Representative Trajectories Research Articles

대용량 GPS 궤적 데이터를 위한 효율적인 클러스터링

Digital road map generation is primarily based on artificial satellite photographing or in-site manual survey work. Therefore, these map generation procedures require a lot of time and a large budget to create and update road maps. Consequently, people have tried to develop automated map generation systems using GPS trajectory data sets obtained by public vehicles. A fundamental problem in this road generation procedure involves the extraction of representative trajectory such as main roads. Extracting a representative trajectory requires the base data set of piecewise line segments(GPS-trajectories), which have close starting and ending points. So, geometrically similar trajectories are selected for clustering before extracting one representative trajectory from among them. This paper proposes a new divide- and-conquer approach by partitioning the whole map region into regular grid sub-spaces. We then try to find similar trajectories by sweeping. Also, we applied the distance measure to compute the similarity between a pair of trajectories. We conducted experiments using a set of real GPS data with more than 500 vehicle trajectories obtained from Gangnam-gu, Seoul. The experiment shows that our grid partitioning approach is fast and stable and can be used in real applications for vehicle trajectory clustering.

Comparing and Combining Time Series Trajectories Using Dynamic Time Warping

This research proposes the application of dynamic time warping (DTW) algorithm to analyse multivariate data from virtual reality training simulators, to assess the skill level of trainees. We present results of DTW algorithm applied to trajectory data from a virtual reality haptic training simulator for epidural needle insertion. The proposed application of DTW algorithm serves two purposes, to enable (i) two trajectories to be compared as a similarity measure and also enables (ii) two or more trajectories to be combined together to produce a typical or representative average trajectory using a novel hierarchical DTW process.Our experiments included 100 expert and 100 novice simulator recordings. The data consists of multivariate time series data-streams including multi-dimensional trajectories combined with force and pressure measurements.Our results show that our proposed application of DTW provides a useful time-independent method for (i) comparing two trajectories by providing a similarity measure and (ii) combining two or more trajectories into one, showing higher performance compared to conventional methods such as linear mean. These results demonstrate that DTW can be useful within virtual reality training simulators to provide a component in an automated scoring and assessment feedback system.

시각 비동기 오차를 고려한 다중 DME 측위 적용 방안 연구

This paper introduces the time-misalignment error between multiple range measurements acquired by an onboard distance measuring equipment (DME) interrogator and proposes an efficient position determination method that can mitigate the negative effects of the time-misalignment error. The introduced time-misalignment error does not occur in conventional utilization of DME combined with VHF omnidirectional range (VOR). The proposed position determination method projects all the DME range measurements acquired irregularly during an interval to the same time instance where the aircraft position is determined. By the simulation utilizing a representative aircraft trajectory, it is shown that it is possible to estimate the horizontal position accurately without any changes of ground DME facilities.

Modeling and Visualizing Regular Human Mobility Patterns with Uncertainty: An Example Using Twitter Data

Traditional space–time paths show the spatiotemporal trajectories of individuals in one to several days. Based on data for such short periods, these space–time paths might not be able to show regular activity patterns, which are pertinent to various types of planning and policy analysis. Travel data gathered for longer periods might capture regular activity patterns, but footprints captured by these data also include irregular activities, introducing noises or uncertainty. Our objective is to determine the representative spatiotemporal trajectories of individuals, accounting for stochastic disturbances and spatiotemporal variability, but using activity data with longer duration. Therefore, we explore using Twitter data, which have relatively low and irregular spatial and temporal resolutions. This article introduces a methodology to construct individual representative space–time paths using various aggregation and spatiotemporal clustering techniques. To depict and visualize spatiotemporal trajectories with uncertain information, we propose space–time cones of variable sizes to reflect the spatial precision of the paths and use colors on the cones to represent the confidence level. To illustrate the proposed methodology, we use the geo-tagged tweets for an extended period. Our analysis indicates that the representative space–time path reasonably describes an individual's regular activity patterns. As visual elements, cones and cone colors effectively show the varying geographical precision along the path and changing certainty levels across different path segments, respectively.

