arise in the elaboration of radiation therapy plans.
In this chapter we first give a small introduction to radiation therapy and explain how the segmentation problem occurs in the determination of the plans. After- wards we present more formally the different exact and approximate versions of the segmentation problem. Finally, we give an outline of the dissertation.
1.1 The elaboration of radiation therapy plans
In the European Union, two deaths out of three are caused by cancer or circulatory diseases. Cancer is responsible of 25% of the deaths and is the first cause of mortality for the people aged between 45 and 64 years1. Nowadays, there exist three main ways to fight cancer: surgery, chemotherapy and radiation therapy.
Radiation therapy consists in sending radiations to a tumor to eradicate it.
Thanks to the many progresses that have been made in medical imaging, physi- cians and radiophysicists have at their disposal highly precise images of the pa- tient’s body. With the help of the resulting 3D model of the patient, radiation can be directed in a better way and hence the efficiency of the treatment is improved.
1data from the website of the European Public Health Alliance, see http://www.epha.org/a/2352
This is why these last years radiation therapy has become one of the most used treatment against cancer. It is prescribed for more than 50% of the cancer pa- tients (see [43]). In addition to medicine and physics, mathematics and computer science are essential tools to elaborate the treatments.
After the cancer has been diagnosed and the radiation therapy sessions have been prescribed, the physician has to locate the tumor as well as the organs situated in the radiation field, called the organs at risk. The physician also has to determine the different dosage he wants to deliver in each of them and has to define a lower bound on the dosage for the tumor (which represents the minimum amount of radiation that is needed to have a sufficient control of the tumor) and an upper bound for each organ at risk (which represents the maximum amount of radiation that an organ can received without damaging). Of course, these bounds usually are conflicting (see Figure 1.1).
Figure 1.1: Prescribed dose bounds for the target (lower bound) and for the organs at risk (upper bounds).
In order to simultaneously send enough radiation to destroy the tumor and to protect the organs at risk, one uses several radiation angles in such a way that the tumor is in the radiation epicenter and the organs located in the radiation field are different for each angle (see Figure 1.2). Radiation is delivered by a linear accelerator (see Figure 1.3) whose arm is capable of doing a complete circle around the patient and hence one can obtain the different radiation angles. Designing a radiation therapy plan that respects the different bounds of dosage given by the physician is a complex optimization problem that is usually tackled in three steps:
Step 1 The first step is to determine the different radiation angles [18].
Figure 1.2: Different angles are used to protect the organs at risk.
Step 2 The second step is to determine the exact intensity function that will be sent to the patient for each angle. This intensity function is encoded as a m×n nonnegative integer matrix A, in which each entry represents an elementary part of the radiation beam, called a bixel or a beamlet, and the value of each entry is the intensity that we want to send through the corresponding bixel. This problem is usually called the intensity or the fluence map optimization problem [27].
Step 3 The final step consists in decomposing the intensity matrix as a linear combination of matrices that represent an intensity that the linear accelera- tor is able to deliver. This last step is called the segmentation problem (see page 17 for a formal definition).
In this thesis we focus on step 3, that is, on the segmentation problem. So, we assume that the radiation angles and the intensity matrices are known. In order to realize each intensity matrix, hospitals are commonly equipped with a complete treatment unit (see Figure 1.3) formed by a couch, a linear accelerator and a so calledmulitleaf collimator (MLC, see Figure 1.4) which is placed between the radiation source and the patient. The MLC is formed by several pairs of metallic leaves (usually between 40 and 60 pairs) which are capable of blocking the radiation. Our aim is to find a set of positions of the leaves of the MLC in such a way that a dosage corresponding to the intensity matrix is delivered.
Mathematically, the segmentation step amounts to decomposing matrixAinto
Figure 1.3: A complete treatment unit (gantry with a linear accelerator, MLC, and couch) at Saint-Luc Hospital (Brussels, Belgium).
Figure 1.4: The multileaf collimator.
a nonnegative integer linear combination of some binary matrices whose structure can be reproduced by the leaves of the MLC (these binary matrices are called segments, see page 14 for a definition and see Figure 1.5 for an illustration).
For each row of such a segment, the MLC has a left and a right leaf. The radiation is blocked at bixels that are covered by a leaf, and so only passes through bixels located between the two leaves. This is why a segment is a binary matrix which has to satisfy the consecutive ones property (see page 14), which means that in each row the ones have to be grouped in a single block.
⇐⇒
0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0
Figure 1.5: Leaves positions for the MLC and the corresponding segment.
Example 1.1. The intensity matrix
A =
5 5 3 3 2 2 5 2 5 3 3 2 2 5 5 3
can be decomposed in the following way:
A = 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
+ 1
1 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1
+ 2
1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0
,
= 2S1+S2+ 2S3.
In picture (white cells correspond to active bixels, that is, bixels where the radi- ation passes, and black cells to inactive bixels; see page 14 for an exact definition of bixel):
S1 = S2 = S3 =
In segmented (or step-and-shoot) intensity modulated radiation therapy, the leaves are never moving while the patient is irradiated. The decomposition of the intensity matrix given by Example 1.1 means that, in practice, when the treatment starts, all the leaves are retracted and all the bixels are exposed (this structure
is given by S1). We irradiate the patient during a time equal to u1 = 2 monitor units (the given units of time). After these two monitor units, the emission of ra- diation is stopped and the leaves move to take the structure given byS2. Once the leaves are in the required positions the patient is irradiated again, during u2 = 1 monitor units, and so on. There also exists a dynamic mode. In this mode, the leaves continuously move during the radiation. We do not consider the dynamic mode in the present thesis.
