Free-breathing whole-heart coronary MRA with 3D radial SSFP and self-navigated image reconstruction.

Respiratory motion is a major source of artifacts in cardiac magnetic resonance imaging (MRI). Free-breathing techniques with pencil-beam navigators efficiently suppress respiratory motion and minimize the need for patient cooperation. However, the correlation between the measured navigator position and the actual position of the heart may be adversely affected by hysteretic effects, navigator position, and temporal delays between the navigators and the image acquisition. In addition, irregular breathing patterns during navigator-gated scanning may result in low scan efficiency and prolonged scan time. The purpose of this study was to develop and implement a self-navigated, free-breathing, whole-heart 3D coronary MRI technique that would overcome these shortcomings and improve the ease-of-use of coronary MRI. A signal synchronous with respiration was extracted directly from the echoes acquired for imaging, and the motion information was used for retrospective, rigid-body, through-plane motion correction. The images obtained from the self-navigated reconstruction were compared with the results from conventional, prospective, pencil-beam navigator tracking. Image quality was improved in phantom studies using self-navigation, while equivalent results were obtained with both techniques in preliminary in vivo studies.

Radiothérapie « flash » à très haut débit de dose : un moyen d'augmenter l'indice thérapeutique par minimisation des dommages aux tissus sains [Ultrahigh dose-rate, "flash" irradiation minimizes the side-effects of radiotherapy].

PURPOSE: Pencil beam scanning and filter free techniques may involve dose-rates considerably higher than those used in conventional external-beam radiotherapy. Our purpose was to investigate normal tissue and tumour responses in vivo to short pulses of radiation. MATERIAL AND METHODS: C57BL/6J mice were exposed to bilateral thorax irradiation using pulsed (at least 40Gy/s, flash) or conventional dose-rate irradiation (0.03Gy/s or less) in single dose. Immunohistochemical and histological methods were used to compare early radio-induced apoptosis and the development of lung fibrosis in the two situations. The response of two human (HBCx-12A, HEp-2) tumour xenografts in nude mice and one syngeneic, orthotopic lung carcinoma in C57BL/6J mice (TC-1 Luc+), was monitored in both radiation modes. RESULTS: A 17Gy conventional irradiation induced pulmonary fibrosis and activation of the TGF-beta cascade in 100% of the animals 24-36 weeks post-treatment, as expected, whereas no animal developed complications below 23Gy flash irradiation, and a 30Gy flash irradiation was required to induce the same extent of fibrosis as 17Gy conventional irradiation. Cutaneous lesions were also reduced in severity. Flash irradiation protected vascular and bronchial smooth muscle cells as well as epithelial cells of bronchi against acute apoptosis as shown by analysis of caspase-3 activation and TUNEL staining. In contrast, the antitumour effectiveness of flash irradiation was maintained and not different from that of conventional irradiation. CONCLUSION: Flash irradiation shifted by a large factor the threshold dose required to initiate lung fibrosis without loss of the antitumour efficiency, suggesting that the method might be used to advantage to minimize the complications of radiotherapy.

Combining T2-preparation with spectral and spatial selectivity for improved coronary MRA

