The crystal growth of poly(ε-caprolactone) can be very effectively directed through the use of small amounts of dibenzylidene sorbitol in conjunction with modest flow fields to yield extremely high levels of the preferred polymer crystal orientation. We show that by introducing small quantities of a terpolymer, based on polyvinyl butyral we can switch the symmetry axis of the final lamellar orientation from parallel to perpendicular to the melt flow direction. During shear flow of the polymer melt, the dibenzylidene sorbitol forms highly extended nanoparticles which adopt a preferred alignment with respect to the flow field and on cooling, polymer crystallisation is directed by these particles. The presence of the terpolymer, based on polyvinyl butyral, limits the aspect ratio of the dibenzylidene sorbitol (DBS) particles, such that the preferred orientation of the particles in the polymer melt changes from parallel to normal to the flow direction. The alignment of lamellar crystals perpendicular to the flow direction has important implications for applications such as scaffolds for tissue engineering and for barrier film properties. When DBS is dispersed in a polymer matrix, the application of modest shear flow leads to a macroscopic alignment of the fibrils [Dibenzylidene sorbitol (DBS) see  [I] is a fibrils ,6,7. Whe fibrils ,6. In thThe properties of polymer based materials depend on the processing procedures used to shape the object as well as the ingenuity of the molecule maker in terms of the chemical configuration of the polymer chains. In the case of a crystallisable polymer such as poly(ε-caprolactone), the properties of the final product are especially dependent on the arrangement of the chain folded lamellar crystals . This isThe role of dibenzylidene sorbitol as a nucleating agent for isotactic polypropylene has been known for some time, although the mechanism is less clear. Smith et al., proposed that it stabilized the helical structure through the cleft in the V shaped nature of the molecular structure . MandelkKey factors with regard to dibenzylidene sorbitol enabling the direction of crystallisation are the formation of highly anisotropic nanoparticles which easily align in low shear flow field . The solThe work reported here is based on the use of self-assembling nanoparticles rather than engineered nanoparticles which have significant environmental impact issues and will limit the recycling or disposal of objects prepared using this approach. As a compound approved for use in food contact applications DBS has none of the potential hazards of engineered nanoparticles such as carbon nanotubes or metallic particles.DBS is widely employed as a clarifying agent with isotactic polypropylene where it yields films with a high level of optical transparency for packaging . It is aw/w dibenzylidene sorbitol  and 3% w/w polyvinyl butyral were dispersed alone and together in PCL   by solution blending using butanone as the co-solvent, cooling the solution to room temperature and allowing the butanone to evaporate under the draught of a fume cupboard. Moulded discs for the shear flow experiments were prepared from these mixtures by melt moulding at 80 °C, followed by cooling to 0 °C to allow the PCL to crystallise fully before warming to room temperature.The polyvinyl butyral based terpolymer  used in this work is a random terpolymer containing ~88% of polyvinyl butyral together with ~0.5% polyvinyl acetate and ~11.5% polyvinyl alcohol units and was obtained from Aldrich . The molecular weight is 90,000 to 120,000 Da. 3% Samples of PCL, PCL/DBS, PCL/PVB, and PCL/DBS/PVB were subjected to controlled flow and thermal profiles using a specially designed parallel plate shear flow system ,24 as sh−1 (|Q| = 4π sin θ/λ where λ is the incident wavelength and 2θ is the scattering angle) with a data accumulation time of 10 s. The sample to detector distance and the detector pixel size were calibrated in terms of geometry using wet collagen.Time resolved X-ray scattering experiments were performed on the fixed wavelength (λ = 1.54 Å) beam-line 2.1  at the D−1 and in steps ∆α = 5°.Wide-angle X-ray Scattering (WAXS) were obtained at room temperature on static samples using a symmetrical transmission diffractometer equipped with a graphite monochromator and a copper X-ray sealed tube source. The intensity was measured as a function of |Q| and α, the angle between the symmetry axis of the sample and Q, in steps of |∆Q| = 0.02 ÅSpecimens for optical microscopy were prepared on glass coverslips in a Metter FP hotstage by melting for 2 min at 150 °C, then cooling rapidly (~30 °C/min) to 60 °C followed by cooling at a rate of 1 °C/min to 30 °C before removing to room temperature. Optical microscopy was performed at room temperature using a Carl Zeiss GFL polarised light microscope.For samples of PCL containing 3% DBS, in-situ SAXS experiments have shown that at 150 °C the DBS will be fully dissolved in the PCL . The PCLThe specimen containing 3% DBS and 3% PVB was prepared by melting at 170 °C before being given the same treatment as the previous samples. This resultant micrograph appears to be the same as shown in −1 for 100 s (shear strain = shear rate x time = 1000 su). After cessation of shearing the sample was cooled to room temperature at rate of 10 °C/min. –1. This is in-line with previous studies and reflects the zero memory of the shear field due to the rapid relaxation of the PCL chains after shearing before crystallisation take place [Samples of PCL, PCL/DBS, PCL/PVB, and PCL/DBS/PVB were processed using a defined thermal and shear profile which consisted of heating to 80 °C at 20 °C/min, holding for 5 min and then subjected to a shear flow field of 10 ske place . The corke place .The WAXS pattern for the PCL/PVB/DBS specimen d is also–1, corresponding to a long period of ~195 Å, arising from the lamellar structure. The constant intensity as a function of the azimuthal angle shows that the lamellae have no preferred orientation with respect to the melt flow direction. A similar pattern is displayed by the PCL/PVB sample Ω= ∫αThe invariant is related to the volume fraction of crystals, at least in the initial stage of crystallisation, as all other things being equal, the invariant would also go through a maximum when the volume occupied by lamellae reaches 50%, and it will decrease slightly on cooling as the electron densities of crystalline and amorphous material approach each other. We have used the plots of invariant versus temperature to identify the temperature at which the maximum rate of crystallisation occurs. The maximum rate of crystallisation for the PCL/DBS/PVB sample is about 5 °C below that observed for the PCL sample. However this is considerably above the peak temperature for PCL/PVB. The addition of the PVB to the PCL/PVB appears to have significantly deactivated the DBS as a nucleant as measured by a shift in the maximum crystallisation temperature of about 10 °C, and this corresponds to the smaller lamellar long period indicated by the SAXS measurements. We emphasise that shear flow is imposed on the polymer melt ~40 °C above the crystallisation temperature of PCL. As the crystallisation of the sheared PCL melt reveals, at the point of crystallisation the melt itself has no memory of the shear flow. The behaviour reported above is the consequence of the distribution and nature of the nanoparticles and their influence on the nucleation of the PCL. There is the possibility that the presence of the slightly higher molecular weight PVB could alter the flow behaviour of the overall system. However, we can discount this possibility as the WAXS b and SAXIt is more probable that the PVB acts to suppress the crystallisation of the DBS. The optical microscopy shows that this is indeed the case at least in part. The morphology shown in Jeffrey showed particles in a fluid during shear flow may exhibit a number of orientation states . A numbeWe propose that the addition of PVB to