PS
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`©
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`INTERNATIONAL JOURNAL OF
`
`
`Wee
`ONCOLOGY
`BIOLOGY*PHYSICS
`
`Co
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`7dae
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`ax
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`VOLUME23, NUMBER1, 1992
`
`ISSN 0360-3016
`
`
`
`The Official Journal of the
`AMERICAN SOCIETY FOR THERAPEUTIC RADIOLOGY AND ONCOLOGY
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`Sponsored bythe
`INTERNATIONALSOCIETY OF RADIATION ONCOLOGY
`CIRCULO DE RADIOTERAPEUTAS IBERO-LATINOAMERICANOS
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`

`

`RADIATION ONCOLOGY
`BIOLOGY:PHYSICS
`
`
`
`VOLUME 23, NUMBER1, 1992
`
`Editor’s Note
`P. Rubin
`
`TABLE OF CONTENTS
`
`@ CLINICAL ORIGINAL CONTRIBUTIONS
`
`Highly Anaplastic Astrocytoma: A Reviewof 357 Patients Treated between 1977 and 1989
`M. D. Prados, P. H. Gutin, T. L. Phillips, W. M. Wara, D. A. Larson, P. K. Sneed, R. L. Davis, D. K. Ahn,
`K. Lamborn andC. B. Wilson
`
`Non-Hodgkin’s Lymphomaof the Brain: Can High Dose, Large Volume Radiation Therapy Improve Survival?
`Report on a Prospective Trial by the Radiation Therapy Oncology Group (RTOG): RTOG 8315
`D. F. Nelson, K. L. Martz. H. Bonner, J. S. Nelson, J. Newall, H. D. Kerman, J. W. Thomson andK. J. Murray
`
`Radiosurgery and Brain Tolerance: An Analysis of Neurodiagnostic Imaging Changes after Gamma Knife Ra-
`diosurgery for Arteriovenous Malformations
`J. C. Flickinger, L. D. Lunsford, D. Kondziolka, A. H. Maitz, A. H. Epstein, S. R. Simons and A. Wu
`
`A Dose Response Analysis of Injury to Cranial Nerves and/or Nuclei Following Proton Beam Radiation Therapy
`M. M. Urie, B. Fullerton, H. Tatsuzaki, S. Birnbaum, H. D. Suit, K. Convery, S. Skates and M. Goitein
`
`The Influence of Dose and Time on Wound Complications Following Post-Radiation Neck Dissection
`J. M.G. Taylor, W. M. Mendenhall, J. T. Parsons and R. S. Lavey
`
`Prostate-Specific Antigen as a Prognostic Factor for Prostate Cancer Treated by External Beam Radiotherapy
`G. K. Zagars
`
`@ BIOLOGYORIGINAL CONTRIBUTIONS
`
`In Vitro Intrinsic Radiation Sensitivity of Glioblastoma Multiforme
`A. Taghian, H. Suit, F. Pardo, D. Gioioso, K. Tomkinson, W. duBois and L. Gerweck
`
`I
`
`3
`
`9
`
`19
`
`5
`27
`
`41
`
`a
`
`55
`
`-
`
`Repopulation Between Radiation Fractions in Human Melanoma Xenografts
`E. K. Rofstad
`
`(Contents continued on page viit)
`
`Index Medicus, MEDLINE, Excerpta Medica, Safety Sci. Abst
`SIS Database,
`Conte:
`tHirent
`Index
`Medicus
`Cery
`INDEXED IN Current Contents, BIOSIS Database,
`Abstr.. Electronics & Commun. Abstr., Computer & Info. Systems Abstr., Cambridge Sci. Abstr.,
`
`This material was copied
`at the NLM and may be
`Subject US Copyright Laws
`
`r., Energy Res. Abstr., Energy Data Base, Toxicology
`and CABS. PASCAL-CNRS Database
`ISSN 0360-3016
`(223)
`
`

