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1. (WO2007008899) VARIABLE FEATHERING FIELD SPLITTING FOR INTENSITY MODULATED FIELDS OF LARGE SIZE
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VARIABLE FEATHERING FIELD SPLITTING FOR INTENSITY MODULATED
FIELDS OF LARGE SIZE

FIELD OF THE INVENTION
[0001] The invention relates to radiation therapy devices, and more particularly, to a system and methods for efficiently and more safely delivering split radiation field treatment to a patient.

BACKGROUND
[0002] A radiation therapy device typically includes a radiation delivery device mounted to a gantry that is swiveled around a horizontal axis of rotation in the course of a radiation therapy treatment. The radiation delivery device generally delivers a high energy radiation beam. During treatment, the radiation beam is directed towards a patient lying in the isocenter of the gantry rotation.

[0003] The device thus normally includes a radiation source, such as a linear accelerator, for supplying the high energy radiation beam. The high energy radiation beam is typically an electron beam or an X-ray beam.
[0004] To control the radiation emitted toward a given object, a beam shielding device, such as a plate arrangement or a collimator, is typically provided in the trajectory of the radiation beam between the radiation source and the patient. A collimator is a computer-controlled mechanical beam shielding device which generally includes multiple leaves, for example, a plurality of relatively thin plates or rods, typically arranged as opposing leaf pairs. The plates are formed from a relatively dense and radiation impervious material and are generally independently positionable to size and shape of the radiation beam. These leaves

move over the tissue being radiated, thus blocking out some areas and filtering others to vary the beam intensity and precisely distributing the radiation dosage.

[0005] A multileaf collimator (MLC) is an example of a multileaf beam shielding device that can accurately and efficiently adjust the size and shape of the radiation beam. The size and shape of a radiation beam is designed during the treatment planning process. This is useful for both intensity modulated radiation treatment (IMRT) and three-dimensional conformal radiation therapy (3D CRT).
[0006] Traditional radiotherapy utilizes uniform beams of radiation, producing a uniform distribution of dose throughout the irradiated volume, which includes the target volume. This ensures the target is adequately covered, but does little or nothing to avoid often critical surrounding structures. With IMRT, the beams of radiation are made to be intentionally nonuniform. In this manner, the dose distribution can be shaped to reduce or eliminate radiation to surrounding structures. As a result, IMRT is increasingly used to treat large volumes because IMRT can deliver more conformal radiation while sparing the surrounding normal tissue.
[0007] Monitor unit (MU) efficiency is a commonly used measure of beam efficiency. MU efficiency is defined as the efficiency with which the incident radiation results in dose being in absorbed in the target region of a patient. A consequence of low MU efficiency is an increase in leakage radiation that reaches the surrounding (normal) tissue of the patient.
[0008] There are several components of a successful IMRT program. The first is a process referred to as "inverse planning." Inverse planning utilizes a mathematical algorithm to optimize the intensity of the various beams. This optimization process typically is highly computer intensive.
[0009] The second component is a process to convert the intensity distributions obtained, often referred to cumulatively as a fluence map, into a series of MLC leaf movements. This is referred to as "leaf sequencing." Many device-specific factors must be accounted for in this process. These factors include radiation leakage through and between the leaves, leaf speed, dose rate, and the "tongue-and-groove" effect.
[0010] EMRT can be performed either while the beam is on, which is referred to as dynamic multileaf collimator (DMLC) delivery, or by turning the beam off while the leaves move to their next position, which is referred to as segmented multileaf collimator (SMLC) delivery. The beam shielding device defines a field on the object to which a prescribed amount of radiation is to be delivered. The usual treatment field shape results in a three-dimensional treatment volume which includes segments of normal tissue, thereby limiting the dose that can be given to the target, such as a tumor. The dose delivered to the tumor can be increased, thereby decreasing the treatment time so that the amount of dose delivered to the normal surrounding tissue is decreased. Although curjent leaf sequencing algorithms have reduced somewhat the radiation level reaching surrounding normal tissue as compared to traditional uniform beams of radiation, these leaf sequences have not provided optimal MU efficiency.
[0011] Most DVlRT treatments are administered with conventional MLC systems that are typically available on commercial linear accelerators. The MLC systems vary in design but each version has certain mechanical limitations, such as maximum leaf over-travel which limits the attainable width of the radiation beam.
[0012] It is sometimes necessary to expose large areas of the body of a patient to radiation. If the size of the required radiation field is larger than the maximum attainable width provided by the radiation delivery system, such as in the case of a large tumor, the entire radiation field cannot be exposed at one time by the radiation system. This necessitates that a large field be split into 'a plurality of abutting field portions, such as 2 or 3 fields portions, where the respective field portions are delivered one at a time. This process is known as feathering.