Cardiovascular disease is the leading cause of death worldwide. Within this subset, coronary artery disease (CAD) is the most prevalent. Magnetic resonance angiography (MRA) is an emerging technique that provides a safe, non-invasive way of assessing CAD progression. To generate contrast between tissues, MR images are weighted according to the magnetic properties of those tissues. In cardiac MRI, T2 contrast, which is governed by the rate of transverse signal loss, is often created through the use of a T2-Preparation module. T2-Preparation, or T2-Prep, is a magnetization preparation scheme used to improve blood/myocardium contrast in cardiac MRI. T2-Prep methods generally use a non-selective +90°, 180°, 180°, -90° train of radiofrequency (RF) pulses (or variant thereof), to tip magnetization into the transverse plane, allow it to evolve, and then to restore it to the longitudinal plane. A key feature in this process is the combination of a +90° and -90° RF pulse. By changing either one of these, a mismatch occurs between signal excitation and restoration. This feature can be exploited to provide additional spectral or spatial selectivity. In this work, both of these possibilities are explored. The first - spectral selectivity - has been examined as a method of improving fat saturation in coronary MRA. The second - spatial selectivity - has been examined as a means of reducing imaging time by decreasing the field of view, and as a method of reducing artefacts originating from the tissues surrounding the heart. Two additional applications, parallel imaging and self-navigation, are also presented. This thesis is thus composed of four sections. The first, "A Fat Signal Suppression for Coronary MRA at 3T using a Water-Selective Adiabatic T2-Preparation Technique", was originally published in the journal Magnetic Resonance in Medicine (MRM) with co-authors Ruud B. van Heeswijk and Matthias Stuber. The second, "Combined T2-Preparation and 2D Pencil Beam Inner Volume Selection", again with co-authors Ruud van Heeswijk and Matthias Stuber, was also published in the journal MRM. The third, "A cylindrical, inner volume selecting 2D-T2-Prep improves GRAPPA-accelerated image quality in MRA of the right coronary artery", written with co-authors Jerome Yerly and Matthias Stuber, has been submitted to the "Journal of Cardiovascular Magnetic Resonance", and the fourth, "Combined respiratory self-navigation and 'pencil-beam' 2D-T2 -Prep for free-breathing, whole-heart coronary MRA", with co¬authors Jerome Chaptinel, Giulia Ginami, Gabriele Bonanno, Simone Coppo, Ruud van Heeswijk, Davide Piccini, and Matthias Stuber, is undergoing internal review prior to submission to the journal MRM. -- Les maladies cardiovasculaires sont la cause principale de décès dans le monde : parmi celles-ci, les maladies coronariennes sont les plus répandues. L'angiographie par résonance magnétique (ARM) est une technique émergente qui fournit une manière sûre, non invasive d'évaluer la progression de la coronaropathie. Pour obtenir un contraste entre les tissus, les images d'IRM sont pondérées en fonction des propriétés magnétiques de ces tissus. En IRM cardiaque, le contraste en T2, qui est lié à la décroissance du signal transversal, est souvent créé grâce à l'utilisàtion d'un module de préparation T2. La préparation T2, ou T2-Prep, est un système de préparation de l'aimantation qui est utilisé pour améliorer le contraste entre le sang et le myocarde lors d'une IRM cardiaque. Les méthodes de T2-Prep utilisent généralement une série non-sélective d'impulsions de radiofréquence (RF), typiquement [+ 90°, 180°, 180°, -90°] ou une variante, qui bascule l'aimantation dans le plan transversal, lui permet d'évoluer, puis la restaure dans le plan longitudinal. Un élément clé de ce processus est la combinaison des impulsions RF de +90° et -90°. En changeant l'une ou l'autre des impulsions, un décalage se produit entre l'excitation du signal et de la restauration. Cette fonction peut être exploitée pour fournir une sélectivité spectrale ou spatiale. Dans cette thèse, les deux possibilités sont explorées. La première - la sélectivité spectrale - a été examinée comme une méthode d'améliorer la saturation de la graisse dans l'IRM coronarienne. La deuxième - la sélectivité spatiale - a été étudiée comme un moyen de réduire le temps d'imagerie en diminuant le champ de vue, et comme une méthode de réduction des artefacts provenant des tissus entourant le coeur. Deux applications supplémentaires, l'imagerie parallèle et la self-navigation, sont également présentées. Cette thèse est ainsi composée de quatre sections. La première, "A Fat Signal Suppression for Coronary MRA at 3T using a Water-Selective Adiabatic T2-Preparation Technique", a été publiée dans la revue médicale Magnetic Resonance .in Medicine (MRM) avec les co-auteurs Ruud B. van Heeswijk et Matthias Stuber. La deuxième, Combined T2-Preparation and 2D Pencil Beam Inner Volume Selection", encore une fois avec les co-auteurs Ruud van Heeswijk et Matthias Stuber, a également été publiée dans le journal MRM. La troisième, "A cylindrical, inner volume selecting 2D-T2-Prep improves GRAPPA- accelerated image quality in MRA of the right coronary artery", écrite avec les co-auteurs Jérôme Yerly et Matthias Stuber, a été présentée au "Journal of Cardiovascular Magnetic Resonance", et la quatrième, "Combined respiratory self-navigation and 'pencil-beam' 2D-T2 -Prep for free-breathing, whole-heart coronary MRA", avec les co-auteurs Jérôme Chaptinel, Giulia Ginami, Gabriele Bonanno , Simone Coppo, Ruud van Heeswijk, Davide Piccini, et Matthias Stuber, subit un examen interne avant la soumission à la revue MRM.

