An analytical model for light ion pencil beam dose distributions: multiple scattering of primary and secondary ions.

An analytical algorithm based on the generalized Fermi-Eyges theory, amended for multiple Coulomb scattering and energy loss straggling, is used for calculation of the dose distribution of light ion beams in water. Pencil beam energy deposition distributions are derived for light ions by weighting a Monte Carlo (MC) calculated planar integral dose distribution with analytically calculated multiple scattering and range straggling distributions. The planar integral dose distributions are calculated using the MC code SHIELD-HIT07, in which multiple scattering and energy loss straggling processes are excluded. The contribution from nuclear reactions is included in the MC calculations. Multiple scattering processes are calculated separately for primary and secondary ions and parameters of the initial angular and radial spreads, and the covariance of these are derived by a least-square parameterization of the SHIELD-HIT07 data. The results from this analytical algorithm are compared to pencil beam dose distributions obtained from SHIELD-HIT07, where all processes are included, as well as to experimental data. The presented analytical approach allows for the accurate calculation of the spatial energy deposition distributions of ions of atomic numbers Z = 1 - 8.

Design and characterization of a tissue-equivalent CVD-diamond detector for clinical dosimetry in high-energy photon beams.

New solid-state detectors, based on chemical vapour deposited (CVD) polycrystalline diamonds produced by hot-filament (HF) or microwave plasma (MW) assisted deposition methods, were constructed for radiation therapy dosimetry. Properties of diamond crystals, such as high radiation sensitivity, resistance to radiation damage and tissue-equivalence giving a low-energy dependence are very advantageous for clinical dosimetry. Therefore the encapsulation was specially designed for these detectors to have as little influence as possible on the radiation response. The prototypes were irradiated with use of a wide range of photon beam qualities ((60)Co gamma-rays, 6 and 18 MV X-rays). The radiation sensitivity varied considerably between samples deposited with HF (9 nC Gy(-1)mm(-3)) and MW (66 and 144 nC Gy(-1)mm(-3)) methods. For all detectors the leakage current was of the order of 10% of the radiation-induced current (bias voltage 100 V, dose rate 0.3 Gy/min). When irradiated with (60)Co gamma-rays, the detectors showed a dose-rate linearity with an exponential Delta parameter close to unity. However, a difference of 8% was found between Delta values for the different beam qualities. A small energy dependence was observed, for which the most probable sources are interface effects due to the silver electrodes and partly the geometry of the encapsulation which needs to be further optimized. Despite some limitations in the performance of present prototype detectors, with an improved CVD technique producing crystals of better electrical and dosimetric properties, and with a well-designed tissue-equivalent encapsulation, CVD-diamonds could serve as very good dosimeters for radiotherapy.

Development of dose delivery verification by PET imaging of photonuclear reactions following high energy photon therapy.

A method for dose delivery monitoring after high energy photon therapy has been investigated based on positron emission tomography (PET). The technique is based on the activation of body tissues by high energy bremsstrahlung beams, preferably with energies well above 20 MeV, resulting primarily in 11C and 15O but also 13N, all positron-emitting radionuclides produced by photoneutron reactions in the nuclei of 12C, 16O and 14N. A PMMA phantom and animal tissue, a frozen hind leg of a pig, were irradiated to 10 Gy and the induced positron activity distributions were measured off-line in a PET camera a couple of minutes after irradiation. The accelerator used was a Racetrack Microtron at the Karolinska University Hospital using 50 MV scanned photon beams. From photonuclear cross-section data integrated over the 50 MV photon fluence spectrum the predicted PET signal was calculated and compared with experimental measurements. Since measured PET images change with time post irradiation, as a result of the different decay times of the radionuclides, the signals from activated 12C, 16O and 14N within the irradiated volume could be separated from each other. Most information is obtained from the carbon and oxygen radionuclides which are the most abundant elements in soft tissue. The predicted and measured overall positron activities are almost equal (-3%) while the predicted activity originating from nitrogen is overestimated by almost a factor of two, possibly due to experimental noise. Based on the results obtained in this first feasibility study the great value of a combined radiotherapy-PET-CT unit is indicated in order to fully exploit the high activity signal from oxygen immediately after treatment and to avoid patient repositioning. With an RT-PET-CT unit a high signal could be collected even at a dose level of 2 Gy and the acquisition time for the PET could be reduced considerably. Real patient dose delivery verification by means of PET imaging seems to be applicable provided that biological transport processes such as capillary blood flow containing mobile 15O and 11C in the activated tissue volume can be accounted for.

