Single-arc dose painting for precision radiation therapy

Abstract

Provided herein are methods and systems for designing a radiation treatment for a subject using single arc dose painting. The methods and systems comprise an algorithm or a computer-readable product having the same, to plan the radiation treatment. The algorithm converts pairs of multiple leaf collimation (MLC) leaves to sets of leaf aperture sequences that form a shortest path single arc thereof where the pairs of MLC leaves each aligned to an intensity profile of densely-spaced radiation beams, and connects each single arc of leaf apertures to form a final treatment single arc. Also provided is a method for irradiating a tumor in a subject using single arc dose painting.

Description

FEDERAL FUNDING LEGEND

This work was supported by National Science Foundation Grant No. CCF-0515203 and by National Institutes of Health Grant No. CA117997. The U.S. Government has certain rights in this invention.

Thus, the subgraph of ^G between any two consecutive vertex layers is a complete bipartite graph. For each edge (u, v) is an element of E(^G), a weight ^ω(u, v)=ω′(π′(u, v))−ω′ (u) is assigned to it, i.e., the weight of a shortest u-to-v path in G′ minus the vertex weight of u. For convenience, ^ω(u, v)^Δ=+∞ if there is no path from u to v in G′.

^G is an edge-weighted DAG and has O(Δ) vertices and O(Δ²) edges (by the definition of L′_i in Example 1). Define the weight ^ω(p) of a path p in ^G as ^ω(p)=Σ_{e is an element of p ^}ω(e). A shortest s-to-t path in G′ is closely related to a shortest s-to-t path in ^G. In fact, it is not difficult to show that if p′=s→u₁→u₂→ . . . →u_n→t is a shortest s-to-t path in G′, then ^p=s→u_lstart→u_rstart→U_lend+1→U_rend+1→t is a shortest s-to-t path in ^G, with ^ω(^p)=ω′(p″) This is due to the optimal-substructure property of shortest paths, i.e., any subpath of a shortest path is also a shortest path. Moreover, if s→v₂→v₃→v₄→t is a shortest s-to-t path in ^G, then s v₁ v₂ v₃ v₄

is a shortest s-to-t path in G′.

Before explaining why the vertex layers L′_lstart, L′_rstart, L′_lend+1, and L′_rend+1, from G′ for the construction of ^G were chosen, Equation (1) must be reviewed. Observe that if i is fixed and l_start, r_{start, lend}, and r_end are temporarily viewed as parameters, then Equation (1) implies that the i-th vertex layer in any G (1≤k≤N) is in exactly one of five possible states, denoted by Λ_i^(I), Λ_i^(II), Λ_i^(III), Λ_i^(IV), and Λ_i^(V), e.g., Λ_i^(I) if 1≤i≤l_start). For a fixed k (1≤k≤N), in the DAG G′_k, the i-th vertex layer L is of type I if L=Λ_i^(I). The type II, III, IV, and V vertex layers are defined similarly.

Clearly, in the DAG G′, all vertex layers between s and L′_lstart, L′_lstart and L_rstart, L″_rstart and L′_lend+1, L′_lend+1, and L′_rend+1, and L′_rend+1 and t are of type I, II, III, IV, and V, respectively (FIG. 2A). This implies an interesting property: For any edge (u, v) of ^G, all vertex layers between u and v in G′ are of the same type. As will be shown, this property plays a key role in speeding up the computation of the weights of all edges in the transformed DAGs ^G₁, ^G₂, . . . ^G_N. It should be pointed out that for one DAG G′, although ^G is of a much smaller size than G′, the weights of the edges of ^G are costly to compute. Thus transforming G′ to ^G will not yield a faster algorithm for solving a single CCPP problem instance.

Computing the Edge Weights for all Transformed DAGs

It is demonstrated herein how to compute the weights of all edges in the transformed DAGs ^G₁, ^G₂, . . . , ^G_N. Recall that the weight of an edge in ^G_k corresponds to a one-pair shortest path problem instance on G′_k. Since every ^G_k has O(Δ²) edges, there are in total O(n⁴Δ²) edges in ^G₁, ^G₂, . . . , ^G_N, corresponding to O(n⁴Δ²) one-pair shortest path problem instances.