A novel radar-absorbing-material based on EBG structure

An innovative phaseshifterless, wideband, micrustrip leaky-wave antenna with an electronically steerable dual-pencil-beam pattern in the H-plane is presented. The log-periodic geometry of the leaky slots of the antenna results in a wide bandwidth of 25.19%. The Jan beam can he steered up to 14° over the wide resonating band of the anteww. The beam is also steerable at a fixed frequency. by reactivelty loading the slots and a maximum steering angle of about 14° is ohserved. for different capacitor values with an improved bandwidth of 33 _i%. This concept is studied using passive components but it can be extended to varactors

A study on orthogonality sampling.

The goal of this paper is to study and further develop the orthogonality sampling or stationary waves algorithm for the detection of the location and shape of objects from the far field pattern of scattered waves in electromagnetics or acoustics. Orthogonality sampling can be seen as a special beam forming algorithm with some links to the point source method and to the linear sampling method. The basic idea of orthogonality sampling is to sample the space under consideration by calculating scalar products of the measured far field pattern , with a test function for all y in a subset Q of the space , m = 2, 3. The way in which this is carried out is important to extract the information which the scattered fields contain. The theoretical foundation of orthogonality sampling is only partly resolved, and the goal of this work is to initiate further research by numerical demonstration of the high potential of the approach. We implement the method for a two-dimensional setting for the Helmholtz equation, which represents electromagnetic scattering when the setup is independent of the third coordinate. We show reconstructions of the location and shape of objects from measurements of the scattered field for one or several directions of incidence and one or many frequencies or wave numbers, respectively. In particular, we visualize the indicator function both with the Dirichlet and Neumann boundary condition and for complicated inhomogeneous media.

Flow in the high latitude ionosphere: measurements at 15s resolution made using the EISCAT ‘Polar’ experiment

We present a first overview of flows in the high latitude ionosphere observed at 15 s resolution using the U.K.-Polar EISCAT experiment. Data are described from experiments conducted on two days, 27 October 1984 and 29 August 1985, which together span the local times between about 0200 and 2130MLT and cover five different regions of ionospheric flow. With increasing local time, these are: the dawn auroral zone flow cell, the dayside region of low background flows equatorward of the flow cells, the dusk auroral zone flow cell, the boundary region between the dusk auroral zone and the polar cap, and the evening polar cap. Flows in both the equatorward and poleward portions of the auroral zone cells appear to be relatively smooth, while in the central region of high speed flow considerable variations are generally present. These have the form of irregular fluctuations on a wide range of time scales in the early morning dawn cell, and impulsive wave-like variations with periods of a few minutes in the afternoon dusk cell. In the dayside region between the flow cells, the ionosphere is often essentially stagnant for long intervals, but low amplitude ULF waves with a period of about 5 min can also occur and persist for many cycles. These conditions are punctuated at one to two hour intervals by sudden ‘flow burst’ events with impulsively generated damped wave trains. Initial burst flows are generally directed poleward and can peak at line-of-sight speeds in excess of 1 km s^{−1} after perhaps 45 s. Flows in the polar cap are reasonably smooth on time scales of a few minutes and show no evidence for the presence of ULF waves. Under most, but not all, of the above conditions, the beam-swinging algorithm used to determine background vector flows should produce meaningful results. Comparison of these flow data with simultaneous plasma and magnetic field measurements in the solar wind, made by the AMPTE IRM and UKS spacecraft, emphasizes the strong control exerted on high latitude flows by the north-south component of the IMF.

Avaliação da percentagem de massa gorda numa população pediátrica: comparação entre os métodos DXA e antropometria