The segmentation problem amounts to finding a set S := {S1, . . . , SK} of binary matrices which respect the consecutives ones property as well as con- straints imposed for dosimetric and/or technical reasons, and corresponding inte- gers u1, . . . , uK ∈Z+ such that
A =
K
X
t=1
utSt
The coefficients are required to be integers because in practice the radiation can only be delivered for times that are integer multiples of monitor units.
1.2 The segmentation problem
We formally explain here what the segmentation problem is and state related definitions and notations. LetA be a nonnegative integer m×n matrix.
Definition 1.2. The entries of the intensity matrix are called the bixels. A bixel is said exposed if the radiation is able to pass through that bixel.
Throughout the thesis, [k] denotes the set {1,2, . . . , k} for an integer k, and [`, r) denotes the set{`, `+ 1, . . . , r−1}for integers ` and r with ` < r. We also allow` =r where [`, r) =∅.
Definition 1.3. Am×n matrixS = (sij)respects theconsecutive ones property if and only if there are integral intervals [`i, ri) for i∈[m] such that
sij =
(1 if j ∈[`i, ri), 0 otherwise.
Definition 1.4. Asegment is a binary matrix that respects the consecutive ones property.
Hence, for each row of a segment, there is an interval (possibly empty) of ones.
For a row i of the segment S we denote by [`i, ri) this interval of ones, where `i denotes the position of the left leaf, and ri the position of the right one (we say that a leaf is in position j if the leaf is located between columns j and j + 1).
Formally,
sij = 1 ⇔`i 6j < ri.
The interval [`i, ri) also gives the position of the first exposed bixel of row i and the position of the first non-exposed bixel after the block of exposed ones. Figure 1.6 gives an example of how we number the positions of the leaves and the bixels.
1 2 3 4 5 6 7 leaf positions:
bixels numbering: 1 2 3 4 5 6
Figure 1.6: A segment formed by a single row with the left leaf in position 4 and the right leaf in position 6. We can rewrite this segment by[4,6). Bixels 4 and 5 are the only exposed bixels.
We can rewrite the segments S1, S2 and S3 from Example 1.1 as:
S1 =
[1,5) [1,5) [1,5) [1,5)
S2 =
[1,5) [3,4) [1,4) [2,5)
S3 =
[1,3) [3,4) [1,2) [2,4)
.
In clinical applications several constraints may arise that reduce the number of segments whose structure can be reproduced by the leaves of the MLC. Since the zeros of the segments are generated by the leaves of the MLC, if a constraint appears on the positions of the leaves, this constraint will lead to a constraint on the segments.
Definition 1.5. A segment is said to be deliverable if it satisfies the considered constraint(s) on the segments.
Definition 1.6. A decomposition of a matrix A is a nonnegative integer linear combination of deliverable segments which gives A.
Now we describe the constraints considered in this thesis.
The interleaf distance constraint (denoted by IDC) requires that the dis- tance between the positions of two left (resp. right) leaves never exceeds a given constantd. In other words, a segment S = ([`i, ri))i∈[m]satisfies the IDC if and only if we have
|`i−`i0| 6 d ∀i, i0 ∈[m]
|ri−ri0| 6 d ∀i, i0 ∈[m].
We can assume that d>1.
The interleaf motion constraint (denoted by IMC) forbids the left leaf of some row to overlap the right leaf of an adjacent row. A segment S = ([`i, ri))i∈[m] satisfies the IMC if and only if we have
`i 6 ri+1 ∀i∈[m−1]
`i 6 ri−1 ∀i∈[2, m].
The tongue-and-groove constraint (denoted by TGC) requires to simultane- ously irradiate the bixels of the same column as much as possible. Mathe- matically, this corresponds to the following two constraints:
aij 6ai−1,j =⇒ sij 6si−1,j
aij >ai−1,j =⇒ sij >si−1,j.
The minimum separation constraint (denoted by MSC ) imposes a mini- mum leaf openingλ∈[n]in each open row of the irradiation field. More for- mally, a deliverable segmentSgiven by its leaf positions([`1, r1), . . . ,[`m, rm)) satisfies the minimum separation constraint if and only if
ri > `i ⇒ri−`i >λ, ∀i∈[m].
In opposition to the IDC, the IMC and the TGC, we cannot decompose every intensity matrix under the MSC for every leaf opening λ >0.
These four constraints are illustrated in Figure 1.7.
Finally we also consider the case where the set of deliverable segments S is explicitly given. Again in this case we cannot decompose any intensity matrix.
We can define the segmentation problem as follows:
< >
6d
S S
S S
S S
SS
S
S S
S S
S SS
S
S S
S S
S SS
The interleaf distance constraint The interleaf motion constraint
< >
>λ
The tongue-and-groove constraint The minimum separation constraint
Figure 1.7: The four constraints on the segments considered in this thesis.
The segmentation problem:
Input: A nonnegative integer matrix A.
Goal: Finding a decomposition ofA.
