Computational ecology is an emerging interdisciplinary discipline founded mainly on modeling and simulation methods for studying ecological systems. Among the existing modeling formalisms, the individual‐based modeling is particularly well suited for capturing the complex temporal and spatial dynamics as well as the nonlinearities arising in ecosystems, communities, or populations due to individual variability. In addition, being a bottom‐up approach, it is useful for providing new insights on the local mechanisms which are generating some observed global dynamics. Of course, no conclusions about model results could be taken seriously if they are based on a single model execution and they are not analyzed carefully. Therefore, a sound methodology should always be used for underpinning the interpretation of model results. The sensitivity analysis is a methodology for quantitatively assessing the effect of input uncertainty in the simulation output which should be incorporated compulsorily to every work based on in‐silico experimental setup. In this article, we present R/Repast a GNU R package for running and analyzing Repast Simphony models accompanied by two worked examples on how to perform global sensitivity analysis and how to interpret the results. It is based on the construction of predictive and explanatory models as well as the quantitative description and analysis of ecological data  are much more fine‐grained than their whole‐population aggregated counterpart, and as a consequence, they tend to be more complex requiring more equations, parameters, and processes in order to represent the same phenomenon. That is, not intrinsically a problem or a quality but simply a constraint imposed by the modeling formalism in use, and it is up to the modelers to find the correct trade‐off between the purpose of the model and the level of details which should be part of the model structure.The AbM is being established progressively as a mainstream and valuable tool for modeling complex adaptive systems in many distinct areas of knowledge, ranging from social science, economics to any flavor of computational and systems science such as biology, ecology, and so on  conjecture, which defines the model purpose and why the model is being developed; (2) design phase, where the model is translated to some runnable implementation; (3) experiment step, which means the execution of model following a well‐established plan oriented to confirm or reject the initial conjecture; and finally, the (4) analysis step, where the data generated in the previous step is analyzed with a sound methodology which will generate new insights, uncover model flaws, and iteratively improve the initial conjecture and design , being η is the average value of response variable considering any specific setting for the input factors xi. Therefore, the value of a single model run is given by y=f+ϵ, where ε is difference between the value of y and the expected value E(y) = η. The error ε is consequence of stochasticity introduced by design in the structure of model to capture the population variability. Finally, recognizing that most real‐world models usually have more than one response variable, the structure of an individual‐based model M can be generalized for n outputs as can be seen belowGenerally speaking, the objective of SA is to understand the effect of varying input factors on the model output  to the output and the reference values, even standalone or in combination with categorical calibration. The most commonly used metric is some of the multiple forms of squared deviation, but any distance function can be alternatively employed as long as two properties hold: d=0 if xi and Ry are equal and d>0 when xi and Ry are not equal.That is, what is known as categorical calibration criteria , being X the parameter vector of size k, the sensitivity index Si is given byThe central point of SA methodology is the estimation of sensitivity indices or coefficients. The sensitivity coefficients allow the quantitative comparison of the contributions from distinct parameters to the model output. In its classical form  are, respectively, the variance contribution attributed to the ith parameter and the total variance. The expression V(Y)−Vi represents the total variance with exception of the variance which is generated by the parameter i. The total‐order index STi is the contribution of all input parameters but one, the ith parameter, and hence estimating the effect of that parameter on the variance reduction  for a model with n input parameters can be expressed as shown in Equation V(Y) the total variance from model output and the components Vi, Vij, and Vijk, respectively, the variance contribution from the parameter i, the variance contribution form input parameters i and j, and the variance contribution form input parameters i, j, and k. Finally, the component V12…n expresses the interactions from all parameters present in the model.The total variance The application of Sobol method, as have been mentioned, can be computationally expensive and sometimes could be useful to reduce the problem dimensionality filtering only the most significant parameters or even simplifying the model structure considering only the parameters accounting for the most of the variability in the model output. It can be accomplished using the Morris screening method to rank the importance of input parameters. The Morris method is an OAT method, meaning that it changes just one factor keeping all other input parameters fixed. The input factors are allowed to vary in discrete levels within the relevant parameter range Morris, . The metμ∗ has been suggested by Campolongo, Cariboni, and Saltelli k input parameters and being X= any value from the region of experimentation Ω, the elementary effects are calculated according to the Equation The original work of Morris k input factors and by the p discrete levels for every parameter. The recommendations for the values of p and Δ are, respectively, that the first should be an even number of levels and the second calculated by the expression Δ = p/(2(p−1))  can detect nonmonotonic behaviors such that the suggested standard calculations are not able to capture otherwise | as described in Campolongo et al. r samples from Fi and Gi distributions according to the Equations The metrics of Morris method are calculated over the μ∗, points that the input factor under scrutiny, possibly has a nonlinear behavior having different signs in function of the system trajectory  the Repast Integration Broker, (2) the Repast Integration Engine, 3) The R Integration wrapper, and finally, (4) the R API for Experiment design. A schematic view of package architecture is shown in Figure  The R InFile Sink is Repast component for saving simulation data to a file. The aggregated datasets use some kind of aggregate operation, such as counting, averaging, summing, or any other used defined aggregate operation . On the other hand, usually the dataset collected from Repast model execution will be a time series where the aggregated measure will be collected at fixed intervals. Therefore, some transformation must be applied in order to obtain a value consistent with the functional definition. In addition, even though the value returned from the Repast model was a scalar one, it would add much more maintainable and flexible to act upon it directly from R without making changes in the Repast code. The objective function is also necessary for calibrating, where the output values are compared to some reference data or even for more complex tasks, such as tuning oscillations in the population output. It is also the place for normalizing he model outputs. The objective function is a required parameter for all methods presented here.The last piece of R/Repast architecture is the definition of the objective function which actually allows the flexible definition of the model analysis target decoupling it from the Repast dataset output. As we have mentioned previously, any model is a functional relationship between a vector of input parameters cbind  will be given for facilitating a general idea about these models. The ODD is a protocol  on the R environment. Once the previous steps have been completed, just copy and paste the examples shown in this article, taking care of changing the references to the model installation directory to the directories where the models have been installed locally.5N of times to achieve stable value on the output variance. The objective of the first example is to show the application of a simple method for finding the minimal number of replications of a simulation model which is required for the variance of response variables become stable, converging to a common value. A straightforward way to determine the output stability has been suggested in Thiele et al. Easy.Stability API call.Normally, one of the advantages of using individual‐based models for biological or ecological processes is the possibility of incorporating variability at an individual level. Therefore, unlike deterministic model, in order to get trustworthy results, the simulation must be repeated a number For this example, the BactoSIM , which determines how many transfers should be performed by a single bacterial cell, and the cell cycle point, which defines the time when the conjugative events are likely to occur.The dynamics of bacterial conjugation is modeled as the execution of following set of cellular processes: the cellular division, the T4SS pili expression, the shoving relaxing which avoid bacterial cells to overlap and allow a more realistic colony growth, and the conjugation process. The state variable update is asynchronous. The order of execution of this process is shuffled to avoid any bias due to a purely sequential execution of model rule base, see Figure costT4SS is the total cost of pili expression. The cost applied for a single pilus expression is costT4SS/param(maxpili). The param(maxpili) is actually a constant having the value of 5 for Escherichia coli. The cellCycle parameter indicates two things: the type of modeling rule and its parameter. A value of −1 set the model to conjugate as soon an infected cell finds a susceptible one. Setting the parameter to 0 will randomize the conjugation time between t0 and G. Finally, using a value >0 indicates the specific point in the cell cycle for conjugation. A polynomial equation is fitted to the experimental data where the dependent variable represents the conjugation rate T/(T + R). Setting isConjugative flag to false creates a simulation where the transconjugant cells are sterile; in other words, they are unable to conjugate. The equation is used only for comparing the quality of simulation output.The model input and initialization requires the parameters shown in Table 5.1.4γ0. This intrinsic rate stands for the number of plasmid transfer events, or conjugations on a cell life‐cycle basis. In addition, the global infective wave speed depends directly from the specific point in the bacterial cell cycle when conjugative event is triggered.Basic Principles: Three models differing in the way the conjugation rule is implemented and their results compared to the available experimental data. The best strategy can be used to build models which could serve as a predictive tool for synthetic biology and to explore some aspects which are hard to observe directly in experimental studies of plasmid spread. The key points of this model lies on the idea of the existence of a local or intrinsic conjugation rate, which has been termed Emergence: The model intends to find out what will be the global outcome arising as function of local rules defining the evolution of the bacterial cells and their interaction with adjacent neighbors. With this objective, the model incorporates the most significant aspects of the spatial structure and the behavior of the cellular processes that are related to the conjugation. Specifically, the values of the generation time of donor and transconjugant cells are one of the emergent properties depending from the metabolic penalizations applied both for conjugation event and for the expression of T4SS genes.Adaptation: All agents adapt their growth according to the local availability of nutrient and space.Fitness: It is considered implicitly to the extent that plasmid‐free individuals will present a better adaptation in terms of growth rate than plasmid bearing cells.Prediction: The model is intended to provide prediction regarding the range of possible values for the number of plasmid transfer events per cell cycle and the cell‐cycle point when conjugative