[0013] Feathering splits the overall field into field portions having equal width. For example, available field-splitting algorithms split the field either near the middle of the field along an arbitrarily-chosen straight line, or with a pre-defined constant overlap region for feathering. Due to concerns of increased whole body dose in IMRT delivery, the problem of MU efficiency in field splitting needs to be addressed.

SUMMARY OF THE INVENTION
[0014] The invention is directed to a radiation delivery system and method which reduces the total monitor units (MUs) used to treat patients requiring large radiation fields. The phrase "large radiation field" is defined herein as a prescribed radiation field width determined by a dose optimization algorithm that exceeds the maximum attainable beam width provided by the radiation delivery system, such as the large field required, for example, in the treatment of certain neck and back tumors.
[0015] A method and associated apparatus for delivering intensity-modulated radiation therapy (IMRT) according to the invention uses variable feathering field splitting for intensity modulated fields of large size. The method includes the steps of providing an intensity matrix for the treatment of a patient, the intensity matrix having a plurality of rows and columns for spanning a prescribed radiation field including a prescribed field width. The intensity matrix is generally determined by a medical professional (radiologist or medical physicist) during the planning step. The prescribed field width is compared to a maximum field width provided by the radiation treatment system. The intensity matrix is split into a plurality of spatially overlapping intensity submatrices when the prescribed width exceeds the maximum field width, wherein the splitting comprises variably feathering the intensity matrix. Radiotherapy is then provided using a leaf sequencing method to generate the submatrices.

BRIEF DESCRIPTION OF THE DRAWINGS
[0016] A better understanding of the present invention can be obtained when the following detailed description is considered in conjunction with the following drawings in which:
[0017] FIG. 1 is a schematic diagram of a system for delivering radiation treatment to a patient, according to one embodiment of the present invention.
[0018] FIG. 2 is a schematic diagram of a beam-shaping device incorporated in a system for delivering radiation treatment to a patient, according to another embodiment of the present invention.
[0019] FIG. 3 is a schematic diagram of a system for delivering radiation treatment to a patient, according to still another embodiment of the present invention.
[0020] FIG. 4 shows a fluence matrix and fields resulting from variably feathering along a non-constant column of the intensity matrix according to the invention.
[0021] FIG. 5 shows (a) a single pair profile, (b) a plan (J1J,.) , for profile / of (a), is obtained using Algorithm SINGLEP AIR and is constructed without taking field width constraints into account. It can be delivered for Iι(xg_w) MUs in the left field (shaded) and the remainder in the right field; (c) the left and right profiles resulting from the split generated by Algorithm S2G . This split is delivered in optimal time using the plan (/,,/,.) ; d) a single pair profile; (e) plan (1,,I1.) , for profile / of (d), is obtained using Algorithm SINGLEP AIR and is constructed without taking field width constraints into account. (/,,/,.) is the modified plan obtained using Algorithm S2G and can be delivered for Iι(xg_w) MUs in the leftfield (shaded) and the remainder in the right field; (f) the left and right profiles resulting from the
SlG . This split is delivered in optimal time using the plan

[0022] FIG. 6 shows a single pair profile.
[0023] FIG. 7 shows pian (/,, /.,.), for profile I of Figure 6, is obtained using Algorithm
SINGLEPAIR and is constructed without taking field width constraints into account. (J," ', !") is the modified plan obtained during an iteration of Algorithm S3G with Xj as shown.

' [0024] FIG. 8 shows The left and right, profiles resulting from the split generated during the iteration of Algorithm S3)Q shown in Figure I;., This split is delivered in optimal time using the plan (Ij", /")