It is stated that the O(n⁴Δ²) edges actually correspond to only O(n²Δ²) distinct one-pair shortest path problem instances. For any k that is an element of {1, 2, . . . , N} and for any edge (u, v) that is an element of ^G_k, denote by SP(u, v, k) the corresponding one-pair shortest path instance, i.e., finding a shortest u-to-v path in G′_k. Considering the indices of the vertex layers in which u and v lie in ^G_k respectively, there are five cases. Only the most typical case, in which u (resp., v) lies in the 2nd (resp., 3rd) layer of ^G_k is discussed. For this case, let i₁=r^(k)_start and i₂=l^(k)_end+1; then u is an element of Λ^II_i1⊂V(G′_k), v is an element of Λ^IV_i2⊂V(G′_k), and all vertex layers between u and v in G′_k are of type III (FIG. 2A). Let G_i1,i2 be a layered DAG whose vertices consist of the vertex layers Λ^II_i1, Λ^III_i1+1, Λ^III_i1+2, . . . , Λ^III_i2−1, and Λ^IV_i2, in this order, and whose edges are defined based on the same domination relation D(•,•).

Clearly, SP(u, v, k) is equivalent to finding a shortest u-to-v path in G_i1,i2. Since i_i (resp., i2) ranges from 1 to n, G_i1,i2 has O(n²) possible choices. Note that u (resp., v) belongs to the layer Λ^II_i1 (resp., Λ^IV_i2), which contains O(Δ) vertices for any i_i (resp., i₂). It follows that SP(u, v, k) has O(n⁴Δ²) possible choices for the case where u (resp., v) lies in the 2nd (resp., 3rd) layer of ^G_k. Other cases can be analyzed similarly. Since there are five cases, SP(u, v, k) has O(n²Δ²)×O(1)=O(n²Δ²) possible choices in total.

Since there are only O(n²Δ²) distinct one-pair shortest path problem instances, it is preferred to implicitly compute and store the weight of each edge of ^G₁, ^G₂, . . . , ^G_N. More specifically, distinct one-pair shortest path instances are solved and the corresponding shortest paths along with their lengths are stored, so that given an edge (u, v) of any ^G_k, its weight can be reported in O(1) time.

It is demonstrated how to solve the O(n²Δ²) distinct one-pair shortest path problem instances in a batch fashion. First a set of one-pair shortest path instances are combined into one single-source shortest path problem instance. Second, a set of single-source shortest path instances are solved in one shot by “merging” the underlying DAGs into one DAG of a comparable size. This approach is illustrated by showing how to solve the one-pair shortest path instances in G_i1,i2 (as defined in the previous paragraph) for all i₁, i₂. Each instance is specified by a source vertex u is an element of Λ^II_i1 and a destination vertex v is an element of Λ^IV_i2. Recall that G_i1,i2 contains the vertex layers Λ^II_i1, Λ^III_i1+1, Λ^III_i1+, . . . , Λ^III_i2−1, and Λ^IV_i2, For a fixed i₁, the layered DAGs G_i1,i2, for all i₂=i₁+1, . . . , n, can be merged into one DAG, simply denoted by G_i1 (FIGS. 5C-5D). For any vertex u is an element of Λ^II_i1, its single-source shortest paths can be computed in G_i1, in O(nHΔ) time as in the CCPP algorithm. Since |Λ^II_i1|=O(Δ) and i₁ ranges from 1 to n, the one-pair shortest path instances in G_i1,i2 can be solved for all i₁, i₂, in O(nHΔ)×O(n)×O(Δ)=O(n²Δ²) time.

Shortest Path Computation on the Transformed DAGs

The algorithm for computing a shortest path in the DAG ^G as defined as defined above is presented. The key idea is to show that ^G satisfies the Monge property [1, 2, 4], and thus a shortest path in ^G can be computed by examining only a small portion of its edges.