In the literature two main objective functions are considered (see [16] for a survey), each of them, of course, has to be minimized:
i. the sum of the coefficients of the decomposition;
ii. the number of used deliverable segments in the decomposition.
Definition 1.7. The sum of the coefficients of a decomposition is called thebeam- on time of that decomposition.
Definition 1.8. The cardinality of a decomposition is the number of used seg- ments in the decomposition.
Since the coefficientutof a segmentStgives the number of monitor units during which we irradiate the patient with the leaves in the positions given by St, the beam-on time of a decomposition corresponds to the total time (in monitor units) during which the radiation is emitted. The problem of finding a decomposition of the intensity matrix that minimizes the beam-on time is called the beam-on time problem (denoted by BOTP) and is stated as follows:
The beam-on time problem (BOTP):
Input: A nonnegative integer matrix A.
Goal: Finding a decomposition A=PK
t=1utSt where ut ∈ Z+ and St is a deliv- erable segment for allt ∈[K], such thatPK
t=1ut is minimized.
Measure: The beam-on time :=PK t=1ut.
If the beam-on time is important to minimize, it could be also desirable to de- crease the time of the radiation therapy session. Indeed, if the radiation therapy sessions last less time this allows the hospital to treat more patients. Moreover, this is also more comfortable for the patient. The problem of finding a decomposi- tion of the intensity matrix that minimizes the cardinality is called thecardinality problem (denoted by CP) and is stated as follows:
The cardinality problem (CP):
Input: A nonnegative integer matrix A.
Goal: Finding a decomposition A=PK
t=1utSt where ut ∈ Z+ and St is a deliv- erable segment for allt ∈[K], such that the cardinalityK is minimized.
Measure: The cardinality :=K.
In the literature the problem of finding a decomposition of the intensity matrix that minimizes first the beam-on time and then the cardinality is also considered.
This problem is called the lex-min problem (denoted by LMP) and is stated as follows:
The lex-min problem (LMP):
Input: A nonnegative integer matrix A.
Goal: Finding a decomposition A = PK
t=1utSt where ut ∈ Z+ and St is a de- liverable segment for all t ∈ [K] whose beam-on time PK
t=1ut equals the minimum beam-on time and such that the cardinality K is minimized.
Measure: The cardinality :=K.
Let us notice that in general, it is not possible to find a decomposition that minimizes the cardinality and the beam-on time simultaneously. Minimizing the beam-on time is given the highest priority.
As we have seen above, there exists some constraints (like the MSC) that do not allow to decompose every intensity matrix A with deliverable segments. In this case, the problem is to find a matrixB that is decomposable and satisfies
kA−Bk∞:= max
i∈[m], j∈[n]|aij−bij|6C, (1.1)
for some given integer constantC (possibly, such a matrixB does not exist), and minimizes
kA−Bk1 := X
i∈[m], j∈[n]
|aij −bij|. (1.2)
The value of kA−Bk1 is called the total change (denoted by TC). The problem of finding a matrixB that is decomposable and that satisfies (1.1) and minimizes (1.2) is called the approximation problem (denoted by APXP). It is stated as fol- lows:
The approximation problem (APXP):
Input: A nonnegative integer matrix A and a set {S1, . . . , Sk} of deliverable segments.
Goal: Finding a matrix B such that B = Pk
t=1utSt where ut ∈ Z+ and St is a deliverable segment for all t∈[k],kA−Bk∞6C and such that kA−Bk1 is minimized.
Measure: The total change TC :=kA−Bk1.
1.3 Organization of the thesis
The thesis is organized as follows: Chapter 2 is devoted to the BOTP. After recalling the results of Baatar, Hamacher, Ehrgott and Woeginger [6] for the BOTP where all the segments are deliverable (in other words we do not consider extra constraints on the segments) we consider the BOTP under the IDC. We present a new formulation of this last problem as an integer program whose matrix constraint is totally unimodular. This leads to the proof that the BOTP remains polynomial under the IDC. We also present a fast algorithm based on the search of a maximal potential in a special network to solve the problem. Afterwards we consider the BOTP under the IMC. After recasting the results of Baataret al. [6]
in our framework we adapt the tools developed for the BOTP under the IDC to the BOTP under the IMC and also present a fast algorithm to solve it. We also indicate how to deal with both the IDC and the IMC at the same time.
In the last section of that chapter we consider the BOTP under the TGC. We have studied the particular case where the intensity matrix is a binary matrix (to this day, the complexity of the general case remains open). We provide for this particular case a polynomial procedure that finds a decomposition of the intensity matrix respecting the TGC with minimum beam-on time.
In Chapter 3 we consider the problem of minimizing the number of used seg- ments in the decomposition (CP and LMP). For both these problems all segments are deliverable. We first start this chapter with two particular cases: the one where the intensity matrix has a single column and the one where the intensity matrix has a single row.
Collins, Kempe, Saia and Young [14] have proved that the single column case of the CP is NP-hard. We present an exact exponential formulation for both the CP and the LMP in that particular case as well as some lower bounds on the number of segments used in the decomposition.
Bansal, Coppersmith and Schieber [8] have proved that in the single row case the CP and the LMP have the same optimal value (this is in general not true).
We give an exact formulation that leads to a polynomial time algorithm to solve the one row case of the CP and the LMP under the assumption that the highest value of the intensity matrix is bounded by a given constantH ∈Z+.
