A finite size pencil beam for IMRT dose optimization

Dose optimization for intensity modulated radiotherapy (IMRT) using small field elements (beamlets) requires the computation of a large number of very small, often only virtual fields of typically a few mm to 1 cm in size. The primary requirements for a suitable dose computation algorithm are (1) speed and (2) proper consideration of the penumbra of the fields which are composed of these beamlets. Here, a finite size pencil beam (fsPB) algorithm is proposed which was specifically designed for the purpose of beamlet-based IMRT. The algorithm employs an analytical function for the cross-profiles of the beamlets which is based on the assumption of self-consistency, i.e. the requirement that an arbitrary superposition of abutting beamlets should add up to a homogeneous field. The depth dependence is stored in tables derived from Monte Carlo computed dose distributions. It is demonstrated that the algorithm produces accurately the output factors and cross-profiles of typical multi-leaf-shaped segments. Due to the accurate penumbra model, the dose distribution features physically feasible gradients at any stage of the iterative optimization, which eliminates the problem of large discrepancies in normal tissue dose due to misaligned gradients between optimized and recomputed treatment plans.

Treatment simulation approaches for the estimation of the distributions of treatment quality parameters generated by geometrical uncertainties

Geometric uncertainties arise during treatment planning and treatment and mean that dose-dependent parameters such as EUD are random variables with a patient specific probability distribution. Treatment planning with highly conformal treatment techniques such as intensity modulated radiation therapy requires new evaluation tools which allow us to estimate this influence of geometrical uncertainties on the probable treatment dose for a planned dose distribution. Monte Carlo simulations of treatment courses with recalculation of the dose according to the daily geometric errors are a gold standard for such an evaluation. Distribution histograms which show the relative frequency of a treatment quality parameter in the treatment simulations can be used to evaluate the potential risks and chances of a planned dose distribution. As treatment simulations with dose recalculation are very time consuming for sufficient statistical accuracy, it is proposed to do treatment simulations in the dose parameter space where the result is mainly determined by the systematic and random component of the geometrical uncertainties. Comparison of the parameter space simulation method with the gold standard for prostate cases and a head and neck case shows good agreement as long as the number of fractions is high enough and the influence of tissue inhomogeneities and surface curvature on the dose is small.

COMPUCELL, a multi-model framework for simulation of morphogenesis

MOTIVATION

CompuCell is a multi-model software framework for simulation of the development of multicellular organisms known as morphogenesis. It models the interaction of the gene regulatory network with generic cellular mechanisms, such as cell adhesion, division, haptotaxis and chemotaxis. A combination of a state automaton with stochastic local rules and a set of differential equations, including subcellular ordinary differential equations and extracellular reaction-diffusion partial differential equations, model gene regulation. This automaton in turn controls the differentiation of the cells, and cell-cell and cell-extracellular matrix interactions that give rise to cell rearrangements and pattern formation, e.g. mesenchymal condensation. The cellular Potts model, a stochastic model that accurately reproduces cell movement and rearrangement, models cell dynamics. All these models couple in a controllable way, resulting in a powerful and flexible computational environment for morphogenesis, which allows for simultaneous incorporation of growth and spatial patterning.

RESULTS

We use CompuCell to simulate the formation of the skeletal architecture in the avian limb bud.

AVAILABILITY

Binaries and source code for Microsoft Windows, Linux and Solaris are available for download from http://sourceforge.net/projects/compucell/

Adapting inverse planning to patient and organ geometrical variation: algorithm and implementation

Image guided radiotherapy has the potential to improve both tumour control and normal tissue sparing by including temporal patient specific geometry information into the adaptive planning process. In this study we present a practical method of image guided adaptive inverse planning based on computed tomography (CT) and portal image feedback during the treatment course. The method is based on a general description of the radiotherapy optimization problem subject to dynamic geometrical variations of the patient/organs. We will demonstrate the feasibility of off-line image feedback into the inverse planning process with the example of three prostate cancer patients. CT and portal images acquired during the early course of the treatment are used to predict the geometrical variation distribution of a patient and to re-optimize the treatment plan accordingly. We will study the convergence of the optimization problem with respect to the number of image measurements and adaptive feedback loops.