DETAILED DESCRIPTION
[0025] FIG. 1 is a schematic diagram a system 100 for delivering radiation treatment to a patient 102 which can be used with the present invention which comprises variable feathering field splitting for intensity modulated fields of large size. The system 100 illustratively includes a radiation source 104 for providing a radiation beam and a beam- shaping device 106 interposed between the radiation source 104 and the patient 102 for shaping the radiation beam.
[0026] The radiation source 104, more particularly, can provide electron, photon, or other radiation useful for treating cancer or other disease. For example, as described in Published U.S. Application No. 20050148841 entitled "LEAF SEQUENCING METHOD
AND SYSTEM" by the present inventors, the radiation source 104 can be an electron accelerator for delivering an electron beam. As illustrated, the radiation source 104 is mounted upon a gantry 108 that rotates upon a fixed axis so as to permit the position of the radiation source to change relative to the patient 102.
[0027] Referring additionally now to FIG. 2, the beam-shaping device 106 interposed between the radiation source 104 and the patient 102 illustratively is shown comprising a plurality of opposing plates or leaves 110a-n that are substantially impervious to the radiation emitted by the radiation source. The leaves 110a-n can be moved by a drive unit (not shown) in a substantially horizontal motion relative to one another and substantially perpendicular to the radiation beam. The movement permits the plurality of leaves 110a-n to be aligned and realigned relative to one another and the radiation beam. Each such alignment comprises a leaf sequence that changes the size and shape of the radiation beam, as further described in Published U.S. Application No. 20050148841. Accordingly, the leaf sequences determine the dimensions of a field on a designated region of the patient 102 to which a prescribed amount of radiation is to be delivered.
[0028] The beam-shaping device 106 can be an MLC. More particularly, the beam-shaping-device can comprise a segmented MLC. Alternatively, the beam-shaping device can comprise a dynamic MLC.
[0029] Referring additionally now to FIG. 3, the system 300 for delivering radiation treatment to a patient 102 illustratively includes a processor or other computing device in communication with the beam-shaping device 106. As described herein, the processor 112 can control the beam-shaping device 106 so that the beam-shaping device splits the radiation beam into a plurality of radiation fields that are delivered to the patient. The radiation beam, more particularly, is split so as to substantially minimize at least one of a total therapy time and a total number of leaf sequences in delivering a predetermined dosage of radiation to the patient. As defined herein, "a substantial minimization of the total therapy time" denotes a reduction of the therapy time to no more 20 percent, and more preferably no more than 10 percent, over an absolute minimum. Similarly, minimization of the total number of leaf sequences denotes no more than 20 percent, and more preferably no more than 10 percent, over the absolute minimum.

[0030] The processor 112 can connect to a standard input-output (I/O) device such as a keyboard. Thus, the processor can be programmed, for example, by a therapist according to instructions dictated by an oncologist or medical physicist. Processor 112 can also control the beam-shaping device 106 by executing and delivering instructions to the drive units (not shown) that align the opposing plates or leaves 110a-n so that different leaf sequences are effected according to the programmed instructions.
[0031] A method and associated apparatus for delivering intensity-modulated radiation therapy (IMRT) using variable feathering field splitting for intensity modulated fields of large size includes the steps of providing an intensity matrix for the treatment of a patient, the intensity matrix having a plurality of rows and columns for spanning a prescribed radiation field including a prescribed field width. The intensity matrix is generally determined by a medical professional during the planning step. The prescribed width is compared to a maximum field width provided by the radiation treatment system. The intensity matrix is split into a plurality of spatially overlapping intensity submatrices when the prescribed width exceeds the maximum field width.
[0032] Unlike previous feathering work which discloses a predefined constant overlap region for feathering, being a constant overlap of up to about a 2 cm width throughout the field, the overlap variable region according to the invention is a non-constant width derived from "variably feathering" the intensity matrix to derive corresponding submatrices. The variable feathering calculation as described herein is preferably based on minimizing MUs and can be implemented by a field splitting module integrated into treatment planning software. Radiotherapy is then provided using a leaf sequencing method to generate the submatrices.
[0033] As noted above, previous feathering work utilized a constant overlap width of no more than about 2 cm. Variable feathering according to the invention can utilize an overlap that ranges from zero to the maximum allowable width of a field (which is currently generally about 14 cm). Suppose the intensity matrix is a large matrix having a width = 20 cm, and the maximum allowed field width is 14 cm. With conventional fixed width feathering of 2 cm, one possible arrangement is to have the first split submatrix have a width of 12 cm, with the second submatrix having a width of width 10 cm, and have their respective two central columns (2 cm) overlap (feathering = 2). If any of the submatrices are made wider, the overlap would have to be wider and this is not allowed in fixed width feathering. In contrast, using variable feathering according to the invention, it is possible for example to split the large intensity matrix into two submatrices each of width 14 cm with an overlap of the 8 central columns (8 cm = 14 cm + 14 cm - 20 cm).
[0034] By allowing wider overlaps according to the invention, the use of smaller overlaps is not discarded, since algorithms according to the invention pick the best overlap for MU efficiency. The best overlap may be wide or narrow. Earlier methods allowed only no or narrow and constant overlaps.
[0035] The ability to utilize wider overlaps provides higher MU efficiency. A simplified explanation for the MU efficiency improvement is that the presence of a wider overlap region gives a bigger time/space "window" of treatment delivery during a transition which may be made from neighboring subfϊelds, such as between the left and right subfield in the case of two subfields. The larger window provided presents greater opportunities for an efficient split. Experimentally, as noted in the Examples, the improvement obtained averaged 19% compared to a commercial system. Earlier methods disclosed by the inventors (straight line split and 2 cm feathering) had given a little over 10% improvement over a conventional t commercial system. Clearly, allowing the wider overlap provides an improvement in MU efficiency.