the PCL/DBS system results in fewer and less anisotropic particles which when subjected to modest shear flow adopt an alignment preferentially parallel to the vorticity axis and normal to the flow direction. We anticipate this is principally due to the reduced aspect ratio in line with the prediction of Leal , but theControlling the morphology of a semi-crystalline polymer can be achieved in a variety of ways yielding major changes to properties. The addition of nucleating agents to polymers may be used to control the crystal structure as in the case of β-nucleating agents for isotactic polypropylene , to contWhen poly(ε-caprolactone) is nucleated by dibenzylidene sorbitol in the presence of a terpolymer based on polyvinyl butyral there is an orthogonal switch in the symmetry axis of the resultant lamellar orientation as compared with PCL nucleated by DBS alone.c-axes parallel to the particles and to the flow direction. We confirm that small quantities of a terpolymer based on polyvinyl butyral inhibit the nucleation in PCL by itself and show that its presence can also modify the behaviour of the DBS. In particular, the presence of the PVB limits the aspect ratio of the DBS particles which form and this alters the preferred orientation of the DBS particles in the sheared polymer melt from parallel to normal to the flow direction. This morphology is suitable for barrier property films and will influence the degradation rates in bioabsorbable biomedical products prepared from PCL.Previously, it has been shown that the aligned extended nanoparticles formed through the use of small amounts of DBS in conjunction with modest flow fields will nucleate lamellae with their


We explore the problem of variable selection in a case‐control setting with mass spectrometry proteomic data consisting of paired measurements. Each pair corresponds to a distinct isotope cluster and each component within pair represents a summary of isotopic expression based on either the intensity or the shape of the cluster. Our objective is to identify a collection of isotope clusters associated with the disease outcome and at the same time assess the predictive added‐value of shape beyond intensity while maintaining predictive performance. We propose a Bayesian model that exploits the paired structure of our data and utilizes prior information on the relative predictive power of each source by introducing multiple layers of selection. This allows us to make simultaneous inference on which are the most informative pairs and for which—and to what extent—shape has a complementary value in separating the two groups. We evaluate the Bayesian model on pancreatic cancer data. Results from the fitted model show that most predictive potential is achieved with a subset of just six (out of 1289) pairs while the contribution of the intensity components is much higher than the shape components. To demonstrate how the method behaves under a controlled setting we consider a simulation study. Results from this study indicate that the proposed approach can successfully select the truly predictive pairs and accurately estimate the effects of both components although, in some cases, the model tends to overestimate the inclusion probability of the second component. The most widely used technology to assess proteomic expression is mass spectrometry that has undergone remarkable evolution over the last 20 years. Particularly ultrahigh‐resolution mass spectrometers (MS) such as Fourier‐transform MS have become the most powerful and efficient tools for the quantitative analysis of complex protein mixtures in biological systems.m/z) range, generated from the detection of ionized molecules. In ultrahigh‐resolution mass spectrometry, each species (such as peptide) is detected and expressed as a “density” of isotope peaks  using knowledge on the properties of the isotope densities . However, the predictive potential of the isotopic cluster information, inherent in such data, has not been fully exploited yet. At date, a common practice for summarizing the proteomic expression (for example so that it can be later used for prediction purposes) is to use information on the amount of expression (intensity).In a recent work, Kakourou, Vach, Nicolardi, van der Burgt, and Mertens  proposedHaving established the presence of an overall isotope cluster effect as well as a shape effect in addition to intensity effect, our aim in this work is twofold: (a) to identify a collection of isotope clusters associated with the disease outcome and (b) to optimally integrate the two sources of information. We wish to address these questions in a way that allows us to assess the added‐value of shape beyond intensity in predicting the class outcome of an individual while maintaining predictive performance.l1‐norm constraint on the vector of regression coefficients that shrinks the parameter estimates toward zero while it induces sparsity in the model. An extension of the lasso, introduced as “group lasso” was proposed  knowledge exists, a more suitable/interesting approach would be to introduce some type of hierarchy in the selection process, for example by prioritizing selection of specific predictors that have already proven their predictive value in previous applications. This would additionally allow us to assess the complementary predictive potential of “secondary” predictors after accounting for “primary” predictors .The problem of assessing the added‐value of a secondary predictor on top of a primary predictor in the high‐dimensional setting has been considered by Boulesteix and Hothorn  and by RIn this paper, we propose an approach that can explore the problem of isotope selection and at the same time address the problem of assessing the additional predictive value of the shape source over and above the intensity source. Our approach uses a Bayesian model formulation that exploits the paired structure of the proteomic data and utilizes prior knowledge on the relative predictive power of each individual source to assess the added‐value of shape, by introducing multiple layers of selection. In doing so, the Bayesian model makes the explicit assumption that the shape source is complementary. In terms of model fitting, this is translated by assuming that a shape measure can be included in the set of predictors on the condition that it is accompanied by/coupled with its corresponding intensity while the reverse does not need to hold. This assumption allows us to make simultaneous inference on which isotope clusters are the most informative with respect to the class outcome and for which isotopes—and most importantly to what extent—shape has a complementary value in separating the two groups.The remainder of this paper is organized as follows: We first introduce the pancreatic cancer data and their paired structure that is a key component of these particular data. The problem of isotope selection, on the one hand, and assessment of the added‐value of shape, on the other, through a Bayesian model formulation is explored next. We then present results from applying the Bayesian model to the pancreatic cancer data consisting of the paired intensity‐shape measurements. Results from the analysis of the fitted model show that most predictive potential is achieved with a subset of just 6 (out of 1289) pairs, which are consistently selected with very high probability of inclusion, while the contribution of the intensity components is much higher than the shape components. We explore the relative performance and the degree of agreement (with respect to the selected isotope clusters) between our method and an alternative approach based on a stability selection strategy utilizing sparse‐group lasso. Additionally, we show a simulation example to demonstrate how the proposed method behaves under a controlled setting. The outcome of the simulation study indicates that the proposed approach can successfully select the truly predictive pairs and accurately estimate the effects of