Consider the vertex layers ^L₂ and ^L₃ of ^G. Clearly, ^L₂=L′_rstart={[α, β]^(rstart)0=α<β<=H−c and |β−α−f (rstart)|≤Δ}={[0, β]^(rstart)|0<β<=H−c and |β−f(rstart)|≤Δ}). Thus ^L₂ can be written as ^L₂={u₁, u₂, . . . , u_|^L2|}, where u_k=[0, β_k]^(rstart)(1≤k≤|^L₂|) and 0<β₁<β2< . . . <β_|^L2|≤H−c. Similarly, write ^L₃ as ^L₃={v₁, v₂, . . . , v_|L3|}, where v_k=[_k, H]^(lend+1)(1≤k≤|^L₃|) and c≤α₁<α₂< . . . <α_|^L3|<H. The following lemma shows the properties of the edges from the vertices on ^L₂ to ^L₃ in ^G.

Lemma: (A) Let u_i (resp., v_j) be a vertex on ^L₂ (resp., ^L₃) of ^G, with 1≤i≤|^L₂| (resp., 1≤j≤|^L₃|). If ^ω(u_i, v_j)=1, then for any j′<j (resp., i′>i), ^ω(u_i, v_j)=∞(resp., ^ω (u_i′, v_j)=∞). (B) Let u_i, u_j′ (resp., v_j, v_j′) be two vertices on ^L₂ (resp., ^L₃) of ^G, with 1≤i<i′≤|^L₂| (resp., 1≤j<j′≤|^L₃|). If both edges e₁=(u_i, v_j′) and e₂=(u_i′,v_j) have finite weights, then edges e₃=(u_i, v_j) and e₄=(u_i′, v_j′) also both have finite weights. Moreover, ^ω(e₃)+^ω(e₄) ^ω(e₁)+^ω(e₂).

The lemma implies that the matrix A of size |^L₂|×|^L₃|, with =^ω(u_i, v_j), is a Monge matrix. Note that for any i and j, ^ω(u_i, v_j) can be reported in O(1) time as described above. Thus in ^G, once the shortest paths from s to all vertices in ^L₂ are computed, using the Monge matrix multiplication algorithms [1, 2, 4], the shortest paths from s to all vertices in ^L₃ can be computed in O(Δ) time. Observe that outside the subgraph of ^G between the vertex layers ^L₂ and ^L₃ (FIG. 2B), there are only O(Δ) edges of ^G. Hence a shortest s-to-t path in ^G can be computed in O(Δ) time. Thus, the following theorem is:

Given a non-negative function ƒ defined on {1, 2, . . . , n} and positive integers c, H, and Δ(Δ≤H), the complete set of the CCPP problem instances is computable in O(n²H²+n4) time.

Claims

1. A method for designing a radiation treatment for a subject using single arc dose painting, comprising:

providing an unconstrained optimization map which supplies intensity profiles of densely-spaced radiation beams;

aligning each intensity profile to a pair of multiple leaf collimation (MLC) leaves;

applying a shortest path algorithm to convert each pair of MLC leaves to a set of leaf aperture sequences, said set of leaf aperture sequences forming a shortest path single arc thereof; and

connecting each single arc of leaf apertures to form a final treatment single arc path, thereby designing the single arc dose painting radiation treatment; and

delivering a treatment dose of radiation to the subject during a single rotation along one or more of the final treatment single arc path.

2. The method of claim 1, further comprising:

delivering wherein the treatment dose is a continuous dose of radiation delivered to the subject through each aperture during a the single rotation along one or more final treatment single arc paths.

3. The method of claim 1, further comprising:

adjusting a shape of the aperture as a radiation dose delivery angle changes along the final treatment arc.

4. The method of claim 1, wherein the paths of more than one single arc are non-coplanar.

5. The method of claim 1, wherein the apertures sweep back and forth along the single arc path during delivery of the radiation dose.

