`
`lN7d§R
`ill
`1992.
`No.1
`[27960000
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`INTERNATIONAL JOURNAL OF
`
`} RADIATION ONCOLOGY
`BIOLOGY-PHYSICS
`
`
`
`VOLUME 23. NUMBER I. 1992
`
`Editor’s Note
`
`P. Rubin
`
`TABLE OF CONTENTS
`
`O (TIN/(Ell. ORIGINAL ('ON'I'Rllj’U'I‘IONS
`
`'
`Highly Anaplastic Astrocytoma: A Review of 357 Patients ’l’reatcd between 1977 and I989
`M. l). l’rados. P. H. (iulin. T. L. Phillips. W. M. Wara. I). A. Larson. P. K. Snead. R. 1.. Davrs, D. K. Ahn.
`K. Lamborn and C. B. Wilson
`
`Non-Hodgkin’s Lymphoma of 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
`l). l“. Nelson. K. l.. Mam. ll. Bonner. .l. S Nelson. J. Ncwall. H. D. Kcrman. J. W. Thomson and K. J. Murray
`
`Radiosurgery and Brain 'l‘olerance: An Analysis of Neurodiagnostic Imaging Changes after Gamma Knife Ra—
`diosurgery for Arteriovenous Malformations
`.l, ('. Flickingcr, 1.. I). Lunsl‘ord. D. Kondziolka. A. ll. 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 'Il‘herapy
`M. M. Eric. B. Fullerton. ll. Tatsuzaki. S. Birnbaum. H. D. Suit. K. Convery. S. Skates and M. (ioncm
`
`The lnlluence of Dose and Time on Wound Complications Following Post—Radiation Neck Dissection
`J, M. (j. 'l uylor. 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
`(i. K. Zagars
`
`O BIOLOGY ORIGIN-1L CONTRIBUTIONS
`
`In Vitra Intrinsic Radiation Sensitivity of Glioblastoma Multiformc
`A. 'l‘aghian. ll. Suit. F. Pardo. D. Gioioso. K. Tomkinson. W. duBois and L. Gerwcck
`
`Repopulalion Between Radiation Fractions in Human Melanoma Xenografts
`lt. K, Rol'stad
`
`(C(mlems continued on page viii)
`
`I
`
`3
`
`9
`
`1‘)
`
`27
`
`4]
`
`47
`
`55
`
`63
`
`[Mil \I I) N ('uiu‘nl ( onlcnts. lilt )SIS Dulalnisc. lndc\ Mctlicus. \1l"|)| lNl-. l'u‘crptu Mctlicu. Saict) Sci. -\|\slr.. l-ncrg} ‘Ris Alain livicrsbllxil'zrBase. [oxlcologx
`‘\l‘\li .
`| lt‘t‘lit‘iltt‘\ & ('ommun.
`\l‘Nli'.. ('ompulcr a Info. Systems Ahxuu (kimhriilgc Sci. Abstr. lll‘\l( ABS.
`1 AM Al -( NR.
`‘11-! MN
`lSSN 0360-30”)
`(223)
`
`This material wascnpiad
`atthE NLM and may be
`Subject usurp-wright: Laws
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`
`
`V5
`
`(Contents (‘()III/lll(t'(/)
`
`Radiosensitivity, Repair Capacity. and Stem Cell Fraction in Human Soft Tissue 'l'umors: An In Vitro Study
`Using Multicellular Spheroids and the Colony Assay
`M. Stuschkc. V. Budach. W. Klacs and H. Sack
`
`0 PHYSICS ORIGINAL CONTRIBUTIONS
`
`Permanent Implants Using All-'98. I’d-103 élfld l-lZS: Radiohiological Considerations Based on the Linear
`Quadratic Model
`C. C. Ling
`
`Random Search Algorithm (RONSC) for Optimization of R
`End Points and (‘onstraints
`A. Nicmierko
`
`adiation Therapy with Both Physical and Biological
`
`Optimization of 3D Radiation Therapy with Both Physical and Biological End Points and Constraints
`A. Nicmierko. M. Uric and M. Goitcin
`
`O H YPER TIIERMI/t ORIGINAL CONTRIBUTIONS
`
`'l‘hermochemotherapy with (,‘isplatin or (‘arboplatin in the BT4 Rat (.‘lioma In Vitro and In Vivo
`B.—C. Schcm, O. Mclla and O. Dahl
`
`Radiation and Heat Sensitivity of Human 'l‘—Lineage Acute Lymphoblastic Leukemia (ALL) and Acute Myth!-
`blastic Leukemia (AMI.) Clones Displaying Multiple Drug Resistance (MDR)
`F. M. Uckun. .l. B. Mitchell. V. Obuz. M. (‘hamlan-Langlie. W. S. Min. S. llaissig and C. W. Song
`
`0 PHASE [/1] CLINICAL TRIALS
`
`6‘)
`
`81
`
`89
`
`99
`
`[09
`
`1 ‘5
`
`Treatment of Non—Small (‘cll Lung (‘ancer with External Beam Radiotherapy and High Dose Rate Brachytherapy