In the next section we generalize this exact formulation to the general case for both the CP and the LMP. Then we give a proof that the CP (and the LMP) is also polynomial when the entries of the intensity matrix are bounded byH, hence generalizing the single row result to arbitrary matrices. This formulation is based on a network with length-vectors on the arcs.
We conclude this section by using this network for a column generation al-
gorithm in order to find lower bounds on the CP and the LMP. We present the results of our implementation and introduce a heuristic based on that implemen- tation.
Finally in Chapter 4 we deal with the APXP. We assume that the set of deliverable segments is explicitly given. After having introduced how we can rephrase the APXP as a special case of a certain closest vector problem (CVP), we prove some general results on this CVP. We start by observing that in the one row case the constraint matrix of the CVP is totally unimodular and the problem can then be solved in polynomial time. We explain that this result implies that, when the segments have to satisfy the MSC, the problem is polynomial. We also provide a direct reduction to the minimum cost flow problem. We afterwards show that the general case is NP-hard to approximate within an(ln 2−ε)mn additive error, for all ε >0 (which in particular implies that the APXP and the CVP are NP-hard). We conclude the section with an analysis of a natural algorithm for the problem based on randomized rounding [39] which provides a matrix B with kA−Bk∞ 6 C +O(√
mnlnmn) and kA−Bk1 6 OPT + O(mn√
mnlnmn) where OPT denotes the minimal total change.
We provide a further hardness of approximation result for the case where the intensity matrix is a 2×n matrix: it is NP-hard to approximate the problem within an additive error of ε n, for some ε > 0 (in particular, the corresponding restriction of the APXP is NP-hard).
We conclude this chapter by generalizing the results to the case where one does not only want to minimize the total change, but a combination of the total change and the beam-on time.
Finally, the Conclusion and outlook are devoted to future research.
The beam-on time problem
This chapter is devoted to the beam-on time problem (BOTP). This problem consists in decomposing a nonnegative integer matrix A of size m ×n into a nonnegative integer linear combination of segments so as to minimize the sum of the coefficients. To recall, a segment is a binary m×n matrix respecting the consecutive ones property. Thus, anm×n matrixS = (sij)is a segment iff there are integral intervals[`i, ri) for i∈[m] such that
sij =
(1 if j ∈[`i, ri), 0 otherwise.
The objective function corresponds to minimizing the total time during which the patient is irradiated, called the beam-on time. Minimizing the beam-on time allows to reduce secondary effects due to diffusion.
The (unconstrained) BOTP is known to be polynomial and efficient methods for solving it have been proposed by several authors (see [1, 6, 17, 28, 30, 32, 44, 47]). In some of the given references, also constrained versions of the BOTP are considered, that is version of the problem where the segments have to satisfy some extra constraints in order to be deliverable.
The approach of Baatar et al. [6] for the unconstrained case roughly consists in first finding different positions in which the leaves have to be placed to decom- pose the intensity matrixA, and afterwards decomposingAwith the help of these
leaves positions with the standard decomposition algorithm (see page 27) in time O(Km), where K is the number of used segments. Let us notice that we do not try to minimize this number K here. The problem of minimizing K is the topic of the next chapter.
We consider extra constraints that reduces the number of deliverable segments.
In particular, we consider the interleaf distance constraint (IDC), the interleaf motion constraint (IMC) and the tongue-and-groove constraint (TGC) (the con- straints are defined on page 16). As for the unconstrained case, Baataret al. have solved the BOTP under the IMC in polynomial time by first looking for positions in which the leaves have to be placed to decompose the intensity matrix Aunder the IMC and afterwards decomposing A with the standard decomposition algo- rithm. They present a O(m2n) algorithm to find the positions of the leaves by solving a sequence ofn multiobjective integer programs. We revisit their method and simplify the proofs and the algorithm by performing a change of variables on their formulation that allows to use potentials in a network. By using this method we obtain anO(mnlogm)algorithm to find the positions of the leaves. Moreover we can adapt this algorithm to consider the IMC and the IDC at the same time and with the same complexity.
In the first section of this chapter we recall the standard decomposition of algorithm of Baatar et al. [6].
In Section 2.2 we present our results for the BOTP under the IDC by first showing that this last problem is polynomial. We then provide an O(mn) al- gorithm that finds the positions of the leaves by solving a maximum potential problem in an acyclic digraph. For the case where A is binary, see also [19].
In Section 2.3 we explain how to adapt our approach for finding the positions of the leaves under the IDC to the same problem under the IMC. We also give the O(mnlogm) algorithm for this last problem. Moreover we explain how to obtain an O(mnlogm) algorithm to find the positions of the leaves for decomposing A with both constraints. The results from Sections 2.2 and 2.3 are joint work with S. Fiorini [20].
We complete this chapter by considering the TGC. We show that the BOTP under the TGC is polynomial when the intensity matrix A is binary. This result was obtained together with A. Kiesel and is the topic of [22].
2.1 The unconstrained case and the standard de- composition algorithm
In this section we recall the approach of Baatar et al. [6] to tackle the uncon- strained BOTP. Since we do not consider any constraints in that section, the notion of segments and deliverable segments coincide.
