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J Van Dyk

Publications and source records attributed to J Van Dyk.

At least 19 recordsLinked to original sources
PubMed2005

Dosimetric impact of image-guided 3D conformal radiation therapy of prostate cancer.

The goal of this work is to quantify the impact of image-guided conformal radiation therapy (CRT) on the dose distribution by correcting patient setup uncertainty and inter-fraction tumour motion. This was a retrospective analysis that used five randomly selected prostate cancer patients that underwent approximately 15 computed tomography (CT) scans during their radiation treatment course. The beam arrangement from the treatment plan was imported into each repeat CT study and the dose distribution was recalculated for the new beam setups. Various setup scenarios were then compared to assess the impact of image guidance on radiation treatment precision. These included (1) daily alignment to skin markers, thus representing a conventional beam setup without image guidance, (2) alignment to bony anatomy for correction of daily patient setup error, thus representing on-line portal image guidance, and (3) alignment to the 'CTV of the day' for correction of inter-fraction tumour motion, thus representing on-line CT or ultrasound image guidance. Treatment scenarios (1) and (3) were repeated with a reduced CTV to PTV margin, where the former represents a treatment using small margins without daily image guidance. Daily realignment of the treatment beams to the prostate showed an average increase in minimum tumour dose of 1.5 Gy, in all cases where tumour 'geographic miss' without image guidance was apparent. However, normal tissue sparing did not improve unless the PTV margin was reduced. Daily realignment to the tumour combined with reducing the margin size by a factor of 2 resulted in an average escalation in tumour dose of 9.0 Gy for all five static plans. However, the prescription dose could be escalated by 13.8 Gy when accounting for changes in anatomy by accumulating daily doses using nonlinear image registration techniques. These results provide quantitative information on the effectiveness of image-guided radiation treatment of prostate cancer and demonstrate that the dosimetric impact is patient dependent.

Artifacts↗
PubMed2005

Validation of contour-driven thin-plate splines for tracking fraction-to-fraction changes in anatomy and radiation therapy dose mapping.

The goal of this study is to validate a deformable model using contour-driven thin-plate splines for application to radiation therapy dose mapping. Our testing includes a virtual spherical phantom as well as real computed tomography (CT) data from ten prostate cancer patients with radio-opaque markers surgically implanted into the prostate and seminal vesicles. In the spherical mathematical phantom, homologous control points generated automatically given input contour data in CT slice geometry were compared to homologous control point placement using analytical geometry as the ground truth. The dose delivered to specific voxels driven by both sets of homologous control points were compared to determine the accuracy of dose tracking via the deformable model. A 3D analytical spherically symmetric dose distribution with a dose gradient of approximately 10% per mm was used for this phantom. This test showed that the uncertainty in calculating the delivered dose to a tissue element depends on slice thickness and the variation in defining homologous landmarks, where dose agreement of 3-4% in high dose gradient regions was achieved. In the patient data, radio-opaque marker positions driven by the thin-plate spline algorithm were compared to the actual marker positions as identified in the CT scans. It is demonstrated that the deformable model is accurate (approximately 2.5 mm) to within the intra-observer contouring variability. This work shows that the algorithm is appropriate for describing changes in pelvic anatomy and for the dose mapping application with dose gradients characteristic of conformal and intensity modulated radiation therapy.

Algorithms↗
PubMed2004

Tracking the dose distribution in radiation therapy by accounting for variable anatomy.

The goal of this research is to calculate the daily and cumulative dose distribution received by the radiotherapy patient while accounting for variable anatomy, by tracking the dose distribution delivered to tissue elements (voxels) that move within the patient. Non-linear image registration techniques (i.e., thin-plate splines) are used along with a conventional treatment planning system to combine the dose distributions computed for each 3D computed tomography (CT) study taken during treatment. For a clinical prostate case, we demonstrate that there are significant localized dose differences due to systematic voxel motion in a single fraction as well as in 15 cumulative fractions. The largest positive dose differences in rectum, bladder and seminal vesicles were 29%, 2% and 24%, respectively, after the first fraction of radiation treatment compared to the planned dose. After 15 cumulative fractions, the largest positive dose differences in rectum, bladder and seminal vesicles were 23%, 32% and 18%, respectively, compared to the planned dose. A sensitivity analysis of control point placement is also presented. This method provides an important understanding of actual delivered doses and has the potential to provide quantitative information to use as a guide for adaptive radiation treatments.

