[Show abstract][Hide abstract] ABSTRACT: Glioblastoma differ from many other tumors in the sense that they grow infiltratively into the brain tissue instead of forming a solid tumor mass with a defined boundary. Only the part of the tumor with high tumor cell density can be localized through imaging directly. In contrast, brain tissue infiltrated by tumor cells at low density appears normal on current imaging modalities. In current clinical practice, a uniform margin, typically two centimeters, is applied to account for microscopic spread of disease that is not directly assessable through imaging. The current treatment planning procedure can potentially be improved by accounting for the anisotropy of tumor growth, which arises from different factors: anatomical barriers such as the falx cerebri represent boundaries for migrating tumor cells. In addition, tumor cells primarily spread in white matter and infiltrate gray matter at lower rate. We investigate the use of a phenomenological tumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates the spatial distribution of tumor cells in normal appearing regions of the brain. The target volume for radiotherapy planning can be defined as an isoline of the simulated tumor cell density. This paper analyzes the model with respect to implications for target volume definition and identifies its most critical components. A retrospective study involving ten glioblastoma patients treated at our institution has been performed. To illustrate the main findings of the study, a detailed case study is presented for a glioblastoma located close to the falx. In this situation, the falx represents a boundary for migrating tumor cells, whereas the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. We further discuss the sensitivity of the model with respect to the input parameters. Correct segmentation of the brain appears to be the most crucial model input. We conclude that the tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.
Physics in Medicine and Biology 02/2014; 59(3):747-70. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Gliomas differ from many other tumors as they grow infiltratively into the brain parenchyma rather than forming a solid tumor mass with a well-defined boundary. Tumor cells can be found several centimeters away from the central tumor mass that is visible using current imaging techniques. The infiltrative growth characteristics of gliomas question the concept of a radiotherapy target volume that is irradiated to a homogeneous dose-the standard in current clinical practice. We discuss the use of the Fisher-Kolmogorov glioma growth model in radiotherapy treatment planning. The phenomenological tumor growth model assumes that tumor cells proliferate locally and migrate into neighboring brain tissue, which is mathematically described via a partial differential equation for the spatio-temporal evolution of the tumor cell density. In this model, the tumor cell density drops approximately exponentially with distance from the visible gross tumor volume, which is quantified by the infiltration length, a parameter describing the distance at which the tumor cell density drops by a factor of e. This paper discusses the implications for the prescribed dose distribution in the periphery of the tumor. In the context of the exponential cell kill model, an exponential fall-off of the cell density suggests a linear fall-off of the prescription dose with distance. We introduce the dose fall-off rate, which quantifies the steepness of the prescription dose fall-off in units of Gy mm(-1). It is shown that the dose fall-off rate is given by the inverse of the product of radiosensitivity and infiltration length. For an infiltration length of 3 mm and a surviving fraction of 50% at 2 Gy, this suggests a dose fall-off of approximately 1 Gy mm(-1). The concept is illustrated for two glioblastoma patients by optimizing intensity-modulated radiotherapy plans. The dose fall-off rate concept reflects the idea that infiltrating gliomas lack a defined boundary and are characterized by a continuous fall-off of the density of infiltrating tumor cells. The approach can potentially be used to individualize the prescribed dose distribution if better methods to estimate radiosensitivity and infiltration length on a patient by patient basis become available.
Physics in Medicine and Biology 02/2014; 59(3):771-89. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: To describe a method for combining sliding window plans [intensity modulated radiation therapy (IMRT) or volumetric modulated arc therapy (VMAT)] for use in treatment plan averaging, which is needed for Pareto surface navigation based multicriteria treatment planning.
The authors show that by taking an appropriately defined average of leaf trajectories of sliding window plans, the authors obtain a sliding window plan whose fluence map is the exact average of the fluence maps corresponding to the initial plans. In the case of static-beam IMRT, this also implies that the dose distribution of the averaged plan is the exact dosimetric average of the initial plans. In VMAT delivery, the dose distribution of the averaged plan is a close approximation of the dosimetric average of the initial plans.