`C Aygun. S. Weincr. A. Scariato. l). Spcai‘man and l.. Stark
`
`[27
`
`
`
`Sequential (‘omparison of Low Dose Rate and llyperl‘ractionated lligh Dose Rate lindobronchial Radiation for
`Malignant Airway Occlusion
`M. Mchta. I). Petcrcit. |.. (‘hosy M. Harmon. J. l’owlcr. S. Shahahi. B. Thomadscn and T. Kinscllu
`
`O BRIEF COMMUNlel TIONS
`
`The Results of Radiotherapy l'or Isolated Elevation of Serum PSA Levels Following Radical Prostatectomy
`S. 1:". Schild. S. J. Buskirk. J. S. Robinow, K. M. ’l omcra. R. (i. l5cr1‘igni and l.. M.
`I‘iriL‘k
`
`Fractionated Radiation Therapy in the 'l‘reatment of Stage III and [V ('e
`Preliminary Results in 20 Cases
`J. Ph. Maire. A. lrloquct. V. Barron/ct. .l. Guérin. J. P. Bcbéar and M, ('audry
`
`rebelIo-Pontine Angle Neurinomas:
`
`0 TE( '1INIC 'A I- INNO l ’xl TIONS .11 NI) NO 'I 'IiS
`
`The Use of Beam‘s Eye View Volumetrics in the Selection of Non-(‘oplanar Radiation Portals
`G. T. Y. Chen. D. R. Spclbring. C. A. Peliuari. J. M. Baltcr. L. C. Myriantlmpoulos. S. Vijayakumul‘ and
`H. Halpern
`
`Ultrasound Directed Extrahepatie Bile Duct lntraluminal Brachytherapy
`B. Minsky, J. Botct. ll. Gerdcs and C. Lightdalc
`
`(Contents continued on page x)
`
`This matarial was copied
`at th a N LM a nd’ may be
`Subject US Cnpyright Laws
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`'33
`
`I41
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`153
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`
`
`Magnetic Resonance Imaging During Intracavitary Gynecologic Brachytherapy
`S. L. Schoeppel. J. H. Ellis. M. L. LaVigne. R. A. Schea and J. A. Roberts
`
`(( '(mlvmx (villi/med)
`
`Development of a Shielded 2“'Am Applicator for Continuous Low Dose Rate Irradiation of Rat Rectum
`R. Nath. S. Rockwell. C. R. King. P. Bongiorni. M. Kelley and I). (‘arter
`
`Reduction of the Dose to the Lens in Prophylactic Cranial Irradiation: A Comparison of Three I)iIl‘erent Treatment
`Techniques and Two Different Beam Qualities
`B. Pakisch. G. Stiicklsehweiger. E. Poicr. C. Urban. W. Kaulfersch. A. Lungmann. (‘. Hauer and A. Hackl
`
`Interstitial Microwave IIyperthermia and Brachytherapy for M
`and Testing of a Modified Intracavitary Ohturator
`T. P. Ryan. J. H. Taylor and C. 'l‘. Coughlin
`
`alignancics ol' the Vulva and Vagina l: Desmn
`
`0 SPECIAL F1114 TURES
`
`An Overview of the First International Consensus Workshop on R
`and Locally Advanced Cancer
`G. E. Hanks. E. J. Maher and L. Coin
`
`adiation Therapy in the Treatment of Metastatlc
`
`The Crisis in Ilealth Care Cost in the U nited 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 IIemihody Irradiation [0
`Standard Fractionated Local Field Irradiation is More Eil'ective 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 COUghlln‘
`I. Weigensbcrg and M. Rotman
`
`Bone Metastasis Consensus Statement
`T. Bates. J. R. Yarnold. P. Blit/cr. O. S. Nelson. P. Rubin and J. Mahcr
`
`A Review of Local Radiotherapy in the Treatment of Bone Metastases and (ford Compression
`T. Bates
`
`A Report of the Consensus Workshop Panel on the Treatment of Brain Metastases
`L. R. Cola. N. Aaronson. R. linggood. J. Loelller a nd 'l‘. J. Priestman
`
`The Role of Radiation Therapy in the Treatment of Brain Metastases
`L. R. Coia
`
`Treatment Strategies in Advanced and Me
`and Europe
`
`tastatic Cancer: Differences in Attitude between the USA. Canada
`
`E. J. Muller. L. Coin. G. Duncan and P.