Humans↗
PubMed2003

The influence of brachytherapy dose heterogeneity on estimates of alpha/beta for prostate cancer.

The sensitivity of estimates of alpha/beta for prostate tumours to dose heterogeneity in 125I brachytherapy implants, as well as to variation in selected radiobiological parameters, is analysed. The tumour control probabilities of brachytherapy and external beam radiotherapy are equated for ranges of alpha, Tpot, RBE and external beam dose. For each combination of parameters, the equality is used to derive the value of alpha/beta. Different clinical (non-uniform) brachytherapy dose distributions, and three uniform brachytherapy dose distributions (120, 144 and 160 Gy) are used. For 'nominal' input parameter values of Tpot = 45 days, alpha = 0.2 Gy(-1), RBE = 1.4, and an external beam dose of 70 Gy, the values obtained for alpha/beta ranged between 2.1 and 12.3 Gy for all of the clinical DVHs, between 2.1 and 3.8 Gy for the better quality clinical implants and between 1.0 and 1.8 Gy for the uniform brachytherapy doses. When only 2% of the volume receiving the lowest dose is omitted from the clinical DVHs, the estimated alpha/beta values ranged between 1.4 and 2.1 Gy. When ranges of input parameters were also considered, the overall range of alpha/beta values for the clinical brachytherapy dose distributions lay between 1.1 and 12.3 Gy for the three best clinical implants, and between 0.7 and 6.3 Gy for uniform doses. We conclude that estimation of alpha/beta without taking into account dose heterogeneity and inter-patient variation may underestimate the actual value alpha/beta.

Brachytherapy↗
PubMed2001

Considerations for the implementation of target volume protocols in radiation therapy.

PURPOSE: Uncertainties in patient repositioning and organ motion are accounted for by defining a planning target volume (PTV). We make recommendations on issues not explicitly discussed in existing protocols for PTV design. METHODS: A quantity called "coverage" is defined to quantify how effectively a PTV encompasses the clinical target volume, and is applied to examine the impact of several factors. A stochastic simulation is used to determine the coverage required for a desirable balance between tumor control probability (TCP) and the irradiated volume. Using a sample anatomy, we assess the importance of the method used to add uncertainties, the shape of the uncertainty distribution, the effect of systematic uncertainties, and the use of nonuniform margins. Additionally, we examine the benefit of patient immobilization techniques. RESULTS: Our example indicates that 95% coverage is a reasonable goal for treatment planning. Using this as a comparison value, our example indicates quadrature addition of uncertainties predicts smaller margins (7 mm) than linear addition (11 mm), Gaussian distribution of uncertainties (7 mm) require the same margin as a uniform distribution (7 mm), systematic uncertainties have a small effect on TCP below a threshold value (4 mm), and nonuniform margins allow only a slight reduction of irradiated volume. CONCLUSION: We recommend that uncertainties should generally be added in quadrature, the exact shape of the uncertainty distribution is not critical, systematic uncertainties should be maintained below some threshold value, and nonuniform margins may be effective when uncertainties are anisotropic.

Clinical Protocols↗
PubMed2001

The effect of seed anisotrophy on brachytherapy dose distributions using 125I and 103Pd.

We have evaluated the effect of the anisotropy of individual seeds on dose distributions for permanent prostate implants using 125I and 103Pd. The dose distributions were calculated for various implants using both the line source and point source calculational formalisms, for two different models of 125I and 103Pd seeds. The dose distributions were compared using cumulative dose volume histograms (DVH) and cumulative difference dose volume histograms (deltaDVH) for the prostate target volume and for the rectum surface. The DVHs could not distinguish between the dose distributions from isotropic and non-isotropic seeds. However, the deltaDVHs were useful in determining the fraction of the target volume for which the difference between the dose distribution for line sources and for point sources exceeded a threshold value. The dose distributions were calculated (1) for all the seeds oriented co-linearly, along either the x-, y-, or z-axis, and (2) for the seeds at randomized orientations, more closely resembling the clinical situation. For all cases, there was a significant difference in the effect of seed anisotropy from the different seed types. For the geometrically simpler test cases with a small number of seeds, the effect of anisotropy on the dose distribution was too large to ignore for any of the seed types investigated. For the idealized pre-plan case, the effect was much smaller. For clinical prostate implants, the calculations done with seeds oriented co-linearly along the z-axis (needle implant axis) were a reasonable approximation for those from simulations of seeds with randomized orientations. Again, the effect of anisotropy varied drastically between different seed models, and also between different clinical cases. However, the effect of anisotropy must be considered in the context of all the other uncertainties in clinical brachytherapy treatments.