[0036] The invention thus treats the most general field splitting problem for segmental MLCs. Ia this generalized model, the only constraint is that the subfields resulting from the split are each required to have a width ≤ w sample points, where w is the maximum allowable field width provided by the radiation system used. Field width is loosely defined as the number of columns over which non-zero fluence values span. There may be bixels (2D pixels) that receive parts of their desired fluence from two subfields.
[0037] Fig. 4 shows a fluence matrix and fields resulting from variably feathering along a non-constant column of the intensity matrix according to the invention. A generalized split with w = 5 is shown. The matrix on the left is split into the two matrices on the right. In the first row there is feathering along column 3. The second row feathers along columns 4 and 5 and the third row feathers along columns 3, 4 and 5. The widths of the fields resulting from the split are shaded in Fig. 4. It can be shown that the inventive algorithm for field splitting is optimal in MU efficiency (See attached papers in Appendix). Tests with 32 clinical matrices showed that the inventive optimal field splitting algorithm reduces total MUs by an average of 19% and up to 45% compared to the algorithm that splits a field in the middle.
[0038] The invention is described in further detail below along with some preferred embodiments for several system arrangements.
1. Single leaf pair profile
[0039] Let J(X1) be the discretized fluence profile obtained from the optimizer that gives the fluence values at sample points xvx2, ... ,xm for a single leaf pair. A profile /(x,) is simply referred to herein as profile / . When the left leaf is placed so that it shields exactly the points xpx2,...,x(. , it will be said that the left leaf is positioned at xM . In particular, the point Λ'/+I is not shielded by it. When the right leaf is placed so that it shields exactly xM, xi+2, ...,xm , it will be said that the right leaf is positioned at xM . The problem of delivering the exact profile / using a single field has been extensively studied. Ma et. al. (Ma L, et. al. An optimized leaf-setting algorithm for beam intensity modulation using dynamic

multileaf collimators. Phys. Med. Biol. 43:1629, 1998) provided an O(m) algorithm for the

problem such that MU is minimized. Kamath et. al. (Kamath S, et. al. Leaf sequencing algorithms for segmented multileaf collimation. Phys. Med. Biol. 48:307, 2003) also described the algorithm (Algorithm SINGLEP AIR) and gave an alternate proof that it obtains

a plan (I1J1 ) with optimal therapy time for / . Here, I1(X1) and I1 (X1) denote, respectively,

the number of MUs after which the left and right leaves pass point X1 during the left to right

sweep. Let S1 = I(xι)-I(xt_λ) , where I(x0) = 0 and /(x,,!+I) = 0 . Let incl,inc2,...,incq be the

indices of the points at which 1(X1) increases, i.e., I(xma) > I(xιna-\) and let

dec\,dec2,.. ,decr be the indices of the points at which 1(X1) decreases, i.e.,

I(xciea) < I(χdea-]) - The optimal therapy time for / , Sl(I) , is given by Lemma 1.

Lemma 1. 51(7) = ∑l δma = -∑^ δ!lea .
The following lemma is useful in proving the optimality of the algorithm developed and is stated without proof.
Lemma 2. The following is true of all treatment plans delivered using one or more fields:

1. If I(x,_ι) > I (X1) , the right leaf must be positioned at X1 for at least

I(x,-\ ) ~ I{χ, ) - ~δ, MUs in every plan for / .

2. If I(xt_}) < I(xt) , the left leaf must be positioned at X1 for at least

I (xt ) - I(x,_x ) = S1 MUs in every plan for / .