`

`
`
`Radiosensitivity, Repair Capacity, and Stem Cell Fraction in Human Soft Tissue Tumors: An In Vitro Study
`Using Multicellular Spheroids and the Colony Assay
`M. Stuschke, V. Budach, W. Klaes and H. Sack
`
`@ PHYSICS ORIGINAL CONTRIBUTIONS
`
`Permanent Implants Using Au-198, Pd-103 and 1-125: Radiobiological Considerations Based on the Linear
`Quadratic Model
`C. C. Ling
`
`Random Search Algorithm (RONSC)for Optimization of Radiation The
`End Points and Constraints
`A. Niemierko
`
`rapy with Both Physical and Biological
`
`Optimization of 3D Radiation Therapy with Both Physical and Biological E nd Points and Constraints
`A. Niemierko, M. Urie and M. Goitein
`
`@ HYPERTHERMIA ORIGINAL CONTRIBUTIONS
`
`Thermochemotherapy with Cisplatin or Carboplatin in the BT, Rat Glioma In Vitro and In Vivo
`B.-C. Schem, O. Mella and O. Dahl
`
`Radiation and Heat Sensitivity of Human T-Lineage Acute Lymphoblastic Leukemia (ALL) and Acute Myelo-
`blastic Leukemia (AML) Clones Displaying Multiple Drug Resistance (MDR)
`F.M. Uckun, J. B. Mitchell, V. Obuz, M. Chandan-Langlie, W. S. Min, S. Haissig and C, W. Song
`
`@ PHASEI/II CLINICAL TRIALS
`
`Treatment of Non-Small Cell Lung Cancer with External Beam Radiotherapy and High Dose Rate Brachytherapy
`C. Aygun, S. Weiner, A. Scariato, D. SpearmanandL. Stark
`
`Sequential Comparison of Low Dose Rate and Hyperfractionated High Dose Rate Endobronchial Radiation for
`Malignant Airway Occlusion
`M. Mehta,D.Petereit, L. Chosy, M. Harmon, J. Fowler, S. Shahabi, B. Thomadsen and T. Kinsella
`
`@ BRIEF COMMUNICATIONS
`
`The Results of Radiotherapy for Isolated Elevation of Serum PSA Levels Following Radical Prostatectomy
`S. E. Schild, S. J. Buskirk, J. S. Robinow, K. M. Tomera, R. G. Ferrigni and L. M. Frick
`
`Fractionated Radiation Therapy in the Treatment of Stage III and IV Ce
`Preliminary Results in 20 Cases
`J. Ph. Maire, A. Floquet, V. Darrouzet, J. Guérin, J. P. Bébéar and M. C
`
`audry
`
`rebello-Pontine Angle Neurinomas:
`
`@ TECHNICAL INNOVATIONS AND NOTES
`
`The Use of Beam’s Eye View Volumetrics in the Selection of Non-Coplanar Radiation Port
`als
`G. T. Y. Chen, D. R. Spelbring, C. A. Pelizzari, J. M. Balter,
`L. C. Myrianthopoulos, S. Vijayakumar and
`H. Halpern
`
`Ultrasound Directed Extrahepatic Bile Duct Intraluminal Brachytherapy
`B. Minsky, J. Botet, H. Gerdes and C. Lightdale
`
`(Contents continued on page x)
`
`This material was copied
`at the NLM and may be
`Subject US Copyright Laws
`
`69
`
`81
`
`89
`
`99
`
`109
`
`Lis
`
`133
`
`141
`
`147
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`165
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`