Anisotropy↗
PubMed2000

In-field and out-of-field effects in partial volume lung irradiation in rodents: possible correlation between early dna damage and functional endpoints.

PURPOSE: Recent observations have shown that there are regional variations in radiation response in mouse lung as measured by functional assays. Furthermore, there are both in-field and out-of-field effects in radiation-induced lung damage as observed by DNA assay in rats. The purpose of this work is: (a) to examine mice lethality data following partial volume lung irradiation to assess the possibility of directional or regional effects, (b) to evaluate the correlation between mice lethality data and DNA damage assayed by micronuclei production in rat lung, and (c) to re-interpret mice lethality considering the existence of directional effects in lung cellular response to partial volume irradiation. METHODS AND MATERIALS: The lethality data for mice, generated at the M. D. Anderson Cancer Center, Houston, and micronuclei yield data for rats obtained at Princess Margaret Hospital, Toronto, were used. A radiobiological model that allows for out-of-field and in-field effects for lung cell damage and lung response was developed. This model is based on the observation of DNA damage in shielded parts of rat lung that was assumed relevant to cell lethality and consequently overall lung response. RESULTS: While the experimental data indicated directional or regional volume effects, the applicability of dose and volume as sole predictors of lung response to radiation was found to be unreliable for lower lung (base) irradiation in mice. This conforms well to rat lung response where micronuclei were observed in shielded apical parts of lung following base irradiation. The radiobiological model, which was specifically developed to account for the lung response outside of primary irradiated volume, provides a good fit to mice lethality data, using parameters inferred from rat micronuclei data. CONCLUSION: Response to lung irradiation in rodents, in particular, elevated sensitivity to base irradiation, can be interpreted with a hypothesis of in-field and out-of-field effects for cellular response. If the existence of these effects for lung is subsequently proven in humans, it will require the incorporation of geometrical and directional information in normal tissue complication probability calculations for lung. These considerations are ignored in present approaches based only on conventional dose-volume histograms.

Animals↗
PubMed2000

Radiation treatment for cervical esophagus: patterns of practice study in Canada, 1996.

PURPOSE: To assess the patterns of practice among Canadian radiation oncologists who treat esophageal cancers, using a trans-Canada survey, completed at the end of 1996. METHODS AND MATERIALS: One of 3 case presentations of different stages of cervical esophageal cancer was randomly assigned and sent to participating radiation oncologists by mail. Respondents were asked to fill in questionnaires regarding treatment techniques and to outline target volumes for the boost phase of radiotherapy. Radiation oncologists from 26 of 27 (96%) of all Canadian centers participated. RESULTS: High-energy X-rays (>/= 10 MV) were employed by 68% of the respondents in part of the treatment course. The majority (83%) of the radiation oncologists used at least two phases of treatment. Very few, 10 of 59 (17%), responses started with multifield treatment. The most frequently used prescription dose was 60 Gy/30 fractions/6 weeks, given with concurrent chemotherapy. Dose prescriptions were to the isocenter in 39 of 48 (81%) or to a particular isodose line in 9 of 48 (19%) of respondents. CONCLUSION: There was a variety of radiation treatment techniques in this trans-Canada survey. The majority of the patients had combined cisplatin-based chemoradiation. The isocenter was not used consistently as a dose prescription point.

Aged↗
PubMed2000

Normal tissue complication probabilities: dependence on choice of biological model and dose-volume histogram reduction scheme.