Consider a fluence profile / (Fig. 5(a)). (I1J1.) is its optimal plan generated using Algorithm

SINGLEP AlE.. Suppose that w < g ≤ 2w, so that the field needs to be split into two. Call the

fields resulting from a split as the left and the right fields, labeling them to denote their respective positions. It is possible that the left field stretches as far to the right and including sample point xw and that the right field stretches as far to the left and including xg_w+l . Examine the unidirectional leaf trajectories in the plan (/,,/,.) (Fig. 5b). Clearly, the plan cannot be delivered as such because of the field width constraint. The strategy according to the invention is to follow the plan (I1J1.) to the maximum extent possible while delivering it using two fields. There are two cases. In the first case, I,(xg_w) ≤ Ir(xw+i ) (as *s me case m Fig. 5(b)), the left field can be treated using the plan (/,,/,.) for It(xg_w) MUs. At the end of this time, the left leaf will be at X1 < xg_w (and can immediately move to xg_w+l) and the right leaf will immediately be positioned in the range [xg_w+vxw+]] (since /r(xH,+l) > I,(xg_w) ). Since both resulting leaf positions are within the range permissible for the right field ([-vi,_ll i.1 ,.Εo] ),treatment using the plan (I1J1.) in the left field is stopped, move to the right field and continue the treatment in the right field. No MU increase is needed due to field splitting.
In the second case, It(xg_w) > Ir(χw+ι) • Fig- 5(d) shows a profile / and Fig. 5(e) shows the plan (IpI1.) for which this is the case. The left field can be treated using the plan (I/, I1.) for I,(xg_w) MUs. The left leaf can move to x> xg_w+] at this time and the remainder of the plan can be delivered using the right field as in the first case. However, the right leaf will have to cross the right end of the left field (xw+l ) when I,.(xw+:) < I/(xg→) MUs have been delivered in the plan (I1J1.) . Since this is not possible, the right leaf is stopped at the point xw+l till I,(xg_w) MUs are delivered in the left field. As a result of this, the right leaf profile will be raised by Il(xg_w) -Il.(xw+]) MUs for x ≥ xw+i . To maintain constant difference between the profiles, the left leaf profile is also raised by Iι(xg→,) -Ir(xw^) MUs for x ≥ xw+i . Call the modified plan (/,,/,.) . When /,(^ ) MUs are delivered, the left leaf can move to the right field and the remainder of the plan (I1, Ir) is delivered using the right field. The plan (I1J1.) , has an increase in total therapy time by Iι(xg→) -Ir(xwH) compared to (I1J1.) . In Fig. 5(e), the horizontal dotted line at //(Λ: ) corresponds to the time at which the transition is made from the left to the right field. Fig. 5(f) shows the fluence profiles delivered in the left and right fields as a result of this split. Algorithm S2G summarizes the general method.

Algorithm S2G
• Find plan (I1J1.) for / using Algorithm SINGLEP AIR, ignoring the field width constraints.
• If I1 (xg_w) > I1. (X+1 ) , raise the left and right leaf profiles by /; (xg_w) - 11. (xw+l ) for x > xw . Otherwise, do not modify the plan. Call the resulting plan (I1J,.) ■
• Treat the left field using the plan (/,,/,.) for It(xg_w) MUs. Then switch to the right field and continue treatment with (I1J1.) ■
The following theorem is stated without proof.
Theorem 1. Algorithm S2G generates generalized field splits that are optimal in total therapy time. The optimal total therapy time of the split generated by Algorithm S2G is iSl(/) + max {0, Iι(xg→) -I,.(xw+ι)} , where Sl(I) is found by ignoring the field width constraints.

2. Multiple leaf pair profiles

Suppose the fluence matrix / consists of n rows and m columns. Denote the rows of / by 1 \ J2, ...Jn . For the case where / is deliverable using one field, the leaf sequencing problem has been well studied. The algorithm that generates optimal therapy time schedules for multiple leaf pairs (Algorithm MULTIP AIR, Kamath S, et. al. Leaf sequencing algorithms for segmented multileaf collimation. Phys. Med. Biol. 48:307, 2003) applies algorithm SINGLEPAIR independently to each row /; of / . Without loss of generality assume that the least column index containing a non zero element in / is 1 and the largest column index containing a non zero element in / is g . If g > w, the profile will need to be split.
Let {(Iυ,Iw),(I2l,I2r),...,(Ilιl,Inr)} be the schedule (set of plans of all leaf pairs) generated by Algorithm MULTIPAIR for delivering the profile / . The points xl 3x2,...,xw, need to be completely treated in the left field. Let k be an index of a leaf pair for which the left leaf is slowest in reaching xg_w+l during the left to right sweep, i.e., Ia(xg-w) = τaaxi≤ISll{Ia(xg_w)} .

