Are You a Good Beam or a Bad Beam Allen Holder Trinity University Mathematics University of Texas Health Science Center at San Antonio

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1 Are You a Good Beam or a Bad Beam University of Texas Health Science Center at San Antonio Joint Work with Matthias Ehrgott and Josh Reese, with help from Bill Salter, Ryan Acosta, and Daniel Nevin

2 Intensity Modulated Radiotherapy (IMRT) The design process selects couch and gantry positions so that the target is treated but surrounding tissues are spared.

3 Model Parameters Optimization models depend on Where dose is calculated, Where the isocenter is placed, What couch angle(s) are allowed, and What gantry angle(s) are used. bold decisions are mandatory, others can be addressed within the optimization model. The examples and images are from the academic treatment system RAD, lagrange.math.trinity.edu/rad.

4 Calculating Dose A naive grid distribution: Dose is calculated on a uniform grid (1 cm spacing). A judicious initial placement: We define a swath to place dose points and then only add points in normal tissue to avoid hot spots.

5 Model Parameters Dose Placement This is the most crucial decision required to build a model. Treatment and technique comparisons are directly related to this decision. Isocenter Placement There is typically a single isocenter placed near the center of mass of the target. There is no reason to believe that this is an optimal placement. Couch Angle(s) Couch Angles should be decided by the optimization model, but is currently not done. We expect the user to define couch angles. Gantry Angles Optimal selection of gantry angles is single biggest problem in the field. Selection is currently required by commercial systems.

6 Notation The following vectors are the minimum needed to form a prescription. T UB - vector of upper bounds on the tumor (target volume). T LB - vector of lower bounds on the tumor. CUB - vector of upper bounds for the critical structures. GUB - vector of upper bounds for the good tissue. The dose to tissue is linear in time, and we let x Ax be the transformation that maps exposure times (x) into deposited dose (Ax). The rows of A are separated as indicated. A = A T A C A G tumor pixels critical structures good tissue.

7 Linear Models The simplest linear models are feasibility models. These models attempt to satisfy A T x T G, A C x CUB, A N x NUB, and x 0. Consistency is not guaranteed because physicians are often overly demanding, and many authors have complained that infeasibility is a shortcoming. In fact, the argument that feasibility alone correctly addresses treatment design is that the region defined by these constraints is relatively small, and hence, optimizing over this region does not provide a significant benefit. Computational Solution of the Inverse Problem in Radiation-Therapy Treatment Planning, Y. Censor, M. Altschuler and W. Powlis, Applied Mathematics and Computation, 1988 (25) Semi-Automatic Radiotherapy Treatment Planning with a Mathematical Model to Satisfy Treatment Goals, W. Powlis, M. Altschuler, Y. Censor and E. Buhle, International Journal of Radiation Oncology, Biology, Physics, 1989 (16)

8 Linear Models An elastic model that removes the threat of infeasibility. min{ω l T α + u T C β + ut N γ : T LB Lα A T x T UB, A C x CUB + U C β, A N x NUB + U N γ, CUB U C β, 0 U N γ, 0 x} (1) This model has a nice theoretical results Theorem The linear model and its dual are strictly feasible, meaning that each of the constraints can simultaneously hold without equality. Theorem Allowing (x (ω), α (ω), β (ω), γ (ω)) to be an optimal solution for a particular ω, we have that l T α (ω) = O(1/ω). Designing Radiotherapy Plans with Elastic Constraints and Interior Point Methods, A. Holder, Health Care and Management Science, 2003 (6) 5-16.

9 Candidate Angles The candidates collection of angles is denoted by A and P(A) is the power set of A. Judgment Function A judgment function is a function f : P(A) IR + with the property that A A implies f(a ) f(a ). Judgment Evaluation For any A P(A), f(a ) = min{z(x) : x X(A )}, where z maps a fluency pattern x X(A ) into IR +. Theorem For all 0 N < A we have min{f(a ) : A P(A), A = N + 1} min{f(a ) : A P(A), A = N}. Beam Selection Problem For a fixed judgment function, the N-beam selection problem is min{f(a ) f(a) : A P(A), A = N} = min{f(a ) : A P(A), A = N} f(a).

10 Beam Selectors Beam Selector The function g : W V is a beam selector if 1. W and V are subsets of P(A) and 2. g(w ) W for all W W. Informed A beam selector is informed if the map requires the evaluation of the judgment function. Weakly Informed A beam selector is weakly informed if it uses the data that describes the judgment function but fall short of evaluating it. Stochastic Selectors A beam selector is stochastic if it depends on a random process.

11 Limited Fluency Beam Selectors Notation We let M be the vector with components M a and define { { A M := A : A P(A), X(A ) x : i x (a,i) M a, a A } }. Limited Fluency Beam Selectors An LF-N beam selector is an N-beam selector g lf : {A M } P(A M ). Limited Fluency Surrogate Instead of solving the full beam selection problem, we solve min { f ( g lf (A M ) ) : g lf F M lf (N)} f(a), where F M lf (N) is the collection of LF-N beam selectors.