PURPOSE: To evaluate the impact of dose-volume histogram (DVH) reduction schemes and models of normal tissue complication probability (NTCP) on ranking of radiation treatment plans. METHODS AND MATERIALS: Data for liver complications in humans and for spinal cord in rats were used to derive input parameters of four different NTCP models. DVH reduction was performed using two schemes: "effective volume" and "preferred Lyman". DVHs for competing treatment plans were derived from a sample DVH by varying dose uniformity in a high dose region so that the obtained cumulative DVHs intersected. Treatment plans were ranked according to the calculated NTCP values. RESULTS: Whenever the preferred Lyman scheme was used to reduce the DVH, competing plans were indistinguishable as long as the mean dose was constant. The effective volume DVH reduction scheme did allow us to distinguish between these competing treatment plans. However, plan ranking depended on the radiobiological model used and its input parameters. CONCLUSIONS: Dose escalation will be a significant part of radiation treatment planning using new technologies, such as 3-D conformal radiotherapy and tomotherapy. Such dose escalation will depend on how the dose distributions in organs at risk are interpreted in terms of expected complication probabilities. The present study indicates considerable variability in predicted NTCP values because of the methods used for DVH reduction and radiobiological models and their input parameters. Animal studies and collection of standardized clinical data are needed to ascertain the effects of non-uniform dose distributions and to test the validity of the models currently in use.

Animals↗
PubMed2000

Polyvinyl alcohol-Fricke hydrogel and cryogel: two new gel dosimetry systems with low Fe3+ diffusion.

Two new Fricke dosimeter gel systems with low diffusion rates have been developed for 3D radiation dosimetry purposes. Both systems consist of a solution of 20% (by weight) polyvinyl alcohol (PVA) in a 50 mM H2SO4 solution with 0.4 mM ferrous ammonium sulphate and xylenol orange (FX). The difference in the two gels is the way that the gelation process was initiated: either by bringing the temperature to (a) +5 degrees C or (b) -20 degrees C before returning them to room temperature. These gels are termed 'hydrogel' and 'cryogel', respectively. The hydrogel is optically transparent, and can be used with either optical or MRI detection methods for dosimetric imaging. The cryogel is rubbery in texture but opaque, so its internal Fe3+ concentration can only be measured with MRI. The hydrogel's optical attenuation coefficient is linear (r2 = 0.99) with dose from 0 to 20 Gy with a sensitivity of 0.106 cm(-1) Gy(-1) (at 543 nm). In terms of MR relaxation rate, the dose response for both the hydrogel and cryogel was linear (r2 = 0.99) with a sensitivity of 0.020 s(-1) Gy(-1) (at 1.5 T). The Fe3+ diffusion coefficient (at 20 degrees C) was measured to be 0.14 mm2 h(-1), which is significantly lower than similar preparations reported for porcine gelatin or agarose. The PVA-FX gels can be stored for long periods of time before exposure to radiation, since the auto-oxidation rate was 10 times less than that of gelatin-Fricke recipes. The new gels developed in this work are a significant improvement on previous Fricke gel systems.

Calibration↗
PubMed1999

A quality assurance phantom for three-dimensional radiation treatment planning.

PURPOSE: Three-dimensional (3D) radiation treatment planning is facilitated through the use of computerized radiation treatment planning systems (RTPSs) and CT simulators (CT-sims). Quality assurance (QA) of these systems is necessary for ensuring that they fulfill their potential. However, comprehensive tools for the systematic QA of these systems have not been developed. We present a phantom that facilitates the evaluation of a large number of nondosimetric functions. These include CT image acquisition and transfer, graphical displays of 3D radiation beams, multiplanar CT image reconstructions, digitally reconstructed radiographs, the representation and manipulation of contoured patient anatomy, dose volume histograms, and the conversion of CT numbers to relative electron densities. METHODS AND MATERIALS: A phantom was constructed which contains materials and geometries that are appropriate for the routine QA of the features described above. The anatomy of the phantom is used as a standard against which the performance of the 3D-RTPS or CT-sim is evaluated. The phantom was used to evaluate three different 3D-RTPSs and a CT-sim at four institutions. RESULTS: Using this phantom, clinically significant errors and limitations in commercially available 3D treatment planning software were discovered. No errors were discovered in the beam display or image reconstructions in the systems examined. Problems were found in the anatomy display, automatic tools, and the CT number to relative electron density conversion data used in some of the systems. CONCLUSION: This phantom is a unique tool designed explicitly for the QA of 3D treatment planning software. Errors and limitations discovered through its use indicate that the QA of commercial treatment planning software is necessary, and that this phantom is an effective device for this task.

Image Processing, Computer-Assisted↗
PubMed1998

Variability of target volume delineation in cervical esophageal cancer.