transfer is most likely to happen.Sensing: All process defined over the agents implicitly sense the local environment and the close neighborhood for their decisions.Interaction: Bacterial cells interact with their nearby individuals for nutrient access, cellular division, mate pair formation, and plasmid transfer.Stochasticity: Stochasticity is introduced at individual level for all cellular process sampling a normal deviate and fitting the value to corresponding process.Collectives: No collectives are taken into account in this model.Observation: The output target variables will be saved at intervals of 1 min of simulated time.5.1.5R) and plasmid‐bearing (D) cells according to input parameters. The agents are placed randomly within a circular surface centered over the lattice central position. The radius of circle where agents are placed is calculated as function of N0 in order to be consistent to the desired initial cell density  heading, (2) mass, (3) division mass, (4) plasmid P1 infection state, and (5) plasmid P1 infection state. The current position of every bacterial cell in the coordinate system is available implicitly through a Repast API call.The model uses two entity types, namely the agents representing the bacterial cells and a 7.2Every bacterial cell in this model is abstracted as the execution of a series of successive processes capturing the basic tenets of bacterial life cycle. These processes are the nutrient uptake, the bacterial cell growth, the division, and the conjugation. The input parameters required for initializing the model are shown in Table 7.2.1P1 and P2 is simulated. The plasmid P1 is a complete conjugative plasmid containing all genes required for horizontal transfer and the plasmid P2 is a cheater, which having lost its conjugative genes, depends on the T4SS system from the plasmid P1. In other words, the model is used to assess how large should be the cost difference required for the lack of conjugative apparatus become a true competitive advantage making P2 dominate over P1.Basic Principles: The plasmid dispersion depends on an intricate balance between metabolic costs associated to horizontal and vertical dispersion strategies. The conjugative proficiency requires the expression of set of transmembrane proteins which are known collectively as type IV secretion systems . This is an expected result as the rule for the conjugative transfer requires the bacterial cells have achieved a value rounding the 70% of cell mass at division. Other interesting aspect to note is the negative values of first‐order indices. Obviously, the sensitivity indices should not be negative. This is a consequence of a small sample size, and to correct the problem, we must increase it. The other important input factors for the plasmid P1 output are, in order of importance, the probability P(γ0), the cost of plasmid P2, and the cost of plasmid P1, both with similar sensitivity indices.The Figure P(γ0), being the sensitivity index of P1 cost, noticeably higher than the other two indices. This could be attributed probably because the plasmid P2 requires a significant cost difference in order to outcompete the plasmid P1 which transfer vertically.The first‐ and total‐order indices for the model output showing average population size of plasmid P2 can be seen in Figure P(γ0).Finally, in the Figure 8The ecological modeling is a complex subject which can be normally perceived as being simpler than it actually is. Specifically, the individual‐based models are subject to many levels of uncertainty, which means that it is hard to get completely fixed the values of model inputs, the model structure, and the outputs. Normally, there is no complete experimental or observational data to construct mechanistic descriptions of individual, and therefore, many assumptions and simplifications must be made in order to implement a model. The same is true regarding the input values, which are particularly critical in the case of the ecology of microorganism, as normally, just very few input parameters are directly observed and the most of them are estimated from whole population experiments. Therefore, it is always important to bearing mind that modeling is an iterative task which must incorporate compulsorily some what‐if analysis of model outputs.Several methods exist for assessing the uncertainty and for estimating the relative importance of input parameters in the model output. We have provided here and overview on those methods which are based on the variance decomposition because they have a wider application scope and are specifically suitable for their use on individual‐based models. These methods, although conceptually simple, are computationally intensive and can be somewhat hard to apply because the required tools are either unavailable or they do not provide an easy integration pattern. Roughly speaking, the sensitivity analysis methods require the generation of large sample of the parameter space and the model evaluation for each of them which, of course, makes the manual execution an infeasible option.The in‐silico experimentation is becoming a vital tool for understanding complex phenomena in a way that cannot be performed without modeling. The effective application of computational ecology methods requires a high level of proficiency in many diverse domains of knowledge which sometimes are neither feasible nor practical. Therefore, it is indispensable to have a ready‐to‐use arsenal of reusable computational tools for modeling and analysis. In this study, we have introduced the R/Repast package and showed how it can help modelers to improve the robustness and quality of individual‐based models results using the functionalities inside the package for analyzing systematically the model outputs. The package can save much effort for modelers by providing simple wrappers for complex methods within a simple and consistent API. We hope that these R/Repast functionalities can facilitate enormously the systematic analysis of individual‐based models implemented in Repast.None declared.


This paper illustrates the need for validating the calculation of monitor units as part of the process of commissioning a photon beam model in a radiation treatment planning system. Examples are provided in which this validation identified subtle errors, either in the dose model or in the implementation of the dose algorithm. These errors would not have been detected if the commissioning process only compared relative dose distributions. A set of beam configurations, with varying field sizes, source‐to‐skin distances, wedges, and blocking, were established to validate monitor unit calculations for two different beam models in two different radiation treatment planning systems. Monitor units calculated using the treatment planning systems were compared with monitor units calculated from point dose calculations from tissue‐maximum ratio (TMR) tables. When discrepancies occurred, the dose models and the code were analyzed to identify the causes of the discrepancies. Discrepancies in monitor unit calculations were both significant (up to 5%) and systematic. Analysis of the dose computation software found: (1) a coordinate system transformation error, (2) mishandling of dose‐spread arrays, (3) differences between dose calculations in the commissioning software and the planning software, and (4) shortcomings in modeling of head scatter. Corrections were made in the beam calculation software or in the data sets to overcome these discrepancies. Consequently, we recommend incorporating validation of monitor unit calculations as part of a photon beam commissioning process.PACS number(s): 87.53.–j, 87.66.–a Beam commissioning is the process by which parameters are determined for a beam model in a radiation treatment planning system and validated to assess how well the model and parameters reproduce measured dose distributions. Depending on the complexity of the beam model, the commissioning process may be simply a verification that the data has been input correctly, or it may be a complicated procedure of parameter fitting. Regardless of the complexity of the process, the major component of beam commissioning compares dose distributions calculated via the treatment planning system with measured dose distributions.A component of the beam commissioning process that is often overlooked is the assessment of how well the treatment planning system calculates monitor units for beam delivery. This is not unusual, because monitor units are often determined outside the treatment planning system and based on the geometric set up of the patient as well as the dose prescription. As long as the treatment planning system accurately calculates relative dose distributions, one typically can obtain an absolute dose from an independent monitor unit calculation.On the other hand, the ability to calculate monitor units accurately should be an essential component of the commissioning of a beam model in a treatment planning system. Good clinical practice requires that monitor unit calculations be checked independently, and the calculation of monitor units by the treatment planning system can provide this independent check. For dynamic therapies such as dynamic multileaf collimation, the calculation by the treatment planning system may be the only source of monitor units. Finally, the calculation of monitor units in a treatment planning system can be a sensitive indicator of errors in the treatment planning system, either in the dose calculation algorithm or in the specific manner in which the algorithm is implemented.This paper addresses the last of these three arguments for validation of monitor unit calculations. In this paper, we illustrate four examples of situations in which discrepancies in monitor unit calculations, determined during the beam commissioning process, led to identification of errors in the computation of dose in the treatment planning system. In one example, using an in‐house treatment planning system, we found that monitor units, correctly computed at a 100‐cm source‐to‐skin distance (SSD), were systematically in error at other SSD values. This discovery led us to identify a problem in the coordinate system in which beam doses were calculated. In a second case, we found that monitor units in a commercial treatment planning system were calculated differently depending on the order in which beams were computed. After the manufacturer was notified of this problem, they identified an error in the manner in which dose‐spread arrays were stored for photon convolution calculations. In a third case, we found a small but systematic discrepancy in the monitor unit calculation. Analysis of the methods in which beams were calculated led us to conjecture that slightly different approximations were used in the calculations used for beam commissioning from approximations used in treatment planning. A small correction in the beam output corrected this discrepancy. A fourth case, examining monitor units for elongated fields, illustrated possible shortcomings in the way a beam model handled head scatter. Modifications were made in the parameters used to describe this beam component. These modifications compromised slightly on the fit of relative dose distributions but provided a much better monitor unit calculation.It should be emphasized that none of the subsequent discussion is meant to reflect poorly on either the algorithms used in the dose modeling or on the treatment‐planning systems. Once the dose computation code had been modified or work arounds had been established, both treatment‐planning systems were used confidently in our clinic. The significant point is that a comparison of monitor unit calculations is a very sensitive tool for identifying errors either in the dose computation algorithm or the implementation of the algorithm. These errors, although small, are nonetheless errors and need to be identified and corrected and validation of monitor unit calculations should be considered an essential part of the commissioning process for a treatment planning system even if the monitor unit calculations are not used clinically.et al.,,et al.,3 , a commercial radiation treatment planning system. Both algorithms are based on the model that the dose can be expressed as the convolution of incident photon fluence with a dose‐spread kernel as indicated in Eq. μ/ρ is the mass attenuation coefficient, dΨ/dE is the energy fluence spectrum at position r′, and A is the convolution kernel. Although both models begin with the same equation, the implementation of the two models is significantly different, as is the method