)
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`INTERNATIONAL JOURNAL OF
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`RADIATION
`ONCOLOGY
`
`BIOLOGY-PHYSICS
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`VOLUME 23, NUMBER 1, 1992
`
`ISSN 0360-3016
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`The Official Journal of the
`AMERICAN SOCIETY FOR THERAPEUTIC RADIOLOGY AND ONCOLOGY
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`Sponsored by the
`INTERNATIONAL SOCIETY OF RADIATION ONCOLOGY
`CIRCUIO DE RADIOTERAI’EUI‘AS IBERO-LA'I‘INOAMERICANOS
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`INTERNATIONAL JOURNAL OF
`
`‘ RADIATION ONCOLOGY
`BIOLOGY-PHYSICS
`
`
`
`VOLUMli 23. NUMBER l. 1992
`
`Editor’s Note
`
`P. Rubin
`
`TABLE OF CONTENTS
`
`O (TIN/(Ell, ORIGIleI. CONTRIBUTIONS
`
`Ilighly Anaplastic Astrocytoma: A Review of 357 Patients Treated between 1977 and [989
`M. I). Prados. P. ll. (lutin. T. L. Phillips. W. M. Want. 1). A. Larson. P. K. Sneed. R. 1.. Davrs. D. K. Ahn.
`K. Lamborn and (‘. B. Wilson
`
`Non-Hodgkin’s Lymphoma of the Brain: (‘an Iligh Dose. Large Volume Radiation Therapy Improve Survival?
`Report on a Prospective Trial by the Radiation Therapy Oncology Group (RTOG): RTOG 83l5
`l). I". Nelson. K. l.. Martx. ll. Bonner. J. S. Nelson. J. Newall. H. D. Kerman. J. W. Thomson and K. J. Murray
`
`Radiosurgery and Brain 'I'olerance: An Analysis of Neurodiagnostic Imaging Changes after Gamma Knife Ra-
`diosurgery for Arteriovenous Malformations
`J. (‘. I’lickingcr. L. I). Lunsl‘ord. I). 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. ll. Tatsuzaki. S. Birnbaum. H. D. Suit. K. Convery. S. Skates and M. (Jonein
`
`The Influence of Dose and Time on Wound Complications Following Post—Radiation Neck Dissection
`J. M. (i. '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 Vitro Intrinsic Radiation Sensitivity of Glioblastoma Multiforme
`A. 'l‘aghian. ll. Suit. F. Pardo. D. Gioioso. K. Tomkinson. W. duBois and L. Geiweck
`
`l
`
`3
`
`9
`
`1‘)
`
`7
`-7
`
`4]
`
`47
`
`55
`
`63
`
`Repopulation Between Radiation Fractions in Human Melanoma Xenografts
`li. K. Rol'stad
`
`(Contents continued on page viii)
`
`l' M ' li ‘u . MliDI lN|-. l'\ "r1121 Medicu. Sat'ct) Sci. MN
`;_ ~,l
`;
`);
`'
`'
`.
`-
`.
`-
`it ihisn m U st
`min \I l) w( mu m ( miunis. RUNS I
`t
`\
`H l
`\bsli .
`l lecliomcs R t‘ommun.
`\bstr.. ('omputcr R Into. Systems Abstr. ('ainhritlgc Sci. Abstr.
`
`This material was. copied
`atthE NLM and may be
`5U bjact U5 Dapvright Laws
`
`it. l-ncrgx Res. Al)$ll'.. lznerg} Data Base. 'l oxiL‘olng)
`and (ABS. PASCAL-(NR5 Database
`ISSN 0360-30 l 6
`(223)
`
`

