Algorithms for Sequencing Multileaf Collimators Srijit Kamath, Sartaj Sahni, Jatinder Palta, Sanjay Ranka and Jonathan Li Department of Computer and nformation Science and Engineering, University of Florida, Gainesville, Florida, USA Department of Radiation Oncology, University of Florida, Gainesville, Florida, USA E-mail: srkamath@cise.ufl.edu
Algorithms for Sequencing Multileaf Collimators 2 1. Problem Description The objective of radiation therapy for cancer treatment is to deliver high doses of radiation to the target volume, while limitting radiation dose on the surrounding healthy tissues. For eample, for head and neck tumors, it is necessary for radiation to be delivered so that the eposure of the spinal cord, optic nerve, salivary glands or other important structures is minimized. n recent years, this has been made possible due to the development of conformal radiation therapy. n conformal therapy, treatment is delivered using a set of radiation beams which are positioned such that the shape of the dose distribution conforms in three dimensions to the shape of the tumor. This is typically achieved by positioning beams of varying shapes from different directions so that each beam is shaped to conform to the projection of the target volume from the beam s eye view and and to avoid the organs at risk in the vicinity of the target. ntensity modulated radiation therapy (MRT) is the state-of-the-art in conformal radiation therpy. MRT permits the intensity of a radiation beam to be varied across a treatment area, thereby improving the dose conformity. Radiation is delivered using a medical linear accelerator (Figure 1). A rotating gantry containing the accelerator structure can rotate around the patient who is positioned on an adjustable treatment couch. Modulation of the beam fluence can be achieved by several techniques. n compensator-based MRT, the beam is modulated with a preshaped piece of material called a compensator (modulator). The degree of modulation of the beam varies depending on the thickness of the material through which the beam is attenuated. The computer determines the shape of each modulator in order to deliver the desired beam. This type of modulation requires the modulator to be fabricated and then manually inserted into the tray mount of a linear accelerator. n tomotherapy-based MRT, the linear accelerator travels in multiple circles all the way around the gantry ring to deliver the radiation treatment. The beam is collimated to a narrow slit and the intensity of the beam is modulated during the gantry movement around the patient. Care must be taken to ensure that adjacent circular arcs do not overlap and thereby do not overdose tissues. This type of delivery is referred to as serial tomotherapy. A modification of serial tomotherapy is helical tomotherapy. n helical tomotherapy, the treatment couch moves linearly (continuously) through the rotating accelerator gantry. So each time the accelerator comes around, it directs the beam on a slightly different plane on the patient. n MLC-based MRT the accelerator structure is equipped with a computer controlled mechanical device called a multileaf collimator (MLC, Figure 2) that shapes the radiation beam, so as to deliver the radiation as prescribed by the treatment plan. The MLC may have up to 120 movable leaves that can move along an ais perpendicular to the beam and can be arranged so as to shield or epose parts of the anatomy during treatment. The leaves are arranged in pairs so that each leaf pair forms one row of the arrangement. The set of allowable MLC leaf configurations may be restricted by leaf movement constraints that are manufacturer and/or model dependent. The first stage in the treatment planning process in MRT is to obtain accurate
Algorithms for Sequencing Multileaf Collimators 3 Figure 1. A linear accelerator (the figure is from http://www.lemed.com/- medical services/mrt.htm) Figure 2. A multileaf collimator (the figure is from http://www.lemed.com/- medical services/mrt.htm)
Algorithms for Sequencing Multileaf Collimators 4 three dimensional anatomical information about the patient. This is achieved using computed tomography (CT) and/or magnetic resonance (MR) imaging. An ideal dose distribution would ensure perfect conformity to the target volume while completely sparing all other tissues. However, such a distribution is impossible to realize in practice. Therefore, doses to targerts and tolerable doses for critical structures are prescribed and an objective function that measures the quality of a plan is developed subject to these dose based constraints. Net, a set of beam parameters (beam angles, profiles, weights) that optimize this objective are determined using a computer program. This method is called inverse planning since resultant dose distribution is first described and the best beam parameters that deliver the distribution (approimately) are then solved for. t is to be noted that inverse planning is a general concept and its implementation details vary vastly among various systems. Following the inverse planning in MLCbased MRT, the delivery of radiation intensity profile for each beam direction is described as a MLC leaf sequence, which is developed using a leaf sequencing algorithm. mportant considerations in developing a leaf sequence for a desired intensity profile include maimizing the monitor unit (MU) efficiency (equivalently minimizing the beam-on time) and minimizing the total treatment time subject to the leaf movement constraints of the MLC model. Finally, when the leaf sequences for all beam directions are determined, the treatment is performed from various beam angles sequentially using computer control. n this chapter, we present an overview of recent advances in leaf sequencing algorithms. 1.1. MLC models and constraints The purpose of the leaf sequencing algorithm is to generate a sequence of leaf positions and/or movements that faithfully reproduce the desired intensity map once the beam is delivered, taking into consideration any hardware and dosimetric characteristics of the delivery system. The two most common methods of MRT delivery with computercontrolled MLCs are the segmental multileaf collimator (SMLC) and dynamic multileaf collimator (DMLC). n SMLC, the beam is switched off while the leaves are in motion. n other words, the delivery is done using multiple static segments or leaf settings. This method is also frequently referred to as the step and shoot or stop and shoot method. n DMLC the beam is on while the leaves are in motion. The beam is switched on at the start of treatment and is switched off only at the end of treatment. The fundamental difference between the leaf sequences of these two delivery methods is that the leaf sequence defines a finite set of beam shapes for SMLC and trajectories of opposing pairs of leaves for DMLC. n practical situations, there are some constraints on the movement of the leaves. The minimum separation constraint requires that opposing pairs of leaves be separated by atleast some distance (S min ) at all times during beam delivery. n MLCs this constraint is applied not only to opposing pairs of leaves, but also to opposing leaves of neighboring pairs. For eample, in Figure 3, L1 and R1, L2 and R2, L3 and R3, L1
Algorithms for Sequencing Multileaf Collimators 5 and R2, L2 and R1, L2 and R3, L3 and R2 are pairwise subject to the constraint. The case with S min = 0 is called interdigitation constraint and is applicable to some MLC models. Wherever this constraint applies, opposite adjacent leaves are not permitted to overlap. Figure 3. nter-pair minimum separation constraint n most commercially available MLCs, there is a tongue-and-groove arrangement at the interface between adjacent leaves. A cross section of two adjacent leaves is depicted in Figure 4. The width of the tongue-and-groove region is l. The area under this region gets underdosed due to the mechanical arrangement, as it remains shielded if either the tongue or the groove portion of a leaf shields it. Radiation l Figure 4. Cross section of leaves Leaf movement Maimum leaf spread for leaves on the same leaf bank is one more MLC limitation, which necessitates a large field (intensity profile) to be split into two or more adjacent abutting sub-fields. This is true for the Varian MLC (Varian Medical Systems, Palo Alto, CA), which has a field size limitation of about 15 cm. The abutting sub-fields are then delivered as separate treatment fields. This often results in longer delivery times, poor MU efficiency, and field matching problems.