by which the reference doses are related to measured quantities.Dose calculation algorithms generally compute relative doses normalized to a reference value. If the dose calculation algorithm can relate the reference dose to a measured dose, it is capable of computing monitor units. This study examines two dose‐calculation algorithms. The first algorithm is the fast Fourier transform (FFT) convolution algorithm of Boyer dmax) and reference SSD  for a set of square fields defined by the collimators. For field sizes intermediate to those used in determining the normalization factors, the in‐air fluence is interpolated between the calculated values. In the presence of blocks, wedges, or other beam modifiers, the in‐air fluence is attenuated as the beam passes through the modifiers. Thus for a specified value of in‐air fluence, the absolute dose delivered to a calculation point will be reduced due to beam attenuation. The in‐air fluence is also modified due to beam divergence, accurately reflecting the distance of the calculation point from the source.A component of the process of commissioning the beam model for the COPPERPlan implementation of the FFT convolution algorithm is to scale the in‐air fluence by a normalization factor. Normalization factors are determined for a set of field sizes by matching calculated photon doses to measured doses at a specified reference depth , the treatment‐planning system returns a quantity termed the “beam weight.” This beam weight is the dose delivered to central‐axis 3 treatment planning system, the calculated dose output (dose per monitor unit) is scaled to match measured dose output at a reference depth for a set of square field sizes. The vendor of the treatment‐planning system recommends a reference depth of 10 cm because at that depth, electron contamination is essentially absent, and the entire dose is due to photons. This is the same depth at which photon beam calibration measurements are taken in the TG51 protocol for clinical dosimetry.10 cm×10 cm field. The ratio of the field‐size‐dependent measured output factor to the calculated output factor is used as a field‐size‐dependent correction factor that relates the calculated dose output to the measured dose output for the reference field size at the reference depth. Separation of the measured output factor into a calculated output factor and a field‐size dependent correction factor is analogous to the separation of the measured output factor into the Sp and Sc factors.In commissioning photon beams computed via the convolution/superposition algorithm, as implemented on the PinnacleIn order to compare monitor units calculated via the radiation treatment‐planning system with monitor units obtained via point dose calculations, an extensive test set was developed. This test set was designed to sample a wide range of field sizes, shapes, and depths for isocentric treatments and is summarized in Table Treatment fields, as identified in Table 12 cm×12 cm field for an 18‐MV photon beam  at an SSD of 80 cm. These central‐axis depth doses were then converted to doses for an SSD of 100 cm using the Mayneord F factor, where they were compared with measured central‐axis depth doses for a 12 cm×12 cm field at an SSD of 100 cm. As can be seen in Fig. In initial tests, monitor units computed using the FFT‐convolution algorithm differed from those computed using central‐axis depth dose data by as much as 2.5%. This difference was greater than our quality assurance acceptance criterion of 2%, and the discrepancy was systematic. This discrepancy was the result of the inability of our system to calculate the dose accurately for SSDs other than 100 cm. This discrepancy can be observed in Fig. Examination of the implementation of the FFT convolution algorithm uncovered an error in the implementation. The error in calculation can be explained by noting that the primary photon fluence was calculated in a divergent fan‐line coordinate system. This coordinate system was selected because in a divergent fan‐line coordinate system, it is easy to calculate the primary fluence via ray tracing the primary fluence travels away from the radiation source in the direction of the fan lines. The dose‐spread kernels are also aligned along the fan lines, ensuring the correct spatial relationship between fluence and fan lines. However, in a divergent coordinate system, the calculation grid points are not equally spaced. Errors were caused in the convolution calculation because the FFT convolution algorithm requires equally spaced grid points.12 cm×12 cm field at 80‐cm SSD, calculated in this manner, and corrected to 100‐cm SSD, was identical to that of the measured dose distribution for a 15 cm×15 cm field. This is precisely the field size that would be seen by the rays that define the primary fluence if the rays were nondivergent.One solution to this problem is to map the fan‐line coordinate system onto a Cartesian coordinate system. Calculation grid points then lie at uniform intervals, and one may then proceed with the FFT convolution. This is, in fact, what has been done to generate the dose distributions whose central‐axis depth doses are illustrated in Fig. In order to correct this problem, the software was modified to calculate the photon fluence in the fan‐line coordinate system but then transform the fluence into a Cartesian coordinate system before convolving it with the dose‐spread kernels. In this coordinate system, the kernels are spatially invariant, so the FFT convolution is accurate. A small error was introduced because the dose‐spread kernels were no longer aligned with the fluence, but this error was found to be negligible.The accuracy of the central‐axis depth‐dose calculation using the transformed fluence is illustrated in Fig. 3 treatment‐planning system, setting up beams for the open fields and each wedge. Beams were queued, and the doses were computed. For most fields, treatment‐planning system calculations agreed with point dose calculations to within our quality assurance criterion of 2%, but discrepancies of up to 5% were observed for wedged fields, especially for 60° wedged fields. In the process of analyzing the cause for the discrepancies, we repeated the dose computation but reversed the order of the beams for calculation, i.e., we began computing with the 60° wedged fields and ended with the open fields. We were surprised to find different values for the monitor units. On this second pass, the monitor units for the 60° wedged fields agreed more closely with the point dose calculations, while monitor units for the open fields showed greater discrepancies. It appeared that the order in which calculations were performed affected the values of the monitor units calculated using the treatment‐planning system. This discrepancy was reported to the vendor of the treatment‐planning system.In the second example, we performed the beam calculations on the PinnacleWithin a week, the vendor identified the cause of the problem. When applying the convolution model to a polyenergetic beam, one properly needs to integrate over the energy spectrum as well as over the three spatial dimensions. In the convolution/superposition algorithm, the energy integration is replaced by energy‐averaging the incident fluence as well as the dose‐spread array.After the vendor corrected this error in the implementation of the convolution/superposition algorithm, we resumed testing of the monitor unit calculations. With beam‐specific energy‐averaged dose‐spread arrays, monitor units were correctly calculated independent of the order in which the beams were computed. For square fields, monitor units calculated using the treatment‐planning system agreed with point dose monitor units to well within the 2% accuracy criterion. However, we still observed a systematic difference between the two sets of monitor unit calculations. In all cases, monitor units computed using the treatment‐planning system were greater than those computed using a point dose calculation, typically by 0.5% to 1.0%. Although the differences were within our accuracy criterion, it was still necessary to investigate a systematic discrepancy.We believe the systematic discrepancy may be due to differences in the manner in which ray tracing is done in the beam‐commissioning mode from the way it is done when doses are computed in the treatment‐planning mode. In the beam‐commissioning mode, ray tracing is done from the true surface of a water phantom to dose calculation points. In the treatment‐planning mode, no phantom surface is explicitly defined; rather, ray tracing is begun at the proximal surface of the first voxel whose CT value corresponded to anything other than air. Because the true surface of the phantom is generally taken to be at the center of the voxel, the increased length of the radiological path caused by tracing from the proximal surface of a voxel led to a small increase in the depth of the dose calculation point and a corresponding increase in monitor units required to deliver the dose prescription. Figure In our clinical treatment planning, we elect to ray‐trace through the three‐dimensional CT array rather than ray‐trace from a presumed surface at central‐axis SSD. Consequently, we increased the value of the machine output (dose per monitor unit) entered into the treatment planning system by approximately 0.5% to remove the systematic discrepancy from the monitor unit calculations.(>2%) than those obtained from point dose calculations. Moreover, the calculated values depended on the orientation of the elongated field to a greater extent than was found by measurement. The calculated values also depended on the location of the collimators. When the collimators defining the narrow dimension of the field were closer to the phantom, the discrepancy between calculation and measurement was greater than when the collimators defining the wide dimension of the field were closer to the phantom. When we set both sets of collimators at the same distance from the source, the orientation dependence was removed. We therefore concluded that the cause of the discrepancy had to do with the modeling of the radiation coming from the collimators of the linear accelerator.Monitor units calculated for elongated fields remained unacceptably greater Within the beam model, the contribution to the dose most directly affected by the geometry of the collimators is the scatter from the flattening filter. In the treatment‐planning system, this scatter is modeled by adding to the in‐air photon fluence a contribution calculated by convolving a Gaussian distribution with a unit mask whose dimensions are identical to those of the collimator‐defined field.In the several examples cited in this paper, we have demonstrated that testing the ability of the treatment‐planning system to calculate monitor units can help us identify subtle errors in the implementation of a dose‐calculation algorithm and in the data sets used in the dose model. In practical applications, the user of the treatment planning system cannot correct errors in the coding of the dose‐calculation algorithm as we were able to do with COPPERPlan. However, by performing an extensive series of tests with a commercial treatment planning system, we were able to identify errors in the system and provide the vendor with significant information to assist them in determining the cause of their coding errors. In reality, no code is completely error free,Even if the implementation and coding of a dose‐calculation model is correct, it must be recognized that any model is an approximation to reality. Fitting model parameters for one aspect of the dose‐calculation model, such as off‐axis profiles, may compromise the accuracy of the dose calculation elsewhere, such as the calculation of monitor units. Furthermore, by understanding differences in the dose computation in various components of the treatment planning system, the user is able to make modifications in the beam data in order to generate a more accurate dose calculation. Commissioning a beam model represents a set of compromises, and all aspects of the beam computation must be analyzed to determine the extent of compromise necessary.The tests that are described in this paper are relatively easy to set up and require approximately four to six hours to compute and analyze. Once the tests are set up, they may be stored and repeated with other beam models when additional beams are being commissioned. The time spent in performing these tests is small compared to the confidence in the dose calculation algorithm that performing these tests can instill in the user.


