Combined effect of heterogeneous target dose and heterogeneous radiosensitivity on tumor control probability for different fractionation regimens.

PURPOSE: To evaluate the combined effect of heterogeneous target dose and heterogeneous radiosensitivity on tumor control probability (TCP) for different number of fractions (Nf). METHODS: The linear-quadratic (LQ) model is employed to study dependence of TCP on Nf under the condition of fixed nominal biologically effective dose (BEDnom). RESULTS: Formula for the optimum target dose which maximizes TCP under the condition BEDnom=const is analytically derived. It is shown that the dependence of TCP on Nfis non-monotonic. In addition, the dependence of TCP on Nf 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 (σD) and standard deviations of parameters alpha (σα) and beta (σß). The findings of this study indicate that hypofractionated regimens for stereotactic body radiation therapy (SBRT) and stereotactic radiosurgery (SRS) with Nf⩽3 can be radiobiologically inferior to the regimens with five or more fractions.

Effect of heterogeneous target dose and radiosensitivity on BED and TCP for different treatment regimens.

Purpose.To evaluate how heterogeneity of the target dose and heterogeneity of intra-tumor radiosensitivity affect biologically effective dose (BED) and tumor control probability (TCP) depending on the number of fractions (Nf).Methods.The dependences of TCP and BED in the planning target volume onNfare studied using the linear-quadratic model. In the considered case, the nominal biologically effective dose(BEDnom)is fixed and the variances of the target dose (σD) and radiosensitivity (σα) are assumed to be small.Results.By using series expansion of the survival probability of malignant cells, it is analytically shown that for smallσDandσαboth BED and TCP increase with increasingNfunder the conditionBEDnom=const.In addition, the dependences of BED and TCP onNffor different values ofσDandσαare studied by using an analytical expression for BED in the case of Gaussian distributions of both target dose and radiosensitivity.Conclusions.Small variations in the absorbed dose and intratumor radiosensitivity can significantly reduce BED and TCP. The decreases in these quantities can be reduced by increasing the number of fractions. The findings of this study indicate that hypofractionated regimens withNf=20and dose per fractiond≤5Gy can lead to higher BED and TCP compared to treatment regimens withNf≤5andd≥10Gy commonly used for stereotactic body radiation therapy (SBRT) and stereotactic radiosurgery (SRS).

Effect of reoxygenation on hypofractionated radiotherapy of prostate cancer.

PURPOSE: To assess the role of reoxygenation of hypoxic tumor cells in hypofractionated radiotherapy of prostate cancer. METHODS: The considered radiobiological model is based on the assumption of two populations (compartments) of cells: oxygenated (aerobic) cells and hypoxic cells. After each fraction of radiation, some of the hypoxic cells reoxygenate while a fraction of initially aerobic cells becomes hypoxic. The kinetics of this process between successive treatments is described by coupled, first-order differential equations. To determine the effect of reoxygenation on cell kill in the treatment target, we utilize the linear-quadratic (LQ) model assuming different radiosensitivities for the aerobic and hypoxic cells. RESULTS: Analytical solutions for the number of surviving malignant cells are obtained for special cases of slow and fast reoxygenation. The radiobiological effect of reoxygenation for different fractionation regimens is also evaluated numerically. CONCLUSIONS: In this study, a radiobiological model for kinetics of reoxygenation in tumors is used to evaluate different fractionation schedules in radiotherapy of prostate cancer. The obtained results indicate that in the case of low alpha/beta ratio for malignant cells (e.g., α / ß  = 1.5 Gy), treatment schedule with 4-10 fractions and dose per fraction >4-5 Gy can result in increased cell kill in the treatment target at the same level of rectal toxicity as compared to conventional fractionation. The findings of this study also suggest that radiotherapy of the prostate with 1-3 fractions can be radiobiologically inferior to treatments with greater number of fractions.

Effect of heterogeneous radiosensitivity on the optimal fractionation in radiotherapy.

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 (BEDtar) under the condition of fixed BED in the affected normal tissue (BEDnt). 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 BEDtar on number of fractions is generally non-monotonic and is affected by the variances σα and σß of the alpha and beta radiosensitivities, respectively. In a particular case when σα and σß are sufficiently small, expression for the optimal target dose which maximizes BEDtar 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 BEDtar depending on the variances σα and σß. Consequently, intratumor heterogeneity is an important factor which can affect radiobiological comparison of different fractionation regimens.

Effect of dose rate in hypofractionated radiotherapy.