Algorithms for Sequencing Multileaf Collimators 6 2. Algorithms for SMLC n this section we study the leaf sequencing problem for SMLC. We first introduce the notation that will be used in the remainder of this chapter. We present the leaf sequencing algorithm for a single leaf pair and subsequently etend it for multiple leaf pairs. 2.1. Single Leaf Pair The geometry and coordinate system used are shown in Figure 5. Consider the delivery of an intensity map produced by the optimizer in the inverse planning stage. t is important to note that the intensity map from the optimizer is always a discrete matri. The spatial resolution of this matri is similar to the smallest beamlet size. The beamlet size typically ranges from 5-10 mm. Let () be the desired intensity profile along the ais. The discretized profile from the optimizer gives the intensity values at sample points 0, 1,..., m. We assume that the sample points are uniformly spaced and that = i+1 i, 0 i < m. () is assigned the value ( i ) for i < i+1, for each i. Now, ( i ) is our desired intensity profile, i.e., ( i ) is a measure of the number of MUs for which i, 0 i < m, needs to be eposed. Figure 6 shows a profile, which is the output from the optimizer at discrete sample points 0, 1,..., m. Figure 5. Geometry and coordinate system 2.1.1. Movement of Leaves n our analysis we assume that the leaves are initially at the left most position 0 and that the leaves move unidirectionally from left to right. Figure 7 illustrates the leaf trajectory during SMLC delivery. Let l ( i ) and r ( i ) respectively denote the amount of Monitor Units (MUs) delivered when the left and right leaves leave position i. Consider the motion of the left leaf. The left leaf begins
Algorithms for Sequencing Multileaf Collimators 7 0 1 m Figure 6. Profile generated by the optimizer at 0 and remains here until l ( 0 ) MUs have been delivered. At this time the left leaf is moved to 1, where it remains until l ( 1 ) MUs have been delivered. The left leaf then moves to 3 where it remains until l ( 3 ) MUs have been delivered. At this time, the left leaf is moved to 6, where it remains until l ( 6 ) MUs have been delivered. The final movement of the left leaf is to 7, where it remains until l ( 7 ) = ma MUs have been delivered. At this time the machine is turned off. The total therapy time, TT( l, r ), is the time needed to deliver ma MUs. The right leaf moves to 2 when 0 MUs have been delivered; moves to 4 when r ( 2 ) MUs have been delivered; moves to 5 when r ( 4 ) MUs have been delivered and so on. Note that the machine is off when a leaf is in motion. We make the following observations: (i) All MUs that are delivered along a radiation beam along i before the left leaf passes i fall on it. The greater the value, the later the left leaf passes that position. Therefore l ( i ) is a non-decreasing function. (ii) All MUs that are delivered along a radiation beam along i before the right leaf passes i, are blocked by the leaf. The greater the value, the later the right leaf passes that position. Therefore r ( i ) is also a non-decreasing function. From these observations we notice that the net amount of MUs delivered at a point is given by l ( i ) r ( i ), which must be the same as the desired profile ( i ). 2.1.2. Optimal Unidirectional Algorithm for One Pair of Leaves When the movement of leaves is restricted to only one direction, both the left and right leaves move along the positive direction, from left to right (Figure 5). Once the desired intensity profile,
Algorithms for Sequencing Multileaf Collimators 8 Figure 7. Leaf trajectory during SMLC delivery ( i ) is known, our problem becomes that of determining the individual intensity profiles to be delivered by the left and right leaves, l and r such that: ( i ) = l ( i ) r ( i ), 0 i m (1) We refer to ( l, r ) as the treatment plan (or simply plan) for. Once we obtain the plan, we will be able to determine the movement of both left and right leaves during the therapy. For each i, the left leaf can be allowed to pass i when the source has delivered l ( i ) MUs. Also, we can allow the right leaf to pass i when the source has delivered r ( i ) MUs. n this manner we obtain unidirectional leaf movement profiles for a plan. From Equation 1, we see that one way to determine l and r from the given target profile is to begin with l ( 0 ) = ( 0 ) and r ( 0 ) = 0; eamine the remaining i s from left to right; increase l whenever increases; and increase r whenever decreases. Once l and r are determined the leaf movement profiles are obtained as eplained in the previous section. The resulting algorithm is shown in Figure 8. Figure 9 shows a profile and the corresponding plan obtained using the algorithm. Ma et. al. (1998) show that Algorithm SNGLEPAR obtains plans that are optimal in therapy time. Their proof relies on the results of Boyer and Strait (1997), Spirou and Chui (1994) and Stein et. al. (1994). Kamath et. al. (2003) provide a much simpler proof. Theorem 1 [Kamath et. al. (2003)] Algorithm SNGLEPAR obtains plans that are optimal in therapy time. Proof: Let ( i ) be the desired profile. Let inc1, inc2,...,inck be the indices of the points at which ( i ) increases. So inc1, inc2,..., inck are the points at which ()
Algorithms for Sequencing Multileaf Collimators 9 Algorithm SNGLEPAR l ( 0 ) = ( 0 ) r ( 0 ) = 0 For j = 1 to m do f (( j ) ( j 1 ) l ( j ) = l ( j 1 ) + ( j ) ( j 1 ) r ( j ) = r ( j 1 ) Else r ( j ) = r ( j 1 ) + ( j 1 ) ( j ) l ( j ) = l ( j 1 ) End for Figure 8. Obtaining a unidirectional plan increases (i.e., ( inci ) > ( inci 1 )). Let i = ( inci ) ( inci 1 ). Suppose that ( L, R ) is a plan for ( i ) (not necessarily that generated by Algorithm SNGLEPAR). From the unidirectional constraint, it follows that L ( i ) and R ( i ) are non-decreasing functions of. Since ( i ) = L ( i ) R ( i ) for all i, we get i = ( L ( inci ) R ( inci )) ( L ( inci 1 ) R ( inci 1 )) = ( L ( inci ) L ( inci 1 )) ( R ( inci ) R ( inci 1 )) L ( inci ) L ( inci 1 ). Summing up i, we get ki=1 [( inci ) ( inci 1 )] k i=1 [ L ( inci ) L ( inci 1 )] = TT( L, R ). Since the therapy time for the plan ( l, r ) generated by Algorithm SNGLEPAR is ki=1 [( inci ) ( inci 1 )], it follows that TT( l, r ) is minimum. Theorem 2 [Kamath et. al. (2003)] f the optimal plan ( l, r ) violates the minimum separation constraint, then there is no plan for that does not violate the minimum separation constraint. 2.2. Multiple Leaf Pairs We use a single pair of leaves to deliver intensity profiles defined along the ais of the pair of leaves. However, in a real application, we need to deliver intensity profiles defined over a 2-D region. Each pair of leaves is controlled independently. f there are no constraints on the leaf movements, we divide the desired profile into a set of parallel profiles defined along the aes of the leaf pairs. Each leaf pair i then delivers the plan for the corresponding intensity profile i (). The set of plans of all leaf pairs forms the solution set. We refer to this set as the treatment schedule (or simply schedule). n this section, we present leaf sequencing algorithms for SMLC with and without constraints. The constraints condidered are (i) minimum separation constraint and (ii) tongue-and-