Objectivo: Proceder à validação da equação de Slaugther e col., (1988), na estimação da percentagem de massa gorda (%MG), em crianças com 9 anos de idade, tendo a DXA como método de referência. Metodologia: A avaliação da composição corporal foi realizada em 450 crianças, das quais 219 eram raparigas (idade: 9.74 ± 0.33 anos; altura: 136.88 ± 6.8 cm; peso: 33.77 ± 8.16 kg; índice de massa corporal (IMC): 17.85 ± 3.16 kg/m2) e 231 eram rapazes (idade: 9.75 ± 0.33 anos; altura: 137.17 ± 6.97cm; peso: 34.3 ± 8.09 kg; IMC: 18.09 ± 3.17 kg/m2), pela DXA (QDR – 1500: Hologic, Waltham, MA, pencil beam mode, software version 5.67 anhanced whole body analisis) e pelas pregas adiposas subcutâneas, cujo os valores das pregas adiposas tricipital e geminal foram utilizados na equação desenvolvida por Slaugther e col., (1988). Na análise estatística, foram utilizadas a comparação de médias, a regressão linear e a concordância entre os métodos. Resultados: A %MG obtida por ambos os métodos apresentou diferenças significativas (p<0.05) entre os géneros, sendo as raparigas as que apresentam, em média, maiores valores de gordura corporal. Tanto para os rapazes como para as raparigas a %MGDXA é superior à %MGSKF. Na predição dos valores de %MG, a equação de Slaugther e col., (1988) tem para ambos os sexos (raparigas: r=0.94; EPE=3.2 e rapazes: r=0.96; EPE=2.7) uma correlação elevada com reduzido erro padrão de estimação com a DXA. Na análise de concordância entre os métodos, os rapazes parecem apresentar uma maior concordância entre o método alternativo e o método de referência, com os limites de concordância a variarem entre -9.26 e 1.57, enquanto nas raparigas variam entre -11.19 e 3.16. Em ambos os sexos é visível a subestimação, em valor médio, do método de referência, com uma média da diferença a situar-se nos -3.8% e -4.0%, respectivamente para rapazes e raparigas. Discussão: A equação de Slaugther e col., (1988) explica, para as raparigas, 87.5% da variância do método de referência, enquanto nos rapazes, 91.3% da variância é explicada.A diferença entre os métodos alternativo e de referência está dependente do nível de adiposidade, sendo que o método alternativo tende a sobrestimar a %MG nas crianças mais magras e a subestimar nas mais gordas. A equação de Slaugther e col., (1988) apresentou uma validade aceitável na avaliação da composição corporal num grupo de crianças, constituindo uma alternativa rápida na estimação da composição corporal numa avaliação inicial em escolas, clubes e estudos de larga escala. Contudo a sua utilidade, para uma correcta intervenção na saúde da criança, pode apresentar uma validade limitada, pelo que pode ser justificada a utilização de um método mais válido em termos clínicos, como a DXA.

Análise da viabilidade de simulações com feixes de prótons de 100, 150, 200 e 250 MeV em alvos heterogêneos

In the recent years, the use of proton beams in radiotherapy has been an outstanding progress (SMITH, 2006). Up to now, computed tomography (CT) is a prerequisite for treatment planning in this kind of therapy because it provides the electron density distribution required for calculation of dose and the interval of doses. However, the use of CT images for proton treatment planning ignores fundamental differences in physical interaction processes between photons and protons and is, therefore, potentially inaccurate (SADROZINSKI, 2004). Proton CT (pCT) can in principle directly measure the density distribution needed in a patient for the dose distribution (SCHULTE, et al, 2004). One important problem that should be solved is the implementation of image reconstruction algorithms. In this sense, it is necessary to know how the presence of materials with different density and composition interfere in the energy deposition by ionization and coulomb excitation, during its trajectory. The study was conducted in two stages, was used in both the program SRIM (The Stopping and Range of Ions in Matter) to perform simulations of the interaction of proton beams with pencil beam type. In the first step we used the energies in the range of 100-250 MeV (ZIEGLER, 1999). The targets were set to 50 mm in length for the beam of 100 MeV, due to its interaction with the target, and short-range, and 70 mm for 150, 200 and 250 MeV The target was composed of liquid water and a layer of 6 mm cortical bone (ICRP). It were made 9 simulations varying the position of the heterogeneity of 5 mm. In the second step the energy of 250 MeV was taken out from the simulations, due to its greater energy and less interaction. The targets were diminished to 50 mm thick to standardize the simulations. The layer of bone was divided into two equal parts and both were put in the ends of the target... (Complete abstract click electronic access below)

Estudo teórico da contribuição a dose de partículas secundárias geradas por prótons de energias entre 100-200 MeV num phantom de osso compacto