both components although, in some cases, the model tends to overestimate the inclusion probability of the second component. We finish with a discussion.2In this paper, we reanalyze data from a case‐control study that was carried out at the Leiden University Medical Centre , given byv, is defined as1,…,J, where pj:=xj∑jxj and x1,…,xJ denote the residuals that measure the deviation of the observed isotopic pattern from the typical pattern. For a more detailed description on how these summary measures were created/derived we refer the interested readers to Kakourou et al. , where y=⊺ is the binary case‐control outcome with yi∈{0,1} for i=1,…,n independent individuals while z=⊺ represents the predictor source with each zi=,…,) consisting of a sequence of paired intensity and shape measurements for p isotope clusters. We consider the binary regression modelpi is the case‐probability for the i‐th observation and β=,…,) represents the vector of paired regression parameters with the first and second elements of each pair corresponding to the effects of the intensity and shape measurements respectively.Let the data be given by 3.2u is expected to carry more information on the class outcome than v. Our main objective is to assess the added‐value of the shape source v on top of the intensity source u in predicting the health status of future patients. Under the assumption that only a set of isotope clusters is predictive of the health status of an individual, the true model is given byz∼=,…,) represents the unknown set of paired measurements that are associated with the class outcome with regression coefficient vector β∼=,…,) that is of unknown dimension k≤p. The isotope dimensionality k is thus considered/treated as a parameter in the model and will be estimated from the data along with the unknown set of intensity‐shape pairs and their corresponding regression coefficients.We assume that only a subset of isotope clusters is relevant for predicting the class outcome and that a=0 then b=0 (if b≠0 then a≠0). The above formulation suggests that shape can be included in the model only on the condition that its corresponding intensity is included as well. Note that the reverse is not true. Hence, rather than forcing mutual selection of both components of the isotope pairs we let the data decide whether the first component (intensity) alone provides all the required information for separating the two groups. With this constraint, kC denotes the dimensionality of the “complete” isotope couples and indicates how many times a shape measure is included in the model in conjunction with its corresponding intensity and kI denotes the dimensionality of intensity singletons so that k=kI+kC.In order to assess the added‐value of shape information over and above intensity information we place a logical constraint on the inclusion of shape information that specifies that if 3.3β0∼N. We specify independent normal priors on the regression parameters a∼j∼N and b∼j∼N, for j=1,…,k and bj≠0, where the variances σa2=τa2ca and σb2=τb2cb control the magnitude of included effects for intensity and shape respectively, τa and τb are known rescaling factors while ca=1/sa and cb=1/sb are randomly distributed scale factors with gamma priors placed on sa and sb. Under the above prior assumption on the regression parameters, the covariance matrix Σ has a block‐diagonal structure with block matrices along the diagonal of the form Σj=[σa200σb2], if bj≠0, and Σj=σa2 otherwise. The prior specification is completed by assigning a prior to the isotope dimension parameter k. We use a discrete uniform prior on the set of integers {0,1,2,…,kmax} with kmax a large positive integer corresponding to the maximum allowed isotope dimension.To complete the model formulation, we have to specify the prior structure for all the model parameters. For the intercept we assume a weakly informative normal prior 3.4We present an adaptation of the reversible jump MCMC implementation described in Mertens  for fittOur MCMC sampler is based on a random choice between three basic move steps: (1) BIRTH, (2) DEATH, and (3) CHANGE. The first two of these steps propose moves between different isotope dimensions while the last one proposes moves within an isotope dimension and between different variable dimensions.bk=1/3 to add a new randomly chosen isotope into the model set. In the DEATH step, we propose with probability dk=1/3 to remove a randomly chosen isotope from the current model set. Shape selection is facilitated by splitting each of these steps into two additional substeps such that, in the case of a BIRTH move proposal, we may choose between proposing to add either the entire pair (couple) into the model, with probability bkC=bk/2, or solely the first component of the pair (intensity), with probability bkI=bk/2. Note that the candidate set from which we may select a new isotope is the set containing all “complete” isotopes that do not currently have any component in the model set. Analogously, in the case of a DEATH move proposal, we may choose between proposing to remove either a complete isotope (couple) from the current set, with probability dkC=dk/2, or an intensity singleton, with probability dkI=dk/2.More specifically, in the BIRTH step, we propose with probability ck=1/3. We do so by introducing a CHANGE step. Again here, within this step, we may choose between two substeps that either change an isotope couple into an intensity singleton that is remove shape from the included isotope pair, with probability ckC→I=ck/2, or change an intensity singleton into an isotope couple that is add shape to the included intensity singleton, with probability ckI→C=ck/2. In this way, we give a second chance to shape selection/deselection by allowing the data to judge whether a shape measure contributes to classification in addition to intensity and thus should join its corresponding intensity or does not provide any additional information over and above intensity and therefore could be omitted from the isotope pair in the model set.Apart from the BIRTH and DEATH steps that propose moves between isotope dimensions by either adding or removing isotopes (singletons or pairs) to and from the current set, we may propose to change the composition of an isotope already included in the model, with probability θ to a new model with parameters θ′ isD the data and θ the vector containing the current values of the model parameters k, σa2 , and σb2, .We chooseNote that with the above prior specification, normal inverse scaled chi‐squared updating can be applied, conditional on the variance hyper‐parameters. This implies that the “conditional” acceptance rates do not require actual calculation of the regression parameters on either the old or newly proposed models, as closed forms are available for the integrated likelihood functions conditional on the above variance terms. This leads to significant gains in computational speed. We use the auxiliary variable construction with Kolmogorov–Smirnow prior on the normal variance to obtain the logistic regression model form.We present the basic structure of the algorithm in form of pseudo code below:Algorithm 1144.1kmax=100. We set the hyperparamaters of the Gamma distribution to α=β=1 for the analysis presented in the paper, which results in a prior mean and variance of 1 for both sa and sb.We applied the isotope variable‐dimension model to the pancreatic cancer data that contained 2578 intensity and shape measurements in total, coupled in 1289 isotope pairs. For the maximum allowed isotope dimensionality we used τa and τb we make use of the knowledge that the variance‐covariance matrix of the regression parameter vector in the logistic regression model is approximately n*(XTX)−1, where n is the sample size and X is the data matrix. We find that the inverse cross‐products of uTu, where u= and vTv, where v=, are of order 10−3 and 10−2 which, for a sample size of n=254, points to a prior guess of 0.25 and 2.5 for τa and