Let ∆ = ∆(A) = (δij) be the m×(n+ 1) matrix defined as δij :=aij−ai,j−1 ∀i∈[m], ∀j ∈[n+ 1],
where we letai,0 =ai,n+1 := 0. If δij >0we know thataij > ai,j−1. Therefore, for at least aij −ai,j−1 = δij monitor units, the radiation has to pass through bixel (i, j)and not through bixel (i, j−1). To achieve this the left leaf in the i-th row has to be placed in positionj for at leastδij monitor units. So, the positive entries of the matrix∆give a lower bound on the time during which the left leaves have to be in a certain position. Similarly, if δij < 0, then aij < ai,j−1. The radiation has to pass through bixel (i, j−1) and not through bixel (i, j) for at least −δij monitor units. We therefore have to place the right leaf of thei-th row in position j for at least −δij monitor units. So, the negative entries of the matrix ∆ give a lower bound on the time during which the right leaves have to be in a certain position. Let ∆+ = δij+
and ∆− = δij−
be the matrices of size m ×(n+ 1) defined by:
δij+:= max{0, δij} and δij−:= max{0,−δij}.
The above discussion immediately leads to the following lower bound on the total beam-on time.
Lemma 2.1 (Baatar et al. [6]). Letting c(A) denote the beam-on time of an optimal solution to the unconstrained BOTP, we have c(A) > max{ci(A) : i ∈ [m]} where
ci(A) :=
n+1
X
j=1
δij+=
n+1
X
j=1
δij− (2.1)
is the complexity of thei-th row. We also call c(A) the complexity of the matrix A.
As observed for example by Baatar et al. [6] and Bortfeld, Kahler, Waldron and Boyer [11], it turns out that the bound given by Lemma 2.1 is exact. Other authors, for example, Engel [17] and Kalinowski [28], have proved the same result.
Since all the rows of A are independent, each row can be decomposed sepa- rately. For the time being, we consider the i-th row of A for some fixed i and seek a decomposition of this row with beam-on-time ci(A). We may regard the matrix ∆+ (resp. ∆−) as a multiset giving the number of unavoidable left (resp.
right) leaf positions; see Figure 2.1 for an illustration. Note that this naturally determines an ordering of the unavoidable leaf positions.
∆+ = (5 0 0 2 0 ,) ∆−= (0 4 0 0 3 )
Figure 2.1: Left and right leaf positions for the row (5 1 1 3).
The idea of the algorithm of Baatar et al. [6] is to match each unavoidable left leaf position with some unavoidable right leaf position in order to obtain a decomposition of the given intensity matrixA. We point out that it is important to carefully pick the matching otherwise one might obtain a matrix different fromA.
A matching that always works is constructed by iteratively associating the first unmatched unavoidable left leaf position with the first unmatched unavoidable right leaf position. Actually, any matching between the left and the right leaf positions works, as long as we always have `i 6 ri for all i ∈ [m], where `i and ri respectively denote the position of the left and the right leaf as illustrated in Figure 1.6 on page 15.
The resulting algorithm can be extended in two ways. First, by considering all rows simultaneously and independently, the algorithm can produce a decomposi- tion of the whole intensity matrixA. Second, we will see in the next section that for the constrained cases it is necessary to replace the pair of matrices ∆+ and
∆− by a more general pair of nonnegative integer matricesD+ and D− such that D+−D−= ∆. After implementing these two extensions, we obtain the standard decomposition algorithm (see Algorithm 1 for a formal description). To solve the BOTP in the unconstrained case, we takeD+ = ∆+ and D−= ∆−.
Algorithm 1Standard decomposition algorithm.
Input: Nonnegative integer matrices D+ and D− of size m×(n+ 1)with D+− D− = ∆ representing leaf positions, given as multisets D1+, . . . , D+m and D1−, . . . , Dm−, one for each row of A.
Output: A decomposition ofA with the prescribed leaf positions.
while there exists some i∈[m] such thatD+i 6=∅do for all i∈[m] such that Di+6=∅ do
`i ←minD+i ; ri ←minD−i
ui ←min{multiplicity of `i in Di+, multiplicity of ri inD−i } end for
for all i∈[m] such that Di+=∅ do
`i ←n+ 1; ri ←n+ 1 ui ←+∞
end for
u←min{u1, . . . , um}
for all i∈[m] such that Di+6=∅ do
remove u copies of `i (resp. ri) from D+i (resp. Di−) end for
outputu times the segment ([`i, ri))i∈[m]
end while
Example 2.2. For the matrix
A =
5 5 3 3 2 2 5 2 5 3 3 2 2 5 5 3
of Example 1.1 (on page 13), the standard decomposition algorithm gives the following matrices and multisets:
D+ = ∆+ =
5 0 0 0 0 2 0 3 0 0 5 0 0 0 0 2 3 0 0 0
D− = ∆− =
0 0 2 0 3 0 0 0 3 2 0 2 0 1 2 0 0 0 2 3
D1+:={1,1,1,1,1} D−1 :={3,3,5,5,5}
D2+:={1,1,3,3,3} D−2 :={4,4,4,5,5}
D3+:={1,1,1,1,1} D−3 :={2,2,4,5,5}
D4+:={1,1,2,2,2} D−4 :={4,4,5,5,5}
and the following decompositon:
A= 2
1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0
+
1 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1
+ 2
1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1
.
Notice that the beam-on time of that decomposition is equal to 5, which is optimal by Lemma 2.1.