`
`A. Lawton
`
`O IN MEMORIAM
`
`In Memoriam: Gilbert IIungerford Fletcher
`L. J. Peters
`
`0 ED] TORI/1 LS
`
`CNS Lymphoma: Back to the Drawing Board
`T. E. Gofiinan and E. Glatstein
`
`169
`
`175
`
`183
`
`189
`
`201
`
`203
`
`207
`
`215
`
`
`
`(Contents continued on page xii)
`
`This material wascnpiedf
`at the NLM and may 1113
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`

`41A
`
`U
`
`Endobronchial Braehytherapy: Wither Prescription l’oint
`M. P. Mchta
`
`Response to Dr. Speiser
`C. Aygun. S. Wciner. A. Scariuto. l). Speurmun and 1.. Stork
`
`The Origins and Basis of the Linear-Quadratic Model
`D. J. Brenner and E. J. Hall
`
`Response to Brenner and ”all
`R. J. Yacs. Y. Maruyama. P. Patel and M. Uruno
`
`Radiotherapy of Graves’ ()phthulmopathy
`I). S. Ellis
`
`In Response to "Radiotherapy of Graves” ()phthalmopathy“
`l. A. Petersen. S, S. Donaldson and l. R. Mchugull
`
`
`
`Iodine-125 Implants for Prostate Cancer
`R. E. Peschel and K.
`[5. Wallner
`
`Response to l)rs. Peschel and Wallner
`(T Koprowski
`
`(‘f—252 Neutron (‘apture 'I‘herapy and 'l elethernm
`Y. Maruyamu. J. Wicr/hicki. M. Ashlzn'i, R. .l. Yum. .l.
`
`l
`
`. Beach. .I. Yunch. R. Zumcnhol‘und (C B. Schroy
`
`O MEETINGS
`
`25l
`
`25]
`
`253
`
`353
`
`_
`233
`
`253
`
`254
`
`:54
`
`355
`
`257
`
`This material wasmpied
`atthe NLM and may be
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`

`l'iii‘», Vol
`lfililmtiun (him/run Hm!
`./
`int
`l'ilnlcd in tlic ll 8 /\ All tights n-wnnl
`
`,‘L up IN Wt
`
`('upyriglu it
`
`.00
`()Ihttllllb/UZ $5.00 t
`IWZ l’crgamon Press Ltd.
`
`0 Physics Original Contribution
`
`
`RANDOM SEARCII ALGORITHM (RONSC) FOR OPTIMIZATION OF RADIATION
`THERAPY WITH BOTH PHYSICAL AND BIOLOGICAL
`
`END POINTS AND CONSTRAINTS
`
`ANDRZliJ NIEMIERKO, PHD.
`
`Division of Radiation Biophysics. Department of Radiation Oncology. Massachusetts General Hospital,
`Boston. MA 021 14. and Harvard Medical School
`
`A new algorithm for the optimization of 3-dimensional radiotherapy plans is presented. The RONSC algorithm
`(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 preperties of the dose distribution and
`derivable information such as dose-volume histograms and calculated estimates of tumor control and normal tissue
`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
`
`()ptimization ofradiation therapy is a very important and.
`at the same time. a very diflicult problem. In planning
`radiation treatments. the principle goal of radiotherapy.
`namely. the complete depletion oftumor cells while pre-
`serving normal struetures. is converted into a few smaller
`and often mutually contradictory subtasks. Because of
`the complex relationship between the dose distribution
`and the outcome of radiotherapy. 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 (l—7.
`l()—l 1.
`I372(). 2344.
`2648. 31. 33). Still. even optimization of the physical
`dose distribution is mathematically a very complex and
`dillicult problem. A variety of optimization models and
`algorithms have been investigated. The algorithms used
`in planning radiotherapy can be grouped into the follow—
`ing categories:
`
`/:‘,\'/mnsliw' .wurc/i techniques
`These brute force techniques evaluate each possible
`combination ol‘tquantized) 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 by setting the score to zero ifthe con-
`straints are not satisfied. Because ofthe truly vast number
`of possible combinations of parameters (e.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 procedure is feasible only
`for very small problems and has been used with simplistic
`dose calculation models for 2-dimensional cases (1 l. 31).
