Approximated MLC shape matrix decomposition with intereaf coision constraint Thomas Kainowski Antje Kiese Abstract Shape matrix decomposition is a subprobem in radiation therapy panning. A given fuence matrix A has to be decomposed into a sum of shape matrices corresponding to homogeneous fieds that can be shaped by a mutieaf coimator (MLC). We sove the probem of minimizing the deivery time for an approximation of A satisfying certain prescribed bounds, under the additiona condition that the used MLC requires the intereaf coision constraint. Key words: Intensity moduated radiation therapy (IMRT); mutieaf coimator; combinatoria optimization; programming invoving graphs 21 MSC: 9C35, 92C5, 9C9 1 Introduction In modern cancer therapy radiation is used to destroy the tumor tissue. At the same time one has to minimize the damage to the heathy tissue, and in particuar to sensibe structures or organs at risk. Intensity moduated radiation therapy was introduced in order to improve the quaity of radiation treatment. In cinica practice it is common to use a inear acceerator which can reease radiation from different directions (Figure 1). In addition, a mutieaf coimator (MLC) (Figure 2) can be used to protect certain parts of the irradiated area. Figure 1: A inear acceerator. Figure 2: Leaf pairs of a mutieaf coimator. For the treatment panning, the first step is to determine a set of directions (typicay 3 9), from which radiation is reeased, given by positions of the isocenter, tabe anges and gantry anges [5, Agorithmic Operations Research 4(1):49 57, 29 1
12]. In a second step, for each direction the fuence distribution is optimized, subject to required dose distribution in the target. The fina step is to determine, for each fuence distribution, a corresponding sequence of MLC eaf positions. Recenty, there have been attempts to formuate the optimization probem more gobay [5, 13], but most of the widey used treatment panning systems mode the three steps independenty. In this paper we consider the ast step for the MLC in the so caed step-and-shoot mode. This means the radiation is switched off whie the eaves are moving, and so the generated intensity moduated fied is just a superposition of finitey many homogeneous fieds which are shaped by the MLC. The two most important objectives in the optimization probem are the tota irradiation time, or deivery time (DT), and the number of used fieds, or decomposition cardinaity (DC). Starting with [2] and [6] there have been proposed severa agorithms for this probem [3, 1, 14, 15], taking into account additiona machine dependent constraints as the intereaf coision constraint [1, 7] or the tongue-and-groove constraint [11] (see [9] or [8] for a survey). A of these agorithms start with the given fuence matrix A and construct a sequence of eaf positions reaizing this matrix. But from a practica point of view there seem to be some doubts if it is reasonabe to consider every entry a ij as fixed once and for a. First, the matrix A is a resut of numerica computations which are based on simpified physica modes of how the radiation passes through the patients body, and second, the representation of A as a superposition of homogeneous fieds is aso based on mode assumptions which are not stricty correct, for instance the dose deivered to an exposed bixe depends on the shape of the fied. So it might be sufficient, to reaize (in our mode) a matrix that is cose to A. It is a natura question, how much the deivery