τb, respectively.For the prior choice on the scaling factors 4.2k, the intensity‐shape pair and intensity singleton dimensionalities kC and kI, the regression parameters as well as the model variances for intensity and shape over the MCMC chains. For posterior inference on the model we combine the two sets of updates into a single set of simulations (1 million samples).To perform the isotope selection, we sampled two sets of 500,000 simulations, after an initial burn‐in of 200,000 samples that were discarded. Convergence was assessed by comparing the last set of simulations with the first through autocorrelation and kernel density plots on the model parameters such as the isotope dimensionality 4.3Previous results on the pancreatic cancer data showed evidence for the presence of predictive information in the isotope clusters that can be used for diagnostic purposes. More specifically, results from using isotope cluster summaries based on either intensity or shape information as input into a ridge logistic model independently, showed that both types of information are predictive of the class outcome, though intensity was found to be more informative than shape. Having established the presence of predictive capacity in the proteomic data, our focus turns toward identifying a collection of isotope clusters associated with the disease as well as investigating the predictive value of shape beyond intensity within the isotope clusters. We begin our exploration by investigating how often an isotope was selected in the model as well as the selection frequency of isotopes consisting of intensity alone and of isotopes consisting of both intensity and shape measures. Figure This tendency of the model is more apparent in Figure m denoting the current model within the set of all simulated models and M denoting the total number of simulated models. Figure The selection effect is reflected, apart from the selection itself, in the calibration of the regression coefficients. We further investigate the effect of selecting intensity or shape and their relative contribution to classification by calculating the marginal mean of regression coefficients asPS), which integrates the information from the effect of inclusion and the calibrated regression effect, as∑j=1k′|βmj| is only across either all intensity or all shapes measures, with k′ the total number of intensity measurements  or the total number of shape measurements  selected in model m∈M, depending on whether PS is calculated for an intensity or a shape measure, respectively. We choose to compute PS separately for intensity and shape measures as a way to account for the different scales of the two predictor sources.Essentially, the selection effect is shared between two types of parameters: (1) the inclusion probability and (2) the regression effect. To get additional insights into the relative contribution of intensity and shape we define a “combined” Posterior Summary (The combined posterior summary measures for intensity (circles) and shape (asterisks) are shown in Figure 4.4We compare results based on our Bayesian variable selection approach against results obtained with an alternative frequentist variable selection approach based on a stability selection strategy that utilizes sparse‐group lasso (SS‐SGL). Stability selection aims to provide a measure of confidence in the selection of variables reported by sparse‐group lasso by adding a resampling step that involves repeating the variable selection procedure over a number of datasets, randomly sampled from the original. We partition the original data into 10, approximately equal sized, subsamples. For each subsample analysis, we discard one subsample in turn and record the components selected by the sparse‐group lasso fitted on the remaining nine subsamples. We define the “selection probabilities” calculated across the results of all subsampled datasets. Intuitively, the selection probabilities can be considered equivalent to the inclusion probabilities generated by our Bayesian variable selection approach and provide a measure of certainty for each selected component since selection of the most informative predictors should be more robust to random perturbations of the data.It should be noted that the selection probabilities resulting from SS‐SGL do not have the same interpretation as posterior probabilities. Moreover, the definition of these probabilities is different from that of the inclusion probabilities because of the conditional nature of the Bayesian selection approach, since it selects shape components on the condition that their corresponding intensity components are included in the model. This asymentrical way of selecting components is introduced due to the underlying assumptions/hypotheses linked with prior knowledge about the relative contribution of shape with respect to intensity. As a result, the model spaces explored by our approach and SS‐SGL are different, leading to interpretational differences. Therefore, a comparative analysis can be carried out solely with respect to the degree of concordance and discordance between the two methods regarding the sets of selected isotopes as well as their relative predictive performance.Table Figure 4.5i=1,…,n, where Pm denotes the a‐posteriori class probability calculated from the m‐th model simulated within the MCMC chain and the sum is across all models simulated. To estimate the error rate we use a cut‐off value of 0.5. We assign an observation as a disease case if the mean a‐posteriori class probability is greater than 0.5 and as a control otherwise. This assignment resulted in a misclassification error rate of 0.086. We also calculate the Brier score, defined asci denotes the true class outcome of the i‐th individual. Finally, the Area Under the Curve (AUC) was found equal to 0.945 . In Figure To assess the predictive performance of the isotope dimension model we use 10‐fold internal cross‐validation. We do so by first partitioning the data into 10 mutually exclusive and exhaustive, equal sized, subsamples. We then run 10 parallel chains in each of which one subsample is used as validation data for evaluating the predictive performance of the model, while the remaining nine subsamples are used to calibrate the Bayesian model. Since the case‐control ratio can greatly affect the selection, to get as consistent variable selection and estimates as possible across the different sets, we partition the original data such that the case‐control ratio in the newly defined calibration and validation data is the same across all subsamples. Evaluation of the model on each subsampled validation set is achieved by applying, within each MCMC step in each chain, the generated rule based on the calibration data to the profiles in the validation set and storing the sequence of validated predictions. To assign observations, we calculate for each observation the mean a‐posteriori class probabilities of group‐membership. That is, we compute the cross‐validated probabilitiesFinally, we investigate the consistency of the model selection across the partitioned datasets and the extent of agreement with the model selection based on the entire data. Table 4.6u and v as N. We simulate the binary outcome data according to the logistic modela and b are 100‐dimensional vectors containing the first and second component effects on the class outcome. To induce associations between the predictor components and the outcome, we draw binary response variables from a Bernoulli distribution withWe use a simplified simulation example to demonstrate how the method behaves in a controlled setting. We generate data to have the same number of patients as in the pancreatic cancer dataset but smaller dimensionality under the independence assumption. We generate 200 variables in total such that half of them represent the first and the other half the second components of 100 paired measurements. For both components of each pair we simulate data from a normal distribution where we specify