Non-coplanar beam direction optimization for intensity-modulated radiotherapy

An algorithm for the optimization of the direction of intensity-modulated beams is presented. Although the global optimum dose distribution cannot be predicted, usually a large number of equivalent beam configurations exists. This degeneracy facilitates beam direction optimization (BDO) through a number of possible approximations and because the target set of good beam configurations is very large. Usually, the target volume is accessible through a finite number of paths of little resistance, which are defined by the properties of the objective function and the global optimum dose distribution. Since these paths can be occupied by a finite number of beams, it is reasonable to assume that a minimum number of beams for a configuration that is degenerate to the global optimum exists. Efficiency of the BDO will be characterized by detecting this degeneracy threshold. Beam configurations are altered by adding and deleting beams. A fast exhaustive (up to 3500 non-coplanar orientations) search finds beam directions that improve a configuration. Redundant beams of a configuration can be identified by a fast criterion based on second-order derivative information of the objective function. This offers a fast means of iteratively substituting redundant beams from a configuration. Inferior stationary states can be evaded by adding more beams than the desired number to the current configuration, followed by the subsequent cancellation of superfluous beams. The significance of BDO is examined in a coplanar and a non-coplanar test case. The existence of a threshold number for the minimum configuration and its dependence on the complexity of the problem are shown. BDO outperforms manual configurations and equispaced coplanar beam arrangements in both example cases.

Potential and limitations of the automatic detection of fiducial markers using an amorphous silicon flat-panel imager

Amorphous silicon electronic portal imaging devices (a-Si EPIDs) allow fast acquisition of high resolution portal images (PI). A visualization of organ movement for adaptive image-guided radiotherapy (IGRT) can be reached by implantation and automatic detection of fiducial markers. A method of automatic detection has been developed for fiducial spherical tungsten markers on PIs, acquired with an a-Si flat-panel imager. The detection method consists of a 2D Mexican hat filter (MHF), whose parameters are tuned to the particular marker signal. The high selectivity of this filter allows a reliable and precise detection of tungsten markers down to a diameter of 1.5 mm. The presented method allows fast, automatic and unsupervised detection of markers. Inevitably, the detection is hampered by image background (bone structures, etc) and noise. A detection success rate higher than 95% was reached, analysing PIs of patients with markers fixed on their skin. Furthermore, this approach to automatic marker detection can also be generalized to elliptic MHFs for the detection of cylindrical markers. The accuracy and detection probability of this method may allow accurate and fast online localization of the considered organ.

On biologically conformal boost dose optimization

A method is described that allows the inclusion of biological imaging data in the optimization of intensity-modulated radiotherapy to produce dose boosts that conform with target subvolumes of potentially reduced radiosensitivity. The biological image (e.g. PET, fMRI, etc) is transformed into a dose efficiency distribution using a piecewise linear calibration function with a prescribed maximum boost factor. Instead of dose alone, the cost function of the optimization algorithm depends on the product of the physical dose times dose efficiency. An example case of a base-of-tongue tumour which was imaged with the hypoxia tracer fluoro-misonidazole is presented, showing the excellent capability of IMRT to produce dose distributions that conform to spatially variable dose prescriptions.

Estimation of a radiation time prolongation factor for intensity-modulated radiotherapy

In general, the deposition of a given target dose requires a longer radiation time for intensity-modulated photon beams (IMBs) than for unmodulated beams. Hence, the routine use of intensity-modulated radiotherapy (IMRT) has repercussions both on the exposure of the patient to scatter and institutional radiation safety. A rule of thumb is presented to assess the maximum prolongation of radiation time for a case class in an idealized setting using static superimposed field segments. The method considers only the degree to which risk structures have to be blocked to meet specified dose restrictions.