Theorem 2.3. The standard decomposition algorithm gives a decomposition ofA whose beam-on time equals the size of the largest multiset Di+ or D−i , i∈[m].
Proof. The key property we use to prove the correctness of the standard decom- position algorithm is the equation D+ − D− = ∆. Letting D+ := (d+ij) and D−:= (d−ij), the latter equation is equivalent to:
j
X
q=1
d+iq−
j
X
q=1
d−iq =aij ∀i∈[m], ∀j ∈[n+ 1]. (2.2)
We claim that as soon Equation (2.2) is satisfied, Algorithm 1 outputs a decom- position of A whose beam-on time equals the size of the largest multiset D+i or Di−.
Recall that we let ai,n+1 = 0by convention. Thus Equation (2.2) forj =n+ 1 implies that the multisets D+i and D−i have the same size. So at the end of the algorithm bothD+i andDi−are empty, for alli. Equation (2.2) for the other values of j implies that we always have `i 6 ri. So the algorithm outputs deliverable segments St for all t ∈ [K]. The sum of all the coefficients ut output by the algorithm is clearly the size of the largest multiset Di+ or D−i . It remains to verify that the sum of the decomposition output by the algorithm isA. Fix some i∈[m] and some j ∈[n]. The bixel (i, j) is irradiated exactly when the position of the left leaf is less than or equal to j (otherwise this bixel is covered by the left leaf) and the position of the right leaf is strictly greater than j (otherwise this bixel is covered by the right leaf). In other words, the total number of intervals [`i, ri) which expose the bixel (i, j) is the number of intervals [`i, ri) such that
`i 6j minus the number of these intervals such that ri 6j. Since the algorithm associates the first unmatched left leaf position with the first unmatched right leaf position, the bixel (i, j) is irradiated during Pj
q=1d+iq −Pj
q=1d−iq = aij monitor units, as required.
As pointed out before, in the unconstrained case we take D+ = ∆+ and D− = ∆−. In this case we have D+ − D− = ∆ and thus Equation (2.2) is satisfied. By what precedes, Algorithm 1 finds a decomposition ofAwhose beam- on time equals the lower bound given by Lemma 2.1, that is, the complexity of A. Therefore, the decomposition of A output by the standard decomposition algorithm is optimal. The complexity of the algorithm is O(Km) whereK is the number of segments output. Note that we have K 6 mn, so in particular the standard decomposition algorithm runs in timeO(m2n). Because the matrices ∆,
∆+ and ∆− and the corresponding multisets can be computed from A in O(mn) time the unconstrained BOTP can be solved inO(mn+Km)time. This concludes our description of the algorithm of Baataret al. for the unconstrained case.
2.2 The interleaf distance constraint
In this section we consider the BOTP under the interleaf distance constraint (IDC). This constraint asks that the distance between the ends of two left (or two right) leaves cannot be bigger than a constant d; see Figure 2.2 for an example with d = 2. A segment S = ([`i, ri))i∈[m] respects the IDC if and only if the
following inequalities hold:
|`i−`i0| 6 d ∀i, i0 ∈[m] (2.3)
|ri−ri0| 6 d ∀i, i0 ∈[m]. (2.4)
ok bad
Figure 2.2: Whend= 2, the left segment respects the IDC while the right segment does not because the left leaf of the first row is in position 4 and the left leaf of the last row is in position 1 and |1−4|= 3 > d= 2.
We prove that the BOTP under the IDC is polynomial for integer matrices.
We also give an algorithm for this problem of complexityO(mn+Km). The idea in this section is to first find the positions of the leaves we need for the decom- position, and afterwards use the standard decomposition algorithm to obtain the decomposition.
In order to motivate our model, we first consider the particular case where the intensity matrixA is such that all its rows have the same complexity, that is, c1(A) = c2(A) = · · · =cm(A) =: T (cf. Equation (2.1)). For q = {1,2, . . . , T} = [T], let `qi (resp. rqi) denote the q-th unavoidable left (resp. right) leaf position in the i-th row of A.
Lemma 2.4. When all rows of the intensity matrixA have the same complexity, say T, there exists a decomposition of A of beam-on time T satisfying the IDC if and only if we have
|`qi −`qi0|6d and |rqi −riq0|6d
for allq ∈[T], and all i, i0 ∈[m]. In particular, we can tell in polynomial time if such decomposition of A exists.
Proof. To show the “if” direction we simply use the standard decomposition al- gorithm (cf. Algorithm 1) with D+ = ∆+ and D− = ∆−. Because the algorithm associates the first unmatched left leaf position with the first unmatched right leaf
position for each row and we have |`qi −`qi0|6 d and |rqi −riq0| 6d for all q, i and i0, all the segments output by the algorithm respect the IDC.
We now prove the “only if” direction. Consider a decomposition ofAsatisfying the constraint with beam-on time T. Then all unavoidable leaf positions and no further positions are used. Suppose, for instance, that there exist row indices i and i0 and some integer q such that`qi +d < `qi0. Since we have `qi0 6`qi for q0 < q and `qi000 > `iq0 for q00 > q, all first q unavoidable left leaf positions in the i-th row are incompatible with all lastT −q+ 1unavoidable left leaf positions in the i0-th row. Hence at most q−1 left leaf positions in the i0-th row are compatible with the firstq unavoidable left leaf positions in thei-th row, a contradiction.