In order to investigate the effect of the number of projections on digital tomosynthesis image quality, images were acquired over a 40 degree arc and sampled into sets of 2 to 41 projections used as input to three different reconstruction algorithms: the shift‐and‐add, the Feldkamp‐Davis‐Kress filtered back projection algorithms, and the simultaneous algebraic reconstruction technique. The variation of several image characteristics, such as in‐plane resolution, contrast to noise ratio, artifact spread, volumetric accuracy, and dose, are investigated based on the reconstruction algorithms used and also the number of projections used as source data. The results suggest that only 11 projections are required since the various parameters checked do not improve much past that number. As a reconstruction algorithm, SART did best but took much longer to reconstruct images. Thus, if reconstruction time is a determining factor, filtered back‐projection looks like a better compromise.PACS number: 87.57.C‐, 87.57.nf, 87.57.Q‐, 87.59.‐e While a lot of work has been done with modified linear accelerators or high efficiency receptors to obtain all the projection images required for a complete CT reconstruction, have shown that images can be acquired with very low doses, but the machine used by these authors had specialized imaging equipment solutions, optimized for imaging at low doses. The present work uses a linear accelerator without such specialized equipment. If one MU is needed for each image, the dose that would be given to the patient for the acquisition of the approximately 100–200 projection images required for a full CT reconstruction becomes too large to be acceptable. The aim of this work is to explore the use of a reconstructed image set which will mimic a CT dataset while only requiring a fraction of the images needed for CT reconstruction.With the advent of Intensity Modulated Radiation Therapy (IMRT) and the ability to produce dose distributions that are highly conformal to the tumor volume, correct patient positioning becomes of prime importance. Solutions to this problem range from invasive techniques where markers or radio frequency beacons are directly placed inside the tumor, to non‐invasive techniques using ionizing radiation as is the case with computed tomography (CT) or using non‐ionizing radiation as exemplified by the use of ultrasound imaging.–6 Wi(–) However, most of this effort concentrates on using data from a kV beam, and only a recent few publications look at the use of mega‐voltage (MV) images for the purposes of digital tomosynthesis. The use of the treatment beam and existing digital receptors to image the patient offers the main advantage that there is no need for further equipment purchase, as would be the case when using kV images that need a kV unit and an additional image receptor. While the two recent publications on MV‐based digital tomosynthesis looked at only one reconstruction algorithm – namely, filtered back‐projection based on the Feldkamp‐Davis‐Kress(FDK) methodology for kV images, the inherent low contrast of MV images warrants this investigation since the results may change due to the quality of the source images. Whereas the present work repeats a small part of the investigation by Descovich et al.,The technique of tomosynthesis was first introduced in 1932 by Ziedses des Plantes.–15 HThis algorithm works by using geometry to line up projection images taken at different angles such that objects on the plane of reconstruction become more prominent while out‐of‐plane objects are smeared out. Objects at different heights above and below the isocenter will cast shadows at different parts of the detector, depending on what angle is used for image acquisition. Therefore, with prior knowledge of the angle at which the projection is taken, and depending on the height of the reconstruction plane relative to the isocenter location, each pixel is shifted by an appropriate amount. This is diagrammatically represented in Fig. As shown on the top row of images, the simple addition of the three acquired images produces an overlay of structures. However, once the appropriate shifts have been applied to each image, addition of the resulting images give rise to tomogram showing objects present at that level in high contrast whereas the out‐of‐plane objects, while still present, show up with much lower contrast.h is the distance of the particular pixel from the origin (at the center of the image) in the reconstructed plane, i is the distance from the origin in the receptor plane, d is the source‐to‐image receptor distance, and α is the angle at which the projection is obtained. The second step shifts each pixel by an amountThe algorithm developed by Kolitsi et al.a.Once all images have been shifted, they are added and normalized to give the final tomogram at level g is the value of the object at location , β is the angle relative to the z‐axis, D is the source to isocenter distance, R is the projection data acquired at angle β,t=xcosβ+ysinβ,s=ycosβ−xsinβ, and  represents the coordinate system of the detector. The algorithm works in three steps:First, the projection data is scaled as if it were measured at the plane containing isocenter.Second, each row of each projection is individually filtered. Typically, a ramp filter (in frequency domain) is used to remove the radial blurring that occurs during the back projection process. The filtering is usually done in the frequency domain where the process involves multiplication of two functions rather than the computationally expensive convolution process that would be required in the Cartesian domain. For this project, a Hamming window was used to filter the images.Finally, the filtered and weighted data is back‐projected over a grid to reconstruct the object.This is one version of the filtered back projection algorithm that is widely used for CT reconstruction. The algorithm was originally proposed by Feldkamp, Davis and KressFor a complete mathematical description of the algorithm, refer to Feldkamp, Davis and Kress.j represents the index of the voxel v under consideration, k is the iteration number, ωij is the contribution of voxel j to pixel i, and Λ is a constant of proportionality called the relaxation parameter, usually chosen to be between 0 and 1, with most studies choosing it to be closer to zero than unity. In this study, the relaxation parameter was set to 0.2. For this project, one iteration was defined to be complete when all projected images acquired had been processed, with the object updated at each step. Thus, iterations were stopped after a preset number of iterations, rather than quantifying the difference between projected data and measured data and stopping once that difference became less than a preset value.This is a member of the family of algebraic reconstruction techniques which uses an iterative method to solve a large system of linear equations. The simultaneous algebraic reconstruction technique is one that updates the value of each voxel one projection at a time. Each voxel of the object to be reconstructed is assigned an initial value (zero in our case). A forward projection is performed and each voxel's value is proportional to the weighted difference between calculated and measured values of all pixels to which that particular voxel contributes. The general equation for the algorithm is512×512 matrix with 12 bit precision. The CCD camera is reported to have a dynamic range of 60 dB and a contrast detectability of at least 1%. For the purposes of this project, the following naming convention is used to denote gantry angles: the zero position (0°) is when the gantry is positioned for an anterior‐posterior field. Positive numbers are used for angles measured clockwise from the zero position, while negative numbers are used for angles measured counterclockwise from the zero position.For all experiments using portal images, the images were acquired using an add‐on CCD camera EPID  mounted on a Varian clinac 600 C/D . According to the manufacturer's technical specifications, the images are acquired in a −20° to 20°. With this acquisition geometry, the detector followed a circular path such that the distance between the source and imager was constant.Based on the investigation of various tomosynthesis angles reported before in literature, a middle value of 40 degrees was chosen for this study. A total of 41 projection images were acquired every degree from The images obtained were sampled such that anywhere from 2 to all 41 projections acquired were used as input to the three reconstruction algorithms. Table Each algorithm was used to reconstruct the same number of slices (50) with a nominal slice separation of 5 mm. The in‐plane voxel dimension was set to 1 mm for the datasets reconstructed with the FDK algorithm and SART. The SAA technique uses the native pixel dimension of the EPID, which was 0.64 mm in this case. All source images were acquired with the EPID positioned 50 cm below the isocenter level. However, the images were resampled to obtain a 1 mm in‐plane voxel dimension for fair comparison with the other datasets reconstructed using the higher dimension.