Algorithms for Sequencing Multileaf Collimators 10 Figure 9. A profile and its plan groove constraint and (optionally) interdigitation constraint. These algorithms are from Kamath et. al. (2003) and Kamath et. al. (2004a). 2.2.1. Optimal Schedule Without The Minimum Separation Constraint Assume we have n pairs of leaves. For each pair, we have m sample points. The input is represented as a matri with n rows and m columns, where the ith row represents the desired intensity profile to be delivered by the ith pair of leaves. We apply Algorithm SNGLEPAR to determine the optimal plan for each of the n leaf pairs. This method of generating schedules is described in Algorithm MULTPAR (Figure 10). Theorem 3 [Kamath et. al. (2003)] Algorithm MULTPAR generates schedules that are optimal in therapy time.
Algorithms for Sequencing Multileaf Collimators 11 Algorithm MULTPAR For(i = 1; i n; i + +) End For Apply Algorithm SNGLEPAR to the ith pair of leaves to obtain plan ( il, ir ) that delivers the intensity profile i (). Figure 10. Obtaining a schedule Proof: Treatment is completed when all leaf pairs (which are independent) deliver their respective plans. The therapy time of the schedule generated by Algorithm MULTPAR is ma{tt( 1l, 1r ), TT( 2l, 2r ),...,TT( nl, nr )}. From Theorem 1, it follows that this therapy time is optimal. 2.2.2. Optimal Algorithm With nter-pair Minimum Separation Constraint The schedule generated by Algorithm MULTPAR may violate both the intra- and inter-pair minimum separation constraints. f the schedule has no violations of these constraints, it is the desired optimal schedule. f there is a violation of the intra-pair constraint, then it follows from Theorem 2 that there is no schedule that is free of constraint violation. So, assume that only the inter-pair constraint is violated. We eliminate all violations of the inter-pair constraint starting from the left end, i.e., from 0. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from 0 along the positive direction looking for the least v at which is positioned a right leaf (say R u ) that violates the inter-pair separation constraint. After rectifying the violation at v with respect to R u we look for other violations. Since the process of eliminating a violation at v may, at times, lead to new violations at j, j < v, we need to retract a certain distance (we will show that this distance is S min ) to the left, every time a modification is made to the schedule. We now restart the scanning and modification process from the new position. The process continues until no inter-pair violations eist. Algorithm MNSEPARATON (Figure 11) outlines the procedure. Let M = (( 1l, 1r ), ( 2l, 2r ),...,( nl, nr )) be the schedule generated by Algorithm MULTPAR for the desired intensity profile. Let N(p) = (( 1lp, 1rp ), ( 2lp, 2rp ),..., ( nlp, nrp )) be the schedule obtained after Step iv of Algorithm MNSEPARATON is applied p times to the input schedule M. Note that M = N(0). To illustrate the modification process we use an eample (see Figure 12). To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs when the right leaf of pair u is positioned at v and the left leaf of pair t, t {u 1, u + 1}, arrives at u, v u < S min. Let u = v S min. To remove this inter-pair separation violation, we modify ( tlp, trp ). The other profiles of N(p) are not
Algorithms for Sequencing Multileaf Collimators 12 Algorithm MNSEPARATON //assume no intra-pair violations eist (i) = 0 (ii) While (there is an inter-pair violation) do (iii) Find the least v, v, such that a right leaf is positioned at v and this right leaf has an inter-pair separation violation with one or both of its neighboring left leaves. Let u be the least integer such that the right leaf R u is positioned at v and R u has an inter-pair separation violation. Let L t denote the left leaf (or one of the left leaves) with which R u has an inter-pair violation. Note that t {u 1, u + 1}. (iv) Modify the schedule to eliminate the violation between R u and L t. (v) f there is now an intra-pair separation violation between Rt and L t, no feasible schedule eists, terminate. (vi) = v S min (vii) End While Figure 11. Obtaining a schedule under the constraint Figure 12. Eliminating a violation modified. The new tlp (i.e., tl(p+1) ) is as defined below. { tlp () 0 < u tl(p+1) () = ma{ tlp (), tl () + } u m where = urp ( v ) tl ( u) = 2 1. tr(p+1) () = tl(p+1) () t (), where t () is the target profile to be delivered by the leaf pair t. Since tr(p+1) differs from trp for u = v S min there is a possibility that N(p + 1) has inter-pair separation violations for right leaf positions u = v S min. Since none of the other right