`

`
`
`(( Vin/unit will/Hum!)
`
`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. Klaes and H. Sack
`
`0 PHYSICS ORIGINAL CONTRIBUTIONS
`
`Permanent lmplants Using Au-l98. I’d-103 and l-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. Uric and M. Goitcin
`
`O H YPER TIIERMIA ORIGINAL CONTRIBUTIONS
`
`'l‘hermochemotherapy with Cisplatin or Carboplatin in the BT4 Rat (llioma In Vitro and In VFW
`B.-C. Schcm. O. Mclla and O. Dahl
`
`Radiation and Heat Sensitivity of Human 'l‘—Lineage Acute Lymphohlastic Leukemia (ALL) and Acute MyCl‘V
`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
`
`0 PIIASE [/1] CLINICAL TRIALS
`
`6‘)
`
`81
`
`89
`
`99
`
`109
`
`115
`
`Treatment of Non—Small (‘ell Lung Cancer with External Beam Radiotherapy and High Dose Rate Brachytherapy
`C. Aygun. S. Weincr, A. Scariato. 1). Spearman and 1.. Stark
`
`127
`
`
`
`Sequential Comparison of Low Dose Rate and llyperl'ractiolmted lligh Dose Rate Endobronchial Radiation for
`Malignant Airway Occlusion
`M. Mchta. I). Petcreit. L. (‘hosy. M. Harmon. J. 1’ow1cr. S. Shahabi. B. Thomadscn and T. Kinsella
`
`O BRIE!" COMMUNK 31 TIONS
`
`The Results of Radiotherapy l'or Isolated l‘llcvation of Serum PSA Levels l<'ollowing Radical Prostatectomy
`S. E. Schild. S. J. Buskirk. J. S. Robinow. K. M. 'l'omcra. R. (i. l’crrigni and 1.. M. Prick
`
`Fractionated Radiation Therapy in the Treatment of Stage III and W ('e
`Preliminary Results in 20 Cases
`J. P1]. Maire. A. Floquct. V. Darrouzet. J. Guérin. J. P. Béhéar and M. (‘
`
`audry
`
`rebello-Pontine Angle Ncurinomas:
`
`0 TECHNICAL INNOVATIONS AND NO'I’ES
`
`The Use of Beam‘s Eye View Volumetrics in the Selection of Non-(‘oplanar Radiation Port
`G. T. Y. Chen. D. R. Spelbring. C. A. Pelizxuri. J. M. Balter.
`1.. C. Myriantlmpoulos. S. Vijayakumar and
`H. Halpern
`
`als
`
`Ultrasound Directed Extrahepatic Bile Duct lntraluminal Brachytherapy
`B. Minsky. J. 80101. H. Gcrdcs and C. Lightdalc
`
`(Contents continued on page x)
`
`This matarial was copied
`at th a- N'LM a nd’ may be
`Su‘ijatt U5 Cuppright Laws
`
`133
`
`141
`
`147
`
`153
`
`165
`
`

`

`
`
`Magnetic Resonance Imaging During Intracavitary Gynecologic Brachytherapy
`S. L. Schoeppel, J. H. Ellis, M. L. LaVigne. R. A. Schea and J. A. Roberts
`
`(Canton/x continued)
`
`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 D. Carter
`
`Reduction of the Dose to the Lens in Prophylactic Cranial Irradiation: A Comparison of Three Different Treatment
`Techniques and Two Different Beam Qualities
`B. Pakisch, G. Stiicklschweiger, E. Poier, C. Urban, W. Katlltet‘scli. A. Langmann, (‘. Hauer and A. Hackl
`
`Interstitial Microwave IIyperthermia and Brachytherapy for Malignaneies of the Vulva and Vagina I: Design
`and Testing of a Modified Intracavitary Obturator
`T. P. Ryan, J. H. Taylor and C. T. Coughlin
`
`0 SPECIAL FEA TURES
`
`An Overview of the First International Consensus Workshop on Radiation Therapy in the Treatment of Metastatlc
`and Locally Advanced Cancer
`G. E. Hanks, E. J. Maher and L. Coia
`
`The Crisis in Health 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 Ilemihody Irradiation I0
`Standard Fractionated Local Field Irradiation is More Effective Than Local Field Irradiation Alone in the
`Treatment of Symptomatic ()sseous Metastases
`C. A. Poulter, D. Cosmatos, P. Rubin, R. Urtasun, J. S. Cooper, R. R. Kuske, N. Hornback. C. Coughlln‘
`l. Weigensberg and M. Rotman
`
`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. Cora, N. Aaronson, R. Linggood, J. Loelller and 'l‘. J. Priestman
`
`The Role of Radiation Therapy in the Treatment of Brain Metastases
`L. R. Coia
`
`Treatment Strategies in Advanced
`and Europe
`E. J. Mahcr, L. Com, G. Duncan and P. A. Lawton
`
`and Metastatic Cancer: Differences in Attitude between the USA, Canada
`
`0 IN MEMORIAM
`
`In Memoriam: Gilbert IIungerford Fletcher
`L. J. Peters
`
`0 ED] TORI/1 LS
`
`CNS Lymphoma: Back to the Drawing Board
`T. E. Goffman and E. Glatstein
`
`(Contents continued on page xii)
`
`Th is material mascopied.‘
`at the NLM and may lJE
`5U hject U’S anrwigh-t Laws‘
`
`I ()9
`
`175
`
`183
`
`189
`
`207
`
`215
`
`
`
`
`