Hemodynamic and thrombogenic analysis of a trileaflet polymeric valve using a fluid–structure interaction approach

Surgical valve replacement in patients with severe calcific aortic valve disease using either bioprosthetic or mechanical heart valves is still limited by structural valve deterioration for the former and thrombosis risk mandating anticoagulant therapy for the latter. Prosthetic polymeric heart valves have the potential to overcome the inherent material and design limitations of these valves, but their development is still ongoing. The aim of this study was to characterize the hemodynamics and thrombogenic potential of the Polynova polymeric trileaflet valve prototype using a fluid–structure interaction (FSI) approach. The FSI model replicated experimental conditions of the valve as tested in a left heart simulator. Hemodynamic parameters (transvalvular pressure gradient, flow rate, maximum velocity, and effective orifice area) were compared to assess the validity of the FSI model. The thrombogenic footprint of the polymeric valve was evaluated using a Lagrangian approach to calculate the stress accumulation (SA) values along multiple platelet trajectories and their statistical distribution. In the commissural regions, platelets were exposed to the highest SA values because of highest stress levels combined with local reverse flow patterns and vortices. Stress-loading waveforms from representative trajectories in regions of interest were emulated in our hemodynamic shearing device (HSD). Platelet activity was measured using our platelet activation state (PAS) assay and the results confirmed the higher thrombogenic potential of the commissural hotspots. In conclusion, the proposed method provides an in depth analysis of the hemodynamic and thrombogenic performance of the polymer valve prototype and identifies locations for further design optimization.

Entrance dynamics of CH4 molecules through a methane–water interface

We studied the entrance mechanism of methane molecules into bulk water by Molecular Dynamics simulation over a broad time window. We corroborated that the presence of absorbed methane, under the studied thermodynamic state (298K and roughly 10MPa), does not influence the molecular configuration of water interface. Some representative interfacial trajectories were analyzed in detail and we propose an entrance mechanism in which interfacial water is not actively involved in the dissolution process. Finally, we described the Helmholtz Free Energy profile through the interface and obtained the dissolution free energy of methane in water.

A general methodology for n-dimensional trajectory clustering

Trajectory data is rich in dimensionality, often containing valuable patterns in more than just the spatial and temporal dimensions. Yet existing trajectory clustering techniques only consider a fixed number of dimensions. We propose a general trajectory clustering methodology which can detect clusters using any arbitrary number of the n-dimensions available in the data. To exemplify our methodology we apply it an existing trajectory clustering approach, TRACLUS, to create the so-called, ND-TRACLUS. Furthermore, in order to better describe the trajectory clusters uncovered when clustering arbitrary dimensions we also introduce, Retraspam, a novel algorithm for n-dimensional representative trajectory formulation. We qualitatively and quantitatively evaluate both our methodology and Retraspam using two real world datasets and find valuable, previously unknown higher dimensional trajectory patterns.

Nonlinear control of an air-breathing engine including its validation with vehicle guidance

An implementable nonlinear control design approach is presented for a supersonic air-breathing ramjet engine. The primary objective is to ensure that the thrust generated by the engine tracks the commanded thrust without violating the operational constraints. An important constraint is to manage the shock wave location in the intake so that it neither gets detached nor gets too much inside the intake. Both the objectives are achieved by regulating the fuel flow to the combustion chamber and by varying the throat area of the nozzle simultaneously. The design approach accounts for the nonlinear cross-coupling effects and nullifies those. Also, an extended Kalman filter has been used to filter out the sensor and process noises as well as to make the states available for feedback. Furthermore, independent control design has been carried out for the actuators. To test the performance of the engine for a realistic flight trajectory, a representative trajectory is generated through a trajectory optimization process, which is augmented with a newly-developed finite-time state dependent Riccati equation technique for nullifying the perturbations online. Satisfactory overall performance has been obtained during both climb and cruise phases.

Characteristics of Binary Tropical Cyclones Observed in the Western North Pacific for 62 Years (1951–2012)

Abstract The statistical and dynamical characteristics of binary tropical cyclones (TCs) observed in the western North Pacific (WNP) for 62 years (1951–2012) are investigated by using best track and reanalysis data. A total of 98 binary TCs occurred with an annual average of 1.58. The occurrence frequency of binary TCs shows significant year-to-year variations and there are two peaks in the mid-1960s and early 1990s. Three-fourths (76.3%) of the binary TCs occurred between July and September, which is consistent with the high activity season of TCs in general. A relatively higher track density for binary TCs is present to the east of the maximum track density for total TCs. This result is likely due to the differences in the locations of genesis and environmental steering flow between binary and total TCs. The poleward steering flow, weaker vertical wind shear, and warmer sea surface temperature are pronounced for binary TCs, and these result in a longer lifetime of TCs, which can increase the chances that they would be detected as binary TCs. By applying the clustering analysis technique, six representative trajectories of the binary TCs are obtained. The transitional speed and recurving location are significantly different with respect to the clustered types. The trajectories of each type are strongly related to the temporal variations in the environmental steering flow and the location of the North Pacific high.