Influence of electrodes on the photon energy deposition in CVD-diamond dosimeters studied with the Monte Carlo code PENELOPE.

A new dosimeter, based on chemical vapour deposited (CVD) diamond as the active detector material, is being developed for dosimetry in radiotherapeutic beams. CVD-diamond is a very interesting material, since its atomic composition is close to that of human tissue and in principle it can be designed to introduce negligible perturbations to the radiation field and the dose distribution in the phantom due to its small size. However, non-tissue-equivalent structural components, such as electrodes, wires and encapsulation, need to be carefully selected as they may induce severe fluence perturbation and angular dependence, resulting in erroneous dose readings. By introducing metallic electrodes on the diamond crystals, interface phenomena between high- and low-atomic-number materials are created. Depending on the direction of the radiation field, an increased or decreased detector signal may be obtained. The small dimensions of the CVD-diamond layer and electrodes (around 100 microm and smaller) imply a higher sensitivity to the lack of charged-particle equilibrium and may cause severe interface phenomena. In the present study, we investigate the variation of energy deposition in the diamond detector for different photon-beam qualities, electrode materials and geometric configurations using the Monte Carlo code PENELOPE. The prototype detector was produced from a 50 microm thick CVD-diamond layer with 0.2 microm thick silver electrodes on both sides. The mean absorbed dose to the detector's active volume was modified in the presence of the electrodes by 1.7%, 2.1%, 1.5%, 0.6% and 0.9% for 1.25 MeV monoenergetic photons, a complete (i.e. shielded) (60)Co photon source spectrum and 6, 18 and 50 MV bremsstrahlung spectra, respectively. The shift in mean absorbed dose increases with increasing atomic number and thickness of the electrodes, and diminishes with increasing thickness of the diamond layer. From a dosimetric point of view, graphite would be an almost perfect electrode material. This study shows that, for the considered therapeutic beam qualities, the perturbation of the detector signal due to charge-collecting graphite electrodes of thicknesses between 0.1 and 700 microm is negligible within the calculation uncertainty of 0.2%.

Assessing the difference between planned and delivered intensity-modulated radiotherapy dose distributions based on radiobiological measures.

AIMS: Because of the highly conformal distributions that can be obtained with intensity-modulated radiotherapy (IMRT), any discrepancy between the intended and delivered distributions would probably affect the clinical outcome. Consequently, there is a need for a measure that would quantify those differences in terms of a change in the expected clinical outcome. MATERIALS AND METHODS: To evaluate such a measure, cancer of the cervix was used, where the bladder and rectum are proximal and partially overlapping with the internal target volume. A solid phantom simulating the pelvic anatomy was fabricated and a treatment plan was developed to deliver the prescribed dose to the phantom. The phantom was then irradiated with films positioned in several transverse planes. The racetrack microtron at 50 MV was used in the treatment planning and delivery processes. The dose distribution delivered was analysed based on the film measurements and compared against the treatment plan. The differences in the measurements were evaluated using both physical and biological criteria. Whereas the physical comparison of dose distributions can assess the geometric accuracy of delivery, it does not reflect the clinical effect of any measured dose discrepancies. RESULTS: It is shown how small inaccuracies in delivered dose can affect the treatment outcome in terms of complication-free tumour cure. CONCLUSIONS: With highly conformal IMRT, the accuracy of the patient set-up and treatment delivery are critical for the success of the treatment. A method is proposed to evaluate the precision of the delivered plan based on changes in complication and control rates as they relate to uncertainties in dose delivery.

Optimized radiation therapy based on radiobiological objectives.

In the broad field of radiation therapy optimization, both simple and complex problems have their origins in the interaction of the radiation beams with the biological structures of normal and malignant tissues of the human body. Therefore, it is no great surprise that many treatment optimization problems are best handled by the use of well-designed radiobiological models. The classic way of quantifying dose-response relations for tumors and normal tissues as well as their cross-correlation with each other and their dependence on the underlying genetic and molecular biology of the cell are first briefly reviewed. Radiobiological objective functions, such as the probability of achieving complication-free cure and its expectation value under influence of stochastic processes during the course of treatment, are defined and shown to solve many of the problems of radiation therapy planning. Finally, it is shown through the use of these quantifiers that, simply by introducing biologically optimal intensity modulated dose delivery, the treatment outcome can be improved by about 20% or more in cases with a complex spread of the disease. Once radiobiological optimal plans have been developed, they can be approximated by ordinary physical planning, but the biological objective functions are still needed to have a figure of merit for the quality of the treatment.