The authors demonstrate the method on three Pareto optimal VMAT plans created for a demanding paraspinal case, where the tumor surrounds the spinal cord. The results show that the leaf averaged plans yield dose distributions that approximate the dosimetric averages of the precomputed Pareto optimal plans well.
The proposed method enables the navigation of deliverable Pareto optimal plans directly, i.e., interactive multicriteria exploration of deliverable sliding window IMRT and VMAT plans, eliminating the need for a sequencing step after navigation and hence the dose degradation that is caused by such a sequencing step.
Medical Physics 02/2014; 41(2):021709. · 2.91 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In concurrent chemoradiotherapy, chemotherapeutic agents are administered
during the course of radiotherapy to enhance the primary tumor control.
However, that often comes at the expense of increased risk of normal-tissue
complications. The additional biological damage is mainly attributed to two
mechanisms of action, which are the independent cytotoxic activity of
chemotherapeutic agents and their interactive cooperation with radiation. The
goal of this study is to develop a mathematical framework to obtain drug and
radiation administration schedules that maximize the therapeutic gain for
concurrent chemoradiotherapy. In particular, we analyze the impact of
incorporating these two mechanisms into the radiation fractionation problem.
Considering each mechanism individually, we first derive closed-form
expressions for the optimal radiation fractionation regimen and the
corresponding drug administration schedule. We next study the case in which
both mechanisms are simultaneously present and develop a dynamic programming
framework to determine optimal treatment regimens. Results show that those
chemotherapeutic agents that interact with radiation may change optimal
radiation fractionation regimens. Moreover, administration of chemotherapeutic
agents possessing both mechanisms may give rise to non-stationary radiation
[Show abstract][Hide abstract] ABSTRACT: We analyze the effect of tumor repopulation on optimal dose delivery in
radiation therapy. We are primarily motivated by accelerated tumor repopulation
towards the end of radiation treatment, which is believed to play a role in
treatment failure for some tumor sites. A dynamic programming framework is
developed to determine an optimal fractionation scheme based on a model of cell
kill due to radiation and tumor growth in between treatment days. We find that
faster tumor growth suggests shorter overall treatment duration. In addition,
the presence of accelerated repopulation suggests larger dose fractions later
in the treatment to compensate for the increased tumor proliferation. We prove
that the optimal dose fractions are increasing over time. Numerical simulations
indicate potential for improvement in treatment effectiveness.
[Show abstract][Hide abstract] ABSTRACT: In multi-stage radiotherapy, a patient is treated in several stages separated
by weeks or months. This regimen has been motivated mostly by radiobiological
considerations, but also provides an approach to reduce normal tissue dose by
exploiting tumor shrinkage. The paper considers the optimal design of
multi-stage treatments, motivated by the clinical management of large liver
tumors for which normal liver dose constraints prohibit the administration of
an ablative radiation dose in a single treatment.
We introduce a dynamic tumor model that incorporates three factors: radiation
induced cell kill, tumor shrinkage, and tumor cell repopulation. The design of
multi-stage radiotherapy is formulated as a mathematical optimization problem
in which the total dose to the liver is minimized, subject to delivering the
prescribed dose to the tumor. Based on the model, we gain insight into the
optimal administration of radiation over time, i.e. the optimal treatment gaps
and dose levels.
We analyze treatments consisting of two stages in detail. The analysis
confirms the intuition that the second stage should be delivered just before
the tumor size reaches a minimum and repopulation overcompensates shrinking.
Furthermore, it was found that, for a large range of model parameters,
approximately one third of the dose should be delivered in the first stage. The
projected benefit of multi-stage treatments depends on model assumptions.
However, the model predicts large liver dose reductions by more than a factor
of two for plausible model parameters.
The analysis of the tumor model suggests that substantial reduction in normal
tissue dose can be achieved by exploiting tumor shrinkage via an optimal design
of multi-stage treatments. This suggests taking a fresh look at multi-stage
radiotherapy for selected disease sites where substantial tumor regression
translates into reduced target volumes.