Algorithms for Sequencing Multileaf Collimators 13 leaf profiles are changed from those of N(p) and since the change in tl only delays the rightward movement of the left leaf of pair t, no inter-pair violations are possible in N(p + 1) for < u = v S min. One may also verify that since tl0 and tr0 are non-decreasing functions of, so also are tlp and trp, p > 0. Theorem 4 [Kamath et. al. (2003)] The following are true of Algorithm MNSEPARATON: (a) The algorithm terminates. (b) When the algorithm terminates in Step v, there is no feasible schedule. (c) Otherwise, the schedule generated is feasible and is optimal in therapy time for unidirectional schedules. 2.2.3. Elimination of Tongue-and-Groove Effect with and without nterdigitation Constraint Figure 13 shows a beams-eye view of the region to be treated by two adjacent leaf pairs, t and t+1. Consider the shaded rectangular areas A t ( i ) and A t+1 ( i ) that require eactly t ( i ) and t+1 ( i ) MUs to be delivered, respectively. The tongueand-groove overlap area between the two leaf pairs over the sample point i, A t,t+1 ( i ), is colored black. Let the amount of MUs delivered in A t,t+1 ( i ) be t,t+1 ( i ). gnoring leaf transmission, the following lemma is a consequence of the fact that A t,t+1 ( i ) is eposed only when both A t ( i ) and A t+1 ( i ) are eposed. t t, t+1 t+1 000 111 000 111 000 111 000 111 000 111 000 111 i 1 i i+1 A A A t t, t+1 t+1 Figure 13. Tongue-and-groove effect Lemma 1 [Kamath et. al. (2004a)] t,t+1 ( i ) min{ t ( i ), t+1 ( i )}, 0 i m, 1 t < n, where m is the number of sample points along each row and n is the number of leaf pairs. Schedules in which t,t+1 ( i ) = min{ t ( i ), t+1 ( i )} are said to be free of tongueand-groove underdosage effects. The following lemma provides a necessary and sufficient condition for a unidirectional schedule to be free of tongue-and-groove underdosage effects. Lemma 2 [Kamath et. al. (2004a)] A unidirectional schedule is free of tongue-andgroove underdosage effects if and only if, (a) t ( i ) = 0 or t+1 ( i ) = 0, or (b) tr ( i ) (t+1)r ( i ) (t+1)l ( i ) tl ( i ), or
Algorithms for Sequencing Multileaf Collimators 14 (c) (t+1)r ( i ) tr ( i ) tl ( i ) (t+1)l ( i ), 0 i m, 1 t < n. Lemma 2 is equivalent to saying that the time period for which a pair of leaves (say pair t) eposes the region A t,t+1 ( i ) is completely contained by the time period for which pair t + 1 eposes region A t,t+1 ( i ), or vice versa, whenever t ( i ) 0 and t+1 ( i ) 0. Note that if either t ( i ) or t+1 ( i ) is zero the containment is not necessary. We will refer to the necessary and sufficient condition of Lemma 2 as the tongue-and-groove constraint condition. Schedules that satisfy this condition will be said to satisfy the tongue-and-groove constraint. van Santvoort and Heijmen (1996) present an algorithm that generates schedules that satisfy the tongue-and-groove constraint for DMLC. The schedule generated by Algorithm MULTPAR (Kamath et. al. 2003) may violate the tongue-and-groove constraint. f the schedule has no tongue-and-groove constraint violations, it is the desired optimal schedule. f there are violations in the schedule, we eliminate all violations of the tongue-and-groove constraint starting from the left end, i.e., from 0. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from 0 along the positive direction looking for the least w at which there eist leaf pairs u, t, t {u 1, u + 1}, that violate the constraint at w. After rectifying the violation at w we look for other violations. Since the process of eliminating a violation at w, may at times, lead to new violations at w, we need to search afresh from w every time a modification is made to the schedule. However, a bound of O(n) can be proved on the number of violations that can occur at w. After eliminating all violations at a particular sample point, w, we move to the net point, i.e., we increment w and look for possible violations at the new point. We continue the scanning and modification process until no tongue-and-groove constraint violations eist. Algorithm TONGUEANDGROOVE (Figure 14) outlines the procedure. Algorithm TONGUEANDGROOVE (i) = 0 (ii) While (there is a tongue-and-groove violation) do (iii) Find the least w, w, such that there eist leaf pairs u, u + 1, that violate the tongue-and-groove constraint at w. (iv) Modify the schedule to eliminate the violation between leaf pairs u and u + 1. (v) = w (vi) End While Figure 14. Obtaining a schedule under the tongue-and-groove constraint Let M = (( 1l, 1r ), ( 2l, 2r ),...,( nl, nr )) be the schedule generated by Algorithm MULTPAR for the desired intensity profile. Let N(p) = (( 1lp, 1rp ), ( 2lp, 2rp ),..., ( nlp, nrp )) be the schedule obtained after Step iv
Algorithms for Sequencing Multileaf Collimators 15 of Algorithm TONGUEANDGROOVE is applied p times to the input schedule M. Note that M = N(0). To illustrate the modification process we use eamples. To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between the leaves of pair u and pair t = u + 1 at w. Note that tlp ( w ) ulp ( w ), as otherwise, either (b) or (c) of Lemma 2 is true. n case tlp ( w ) > ulp ( w ), swap u and t. Now, we have tlp ( w ) < ulp ( w ). n the sequel, we refer to these u and t values as the u and t of Algorithm TONGUEANDGROOVE. From Lemma 2 and the fact that a violation has occurred, it follows that trp ( w ) < urp ( w ). To remove this tongue-and-groove constraint violation, we modify ( tlp, trp ). The other profiles of N(p) are not modified. The new plan for pair t, ( tl(p+1), tr(p+1) ) is as defined below. f ulp ( w ) tlp ( w ) urp ( w ) trp ( w ), then { tlp () 0 < w tl(p+1) () = (2) tlp () + w m where = ulp ( w ) tlp ( w ). tr(p+1) () = tl(p+1) () t (), where t () is the target profile to be delivered by the leaf pair t. Otherwise, { trp () 0 < w tr(p+1) () = trp () + (3) w m where = urp ( w ) trp ( w ). tl(p+1) () = tr(p+1) () + t (), where t () is the target profile to be delivered by the leaf pair t. The former case is illustrated in Figure 15 and the latter is illustrated in Figure 16. Note that our strategy for plan modification is similar to that used by van Santvoort and Heijmen (1996) to eliminate a tongue-and-groove violation for dynamic multileaf collimator plans. Since ( tl(p+1), tr(p+1) ) differs from ( tlp, trp ) for w there is a possibility that N(p+1) is involved in tongue-and-groove violations for w. Since none of the other leaf profiles are changed from those of N(p) no tongue-and-groove constraint violations are possible in N(p + 1) for < w. One may also verify that since tl0 and tr0 are non-decreasing functions of, so also are tlp and trp, p > 0. Theorem 5 [Kamath et. al. (2004a)] Algorithm TONGUEANDGROOVE generates schedules free of tongue-and-groove violations that are optimal in therapy time for unidirectional schedules. The elimination of tongue-and-groove constraint violations does not guarantee elimination of interdigitation constraint violations. Therefore the schedule generated by Algorithm TONGUEANDGROOVE may not be free of interdigitation violations. The algorithm we propose for obtaining schedules that simultaneously satisfy both constraints, Algorithm TONGUEANDGROOVE-D, is similar to Algorithm TONGUEANDGROOVE. The only difference between the two algorithms lies in the definition of the constraint condition. To be precise we make the following definition.