The contribution of the total dose due to deposition of secondary energy particles caused by nuclear inelastic processes (n, 2H, 3H, 3He and  ) in proton therapy is an opened problem and in discussion. In the calculations of plans implemented for routine treatment, the paid dose is calculated whereas that the proton loses energy by ionization and or coulomb excitement. The contribution of inelastic processes associated with nuclear reactions is not considered, mainly due to the difficulty of processing targets consisting of various materials. In this sense, there are only estimates for pure materials or simple composition (water, for example).This work presents the results of simulations by the Monte Carlo method employing the code MCNPX v2.50 (Monte Carlo N-Particle eXtended) of the contribution to the total dose of secondary particles. The study was implemented in a cylindrical phantom composed by compact bone, for monochromatic beams of protons between 100 and 200 MeV with pencil beam form

Treatment planning aspects and Monte Carlo methods in proton therapy

Over the last years, the interest in proton radiotherapy is rapidly increasing. Protons provide superior physical properties compared with conventional radiotherapy using photons. These properties result in depth dose curves with a large dose peak at the end of the proton track and the finite proton range allows sparing the distally located healthy tissue. These properties offer an increased flexibility in proton radiotherapy, but also increase the demand in accurate dose estimations. To carry out accurate dose calculations, first an accurate and detailed characterization of the physical proton beam exiting the treatment head is necessary for both currently available delivery techniques: scattered and scanned proton beams. Since Monte Carlo (MC) methods follow the particle track simulating the interactions from first principles, this technique is perfectly suited to accurately model the treatment head. Nevertheless, careful validation of these MC models is necessary. While for the dose estimation pencil beam algorithms provide the advantage of fast computations, they are limited in accuracy. In contrast, MC dose calculation algorithms overcome these limitations and due to recent improvements in efficiency, these algorithms are expected to improve the accuracy of the calculated dose distributions and to be introduced in clinical routine in the near future.

Monte Carlo and analytical dose calculations for ocular proton therapy

Uveal melanoma is a rare but life-threatening form of ocular cancer. Contemporary treatment techniques include proton therapy, which enables conservation of the eye and its useful vision. Dose to the proximal structures is widely believed to play a role in treatment side effects, therefore, reliable dose estimates are required for properly evaluating the therapeutic value and complication risk of treatment plans. Unfortunately, current simplistic dose calculation algorithms can result in errors of up to 30% in the proximal region. In addition, they lack predictive methods for absolute dose per monitor unit (D/MU) values. ^ To facilitate more accurate dose predictions, a Monte Carlo model of an ocular proton nozzle was created and benchmarked against measured dose profiles to within ±3% or ±0.5 mm and D/MU values to within ±3%. The benchmarked Monte Carlo model was used to develop and validate a new broad beam dose algorithm that included the influence of edgescattered protons on the cross-field intensity profile, the effect of energy straggling in the distal portion of poly-energetic beams, and the proton fluence loss as a function of residual range. Generally, the analytical algorithm predicted relative dose distributions that were within ±3% or ±0.5 mm and absolute D/MU values that were within ±3% of Monte Carlo calculations. Slightly larger dose differences were observed at depths less than 7 mm, an effect attributed to the dose contributions of edge-scattered protons. Additional comparisons of Monte Carlo and broad beam dose predictions were made in a detailed eye model developed in this work, with generally similar findings. ^ Monte Carlo was shown to be an excellent predictor of the measured dose profiles and D/MU values and a valuable tool for developing and validating a broad beam dose algorithm for ocular proton therapy. The more detailed physics modeling by the Monte Carlo and broad beam dose algorithms represent an improvement in the accuracy of relative dose predictions over current techniques, and they provide absolute dose predictions. It is anticipated these improvements can be used to develop treatment strategies that reduce the incidence or severity of treatment complications by sparing normal tissue. ^