Physics in Medicine and Biology 11/2013; 59(12). · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Purpose: The paper considers the fractionation problem in intensity modulated proton therapy (IMPT). Conventionally, IMPT fields are optimized independently of the fractionation scheme. In this work, we discuss the simultaneous optimization of fractionation scheme and pencil beam intensities.Methods: This is performed by allowing for distinct pencil beam intensities in each fraction, which are optimized using objective and constraint functions based on biologically equivalent dose (BED). The paper presents a model that mimics an IMPT treatment with a single incident beam direction for which the optimal fractionation scheme can be determined despite the nonconvexity of the BED-based treatment planning problem.Results: For this model, it is shown that a small α∕β ratio in the tumor gives rise to a hypofractionated treatment, whereas a large α∕β ratio gives rise to hyperfractionation. It is further demonstrated that, for intermediate α∕β ratios in the tumor, a nonuniform fractionation scheme emerges, in which it is optimal to deliver different dose distributions in subsequent fractions. The intuitive explanation for this phenomenon is as follows: By varying the dose distribution in the tumor between fractions, the same total BED can be achieved with a lower physical dose. If it is possible to achieve this dose variation in the tumor without varying the dose in the normal tissue (which would have an adverse effect), the reduction in physical dose may lead to a net reduction of the normal tissue BED. For proton therapy, this is indeed possible to some degree because the entrance dose is mostly independent of the range of the proton pencil beam.Conclusions: The paper provides conceptual insight into the interdependence of optimal fractionation schemes and the spatial optimization of dose distributions. It demonstrates the emergence of nonuniform fractionation schemes that arise from the standard BED model when IMPT fields and fractionation scheme are optimized simultaneously. Although the projected benefits are likely to be small, the approach may give rise to an improved therapeutic ratio for tumors treated with stereotactic techniques to high doses per fraction.
Medical Physics 09/2013; 40(9):091702. · 2.91 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The main approach to smooth Pareto surface navigation for radiation therapy
multi-criteria treatment planning involves taking real-time averages of
pre-computed treatment plans. In fluence-based treatment planning, fluence maps
themselves can be averaged, which leads to the dose distributions being
averaged due to the linear relationship between fluence and dose. This works
for fluence-based photon plans and proton spot scanning plans. In this
technical note, we show that two or more sliding window volumetric modulated
arc therapy (VMAT) plans can be combined by averaging leaf positions in a
certain way, and we demonstrate that the resulting dose distribution for the
averaged plan is approximately the average of the dose distributions of the
original plans. This leads to the ability to do Pareto surface navigation, i.e.
interactive multi-criteria exploration of VMAT plan dosimetric tradeoffs.
[Show abstract][Hide abstract] ABSTRACT: We propose a novel optimization model for volumetric modulated arc therapy
(VMAT) planning that directly optimizes deliverable leaf trajectories in the
treatment plan optimization problem, and eliminates the need for a separate
arc-sequencing step. In this model, a 360-degree arc is divided into a given
number of arc segments in which the leaves move unidirectionally. This
facilitates an algorithm that determines the optimal piecewise linear leaf
trajectories for each arc segment, which are deliverable in a given treatment
time. Multi-leaf collimator (MLC) constraints, including maximum leaf speed and
interdigitation, are accounted for explicitly. The algorithm is customized to
allow for VMAT delivery using constant gantry speed and dose rate, however, the
algorithm generalizes to variable gantry speed if beneficial. We demonstrate
the method for three different tumor sites: a head-and-neck case, a prostate
case, and a paraspinal case. For that purpose, we first obtain a reference plan
for intensity modulated radiotherapy (IMRT) using fluence map optimization and
20 equally spaced beam directions. Subsequently, VMAT plans are optimized by
dividing the 360-degree arc into 20 corresponding arc segments. Assuming
typical machine parameters (a dose rate of 600 MU/min, and a maximum leaf speed
of 3 cm/sec), it is demonstrated that the quality of the optimized VMAT plans
approaches the quality of the IMRT benchmark plan for delivery times between 3
and 4 minutes.