For each leaf pair i , compare Iir(xw+1) with /w(xg_,v) . If /H(xg_w ) < /.,. (xw+1) , then the profile of leaf pair / remains unaltered. If Ikl(xg_M) > Iir (xw+] ) , then the right leaf of pair i will have to stop at xw+1 till the left leaf of pair k arrives at xg_w+1 • As a result, the left and right leaf profiles of pair / get raised by Ik,(xg→.) -Iir(xw+]) for x > xw . Call the resulting schedule

{(/'i/,/'ir)»(/'2/»/'2r)»-..,(/',,/,/'»r)} - When the left leaf of pair k reaches xg_w+l in this schedule, stop treatment of the left field and move to the right field. The remainder of the schedule is delivered in the right field. The method is described in Algorithm M2G . The optimal total therapy time for the split generated by Algorithm M2G is max j [Sl(Ij) + max {0, Ia (xg_J - 1 jr (xw+i )} } .
Algorithm MlG
• Find the schedule {(Iu,I]r),(I2l,I2r),...,(Inl,Inr)} for / using Algorithm MULTIP APR, ignoring the field width constraints.
Let lkl (xg_w) = max1≤/≤,, {/, (χg_ J} .
• For each leaf pair i do step 4.

• If Λ/CVw)>Λ(Λv+i)> raise me ^ ^d right profiles of pair i by hi (xg-w) ~ Iu (xw+\ ) ^or x > xw ■ Otherwise, do not modify the plan for pair i .
Call the resulting schedule {(/„ , Ih ), (I21 ,I2r),..., (I111 , In, ) } .
Treat the left field using the schedule {(/„,/„ ),(I2l,I2r),...,(Inl,Im)} for Λ/(xg-u) MUs- Then switch to the right field and continue treatment with this schedule.
Lemma 3. MlG(I) > max, [Sl(I1 ) + max {0, Iu (xg_w) - Ijr OH,+I )} } , where Sl(I1 ) is found by ignoring the field width constraints and k is as in Algorithm M2G .
3. For each row j , it is shown that
MlG(T) ≥ Sl(IJ) + msx.{0,Ia(xg_w)-Ijr(xwH)} . It follows that
M2G(/)>mαx/{)S1l(/y) + max{0,/ϋ(Λ:g_J-///(x11+1)}}.



In this case max{0,Ikl(xg_w) -Ijt (xH+1)} = 0. It is easy to see that, MlG(I) > Sl(Ij) .



Let SJi = I1(X1)- I1(X1^1). Let I and R , respectively, denote the left and right profiles resulting from a generalized split. Let L} denote the j th row of L and let R} denote the j th row of R. Let δlμ =Lj (JC, ) - Z, (JCM ) and let δlβ = Rj(X1) -Rj(X1^) . The optimal total therapy time of the split is Ml(I) +Ml(R). Due to the field width constraint, the points X1, X2,... ,xg_w, can only be exposed in L. So, Lj(xt) = Ij(xt), l≤i≤g-w, and therefore, SIJ1=SJ1, l≤i≤g-w . Similarly, SIJ1=SJ1, w+l≤i≤g . Let the number of MUs for which the left leaf of leaf pair j stops at point X1 in optimal schedules for Z and R , respectively, be I11(X1) and J^O1) and the number of MUs for which the right leaf stops at

point X1 in optimal schedules for L and R , respectively, be Lj,(x,) and j§ r(xt) .

Since the points 1, 2,..., g — w , can only be exposed in L , for each j ,

Ml(L) ≥ Y £ (jf ) ≥ Y Λ,, (Lemma 2)

~ Zjώfa>0,l<ι<g-w *'
-" J> ~ kl \Xg-w) ~ 1Jr \Xw+\ ) + 2-1 δ, <0,ι≤w+l ^~ J' ' '

Similarly, from Lemma 1 and the fact that the points w+ l,w+2,...,g , can only be exposed in

R1 , for each j , Ml(R) ≥ Sl(R1) ≥ ∑_+| R11(X1) ≥ ∑^<0,>u,+I (~δ2β) (Lemma 2)



Ml(L) + Ml(R)

. Therefore, M2G(/) > 61I(Z7 ) + Iu (Xg_u ) - 11, (xv,,+l ) .

Theorem 2. Algorithm M2G generates generalized field splits that are optimal in total therapy time.

5. Follows from Lemma 3 and the fact that the total therapy time of the split

generated by Algorithm MlG is
.