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 (BEDnt) are studied as a function of number of fractions and dose rate under the condition of fixed BED in the treatment target (BEDtar). RESULTS: In this study we demonstrate that compared to standard fractionation, hypofractionation can either decrease or increase BEDnt depending on the average dose rate. In the considered examples, maximum value of BEDnt in the spinal cord varies monotonically with number of fractions (Nf) 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, BEDnt in the cord can vary non-monotonically with Nf. In the later case, there exists optimum number of fractions which corresponds to the minimum BEDnt. 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 BEDnt. CONCLUSION: We have demonstrated that, as compared to standard fractionation, hypofractionation can either increase or decrease BEDnt 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.

Effect of intratumor heterogeneity on BED for hypofractionated dose regimens.

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.

Effect of radiation protraction on BED in the case of large fraction dose.

PURPOSE: To investigate the effect of radiation protraction on biologically effective dose (BED) in the case when dose per fraction is significantly greater than the standard dose of 2 Gy. METHODS: By using the modified linear-quadratic model with monoexponential repair, the authors investigate the effect of long treatment times combined with dose escalation. RESULTS: The dependences of the protraction factor and the corresponding BED on fraction time were determined for different doses per fraction typical for stereotactic radiosurgery (SRS) and stereotactic body radiation therapy (SBRT). In the calculations, the authors consider changes in the BED to the normal tissue under the condition of fixed BED to the target. CONCLUSION: The obtained results demonstrate that simultaneous increase in fraction time and dose per fraction can be beneficial for SRS and SBRT because of the related decrease in BED to normal structures while BED to the target is fixed.

Use of radiation protraction to escalate biologically effective dose to the treatment target.

PURPOSE: The aim of this study is to evaluate how simultaneously increasing fraction time and dose per fraction affect biologically effective dose for the target (BED(tar)) while biologically effective dose for the normal tissue (BED(nt)) is fixed. METHODS: In this investigation, BED(tar) and BED(nt) were studied by assuming mono-exponential repair of sublethal damage with tissue dependent repair half-time. RESULTS: Our results demonstrate that under certain conditions simultaneously increasing fraction time and dose per fraction result in increased BED(tar) while BED(nt) is fixed. The dependence of biologically effective dose on fraction time is influenced by the dose rate. In this investigation we analytically determined time-varying dose rate R̃ which minimizes BED. Changes in BED with fraction time were compared for constant dose rate and for R̃. CONCLUSIONS: A number of recent experimental and theoretical studies have demonstrated that slow delivery of radiation (known as radiation protraction) leads to reduced therapeutic effect because of increased repair of sublethal damage. In contrast, our analysis shows that under certain conditions simultaneously increasing fraction time and dose per fraction are radiobiologically advantageous.

General properties of different models used to predict normal tissue complications due to radiation.

In the current study the author analyzes general properties of three different models used to predict normal tissue complications due to radiation: (1) Surviving fraction of normal cells in the framework of the linear quadratic (LQ) equation for cell kill, (2) the Lyman-Kutcher-Burman (LKB) model for normal tissue complication probability (NTCP), and (3) generalized equivalent uniform dose (gEUD). For all considered cases the author assumes fixed average dose to an organ of interest. The author's goal is to establish whether maximizing dose uniformity in the irradiated normal tissues is radiobiologically beneficial. Assuming that NTCP increases with increasing overall cell kill, it is shown that NTCP in the LQ model is maximized for uniform dose. Conversely, NTCP in the LKB and gEUD models is always smaller for a uniform dose to a normal organ than that for a spatially varying dose if parameter n in these models is small (i.e., n <1). The derived conflicting properties of the considered models indicate the need for more studies before these models can be utilized clinically for plan evaluation and/or optimization of dose distributions. It is suggested that partial-volume irradiation can be used to establish the validity of the considered models.

Effect of radiation protraction in intensity-modulated radiation therapy with direct aperture optimization: a phantom study.

The effect of radiation protraction in step-and-shoot IMRT is investigated for treatment plans created with the help of direct aperture optimization. The latter approach can be used during inverse planning for all clinical linear accelerators with conventional MLC. Direct aperture optimization significantly shortens fraction time for IMRT plans as compared to that for plans obtained by using the conventional inverse planning approach. By analyzing several IMRT plans obtained with direct aperture optimization we found that for alpha/beta ratio of 10 Gy (characteristic of fast growing tumors) the protraction effect is probably clinically insignificant for both conventional and large fraction sizes of 1.9 Gy and 5.7 Gy, respectively. For small alpha/beta of 1-1.5 Gy and conventional fraction size the effect of protraction is still small; however, this effect can be significant for hypofractionated treatments. Based on the obtained results it is recommended that, when possible, IMRT for slow growing prostate cancers be performed with small number of beams (e.g., 5) and small number of segments (e.g., 5-7 segments per beam) to reduce delivery time and, as a result, the associated effect of radiation protraction.