Algorithms for Sequencing Multileaf Collimators 16 tl(p+1) ulp urp tr(p+1) tlp trp w Figure 15. Tongue-and-groove constraint violation: case1 ulp tl(p+1) tr(p+1) tlp trp urp w Figure 16. Tongue-and-groove constraint violation: case2 (close parallel dotted and solid line segments overlap, they have been drawn with a small separation to enhance readability)
Algorithms for Sequencing Multileaf Collimators 17 Definition 1 [Kamath et. al. (2004a)] A unidirectional schedule is said to satisfy the tongue-and-groove-id constraint if (a) tr ( i ) (t+1)r ( i ) (t+1)l ( i ) tl ( i ), or (b) (t+1)r ( i ) tr ( i ) tl ( i ) (t+1)l ( i ), for 0 i m, 1 t < n. The only difference between this constraint and the tongue-and-groove constraint is that this constraint enforces condition (a) or (b) above to be true at all sample points i including those at which t ( i ) = 0 and/or t+1 ( i ) = 0. Lemma 3 [Kamath et. al. (2004a)] A schedule satisfies the tongue-and-grooveid constraint iff it satisfies the tongue-and-groove constraint and the interdigitation constraint. Theorem 6 [Kamath et. al. (2004a)] Algorithm TONGUEANDGROOVE-D generates schedules free of tongue-and-groove-id violations that are optimal in therapy time for unidirectional schedules. 3. Algorithms for DMLC 3.1. Single Leaf Pair 3.1.1. Movement of leaves We assume that ( 0 ) > 0 and ( m ) > 0 and that when the beam delivery begins the leaves can be positioned anywhere. We also assume that the leaves can move with any velocity v, v ma v v ma, where v ma is the maimum allowable velocity of the leaves. Figure 17 illustrates the leaf trajectory during DMLC delivery. l ( i ) and r ( i ), respectively, denote the amount of Monitor Units (MUs) delivered when the left and right leaves leave position i. The total therapy time, TT( l, r ), is the time needed to deliver ma MUs. Note that the machine is on throughout the treatment. All MUs that are delivered along a radiation beam along i before the left leaf passes i fall on it and all MUs that are delivered along a radiation beam along i before the right leaf passes i, are blocked by the leaf. So the amount of MUs delivered at a point is given by l ( i ) r ( i ), which must be the same as ( i ). 3.1.2. Maimum Velocity Constraint As noted earlier, the velocity of leaves cannot eceed some maimum limit (say v ma ) in practice. This implies that the leaf profile cannot be horizontal at any point. From Figure 17, observe that the time needed for a leaf to move from i to i+1 is ( i+1 i )/v ma. f Φ is the flu density of MUs from the source, the number of MUs delivered in this time along a beam is Φ ( i+1 i )/v ma. So, l ( i+1 ) l ( i ) Φ ( i+1 i )/v ma = Φ /v ma. The same is true for the right leaf profile r.
Algorithms for Sequencing Multileaf Collimators 18 ma ( ) l 6 ( ) l 5 l r ( ) l ( ) l 3 1 ( ) l 0 0 1 2 3 4 5 6 7 8 9 10 Figure 17. Leaf trajectory during DMLC delivery 3.1.3. Optimal Unidirectional Algorithm for one Pair of Leaves As in the case of SMLC, the problem is to find plan ( l, r ) such that: ( i ) = l ( i ) r ( i ), 0 i m (4) Of course, l and r are subject to the maimum velocity constraint. For each i, the left leaf can be allowed to pass i when the source has delivered l ( i ) MUs and the right leaf can be allowed to pass i when the source has delivered r ( i ) MUs. n this manner we obtain unidirectional leaf movement profiles for a plan. Similar to the case of SMLC, one way to determine l and r from the given target profile is to begin from 0 ; set l ( 0 ) = ( 0 ) and r ( 0 ) = 0; eamine the remaining i s to the right; increase l at i whenever increases and by the same amount (in addition to the minimum increase imposed by the maimum velocity constraint); and similarly increase r whenever decreases. This can be done till we reach m. This yields Algorithm DMLC-SNGLEPAR. Note that we move the leaves at the maimum velocity v ma whenever they are to be moved. The resulting algorithm is shown in Figure 18. Figure 17 shows a profile and the corresponding plan ( l, r ) obtained using Algorithm DMLC-SNGLEPAR. Ma et. al. (1998) show that Algorithm DMLC- SNGLEPAR obtains plans that are optimal in therapy time. Their proof relies on the results of Boyer and Strait (1997), Spirou and Chui (1994) and Stein et. al. (1994). Kamath et. al. (2004) provide a much simpler proof. Theorem 7 [Kamath et. al. (2004)] Algorithm DMLC-SNGLEPAR obtains plans that are optimal in therapy time. Proof: Let ( i ) be the desired profile. Let 0 = inc0 < inc1 <... < inck be the indices of the points at which ( i ) increases. So inc0, inc1,..., inck are the points
Algorithms for Sequencing Multileaf Collimators 19 Algorithm DMLC-SNGLEPAR l ( 0 ) = ( 0 ) r ( 0 ) = 0 For j = 1 to m do f (( j ) ( j 1 )) l ( j ) = l ( j 1 ) + ( j ) ( j 1 ) + Φ /v ma r ( j ) = r ( j 1 ) + Φ /v ma Else r ( j ) = r ( j 1 ) + ( j 1 ) ( j ) + Φ /v ma l ( j ) = l ( j 1 ) + Φ /v ma End for Figure 18. Obtaining a unidirectional plan at which () increases (i.e., ( inci ) > ( inci 1 ), assume that ( 1 = 0)). Let i = ( inci ) ( inci 1 ), i 0. Suppose that ( L, R ) is a plan for ( i ) (not necessarily the plan generated by Algorithm DMLC-SNGLEPAR). Since ( i ) = L ( i ) R ( i ) for all i, we get i = ( L ( inci ) R ( inci )) ( L ( inci 1 ) R ( inci 1 )) = ( L ( inci ) L ( inci 1 )) ( R ( inci ) R ( inci 1 )) = ( L ( inci ) L ( inci 1 ) Φ /v ma ) ( R ( inci ) R ( inci 1 ) Φ /v ma ) Note that from the maimum velocity constraint R ( inci ) R ( inci 1 ) Φ /v ma, i 1. So R ( inci ) R ( inci 1 ) Φ /v ma 0, i 1, and i L ( inci ) L ( inci 1 ) Φ /v ma. Also, 0 = ( 0 ) ( 1 ) = ( 0 ) L ( 0 ) L ( 1 ), where L ( 1 ) = 0. Summing up i, we get ki=0 [( inci ) ( inci 1 )] k i=0 [ L ( inci ) L ( inci 1 )] k Φ /v ma. Let S 1 = k i=0 [ L ( inci ) L ( inci 1 )]. Then, S 1 k i=0 [( inci ) ( inci 1 )]+k Φ /v ma. Let S 2 = [ L ( j ) L ( j 1 )], where the summation is carried out over indices j (0 j m) such that ( j ) ( j 1 ). There are a total of m+1 indices of which k+1 do not satisfy this condition. So there are m k indices j at which ( j ) ( j 1 ). At each of these j, L ( j ) L ( j 1 )+Φ /v ma. Hence, S 2 (m k) Φ /v ma. Now, we get S 1 +S 2 = m i=0 [ L ( i ) L ( i 1 )] k i=0 [( inci ) ( inci 1 )]+m Φ /v ma. Finally, TT( L, R ) = L ( m ) = L ( m ) L ( 1 ) = m i=0 [ L ( i ) L ( i 1 )] ki=0 [( inci ) ( inci 1 )] + m Φ /v ma = TT( l, r ). Hence, the treatment plan ( l, r ) generated by DMLC-SNGLEPAR is optimal in therapy time. 3.2. Multiple Leaf Pairs We present multiple leaf pair sequencing algorithms for DMLC without constraints and with the interdigitation constraint. These algorithms are from Kamath et. al (2004).