On the degeneracy of the IMRT optimization problem

One approach to the computation of photon IMRT treatment plans is the formulation of an optimization problem with an objective function that derives from an objective density. An investigation of the second-order properties of such an objective function in a neighborhood of the minimizer opens an intuitive access to many traits of this approach. A general finding is that only a small subset of the parameter space has nonzero curvature, while the objective function is entirely flat in a neighborhood of the minimizer in most directions. The dimension of the subspace of vanishing curvature serves as a measure for the degeneracy of the solution. This finding is important both for algorithm design and evaluation of the mathematical model of clinical intuition, expressed by the objective function. The structure of the subspace of great curvature is found to be imposed on the problem by conflicts between objectives of target and critical structures. These conflicts and their corresponding modes of resolution form a common trait between all reasonable treatment plans of a given case. The high degree of degeneracy makes the use of a conjugate gradient optimization algorithm particularly favorable since the number of iterations to convergence is equivalent to the number of different eigenvalues of the curvature tensor and is hence independent from the number of optimization parameters. A high level of degeneracy of the fluence profiles implies that it should be possible to stipulate further delivery-related conditions without causing severe deterioration of the dose distribution.

Tools for the analysis of dose optimization: II. Sensitivity analysis

Dose optimization requires that the treatment goals be specified in a meaningful manner, but also that alterations to the specification lead to predictable changes in the resulting dose distribution. Within the framework of constrained optimization, it is possible to devise a tool that quantifies the impact on the objective of target volume coverage of any change to a dosimetric constraint of normal tissue or target dose homogeneity. This sensitivity analysis relies on properties of the Lagrange function that is associated with the constrained optimization problem, but does not depend on the method used to solve this problem. It is useful particularly in cases with multiple target volumes and critical normal structures, where constraints and objectives can interact in a non-intuitive manner.

Tools for the analysis of dose optimization: I. Effect-volume histogram

With the advent of dose optimization algorithms, predominantly for intensity-modulated radiotherapy (IMRT), computer software has progressed beyond the point of being merely a tool at the hands of an expert and has become an active, independent mediator of the dosimetric conflicts between treatment goals and risks. To understand and control the internal decision finding as well as to provide means to influence it, a tool for the analysis of the dose distribution is presented which reveals the decision-making process performed by the algorithm. The internal trade-offs between partial volumes receiving high or low doses are driven by functions which attribute a weight to each volume element. The statistics of the distribution of these weights is cast into an effect-volume histogram (EVH) in analogy to dose-volume histograms. The analysis of the EVH reveals which traits of the optimum dose distribution result from the defined objectives, and which are a random consequence of under- or misspecification of treatment goals. The EVH can further assist in the process of finding suitable objectives and balancing conflicting objectives. If biologically inspired objectives are used, the EVH shows the distribution of local dose effect relative to the prescribed level.

Optimization of intensity modulated radiotherapy under constraints for static and dynamic MLC delivery

Multi-leaf collimators (MLCs) are emerging as the prevalent modality to apply intensity modulated radiotherapy (IMRT). Both the principle and the particular design of MLCs stipulate complex constraints on the practically applicable intensity modulated radiation fields. Most consequentially, the distribution of exposure times across the maximum field outline is either a piecewise constant function in the static mode or a piecewise linear function in the dynamic mode of driving an MLC. In view of clinical utility, the total leaf movement should be minimized, which requires that MLC-related constraints be considered in the dose optimization process. A method is proposed to achieve this for both static MLC fields and dynamic leaf close-in application. The method is an amendment to a generic gradient-based IMRT dose optimization algorithm and solves numerical problems related to the non-convexity of the MLC constraints, which can cause erratic behaviour of a gradient-based algorithm. It employs bistable penalty functions to select preferrable leaf configurations from the configuration space of the MLC, which is limited by specific design features. Together with an 'annealing' escape mechanism from local minima, the algorithm is capable of finding the optimum of an IMRT problem as leaf sequences with minimized leaf travel. In particular, the efficiency of static IMRT can be raised to the levels of unmodulated fields with very few field segments, thereby increasing the utility of IMRT in clinical practice.