Influence of nuclear reaction rates on the nucleosynthesis in neutrino-driven winds

We study the sensitivity of the νp-process abundance pattern to (n,p), (p,γ), and (n,γ) rates for nuclei between Ni and Sn. We illustrate our findings for two different initial electron fractions and a representative trajectory. We discuss how these rates influence the abundance pattern and the nuclear flow. Core-collapse supernovae are important contributors to the chemical enrichment of galaxies. They synthesize heavy elements and eject these newly synthesized elements (iron and heavier elements) into the interstellar medium to- gether with elements produced during the star's lifetime. After the explosion, the nascent, hot neutron star cools by emitting neutrinos. These neutrinos interact with the stellar matter and deposit energy, resulting in a supersonic outflow, the neutrino-driven wind. The conditions in the winds are not suitable for the synthesis of the heaviest ele- ments by the r-process. However, lighter heavy elements, such as Sr, Y, Zr, can be formed. Recent hydrodynamic models suggest that very outward in a neutrino wind, the large proton density allows for a small number of neutrons to be produced by νe cap- tures on protons. This permits neutron-induced reactions - such as (n,γ) and (n,p) - to compete with proton cap- tures, electron captures, and β + -decays. Especially (n,p) reactions on proton-rich nuclei have cross sections 10 − 12 orders of magnitude larger than proton captures. There- fore, already a tiny neutron abundance can cause neutron- induced reaction flows competing with proton captures. The (n,p) reactions in the νp-process overcome the wait- ing points in the rp-process which are characterized by slow proton capture rates and long β-decay (electron cap- ture) lifetimes. Depending on the neutron density ρn, the (n,p) reactions regulate the flow from one isotonic chain to the next higher as they are faster than the β + -decays or electron captures. The path of the νp-process lies initially close to the N = Z line and moves towards stability at higher masses.

Numerical Study of a Bi-disperse Gas-solid Fluidized Bed Using an Eulerian and Lagrangian Hybrid Model

In this paper, we present a hybrid model for the numerical assessment of poly-disperse gas-solid fluidized beds. The main idea of such a modeling strategy is to use a combination of a Lagrangian discrete phase model (DPM) and a kinetic theory based TFM to take advantage of the benefits of those two different formulations. On the one hand, the local distribution of the different particle diameters, which is required for the gas-solid drag force, can be obtained by tracking statistically representative particle trajectories for each particle diameter class. On the other hand, the contribution from the inter-particle stresses, i.e. inter-particle collisions, can be deduced from the TFM solution. These then appear as additional body force in the force balance of the DPM. Note that in a first step we solely consider diameter averaged solids stresses since the drag force is at least on order of magnitude higher than the solids stresses in fluidized beds. Finally, the numerical model is applied to a fluidized bed of a bi-disperse mixture of glass particles (0.5mm and 2.5mm particles) and with a cross-section of 0.15 m x 0.02 m. The results are then analyzed with respect to experimental data of Puttinger et al [1]. This comparison demonstrates that the computed bed hydrodynamics is in fairly good agreement with the experiment. However, the results also suggest that sub-grid drag corrections [2–4] for poly-disperse fluidized beds are required to make the numerical investigation of industrial scale fluidized bed units accessible.