(2.5cm×5cm) of aluminum and nylon sheets were used to create individual patterns representing patterns with one line pair every 2 mm, 4 mm, 6 mm, 8 mm and 10 mm. Five individual bar patterns were created, each containing four aluminum ‘bars’ separated by nylon ‘bars’ to create 3.5 line pairs per set. These blocks were placed on a block of Styrofoam and positioned in a water‐filled container and the phantom was positioned such that isocenter was at the center of the bar patterns as show in Fig. An in‐house phantom was developed to visually assess the spatial resolution. Small sections Since the visual inspection is highly subjective, a second experiment was performed to measure the modulation transfer function (MTF) for each reconstructed dataset. The phantom in this case was a 6 mm‐thick slab of lead used to create a straight edge in the reconstructed datasets. The MTF was calculated as the Fourier transform of the gradient of a line profile taken through the edge.0.28g/cm3 and 1.69g/cm3 were positioned in a water‐filled container such that their long axis was parallel to the gantry's rotational axis. The phantom was positioned such that the isocenter was between the two cylinders as shown in Fig. Two cylindrical inserts  with relative electron densities of 15×15mm2 region of interest (ROI) to sample the same region in the same slice for each dataset. For each cylindrical insert, the mean signal strength was extracted from the ROI placed in the middle of the cylinder. For water measurements, the ROI was placed in a region away from the cylinder location on the isocenter slice. The contrast was characterized by the signal difference to noise ratio (SDNR) defined asμobject is the mean signal in the ROI within the cylinder being considered, μwater is the mean ROI value in the water region immediately adjacent to that cylinder and σwater is the standard deviation of the ROI in that water region.Datasets were reconstructed using all three algorithms and imported into ImageJ. A macro was created to use a z0 is the depth of the actual feature, z is the depth of an out‐of‐focus plane containing the artifact, μ¯Feature(z0) and μ¯BG(z0) are, respectively, the mean pixel value within the feature and in the background at depth Z0, and μ¯Artifact(z) and μ¯BG(z0) are, respectively, the mean pixel value within the ghost image and background at depth z.Due to the use of projection images acquired over a limited angle, all CBDT reconstructions contain ghosting artifacts which are more pronounced in the reconstructed slices immediately adjacent to the plane on which the object is present. Wu et al.To test for the variation in ASF with depth, a quarter was suspended inside a water‐filled container which was positioned on the couch so that the coin was at isocenter level as shown in Fig. 100×100×100mm3 virtual water phantom was created using MatLab , with voxel size of 1×1×1mm3. Within this phantom, a virtual cube, sphere, and cylinder of bony material were inserted. The phantom was imported into the Pinnacle treatment planning system  as a CT set and the forward projection capabilities of the system was used to export digitally reconstructed radiographs (DRRs) every degree for gantry positions from −20° to 20°. The mean energy used for the DRR calculation was set to 2 MeV to ensure MV‐quality images. These DRRs were sampled as shown in Table In order to check for volumetric accuracy, a All calculations were performed on a dual‐core AMD Turion machine with a nominal processor speed of 1.9 GHz and 2 GB of RAM. All algorithms were programmed in‐house using Visual C++ . Table 3mm×3mm×1mm lithium fluoride (LiF) thermoluminescent dosimeters (TLDs) chips were positioned at different points within the Rando pelvic phantom . The TLDS were batched with responses within ±5%. The calibration was done by exposing one packet of TLDs to a known dose at the depth of maximum dose in solid water and then using that calibration factor to obtain the doses from the other packets used in the experiment.To determine the dose delivered to different points during image acquisition, Three points within the phantom were identified as locations of the bladder, prostate, and rectum. One packet of three TLDs was positioned at each point and images were acquired for 7, 11, and 14 projections over the 40° span. The minimum of 1 MU per image was used to acquire each projection. A CT study of the Rando phantom was also obtained and imported into Pinnacle3. The expected doses were calculated for the same three points using the treatment planning software and compared to the measured doses when these were available.Figure Figure Figure Figure Figure The spatial resolution experiment shows that the limiting resolving power of 2 mm is achieved almost immediately regardless of the algorithm used for reconstruction. It should be made clear that all of these resolution measurements are for the in‐plane measurements, since the out‐of‐plane resolution is expected to be worse for a limited‐angle scan. If only these results were considered, it seems like there is no advantage of any one algorithm over another once the minimum of three projections is met. However, combining the contrast to noise test, some algorithms start to stand out. From Fig. It is not surprising that ASF is worst for lower number of projections used as input data. It was already stated that the out‐of‐plane objects are still partially present in every slice. The ghost signals are actually smeared along the planes not containing the object. When only two projections are used, then the ghosts actually form two distorted images, a doublet. The ghost intensity is shared between these two artifacts. As more and more projections are used, the intensity is actually smeared across a larger area. Since ASF looks at a ratio of ghost contrast to object contrast, the ASF gets smaller in magnitude when more projections are used since contrast of the ghosts gets smaller.The volumetric accuracy test showed that the calculated volumes always trended towards a constant value with an increasing number of projections used for the dataset calculation. For the two objects that showed constant cross‐sections in the DTS slices – namely the cube and cylinder – the calculated volumes got very close to the true value as the number of projections used increased. For objects with a varying cross‐section, the calculated values were always overestimated and, while there was convergence in volume, it was towards a value that was more than twice the true volume. With prior knowledge of what the objects looked like, it was easy to determine how to contour objects which had a constant cross‐section, and also to determine when the object was not actually present on a slice because the signal was only partially overlapping see Fig. . For theHowever, for the spherical object, the situation is more complex. Unlike an object with constant cross‐section, incomplete overlap of signals does not mean absence of the object from the slice but, instead, may mean that the object is only partially present. Figure An effect also visible in all reconstructions using very few projections is that the ghost images sometimes actually appear as a ringing artifact. This effect is illustrated in Fig. Distortions in the shape of objects are inherent with the technique of digital tomosynthesis due to the limitation of imaging only over a short arc and never seeing the whole object. This, in turn, means that DTS‐based datasets may not be used for such tasks as treatment planning in their native form where a correct estimation of volumes is very important. However, the spatial accuracy of reconstructed images was also investigated. This was done by measuring the distance between the various bar patterns used in the spatial resolution phantom. When the image‐based distances were compared against those physical measurements, all agreed within 1 mm (or one pixel). This led to the conclusion that the reconstructions are accurate in terms of spatial accuracy.The dose measurements show the expected increase in dose to all three points considered with an increasing number of projections used. When compared with the doses each point gets from the current portal imaging protocol, it is clear that approximately the same dose is delivered to acquire the portal image set as would be given during the acquisition of 11 projections.The purpose of this project was to determine the effect that the number of projection images used to reconstruct MV‐CBDT datasets had on various aspects of these reconstructions. The current imaging protocols for portal images gives the same amount of dose that would be given to acquire 11 projection images. Keeping this number of projections in mind, as far as the spatial resolution is concerned, all of the bar patterns are already resolved, except for the 1 mm pattern which never is. For all algorithms, the SDNR and reconstructed volume values seem to have already reached a plateau. ASF values will increase slightly when more projections are used. Since reconstruction times are linear with number of projections, the reconstruction times are on the order of less than 1 minute for SAA and FDK to 21 minutes for one iteration of SART.The authors recognize that this evaluation is purely qualitative but also feel that, when all the tests are evaluated together, 11 projections seem like a good number of projections to aim for, especially because some of the tests – specifically, the SNR and volumetric accuracy investigations – seem to converge once this number of projections has been reached. For the ASF evaluation, there is a very small improvement when more than 11 projections are used but, given the increased dose with more projections, the authors do not believe the gain is worth the risk from increased radiation.As it stands, the SAA algorithm does not look like a good candidate for reconstruction since artifact spread is so high compared to the other algorithms. While FDK produces noisy images, reconstruction times are much lower than SART, even for a single iteration. Both SART and FDK produce comparable artifact spread when 11 projections are used. The choice of algorithm between SART and FDK depends on the user. If this is to be clinically used, then the current reconstruction times for SART makes the algorithm impractical for the purposes of image reconstruction. Therefore, for clinical use, it would seem like FDK would be a better choice, if the end user is willing to live with the noisier images. The FDK images may be improved in certain cases where spatial resolution is not as important as contrast and a more aggressive filter is used for the purposes of noise removal. If, on the other hand, the problem of reconstruction times is solved either with better software implementation or with hardware acceleration, then SART would certainly be the algorithm of choice.It should be pointed out that this study's conclusions are specific to digital tomosynthesis with projections acquired using a CCD‐based camera and based on the phantom geometries investigated. Artifact spread is a known problem with digital tomosynthesis and the contrast and resolution of images will be affected by out‐of‐plane structures, something that will change between subjects. However, this is the first study, to the best of the authors’ knowledge, investigating the effect of number of projections and reconstruction algorithm on the final quality of images for megavoltage digital tomosynthesis. The authors are confident that the conclusions, demonstrating that MV‐CBDT reconstruction does not require the use of projections acquired every degree, will remain valid for other phantom geometries and image acquisition equipment since most parameters end up reaching a plateau after a critical number of projections is used. Future work should repeat the current experiment while varying the tomosynthesis angle, and also investigate whether imagers with different sensitivities produce different results.