A variable fluence step clustering and segmentation algorithm for step and shoot IMRT

A step and shoot sequencer was developed that can be integrated into an IMRT optimization algorithm. The method uses non-uniform fluence steps and is adopted to the constraints of an MLC. It consists of a clustering, a smoothing and a segmentation routine. The performance of the algorithm is demonstrated for eight mathematical profiles of differing complexity and two optimized profiles of a clinical prostate case. The results in terms of stability, flexibility, speed and conformity fulfil the criteria for the integration into the optimization concept. The performance of the clustering routine is compared with another previously published one (Bortfeld et al 1994 Int. J. Radiat. Oncol. Biol. Ph.vs. 28 723-30) and yields slightly better results in terms of mean and maximum deviation between the optimized and the clustered protile. We discuss the specific attributes of the algorithm concerning its integration into the optimization concept.

Strain-specific differentiation of environmental Escherichia coli isolates via denaturing gradient gel electrophoresis (DGGE) analysis of the 16S-23S intergenic spacer region

Denaturing gradient gel electrophoresis (DGGE) was applied to the 16S-23S rRNA intergenic spacer region (ISR) as a means to evaluate strain level differences in Escherichia coli. The ISRs of 81 environmental E. coli isolates obtained from bovine, poultry, and human sources yielded a total of 41 unique DGGE banding patterns, with identical patterns and common bands within each source and no overlapping patterns among sources. An additional 51 isolates from two nearby streams yielded 45 unique banding patterns with no overlap between sites. However, two of the isolates from the streams had identical banding patterns to those from two of the source isolates, resulting in a total of 84 unique DGGE banding patterns out of 132 isolates identified in this study. These results revealed high diversity among environmental E. coli isolates, which made it difficult to unambiguously ascribe strains found in water samples to specific host organisms.

A representation of an NTCP function for local complication mechanisms

A mathematical formalism was tailored for the description of mechanisms complicating radiation therapy with a predominantly local component. The functional representation of an NTCP function was developed based on the notion that it has to be robust against population averages in order to be applicable to experimental data. The model was required to be invariant under scaling operations of the dose and the irradiated volume. The NTCP function was derived from the model assumptions that the complication is a consequence of local tissue damage and that the probability of local damage in a small reference volume is independent of the neighbouring volumes. The performance of the model was demonstrated with an animal model which has been published previously (Powers et al 1998 Radiother. Oncol. 46 297-306).

Monte Carlo dose computation for IMRT optimization

A method which combines the accuracy of Monte Carlo dose calculation with a finite size pencil-beam based intensity modulation optimization is presented. The pencil-beam algorithm is employed to compute the fluence element updates for a converging sequence of Monte Carlo dose distributions. The combination is shown to improve results over the pencil-beam based optimization in a lung tumour case and a head and neck case. Inhomogeneity effects like a broader penumbra and dose build-up regions can be compensated for by intensity modulation.

Intensity modulated photon beams subject to a minimal surface smoothing constraint

A method to smooth intensity-modulated photon beams is presented which can be applied in conjunction with any optimization algorithm. The method employs an additional soft constraint to minimize the area of the surface defined by the photon fluence.

An objective function for radiation treatment optimization based on local biological measures

The implementation of biological optimization of radiation treatment plans is impeded by both computational and modelling problems. We derive an objective function from basic model assumptions which includes the normal tissue constraints as interior penalty functions. For organs that are composed of parallel subunits, a mean response model is proposed which leads to constraints similar to dose-volume constraints. This objective function is convex in the case when no parallel organs lie in the treatment volume. Otherwise, an argument is given to show that a number of local minima may exist which are near degenerate to the global minimum. Thus, together with the measure quality of the objective function, highly efficient gradient algorithms can be used. The number of essential biological model parameters could be reduced to a minimum. However, if the optimization constraints are given as TCP/NTCP values, Lagrange multiplier updates have to be performed by invoking comprehensive biological models.