Individualizing cancer treatment: biological optimization models in treatment planning and delivery.

PURPOSE: During the last 30 years radiation therapy has developed from classical rectangular beams via conformation therapy with largely uniform dose delivery, but irregular field shapes, to fully intensity modulated dose delivery where the total dose distribution in the tumor can be fully controlled in three dimensions. This last step has been developed during the last 15-20 years and has opened up the possibilities for truly optimized radiation therapy. METHODS AND MATERIALS: Today it is not only possible to produce almost any desired dose distribution in the tumor volume. It is also possible to deliver the dose distribution, which has the highest probability to cure the patient without inducing severe complications in normal tissues. To fully exploit the advantages of intensity-modulated radiation therapy, quality of life or radiobiologic objectives have to be used, preferably combined with predictive assay of radiation sensitivity. RESULTS: This article will briefly discuss the biologic objective functions and the associated advantages in the treatment outcome using new approaches such as consideration of stochastic variations in sensitivity and optimization of the angle of incidence and fractionation schedule with intensity-modulated beams. Finally, different possibilities for realizing general three-dimensional intensity-modulated dose delivery will be discussed. CONCLUSIONS: Once accurate genetically and/or cell survival based predictive assays become available, radiation therapy will become an exact science allowing truly individual optimization considering also the panorama of side-effects that the patient is willing to accept.

Which is the most suitable number of photon beam portals in coplanar radiation therapy?

PURPOSE: Computer-controlled milling machines for compensator manufacture, dynamic multileaf collimators, and narrow scanned electron or bremsstrahlung photon beams have opened up new possibilities to shape nonuniform fluence profiles and have thus, paved the road for truly three dimensional (3D) dose delivery. The present paper investigates the number of beam portals required to optimize coplanar radiation therapy using uniform and nonuniform dose delivery. METHODS AND MATERIALS: A recently developed algorithm has been used for optimization of the dose delivery in such a way that the probability of achieving tumor control without causing severe reactions in healthy normal tissues becomes as high as possible. This method has been used to optimize the delivered dose distribution for an increasing number of beam portals to determine how many coplanar beam portals are desirable to safely achieve a good treatment outcome. Target volumes in the head and neck, thorax, and abdomen have been investigated. RESULTS: Nonuniform dose delivery allows a considerable improvement in the treatment outcome compared to uniform dose delivery. This is evident both from the probability of achieving complication-free tumor control and the value of relevant properties of the dose distribution, such as the mean value and the standard deviation of the mean dose to target volume and organs at risk. The results also show a close relationship between the dose distribution parameters and the probability of achieving complication-free tumor control. The level of complication-free tumor control first increases rapidly when the number of beam portals is increased, but already reaches a level of saturation after three to five beam portals. When the saturation level has been reached, the standard deviation of the mean dose to the target volume is around 3%. CONCLUSIONS: To achieve optimal expectation value of the treatment outcome, within an accuracy of a few percent as measured by the probability of achieving complication-free tumor control, it is generally sufficient to use three nonuniform beam portals. A very large number of coplanar beams may only raise the probability of achieving complication-free tumor control by 1 to 2%. However, good treatment outcome with three beam portals requires that the directions of incidence of the coplanar nonuniform beams are optimally selected. If, on the other hand, the treatment is performed using uniform beams, it is not possible, even with an infinite number of fields, to obtain as high a level of complication-free tumor control as with a few nonuniform beams. From an optimization point of view, it is sufficient to reach a relative standard deviation of the mean dose to the target volume of around 3%. Improved dose homogeneity beyond this level will, in general, not significantly improve the complication-free tumor control.

Optimization of uncomplicated control for head and neck tumors.