Algorithms for Sequencing Multileaf Collimators 20 3.2.1. Optimal Schedule Without Constraints For sequencing of multiple leaf pairs, we apply Algorithm DMLC-SNGLEPAR to determine the optimal plan for each of the n leaf pairs. This method of generating schedules is described in Algorithm DMLC- MULTPAR (Figure 19). Note that since 0, m are not necessarily non-zero for any row, we replace 0 by l and m by g in Algorithm DMLC-SNGLEPAR for each row, where l and g, respectively, denote the first and last non-zero sample points of that row. Also, for rows that contain only zeroes, the plan simply places the corresponding leaves at the rightmost point in the field (call it m+1 ). Algorithm DMLC-MULTPAR For(i = 1; i n; i + +) End For Apply Algorithm DMLC-SNGLEPAR to the ith pair of leaves to obtain plan ( il, ir ) that delivers the intensity profile i (). Figure 19. Obtaining a schedule Theorem 8 [Kamath et. al. (2004)] Algorithm DMLC-MULTPAR generates schedules that are optimal in therapy time. 3.2.2. Optimal Algorithm With nterdigitation Constraint The schedule generated by Algorithm DMLC-MULTPAR may violate the interdigitation constraint. Note that no intra-pair constraint violations can occur for S min = 0. So the interdigitation constraint is essentially an inter-pair constraint. f the schedule has no interdigitation constraint violations, it is the desired optimal schedule. f there are violations in the schedule, we eliminate all violations of the interdigitation constraint starting from the left end, i.e., from 0. To eliminate the violations, we modify those plans of the schedule that cause the violations. We scan the schedule from 0 along the positive direction looking for the least v at which is positioned a right leaf (say R u ) that violates the interpair separation constraint. After rectifying the violation at v with respect to R u we look for other violations. Since the process of eliminating a violation at v, may at times, lead to new violations involving right leaves positioned at v, we need to search afresh from v every time a modification is made to the schedule. We now continue the scanning and modification process until no interdigitation violations eist. Algorithm DMLC-NTERDGTATON (Figure 20) outlines the procedure. Let M = (( 1l, 1r ), ( 2l, 2r ),...,( nl, nr )) be the schedule generated by Algorithm DMLC-MULTPAR for the desired intensity profile. Let N(p) = (( 1lp, 1rp ), ( 2lp, 2rp ),..., ( nlp, nrp )) be the schedule obtained after Step iv of Algorithm DMLC-NTERDGTATON is applied p times to the input schedule M. Note that M = N(0). To illustrate the modification process we use eamples. There are two types of violations that may occur. Call them Type1 and Type2 violations and call the
Algorithms for Sequencing Multileaf Collimators 21 Algorithm DMLC-NTERDGTATON (i) = 0 (ii) While (there is an interdigitation violation) do (iii) Find the least v, v, such that a right leaf is positioned at v and this right leaf has an interdigitation violation with one or both of its neighboring left leaves. Let u be the least integer such that the right leaf R u is positioned at v and R u has an interdigitation violation. Let L t denote the left leaf with which R u has an interdigitation violation. Note that t {u 1, u + 1}. n case R u has violations with two adjacent left leaves, we let t = u 1. (iv) Modify the schedule to eliminate the violation between R u and L t. (v) = v (vi) End While Figure 20. Obtaining a schedule under the constraint corresponding modifications Type1 and Type2 modifications. To make things easier, we only show two neighboring pairs of leaves. Suppose that the (p + 1)th violation occurs between the right leaf of pair u, which is positioned at v, and the left leaf of pair t, t {u 1, u + 1}. n a Type1 violation, the left leaf of pair t starts its sweep at a point Start(t, p) > v (see Figure 21). To remove this interdigitation violation, modify ( tlp, trp ) to ( tl(p+1), tr(p+1) ) as follows. We let the leaves of pair t start at v and move them at the maimum velocity v ma towards the right, till they reach Start(t, p). Let the number of MUs delivered when they reach Start(t, p) be 1. Raise the profiles tlp () and trp (), Start(t, p), by an amount 1 = Φ (Start(t, p) v )/v ma. We get, { Φ ( v )/v ma v < Start(t, p) tl(p+1) () = tlp () + 1 Start(t, p) tr(p+1) () = tl(p+1) () t (), where t () is the target profile to be delivered by the leaf pair t. A Type2 violation occurs when the left leaf of pair t, which starts its sweep from v, passes v before the right leaf of pair u passes v (Figure 22). n this case, tl(p+1) is as defined below. { tlp () < v tl(p+1) () = tlp () + v where = urp ( v ) tlp ( v ) = 3 2. Once again, tr(p+1) () = tl(p+1) () t (), where t () is the target profile to be delivered by the leaf pair t. n both Type1 and Type2 modifications, the other profiles of N(p) are not modified. Since tr(p+1) differs from trp for v there is a possibility that N(p + 1) has interpair separation violations for right leaf positions v. Since none of the other right leaf profiles are changed from those of N(p) and since the change in tl only delays the