Medical Physics 06/2013; 40(6). · 2.91 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Navigation-based multi-criteria optimization has been introduced to radiotherapy planning in order to allow the interactive exploration of trade-offs between conflicting clinical goals. However, this has been mainly applied to fluence map optimization. The subsequent leaf sequencing step may cause dose discrepancy, leading to human iteration loops in the treatment planning process that multi-criteria methods were meant to avoid. To circumvent this issue, this paper investigates the application of direct aperture optimization methods in the context of multi-criteria optimization. We develop a solution method to directly obtain a collection of apertures that can adequately span the entire Pareto surface. To that end, we extend the column generation method for direct aperture optimization to a multi-criteria setting in which apertures that can improve the entire Pareto surface are sequentially identified and added to the treatment plan. Our proposed solution method can be embedded in a navigation-based multi-criteria optimization framework, in which the treatment planner explores the trade-off between treatment objectives directly in the space of deliverable apertures. Our solution method is demonstrated for a paraspinal case where the trade-off between target coverage and spinal-cord sparing is studied. The computational results validate that our proposed method obtains a balanced approximation of the Pareto surface over a wide range of clinically relevant plans.
Physics in Medicine and Biology 01/2013; 58(3):621-639. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We propose an algorithm for aperture shape optimization (ASO) for step and shoot delivery of intensity-modulated radiotherapy. The method is an approach to direct aperture optimization (DAO) that exploits gradient information to locally optimize the positions of the leafs of a multileaf collimator. Based on the dose-influence matrix, the dose distribution is locally approximated as a linear function of the leaf positions. Since this approximation is valid only in a small interval around the current leaf positions, we use a trust-region-like method to optimize the leaf positions: in one iteration, the leaf motion is confined to the beamlets where the leaf edges are currently positioned. This yields a well-behaved optimization problem for the leaf positions and the aperture weights, which can be solved efficiently. If, in one iteration, a leaf is moved to the edge of a beamlet, the leaf motion can be confined to the neighboring beamlet in the next iteration. This allows for large leaf position changes over the course of the algorithm. In this paper, the ASO algorithm is embedded into a column-generation approach to DAO. After a new aperture is added to the treatment plan, we use the ASO algorithm to simultaneously optimize aperture weights and leaf positions for the new set of apertures. We present results for a paraspinal tumor case, a prostate case and a head and neck case. The computational results indicate that, using this approach, treatment plans close to the ideal fluence map optimization solution can be obtained.
Physics in Medicine and Biology 12/2012; 58(2):301-318. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider the fractionation problem in radiation therapy. Tumor sites in which the dose-limiting organ at risk (OAR) receives a substantially lower dose than the tumor, bear potential for hypofractionation even if the α/β-ratio of the tumor is larger than the α/β-ratio of the OAR. In this work, we analyze the interdependence of the optimal fractionation scheme and the spatial dose distribution in the OAR. In particular, we derive a criterion under which a hypofractionation regimen is indicated for both a parallel and a serial OAR. The approach is based on the concept of the biologically effective dose (BED). For a hypothetical homogeneously irradiated OAR, it has been shown that hypofractionation is suggested by the BED model if the α/β-ratio of the OAR is larger than α/β-ratio of the tumor times the sparing factor, i.e. the ratio of the dose received by the tumor and the OAR. In this work, we generalize this result to inhomogeneous dose distributions in the OAR. For a parallel OAR, we determine the optimal fractionation scheme by minimizing the integral BED in the OAR for a fixed BED in the tumor. For a serial structure, we minimize the maximum BED in the OAR. This leads to analytical expressions for an effective sparing factor for the OAR, which provides a criterion for hypofractionation. The implications of the model are discussed for lung tumor treatments. It is shown that the model supports hypofractionation for small tumors treated with rotation therapy, i.e. highly conformal techniques where a large volume of lung tissue is exposed to low but nonzero dose. For larger tumors, the model suggests hyperfractionation. We further discuss several non-intuitive interdependencies between optimal fractionation and the spatial dose distribution. For instance, lowering the dose in the lung via proton therapy does not necessarily provide a biological rationale for hypofractionation.