time can be reduced by giving ony an approximate representation of A satisfying certain minimum and maximum dose constraints. As an immediate consequence, the next probem arises: find an approximation with this optima DT which is as cose as possibe to A. These questions have been answered for unconstrained MLCs in [4], and in the present paper we generaize the ideas from this reference to MLCs with intereaf coision constraint. In Section 2 we give an precise statement of the probem, Section 3 reviews an exact agorithm for shape matrix decomposition with intereaf coision constraint, in Section 4 we present our graphtheoretica characterization of the minima DT of an approximation with a constructive proof, in Section 5 we show how the tota change can be reduced heuristicay, and the fina Section 6 contains some test resuts. 2 Notation and probem formuation Throughout the rest of the paper, for a natura number n, [n] denotes the set {1, 2,..., n} and for integers m < n, [m, n] denotes the set {m, m + 1,..., n}. For integers a, we aso use the notation a + for the nonnegative part, defined by { a if a, a + = otherwise. Our starting point is an m n matrix A with nonnegative integer entries. The entry a ij represents the desired fuence at bixe (i, j). In addition, for each entry (i, j) we have ower and upper bounds a ij and a ij, such that a ij a ij a ij. Definition 1 (Feasibe Approximation). Any integer matrix B with a ij b ij a ij 2
is caed feasibe approximation of A. The tota change T C(B) of a feasibe approximation B is defined by m n T C(B) = b ij a ij. i=1 j=1 The homogeneous fieds that can be shaped by the MLC are described by binary matrices of size m n which we ca shape matrices. Definition 2 (Shape matrix). An m n matrix S is a shape matrix if there are pairs of integers ( i, r i ) (i = 1,..., m), such that the foowing conditions are satisfied: { 1 if i < j < r i, 1. s ij = otherwise. 2. i < r i+1 and r i > i+1 for a i [m 1]. The second condition in Definition 2 is caed intereaf coision constraint (ICC). It ensures the eft eaf of row i and the right eaf of row i±1 do not overap, which is required by some widey used MLCs, for instance the Eekta MLC. An MLC eaf sequence for A corresponds to a representation of A as a weighted sum of shape matrices. Definition 3 (Shape matrix decomposition). A shape matrix decomposition of A is a representation of A as a positive integer combination of shape matrices k A = u t S (t). The deivery time (DT) of this decomposition is just the sum of the coefficients, k DT = u t. Exampe 1. For the shape matrix decomposition 1 3 3 1 1 1 1 1 2 4 1 1 1 4 4 = 2 1 1 1 + 1 1 1 1 1 1 + 1 1 1 1 1 3 3 1 1 1 1 1 1 we have DT = 4. t=1 Now we formuate three optimization probems. MinDT. Find a shape matrix decomposition A = k t=1 u ts (t) such that DT = k t=1 u t is minima. Approx-MinDT. Find a feasibe approximation B and a shape matrix decomposition B = k t=1 u ts (t) such that DT = k t=1 u t is minima. Approx-MinDT-TC. Find a feasibe approximation B and a shape matrix decomposition B = k t=1 u ts (t) such that DT = k t=1 u t is minima, and under this condition T C(B) is minima. The first probem MinDT is the exact decomposition probem which can be soved by severa efficient agorithms [1, 7, 1]. The idea underying one of these agorithms is reviewed in the next section because it is the basis for our approach to the second probem Approx-MinDT. Finay, we observe that the second part of each of the probems Approx-MinDt and Approx-MinDT-TC, the search for the shape matrix decomposition, can be ignored safey, because, once the matrix B is fixed, we can appy any exact decomposition agorithm to compete the task. 3 t=1