Almost 200 patients have been treated for head and neck tumors at two different dose levels. Based on the clinically observed probabilities for tumor control and fatal normal tissue complications at the two dose levels, the dose giving maximum uncomplicated control has retrospectively been calculated and compared with the clinical data. A Poisson statistical model for control and complications has been used including a correlation parameter, delta, to describe the fraction of patients where control and complications are statistically independent. The clinically observed probability of uncomplicated tumor control, P+, is consistent with only a small fraction of the patients treated being statistically independent (delta = 0.2 or 20%). Customarily, 100% of the patients are assumed to be statistically independent with regard to tumor control and normal tissue complications. More precisely, the clinical data are consistent, with almost 20% of the patients being significantly more sensitive to radiation since they gain local tumor control but simultaneously suffer fatal complications. An even larger fraction of the patients (almost 30%) seemed to be more resistant to radiation, showing neither serious treatment complications nor control of the local tumor growth. It is suggested that if these patient groups could be identified by a predictive assay for the radiation sensitivity of their normal tissues and preferably also for their tumors, the uncomplicated tumor control could be increased by about 20%. This figure is based on the actuarial survival of the patients and has been corrected for the inevitable uncertainty in dose delivery. It is also pointed out that about 20% of the patients can never be saved by a predictive assay because of the considerable statistical variance associated with the Poisson process and the eradication of the last clonogenic tumor cell. Finally, note that the possible existence of radiation sensitive and resistant patient groups is consistent with known genetic deficiencies such as ataxia telangiectasia for the sensitive patients and the existence of repair efficient head and neck tumors that are unusually efficient in repairing double strand breaks. If such sensitive and resistant patient groups do exist, it should be sufficient to perform a predictive assay on normal tissues alone avoiding the often impossible task of sampling the most radiation resistant tumor cell line.

Radiotherapeutic computed tomography with scanned photon beams.

Radiotherapeutic computed tomography is a powerful technique to generate anatomical transversal tomograms of the patient in treatment position by using the therapy beam from the treatment unit. For this purpose the treatment unit has to be equipped with a detector array that can detect the beam transmitted through the patient and a computer that analyzes the data and performs the back projection. When the treatment unit uses scanned elementary photon beams, the only practical technique available for generating high quality high energy photon beams, the operation principle and, to some extent, the image quality is similar to that of a 3rd generation CT-scanner. The optimum choice of detection geometry and type of radiation detectors for radiotherapeutic computed tomography particularly at high photon energies are discussed indicating the merits of BGO (bismuthgermanate) or CWO (cadmiumtungstate) photodiod arrays. The first tomographic images of a thorax phantom at an acceleration potential of 50 MV using such detectors are presented. The image contrast is similar to that for 300 kV X rays mainly because the considerable influence of pair production at 50 MV. Line spread and modulation transfer functions are presented indicating a resolution of the order of two millimeters using a crystal thickness of 5 mm. The advantages with radiotherapeutic computed tomography, beside forming a new general communication channel between different diagnostic techniques, dose planning, and radiation delivery, are the elimination of position errors and the provision of exact attenuation data for dose planning.

Optimization of stationary and moving beam radiation therapy techniques.

A new approach is suggested for the optimization of stationary and more general moving beam type of irradiations. The method reverses the order of conventional treatment planning as it derives the optimum incident beam dose distributions from the desired dose distribution in the target volume. It is therefore deterministic and largely avoids the trial and error approach often applied in treatment planning of today. Based on the approximate spatial invariance of the convergent beam point irradiation dose distribution, the desired dose distribution in the target volume is analyzed in terms of the optimum density of such point irradiations. Since each point irradiation distribution is optimal for the irradiation of a given point and due to the linearity of individual energy depositions or absorbed dose contributions, the resultant point irradiation density will also generate the best possible irradiation of an extended target volume when the maximum absorbed dose at a certain distance from the target should be minimized. The optimum shape of the incident beam for each position of the gantry is obtained simply by inverse back projection of the point irradiation density on the position of the radiation source for that orientation of the incident beam.

Current algorithms for computed electron beam dose planning.

The field of electron beam dose planning has been in a state of very rapid development during the last decade. The essentially one-dimensional manual corrections used since the beginning of the sixties, has been replaced by at least two- and sometimes three-dimensional computer algorithms capable of taking all irregularities of the body cross-section and the properties of the various tissues into account. This is achieved by dividing the incoming broad beams in a number of narrow pencil beams, the penetration of which can be described by essentially one-dimensional formalisms. The constituent pencil beams are most often described by Gaussian, experimentally or theoretically derived distributions. The accuracy of different dose planning algorithms is discussed in some detail based on their ability to take the different physical interaction processes of high energy electrons into account. It is shown that those programs that take the deviations from the simple Gaussian model into account give the best agreement with experimental results. With such programs a dosimetric relative accuracy of about 5% is generally achieved except in the most complex inhomogeneity configurations. Finally, the present limitations and possible future developments of electron dose planning are discussed.