PURPOSE: Three-dimensional (3D) conformal radiation therapy (CRT) assumes and requires the precise delineation of the target volume. To assess the consistency of target volume delineation by radiation oncologists, who treat esophageal cancers, we have performed a transCanada survey. MATERIALS AND METHODS: One of three case presentations, including CT scan images, of different stages of cervical esophageal cancer was randomly chosen and sent by mail. Respondents were asked to fill in questionnaires regarding treatment techniques and to outline boost target volumes for the primary tumor on CT scans, using ICRU-50 definitions. RESULTS: Of 58 radiation oncologists who agreed to participate, 48 (83%) responded. The external beam techniques used were mostly anterior-posterior fields, followed by a multifield boost technique. Brachytherapy was employed by 21% of the oncologists, and concurrent chemotherapy by 88%. For a given case, and the three volumes defined by ICRU-50 (i.e., gross tumor volume [GTV], clinical target volume [CTV], and planning target volume [PTV]) we determined: 1. The total length in the cranio-caudal dimension; 2. the mean diameter in the transverse slice that was located in a CT slice that was common to all participants; 3. the total volume for each ICRU volume; and 4. the (5, 95) percentiles for each parameter. The PTV showed a mean length of 14.4 (9.6, 18.0) cm for Case A, 9.4 (5.0, 15.0) cm for Case B, 11.8 (6.0, 16.0) cm for Case C, a mean diameter of 6.4 (5.0, 9.4) cm for Case A, 4.4 (0.0, 7.3) cm for Case B, 5.2 (3.9, 7.3) cm for Case C, and a mean volume of 320 (167, 840) cm3 for Case A and 176 (60, 362) cm3 for Case C. The results indicate variability factors (95 percentile divided by 5 percentile values) in target diameters of 1.5 to 2.6, and in target lengths of 1.9 to 5.0. CONCLUSION: There was a substantial inconsistency in defining the planning target volume, both transversely and longitudinally, among radiation oncologists. The potential benefits of 3D treatment planning with high-precision dose delivery could be offset by this inconsistency in target-volume delineation by radiation oncologists. This may be particularly important for multicenter clinical trials, for which quality assurance of this step will be essential to the interpretation of results.

Aged↗
PubMed1998

Partial volume rat lung irradiation: an evaluation of early DNA damage.

PURPOSE: 1. To investigate early DNA damage induced in rat lung cells following single-dose, partial-volume irradiation (lung base and lung apex). 2. To determine the variation in DNA damage in different lung regions. 3. To investigate the possible mechanisms associated with early DNA damage after lung irradiation. METHODS AND MATERIALS: The whole lung or the upper or lower half of the entire lung of Sprague-Dawley rats was exposed to 10 Gy 60Co gamma rays. The animals were sacrificed at various times up to 42 h after irradiation. A trypsin-digested lung cell suspension was prepared and cells that attached to slides in the initial 24-h period were then grown in the presence of culture medium with cytochalasin-B for a further 72 h. Radiation-induced DNA damage was quantified in the cells (primarily fibroblasts) from both irradiated and unirradiated lung regions by using a well-characterized micronucleus assay. RESULTS: When the lungs were removed at 16-18 h after whole-lung irradiation, about 0.85 micronuclei (MN) per binucleate cell (BNC) were observed in the lung cells of the irradiated animals, compared to 0.02 MN/BNC in the lung cells of the controls. When only the lung base was irradiated, the frequency of micronuclei was 0.85 MN/BNC compared to 0.43 MN/BNC observed in cells from the irradiated lung apex. Of particular interest was the finding that the unirradiated lung apex also showed a large frequency of micronuclei (0.43 MN/BNC) after the irradiation of the lung base, whereas the unirradiated lung base showed only a marginal (approximately 2-fold) increase relative to the spontaneous frequency following irradiation of the lung apex. The changes in the frequency of micronuclei varied with the time at which the lungs were removed from the rats for early times, but had stabilized by 18 h after irradiation. Normal (unirradiated) cells grown in filtered or unfiltered conditioned media obtained from irradiated cell cultures showed an insignificant marginal increase in the number of micronuclei relative to the spontaneous frequency. Lung cells obtained from the lung base or the lung apex of healthy controls and irradiated separately in vitro showed no regional differences in the induction of micronuclei. Cells from the lungs of rats injected with superoxide dismutase, within 1 h prior to irradiation of the lung base, and processed 16-18 h after irradiation showed a reduction in the number of MN in the shielded lung apex, indicating the possible involvement of oxygen radicals. CONCLUSIONS: These data indicate that cells in the lung base sustain more DNA damage than those in the lung apex when either region is irradiated; however, when the whole lung, is irradiated, the lung damage observed is similar in the two regions. Also, out-of-field effects are observed for the lung apex but not the lung base. Possible mechanisms include a clastogenic (chromosome damaging) factor produced in the plasma following irradiation and/or the production of oxygen radicals by activated lymphocytes/monocytes. The partial blocking of the DNA damage, observed in the unirradiated lung apex following irradiation of the lung base, by superoxide dismutase, suggests that oxygen radicals are involved in this out-of-field effect. These radicals are likely produced as a result of the induction of inflammatory cytokines, such as tumor necrosis factor (TNF) and interleukin-1 (IL-1) by the irradiation. The reason for the lack of an out-of-field effect in the lung base when the lung apex is irradiated is unknown, but may be due to the greater volume of lung irradiated in the lower lung field, because this is likely to affect the level of cytokines produced. Alternatively, it may reflect cytokines produced as a result of the partial liver irradiation that occurs with the lower lung field.