`

`
`
`Magnetic Resonance Imaging During Intracavitary Gynecologic Brachytherapy
`S. L. Schoeppel, J. H. Ellis, M. L. LaVigne, R. A. Schea andJ. A. Roberts
`
`(Contents continued)
`
`Development of a Shielded *'Am Applicator for Continuous Low Dose Rate Irradiation of Rat Rectum
`R. Nath, S. Rockwell, C. R. King, P. Bongiorni, M. Kelley and D. Carter
`
`Reduction of the Dose to the Lens in Prophylactic Cranial Irradiation: A Comparisonof Three Different Treatment
`Techniques and Two Different Beam Qualities
`B. Pakisch, G.Stiicklschweiger, E. Poier, C. Urban, W. Kaulfersch, A. Langmann, C. Hauer and A. Hack!
`
`Interstitial Microwave Hyperthermia and Brachytherapy for Malignancies of the Vulva and Vagina I: Design
`and Testing of a Modified Intracavitary Obturator
`T. P. Ryan, J. H. Taylor and C. T. Coughlin
`
`@ SPECIAL FEATURES
`
`An Overview of the First International Consensus Workshopon Radiation Therapy in the Treatment of Metastatic
`and Locally Advanced Cancer
`G. E. Hanks, E. J. Maherand L. Coia
`
`The Crisis in Health Care Cost in the United States: Some Implications for Radiation Oncology
`
`G. E. Hanks
`
`A Report of RTOG 8206: A Phase III Study of Whether the Addition of Single Dose Hemibody Irradiation to
`Standard Fractionated Local Field Irradiation is More Effective Than Local Field Irradiation Alone in the
`Treatment of Symptomatic Osseous Metastases
`C. A. Poulter, D. Cosmatos, P. Rubin, R. Urtasun, J. S. Cooper, R. R. Kuske, N. Hornback, C. Coughlin,
`I. Weigensberg and M. Rotman
`
`169
`
`175
`
`183
`
`189
`
`201
`
`203
`
`207
`
`aie
`
`239
`
`Bone Metastasis Consensus Statement
`T. Bates, J. R. Yarnold, P. Blitzer, O. S. Nelson, P. Rubin and J. Maher
`A Review of Local Radiotherapy in the Treatment of Bone Metastases and Cord Compression
`
`T. Bates
`
`A Report of the Consensus Workshop Panel on the ‘Treatment of Brain Metastases
`L. R. Coia, N. Aaronson, R. Linggood, J. Loeffler and T. J. Priestman
`The Role of Radiation Therapy in the Treatment of Brain Metastases
`L. R. Coia
`
`Treatment Strategies in Advanced
`and Metastatic Cancer: Differences in Attitude between the USA, Canada
`and Europe
`E. J. Maher, L. Coia, G. Duncan andP. A. Lawton
`
`@ IN MEMORIAM
`
`In Memoriam: Gilbert Hungerford Fletcher
`L. J. Peters
`
`@ EDITORIALS
`
`CNS Lymphoma:Backto the Drawing Board
`T. E. Goffman and E. Glatstein
`
`(Contents continued on pagexii)
`
`
`
`
`This material was copied
`at the NLM and may be
`Subject US Copyright LawsNT—_ee
`
`

`

`oS.
`
`L
`
`|
`
`Endobronchial Brachytherapy: Wither Prescription Point
`M. P. Mehta
`
`Response to Dr. Speiser
`C. Aygun, S. Weiner, A. Scariato, D. SpearmanandL. Stark
`
`The Origins and Basis of the Linear-Quadratic Model
`D. J. BrennerandE. J. Hall
`
`Response to Brenner and Hall
`R. J. Yaes, Y. Maruyama, P. Patel and M. Urano
`
`Radiotherapy of Graves’ Ophthalmopathy
`D. S. Ellis
`
`In Response to “Radiotherapy of Graves’ Ophthalmopathy”
`I. A. Petersen, S. S. Donaldson and I. R. McDougall
`
`
`
`Iodine-125 Implants for Prostate Cancer
`R. E. Peschel and K. E. Wallner
`
`Response to Drs. Peschel and Wallner
`C. Koprowski
`
`Cf-252 Neutron Capture Therapy and Teletherapy
`Y. Maruyama, J. Wierzbicki, M. Ashtari, R. J. Yaes, J. L. Beach, J. Yanch, R. ZamenhofandC. B. Schroy
`
`@ MEETINGS
`
`251
`
`251
`
`252
`
`252
`
`;
`253
`
`253
`
`254
`
`254
`
`255
`
`257
`
`This material was copied
`at the NLM and may be
`Subject US Copyright Laws
`
`