Electron contamination from photon beam collimators.

Photon beam collimators are a source of secondary electrons that contaminate the beams and increase the surface dose. Calculations and measurements of this contamination have been made for 6 MV and 21 MV X-ray beams. Calculations are made using an electron transport model including the production of electrons by the photons and their transport and multiple scatter, both in the collimator and in the air. The variation with collimator material and geometry is investigated. The lowest contamination is obtained for high density and atomic numbers because of the small lateral electron range and large linear stopping power. The collimator geometry is also very important. If the collimator surfaces are hit by the photons, the collimator-produced electrons reaching the phantom may be increased by a factor of 2 or more. Calculations show that the surface dose from these electrons is less than 5% for 21 MV X-rays. This means that the dominating source of contaminating electrons at this energy is the beam flattening filter.

Computer assisted dosimetry of scanned electron and photon beams for radiation therapy.

A computer controlled beam forming system for energies up to 50 MeV has been developed in order to produce high quality electron and photon beams for radiation therapy. The desired radiation field shape and dose distribution are achieved by programming the scanning pattern of a narrow and unfiltered electron or photon beam. The computer that controls the scanning pattern also performs dosimetric analyses in the resultant radiation beams. The system allows real time display of the measured dose distributions at a rate of up to five discrete dose values per second for a 15 cm square field. Measurements in scanned as well as in stationary electron and photon beams at energies of 10, 20 and 50 MeV are presented. Finally, the consequences of photon generated electrons in the very broad high energy photon beams that can be produced by a scanning system are illustrated and discussed.

The clinical value of different treatment objectives and degrees of freedom in radiation therapy optimization.

With inverse radiation therapy planning methods, both biological and physical objective functions can be used to perform the optimization. A biological objective function, namely the probability of achieving tumor control without causing severe complications in normal tissues, P+, has been used to evaluate six different optimization methods. All six methods have been tested on two different clinically relevant treatment geometries. The results show that optimization with a physical objective function which gives the best possible agreement with the desired dose distribution in the least squares sense, may result in severe loss of complication-free tumor control due to insufficient consideration of the organs at risk. It is generally better to use a physical objective function which minimizes the over-dosage when the desired dose distribution can not be exactly reproduced. In all cases the use of physical objective functions results in a lower probability of controlling the tumor without causing severe normal tissue reactions than if the biological objective function, P+, is used. However, the results also show the importance of accurately accounting for beam divergence, dose build-up, beam attenuation, and lateral scatter during the optimization procedure, particularly when the biological objective function is used. The loss in P+ by assuming that all energy deposition kernels are identical and that all the constituent beams have fixed relative weights can be 15% or more. When lateral scatter is not accounted for during the optimization, serious injury to organs at risk may result. This problem is specially severe for organs that are partly or totally encapsulated by the target volume. For superficial target volumes accurate consideration of the dose build-up of the incident pencil beams is fundamental.

Neutron beam characteristics from 50 MeV protons on beryllium using a continuously variable multi-leaf collimator.

The dose distributional properties of a p(50) Be neutron beam using a continuously variable multi-leaf collimator are presented and compared with a 6 MV photon beam. The differences in physical dose delivery between these two radiation modalities are generally insignificant for radiation therapy, and stringent comparisons of neutron and photon treatments should therefore be possible. The flexibility in field shaping with the multi-leaf collimator opens new possibilities in the treatment of complex irregular target volumes. The collimator consists of 40 wedge-shaped leaves that are independently moved under computer control with their collimating surfaces always aligned with the effective radiation source to minimize the penumbra. The leaf collimator eliminates the need for handling of heavy insert collimators and beam blocks at the same time that it allows dynamic conformation therapy with neutrons.

Influence of microdosimetric quantities on observed dose-response relationships in radiation therapy.