12 Limited Fluency Beam Selectors Theorem There exists ˆM 0 such that the LF-N beam selection problem and the beam selection problem have the same optimal value. Also, if M j is a nondecreasing sequence in IR A such that lim j M j > ˆM, then f(ĝ lf (A M j )) f(a ), where A is an optimal solution to the beam selection problem and ĝ lf is any LF-N-beam selector that satisfies ĝ lf (A ˆM ) argmin{f(a ) : A P(A ˆM ), A = N}. Theorem For any M, the search space of the limited fluency problem is no greater than the search space of the beam selection problem. The two problems are equivalent for sufficiently large M if and only if for every a A, there is an A P(A) such that a A, A = N, and X(A ).

13 Set Cover Beam Selectors Notation An angle a covers the dose-point k if d i (k,a,i) ε, and for each k T, let A ε k = {a A : a covers dose-point k}. It is important to notice that A ε k = A for all k T if and only if 0 ε ε := min{ d i (k,a,i) : k T, a A}. We define W ε sc = {Aε k : k T }. Set Cover Beam Selector An SC-N-beam selector is an N-beam selector having the form ) g sc : W ε sc (P(A ε k )\. Set Cover Surrogate Allowing F ε sc (N) to be the collection of SC-N-beam selectors, the set covering approach is min f W W ε sc g sc (W ) k T : gsc F ε sc (N) f(a).

14 Set Cover Beam Selectors Theorem If 0 ε ε, the set covering problem is equivalent to the beam selection problem. Implementation Define angle scores by c a = k C i (u C ) k q (k,a,i) CUB k, C, 0, C =, Then solve min { a c a y a : a q (k,a) y a 1, k T, a y a = N, y a {0, 1} }

15 Set Cover Beam Selectors Theorem Assume that 0 ε ε, and order the angles so that c a1 c a2... c a A. Let j and j be the smallest and largest values, respectively, such that c aj =... = c an =... = c aj. Then, y is an optimal in the set cover problem if and only if y aj = 1 for 1 j < j, y aj = N j, and j j j y aj = 0 for j > j. Corollary If 0 ε ε, the set cover technique has ( j j +1 N j +1) optimal solutions.

16 Scoring Beam Selectors Definition An S-N-beam selector is an N-beam selector such that g s : {A} P(A). Scoring Surrogate Allowing F s (N) to be the collection of S-N-beam selectors, we immediately have by definition that min{f(a ) : A A, A = N} f(a) = min{f(g s (A)) : g s F s (N)} f(a). So, the scoring method is the same as the beam selection problem. Theorem If 0 ε ε, then F ε sc (N) = F s(n).

17 Scoring Beam Selectors Example Score each angle by where c a = 1 T k T i ( d(k,a,i) ˆx (a,i) ˆx (a,i) = min{min{cub k /d (k,a,i) : k C}, min{nub k /d (k,a,i) : k N}}. T G ) 2 Example Define the entropy of an angle by e a := i x (a,i) ln x (a,i) score of a is c a = 1 e a min{e a : a A}. max{e a : a A} and the

18 Vector Quantization Selectors Contiguous Partition We say that A is a contiguous subset of A if A is an ordered subset of the form {a j, a j+1,..., a j+r }. A contiguous partition of A is a collection of contiguous subsets of A that partition A, and we let W vq (N) be the collection of N element contiguous partitions of A. Vector Quantization Beam Selector A VQ-N-beam selector is a function of the form g vq : {W j : j = 1, 2,..., N} {{a j } : a j W j }, where {W j : j = 1, 2,..., N} W vq (N). Vector Quantization Surrogate The vector quantization approach to the beam selection problem is { ( N ) } min f g vq (W j ) : g vq F vq (N) f(a), j=1 where F vq (N) is the collection of VQ-N-beam selectors.

19 Vector Quantization Selectors Fact The search space for vector quantization technique is larger than the search space for the beam selection problem. Distortion The distortion of a quantizer is N j=1 a W j α(a) m(a, g vq(w j )), Theorem The VQ-N-beam selector g vq minimizes distortion for the contiguous partition {W j : j = 1, 2,..., N} if and only if g vq satisfies N j=1 a W j α(a) m(a, g vq(w j )) N j=1 min a W j α(a) m(a, a ) : a W j.

20 Vector Quantization Selectors Notation The elements of F vq (N) that satisfy the condition are g opt vq, and we let Fvq opt (N) be the collection of these functions. Discrete Interpretation In the discrete setting, the interpretation of the result is that the image set is the centers of mass. Theorem If 0 ε ε, then g vq opt F vq (N) g opt vq (Wopt vq ) F opt vq (N) F ε sc (N) = F s(n) F vq (N), where W opt vq is the domain of g opt vq.

21 Vector Quantization Selectors Implementation To remove the dependency on a particular solver, we consider generating a balanced probability distribution ( ( )) lexmin z(x), sort x (a,i) : a A. where sort is a function mapping IR A into IR A that reorders the components of the vector ( ) x i (a,i) a A i in a nonincreasing order.