We propose a novel LASSO (least absolute shrinkage and selection operator) penalized regression method used to analyze samples consisting of  related individuals. Developed in the context of linear mixed models, our method models the relatedness of individuals in the sample through a random effect whose covariance structure is a linear function of known matrices with elements combinations of the condensed coefficients of identity between the individuals in the sample. We implement our method to analyze the simulated family data provided by the 19th Genetic Analysis Workshop in an effort to identify loci regulating the simulated trait of systolic blood pressure. The analyses were performed with full knowledge of the simulation model. Our findings demonstrate that we can significantly reduce the rate of false positive signals by incorporating the relatedness of the study participants. Current association methods for quantitative phenotypes are mostly designed for use with unrelated individuals. However, often studies include information on families or related individuals. Also, many times samples assumed to include only independent individuals may suffer from underlying cryptic relatedness among the participants. Using related individuals in statistical analyses has many advantages, such as increased ability for quality control and robustness of the results with respect to population stratification. Ignoring relatedness between study participants can have significant impact on the study results and increase false positive rates . Linear Because of the computational intensity involved in the estimation of the parameters of the LMM, most methods perform single marker analysis . As GWASth Genetic Analysis Workshop (GAW19) [We have developed a new association method that combines LMM and LASSO to capitalize on the benefits of both approaches. Our method handles multiple markers and models  common genetic background among individuals through the use of a random polygenic effect. It employs a LASSO algorithm  to estimshop GAW1 [9] to sn  related individuals. Let y =  be the vector of the phenotypic values of the quantitative trait of interest. We assume that for each person i the value of his/her phenotype is determined  by a number, say S, of  loci/SNPs, hereafter referred to as quantitative trait loci (QTLs). Furthermore, the trait is also influenced  by K known covariates Xi, i = 1,…, K, . In addition, there is some environmental effect or residual, denoted as ei. Finally, we suppose the phenotype is affected by a number of other loci whose collective (random) polygenic effect on the trait is denoted as ui. Assuming additivity across all effects, the overall phenotypic value of a person isβ =  is a vector of unknown coefficients for the covariate fixed effects, γ =  is a vector of unknown coefficients of the fixed effect of the SNPs, Zi =  a vector of 0, 1, or 2 s, denoting the number of copies of the disease (minor) allele carried by the individual at each of the loci. Note that, if a SNP does not contribute to the trait levels, we expect its corresponding coefficient γs to be equal to zero. We assume that the vector of random polygenic effects u =  follows a multivariate normal distribution with mean 0 and covariance matrix Ω, while the errors eis are independent and identically distributed coming from a normal distribution with mean 0 and variance. Finally, the two random effects, ui and ei, are considered independent of each other and the covariates. Under these assumptions, the joint distribution of the yi’s is a multivariate normal distribution with mean equal to βX + γZ and variance covariance matrixXT = , ZT = , and I is the n × n identity matrix. If there is minimal or no inbreeding among the individuals in the sample, then the matrix Ω is approximately equal toΦ and Δ are known matrices whose elements are functions of the condensed coefficients of identity between the individuals in the sample, and σap2 and σdp2 are the additive and dominance polygenic variance of the trait [Consider a sample of he trait , 10.γs are expected to be negligible, a LASSO method may be useful as it generally forces most of the parameters in the model to shrink toward zero with only a small number of the parameters remaining significantly different from zero. In our case, the penalized likelihood function that needs to be maximized to estimate the necessary parameters is given by the following equation:θ =  and λ is a nonnegative penalty parameter. Maximization of the above function can be achieved using an algorithm similar to that described by Schelldorfer et al. [Estimation of the model parameters is done by maximizing the corresponding multivariate normal likelihood. The fitting of the model, though, is computationally intense and plagued by many issues because of the high dimensionality of the parameter space and the small available sample size. However, as most r et al. .For our analyses, we used the family-simulated data sets from GAW19 [We used the SBP at the last known appointment as the phenotype of interest and the imputed genotypes provided as those were used in the generation of the simulated phenotypes. Because of the large number of SNPs available, greater than 8,000,000, we preprocessed the genotypes to reduce the number of SNPs used in the analyses to reduce the computational burden. We excluded SNPs with minor allele count less than 10 copies in the entire data set  to avoid potential problems with unstable estimates of the model parameters in equation  were used to fit the LASSO model. In all analyses in phase I or II, phenotypes were adjusted for gender, age, smoking status, and blood pressure medication.The number of SNPs on the map was prohibitively large for directly implementing our approach. So, we implemented a 2-step scheme to reduce the number of SNPs used when fitting the penalized model. Two-step methods are routinely employed when the computational intensity of a method prohibits direct use of the method on a GWAS . In phasp <1E-05. A SNP was called as “significant” if its p value was less than the given threshold  or if its coefficient was different from zero in the optimal LASSO model. We present the results for 3 LASSO analyses. The first, Add, represents an analysis in which the random polygenic effect was assumed to consist of only additive genetic variance, that is, the covariance structure was assumed to be Ω = 2Φσa2. The second model, Add-Dom, assumed that the trait had also a dominance effect and thus the matrix Ω was set to have the form Ω = 2Φσa2 + Δσd2. Finally, for comparison, we also implemented the standard LASSO method that assumes that all individuals in the study are unrelated, denoted as Ind. The standard LASSO approach is a subcase of our approach where matrix Ω is the 0 matrix.The results from the analyses of all replicates are summarized in Table The first group of columns in the table, “Significant SNPs,” summarizes the number of SNPs declared “significant” by each method. The second section, “QTLs discovered,” summarizes the number of QTLs declared “significant” and is used to gauge the TDR of the method. This column uses a very strict definition of what constitutes “true” hit. A more relaxed rule for what constitutes a true signal considered as a true hit any SNP that resided within (an arbitrary) 2 kbp from any QTL. There were 2248 SNPs on the map that were within 2 kbp from a simulated QTL. The results using this definition are summarized in the “SNPs within 2 kbp from QTL” section of the table.From Table The results from the LASSO analyses show that modeling the relatedness of the individuals in the sample pays off, regardless of whether we did (Add-Dom) or did not (Add) model the dominance effect. The LASSO method for related individuals, on average, reduced the number of SNPs with nonzero coefficients in the final model relatively to the corresponding number from the standard LASSO method (Ind). For example, when the Bayesian information criterion (BIC) was used to select the best model, the Ind model resulted in 15 significant SNPs, whereas both the Add and the Add-Dom models included, on average, approximately 7 “significant” SNPs,, marking a reduction of approximately 53 % in the number of “significant” SNPs. For this particular data, it appears that using the more complex modeling of the covariance structure of the random effect had no significant effect on the performance of the method, as the results from both the additive (Add) and the additive-dominance (Add-Dom) models were almost identical. This was expected as the particular phenotype (SBP) was generated using a model with only additive effects. The choice of the information criterion was also not surprising. BIC being stricter resulted in models that, on average, included fewer SNPs with nonzero coefficients, 7.1 from 137.1, than when Akaike information criterion (AIC) was used. The actual TDR of the method, though, appears very low (approximately 1 %). This is not surprising as the TDR of the method was affected by the low discovery rate of the PLINK analysis.p <1E-04, for declaring significant SNPs with the PLINK analysis in hopes of increasing the number of actual simulated QTLs “identified” in the phase I analyses. For the LASSO model, we opted not to fit the Add-Dom model as it significantly increases the analysis time and, based on the analysis above, we did not anticipate the results of the 2 models to significantly differ. Table To further study the TDR of the LASSO method, we reanalyzed the data using a less-strict threshold, p value threshold of 1E-04 for the GEMMA analyses.In an effort to increase the number of true QTLs identified, we repeated the analyses, using the genome-wide efficient mixed-model analysis (GEMMA) software  in phaseTable We have presented a 2-stage approach that uses a LASSO method for analyzing quantitative phenotypes of  related individuals. We first perform a GWAS to uncover SNPs potentially associated with the regulation of the phenotype. Next, a multivariate LASSO approach that accounts for relevant covariates and relatedness among the study participants is used to further weed out additional SNPs. The method models relatedness of individuals in the sample through inclusion of a random polygenic effect whose covariance structure reflects relatedness of the individuals. We used the GAW19 simulated family data to study the properties of the method and to compare its performance to the standard LASSO for unrelated individuals.In application to the GAW19 simulated data, our results indicate that incorporating the relatedness of individuals in the LASSO model can help significantly reduce the number of non–trait-regulating SNPs falsely inferred as associated with the trait relatively to the standard LASSO approach, which assumes unrelated individuals. Furthermore, for these particular data, it appears that including a dominance genetic effect in the model has no significant effect other than to increase the computational intensity of the method. This was expected as the particular data were generated without dominance effects.The power of the LASSO method appears to be small, mainly as a result of the inability of the initial GWAS scan to identify QTLs to take forward for further analysis. In an attempt to increase the number of QTLs selected for the phase II analysis, we explored 2 methods for the GWAS: (a) using PLINK on only the founders of the pedigrees, and (b) using GEMMA to allow for related individuals and use of all study participants. Unfortunately, even though the GEMMA analysis yielded a slightly larger number of true signals, neither of the 2 methods was able to advance to the next stage a significant number of true QTLs. This behavior could potentially be a result of the low contribution of the simulated QTLs to the phenotype. Perhaps using other methods in stage I, such as the transmission disequilibrium test or family-based association test, on an appropriate subset of the data, may increase the number of QTLs that are carried forward, thereby potentially increasing the TDR of the method.