the independent normal random variables kmax=50, α=β=1 for the hyperparameters of the Gamma distribution and τa=τb=1 while we vary the values of the nonzero elements of β from 1 to 3.5.We consider three different scenarios in which we vary the number of true, nonzero effects, as well as their magnitude. In all three scenarios we use In the first scenario, we select solely the first component of the last pair to have a nonzero effect and we set its value to be equal to 3.5. We wish to test whether the model can identify the single predictive pair—and more specifically the single predictive component within the pair, on the one hand, and estimate its true effect, on the other. The top three plots of Figure In the second scenario, apart from the first component of the last pair, we assign nonzero effects to both components of the first pair. We set the effect for the two components of the first pair equal to 2.5 and the effect of the first component of the last pair equal to 1.5. Looking at the fist plot in the middle row of Figure ak≠0 for k=1,30,50,80,100 and bk≠0 for k=1,30. We also assume smaller effects for the second components than the first, that is we assume that the first components are more informative than the second. In particular, we set a1=3.5, a30=2, a50=a80=2.5, a100=1.5 and b1=1.5, b30=1. From the first plot at the bottom of Figure  and , the probability that the entire pair is selected is overestimated, yet smaller than the probability than only the first component is selected. The same is not true for pair  of which the probability of selecting the entire pair exceeds, even though marginally, the probability of selecting the single component. This overestimation of the inclusion probabilities for the noninformative components is counterbalanced by assigning them a zero effect, as can be seen from the marginal effects plot at the bottom of Figure PS provides an adequate summary for assessment of the true impact of the component variables in the model. Results from extending this scenario to the situation where we additionally have correlations between the predictive pairs are presented in the Appendix.In the third scenario, we increase the number of pairs with both components discriminative to two and the number of pairs with only the first component discriminative to three such that 5In this work, we addressed the problem of isotope cluster selection through a Bayesian model formulation. Results from applying the Bayesian selection model to the pancreatic cancer data showed rather strong evidence in favor of specific isotope clusters being associated with the class outcome while most often a limited number of about 15 isotope clusters was selected by the model as a sufficient subset for separating the two groups. In addition to the isotope selection, the model formulation allows for assessment of the added‐value of the shape source over and above the intensity source through the employment/application of an additional layer of shape selection within the isotope cluster selection. To perform the shape selection we make the explicit assumption that this type of information is complementary to the intensity information which, in the model fitting process, can be formulated as shape being included in the model on the condition that its corresponding intensity is already selected or will be selected to the model alongside with shape. This assumption, and hence this “asymmetrical” manner in which we select/deselect shape, is based on our prior knowledge from previous work that the overall intensity level is a superior source of predictive information as compared to shape. The main outcome with regards to assessing the added‐value of shape was that most shape measures were associated with effects of much smaller magnitude as compared to intensity , regardless of their inclusion or not in the model, suggesting that intensity alone provides all the required information for separating the two groups.We compared results from our Bayesian variable selection approach with results obtained with an alternative frequentist variable selection approach based on a stability selection strategy that utilizes sparse‐group lasso (SS‐SGL). We describe how stability selection can be employed in order to obtain “selection probabilities” that may serve as a measure of confidence for the variables selected by sparse‐group lasso. Intuitively, the selection probabilities can be considered equivalent to the inclusion probabilities generated by our Bayesian variable selection approach. It should be noted however that the Bayesian variable selection method is structurally—and in terms of objectives—different from SS‐SGL due to the underlying assumptions linked with prior knowledge on the relative predictive power of each individual source. Because of this structural difference, the selection probabilities resulting from SS‐SGL do not have the same interpretation as posterior probabilities and, moreover, the definition of the selection probabilities is different from that of the inclusion probabilities generated with the Bayesian approach due to the statistics being different. To overcome the interpretational differences, we restricted our comparison to the evaluation of results obtained using each method with respect to the achieved isotope selection and classification performance. Results from the real data analysis showed that the Bayesian approach selects lower dimensional models to describe the data and leads to a more consistent variable selection compared to SS‐SGL. Moreover, classification results suggested that the Bayesian approach can lead to greater accuracy  compared to SS‐SGL, justifying thus the choice of incorporating prior knowledge in the analysis by making problem‐specific assumptions.γj in the likelihood so that the variables that do not contribute to the linear predictor, which is now of the form β0+∑j=1kγjβjXj, drop off. In our Bayesian selection approach we used a reversible‐jump implementation in which the level of sparsity is controlled through a prior on the model (isotope) dimension. This allows us, on the one hand, to estimate the optimal dimensionality of the isotope clusters predictor set, and on the other, to assess the additional value of shape at the isotope cluster level, through the Metropolis–Hastings acceptance ratios, which account for inclusion or exclusion of shape to or from the model.Variable selection methods have seen many applications within the Bayesian statistics field. Among these, a popular method for performing variable selection is via spike‐and‐slab priors  proposed a frequentist two‐step approach for assessing the augmented predictive value of a secondary source on top of a primary source based on a sequential double cross‐validation procedure. Apart from the selective nature of our Bayesian model, a key difference between this sequential approach and our Bayesian selection method is that the latter takes explicitly into account the pairing structure of our data and could be extended to deal also with more complex grouping data structures such as triplets, quadruples etc. Moreover, in contrast to the sequential approach, our Bayesian approach can be used not solely to evaluate whether there is additional predictive information in the secondary source (shape) after correcting for the primary (intensity) source but also to address the question of where this extra information, if any, comes from. This is essentially achieved through the “asymmetrical” layer of shape selection within the isotope cluster selection that evaluates—and estimates—for each unique isotope cluster the relative contribution of each shape measure to classification.