`

`J. Radiation Oncology Biol. Phys., Vol. 23, pp. 89-98
`Int.
`Printed in the U.S.A. All nights reserved
`
`.00
`$5.00 +
`0360-3016/92
`Copyright ©
`1992 PergamonPress Ltd.
`
`@ Physics Original Contribution
`
`
`RANDOM SEARCH ALGORITHM (RONSC) FOR OPTIMIZATION OF RADIATION
`THERAPY WITH BOTH PHYSICAL AND BIOLOGICAL
`END POINTS AND CONSTRAINTS
`
`ANDRZEJ NIEMIERKO, PH.D.
`
`Division of Radiation Biophysics, Department of Radiation Oncology, Massachusetts General Hospital,
`Boston, MA 02114, and Harvard Medical School
`
`A new algorithm for the optimization of 3-dimensional radiotherapy plans is presented. The RONSCalgorithm
`(Random Optimization with Non-linear Score functions and Constraints) is based on the idea of random search in
`the space of feasible solutions. RONSC takes advantage of some specific properties of the dose distribution and
`derivable information such as dose-volume histograms andcalculated estimates of tumor control and normaltissue
`complication probabilities. The performance of the algorithm for clinical and test cases is discussed and compared
`with the performance of the simulated annealing algorithm, which is also based on the idea of random search.
`
`Optimization, Modeling, Treatment planning.
`
`INTRODUCTION
`
`Optimization ofradiation therapyis a very important and,
`at the same time, a very difficult problem. In planning
`radiation treatments, the principle goal of radiotherapy,
`namely, the complete depletion of tumor cells while pre-
`serving normal structures, is converted into a few smaller
`and often mutually contradictory subtasks. Because of
`the complex relationship between the dose distribution
`and the outcome ofradiotherapy, most investigators have
`concentrated on optimization of the dose distribution
`based on end points and/or constraints that are stated in
`terms of the physical dose (1-7, 10-11, 13-20, 23-24,
`26-28, 31, 33). Still, even optimization of the physical
`dose distribution is mathematically a very complex and
`difficult problem. A variety of optimization models and
`algorithms have been investigated. The algorithms used
`in planning radiotherapy can be groupedintothe follow-
`ing categories:
`
`Exhaustive search techniques
`These brute force techniques evaluate each possible
`combinationof(quantized) treatment parameters. A score
`is calculated for each analyzed set of treatment plan pa-
`rameters, possibly derived from sub-scores combined with
`subjectively chosen weight factors. Constraints can be
`
`taken into account bysetting the score to zero if the con-
`straints are not satisfied. Because of the truly vast number
`of possible combinations of parameters (c.g.. for four
`beams with four possible wedges, 20 quantized weights
`and only 36 orientations of each beam, the number of
`possible plans exceeds 10!>), this procedureis feasible only
`for very small problemsand has been used with simplistic
`dose calculation models for 2-dimensionalcases (11, 31).
`
`Linear and quadratic mathematical programming
`If the scoring function and constraints can be written
`as a linear (or quadratic) function of the plan parameters,
`then very elegant mathematical techniques can be applied
`which are guaranteed to find the optimum score (9, 14,
`18, 20, 26-27). The most popular techniquesare the Sim-
`plex algorithm (for linear problems) and the Wolfe or
`Beale algorithms(for quadratic objective functions with,
`nevertheless,
`linear constraints). These techniques can
`solve, with reasonable speed, relatively small problems
`(say, up to 200 constraints with up to 20 variables). A
`combinatorial linear programming algorithm has also
`been investigated and successfully applied for problems
`with a few hundred constraints (15). The combinatorial
`algorithm allows someofthe variables to have only integer
`(or, in general, discrete) values but otherwise suffers from
`
`
`
`Reprint requests to: Andrzej Niemierko, Ph.D., Department
`of Radiation Oncology, Massachusetts General Hospital, Boston,
`MA 02114.
`Acknowledgements—Vhe author would like to thank Michael
`Goitein and Marcia Urie for helpful discussions.
`
`Supported in part by Grants CA 50628 and CA 21239 from
`the National Cancer Institute, DHHS.
`Accepted for publication 7 October 1991.
`
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`at the NLM and may be
`Subject US Copyright Laws
`
`