3 Review of the exact decomposition The basis of our approach is a characterization of the minima DT of a decomposition with ICC as the maxima weight of a q s path in the foowing digraph G = (V, E) [7, 8]. V = {q, s} [m] [, n + 1], E = {(q, (i, )) : i [m]} {((i, n + 1), s) : i [m]} {((i, j), (i, j + 1)) : i [m], j [, n]} {((i, j), (i + 1, j)) : i [m 1], j [n]} {((i, j), (i 1, j)) : i [2, m], j [n]}. In order to avoid case distinctions, we add two coumns to our matrix and put Now we can define arc weights by a i = a i,n+1 = w(q, (i, )) = w((i, n + 1), s) = (i [m]). (i [m]) w((i, j 1), (i, j)) = max{, a i,j a i,j 1 } (i [m], j [n + 1]) w((i, j), (i + 1, j)) = a ij w((i, j), (i 1, j)) = a ij (i [m 1], j [n]) (i [2, m], j [n]). We ca this graph the DT-ICC-graph for A. Figure 3 shows the DT-ICC-graph for the matrix 4 5 1 4 5 A = 2 4 1 3 1 4 2 3 2 1 2 4. 5 3 3 2 5 3 4 1 1 3 1 2 4 4 5 1 3 1 1 4 4 5 2 2 2 3 q 2 2 3 4 2 1 1 3 2 1 4 4 2 1 1 2 s 5 2 3 3 3 2 2 1 5 2 3 4 5 3 Figure 3: The DT-ICC-Graph for matrix A. Definition 4. Let A be an intensity matrix, and et G be the DT-ICC-graph for A. The maxima weight of a q s path in G is caed ICC-compexity of A and denoted by c(a). More formay, c(a) = max{w(p ) : P is a q s path in G.}. Using this definition the main resut of [7] can be formuated as foows. Theorem 1. The minima DT of a decomposition of A with ICC equas c(a). 4
4 Approximation To simpify our notation, for each (i, j) [m] [n] we introduce the interva of acceptabe fuence vaues ] I ij = [a ij, a ij, a ij a ij a ij. We want to find a matrix B such that b ij I ij for (i, j) [m] [n] and c(b) min. We foow an approach from [4] and repace every vertex (i, j) [m] [n] by I ij copies, i.e. by the set V ij = {(i, j)} I ij. In order to avoid case distinctions in the discussion beow we aso repace the vertices in coumns and n + 1 by V i = {(i,, )} and V i,n+1 = {(i, n + 1, )}. An arc ((i, j), (i, j + 1)) in the DT-ICC-graph G is repaced by the compete bipartite graph V ij V i,j+1, and simiary for the arcs ((i, j), (i ± 1, j)). The weights of the arcs ((i, j, k), (i, j + 1, )) shoud mode the approximation matrix B if we choose b ij = k and b i,j+1 =, and simiary for the other arc types. Hence we define the arc weights by w(q, (i,, )) = w((i, n + 1, ), s) = (i [m]), (i [m]), w((i,, ), (i, 1, k)) = k (i [m], k I i1 ), w((i, n, k), (i, n + 1, )) = (i [m], k I in ), w((i, j 1, k), (i, j, )) = ( k) + (i [m], j [n], k I i,j 1, I ij ), w((i, j, k), (i + 1, j, )) = k (i [m 1], j [n], k I ij, I i+1,j ), w((i, j, k), (i 1, j, )) = k (i [2, m], j [n], k I ij, I i 1,j ). In order to determine the minima compexity of an approximation matrix we compute numbers W (i, j, k) such that W (i, j, k) = max { min W (i, j 1, ) + (k ) +, min W (i 1, j, ), min W (i + 1, j, ) }. The intuitive idea is that for every feasibe approximation B with b ij = k, the maxima weight of a q (i, j) path in the DT-ICC-graph for B is at east W (i, j, k). The numbers W (i, j, k) can be computed efficienty (compexity O(m 2 n 2 ), where denotes any upper bound for I ij ) as described in Agorithm 1. Again, in order to avoid case distinctions at the boundaries, we add the vaues W (, j, ) = W (m + 1, j, ) = a j = a m+1,j = (j [n]). Definition 5. The ICC-approximation compexity of A (with respect to the given intervas I ij ) is defined by c(a) = max min W (i, n, k). i k 5