{−1,0,1}. In this way, we replace calibration of the regression coefficients with selection by restricting the effect to be absent, present‐negative or present‐positive. Moreover, by assuming prespecified values for the magnitude of the regression effects, no further assumptions, for instance on the distribution of the variances for the intensity and shape effects, are needed. A limitation of this particular prior specification is that simple conjugate updating rules are no longer applicable. This suggests that calculation of the acceptance probabilities would require actual estimation of the integrated likelihood functions of the old and newly proposed models that could increase the computational time considerably.An alternative way of implementing the double‐layered selection of isotope clusters, on the one hand, and shape measures within the isotope cluster configuration, on the other, would be to restrict the parameter space by modifying the prior specification on the regression coefficient parameters. More specifically, rather than assuming Normal priors for the intensity and shape effects we may constrain the regression coefficients to be within the set of values We restricted our discussion to intensity‐shape combinations in which we considered simple linear effects for both intensity and shape summary measures, mainly due to the fact that such linear rules can be calibrated and interpreted with relative ease in practical application. In reality however the relationship between the class outcome and the two predictor sources—as well as the relationship between the intensity source and the shape source —may be more complex. In fact, it would be interesting to consider more complex effect structures, for example by including also quadratic terms for intensity and shape as well as interactions between intensity and shape into the model in addition to the linear effects. Allowing for this flexibility in the model could potentially improve the predictive capacity of intensity and shape integration and give us the chance to learn more about the interplay between these two different types of information. The idea of considering more complex structures in order to capture the true relationships in the data, in the hope that this will result in improved predictions and/or more thorough inference, is a promising topic for future research. Preliminary results from an exploratory analysis showed that including quadratic as well as interaction terms in a univariate model , could lead to improvement of the model fit, as compared to univariate models with only linear terms, for a considerable number of clusters.An undesired feature of the proposed Bayesian selection approach is that it tends to overselect shape components even in the cases where there is no true effect. This tendency of the model is more pronounced for isotope clusters that are selected with high overall isotope probability of inclusion. The overestimation of the inclusion probabilities for the non‐informative shape components is partly counterbalanced by the calibration of the marginal regression coefficients for shape that were effectively zero for almost all shape measures that were included to the model with high frequency. Overestimation of inclusion probabilities is actually a known deficiency of reversible‐jump estimation procedures for (linear) models, yet not discussed/identified in the related literature.σb, the acceptance probability for adding a shape measure becomes smaller while the acceptance probability for removing a shape measure becomes larger. The drawback however with this solution is that conjugate updating is not applicable for this family of distributions. Another alternative could be to use a more general and flexible prior specification for the variance of the intensity and shape effects so that the model can adapt to the data at hand. A solution toward this direction could be to assume independent—across isotopes—Inverse Gamma priors with different hyperparameters for the intensity and shape variances with additional priors placed on these hyperparameters. This prior specification could give us more flexibility and control in borrowing strength across the shape measures as compared to setting the hyperparameters of the Inverse Gamma priors to fixed values.For posterior inference, we proposed a posterior summary that combines the information that can be extracted from the inclusion probability estimates and the marginal regression effect estimates as a way to deal with the “inflated” shape selection. Results from both the real data analysis and a simulation study showed that the “combined” posterior summary gives a reasonable assessment of the relative contribution of shape in separating the two groups when combined with intensity. A more formal solution to this problem would be to modify the assumptions on the distribution of the variance for the shape effects . For example, instead of an Inverse Gamma distribution for the shape variance we could assume a Uniform distribution in order to shift the values of the variance distribution to larger values. This would result in more conservative shape selection as, for higher values of The authors have declared no conflict of interest.AppendixClick here for additional data file.AppendixClick here for additional data file.Supplementary MaterialClick here for additional data file.


Pop‐Inference is an educational tool designed to help teaching of hypothesis testing using populations. The application allows for the statistical comparison of demographic parameters among populations. Input demographic data are projection matrices or raw demographic data. Randomization tests are used to compare populations. The tests evaluate the hypothesis that demographic parameters differ among groups of individuals more that should be expected from random allocation of individuals to populations. Confidence intervals for demographic parameters are obtained using the bootstrap. Tests may be global or pairwise. In addition to tests on differences, one‐way life table response experiments (LTRE) are available for random and fixed factors. Planned (a priori) comparisons are possible. Power of comparison tests is evaluated by constructing the distribution of the test statistic when the null hypothesis is true and when it is false. The relationship between power and sample size is explored by evaluating differences among populations at increasing population sizes, while keeping vital rates constant. The application was written, and has been used, as an educational tool in postgraduate courses of population biology at the University of Oviedo, Spain. Students are often puzzled by the possibility of comparing population parameters, such as the population growth rate, using one single observation of the parameter per population. They feel that comparing just single numbers is wrong and that they need some estimation of the variability of the demographic parameters. Even among PhD students, it is not rare the feeling that several replicate estimates of the parameter should be independently obtained to compare populations. However, the comparison of population using single observations of the growth rates is common  the analysis of the distributions of the test statistics under true and false null hypotheses and (ii) the analysis of the effects of increased densities on power, p‐values, and confidence intervals. Pop‐Inference may be used by students in a dual way. At a first stage, to learn how to analyze and interpret differences among populations and then to analyze differences among populations using data from their own projects.The interest of the application is not limited to the numerical output of inferential tests, the Tests on differences in demographic parameters, construction of confidence intervals, and life table response experiments (LTRE) are routinely performed using both randomization tests and the bootstrap  and survival. As a simplification, it is often assumed that data are collected immediately before (prebreeding) or after the reproductive season (postbreeding censuses). See Akçakaya, Burgman, and Ginzburg  for the 3Descriptive information for each population includes six demographic parameters that will be used to compare populations and sensitivity and elasticity matrices. Except