`

`90
`
`I. J. Radiation Oncology @ Biology @ Physics
`
`Volume 23, Number |, 1992
`
`the same limitations as linear or quadratic programming,
`algorithms.
`
`Non-linear mathematical programming
`If the scoring function or constraints are notlinear or
`quadratic in the parameters of interest, then non-linear
`search techniques have to be used (5, 16, 20). Their lim-
`itations are that, with the exception of some unimodal
`functions, they are susceptible to getting trapped inalocal
`extremum ofthe score function, they are sensitive to the
`starting conditions, and their performance dramatically
`decreases as problems becomelarger. The non-linearal-
`gorithms with the best performance require calculations
`of the first or even higher derivatives of the objective
`function and constraints and, in general, belong to one
`of two families of algorithms: conjugate gradient algo-
`rithms and variable metric algorithms. Because ofthe large
`size and mathematical difficulties of practical clinical
`problems, noneofthe standard non-linear programming
`algorithms have so far been found clinically useful.
`
`Finding a feasible solution
`Oneclass ofsolutions that has been proposed involves
`stating the problem asa set of a dose constraints without
`an objective function. An iterative approach thensolves
`the possibly many thousands oflinear inequalities (con-
`straints) (4, 24, 28) and the solution is the first case en-
`countered that satisfies all the constraints. In this approach
`(whichis not, as such, an optimization algorithm because
`nothing is maximized or minimized) it
`is assumed that
`every feasible solution is clinically satisfactory andthat
`all feasible solutions are of more or less equal quality.
`This approachrequires constraints to be defined in a such
`way that the space offeasible solutions is relatively small
`orflat. This is possible when the planneror the clinician
`designing treatment plan has a knowledge about
`the
`physically obtainable optimal dose distribution, andset
`ups constraints (i.e., dose limits) which quite closely define
`this optimal distribution. If the constraints are too tight
`(which is not known a priori) there is no solution. Ifthe
`constraints are too loose, there is an infinite space ofso-
`lutions and the probability that the first solution foundis
`the best one is equal to probability that this solution is
`the worst one (ofall feasible solutions).
`
`Inversesolution
`A different approach, which addresses the so-called in-
`verse problem in radiation therapy(as opposed to forward
`approaches described above), has recently been investi-
`gated (1-3, 6, 10, 13, 17, 30, 33). The inverse approach
`posits an ideal dose distribution and attempts to determine
`beam weights and compensator shapes that
`lead to a
`physicalsolution thatis “‘as close as possible”’ to the ideal.
`The idea is similar to the problem ofreconstructing a
`tomographic image from projections at many angles. In
`principle, there are some one-pass solutions to this prob-
`lem but, in practice, the algorithms used to solve inverse
`
`problems tendtobe iterative in nature and, therefore, not
`self-evidently faster than otheriterative search techniques.
`Besides the well known problems with the mathematics
`of deconvolution (for example, the convolution kernel 1s
`assumed to be spatially invariant—whichis not the case
`in radiation therapy for inhomogeneous media and with
`scatter effects taken into account), there are other, more
`fundamental problems.
`It has not been proved, nor do
`there seem to be mathematical groundsfor the assertion.
`that the truncation of negative weights (whicharethere-
`sult of an unconstrained deconvolution) gives the ‘‘closest”
`physically obtainable solution to the ideal solution. In-
`deed, the concept ofthe “closest” solution is not rigorously
`defined. The physical solution obtained by truncation of
`negative beam intensities does not satisfy the ideal pre-
`scription and does not appear to maximize or minimize
`any score ofclinical interest (e.g., 1t does not minimize
`the integral dose outside the target volume (1 3)).
`Apart from theoretical issues, the real concern with the
`inverse approach is in the way the problem is defined. It
`is not, a priori, possible, to prescribe (i.e., to define using
`equalities) a “‘best” physically obtainable dose distribution.
`Practical dose distributions are always non-uniform(often
`for good reasons) and, contrary to the problem ofrecon-
`structing tomographic images, have regions that are clin-
`ically more important than others. It is easy to show that
`the idea of matching the dose distribution to a specified
`one rejects, as worse, solutions (1.e., dose distributions)
`that by any clinically sound measure are superior to the
`prescribed one. For example, of two solutions with the
`same dose to the target but with different doses to an
`organ at risk, the solution with higher dose to the organ
`at risk will be judged by the algorithmas the better ifits
`doseis closer to the prescribed dose. All this having been
`said,
`the dose distributions developed in the ‘inverse
`problem” papers are undoubtedly interesting. Perhaps
`their main interest is in showing the advantages that may
`accrue from designing non-uniform beamprofiles (8).
`Recently, Webb has proposed solving the ‘inverse
`problem” for conformal radiotherapy using the simulated
`annealing algorithm (33). Simulated annealing is a heu-
`ristic combinatorial approach based on an analogy with
`the way that liquids crystallize—that is, the way liquids
`reach a state of minimum energy. In the case of planning
`radiotherapy, energy is equated with some objective func-
`tion which, in ref. (33), is the accuracy with which doses
`at all pixels of the plan (or over the certain limited regions)
`are matched to the prescribed “optimal” doses. Webb's
`selection of the objective function shares the difficulties
`of the inverse approach in the way the problemis posed—
`as mentioned above. It also suffers from the basic feature
`of simulated annealing, namely that the system must be
`cooling (i.e., converging to the optimum) slowly. As a
`result, even for the 2D cases investigated, and with a sim-
`plified dose model, the optimization required 12 or more
`hours of VAX 750 CPU time(33). The resulting ‘topt-
`mized”’ dose distributions confirm that the simulated an
`
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`Subject US Copyright Laws
`
`