Statistical methods constitute a useful approach to understand and quantify the uncertainty that governs complex tsunami mechanisms. Numerical experiments may often have a high computational cost. This forms a limiting factor for performing uncertainty and sensitivity analyses, where numerous simulations are required. Statistical emulators, as surrogates of these simulators, can provide predictions of the physical process in a much faster and computationally inexpensive way. They can form a prominent solution to explore thousands of scenarios that would be otherwise numerically expensive and difficult to achieve. In this work, we build a statistical emulator of the deterministic codes used to simulate submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean, in two stages. First we calibrate, against observations of the landslide deposits, the parameters used in the landslide simulations. This calibration is performed under a Bayesian framework using Gaussian Process (GP) emulators to approximate the landslide model, and the discrepancy function between model and observations. Distributions of the calibrated input parameters are obtained as a result of the calibration. In a second step, a GP emulator is built to mimic the coupled landslide-tsunami numerical process. The emulator propagates the uncertainties in the distributions of the calibrated input parameters inferred from the first step to the outputs. As a result, a quantification of the uncertainty of the maximum free surface elevation at specified locations is obtained. MarinThe numerical codes used for landslide and tsunami modelling may vary depending on several features like the physics, the numerical scheme and the dimensions . The shaTo account for dispersive effects during landslide tsunami propagation, the Boussinesq approximation can be employed and state-of-art Boussinesq solvers have been used for operational tsunami research ,17,23,24In cases where large numbers of numerical simulations become cumbersome, statistical techniques can be employed to shed light on the way that the mechanical processes can influence the results. Up to present, various statistical methods have been implemented to quantify and minimize the degree of uncertainty in tsunami science . The useet al. p, and the design space for the calibration parameters is scaled to l. To complete the Bayesian formulation, independent prior distributions π(.) are specified for each of the parameters of  and similarly for δ(.). A MCMC method can then be employed to explore the posterior distributions and estimate the calibration parameters. The model makes use of the Metropolis–Hastings algorithm which generates a sequence of samples T and y=[y1…ym]T responses with xi∈Rn, yi∈Rq. Using the design points in the input space and the corresponding response, a GP is tuned to mimic the input–output relationship and the kriging model is able to calculate predictions anywhere in the input space using the GP structure. Both the mean function  and the correlation function have to be set to fix the GP. However, the mean is often set to zero when there is not enough evidence of a link to other functions; in that case the correlation satisfactorily captures the variability. The key objective is, based on the provided information, to build a GP approximation model which can be later used to predict the physical process using kriging. We give a brief description of the approach. Given a n-dimensional input x, subset of Rn, the deterministic response y(x)∈R can be written asF=[f(x)]Tβ,f(x)=[f1(x)…fp(x)] is the regression model. The approximation error is denoted by z. The approximation error is assumed to behave as a GP in the region of interest; its probability distribution has zero mean and finite variance with a particular covariance structure. The approximation model of the fitted response F and a fitted random process zF=0 is chosen here: no regression term F is present. Indeed, we found that trying to include such terms was not beneficial here; note that it could be beneficial in other settings T design sites and Y responses, the vector of correlations between inputs and the predicted location x becomesYm responses. The set of the inputs consists of the values of the six parameters ρ) used in the simulation of the numerical experiments and the Bayesian calibration. To build the predictor, a correlation model and the appropriate choice of correlation function parameters ϑ are required. We select a Gaussian correlation model, i runs across six dimensions of inputs. The model was chosen for its flexibility, broad generality, and evidence of a satisfactory behaviour in various contexts. Our preliminary analyses showed little influence of these choices here. In our case with no regression function, the kriging predictor is simply given byϑ are selected to be 0.001 and 10 . We use a maximum-likelihood (ML) approach to estimate ϑ. As numerical maximization can be challenging, we carry out this maximization for several initial starting points. To homogeneously draw the initial starting points for ϑ, we use a small LHS to create a different set of 20 initial scenarios. The comparisons show that there is some variability in the absolute difference between the predictions and the true response influenced by these starting points for ϑ if we had chosen these starting points as guesses.For the set of ϑ* repeatedly picking the 20 starting points ϑ, in order to allow the maximization under an exploration–exploitation paradigm. After a limited number of runs, the final estimate ϑ* converges towards a certain value for all starting points . To evaluate each of these final estimates, we do cross-validation tests with the implementation of LOO diagnostics. The governing principle in LOO is to estimate ϑ* each time neglecting one out of 100 design sites and compare the stochastic predictions with the true deterministic response of the process. We compute our LOO diagnostics by comparing the absolute difference between the predictions and the true process. We select the value of ϑ* that results in the best LOO prediction  among our 20 possible values. We eventually pick ϑ=.Hence, we run ML to obtain a final estimate for t≈1 s) to yield 4500 expected maximum tsunami elevations at each desired wave gauge. The results constitute a probabilistic statement about the maximum water surface elevation at this specific location. The distributions of the predictions are presented in the form of histograms for the offshore gauge 01 and the onshore gauge 62 . The highest predicted tsunami amplitudes may exceed 15 m in more infrequent cases. At gauge 62, the most probable maximum run-up is approximately 8 m. The majority of the predicted maximum run-up heights range between 7 and 10 m and they rarely exceed a value of 16 m (b). Following the same process, predictions and assumptions for the rest of the gauges can be made with the aid of the statistical emulator.The histograms of the maximum free surface elevation in gauges 01 and 62 exhibit a positively skewed probabilistic distribution. At gauge 01, the majority of the predictions give maximum tsunami amplitudes ranging between 7 and 12 m a. The hi of 16 m b. Followa,b.The emulation allows some further exploration of the sensitivity of the maximum tsunami amplitudes to the varying input parameters due to the possibility of running the model at many more locations in the input space. Gauge 01 is selected to illustrate the sensitivity analysis . The maxfigure 11f). However, the relationship between the maximum free surface elevation and the turbulence coefficient appears to be slightly nonlinear (c). We note that the tsunami amplitude increases when The plots show an almost linear relationship between the maximum free surface elevation and the maximum landslide thickness f. Howeveonlinear c. We notandslide . The res8.In this study, we implemented a Bayesian calibration to explore the optimal values of the landslide parameters used in the numerical modelling of submarine sliding and tsunami generation in the RBSC. A comprehensive statistical emulation of the one-way coupled numerical process was then employed to explore the probabilistic maximum tsunami elevation at specific locations. The possible tsunami amplitudes generated by a submarine slope collapse in the lower slope region of the Rockall Bank were estimated with the use of the emulator.T0<1000 Pa and ϕbed∼0.2°) in order to match the observed run-out. The density plots show an inversely proportional relationship between the yield strength and the basal friction. The calibrated maximum slide thickness is likely to be within the higher values of the initial range: hmax>100 m are likely to help match modelled landslides to the desired run-out length. The apparent density is also negatively skewed with optimal values higher than 1000 kg m−3. The posterior distributions of the turbulence coefficient and the dynamic viscosity show a relatively uniform distribution. No significant effect of the two parameters on the simulated run-out of the flow can be deduced from the calibration.The posterior distributions of the parameters, resulting from the calibration, show that the yield strength and the basal friction of the landslide can affect critically the run-out length of the flow. The optimal values of both parameters appear to be very constrained within the initially attributed range (t≈1 s). The time of the predictions is excessively fast when compared to the computational speed of one run of the combined models (roughly 11–13 h for each VOLNA run with 2 GPUs). The probabilistic distributions of the predicted maximum tsunami amplitudes were presented for two locations: an offshore gauge close to the generation region (gauge 01) and an onshore gauge in the Mullet Peninsula (gauge 62). The predictions at gauge 01 show that the tsunami amplitudes of a similar landslide would more likely vary between 7 and 12 m. Likewise, at gauge 62 the predictions show that although cases of higher tsunami run-up (>10 m) are not excluded, they exhibit a less common occurrence rate in comparison to lower run-up heights (between 7 and 10 m). Similar predictions can be made for the rest of the gauges.Within the limitations of our modelling assumptions, we employed the emulator to make predictions of the maximum tsunami amplitudes using the posterior distributions of the calibration parameters. In total, predictions of 4500 cases were made in only a few moments of time  do not appear to have a significant effect on the magnitude of the tsunami amplitudes. As so, to achieve the optimal run-out length of the flow the yield strength and the basal friction are critical whereas the turbulence coefficient does not hold such an important role. Different rheological parameters can thus affect the magnitude of different processes and their influence should not be disregarded.We note that building a statistical surrogate of a deterministic code is not a replacement for scientifically sound simulations of a physical process in the first place. It forms a verified, fast and inexpensive technique to make predictions of the process and quantify uncertainty, particularly in cases where a large number of simulations has to be considered and it is difficult or computationally expensive to run the codes. In most instances, the results are only applicable for the specific case examined. A diverse parameter selection, or an alternative scientific approach to the problem, will possibly generate different results and another calibration and emulation should be performed on a case-by-case situation.To perform the calibration and the emulation, we used data from the RBSC which were available to us during the process. As scientific research advances day by day with technological progress and the results of new studies, new information for the event can be revealed. Ongoing research in the subject shows that the deposits of the most recent event have a slightly different shape than the one we used in this study . As the This work forms one of the first attempts to study numerically slope failure in the RBSC, and built a statistical emulator of a real case study for uncertainty quantification and sensitivity analysis. Driven by this, future research endeavours may focus on different approaches to study parametric variability. In addition, to adequately account for the physics of the phenomenon, a comparison and verification of the modelling strategy employed here with the results of fully coupled modelling of the event, including incorporation of the landslide tsunami dispersive features, should also be considered. The construction of extended hazard maps for the area could also be useful for future hazard assessments. To do so, a slope stability assessment and a different meshing strategy in the modelling, with mesh refinement on the coastal areas, would be of critical importance. Overall, the approach followed in this study forms a tool to study a certain parametric variability to understand the uncertainty governing submarine sliding and tsunami generation in the RBSC. It can be used as the basis for future probabilistic assessments of landslide-generated tsunamis in other regions of the world prone to such events.