The steepness of dose-response curves in radiation therapy depends to a large extent on the statistics of cell killing. This is so if the last few clonogenic tumor cells have to be hit or eradicated by other means to cure the patient. The steepness is dependent on the number of clonogenic cells in the tumor and the possible variation in their sensitivity. However, the uniformity of the dose distribution is also important and a decreased slope may result when the delivery of the dose is nonuniform or statistically uncertain. The variance in the energy imparted at the microdosimetric level to individual cell nuclei constitutes the ultimate limit of the variance in delivered dose at a given mean tumor dose. Considering all dosimetric variances it is shown that for low-LET beams the conventional microdosimetric variance will dominate, while in neutron and high-LET beams in general the microdosimetric variance may contribute significantly to the observed dose-response relationship. As a result the normalized slope of the dose-response curve for tumor control and normal tissue complications with neutrons and other high-LET beams will be reduced compared to that with photons. This conclusion is found to be in quantitative agreement with available data from clinical trials with neutron therapy. Finally, it is pointed out that for beams with a very high RBE and LET it may be favorable to deliver a fraction of the total dose in the form of conventional low-LET radiation. This addition of low-LET radiation may be desirable to ensure a dose to all clonogenic tumor cell nuclei that is sufficiently high and uniform to achieve a high probability of tumor control.

Repairable-conditionally repairable damage model based on dual Poisson processes.

The advent of intensity-modulated radiation therapy makes it increasingly important to model the response accurately when large volumes of normal tissues are irradiated by controlled graded dose distributions aimed at maximizing tumor cure and minimizing normal tissue toxicity. The cell survival model proposed here is very useful and flexible for accurate description of the response of healthy tissues as well as tumors in classical and truly radiobiologically optimized radiation therapy. The repairable-conditionally repairable (RCR) model distinguishes between two different types of damage, namely the potentially repairable, which may also be lethal, i.e. if unrepaired or misrepaired, and the conditionally repairable, which may be repaired or may lead to apoptosis if it has not been repaired correctly. When potentially repairable damage is being repaired, for example by nonhomologous end joining, conditionally repairable damage may require in addition a high-fidelity correction by homologous repair. The induction of both types of damage is assumed to be described by Poisson statistics. The resultant cell survival expression has the unique ability to fit most experimental data well at low doses (the initial hypersensitive range), intermediate doses (on the shoulder of the survival curve), and high doses (on the quasi-exponential region of the survival curve). The complete Poisson expression can be approximated well by a simple bi-exponential cell survival expression, S(D) = e(-aD) + bDe(-cD), where the first term describes the survival of undamaged cells and the last term represents survival after complete repair of sublethal damage. The bi-exponential expression makes it easy to derive D(0), D(q), n and alpha, beta values to facilitate comparison with classical cell survival models.

Mean energy in electron beams.

The mean energy of the energy spectrum is an essential parameter for the dosimetry of therapeutic electron beams. Frequently it is assumed that the mean energy of such beams remains constant across the beam and only its degradation with depth is considered. The present work analyzes the variation of the mean energy of primary electrons with depth and lateral position in an electron beam using the Monte Carlo method. Results are compared with relations commonly employed for determination of mean energy at a depth. For the variation of the mean electron energy with depth in broad beams, good agreement was found between Monte Carlo results and an analytic continuous slowing down expression, which takes the variation of radiation stopping power with depth into account. Due to the calculated lateral variation of the mean energy, the relative absorbed dose profile determined with an air ionization chamber in a clinical beam should differ by less than 1% from the measured ionization profile.

Optimization of the dose delivery in a few field techniques using radiobiological objective functions.

A method for finding optimal primary fluence profiles for multiple field external beam radiation therapy techniques has been developed using a radiobiologically based objective function that quantifies the probability of achieving complication-free tumor control, P+. The objective function P+ has the valuable property of giving the highest possible dose to the tumor without causing severe damage to normal tissues at risk. This radiobiologically based objective function selects suitable dose levels but also takes into account the dose homogeneity in the target volume to the extent that it is not causing an excessive risk of local recurrence or damage to surrounding normal tissues. The biological parameters used can either be patient specific, as determined by a predictive assay on biopsy specimens, or taken from a library of radiobiological parameter values characteristic for different tissue types of a reference patient. In its present form the method can be used to determine the optimum incident photon fluence profiles for each beam. The method has been used to investigate for a given target volume a large number of combinations of beam entry directions to find the best beam orientations with respect to the probability of achieving complication-free tumor control. It is demonstrated that when nonuniform dose delivery is available it is unsuitable to combine parallel opposed beams in two-beam techniques and to a lesser extent also to use perpendicular beams. In two-beam techniques the best angle between the beams is generally in the 100 degrees-120 degrees range. The major symmetry characteristics of the P+ phase space for two-beam techniques are also identified. The method can easily be extended from two to three dimensions and noncoplanar geometry, but it is presented here in its two-dimensional form for clarity and speed of calculation.