`

`Random search optimization algorithm @ A. NIEMIERKO
`
`91
`
`is supposed to do: ex-
`nealing approach performs as it
`tremely slowly but surely system goes to the state with the
`extremal value of the objective function. However, the
`resulting dose distributions that have been reported seem
`clinically sub-optimal. For example in one case (33), de-
`spite using 128 beams, the meantarget dose was found
`to be over 20% higher or (in another case) over 10%
`smaller than the prescribed target dose, with the dose in-
`homogeneity within the target (expressed as the standard
`deviation of the mean target dose) reaching 15%. These
`results would seemtobe easy for an experienced planner
`to beat using a few conventional beams. However, the
`simulated annealing algorithm is a promising and a pow-
`erful tool for optimization oflarge and difficult problems.
`The algorithm has been a subject ofintensive research
`and its performance has recently been significantly im-
`proved (29).
`
`Artificial intelligence
`Another interesting optimization approachthat seems
`to be potentially useful in radiotherapy treatment planning
`is based on artificial intelligence (Al) or, more precisely,
`the use of knowledge-based systemsthat represent in the
`computer the knowledge of‘“‘experts” in radiotherapy (23,
`34). Some techniques ofAl, especially these concerning
`the problem of exploring alternatives (e.g., alpha-beta
`pruning or branch and bound methods), can be also used
`in some mathematical programming techniques, partic-
`ularly those that use a heuristic methodology.
`The “natural intelligence” approach (as opposed toar-
`uficial intelligence) has not yet been explored in radio-
`therapy planning. It
`is represented by the genetic algo-
`rithm—that is,
`the algorithm which has been used by
`Mother Nature in the evolution process to produce (by
`reproduction and mutation) species able to thrive in a
`particular environment (32). The idea ofa genetic algo-
`rithm (namely, the ability to “learn”—adapt to changes
`in its environment) has been explored, for example, to
`design very-large-scale integrated (VLSI) computer chips
`and in pattern recognition systems—and, of course, has
`proveditself in field tests for 3.5 billion years.
`In spite of sometimes using very sophisticated algo-
`rithms, optimization oftreatment plans in radiation ther-
`apy has not met with broad clinical acceptance. As we
`mentioned in a companion paper(22), it seems to us that
`one reason optimization attempts have not been successful
`is that previous investigations,
`in order to reduce the
`mathematical difficulties of the problem, have short-
`changed the extremely difficult problem of computing
`clinically relevant objective functions.
`Since a reasonable description of a 3-dimensional
`treatment requires at least hundreds, or more likely thou-
`sands, of (possibly) non-linear (in)equalities, and since
`any reasonable clinically relevant objective function is a
`non-linear and sometimes multi-modal and non-contin-
`uous function of many variables, we concluded that we
`needa veryfast search algorithm that can accommodate
`
`an objective function that is non-linear in the plan pa-
`rameters. No existing algorithm seemed to meet these re-
`quirements, so we were led to develop a new approach.
`Wewished the algorithm to be able toreflect, as closely
`as possible, the main goal of radiotherapy, that is, eradi-
`cation of the tumor tissue while the normal tissues are
`spared.
`Webased our approach on our previous experience
`with mathematical programming algorithms (7, 19, 20),
`especially with the refreshing idea of simulated annealing
`(12), and on the encouraging results of our random sam-
`pling approach for evaluating treatment plans (21). We
`term the algorithm RONSC which stands for: Random
`Optimization with Non-linear Score functions and Con-
`straints.
`
`METHODS AND MATERIALS
`
`The optimization model was formulated in the classic
`form used in mathematical programming techniques: an
`objective function which scores the plan is maximized
`subject to a set of constraints, that is, inequalities and
`equalities defining the space offeasible solutions. Math-
`ematically, the optimization goalis to find a solution (a
`vectorof variables of the model, .*), which maximizes the
`objective function /(x) in the space offeasible solutions
`(2p (i.e., solutions whichsatisfy all constraints):
`
`S(X) = max f(x)
`XEN
`
`(1)
`
`where Xis the desired optimum.
`The space offeasible solutions Q9 is defined by con-
`straints as follows (Figure | shows a 2-dimensional ex-
`ample):
`
`Qo = {x ECOG R":2(X) < C.,
`h(x) = C;, and logic of(g, h)}
`
`(2)
`
`and
`
`fiR" > R', e:R"> Rs,
`
`h.R" > R™,
`
`(3)
`
`R" is an n-dimensional space of real numbers. g and
`h are matrixesofcoefficientsof, respectively, /, inequality
`and m), equality constraints. C, and C), are corresponding
`vectors of constraint limits. “Logic of (g, 1)” denotes con-
`straints being the logical combinationsof constraints (e.g.,
`maximum dose to the spinal cord <55 Gy OR dose to
`80%of volumeof the spinal cord <50 Gy AND compli-
`cation probability for the spinal cord <2%) (22).
`It can easily be shown that minimization (as opposed
`to maximization) of an objective function and the use of
`constraints with the opposite direction of inequality to
`that used in expressions | or 2, can be resolved into the
`general form expressed in equations 1-2.
`
`This material was copied
`at the NLMand may be
`Subject US Copyright Laws
`
`