`
`Linear unn’ quadratic nnn/rcmulim/ programming
`lfthe scoring function and constraints can be written
`as a linear (or quadratic) function 0fthe 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 techniques are 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
`cmnin'naloria/ linear programming algorithm has also
`been investigated and successfully applied for problems
`with a few hundred constraints (15). The combinatorial
`algorithm allows some ofthe variables to have only integer
`(or. in general, discrete) values but otherwise suffers from
`
`
`
`Reprint requests to: Andr/cj Nicmicrko. PhD. Department
`of Radiation Oncology. Massachusetts General Hospital. Boston.
`MA (ill [4.
`Iii.miit'it‘iiet'iiii'iil,v—'l'hc author would like to thank Michael
`(ioitcin and Marcia Uric 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.
`
`This matéi'ial wasmpiE-d
`atthe NLM and may be
`Subject US Copyright Laws
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`

`90
`
`l.
`
`.l. Radiation Oncology 0 Biology. Physics
`
`Volume 31. Number 1. I093
`
`the same limitations as linear or quadratic programming
`algorithms.
`
`Non-linear mathematical programming
`lfthe scoring function or constraints are not linear or
`quadratic in the parameters of interest. then non-linear
`search techniques have to be used (5.
`lo. 20). Their lim-
`itations are that. with the exception of some unimodal
`functions. they are susceptible to getting trapped in a local
`extremum of the score function. they are sensitive to the
`starting conditions. and their performance dramatically
`decreases as problems become larger. The non-linear al—
`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. none of the standard non-linear programming
`algorithms have so far been found clinically useful.
`
`Finding a ‘leasih/z' solution
`One class of solutions that has been proposed involves
`stating the problem as a set ofa dose constraints without
`an objective function. An iterative approach then solves
`the possibly many thousands of linear inequalities (con-
`straints) (4. 24. 28) and the solution is the first case en—
`countered that satislies all the constraints. In this approach
`(which is not. as such. an optimization algorithm because
`nothing is maximized or minimized) it
`is assumed that
`every feasible solution is clinically satislactory and that
`all feasible solutions are of more or less equal quality.
`This approach requires constraints to be defined in a such
`way that the space of feasible solutions is relatively small
`or flat. This is possible when the planner or the clinician
`designing treatment plan has a knowledge about
`the
`physically obtainable optimal dose distribution. and set
`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. lfthe
`constraints are too loose. there is an infinite space of so—
`lutions and the probability that the first solution found is
`the best one is equal to probability that this solution is
`the worst one (of all feasible solutions).
`
`Inverse .rolzll inn
`
`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
`physical solution that is “as close as possible" to the ideal.
`The idea is similar to the problem of reconstructing 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 tend to be iterative in nature and. therefore. not
`self—evidently faster than other iterative search techniques.
`Besides the well known problems with the mathematics
`ofdeconvolution (for example. the convolution kernel is
`assumed to be spatially invariant—which is not the case
`in radiation therapy for inhomogeneous media and with
`scatter effects taken into account). there are other. morc
`fundamental problems.
`It has not been proved. nor do
`there seem to be mathematical grounds for the assertion.
`that the truncation of negative weights (which are the re—
`sult ofan 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 minimilc
`any score of clinical interest (e.g..
`it does not minimize
`the integral dose outside the target volume (l3)).
`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 of recon-
`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 (i.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 algorithm as the better ifits
`dose is 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 beam profiles (8).
`Recently. Webb has proposed solving the “inverse
`problem“ for conformal radiotherapy using the siinu/um/
`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 ofminimum 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 ofthc plan (or over the certain limited regions)
`are matched to the prescribed “optimal" doses. Webb‘s
`selection of the objective function shares the ditlicultics
`ofthe inverse approach in the way the problem is posedh
`as mentioned above. It also suffers from the basic feature
`of simulated annealing, namely that the system "INN! be
`cooling (i.e.. converging to the optimum) slowly. As a
`result. even for the 21) cases investigated. and with a Sllllr
`plified dose model. the optimization required l2 or mote
`hours of VAX 750 ('l’ll time (33). The resulting “optir
`mired” dose distributions conlirm that the simulated an
`
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`Random search optimi/alion algorithm 0 A. Nlt-MIHtM)
`
`91
`
`is supposed to do: ex-
`nealing approach perlorms 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 mean target 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 seem to be 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 of intensive research
`and its performance has recently been significantly im—
`proved (29).