6. The method of claim 1, wherein sequencing leaf apertures comprises one or more of a line segment approximation on component intensity profiles leaf position, weight optimization of apertures and optimization of leaf position and aperture weight.

7. The method of claim 1, wherein the leaf aperture sequences in each set have one or both of a different starting or ending leaf aperture.

8. The method of claim 7, wherein a starting and ending position of a leaf aperture trajectory are fixed.

9. The method of claim 1, wherein multiple leaf collimation is dynamic.

10. The method of claim 1, further comprising

irradiating a tumor in a subject with the continuous dose of radiation through sets of multiple leaf collimation (MLC) aperture sequences during a single rotation along one or more of the treatment single arc paths.

11. The method of claim 10, further comprising:

adjusting a shape of the aperture as a radiation dose delivery angle changes along the treatment single arc path.

12. The method of claim 10, further comprising:

repeating the irradiation step during another rotation along the treatment single arc path(s).

13. The method of claim 10, wherein each set of MLC aperture sequences form a shortest path single arc thereof, said sets connected to form a shortest path treatment single arc.

14. A system for delivering radiation treatment using single arc dose painting, comprising in a radiation delivery device:

a radiation source for generating a radiation beam;

a multiple leaf collimator having a plurality of leafs for shaping the a radiation beam;

structure for generating tangibly storing an algorithm enabling processor-executable instructions to generate an unconstrained optimization map of intensity profiles of densely-spaced radiation beams;

structure for aligning tangibly storing an algorithm enabling processor-executable instructions to align each intensity profile to a pair of multiple leaf collimation (MLC) leaves; and

structure for applying tangibly storing a shortest path algorithm, said shortest path algorithm converting enabling processor-executable instructions to convert each pair of MLC leaves to a set of leaf aperture sequences forming a shortest path single arc thereof, said shortest path algorithm and to further connecting connect each single arc of leaf apertures to form a final treatment single arc effective for single arc dose painting; and

a source of a dose of continuous radiation beams, varying radiation beams or both deliverable to the subject through each aperture during a single rotation along the final treatment single arc path.

15. The system of claim 14, the shortest path algorithm further adjusting enabling processor-executable instructions to adjust a shape of the leaf aperture as a radiation dose delivery angle changes along the final treatment single arc.

16. The system of claim 14, wherein the shortest path algorithm sequences enables processor-executable instructions to sequence leaf apertures via one or more of a line segment approximation on component intensity profiles leaf position, weight optimization of apertures or optimization of leaf position and aperture weight.

17. The system of claim 14, wherein the multiple leaf collimator is dynamic.

18. A computer-readable medium tangibly storing an algorithm to determine a final single arc path for a single arc dose painting radiation treatment, said algorithm enabling processor-executable instructions to:

convert pairs of multiple leaf collimation (MLC) leaves to sets of leaf aperture sequences that form a shortest path single arc thereof, said pairs of MLC leaves each aligned to an intensity profile of densely-spaced radiation beams; and

connect each single arc of leaf apertures to form a final treatment single arc.

19. The computer-readable medium of claim 18, said algorithm further enabling instructions to:

adjust a shape of the leaf aperture as a radiation dose delivery angle changes along the final treatment single arc.

20. The computer-readable medium of claim 18, wherein the algorithm sequences leaf apertures via one or more of a line segment approximation on component intensity profiles leaf position, weight optimization of apertures or optimization of leaf position and aperture weight.

21. A method for designing a radiation treatment using a treatment arc comprising:

accessing an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to a plurality of radiation beams;

aligning specific ones of the intensity profiles to corresponding pairs of multiple leaf collimator (MLC) leaves;

determining via a shortest path algorithm, for at least one of the pairs of MLC leaves, a plurality of leaf aperture sequences corresponding to angular intervals as each comprise a part of the treatment arc and combining the plurality of leaf aperture sequences over the angular intervals to form a treatment arc;

developing a final treatment arc using the treatment arc; and

delivering a dose of radiation to a subject based upon the leaf aperture sequences while traversing the final treatment arc; wherein the dose of radiation comprises a continuous dose of radiation, a varying dose of radiation or both while traversing the final treatment arc.