Ceary, c(a) is a ower bound for the ICC-compexity of a feasibe approximation of A. We wi show that this bound is sharp by an expicit construction of an approximation matrix B. For the ast coumn we put { a in if W (i, n, a in ) c(a), b in = max{k : W (i, n, k) c(a)} otherwise. For j < n, we assume that the entries b i,j+1 are aready determined, and put b ij = max { k : W (i, j, k) + (b i,j+1 k) + W (i, j + 1, b i,j+1 ) }. Exampe 2. We consider the foowing fuence matrix A with c(a) = 8. ( ) 4 A = 4 We choose the upper and ower bound such that b ij a ij 1 for every (i, j). The intervas and an optima approximation are ( ) ( ) [3, 5] [, 1] [, 1] 3 1, B = [, 1] [, 1] [3, 5] 1 1 3 with c(b) = 4. Our agorithm obtains matrix B as foows. First we compute the numbers W (i, j, k), and obtain, for each (i, j), a vector (W i,j,aij, W i,j,aij +1,..., W i,j,aij ). These vectors are coected in the foowing array. Thus the optima DT is (3, 4, 5) (3, 3) (3, 3) (, 1) (2, 2) (4, 5, 6). max{min{3, 3}, min{4, 5, 6}} = 4. For the third coumn we choose b 13 = and b 23 = 3. For the entry (1, 2) we have W (1, 2, ) + w((1, 2, ), (1, 3, )) = W (1, 2, 1) + w((1, 2, 1), (1, 3, )) = W (1, 3, ). We choose the maxima possibe vaue b 12 = 1. Observe that b 12 = is indeed not possibe, since it eads to an increased DT. For entry (2, 2) we have W (2, 2, ) + w((2, 2, ), (2, 3, 3)) = 2 + 3 > W (2, 3, 3), so here b 22 = 1 is the ony possibe choice. Simiary, we get b 11 = 3 and b 21 = 1. Ceary, the atter one can be repaced by. In order to prove that our method is correct, we need some simpe properties of the numbers W (i, j, k). 6
Lemma 1. For every (i, j) [m] [n] and every k such that (i, j, k), (i, j, k + 1) V ij we have Furthermore, W (i, j, k + 1) = W (i, j, k) + 1 iff for some I i,j 1 with k. Proof. Since W (i, j, k) W (i, j, k + 1) W (i, j, k) + 1. (1) W (i, j, k) = W (i, j 1, ) + (k ) + W (i, j 1, ) + (k ) + W (i, j 1, ) (k + 1 ) + and using the definition of the W (i, j, k), we concude W (i, j, k) W (i, j, k + 1). On the other hand, we have W (i, j, k) = max { min W (i, j 1, ) + (k ) +, min W (i 1, j, ), min W (i + 1, j, ) } max { min W (i, j 1, ) + (k + 1 ) +, min W (i 1, j, ), min W (i + 1, j, ) } 1 = W (i, j, k + 1) 1, where equaity occurs iff W (i, j, k) = W (i, j 1, ) + (k ) + and k. The next emma is the key step of our argument. It asserts that the chosen b ij do not ead to conficts inside the coumns. Lemma 2. For a j and a i [m 1], we have and for a j and a i [2, m], we have W (i, j, b ij ) b ij W (i + 1, j, b i+1,j ), W (i, j, b ij ) b ij W (i 1, j, b i 1,j ). Proof. We ony show the first statement, since the second one can be proved simiary. Suppose the statement is fase, i.e. W (i, j, b ij ) b ij > W (i + 1, j, b i+1,j ). By construction, there is some k I ij such that W (i, j, k) k W (i + 1, j, b i+1,j ). Case 1. k < b ij. Let δ = b ij k >. By Lemma 1 we have But now we obtain W (i, j, k) W (i, j, b ij ) δ. W (i, j, k) k (W (i, j, b ij ) δ) (b ij δ) > W (i + 1, j, b i+1,j ), and this is the required contradiction. 7