`
`,Ail‘fi/lt'l'a/ intelligence
`Another interesting optimization approach that seems
`to be potentially useful in radiotherapy treatment planning
`is based on artificial izzlt'l/igcm'e (Al) or. more precisely,
`the use of lamw/ccice»bu.s‘ec/ systems that represent in the
`computer the knowledge of"experts" in radiotherapy (23.
`34). Some techniques of A1. especially these concerning
`the problem of exploring alternatives (cg. 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 to ar-
`tificial
`intelligence) has not yet been explored in radio-
`therapy planning.
`lt is represented by the gcnclic alco-
`ri'l/Im—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 (VLSl) computer chips
`and in pattern recognition systems—and. of course. has
`proved itselfin 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-dimensi0na1
`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 multismodal and non—contin—
`uous function of many variables. we concluded that we
`need a very fast 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.
`We wished the algorithm to be able to rellect. as closely
`as possible. the main goal of radiotherapy, that is. eradi-
`cation of the tumor tissue while the normal tissues are
`spared.
`We based our approach on our previous experience
`with mathematical programming algorithms (7. 19. 20).
`especially with the refreshing idea ofsimulated annealing
`(12). and on the encouraging results ofour 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 of feasible solutions. Math-
`ematically. the optimization goal is to find a solution (a
`vector of variables ofthc model. I). which maximizes the
`objective function _/'(.\") in the space of feasible solutions
`(20 (i.e.. solutions which satisfy all constraints):
`
`f0") : max/(f)
`tea”
`
`(1)
`
`where .\”' is the desired optimum.
`The space of feasible solutions $20 is defined by con-
`straints as follows (Figure 1 shows a 2—dimensional ex—
`ample):
`
`(2“ = l_\_' E Q C R'lifffi S (1.,
`
`710?) f C, and logic of (t7. 71)}
`
`(2)
`
`and
`
`j‘zRIi_>R1‘ g:Rn_) Rm“,
`
`TERM—>Rnth'
`
`(3)
`
`R” is an n-dimensional space of real numbers. g and
`[—1 are matrixes of coefficients of. respectively. my inequality
`and m), equality constraints. Q and C, are corresponding
`vectors ofconstraint limits. “Logic of(§, h)“ denotes con-
`straints being the logical combinations of constraints (cg.
`maximum dose to the spinal cord <55 Gy OR dose to
`80% of volume of 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 1 or 2. can be resolved into the
`general form expressed in equations 1—2.
`
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`at the M’LM and may be
`Subject: US Copyright: Laws
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`

`l. J. Radiation Oncology 0 Biology. Physics
`
`Volume 2}. Number |. 1001
`
`|
`
`1..
`
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`to
`
`l. The 2-dimensional space of feasible solutions (dashed
`Fig.
`area) defined by two linear and one non-linear inequality con—
`straints.
`
`livery generated solution is projected onto the hyper
`Hg. 3.
`surface of the most demanding constraints (thick solid line!
`through renormali/ation by a factor I". 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 we have restricted ourselves
`to this case in this paper and the companion paper (22).
`When this 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 fol10ws:
`
`l. The parameters of the model (i.e.. beam weights) are
`non—negative (this characteristic. alone. reduces the
`space of feasible solution by a factor 2” where H is the
`number of parameters—see Figure l).
`The coefficients of the objective function and con—
`straints are non-negative (this characteristic makes ev-
`ery feasible solution from the dashed region in Figure
`2 not worse than that in the lower-left corner [point
`I] ofthat region).
`Objective functions and constraints are generally well
`behaved and. although some of them are non-linear
`
`b.)
`
`4.0
`
`3.0
`
`2.0
`
`1.0 0.0
`
`0.0
`
`1.0
`
`2.0
`
`3.0
`
`Fig. 2. For objective functions with non-negative coetlicients.
`every set of parameters (X. > x}. .\'g > Kl) (dashed area) gives as
`good or better value ofthe objective function than the parameters
`corresponding to the lower-left hand corner ofthat area.
`
`or discrete. they are usually monotonic functions of
`the beam parameters.
`4. The order of magnitude and the range of possible val-
`ues of the beam parameters are known a prinri.
`5. Because ofthe physical properties ofdose distributions.
`the objective functions we have considered are rela-
`tively slowly varying functions of the chosen param-
`eters and. therefore, the space o