Although the invention has been described above relative to splitting a profile into two (2) intensity matrices m an optimal manner, the invention more generally splits a profile into a plurality (two or more) intensity matrices in an optimal manner. Splitting a profile into three (3) intensity matrices is described below.

Splitting a profile into three
Consider the problem of splitting a single leaf pair profile I (Figure 6) inr.o three fields. In the discussion below, the indices aie calculated assuming that 2w < g < 3w. However, the method can easily be used for g < 2tι> with some modifications. The method we describe is an extension of the method used for splits into two. Denote the three fields resulting from the split as left, middle and right fields. As in case of the split, into two, the left field can extend over the points Z1, X2, - . - , xw, and the right field can extend over xg-.w+ι, xg-w+2, - . - , Xg- For the position of the middle profile, there are several possibilities within a range. We examine each one of these and select the best.
The left most sample point that can be exposed in the middle field is a-g-^+i- When zs-2u)+i is included in the middle field, the middle field can extend over xg-2w+i, Xg-2w+2, • • ■ , χg-w- In this case, the w points xg→+i, xg→+2, • • ■ , xg, are treated by the right profile and so the middle profile cannot be any further to the left, without leaving at least the point xg-w not treated. Shifting the left boundary of the middle field one sample point to the right, the middle profile can extend over %g-2w+2, Xg-2w+3, ■ ■ • > Xg-w+i- Proceeding in this manner, it is clear that the left most position included in the middle profile has to be one of the following: xg..w+i, xg-w±2, . . . , sm+i. Algorithm 53G determines the optimal total therapy time separately assuming each one of these is necessarily the left most in the middle profile. The global optimum will be the least among these times. Next we explain how we find the optimal therapy time when a point Xj is the left most point in the middle profile.
Suppose that the left most point of the middle profile is XJ, i.e., Xj is necessarily part of the middle profile. First construct the trajectories for the left and right leaves assuming J is treated using one field (Figure 7). Next, examine this plan (Jj, Ir) and determine whether J/fø-i) < Ir(xw+\). If this is the case, start treatment of the left field with both leaves at the extreme left and move to the middle field when the left leaf reaches Xj in the left to right sweep. Otherwise, stop the right leaf at xw+\ till the left leaf reaches Xj. As a result of this action, the right leaf profile gets raised by an amount J/ (ZJ_I) — Ir(xw+ι) for x > xw+i. Raise the left leaf profile by the same amount for x > xw+χ to account for the difference between the profiles. Gall this modified plan (I[, Ir'). In the plan (II', ^) see if J[(:c5_ω) < Ir'{xj+W). If this is the case, stop treating the middle field and move to the right field when the left leaf reaches xg-w+\ during the sweep. Otherwise, stop the right leaf at sample point Xj+w till the left leaf reaches Xg-w+ι. The right leaf profile Ir' gets raised by an amount J/(xs_ω) - IT' (XJ+VJ) for x > Xj+W. The left leaf profile I[ is also raised by I[{xg_w) - Ir'{xj+W_ι) for x > Xj+W, The resulting plan is denoted by (I", I"). We show that the split generated as a result of this method is optimal in total therapy time for all cases where the middle profile has Xj as its left most point. The split generated for the profile of Figure 6 with the Xj as in Figure 7 is shown in Figure 8.
Algorithm 53G
(1) Find plan (J;, Jr) for I using Algorithm SINGLEPAIR, ignoring the field width constraints.

(2) For j = g - 2w +.1 to w + 1 do steps 3 through 5.
(3) If h{xj-i) > Ir{xw+i), raise the left and right leaf profiles by h{xj-ι) - Ir(xw+i) for x > xw. Otherwise, donot modify the plan. Call the resulting plan {I{, Ir').

(4) If Il{xg-ω) > IΛ' ^J+VJ), raise the left and right leaf profiles by Ii(xg-W) - IΛ' xj+ω) for x ≥ xJ+ul. Otherwise, donot modify the plan. Call the resulting plan (I/', i?)-

(5) If TT(Ij1', I") is the least among all j so far, set j' = j.

(6) Treat the profile using the plan (I", I") obtained using j - f. Treat the left field for the first I"(x,-i) MUs; then move to the middle field; finally, switch to the right field when I['(xg→) MUs have been delivered.