`

`92
`
`L. J. Radiation Oncology @ Biology @ Physics
`
`Volume 23, Number 1, 1992
`
`f
`
`1
`i”
`al
`f,
`
`F=
`
`gy (Xy, XQ)
`Cc
`
`1,2
`
`found using golden section search
`1
`1
`ol
`imax (fy, ff)
`
`}
`
`linear constraints
`
`—— non-linear constraint 00
`
`10
`
`2.0
`
`40
`
`40
`
`
`
`
`
`%
`
`linear constraints
`—— non-linear constraint
`
`- space of
`
`feasible solutions
`
` }
`
`Fig. 1. The 2-dimensional space of feasible solutions (dashed
`area) defined by two linear and one non-linear inequality con-
`straints.
`
`Fig. 3. Every generated solution is projected onto the hyper-
`surface of the most demanding constraints (thick solid line)
`through renormalization by a factor /. The most demanding
`constraint is the one for which the ratio fis the greatest.
`
`The optimization problem, as described in the accom-
`panying paper (22), is computationally very demanding.
`However,clinically useful results can be obtained by lim-
`iting the optimization to a subset of possible parameters,
`namely the beam weights and wehaverestricted ourselves
`to this case in this paper and the companion paper(22).
`Whenthis limitation is imposed, the problem has some
`characteristic properties which, when properly taken into
`account, can substantially reduce the calculational burden
`of optimization. They are as follows:
`
`1. The parameters of the model (i.e., beam weights) are
`non-negative (this characteristic, alone, reduces the
`space offeasible solution by a factor 2” where nis the
`number of parameters—see Figure 1).
`2. The coefficients of the objective function and con-
`straints are non-negative (this characteristic makes ev-
`ery feasible solution from the dashedregion in Figure
`2 not worse than that in the lower-left corner [point
`1] of that region).
`Objective functions and constraints are generally well
`behaved and, although some of them are non-linear
`
`XQ
`
`Xy
`
`Fig. 2. For objective functions with non-negative coefficients,
`every set of parameters (x, > Xx}, %2 > x}) (dashedarea) gives as
`goodorbetter value of the objective function than the parameters
`corresponding to the lower-left hand cornerofthat area.
`
`or discrete, they are usually monotonic functions of
`the beam parameters.
`4. The order of magnitude and the range ofpossible val-
`ues of the beam parameters are known apriori.
`5. Because ofthe physical properties of dose distributions,
`the objective functions we have considered are rela-
`tively slowly varying functions of the chosen param-
`eters and, therefore, the space offeasible solutions is
`relatively “flat” around the true mathematical opti-
`mum, which,
`in turn, means that many solutions
`“near” the true mathematical optimum maybeclin-
`ically indistinguishable from the mathematically ex-
`tremal solution.
`
`It can be shownthat the non-negativity ofall parameters
`and coefficients of the model expressed in equations 1-2
`(which limits all constraints to have the same sign ofin-
`equality) forces solutions to be on a hyper-surface defined
`by one ofthe constraints (Figure 3—bold border of the
`space offeasible solutions). In other words, a solution1s
`always driven by the most demanding constraint (con-
`straint #2 in Figure 3) or,
`less likely, more than one, if
`they cross eac