Physics in Medicine and Biology 12/2012; 58(1):159-167. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Purpose: We present a method to include robustness in a multi-criteria optimization (MCO) framework for intensity-modulated proton therapy (IMPT). The approach allows one to simultaneously explore the trade-off between different objectives including robustness and nominal plan quality. Methods: A database of plans each emphasizing different treatment planning objectives, is pre-computed to approximate the Pareto surface. An IMPT treatment plan that strikes the best balance between the different objectives can be selected by navigating on the Pareto surface. We integrate robustness into MCO by adding robustified objectives and constraints. Uncertainties are modeled by pre-calculated dose-influence matrices for a nominal scenario and a number of pre-defined error scenarios (shifted patient positions, proton beam undershoot and overshoot). A robustified objective represents the worst objective function value that can be realized for any of the error scenarios and thus provides a measure of plan robustness. The optimization method uses a linear projection solver and is capable of handling large problem sizes resulting from a fine dose grid resolution, many scenarios, and a large number of proton pencil beams. Results: A base-of-skull case is used to demonstrate that the robust optimization method reduces the sensitivity of the treatment plan to setup and range errors to a degree that is not achieved by a safety margin approach. A chordoma case is analyzed in more detail to demonstrate the involved trade-offs between target underdose and brainstem sparing as well as robustness and nominal plan quality. The robust optimization for each Pareto optimal plan takes less than 5 min on a standard computer. Conclusions: The uncertainty pertinent to the IMPT procedure can be reduced during treatment planning by optimizing a database of plans that emphasize different treatment objectives, including robustness. The planner can then interactively explore all convex combinations of database plans to decide on the most-preferred trade-off.
Medical Physics 06/2012; 39(6):3981. · 2.91 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Purpose: We study a phenomenological tumor growth model for improved target volume definition for radiotherapy of glioblastoma. Currently, an isotropic margin is added to the visible tumor to account for microscopic tumor cell infiltration in normal appearing brain tissue. However, it is known that glioma growth is not isotropic. The ventricular system and the dura, including falx cerebri and tentorium cerebelli, represent anatomical barriers for migrating cells. Such anatomical constraints are currently not consistently and quantitatively incorporated in target delineation. Methods: We assume that tumor growth is characterized by local proliferation of tumor cells and diffusion into neighboring tissue. Mathematically, this can be described via a partial differential equation of reaction-diffusion type, the Fisher-Kolmogorov-Equation. Anatomical constraints are modeled via no- flux boundary. Solving the model equations provides a three-dimensional distribution of the tumor cell density. The radiotherapy target can be defined as an iso-line of the cell density. Results: Two specific questions were investigated: First, tumor locations in which the model may be particularly useful have been identified; and second, a sensitivity analysis with respect to the model inputs was performed. Tumors located in proximity to the falx cerebri may benefit: The model is able to describe both the falx as a boundary and the corpus callosum, which provides a route for tumor cells to spread to the contralateral hemisphere. This effect is often not consistently accounted for in manual target delineation. Among all model parameters, correct segmentation of the brain is a critical prerequisite. Conclusions: The tumor growth model represents a tool to objectively create target volumes for radiotherapy of glioblastoma by consistently accounting for known growth patterns. All model inputs can be linked to model outputs, making it possible to assess the impact of uncertainties - an essential step for an application in clinical practice.