Optimal Generalized Split for Splitting a profile into three

Again, we compute the indices assuming that 2w < g < 3w. In general the method can also be used for g < Iw. As in the case of single leaf pair, the left most position included in the middle profile has to be one of the following: xs-+i, Xg-2w+2, ■ • ■ xw+ι- The optimal total therapy time is separately found assuming each one of these points is necessarily the left most in the middle profile. The optimal total theiapy tune will be the least among these.
Assume that the left most point of the middle profile is X3 , g — 2w + 1 < j < w + 1. The points j-i, X2, , Lj-i . need to be completely treated in the left field. Let k be an index of a left pair for which the left loaf is slowest in ieaching T7 dm ing the left to right sweep, i.e., 7^(xj_i) = maxi<j<ra{/,((a;j_i)}. For each leaf pair τ, compare I1, (xw+ι) with Iu(xj-i) If Iki(xj-i) < Iir(xui+i), then the profile of leaf pan ? remains unaltered On the other hand, if Iki(xj-i) > Iιr{xw+i), then the right leaf of pair i will have to stop at xw+χ till the left leaf of pair k arrives at Xj . As a result, the left and right ledi μiυliLut. υl pan i get iaibed by Iki{x3-ι) - I(XUJ+I) for x > xw. Call this modified schedule {{Il'l , Il'r), ULu
Move to treatment of the middle field when the left leaf of pair k arrives at x? in this schedule. Note that the middle field can extend up to Xj+w-i on the right and that the left most point of the right field is xg-w+1. Modify the schedule {(l[ι, I[r), {I^i J^' ), ■ ■ ■ , (^ O } ^ befoie so that the treatment of the right field begins when the slowest left leaf ieaches xfl_ω+1. The final schedule is {(/^, I{'r), (/£,, I{'r), . . . , (/^, l£r)}. The split generated as a result of this method is optimal in total therapy time for all cases where the middle profile has X1 as its left most point. Algorithm M3G varies j over g - 2w + 1, g - 2w + 2, . . . , w + I, and finds the best split.
Algorithm M3G

( 1 ) Fmd the schedule { (i,/, /lr). (I21 , 12, ), . , (InJ . Inr) } for / using Algorithm MULTIPAIR, ignoring the field width constraints.

(2) Foi j - g - Iw + 1 to w + 1 do steps 3 through 8.

(3) Let hiix j-]) = ma.xι<ι≤,,{I,ι(xj-.ι)}.

(4) For each leaf pair i do step 5.

(5) Ii /^(.ij-i) > Λ, (XuH-l)i raise the left and right piofiles of pair % by Ikl{Xj-i) -Iιr(xω+i) f°r x > Xw Otherwise, donot modify the plan for pair i.

(6) Call the resulting schedule {(Iu' , Il'r), {I2'lJ2'r), . . . , {In' l, 1n'r)}.

(7) Foi each leaf paii i do step 8
(8) If Ik'ι(xg→,) > Jln(Z7+U,), raise the left, and right piofiles of pair i by Ik'l(xg^ω) - rιr(xJ+ω) for ,τ > .Ty-H,. Otherwise, donot modify the plan for pair ι.

(9) Call the resulting schedule {(!{'„ 1^), VZ1J(Ir), • • • , (Iή'lJnr)}- (10) If TTi[I1I1Jl), Vi1J^), . . . , rø, C)} is the least among all j so far, j = /.
(11) Treat the left field using the schedule {(/{',, I{'r), (I%h Ii7,), . . . , (J^, /£r)}, which is obtained using j = j' for Il'ι{xj-ι) MUs. Then switch to the right field and continue treatment with this schedule till Ik'l(xg-W) MUs. Finally, move to the light field and complete the tieatment.

EXAMPLES
[0040] The present invention is further illustrated by the following specific examples, which should not be construed as limiting the scope or content of the invention in any way.
[0041] Results based on variably feathering according to the invention is described below. The performance of the Algorithms M2G was tested using 32 clinical fluence matrices, each of which exceeded the maximum allowable field width w . The fluence matrices were generated with a commercial inverse treatment planning system (CORVUS v5.0). The percent decrease in MUs as a result of optimal field splitting over the split generated by the commercial system were computed. The average decrease in MUs is 19% for the 32 fluence matrices. The maximum decrease in MUs is 45%. All the subfields overlap to various degrees, creating a natural feathering area which is clinically desirable.
[0042J This invention has been described herein in considerable detail to provide those skilled in the art with information relevant to apply the novel principles and to construct and use such specialized components as are required. However, it is to be understood that the invention can be carried out by different equipment, materials and devices, and that various modifications, both as to the equipment and operating procedures, can be accomplished wilhυul departing from the scope of the invention itself.