22. The method of claim 21 wherein the plurality of radiation beams comprises a plurality of densely-spaced radiation beams.

23. The method of claim 22 wherein the plurality of densely-spaced radiation beams are spaced no less than about ten degrees from one another.

24. The method of claim 21, wherein delivering a continuous dose of radiation to the subject comprises, at least in part, sweeping MLC apertures back and forth along the final treatment arc.

25. The method of claim 21 wherein determining a plurality of leaf aperture sequences comprises, at least in part, adjusting MLC aperture shapes as a radiation dose delivery angle changes along the treatment arc.

26. The method of claim 21 wherein determining the plurality of leaf aperture sequences comprises one or more of a line segment approximation on component intensity profiles leaf position, weight optimization of apertures, and optimization of leaf position and aperture weight.

27. The method of claim 21 wherein the plurality of leaf aperture sequences provide for dynamic multiple leaf collimation over the treatment arc.

28. The method of claim 21 wherein the optimization map comprises, at least in part, fluence distribution.

29. A method for designing a radiation treatment comprising:

accessing an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to a plurality of radiation beams;

using the intensity profiles to develop a plurality of multiple leaf collimator aperture settings for use at various angles along a treatment arc;

determining via a shortest path algorithm a plurality of leaf aperture sequences from the plurality of multiple leaf collimator aperture settings;

using the plurality of leaf aperture sequences to form a radiation treatment plan to deliver a dose of radiation to a subject while dynamically adjusting a corresponding leaf collimator aperture based upon the plurality of leaf aperture sequences while traversing the treatment arc; and

delivering a dose of radiation to the subject while traversing the treatment arc using the radiation treatment plan; wherein the dose of radiation comprises a continuous dose of radiation, a varying dose of radiation or both while traversing the treatment arc via the radiation treatment plan.

30. The method of claim 29, wherein delivering the dose of radiation to the subject while dynamically adjusting a corresponding leaf collimator aperture comprises, at least in part, sweeping MLC apertures back and forth along the treatment arc.

31. The method of claim 29 wherein determining a plurality of leaf aperture sequences comprises, at least in part, adjusting MLC aperture shapes as a radiation dose delivery angle changes along the treatment arc.

32. A method for designing a radiation treatment using a treatment arc comprising:

accessing an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to a plurality of radiation beams;

determining via a shortest path algorithm a plurality of multiple leaf collimator aperture sequences corresponding to angular intervals as each comprise a part of the treatment arc and forming the treatment arc based at least in part on the plurality of leaf aperture sequences over the angular intervals to form a treatment arc that provides for sweeping a multiple leaf collimator aperture back and forth along a treatment arc; and

delivering a dose of radiation to a subject while sweeping the multiple leaf collimator aperture back and forth along the treatment arc; wherein the dose of radiation comprises a continuous dose of radiation, a varying dose of radiation or both along the treatment arc during delivery.

33. The method of claim 32 wherein the plurality of radiation beams comprises a plurality of densely-spaced radiation beams.

34. The method of claim 33 wherein the plurality of densely-spaced radiation beams are spaced no less than about ten degrees from one another.

35. A planning system for developing a radiation treatment to be administered via a radiation source for generating a plurality of radiation beams and a multiple leaf collimator having a plurality of leafs for shaping the radiation beams, the planning system comprising:

structure tangibly storing an algorithm enabling processor-executed instructions to access an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to the plurality of radiation beams;

structure tangibly storing an algorithm enabling processor-executed instructions to align specific ones of the intensity profiles to corresponding pairs of multiple leaf collimator (MLC) leaves;

structure tangibly storing a shortest path algorithm enabling processor-executed instructions to:

determine, for at least one of the pairs of MLC leaves, a plurality of leaf aperture sequences corresponding to angular intervals as each comprise a part of the treatment arc;

combine the plurality of leaf aperture sequences over the angular intervals to form a treatment arc; and

develop a final treatment arc using the treatment arc;

a source of a continuous dose of radiation deliverable to a subject based upon the leaf aperture sequences while traversing the single treatment arc; and

a source of a varying dose of radiation deliverable to a subject based upon the leaf aperture sequences while traversing the single treatment arc.