Quantifying Macromolecular Conformational Transition Pathways

Diverse classes of proteins such as molecular motors, enzymes, and active transmembrane transporters function through large-scale conformational changes. Computer simulations of these conformational transitions are challenging. A range of coarse-grained and biased simulation techniques have been used to generate individual transitions or ensembles of transition pathways but it has been difficult to compare pathways produced by different methods and so to assess their relative strengths. We introduce a comprehensive method (pathway similarity analysis, PSA) for quantitatively characterizing and comparing macromolecular pathways. The Hausdorff and Frechet metrics (known from computational geometry) are used to quantify the degree of similarity between piecewise-linear curves in configuration space. We tested PSA on a toy system to study the effect of temperature fluctuations (path roughness) and dimensionality. We compare a sample of publicly accessible transition pathway simulation servers and our own dynamic importance sampling (DIMS) MD method for the closed-to-open transitions of the apo enzyme adenylate kinase (AdK). PSA was applied to ensembles of hundreds of trajectories of the conformational transitions of the transporter Mhp1 and of AdK and diphtheria toxin, which were produced by DIMS MD and the Geometrical Pathways algorithm. Clustered PSA enabled the selection of a small subset of representative trajectories for further analysis. A strength of PSA is its use of the full information available from the 3N-dimensional configuration space trajectory, without requiring additional specific knowledge about the system. We show how trajectory analysis methods relying on pre-defined collective variables such as native contacts or geometric quantities can be used synergistically with PSA. We discuss the method's potential to enhance our understanding of transition path sampling methods, validate them, and ultimately help guide future research toward deeper physical insights into conformational transitions.

Fast motion-including dose error reconstruction for VMAT with and without MLC tracking

Multileaf collimator (MLC) tracking is a promising and clinically emerging treatment modality for radiotherapy of mobile tumours. Still, new quality assurance (QA) methods are warranted to safely introduce MLC tracking in the clinic. The purpose of this study was to create and experimentally validate a simple model for fast motion-including dose error reconstruction applicable to intrafractional QA of MLC tracking treatments of moving targets.MLC tracking experiments were performed on a standard linear accelerator with prototype MLC tracking software guided by an electromagnetic transponder system. A three-axis motion stage reproduced eight representative tumour trajectories; four lung and four prostate. Low and high modulation 6 MV single-arc volumetric modulated arc therapy treatment plans were delivered for each trajectory with and without MLC tracking, as well as without motion for reference. Temporally resolved doses were measured during all treatments using a biplanar dosimeter. Offline, the dose delivered to each of 1069 diodes in the dosimeter was reconstructed with 500 ms temporal resolution by a motion-including pencil beam convolution algorithm developed in-house. The accuracy of the algorithm for reconstruction of dose and motion-induced dose errors throughout the tracking and non-tracking beam deliveries was quantified.Doses were reconstructed with a mean dose difference relative to the measurements of −0.5% (5.5% standard deviation) for cumulative dose. More importantly, the root-mean-square deviation between reconstructed and measured motion-induced 3%/3 mm γ failure rates (dose error) was 2.6%. The mean computation time for each calculation of dose and dose error was 295 ms.The motion-including dose reconstruction allows accurate temporal and spatial pinpointing of errors in absorbed dose and is adequately fast to be feasible for online use. An online implementation could allow treatment intervention in case of erroneous dose delivery in both tracking and non-tracking treatments.

Quality assurance for the clinical implementation of kilovoltage intrafraction monitoring for prostate cancer VMAT.

Kilovoltage intrafraction monitoring (KIM) is a real-time 3D tumor monitoring system for cancer radiotherapy. KIM uses the commonly available gantry-mounted x-ray imager as input, making this method potentially more widely available than dedicated real-time 3D tumor monitoring systems. KIM is being piloted in a clinical trial for prostate cancer patients treated with VMAT (NCT01742403). The purpose of this work was to develop clinical process and quality assurance (QA) practices for the clinical implementation of KIM. Informed by and adapting existing guideline documents from other real-time monitoring systems, KIM-specific QA practices were developed. The following five KIM-specific QA tests were included: (1) static localization accuracy, (2) dynamic localization accuracy, (3) treatment interruption accuracy, (4) latency measurement, and (5) clinical conditions accuracy. Tests (1)-(4) were performed using KIM to measure static and representative patient-derived prostate motion trajectories using a 3D programmable motion stage supporting an anthropomorphic phantom with implanted gold markers to represent the clinical treatment scenario. The threshold for system tolerable latency is <1 s. The tolerances for all other tests are that both the mean and standard deviation of the difference between the programmed trajectory and the measured data are <1 mm. The (5) clinical conditions accuracy test compared the KIM measured positions with those measured by kV/megavoltage (MV) triangulation from five treatment fractions acquired in a previous pilot study. For the (1) static localization, (2) dynamic localization, and (3) treatment interruption accuracy tests, the mean and standard deviation of the difference are <1.0 mm. (4) The measured latency is 350 ms. (5) For the tests with previously acquired patient data, the mean and standard deviation of the difference between KIM and kV/MV triangulation are <1.0 mm. Clinical process and QA practices for the safe clinical implementation of KIM, a novel real-time monitoring system using commonly available equipment, have been developed and implemented for prostate cancer VMAT.