Now let us consider the general case. Consider a decomposition of A. Just as we defined the difference matrix∆ = ∆(A)of the intensity matrixAwe can define a difference matrix ∆(S) for any segment S. This allows us to “differentiate” the considered decomposition ofA in the following way:
A=
K
X
t=1
utSt =⇒ ∆(A) =
K
X
t=1
ut∆(St).
The last equation implies for the positive and negative parts of the considered matrices:
K
X
t=1
ut∆+(St) = ∆+(A) +X and
K
X
t=1
ut∆−(St) = ∆−(A) +Y
for some nonnegative integer matrices X and Y of size m× (n + 1) (here, we use the subadditivity of the functions x 7−→ x+ = max{0, x} and x 7−→ x− = max{0,−x}). We interpret any entry xij of X as the number of extra left leaf positions equal toj in the i-th row, and similarly for the matrixY. Now we have 0 =
K
X
t=1
ut∆(St)−∆(A) =
K
X
t=1
ut(∆+(St)−∆−(St))−(∆+(A)−∆−(A)) = X−Y, so the matrices X and Y are equal.
We obtain that the number of extra leaf positions does not depend on the side (left or right). In conclusion, any decomposition of Adetermines a unique matrix W describing the extra leaf positions.
Conversely, if we have a nonnegative integer matrix W of size m×(n + 1) we can infer a decomposition of A as follows. We consider that the matrices
∆+ +W and ∆−+W respectively prescribe the left and right leaf positions of some decomposition. We then apply the standard decomposition algorithm with
D+= ∆++W and D− = ∆−+W. This produces a decomposition ofA because we have D+−D−= (∆++W)−(∆−+W) = ∆.
In conclusion, the matrixW implicitly defines a decomposition. We can require without loss of generality that ∆+ +W (or ∆−+W) has constant row sums.
Otherwise, row sums are not constant, which means that in the last segments produced by the standard decomposition algorithms, some rows will be closed.
That is, we will have `i =ri for some i. By default, the standard decomposition algorithm will set`i =ri =n+ 1in this case. Asking that row sums are constant amount to force us to completely specify all leaf positions, even for the rows that are closed. This amounts to asking that the number of left (or right) leaf positions is the same for all rows. Therefore, all segments used in the decomposition have well defined left and right leaf positions for each row if we use the standard decomposition Algorithm 1. Observe that the beam-on time is then simply the row sum of any row of∆++W (or ∆−+W).
We can now state and prove a result generalizing Lemma 2.4. As a direct consequence, we obtain an integer programming formulation of BOTP under the IDC.
Proposition 2.5. Let ∆+, ∆− be defined as above. Consider a matrix W of extra leaf positions such that ∆+ +W (and hence ∆− +W) has constant row sums. Then W induces a decomposition of A satisfying the IDC if and only if, for all i, i0 ∈[m], we have
j
X
q=1
(δiq++wiq)6
j+d
X
q=1
(δ+i0q+wi0q) ∀j ∈[n−d+ 1]; (2.5)
j
X
q=1
(δiq−+wiq)6
j+d
X
q=1
(δ−i0q+wi0q) ∀j ∈[n−d+ 1]. (2.6) The beam-on time of the decomposition induced by W is the row sum of any row of ∆++W.
Before proving the proposition, we offer an interpretation of Equations (2.5) and (2.6). Equation (2.5) says that the number of left leaf positions smaller than or equal to j in the i-th row does not exceed the number of left leaf positions smaller than or equal to j +d in the i0-th row. The interpretation of (2.6) is similar.
Proof. Let T denote the row sum of any row of ∆+ +W, and let `qi (resp. riq) denote the q-th left (resp. right) leaf position for the i-th row, as described in
∆++W (resp.∆−+W). Then Equation (2.5) is equivalent to
|`qi −`qi0|6d ∀q∈[T] (2.7)
and Equation (2.6) is equivalent to
|riq−rqi0|6d ∀q∈[T] (2.8) Indeed, Equation (2.5) implies (2.7) because `qi +d < `qi0 for some q implies, for j =`qi,
j
X
q=1
(δiq++wiq)>
j+d
X
q=1
(δi+0q+wi0q).
Conversely, the latter inequality implies that there exists some q such that `qi + d < `qi0. A similar argument yields the equivalence of Equations (2.6) and (2.8).
Now we can readily follow the proof of Lemma 2.4 to conclude the proof of the result.
Thanks to Proposition 2.5, we can model the BOTP under IDC as follows:
(BOTP-IDC) min T s.t.
n+1
X
q=1
(δiq++wiq) =T ∀i; (2.9)
j
X
q=1
(δiq++wiq)6
j+d
X
q=1
(δi+0q+wi0q) ∀j, i, i0 (i6=i0); (2.10)
j
X
q=1
(δiq−+wiq)6
j+d
X
q=1
(δi−0q+wi0q) ∀j, i, i0 (i6=i0); (2.11) wij >0 ∀i, j;
wij ∈Z ∀i, j. (2.12)
In order to solve the above IP we first rewrite it by considering new variablesπ0, πij for i ∈ [m] and j ∈ [n+ 1] and πn+2. The relationship between the old and the new variables is as follows:
πij −π0 := −
j
X
q=1
wiq ∀i∈[m], j ∈[n+ 1], πn+2−π0 := −T.