for the composition of histories, for each parameter, the application calculates the observed value, the 90% and 95% confidence intervals, and the simulated median. Confidence intervals are calculated using bootstrap techniques , defined as the average number of new individuals produced by a newly born individual during its lifetime. R0 is calculated as the dominant eigenvalue of the matrix R, which is obtained as R = FN, where N is the fundamental matrix whose elements are the expected times spent at each stage by individuals in the population, and F is a fecundity matrix with the expected number of recruits of each class produced by time step  is the time needed by the population to increase by a factor of R0. Because of λT = R0, the generation time is estimated as T = ln R0/ln λ  gives the proportion of individuals in each stage that any initial distribution of individuals converges to  is the Fisherian concept of age‐specific reproductive value generalized by Goodman . IntuitiThe collection of individual histories. An individual history is a summary of the events of the life cycle associated with the individuals in the population. The individual history describes the pattern of survival, growth, and reproduction affecting each individual. The history specifies whether the individual survives or not and the destination class if it does and gives the number of recruits produced during the projection interval and the stages to which new individual recruit. To implement the bootstrap and randomization tests, Caswell  suggesteSensitivity and elasticity matrices describe the changes in the asymptotic population growth rate, λ, as a consequence of changes in matrix entries Caswell, . SensitiPop‐Inference may also perform life table response experiments (LTRE). LTRE evaluate how variability in vital rates contributes to the observed variability in λ. LTRE for both random and fixed effects are available, but only for one‐way designs. An example of the use of LTRE is in Zuidema, De Kroon, and Werger . A significant global test indicates that at least one of the populations differ from the others, as the observed differences are beyond the expected random allocation of individuals to populations, but it does not identify the population(s) differing.Nonoverlapping confidence intervals have been used to evaluate differences in demographic parameters . The four test statistics below are different ways to express differences among a set of values. A survey of distance indices with comments on their relevance in different scenarios is given in Cha . The λ, n in Cha .x1 and x2 are the parameter values for populations 1 and 2, respectively. For global comparisons, the test statistic is the sum of squares of the parameter,n is the number of populations, xi is the parameter value for population i and x¯ is the average parameter across populations.For scalar population parameters in pairwise comparisons, the test statistic is the absolute difference between the two magnitudes,m is the number of stages if SSS or RV are being compared among populations or the number of distinct individual histories, pj1 and p2j are the proportions for stage or history j in populations 1 or 2. Dw is a standard measure of the difference of two probability vectors and also is the Keyfitz's Delta   to the populations and the test statistic calculated again. The procedure is repeated a large number of times . From the distribution of simulated values, a critical value of the test statistic (leaving out 5% of largest observations) is obtained. Now, the application obtains the distribution of the test statistic assuming that the null hypothesis is false. Populations are independently resampled with replacement  and the test statistics computed again from the bootstrap samples. The proportion of values from this distribution larger than the critical value obtained assuming a true null hypothesis is an estimation of power  does not mean that populations are identical or are under identical environmental constraints  are obtained after some working out of the life histories stored in auxiliary matrices. Conceptually, that is not very different to the extraction of well‐known parameters such as a variance.Due to the lack of tabulated test statistics to compare demographic parameters, the need for randomization tests is well understood by the students. Randomization tests are made very intuitive by graphing the distribution of the test statistic under the true  null hypothesis. Understanding the nature and interpretation of the randomization tests has a beneficial side effect. They reveal the importance of sample size and how individuals are sampled from populations. Students are given the keys to design better sampling programs to study demography of wild populations.None declared.This article has a single author who is responsible for the whole content.https://osf.io/2ea6j/.The application may be accessed from the Open Science Framework at  Click here for additional data file. Click here for additional data file. Click here for additional data file.


MEDLINE and Web of Science identifying relevant reports published up to November 2018. Population hemoglobin/erythropoietin level correlations were retrieved, and these empirically derived correlations were compared with the positive correlation implied by a set point model of regulation. Authors of papers containing potentially suitable data were contacted with requests for further analyses, and a meta‐analysis was performed. Twelve correlations between hemoglobin and erythropoietin levels from eleven papers were analyzed. None of these correlations were significantly positive, three, restricted to the normal range of hemoglobin, were significantly negative. All but one of the other correlations showed a negative trend. New analyses of previously published data sets resulted in similar findings. In particular a new analysis of large data sets of males (n = 2417) and females (n = 2592) with normal range hemoglobin levels, revealed significantly negative correlations. A meta‐analysis of our results indicated that the data overall are not consistent with a positive relationship between hemoglobin and erythropoietin (P < 0.0001). Population data indicate that individuals do not have set point levels of hemoglobin.Hemoglobin levels are believed to be regulated as per a set point model of regulation. This model of regulation, by which specific levels of a parameter are targeted and defended by physiological systems, implies a particular population correlation between the parameter and its controlling hormone. Empirical population correlations of other parameters and their controlling hormones, have denied the presence of such set point‐based regulation. To assess if hemoglobin is regulated according to a set point model we performed a systematic search of PubMed/ Reports were excluded if data was not to be considered a reflection of, at least in part, a normal population , or the studied population was less than 20 individuals. Review articles, editorials, and meeting abstracts were also excluded.We extracted from each study the empirically derived associations between hemoglobin and erythropoietin levels, the form of the expression of this association , and the statistical significance of any association. The following information was also extracted from each study: first author, number of individuals in the population, sex, and the prevalence of anemia. We did not address the individual methodology of hemoglobin and erythropoietin measurement on the grounds that any correlations would remain similar with any valid laboratory method.www.ohri.ca/programs/clinical_epidemiology/oxford.asp) was considered inappropriate because these instruments had not been developed for this setting. The PRISMA  guidelines were followed  and a forest plot to test whether the collection of published results were consistent with a positive relationship between hemoglobin and erythropoietin. In order