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Purpose: To evaluate the combined effect of heterogeneous target dose and heterogeneous radiosensitivity on tumor control probability (TCP) for different number of fractions (N f )., Methods: The linear-quadratic (LQ) model is employed to study dependence of TCP on N f under the condition of fixed nominal biologically effective dose (BED nom )., Results: Formula for the optimum target dose which maximizes TCP under the condition BED nom =const is analytically derived. It is shown that the dependence of TCP on N f is non-monotonic. In addition, the dependence of TCP on N f for different variances of the target dose and radiosensitivity of malignant cells is demonstrated by using numerical computations., Conclusions: It is shown that the optimum mean dose in the target is defined by the standard deviations of the target dose (σ D ) and standard deviations of parameters alpha (σ α ) and beta (σ β ). The findings of this study indicate that hypofractionated regimens for stereotactic body radiation therapy (SBRT) and stereotactic radiosurgery (SRS) with N f ⩽3 can be radiobiologically inferior to the regimens with five or more fractions., (Copyright © 2021 Associazione Italiana di Fisica Medica e Sanitaria. Published by Elsevier Ltd. All rights reserved.)

Purpose: to determine optimal fractionation schedule which provides maximum biologically effective dose (BED) in the targeted tumor in the case of heterogeneous radiosensitivity of malignant cells in combination with a spatially varying sparing factor for the organ at risk., Methods: the mathematical framework used in the current study is based on the linear-quadratic (LQ) model with heterogeneous parameters alpha and beta for malignant cells. To determine optimal fractionation, we consider changes in BED in the treatment target (BED tar ) under the condition of fixed BED in the affected normal tissue (BED nt )., Results: we demonstrate both analytically and via numerical calculations that there exists an optimal number of fractions for which biologically effective dose in the uniformly irradiated target is at maximum. It is also shown that dependence of BED tar on number of fractions is generally non-monotonic and is affected by the variances σ α and σ β of the alpha and beta radiosensitivities, respectively. In a particular case when σ α and σ β are sufficiently small, expression for the optimal target dose which maximizes BED tar was derived analytically., Conclusion: the obtained results demonstrate that in the presence of heterogeneous alpha and beta in the tumor, hypofractionation can either increase or decrease BED tar depending on the variances σ α and σ β . Consequently, intratumor heterogeneity is an important factor which can affect radiobiological comparison of different fractionation regimens., (Copyright © 2019 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.)

Purpose: The purpose of the study was to evaluate changes in biologically effective dose (BED) in the targeted tumor due to varying number of treatment fractions in the case when alpha and beta radiosensitivities of malignant cells are heterogeneous., Methods: The approach used in the current study relies on the linear-quadratic (LQ) model. Within the framework of this model, we consider changes in the biologically effective dose in the treatment target ( B E D tar ) caused by varying number of fractions under the condition of fixed BED in the affected normal tissue ( B E D nt )., Results: In this study, we analytically derive the necessary and sufficient condition which ensures that, compared to standard fractionation, hypofractionation increases B E D tar in the case of heterogeneous radiosensitivities in the treatment target. We also derive expression for dose per fraction which maximizes B E D tar . In addition, variations in B E D tar with number of fractions were determined numerically for several clinical cases by using spinal cord as an example of serial organ at risk (OAR)., Conclusions: The results from this study demonstrate that intratumor heterogeneity influences radiobiological properties of different fractionation regimens as follows: (a) variations in B E D tar caused by varying number of fractions can be nonmonotonical and (b) there exist optimum dose per fraction and number of fractions which maximize B E D tar under the condition of fixed biologically effective dose in the affected OAR., (© 2019 American Association of Physicists in Medicine.)

Purpose: To evaluate how dose rate affects radiobiological properties of hypofractionated radiotherapy., Methods: This study is based on the linear-quadratic (LQ) model used to determine biologically effective dose (BED). Changes in the biologically effective dose in normal tissue (BED nt ) are studied as a function of number of fractions and dose rate under the condition of fixed BED in the treatment target (BED tar )., Results: In this study we demonstrate that compared to standard fractionation, hypofractionation can either decrease or increase BED nt depending on the average dose rate. In the considered examples, maximum value of BED nt in the spinal cord varies monotonically with number of fractions (N f ) when dose rate is sufficiently high so that the corresponding fraction time is much smaller than characteristic repair half-lives for malignant and normal cells. In contrast, in the case of a lower dose rate of 300 MU/min, BED nt in the cord can vary non-monotonically with N f . In the later case, there exists optimum number of fractions which corresponds to the minimum BED nt . It is shown that in the case when radiation induced sublethal damage is repaired faster in the target than in the affected organ at risk (OAR), increasing dose rate helps lower BED nt ., Conclusion: We have demonstrated that, as compared to standard fractionation, hypofractionation can either increase or decrease BED nt in the OAR depending on the utilized dose rate. Consequently, radiobiological assessment of hypofractionation should take into account dose rate as well as repair rates in the target and OAR., (Copyright © 2019 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.)

An analytical model is presented that describes the effects of an applied longitudinal magnetic field and energy loss on the transport of relativistic electrons in a medium. By using a modified Fermi equation we derive analytical solutions for the distribution function of electrons. The obtained results can be used to predict the influence of longitudinal magnetic fields on the lateral spread of electrons in both coordinate space and velocity space.

Locally focused magnetic resonance imaging (LF MRI) allows imaging with variable spatial resolution within the field of view (FOV). Because LF MRI uses a priori information to provide locally high resolution in regions with rapid spatial variations in intensity (e.g., blood/tissue interface), it allows accurate reproduction of intense sharp edges in the specimen without blurring and truncation artifacts. This study employs LF MRI for 3D imaging of stationary and pulsatile flow. In the implemented version of LF MRI analytically defined basis functions are used to determine image intensity in regions depicted with low or high resolution. It is demonstrated that LF MRI of flow allows a significant (i.e. 3-4 times) reduction in scan time as compared to conventional FT MRI. It is also shown that LF images of pulsatile flow have a decreased appearance of ghosting artifacts as compared to the images reconstructed by using the conventional method.

Vibrations in a granular material can spontaneously produce convection rolls reminiscent of those seen in fluids. Magnetic resonance imaging provides a sensitive and noninvasive probe for the detection of these convection currents, which have otherwise been difficult to observe. A magnetic resonance imaging study of convection in a column of poppy seeds yielded data about the detailed shape of the convection rolls and the depth dependence of the convection velocity. The velocity was found to decrease exponentially with depth; a simple model for this behavior is presented here.