Furthermore, we replace (2.9) by two inequalities. We thus obtain the following IP:
(BOTP-IDC’) max πn+2−π0
s.t. πn+2−πi,n+1 6 −
n+1
X
q=1
δiq+ ∀i∈[m];
πi,n+1−πn+2 6
n+1
X
q=1
δiq+ ∀i∈[m];
πi0,j+d−πij 6
j+d
X
q=1
δi+0q−
j
X
q=1
δiq+ ∀j, i, i0 (i6=i0); (2.13) πi0,j+d−πij 6
j+d
X
q=1
δi−0q−
j
X
q=1
δiq− ∀j, i, i0 (i6=i0); (2.14)
πi1−π0 6 0 ∀i∈[m];
πij −πi,j−1 6 0 ∀i, j >1;
πij ∈ Z ∀i, j. (2.15)
Notice that the objective value of (BOTP-IDC) is the negative of the objective value of (BOTP-IDC’).
The above formulation involves a system of difference constraints. It is well known that such a system can be represented by a network D = (V, E) called the constraint graph (see Ahuja, Magnanti and Orlin [2, p. 103]). The constraint graph has one vertex i for each variable xi and one arc (i, j) of length b for each constraint of the type xj −xi 6b. In our case, the set of vertices is
V :={0} ∪ {(i, j) :∀i∈[m],∀j ∈[n+ 1]} ∪ {n+ 2}.
We denote the set of arcs by E and define it as:
E := {(0,(i,1)) : ∀i∈[m]}
∪ {((i, j),(i, j+ 1)) : ∀i∈[m], ∀j ∈[n]}
∪ {((i, j),(i0, j+d)) : ∀i6=i0 ∈[m], ∀j ∈[n+ 1−d]}
∪ {((i, n+ 1), n+ 2) : ∀i∈[m]}
∪ {(n+ 2,(i, n+ 1)) : ∀i∈[m]}
The length of each arce∈E is denoted by `(e). This length function is defined as
`(0,(i,1)) := 0,
`((i,j),(i,j+1)) := 0,
`((i,j),(i0,j+d)) := min{
j+d
X
q=1
δi+0q−
j
X
q=1
δiq+,
j+d
X
q=1
δi−0q−
j
X
q=1
δiq−},
`((i,n+1),n+2) := −
n+1
X
q=1
δiq+
`(n+2,(i,n+1)) :=
n+1
X
q=1
δiq+
forall i, i0 ∈ [m] and j ∈ [n + 1−d]. To avoid parallel arcs, we create only one arc for each pair of constraints (2.13) and (2.14) with the same parameters; the length of this arc is the minimum of the two upper bounds.
The new variables determine a potential in the network D = (V, E) with respect to the length function ` : E → Z. Thus solving BOTP under the IDC amounts to finding a potentialπ whose valueπn+2−π0 is maximum. Let us recall that apotential for a networkD(and length function`) is a functionpot:V −→R such that
pot(w)6pot(v) +`(v,w),
for all vertices v and w such that (v, w) is an arc. The optimal beam-on time is simply the negative of the distance (that is, the length of a shortest path) from vertex 0 to vertexn+ 2 in the network.
The following remarks are in order. Note that the only (simple) cycles in the network D contain two opposite arcs and have length zero. So there are no neg- ative length cycles in the network. In fact D is essentially acyclic. Because all lengths are integral the integrality constraint (2.15) can be removed from problem (BOTP-IDC’). Indeed, the constraint matrix of (BOTP-IDC’) is totally unimod- ular since each row of the constraint matrix has exactly one 1, one −1 and 0 everywhere else, and all right hand sides are integral. It follows that the integral- ity constraint (2.12) can also be removed from problem (BOTP-IDC).
Because an optimal potential can be computed in polynomial time, for exam- ple, via the Bellman-Ford method [9, 35], we conclude that the BOTP under the IDC is a polynomial problem.
Figure 2.3: Example of network D for the IDC.
Example 2.6. We briefly discuss an example ford = 2. LetA be the following intensity matrix:
A=
0 0 3 4 2 1 2 2
.
We want to find an optimal decomposition ofAwhich respects the IDC for d= 2.
The network corresponding to A is given in Figure 2.3. After computing an optimal potential, we obtain
W =
0 0 0 1 0 0 0 0 0 2
, which yields the following optimal decomposition:
A=
0 0 1 0 1 0 0 0
+
0 0 1 1 1 1 1 1
+
0 0 1 1 0 0 1 1
+ 2
0 0 0 1 0 0 0 0
.
Now we present a O(mn +Km) algorithm to solve the problem. We have just seen that the optimal beam-on time is exactly the negative of the length of a shortest path from vertex 0 to vertex n+ 2in the network. As the network D is essentially acyclic we can adapt the standard dynamic programming algorithm due to Morávek [38] to compute an optimal potential in time O(|E|) = O(m2n).
Indeed, this dynamic programming algorithm computes for each vertex v ∈V of a network the length of the shortest path between the source andv. Let us denote by M(v) the length of the shortest path from the source s to v. The dynamic program of Morávek computes M(v)as follows:
M(v) := min{M(u) +`(u,v): (u, v)∈E}.