RADIANCE-A planning software for intra-operative radiation therapy

In the last decades accumulated clinical evidence has proven that intra-operative radiation therapy (IORT) is a very valuable technique. In spite of that, planning technology has not evolved since its conception, being outdated in comparison to current state of the art in other radiotherapy techniques and therefore slowing down the adoption of IORT. RADIANCE is an IORT planning system, CE and FDA certified, developed by a consortium of companies, hospitals and universities to overcome such technological backwardness. RADIANCE provides all basic radiotherapy planning tools which are specifically adapted to IORT. These include, but are not limited to image visualization, contouring, dose calculation algorithms-Pencil Beam (PB) and Monte Carlo (MC), DVH calculation and reporting. Other new tools, such as surgical simulation tools have been developed to deal with specific conditions of the technique. Planning with preoperative images (preplanning) has been evaluated and the validity of the system being proven in terms of documentation, treatment preparation, learning as well as improvement of surgeons/radiation oncologists (ROs) communication process. Preliminary studies on Navigation systems envisage benefits on how the specialist to accurately/safely apply the pre-plan into the treatment, updating the plan as needed. Improvements on the usability of this kind of systems and workflow are needed to make them more practical. Preliminary studies on Intraoperative imaging could provide an improved anatomy for the dose computation, comparing it with the previous pre-plan, although not all devices in the market provide good characteristics to do so. DICOM.RT standard, for radiotherapy information exchange, has been updated to cover IORT particularities and enabling the possibility of dose summation with external radiotherapy. The effect of this planning technology on the global risk of the IORT technique has been assessed and documented as part of a failure mode and effect analysis (FMEA). Having these technological innovations and their clinical evaluation (including risk analysis) we consider that RADIANCE is a very valuable tool to the specialist covering the demands from professional societies (AAPM, ICRU, EURATOM) for current radiotherapy procedures.

Investigations into a wideband spatial beamformer employing a rectangular array of planar monopoles

This paper presents a rectangular array antenna with a suitable signal-processing algorithm that is able to steer the beam in azimuth over a wide frequency band. In the previous approach, which was reported in the literature, an inverse discrete Fourier transform technique was proposed for obtaining the signal weighting coefficients. This approach was demonstrated for large arrays in which the physical parameters of the antenna elements were not considered. In this paper, a modified signal-weighting algorithm that works for arbitrary-size arrays is described. Its validity is demonstrated in examples of moderate-size arrays with real antenna elements. It is shown that in some cases, the original beam-forming algorithm fails, while the new algorithm is able to form the desired radiation pattern over a wide frequency band. The performance of the new algorithm is assessed for two cases when the mutual coupling between array elements is both neglected and taken into account.

Proton radiotherapy: a therapeutic opportunity for pediatric brain tumor patients

Proton radiation therapy is a form of external radiation that uses charged particles which have distinct physical advantages to deliver the majority of its dose in the target while minimizing the dose of radiation to normal tissues. In children who are particularly susceptible even at low and medium doses of radiation, the significant reduction of integral dose can potentially mitigate the incidence of side effects and improve quality of life. The aim of the first part of the thesis is to describe the physical and radiobiological properties of protons, the Proton Therapy Center of Trento (TCPT) active for clinical purpose since 2014, which use the most recent technique called active pencil beam scanning. The second part of the thesis describes the preliminary clinical results of 23 pediatric patients with central nervous system tumors as well as of two aggressive pediatric meningiomas treated with pencil beam scanning. All the patients were particularly well-suited candidates for proton therapy (PT) for possible benefits in terms of survival and incidence of acute and late side effects. We reported also a multicentric experience of 27 medulloblastoma patients (median age 6 years, M/F ratio 13/14) treated between 2015 and 2020 at TPTC coming from three Pediatric oncology centers: Bologna, Florence, and Ljubljana, with a focus on clinical results and toxicities related to radiotherapy (RT). Proton therapy was associated with mostly mild acute and late adverse effects and no cases of CNS necrosis or high grade of neuroradiological abnormailities. Comparable rates of survival and local control were obtained to those achievable with conventional RT. Finally, we performed a systematic review to specifically address the safety of PT for pediatric CNS patients, late side effects and clinical effectiveness after PT in this patient group.

A convolution model for energy transport in a therapeutic fast neutron beam

A three-dimensional model has been proposed that uses Monte Carlo and fast Fourier transform convolution techniques to calculate the dose distribution from a fast neutron beam. This method transports scattered neutrons and photons in the forward, lateral, and backward directions and protons, electrons, and positrons in the forward and lateral directions by convolving energy spread kernels with initial interaction available energy distributions. The primary neutron and photon spectrums have been derived from narrow beam attenuation measurements. The positions and strengths of the effective primary neutron, scattered neutron, and photon sources have been derived from dual ion chamber measurements. The size of the effective primary neutron source has been measured using a copper activation technique. Heterogeneous tissue calculations require a weighted sum of two convolutions for each component since the kernels must be invariant for FFT convolution. Comparisons between calculations and measurements were performed for several water and heterogeneous phantom geometries. ^