Medical Physics 06/2012; 39(6):3602. · 2.91 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a method to include robustness in a multi-criteria optimization (MCO) framework for intensity-modulated proton therapy (IMPT). The approach allows one to simultaneously explore the trade-off between different objectives as well as the trade-off between robustness and nominal plan quality. In MCO, a database of plans each emphasizing different treatment planning objectives, is pre-computed to approximate the Pareto surface. An IMPT treatment plan that strikes the best balance between the different objectives can be selected by navigating on the Pareto surface. In our approach, robustness is integrated into MCO by adding robustified objectives and constraints to the MCO problem. Uncertainties (or errors) of the robust problem are modeled by pre-calculated dose-influence matrices for a nominal scenario and a number of pre-defined error scenarios (shifted patient positions, proton beam undershoot and overshoot). Objectives and constraints can be defined for the nominal scenario, thus characterizing nominal plan quality. A robustified objective represents the worst objective function value that can be realized for any of the error scenarios and thus provides a measure of plan robustness. The optimization method is based on a linear projection solver and is capable of handling large problem sizes resulting from a fine dose grid resolution, many scenarios, and a large number of proton pencil beams. A base-of-skull case is used to demonstrate the robust optimization method. It is demonstrated that the robust optimization method reduces the sensitivity of the treatment plan to setup and range errors to a degree that is not achieved by a safety margin approach. A chordoma case is analyzed in more detail to demonstrate the involved trade-offs between target underdose and brainstem sparing as well as robustness and nominal plan quality. The latter illustrates the advantage of MCO in the context of robust planning. For all cases examined, the robust optimization for each Pareto optimal plan takes less than 5 min on a standard computer, making a computationally friendly interface possible to the planner. In conclusion, the uncertainty pertinent to the IMPT procedure can be reduced during treatment planning by optimizing plans that emphasize different treatment objectives, including robustness, and then interactively seeking for a most-preferred one from the solution Pareto surface.
Physics in Medicine and Biology 02/2012; 57(3):591-608. · 2.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: PURPOSE: Dose-volume histograms (DVH) are the most common tool used in the appraisal of the quality of a clinical treatment plan. However, when delivery uncertainties are present, the DVH may not always accurately describe the dose distribution actually delivered to the patient. We present a method, based on DVH formalism, to visualize the variability in the expected dosimetric outcome of a treatment plan. METHOD: For a case of chordoma of the cervical spine, we compared two intensity-modulated proton therapy plans. Treatment Plan A was optimized based on dosimetric objectives alone (i.e., desired target coverage, normal tissue tolerance). Plan B was created employing a published probabilistic optimization method that considered the uncertainties in patient set-up and proton range in tissue. Dose distributions and DVH for both plans were calculated for the nominal delivery scenario, as well as for scenarios representing deviations from the nominal set-up, and a systematic error in the estimate of range in tissue. The histograms from various scenarios were combined to create DVH-bands to illustrate possible deviations from the nominal plan for the expected magnitude of set-up and range errors. RESULTS: In the nominal scenario, the DVH from Plan A showed superior dose coverage, higher dose homogeneity within the target, and improved sparing of the adjacent critical structure. However, when the dose distributions and DVH from plans A and B were recalculated for different error scenarios (e.g., proton range underestimation by 3 mm), the plan quality, reflected by DVH, deteriorated significantly for Plan A, while Plan B was only minimally affected. In the DVH-band representation, Plan A produced wider bands, reflecting its higher vulnerability to delivery errors, and uncertainty in the dosimetric outcome. CONCLUSIONS: The results illustrate that comparison of DVH for the nominal scenario alone does not provide any information about the relative sensitivity of dosimetric outcome to delivery uncertainties. Thus, such comparison may be misleading, and may result in the selection of an inferior plan for delivery to a patient. A better-informed decision can be made, if additional information about possible dosimetric variability is presented, e.g., in the form of DVH bands.
[Show abstract][Hide abstract] ABSTRACT: In this paper we look at the development of radiation therapy treatment planning from a mathematical point of view. Historically, planning for Intensity-Modulated Radiation Therapy (IMRT) has been considered as an inverse problem. We discuss first the two fundamental approaches that have been investigated to solve this inverse problem: Continuous analytic inversion techniques on one hand, and fully-discretized algebraic methods on the other hand. In the second part of the paper, we review another fundamental question which has been subject to debate from the beginning of IMRT until the present day: The rotation therapy approach versus fixed angle IMRT. This builds a bridge from historic work on IMRT planning to contemporary research in the context of Intensity-Modulated Arc Therapy (IMAT).