Animals↗
PubMed1997

Refinements of the finite-size pencil beam model of three-dimensional photon dose calculation.

Modern three-dimensional (3-D) photon dose calculation algorithms need to be fast and accurate if they are to be practical for treatment optimization. Refinements to a previously proposed finite-size pencil beam (FSPB) method are presented in order to fulfill these needs. Specifically, a fast Fourier transform (FFT) convolution technique is used to speed calculation of the FSPB; the fluence spectrum is modeled, and the effects of finite source size, a Gaussian x-ray source intensity profile and partial transmission through a multileaf collimator (MLC) leaf are approximated. The use of FFT techniques in the calculation of small diverging fields involves approximations that are investigated for a 6 MV beam and shown to introduce errors that vary with energy but do not exceed 0.7% on the central axis. Dose distributions calculated by FSPB superposition are in excellent agreement with those calculated by full field FFT convolution. Two key advances over the original implementation of the FSPB model are demonstrated: the fast calculation of the FSPB facilitates development, and the incorporation of realistic beam parameters enables accurate modeling of clinical beams.

Algorithms↗
PubMed1997

Lateral electron transport in FFT photon dose calculations.

Previously published FFT dose calculations and their verifications have been performed for field sizes greater than 10 cm x 10 cm with photon energies up to 18 MV [Zhu and Boyer, Phys. Med. Biol. 35, 351-368 (1990)]. With a smaller field size at megavoltage energies, lateral electronic disequilibrium exists in low density inhomogeneities which is not handled properly by the FFT calculations. In this paper, we extend FFT dose calculations to address this problem without significantly increasing the computational time. We employ lateral scaling of the field size at each depth along with an inverse scaling of the resultant dose distribution. The concept of effective density in introduced for the lateral and inverse scaling. We demonstrate that electron transport can be adequately captured in the presence of inhomogeneities, thus improving 3D FFT dose calculations in inhomogeneous media.

Calibration↗
PubMed1996

Theoretical developments on fast Fourier transform convolution dose calculations in inhomogeneous media.

A theory is presented on dose calculations in inhomogeneous media that takes advantage of fast Fourier transform (FFT) convolution for practical three-dimensional treatment planning using photon beams. While the initial work of Boyer and Mok [Med. Phys. 13, 503-509 (1986)] provided a theory which is based on first principles, it failed to give satisfactory predictions inside inhomogeneities. Subsequently, Zhu and Boyer [Phys. Med. Biol. 35, 351-368 (1990)] showed that their formulas agreed well with measured data, but these formulas were empirically altered from Boyer and Mok's. In this work, Boyer and Mok's first-order theory is extended to include second-order inhomogeneity effects. A new correction dose formula is derived which corrects the first scattered dose due to the presence of inhomogeneities. This correction dose formula works better than Zhu and Boyer's empirical correction dose formula. Furthermore, the primary dose formula used by Zhu and Boyer, which was empirically modified from Boyer and Mok's, is justified theoretically. Clear statements are made about the assumptions and the approximations that enter into the derivation which in turn uncover the limitations of this FFT convolution dose calculation.

Biophysical Phenomena↗