Case 2. k > b ij. Let δ = k b ij >. By construction of the numbers b ij, W (i, j, b ij ) + (b i,j+1 b ij ) + W (i, j + 1, b i,j+1 ), W (i, j, b ij + 1) + (b i,j+1 (b ij + 1)) + > W (i, j + 1, b i,j+1 ). Using Lemma 1, this is possibe ony if Using Lemma 1 repeatedy, we obtain W (i, j, b ij + 1) = W (i, j, b ij ) + 1. W (i, j, k) = W (i, j, b ij ) + δ. But together this impies W (i, j, k) k = W (i, j, b ij ) b ij, which is a contradiction. Now et G be the DT-ICC-graph for B. Denote by α 1 (i, j) the maxima weight of a q (i, j) path in G. Note that the numbers α 1 (i, j) can be computed simiary to the numbers W (i, j, k). Ceary, α 1 (i, 1) = b i1, and the procedure for coumn j > 1 is described in Agorithm 2. Lemma 3. For a (i, j) we have α 1 (i, j) W (i, j, b ij ). Proof. We use induction on j. For j = 1 the caim is obvious: α 1 (i, 1) = W (i, 1, b i1 ) = b i1. Now et j > 1. After the initiaization of the numbers α 1 (i, j) in the first oop of Agorithm 2 we obtain for every i, α 1 (i, j) = α 1 (i, j 1) + (b ij b i,j 1 ) + W (i, j 1, b i,j 1) + (b ij b i,j 1) + W (i, j, b ij). We just have to check that this inequaities remain vaid in every updating step. Suppose the first vioation occurs when we repace α 1 (i, j) by α 1 (i ± 1, j) b i±1,j. In this case, α 1 (i, j) = α 1 (i ± 1, j) b i±1,j W (i ± 1, j, b i±1,j ) b i±1,j W (i, j, b ij ), where the ast inequaity is Lemma 2. So the statement of the emma remains vaid. By Lemma 3 (and Theorem 1), matrix B aows a decomposition with DT c(a) and this impies the foowing theorem. Theorem 2. The minima DT of a decomposition of a feasibe approximation of A equas c(a) and an approximation matrix B reaizing this DT can be constructed as described above in time O(m 2 n 2 ). Proof. The ony thing that is eft to prove is the compexity statement. For this it is sufficient to note that the computation of the numbers W (i, j, k) dominates the computation time, since this has compexity O(m 2 n 2 ) as can be seen immediatey from Agorithm 1. But after the numbers W (i, j, k) have been computed we ook at every entry (i, j) ony once and in order to fix b ij we have to do at most I ij comparisons. So the matrix B is determined in time O(mn ) and this concudes the proof. 8
Agorithm 1 Computation of the numbers W (i, j, k) for i [m] do W (i,, ) = for j = 1 to n do for i [m] do for a k do W (i, j, k) = min W (i, j 1, ) + (k ) + for i = 2 to m do for a k do W (i, j, k) = max { W (i, j, k), min W (i 1, j, ) } for i = i 1 downto 1 do for a k do W (i, j, k) = max { W (i, j, k), min W (i + 1, j, ) } Agorithm 2 Computation of the numbers α 1 (i, j) for fixed j for i [m] do α 1 (i, j) = α 1 (i, j 1) + (b ij b i,j 1 ) + for i = 2 to m do α 1 (i, j) = max {α 1 (i, j), α 1 (i 1, j) b i 1,j } for i = i 1 downto 1 do α 1 (i, j) = max { α 1 (i, j), α 1 (i } + 1, j) b i +1,j 9
5 Reducing the tota change The construction described in Section 4 eads to an approximation B with minima deivery time, but a arge tota change T C(B). The reason is, that we put b ij = max {k : W (i, j, k) + (b i,j+1 k) W (i, j + 1, b i,j+1 )}, even if none of the vertices (i, j, k) is critica, i.e. part of a q-s-path of maxima weight in the DT-ICC-graph of a feasibe approximation of A. Thus, the aim is to find an approximation with the same deivery time, but smaer tota change. Ceary, we can repace b ij by a vaue b ij with b ij < b ij a ij in the case b ij < a ij, respectivey with a ij b ij > b ij in the case a ij > b ij, if this decision does not increase the maxima weight of a q-s-path in the DT-ICC-graph. Let therefore G be the DT-ICC-graph of B and et α 1 (i, j) denote the maxima weight of a q-(i, j)-path in G. Simiary, et α 2 (i, j) denote the maxima weight of an (i, j)-s-path in G. The vaues α 2 (i, j) can be computed simiary as the numbers α 1 (i, j). Definition 6. Let B be a feasibe approximation of A. For (i, j) [m] [n], an integer b is caed (i, j) feasibe (with respect to B) if the foowing conditions are satisfied. 