Improving delivery of segments with small MU in step-and-shoot IMRT.

The purpose of this study is to describe four new delivery schemes for intensity-modulated radiation therapy (IMRT). In the first two schemes the order in which segments are delivered is varied from fraction to fraction. The last two delivery schemes employ fixed order of segments. The obtained results indicate that the suggested approaches can significantly reduce the so-called "overshoot" and "undershoot" phenomena and the associated discrepancies between planned and delivered monitor units.

Electron beam transport in the presence of a strong, axial magnetic field.

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.

MR imaging of flow with locally high spatial resolution.

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.

Locally focused contrast-enhanced carotid MRA.

With conventional Fourier transform (FT) magnetic resonance imaging (MRI), it is difficult to perform contrast-enhanced three-dimensional (3D) MR angiography (MRA) with the temporal and spatial resolution necessary to depict the carotid arteries. However, locally focused (LF) MRI is a more efficient method that utilizes prior knowledge of the image content to reconstruct images from sparse k-space samples. In this paper, we show how LF MRI can be used to perform high-resolution gadolinium (Gd)-enhanced 3D carotid MRA in less than 10 seconds. First, the accuracy of the technique was demonstrated by comparing LF and conventional (FT) images of a vascular phantom. Then the method was used to perform Gd-enhanced 3D MRA of a patient's carotid arteries. Instead of using bolus timing, the arterial phase was retrospectively identified in a consecutive series of images, just as in X-ray angiography.

Differentiation between the effects of T1 and T2* shortening in contrast-enhanced MRI of the breast.

The purpose of this study is to describe a technique for magnetic resonance imaging (MRI) that can potentially improve identification of malignant tissue in the human breast. The suggested MRI technique is based on the differentiation between two competing effects leading to opposite changes in image intensity, namely, T1 and T2* shortening caused by administration of gadolinium chelate. The proposed approach also allows calculation of changes in the R2* relaxation rate in breast tissue. The feasibility of the technique for in vivo MRI and increased lesion contrast is demonstrated. The results indicate that this technique may improve detection of malignant breast tissue.

Locally focused MRI of interventions.

Certain interventional MR procedures would benefit from T2-weighted imaging because of the sensitivity of T2-weighted images to tissue damage and target lesion contrast. To acquire such images with reasonable temporal resolution, a single-shot acquisition should be used because of the inherently long TR needed for T2 weighting. Unfortunately, most scanners require long readout times (eg, greater than 150 msec) and high bandwidths (eg, greater than 120 kHz) to perform conventional single-shot imaging with high spatial resolution. The resulting images are thus degraded by unacceptable artifacts and noise levels. This study illustrates how to create locally focused MR images that have high spatial resolution in a region of interest and lower spatial resolution elsewhere in the image. Because these images can be created from sparse k-space data, a scanner with modest gradients (eg, 10 mT/m maximal amplitude, 500 microsec minimal rise time) can acquire them after a single excitation with relatively short readout time and low bandwidth. This technique may make it practical to monitor interventions with T2-weighted imaging. The method was illustrated by reconstructing dynamic changes, which were simulated experimentally by moving objects in the vicinity of a normal human head.

Differentiating between T1 and T2* changes caused by gadopentetate dimeglumine in the kidney by using a double-echo dynamic MR imaging sequence.

Dynamic MR images of the passage of gadopentetate dimeglumine through the kidneys of normal rats are obtained using a dual gradient-echo sequence. The amplitudes of gradient echoes are defined by local T1 and T2* values in the tissue. The ratio of these amplitudes, primarily defined by local T2*, can be used to differentiate between T1 and T2* effects. This is particularly important with regard to renal studies because, due to a highly inhomogeneous distribution of gadopentetate dimeglumine in the kidney, T2* shortening can impede MR data analysis. To study changes in the observed signal caused by gadopentetate dimeglumine, curves of MR renal intensity versus time were obtained in the cortex and medulla after administration of the contrast agent. Using T2* compensation, distinct temporal peaks were observed in the cortex and outer medulla, indicating a high concentration of gadopentetate dimeglumine in the vascular phase. The authors conclude that this technique can be a useful tool for studying renal function noninvasively.

Granular convection observed by magnetic resonance imaging.

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.