36. The planning system of claim 35 wherein the plurality of radiation beams comprises a plurality of densely-spaced radiation beams.

37. The planning system of claim 36 wherein the plurality of densely-spaced radiation beams are spaced no less than about ten degrees from one another.

38. The planning system of claim 35, wherein the source of the continuous dose of radiation is configured to deliver the continuous dose of radiation by, at least in part, sweeping MLC apertures back and forth along the final treatment arc.

39. The planning system of claim 35 wherein the shortest path algorithm enables processor-executed instructions to, at least in part, adjust MLC aperture shapes as a radiation dose delivery angle changes along the treatment arc.

40. The planning system of claim 35 wherein the shortest path algorithm enables processor-executed instructions to use one or more of a line segment approximation on component intensity profiles leaf position, weight optimization of apertures, and optimization of leaf position and aperture weight.

41. The planning system of claim 35 wherein the plurality of leaf aperture sequences provide for dynamic multiple leaf collimation over the treatment arc.

42. The planning system of claim 35 wherein the optimization map comprises, at least in part, fluence distribution.

43. A planning system for developing a radiation treatment to be administered via a radiation source for generating a plurality of radiation beams and a multiple leaf collimator having a plurality of leafs for shaping the radiation beams, the planning system comprising:

structure tangibly storing an algorithm enabling processor-executed instructions to access an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to a plurality of radiation beams;

structure tangibly storing an algorithm enabling processor-executed instructions to use the intensity profiles to develop a plurality of multiple leaf collimator aperture settings for use at various angles along a treatment arc;

structure tangibly storing a shortest path algorithm enabling processor-executed instructions to determine a plurality of leaf aperture sequences using the plurality of multiple leaf collimator aperture settings and to use the plurality of leaf aperture sequences to form a radiation treatment plan to deliver a dose of radiation to a subject while dynamically adjusting a corresponding leaf collimator aperture based upon the plurality of leaf aperture sequences while traversing the single treatment arc; and

a source of a continuous dose of radiation, a varying dose of radiation or both deliverable to the subject while traversing the single treatment arc.

44. The planning system of claim 43 wherein the shortest path algorithm enables processor-executed instructions to deliver the dose of radiation to the subject by, at least in part, sweeping MLC apertures back and forth along the single treatment arc.

45. The planning system of claim 43 wherein the algorithm for determining a plurality of leaf aperture sequences determines the plurality of leaf aperture sequences by, at least in part, adjusting MLC aperture shapes as a radiation dose delivery angle changes along the treatment arc.

46. A planning system for developing a radiation treatment to be administered via a radiation source for generating a plurality of radiation beams and a multiple leaf collimator having a plurality of leafs for shaping the radiation beams, the planning system comprising:

structure tangibly storing an algorithm enabling processor-executed instructions to access an optimization map that supplies intensity profiles wherein at least some of the intensity profiles differ from one another with respect to a plurality of radiation beams;

structure tangibly storing a shortest path algorithm enabling processor-executed instructions to:

determine a plurality of multiple leaf collimator aperture sequences corresponding to angular intervals as each comprise a part of the treatment arc; and

form the treatment arc based at least in part on the plurality of leaf aperture sequences over the angular intervals to form a treatment arc that provides for sweeping a multiple leaf collimator aperture back and forth along the treatment arc; and

a source of a continuous dose of radiation, a varying dose of radiation or both deliverable to a subject while sweeping the multiple leaf collimator aperture back and forth along the treatment arc.

47. The planning system of claim 46 wherein the plurality of radiation beams comprises a plurality of densely-spaced radiation beams.

48. The planning system of claim 47 wherein the plurality of densely-spaced radiation beams are spaced no less than about ten degrees from one another.