22 Vector Quantization Selectors Algorithm Let z 0 = f(a). Initialize ˆx to be the zero vector of length A, t = 1, B t =, and U t = {j : j = 1, 2,..., A }. Do until B t = A : Let x U t solve z t = min{ x U t : f(a) = z 0, x Bt = ˆx Bt, x Ut 0}. Let Set α = (1/ ˆx )ˆx. B t = {j : (x U t ) j must be z t } B t+1 = B t B t U t+1 = U t \ B t ˆx Bt = z t t = t + 1

23 Vector Quantization Selectors Theorem With the notation of the algorithm, assume that f is a judgment function with the property that x U t solves z t = min{ x Ut : f(a) = z 0, x Bt = ˆx Bt, x Ut 0} = min{ x Ut : x X, x Bt = ˆx Bt, x Ut 0}. Then, the algorithm terminates with an α in the (t + 1)st iteration having the property that α(a) = 0 for some a A if and only if either D (T,Bt ) ˆx Bt l or [ D (T,Bt ) D (C N,Bt ) ] ˆx Bt ( ) u 1, u 2 Moreover, α(a) = 0 implies a B t+1. [ D (T,Bt ) D (C N,Bt ) ] ˆx Bt ( ) u 1. u 2

24 Deterministic Iterative Beam Selectors Basic Idea Define a neighborhood and select a best angle over this neighborhood. Notice this can be interpreted as a scoring technique or as a vector quantizer. Seeding Select angles via one of the earlier selectors and then adjust angles locally i.e. try to locally improve. Alternatively, use equally spaced angles. Novelty One of the suggested techniques uses a deleted neighborhood in an attempt to spread energy.

25 Stochastic Iterative Beam Selectors Stochastic-N-Beam Selector A stochastic-n-beam selector is an N-beam selector such that g st : {A} P(A), where g st (A) is a random set according to some distribution. Implementations Stochastic beam selectors are implemented as simulated annealing or genetic algorithms. These differ from the deterministic methods in that they operate over the entire search space. Note on SA Simulated is typically described but implemented with a temperature of 0 -i.e pure random search is used.

26 Stochastic Iterative Beam Selectors Example GA (A 1, A 2 ) {a 1 i + γ(a2 i a1 i ) : i = 1,..., N} where γ [ 0.25, 1.25] is a random number A 1 (A 1 \ {a i }) {a i + δ} where i {1,..., N} and δ [ π/12, π/12] are random A 1 (A 1 \ {a i }) {a} where i {1,..., N} and a A are randomly chosen A 1 (A 1 \ {a i }) {a i + π} where i {1,..., N} is randomly chosen A 1 {a i + (2π/N)k : k = 0,..., N 1} where i {1,..., N} is randomly chosen.

27 Experimental Design Experimental Design We had 5, 2D examples: Candidate Angles Candidate set A is {2πi/72 : i = 0, 1,..., 71}

28 Experimental Design A taxonomy of different selectors generated 5, 9 and 14 beam treatments. SC VQ Scoring GA SA LS Angle Sets Examples Different Types Angle Assignments N/A N/A N/A Total

29 Angle Assignments The angle values on the fictitious example: BalancedAvg, BalancedMax, DualAvg, DualMax, EntropyAvg, InteriorAvg, InteriorMax, PrimalAvg, PrimalMax, Scoring

30 Numerical Results A Fictitious but Important Example Rounded variances for the deterministic selectors on the fictitious example. Ideal value is 0. N BA BM PA PM DA DM IA IM S Score avg VQ avg SC avg

31 Stochastic Iterative Beam Selectors A Fictitious but Important Example Rounded variances for the iterative techniques on the fictitious example (again, ideal value is 0). N LS1 LS2 SA1 SA2 SA3 GA1 GA avg

32 Example Treatments Below are treatments from a VQ-14 beam selector with the BalancedAvg angle values and from a SC-5 beam selector (with cover tolerance of 0.5 max dose) with the same angle values

33 Average Run Times S VQ SC LS SA GA BA.43 BA.74 BA.55 LS SA GA BM.43 BM.74 BM.55 LS SA GA PA.46 PA.74 PA.58 SA PM.46 PM.74 PM.58 DA.46 DA.73 DA.58 DM.47 DM.74 DM.58 IA.47 IA.75 IA.58 IM.47 IM.75 IM.61 SC1.36 SC1.63 SC1.49 SC2.37 SC2.63 SC2.49 S.47 S.76 S.58 ENT.36 ENT.58 ENT.48

34 Representative Examples

35 General Trends The vector quantizer with the balanced-avg angle assignments consistently produced quality treatments. The balanced and interior (both average and maximum) outperformed the primal and dual angle assignments, with the dual method often leading to poor quality angle collections. Scoring methods were erratic. The local search algorithms worked well in general but were not as fast as vector quantization. The second and third simulated annealing selectors consistently produced quality treatments but were the most computationally heavy, often requiring hours to select beams. The genetic algorithm approaches did as well as simulated annealing in about half of the time.

36 Thank you for your time. Please ask questions.