Algorithms for Sequencing Multileaf Collimators 22 ulp urp tl(p+1) tlp tr(p+1) 1 trp v Start(t,p) Figure 21. Eliminating a Type1 violation tl(p+1) ulp 3 2 urp tr(p+1) trp tlp v Figure 22. Eliminating a Type2 violation (close parallel dotted and solid line segments overlap, they have been drawn with a small separation to enhance readability)
Algorithms for Sequencing Multileaf Collimators 23 rightward movement of the left leaf of pair t, no interdigitation violations are possible in N(p + 1) for < v. One may also verify that since tl0 and tr0 are feasible plans that satisfy the maimum velocity constrains, so also are tlp and trp, p > 0. Theorem 9 [Kamath et. al. (2004)] Algorithm DMLC-NTERDGTATON generates DMLC schedules free of interdigitation violations that are optimal in therapy time for unidirectional schedules. 4. Field splitting without feathering n this section we deviate slightly from our earlier notation and assume that the sample points are 1, 2,..., m rather than 0, 1,..., m. All other notation remains unchanged. The notation and algorithms are from Kamath et. al. (2004b). 4.1. Optimal field splitting for one leaf pair 4.1.1. Delivering a profile using one field An intensity profile can be delivered in optimal therapy time using the plan generated by Algorithm SNGLEPAR. The optimal therapy time is given by the following lemma. Lemma 4 [Kamath et. al. (2004b)] Let inc1, inc2,...,incq be the indices of the points at which ( i ) increases, i.e., ( inci ) > ( inci 1 ). The therapy time for the plan ( l, r ) generated by Algorithm SNGLEPAR is q i=1[( inci ) ( inci 1 )], where ( inc1 1 ) = 0. Algorithm SNGLEPAR can be directly used to obtain plans when is deliverable using a single field. Let l be the least inde such that ( l ) > 0 and let g be the greatest inde such that ( g ) > 0. We will assume without loss of generality that l = 1. So the width of the profile is g sample points, where g can vary for different profiles. Assuming that the maimum allowable field width is w sample points, is deliverable using one field if g w; requires at least two fields for g > w; requires at least three fields for g > 2w. The case where g > 3w is not studied as it never arises in clinical cases. The objective of field splitting is to split a profile so that each of the resulting profiles is deliverable using a single field. Further, it is desirable that the total therapy time is minimized, i.e., the sum of optimal therapy times of the resulting profiles is minimized. We will call the problem of splitting the profile of a single leaf pair into 2 profiles each of which is deliverable using one field such that the sum of their optimal therapy times is minimized as the S2 (single pair 2 field split) problem. The sum of the optimal therapy times of the two resulting profiles is denoted by S2(). S3 and S3() are defined similarly for splits into 3 profiles. The problem S1 is trivial, since the input profile need not be split and is to be delivered using a single field. Note that S1() is the optimal therapy time for delivering the profile in a single field. From Lemma 4 and the fact that the plan generated using Algorithm SNGLEPAR is optimal in therapy time, S1() = q i=1[( inci ) ( inci 1 )].
Algorithms for Sequencing Multileaf Collimators 24 4.1.2. Splitting a profile into two Suppose that a profile is split into two profiles. Let j be the inde at which the profile is split. As a result, we get two profiles, P j and S j. P j ( i ) = ( i ), 1 i < j, and P j ( i ) = 0, elsewhere. S j ( i ) = ( i ), j i g, and S j ( i ) = 0, elsewhere. P j is a left profile and S j is a right profile of. Lemma 5 Let S1(P j ) and S1(S j ) be the optimal therapy times, respectively, for P j and S j. Then S1(P j ) + S1(S j ) = S1() + Î( j), where Î( j) = min{( j 1 ), ( j )}. We illustrate Lemma 5 using the eample of Figure 23. The optimal therapy time for the profile is the sum of increments in intensity values of successive sample points. However, if is split at 3 into P 3 and S 3, an additional therapy time of Î( 3 ) = min{( 2 ), ( 3 )} = ( 3 ) is required for treatment. Similarly, if is split at 4 into P 4 and S 4, an additional therapy time of Î( 4) = min{( 3 ), ( 4 )} = ( 3 ) is required. Lemma 5 leads to the following O(g) algorithm for S2. Algorithm S2 (1) Compute Î( i) = min{( i 1 ), ( i )}, for g w < i w + 1. (2) Split the field at a point j where Î( j) is minimized for g w < j w + 1. t is evident from Lemma 5 that if the width of the profile is less than the maimum allowable field width (g w), the profile is best delivered using a single field. f g > 2w two fields are insufficient. So it is useful to apply Algorithm S2 only for w < g 2w. Once the profile is split into two as determined by Algorithm S2, the left and right profiles are delivered using separate fields. The total therapy time is S2() = S1(P j ) + S1(S j ), where j is the split point. 4.1.3. Splitting a profile into three Suppose that a profile is split into three profiles. Let j and k, j < k, be the indices at which the profile is split. As a result we get three profiles P j, M (j,k) and S k, where P j ( i ) = ( i ), 1 i < j, M (j,k) ( i ) = ( i ), j i < k, and S k ( i ) = ( i ), k i g. P j, M (j,k) and S j are zero at all other points. P j is a left profile, M (j,k) is a middle profile of and S k is a right profile. Lemma 6 [Kamath et. al. (2004b)] Let S1(P j ), S1(M (j,k) ) and S1(S k ) be the optimal therapy times, respectively, for P j, M (j,k) and S k. Then S1(P j ) + S1(M (j,k) ) + S1(S k ) = S1() + min{( j 1 ), ( j )} + min{( k 1 ), ( k )} = S1() + Î( j) + Î( k). Proof: Similar to that of Lemma 5 Lemma 6 motivates the following algorithm for S3. Algorithm S3 (1) Compute Î( i) = min{( i 1 ), ( i )}, for 1 < i w + 1, g w < i g. (2) Split the field at two points j, k such that 1 j w + 1, g w < k g, 0 < k j w, and Î( j) + Î( k) is minimized.