1. b I ij. 2. α 1 (i, j 1) + (b b i,j 1 ) + + (b i,j+1 b) + + α 2 (i, j + 1) c(a). 3. i = 1 or α 1 (i, j 1) + (b b i,j 1 ) + b + α 2 (i 1, j) c(a). 4. i = m or α 1 (i, j 1) + (b b i,j 1 ) + b + α 2 (i + 1, j) c(a). 5. i = 1 or α 1 (i 1, j) b i 1,j + (b i,j+1 b) + + α 2 (i, j + 1) c(a). 6. i = m or α 1 (i + 1, j) b i+1,j + (b i,j+1 b) + + α 2 (i, j + 1) c(a). 7. i {1, m} or α 1 (i 1, j) b i 1,j b + α 2 (i + 1, j) c(a). 8. i {1, m} or α 1 (i + 1, j) b i+1,j b + α 2 (i 1, j) c(a). In other words, b is (i, j) feasibe iff we can repace b ij by b without destroying the DT optimaity of B. Fig 4 iustrates the different possibiities for a path to pass through vertex (i, j). Each of these possibiities corresponds to one of the conditions 2 through 8 in Definition 6. We propose a heuristic, formay described in Agorithm 3, to reduce the tota change. Ceary, the appication of this agorithm can be iterated unti no more changes occur. Agorithm 3 Heuristic for tota change minimization for j = 1 to n do for i = 1 to m do if b ij < a ij and b ij + 1 is (i, j) feasibe then b ij + + if b ij > a ij and b ij 1 is (i, j) feasibe then b ij Update the numbers α 1 (k, ) and α 2 (k, ) 1
(i 1, j) (i, j 1) (i, j) (i, j 1) (i, j) (i, j + 1) (i, j 1) (i, j) (i + 1, j) (i 1, j) (i, j) (i, j + 1) (i 1, j) (i 1, j) (i, j) (i, j) (i, j) (i, j + 1) (i + 1, j) (i + 1, j) (i + 1, j) Figure 4: The seven different types of paths that are affected by the choice of b ij. 6 Test Resuts In this section we demonstrate the DT-reduction obtained by the methods from Section 4 and the tota change reduction using the heuristic approach from Section 5. We use matrices of size 15 15 and 3 3 with random entries a ij {, 1,..., L} for L {8, 12, 16}. In our tests we choose the upper and ower bounds for the entries such that each entry is changed by at most 2, i.e. we put a ij = (a ij 2) +, a ij = a ij + 2. For each L, we construct decompositions of 1 matrices, and compute the average minima deivery time c(a) and the tota change according to our agorithm from Section 4. Finay, we anayse the tota change reduction, that can be achieved using Agorithm 3. The resuts are shown in Tabe 1 and 2. For comparison we incude the minima DT for exact decomposition with ICC [7]. Coumns DT 1 and DT 2 contain the average deivery times for the exact and for the approximated decomposition, respectivey. Coumns T C 1 and T C 2 contain the tota change vaues before and after the appication of Agorithm 3. Our agorithms are competey practicabe. On a 3GHz L DT 1 DT 2 T C 1 T C 2 8 35.7 14.6 329.1 188.7 12 51.8 29.2 358.3 14.8 16 67.7 44.6 373.9 112.8 Tabe 1: Test resuts for m = n = 15. L DT 1 DT 2 T C 1 T C 2 8 67.7 24.5 136. 837.2 12 97.9 51.4 1484.3 651.3 16 127.7 79.9 1546.2 55.4 Tabe 2: Test resuts for m = n = 3. workstation, the computations for the ast row, i.e. for the decomposition of 1 matrices of size 15 15 with entries from {, 1,..., 16} took ony 5 seconds for m = n = 15 and ess than a minute for m = n = 3. Basicay, we can draw two concusions from our resuts. 1. The approximation approach eads to an significant DT-reduction: for L = 16, aowing a change of at most 2 for each entry reduces the DT by more than 3%. 2. Our heuristic eads to a arge tota change reduction: for L = 16 the tota change can be reduced by amost 6%. 11
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