`

`A '
`
`/
`
`Ll
`
`Endobronehial Braehytherapy: Wither Prescription l’oint
`M. P. Mehta
`
`Response to Dr. Speiser
`C. Aygun. S. Weincr. A. Seariuto. l). Spearmun and L. Stark
`
`The Origins and Basis of the Linear-Quadratic Model
`D. J. Brenner and E. J. Hall
`
`Response to Brenner and ”all
`R. J. Yaes. Y. Muruyamu. P. Patel and M. Uruno
`
`Radiotherapy of Graves’ ()phthalmopathy
`I). 5. Ellis
`
`In Response to “Radiotherapy of Graves~ ()phthalmoputhy“
`l. A. Petersen. S. S. Donaldson and l. R. MCDOng‘dll
`
`
`
`Iodine-125 Implants for Prostate (‘aneer
`R. E. Pesehel and K. 15. Wallner
`
`Response to l)rs. Pesehel and Wallner
`C. Koprowski
`
`(‘f—252 Neutron (‘apture 'l‘herapy and 'l'eletherapy
`Y. Maruyamu, J. Wier/hicki. M. Ashturi. R. J. Yates. J.
`
`l., Bench. J. Ynneh. R. Zumenhol‘und (I B. Sehmy
`
`O MEETINGS
`
`25l
`
`251
`
`253
`
`253
`
`253
`
`253
`
`254
`
`254
`
`355
`
`257
`
`This material was cnpiad
`atthe NLM and may be
`
`Sub'Ect US-E‘nflriwiht Laws
`
`

`

`I'Itiw, Vol
`./ Harlin/tun ()IIKH/il‘tl’l‘ Iim/
`Int
`l'ilnlcd In the I] S A All tights Ii-scnnl
`
`3 I. pp K9 9%
`
`.00
`$5.00 l
`(IMO-Itilb/‘il
`('upyrighl it
`1992 l’ergamon Press Ltd.
`
`0 Physics Original (,‘antriinilian
`
`
`RANDOM SEARCH ALGORITHM (RONSC) FOR OPTIMIZATION OF RADIATION
`THERAPY WITH BOTH PHYSICAL AND BIOLOGICAL
`
`ENI) POINTS AND CONSTRAINTS
`
`ANDRZISJ NlEMlERKO, 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 properties 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
`
`Optimization of radiation 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 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—1 1.
`l3—20. 23—24.
`26—28. 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:
`
`l:'.\'/Iait.s'tive search techniques"
`These brute force techniques evaluate each possible
`combination of(quantized) treatment parameters. A score
`is calculated for each analyzed sct oftreatment 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 and quadratic mathematical programming
`If the scoring function and constraints can be written
`as a linear (or quadratic) function ofthe 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
`combinatorial 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 Niemicrko. l’h.D.. Department
`of Radiation Oncology, Massachusetts General Hospital. Boston.
`MA (BI 14.
`
`.lt-lrntneierieementx—'l'lie author would like to thank Michael
`(tOllClH 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 was cupied
`at the MLl'u'l and may be
`Subject US Cum-right Laws
`
`