Monte Carlo simulation of aerosol evolution in a planar mixing layer

Purpose – The purpose of this paper is to investigate aerosol evolution in a planar mixing layer from a Lagrangian point of view. After using Monte Carlo (MC) method to simulate the evolution of aerosol dynamics along particles trajectories, the particles size distributions are obtained, which are unavailable in mostly used methods of moments. Design/methodology/approach – Nucleation and growth of dibutyl phthalate (DBP) particles are simulated using the quadrature method of moments in a planar mixing layer, where a fast hot stream with DBP vapor is mixing with a slow cool stream without vapor. Trajectories of aerosol particles are recorded. MC method is used to simulate the aerosol evolution along trajectories. Findings – Investigation on aerosol evolution along the trajectories prompts to classify these trajectories into three groups: first, trajectories away from the active nucleation zone; second, trajectories starting from the active nucleation zone; and third, trajectories crossing over the active nucleation zone. Particle size distributions (psds) along selected representative trajectories are investigated. The psd evolution exhibits interesting behavior due to the synthetic effects of nucleation and condensation. Condensation growth tends to narrow down the psd, and form a sharp front on the side of big particle size. Nucleation is able to broaden the psd through generating the smallest particles. The duration and strength of nucleation have significant effect on the shape of psd. Originality/value – As far as the authors knowledge, it is the first simulation of aerosol evolution that takes a Lagrangian point of view, and uses MC simulation along particles trajectories to provide the particles size distribution.

Aeroheating Uncertainties in Uranus and Saturn Entries by the Monte Carlo Method

The 2013–2022 decadal survey (“Vision and Voyages for Planetary Science in the Decade 2013-2022,” National Research Council Rept., National Academy Press, Washington, D.C., 2011) for planetary exploration has identified probe missions to Uranus and Saturn as high priorities. This work endeavors to examine the uncertainty for determining aeroheating in such entry environments. Representative entry trajectories are constructed using the TRAJ software. Flowfields at selected points on the trajectories are then computed using the data parallel line relaxation computational fluid dynamics code. A Monte Carlo study is performed on the data parallel line relaxation input parameters to determine the uncertainty in the predicted aeroheating, and correlation coefficients are examined to identify which input parameters show the most influence on the uncertainty. A review of the present best practices for input parameters (for example, transport coefficient and vibrational relaxation time) is also conducted. It is found that the -uncertainty for heating on the Uranus entry is no more than 2.1%, assuming an equilibrium catalytic wall, with the uncertainty being determined primarily by diffusion and an recombination rate within the boundary layer. However, if the wall is assumed to be partially catalytic or noncatalytic, this uncertainty may increase to as large as 18%. The catalytic wall model can contribute over three times the change in heat flux and a 20% variation in film coefficient. Therefore, coupled material response/fluid dynamic models are recommended for this problem. It was also found that much of this variability is artificially suppressed when a constant Schmidt number approach is implemented. Because the boundary layer is reacting, it is necessary to employ self-consistent effective binary diffusion to obtain a correct thermal transport solution. For Saturn entries, the uncertainty for convective heating was less than 3.7%. The major uncertainty driver was dependent on shock temperature/velocity, changing from boundary-layer thermal conductivity to diffusivity and then to shock-layer ionization rate as velocity increases. While radiative heating for Uranus entry was negligible, the nominal solution for Saturn computed up to 20% radiative heating at the highest velocity examined. The radiative heating followed a nonnormal distribution, with up to a three-time variation in magnitude. This uncertainty is driven by the dissociation rate, as that persists in the hot nonequilibrium zone contributes significantly to radiation.

Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models

Self-consistent N-body simulations are efficient tools to study galactic dynamics. However, using them to study individual trajectories (or ensembles) in detail can be challenging. Such orbital studies are important to shed light on global phase space properties, which are the underlying cause of observed structures. The potentials needed to describe self-consistent models are time-dependent. Here, we aim to investigate dynamical properties (regular/chaotic motion) of a non-autonomous galactic system, whose time-dependent potential adequately mimics certain realistic trends arising from N-body barred galaxy simulations. We construct a fully time-dependent analytical potential, modeling the gravitational potentials of disc, bar and dark matter halo, whose time-dependent parameters are derived from a simulation. We study the dynamical stability of its reduced time-independent 2-degrees of freedom model, charting the different islands of stability associated with certain orbital morphologies and detecting the chaotic and regular regions. In the full 3-degrees of freedom time-dependent case, we show representative trajectories experiencing typical dynamical behaviours, i.e., interplay between regular and chaotic motion for different epochs. Finally, we study its underlying global dynamical transitions, estimating fractions of (un)stable motion of an ensemble of initial conditions taken from the simulation. For such an ensemble, the fraction of regular motion increases with time.

Estimating the effects of age on NHL player performance

Using NHL data for the 1997–1998 through 2011–2012 seasons, we examine the effect of age on scoring performance and plus-minus for NHL skaters (non-goalies) and on save percentage for goaltenders. We emphasize fixed-effects regression methods that estimate a representative age-performance trajectory. We also use a method based on the best performances over time, a method based on the age distribution of NHL players, and a “naïve” specification that does not correct for selection bias. In addition we estimate individual age-performance relationships to obtain a distribution of peak ages. All methods provide similar results (with small but understandable differences) except the naïve specification, which yields implausible results, indicating that correcting for selection bias is very important. Our best estimate of the scoring peak age is between 27 and 28 for forwards and between 28 and 29 for defencemen. Both forwards and defencemen exhibit near-peak performance over a wide range, going from about 24 to 32 and 24 to 34, respectively. Goaltenders display little systematic performance variation over most of the age range from the early 20s to late 30s.

Dynamic Modelling of a 3DOF Medical Parallel Robot with One Decoupled Motion

The paper presents the dynamic modeling of a 3DOF parallel robot of 1PRRR+2PRPaR type using the Lagrange multipliers method in the rigid link assumption. Numerical simulations of the analytical dynamic model, developed using Maple software, on representative trajectories are carried out and these results are finally validated trough numerical simulations in the MBS ADAMS software. Final conclusions are drawn, useful for researchers and practitioners in the robotic field.

Time‐resolved dose distributions to moving targets during volumetric modulated arc therapy with and without dynamic MLC tracking

The highly conformal doses delivered by volumetric modulated arc therapy (VMAT) may be compromised by intrafraction target motion. Although dynamic multileaf collimator (DMLC) tracking can mitigate the dosimetric impact of motion on the accumulated dose, residual errors still exist. The purpose of this study was to investigate the temporal evolution of dose errors throughout VMAT treatments delivered with and without DMLC tracking. Tracking experiments were performed on a linear accelerator connected to prototype DMLC tracking software. A three-axis motion stage reproduced representative clinical trajectories of four lung tumors and four prostates. For each trajectory, two VMAT treatment plans (low and high modulation) were delivered with and without DMLC tracking as well as to a static phantom for reference. Dose distributions were measured continuously at 72 Hz using a dosimeter with biplanar diode arrays. During tracking, the MLC leaves were continuously refitted to the 3D target position measured by an electromagnetic transponder at 30 Hz. The dosimetric errors caused in the 32 motion experiments were quantified by a time-resolved 3%/3 mm γ-test. The erroneously exposed areas in treatment beam's eye view (BEV) caused by inadequate real-time MLC adaptation were calculated and compared with the time-resolved γ failure rates. The transient γ failure rate was on average 16.8% without tracking and 5.3% with tracking. The γ failure rate correlated well with the erroneously exposed areas in BEV (mean of Pearson r = 0.83, p < 0.001). For the final accumulated doses, the mean γ failure rate was 17.9% without tracking and 1.0% with tracking. With tracking the transient dose errors tended to cancel out resulting in the low mean γ failure rate for the accumulated doses. Time-resolved measurements allow pinpointing of transient errors in dose during VMAT delivery as well as monitoring of erroneous dose evolution in key target positions. The erroneously exposed area in BEV was shown to be a good indicator of errors in the dose distribution during treatment delivery.