Algorithms for Sequencing Multileaf Collimators 25 1 2 3 4 P 3 () P () 4 (a) 5 6 S 3() S () 4 ( 2) ( 3) ( 1) ^ ( 3 ) 1 2 3 P 3 () (b) 3 4 5 6 S 3() (c) ( 2) ( ) 4 ( 3) ( 1) ^ ( 4 ) 1 2 3 4 P 4 () 4 5 6 S 4() (d) (e) Figure 23. Splitting a profile (a) into two. (b) and (c) show the left and right profiles resulting from a split at 3 ; (d) and (e) show the left and right profiles resulting from a split at 4
Algorithms for Sequencing Multileaf Collimators 26 Note that for Algorithm S3 to split into three profiles that are each deliverable in one field, it must be the case that g 3w. Once the profile is split into three as determined by Algorithm S3, the resulting profiles are delivered using separate fields. The minimum total therapy time is S3() = S1(P j ) + S1(M (j,k) ) + S1(S k ). Algorithm S3 eamines at most g 2 candidates for (j, k). So the compleity of the algorithm is O(g 2 ). 4.1.4. Bounds on optimal therapy time ratios The following bounds have been proved on ratios of optimal therapy times. Lemma 7 [Kamath et. al. (2004b)] (a) 1 S2()/S1() 2 (b) 1 S3()/S1() 3 (c) 0.5 < S3()/S2() < 2 Lemma 7 tells us that the optimal therapy times can at most increase by factors of 2 and 3, respectively, as a result of a splitting a single leaf pair profile into 2 and 3. Also, the optimal therapy time for a split into 2 can be at most twice that for a split into 3 and vice versa. 4.2. Optimal field splitting for multiple leaf pairs The input intensity matri (say ) for the leaf sequencing problem is obtained using the inverse planning technique. The matri consists of n rows and m columns. Each row of the matri specifies the number of monitor units (MUs) that need to be delivered using one leaf pair. Denote the rows of by 1, 2,..., n. For the case where is deliverable using one field, the leaf sequencing problem has been well studied in the past. The algorithm that generates optimal therapy time schedules for multiple leaf pairs (Algorithm MULTPAR) applies algorithm SNGLEPAR independently to each row i of. Without loss of generality assume that the least column inde containing a non zero element in is 1 and the largest column inde containing a non zero element in is g. f g > w, the profile will need to be split. We define problems M1, M2 and M3 for muliple leaf pairs as being analogous to S1, S2 and S3 for single leaf pair. The optimal therapy times M1(), M2() and M3() are also defined similarly. 4.2.1. Splitting a profile into two Suppose that a profile is split into two profiles. Let j be the column at which the profile is split. This is equivalent to splitting each row profile i, 1 i n, at j as defined for single leaf pair split. As a result we get two profiles, P j (left) and S j (right). P j has rows Pj 1, P j 2,...,P j n and S j has rows Sj 1, S2 j,...,sn j. Lemma 8 [Kamath et. al. (2004b)] Suppose is split into two profiles at j. The optimal therapy time for delivering P j and S j using separate fields is ma i {S1(P i j )} + ma i {S1(S i j )}.
Algorithms for Sequencing Multileaf Collimators 27 Proof: The optimal therapy time schedule for P j and S j are obtained using Algorithm MULTPAR. The therapy times are ma i {S1(Pj i)} and ma i{s1(sj i )} respectively. So the total therapy time is ma i {S1(Pj i)} + ma i{s1(sj i)}. From Lemma 8 it follows that the M2 problem can be solved by finding the inde j, 1 < j g such that ma i {S1(P i j)} + ma i {S1(S i j)} is minimized (Algorithm M2). Algorithm M2 (1) Compute ma i {S1(Pj i)} + ma i{s1(sj i )} for g w < j w + 1. (2) Split the field at a point j where ma i {S1(Pj)} i + ma i {S1(Sj)} i is minimized for g w < j w + 1. From Lemma 4, S1(Pj i) = inci j[( inci ) ( inci 1 )]. For each i, S1(P1 i), S1(P 2 i),..., S1(Pg i ) can all be computed in a total of O(g) time progressively from left to right. So the computation of S1s (optimal therapy times) of all left profiles of all n rows of can be done in O(ng) time. The same is true of right profiles. Once these values are computed, step (1) of Algorithm M2 is applied. ma i {S1(Pj)} i + ma i {S1(Sj)} i can be found in O(n) time for each j and hence in O(ng) time for all j in the permissible range. So the time compleity of Algorithm M2 is O(ng). 4.2.2. Splitting a profile into three Suppose that a profile is split into three profiles. Let j, k, j < k, be the indices at which the profile is split. Once again, this is equivalent to splitting each row profile i, 1 i n at j and k as defined for single leaf pair split. As a result we get three profiles P j, M (j,k) and S k. P j has rows P 1 j, P 2 j,...,p n j, M (j,k) has rows M 1 (j,k), M2 (j,k),...,mn (j,k) and S k has rows S 1 k, S 2 k,...,s n k. Lemma 9 [Kamath et. al. (2004b)] Suppose is split into three profiles by splitting at j and k, j < k. The optimal therapy time for delivering P j, M (j,k) and S k using separate fields is ma i {S1(P i j )} + ma i{s1(m i (j,k) )} + ma i{s1(s i k )}. Proof: Similar to that of Lemma 8. Algorithm M3 solves the M3 problem. Algorithm M3 (1) Compute ma i {S1(P i j)} + ma i {S1(M i (j,k) )} + ma i{s1(s i k)} for 1 < j w + 1, g w < k g, 0 < k j w. (2) Split the field at two points j, k, such that 1 < j w + 1, g w < k g, 0 < k j w, and ma i {S1(Pj i)}+ma i{s1(m(j,k) i )}+ma i{s1(sk i )} is minimized. The compleity analysis is similar to that of Algorithm M2. n this case though, O(g 2 ) pairs of split points have to be eamined. t is easy to see that the time compleity of Algorithm M3 is O(ng 2 ).