`

`90
`
`l.
`
`.l. Radiation Oncology 0 Biology. Physics
`
`Volume 23. Number 1. I992
`
`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. 16. 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/casih/t' 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 satisfies 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 satisfactory and that
`all feasible solutions are of more or less equal quality.
`This approach requires constraints to be delined 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).
`
`In verse solution
`
`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. more
`
`It has not been proved. nor do
`fundamental problems.
`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"
`
`In—
`physically obtainable solution to the ideal solution.
`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 of clinical interest (e.g..
`it does not minimize
`the integral dose outside the target volume (13)).
`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 ma)
`accrue from designing non-uniform beam profiles (8).
`Recently. Webb has proposed solving the “inverse
`problem“ for conformal radiotherapy using the simulated
`annealing algorithm (33). Simulated annealing is a heu~
`ristie 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 ofthe plan (or over the certain limited regionS)
`are matched to the prescribed “optimal“ doses. Webb‘s
`selection of the objective function shares the difficulties
`ofthe inverse approach in the way the problem is poscdr'
`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 21) cases investigated. and with a sinii
`plified dose model. the optimization required ll or more
`
`hours of VAX 750 (‘Pll time (33). The resulting “opltr
`mized” dose distributions conlirm that the simulated an
`
`This material was copied
`atfihe N‘LMandl may be
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`

`

`Random search optimization algorithm. A. Nit-MILRKo
`
`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 I28 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).
`
`,‘lrli/icia/ inIe/ligena'
`
`Another interesting optimization approach that seems
`to be potentially useful in radiotherapy treatment planning
`is based on artificial inlt'l/igena' (Al) or. more precisely,
`the use of know/vcige-ltascz/ sits/wilt" that represent in the
`computer the knowledge of“experts" in radiotherapy (23.
`34). Some techniques of Al, 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 to ar-
`tificial intelligence) has not yet been explored in radio-
`therapy planning.
`lt is represented by the gene/iv 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
`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-dimensi0nal
`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
`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 reflect, 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 ofthe model, E), which maximizes the
`objective function _/'(.f) in the space of feasible solutions
`(20 (i.e., solutions which satisfy all constraints):
`
`f0") : max ./'(.\_‘)
`icon
`
`(1)
`
`where .\" is the desired optimum.
`The space of feasible solutions S20 is defined by con-
`straints as follows (Figure 1 shows a 2—dimensional ex-
`ample):
`
`S2“ = l.\_ E Q C R'lififf) S (1%"
`
`71(93) : (1,, and logic of (g. 71)}
`
`(2)
`
`and
`
`_/VIR"—>Rl,
`
`§:Rn_+ RINK,
`
`l-IIRnale'.
`
`(3)
`
`R” is an n-dimensional space of real numbers. 1,7 and
`1—1 are matrixes of coefficients of, respectively. my inequality
`and m,, equality constraints. Q and C), are corresponding
`vectors of constraint limits. “Logic of (g, h)” denotes con-
`straints being the logical combinations of constraints (e.g.,
`maximum dose to the spinal cord <55 Gy OR dose to
`80% of volume ofthe 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) ofan 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.
`
`This :matiEiri al was mpied
`at th-r.l MLM and may like
`5U inject US Copyright Laws
`
`

`

`I
`
`ltucttt t'ottsttattth
`
`I. J. Radiation Oncology 0 Biology 0 Physics
`
`Volume 23. Number I. 1001
`
`l
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`
`
`
`:20
`
`linear constraints
`norrlinettr constraint
`
`. since (11
`t
`
`feasible solutrous
`
` }
`
`-1 ll
`
`g‘ (Kl. K1)
`(,
`
`V
`
`,
`1..
`
`I
`
`1
`
`“I
`1‘
`
`lotmtl using gnlrtrn settron search
`1
`(ll
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`1' m... (1,. t... I.
`
`I
`
`non hnertt constraint
`
`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 renormalization 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 follows:
`
`1. 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 n is the
`number of parameters—see Figure 1).
`. 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] of that 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
`
`40
`
`X1
`
`Fig. 2. For objective functions with non-negative coellicients.
`every set of parameters (.\'| > X}. .\'g > XE) (dashed area) gives as
`good or better value ofthe objective function than the parameters
`corresponding to the lower-left hand corner ofthat area.
`
`Kl}
`
`or discrete. they are usually monotonic functions of
`the beam parameters.
`. The order of magnitude and the range of possible val-
`ues of the beam parameters are known a priori.
`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 of feasible solutions is
`relatively “flat“ around the true mathemat