Metamaterials are engineered materials that offer the flexibility to manipulate the incident waves leading to exotic applications such as cloaking, extraordinary transmission, sub-wavelength imaging and negative refraction. These concepts have largely been explored in the context of electromagnetic waves. Acoustic metamaterials, similar to their optical counterparts, demonstrate anomalous effective elastic properties. Recent developments have shown that coiling up the propagation path of acoustic wave results in effective elastic response of the metamaterial beyond the natural response of its constituent materials. The effective response of metamaterials is generally evaluated using the ‘S’ parameter retrieval method based on amplitude of the waves. The phase of acoustic waves contains information of wave pressure and particle velocity. Here, we show using finite-element methods that phase reversal of transmitted waves may be used to predict extreme acoustic properties in space coiling metamaterials. This change is the difference in the phase of the transmitted wave with respect to the incident wave. This method is simpler when compared with the more rigorous ‘S’ parameter retrieval method. The inferences drawn using this method have been verified experimentally for labyrinthine metamaterials by showing negative refraction for the predicted band of frequencies. These are envisioned to have several important applications such as surface wave manipulation, sub-wavelength focusing –5, extraThis idea of electromagnetic wave manipulation using metamaterials was soon adapted and extended to manipulate acoustic wave propagation. This enabled achieving energy flow in the direction opposite to that of sound propagation. Sound wave propagation can be controlled by mass density and bulk modulus of the material, which are both positive in conventional materials. One of the first demonstrations of acoustic metamaterials  used locet al. [The properties of the metamaterials can be evaluated using several methods based on measurements or simulations of reflection and transmission coefficients. The ‘S’ parameter retrieval method was proposed by Nicolson and Ross in 1970  and is tet al.  for elec2.–3 and speed of sound as 6420 m s–1. The density for hypothetical high refractive index medium was taken to be approximately 84 kg m−3 and the speed of sound in this medium was taken to be approximately 42 m s–1. Inhomogeneous Helmholtz equation was used as acoustic wave equation. Plane wave with a pressure amplitude of 1 Pa was used as the incident wave. Periodic boundary conditions are applied normal in the direction of propagation of the wave. Sound soft conditions were used to avoid internal reflections.Finite-element method was used to simulate 2D and 3D labyrinthine structures in a rectangular acoustic waveguide to obtain the transmission wave properties. These calculations were also used to calculate the ‘S’ parameters to determine the refractive index and acoustic impedance explicitly. Material properties were chosen to be isotropic. The equivalent structure was made of aluminium with channels containing hypothetical material with an effective high refractive index. The density of aluminium was taken to be 2700 kg mExperiments were performed to observe negative refraction in the predicted band of frequencies. The 2D and 3D labyrinthine unit cells were fabricated and arranged in a prismatic arrangement. Sound was generated using a 5 Ω speaker. An array of microphones were arranged at equal separation  from the prismatic arrangement of metamaterial unit as shown in 3.Phase of a longitudinal wave contains information about its wave pressure and particle velocity. Thus, it can be used to engineer the wave propagation characteristics by modifying the geometry of the medium through which the wave traverses. To understand the correlation of the transmitted wave with extreme properties, the transmitted wave characteristics have been investigated for 2D and 3D labyrinthine structures.a. A monitor was placed at the right end of the waveguide to determine the transmitted wave characteristics. Transmitted wave characteristics for the surrounding shell (panel 1 in figure 2a), the complete outer shell (panel 2 in figure 2a) and the meta unit (panel 3 in figure 2a) are shown in figure 2b–d, respectively. Figure 2b shows that the surrounding shells participate insignificantly in reflection of the incident wave. Amplitude of the wave remains almost unchanged and the wave gets transmitted almost without attenuation as expected. However, as soon as the front and rear surfaces are incorporated, a significant decrease in amplitude of the transmitted wave is observed as in figure 2c. This happens due to the impedance mismatch at the interface. The inclusion of space coiling structure introduces many such interfaces and transmitted wave develops an overall phase difference while exiting the meta unit as seen in figure 2d. The schematic cross sections of the 2D and 3D labyrinthine structures are shown in figure 2e, f, respectively. Transmitted wave profiles obtained were fitted with equation of sine wave to determine the phase difference developed during the transition. The relative phase of the transmitted wave was obtained by taking the difference of the phase of the incident wave from that of the transmitted wave.Sine waves with frequencies in the range of 1 kHz to 4 kHz were incident from the left end of the waveguide on the unit cell placed inside the waveguide as shown in figure 2a,b. Amplitude of the total acoustic pressure changes from positive to negative as seen in figure 3a at approximately 3.4 kHz for 2D maze.The change in phase of wave depends on the frequency as seen in figure 3a,b. This change of phase of transmitted wave remains negative suggesting anomalous behaviour of acoustic wave propagation. The propagation of acoustic waves in space coiling labyrinthine structures resembles motion of ultraslow fluid particles compared to the background. This enables extension of Mie theory to understand the anomalous behaviour in space coiling structures for acoustic waves. This extension to acoustic waves relates monopolar Mie resonance to the negative bulk modulus, while the dipolar resonances to negative mass density [The slope of the change of phase turns negative at approximately 3.3 kHz for the 2D and approximately 3.7 kHz for the 3D structures as seen in figure 4 density .Figure The pressure difference force is related to the pressure gradient. The change in phase from positive to negative would correspond to negative density gradient and hence this change in phase is expected to act as a marker for the negative properties. Similar observations have been made in the case of other metamaterials  and optia,b were constructed. The path AOB (inset in figure 4a) represents the path of acoustic waves travelling through the medium in the 2D structures. A'OB' (inset in figure 4b) represents similar acoustic wave path in the case of 3D structures. The bluish region represents the material used for the construction of metamaterial structures. The refractive index of the white region, which is a hypothetical high refractive index material, provides a measure of coiling up of space in the space coiling structure. This is obtained by dividing the total path length in the actual structure by the path length in the equivalent cell. The refractive index (n) and impedance (Z) can be expressed in terms of the reflection (S11) and transmission (S21) coefficients by the following relations:a for 2D maze structure. Similar calculations were performed for 3D structures and have been plotted in figure 4b.To calculate the effective refractive index of the metamaterial structures, equivalent cells as shown in figure 4a in the range of frequencies predicted by transmitted phase analysis.As a final check, experiments were performed to detect the refracted wave in the negative refractive index regime of a prismatic arrangement of 2D and 3D space coiling structures as shown in figure 5b, the transmitted wave amplitude lies predominantly in the positive refractive index region, while for all frequencies below down to 3.1 kHz, the transmitted waves propagate predominantly in the region of negative refractive index. Thus the polar plot provides a direct experimental verification of negative refraction in the range of frequencies as predicted by the phase analysis of transmitted wave.Total acoustic pressure was measured at various angles using an array of condenser microphones and a polar plot was constructed by representing the radial direction as the amplitude of the transmitted acoustic pressure. It is observed that at approximately 3.8 kHz in figure 54.Finite-element methods have been used to calculate the transmitted waves profile of acoustic waves passing through 2D and 3D space coiling metamaterials. The analysis of the transmitted waves shows that phase is developed while traversing through the metamaterial unit. This may be used to predict negative refraction in space coiling structures. This approach reduces the computational complexity for estimating extreme properties when compared with the ‘S’ parameter retrieval method. Our investigation suggests that the phase of the transmitted wave and hence the phase velocity undergoes a reversal while travelling through the space coiling structures when compared with normal propagation of acoustic waves. Experimental data show negative refraction of sound waves in the range of frequencies predicted using phase analysis of transmitted wave. To conclude, the change in the phase of the transmitted wave can be used as a marker for qualitatively predicting negative refraction in space coiling structures.