to include the small studies that did not provide Pearson correlation coefficients we also conducted a more general meta‐analysis that did not require Pearson correlation coefficients per se, merely the sign of the slope of the linear relationship and its statistical significance. To do this we classified all outcomes in one of four categories:positive relationship that is statistically significant,positive relationship that is not statistically significant,negative relationship that is not statistically significant, andnegative relationship that is statistically significant.We performed a meta‐analysis of Pearson correlation coefficients  and regarded the different papers as independent observations of this relationship. As all the papers had used a significance level of 5%, the expected proportion of results to fall in these four categories should be 2.5%, 47.5%, 47.5%, and 2.5%, respectively. We therefore conducted exact multinomial tests (using T Menzel ) on all As our data included analyses which were not restricted to the normal range of hemoglobin, we also conducted the same process on the restricted set of analyses that had been classified as only considering the normal range. When we in addition obtained new analyses, conducted on the original data with additional restrictions as to the normality of subjects, we tested these analyses and also tested the combination of the results from the systematic review, restricted to the normal range of hemoglobin, and the results of the new analyses.The systematic search identified 144 potentially relevant records. Another study was identified through review of the reference lists. After removal of duplicates, 104 records were screened for eligibility, identifying eleven data sets suitable for analysis  specifically examining the normal range, or the six studies examining the normal range in conjunction with the anemic range, revealed a significantly positive correlation between hemoglobin and erythropoietin. Significantly negative correlations were found. None of the studies therefore support the existence of a set point for hemoglobin and each of the data sets, to a greater or lesser extent, are in fact inconsistent with the existence of such a set point. By contrast, all of the data sets are consistent with a balance point model of hemoglobin regulation.We were able to have further analyses performed on three of the above‐mentioned studies by the original authors Table . The resP < 0.001) negative correlations between hemoglobin and erythropoietin in individuals of both sexes with normal range haemoglobin levels , is less than the population variation in the parameter physiology  . In particular our work does not contradict the evidence that erythropoietin levels rise in the event of a drop in hemoglobin levels such that a stable level of hemoglobin is restored. Our work does aid physiological research as it explains the difficulties finding the physiological substrates of set points (Modell et al. Further practical consequences of this advance in the comprehension of the fundamental outline of physiological regulation may emerge.There were no conflicts of interest for any of the authors in regard with this work.


The present data article is based on the research work which investigates the influence of blends of diesel and renewable fuels on compression ignition engine emissions. In this experimental work, a 2.4 L, turbocharged, direct injection compression ignition engine and water brake dynamometer were used. Different ternary blends were created by mixing diesel, biodiesel and ethanol together in accordance with a mixture design of experiments. The homogeneity of each ternary blend was qualitatively checked by observing the samples for 24 hours for visible separation.The engine was run over the WLTP drive cycle for each individual ternary blend and the exhaust emissions were recorded. NOVA 7466K and TESTO 350 gas analysers were used to record the exhaust emissions. A factory standard MAF sensor was used to record the inlet air mass flow and an aftermarket ECU was used to determine the fuel flow. The ternary blends were blended using standard laboratory measuring equipment. This was done to ensure the highest accuracy in the measurement, as the two gas analysers have different accuracy levels for different exhaust gasses. Both were located upstream of any exhaust after treatment systems. A summary of the analyser accuracy and measurement method is presented in The engine used for the data collection is a 2.4 L Euro IV turbocharged compression ignition engine with an aftermarket ECU. The engine parameters are listed in The factory fitted mass airflow sensor (MAF) was calibrated with a Superflow SF-120 flow bench and was used to measure the intake mass air flow.The fuel consumption was determined by weighing the fuel tank before and after the test run and dividing the difference by the distance travelled over the WLTP. A Kern EMB 5.2K1 scale was used to weigh the fuel tank with an accuracy of The experimental response of 14 runs in the design matrix along with their corresponding points of the fitted mixture design are shown in 2In order to ensure that the old fuel blend from the previous test does not influence the next test, the fuel system was flushed with the next test's blend of fuel before formal testing began.It was necessary to determine the amount of flushes required that will successfully remove all remaining fuel blend from the previous test. This was done using red fuel dye and conventional diesel fuel. A sample of the clean diesel fuel was photographed using a Canon EOS 700D under homogeneous light conditions. After the dye was added to the fuel tank, the following procedure was followed using the engine's fuel delivery system:Step 1: Run engine with fuel which contains the red dye.Step 2: Use fuel primer pump to pump out all fuel from the fuel system.Step 3: Replace current fuel filter with an empty fuel filter.Step 4: Replace fuel in the fuel tank with clean fuel and run the fuel primer pump for 5 minutes.Step 5: Idle engine for 5 minutes.Step 6: Run engine at 2500 rpm for one minute.Step 7: Use fuel primer pump to pump out all the fuel from the fuel system.Img_base = imread('base.jpg');Img_i = imread('i.jpg');where base refers to the image of the fuel sample with no dye added to it and i refers to the fuel sample taken after each iteration of the engine being flushed with clean fuel. A rectangular area was then determined that has a uniform distribution of colour for all the images. The coordinates of the rectangle were inputted as a column matrix into the software. New image files were created by cutting out the rectangle from the original photos of the fuel samples:Img_base_crop = imcrop;Img_crop_i = imcrop;After the new images were created, the red spectrum of the base fuel's cropped image (clean fuel sample) was subtracted from the red spectrum of each iteration's cropped image. This was done to eliminate any red colour that is present in the clean fuel sample. As a result the red spectrum present in the cropped image is that of the fuel dye only.Img_final_i = Img_crop_i  - Img_base_crop ;The red spectrum present in the fuel samples after each flush iteration was shown visually with a histogram. The histogram graphs were used qualitatively to determine after which flush the majority of the red dye has been removed. This can be seen when the histogram shows that the majority of the red spectrum equals zero (no red colours present in that pixel).A sample of the fuel in the fuel tank was taken after the steps were followed and a photo was taken of the fuel sample. Steps 3–7 were then repeated and another photo was taken of the fuel sample. After sufficient iterations were performed, the image files were imported to Matlab:It is important to ensure that when taking pictures of the fuel samples, the camera is mounted on a tripod and all samples are placed in the same position to ensure that the photographs of the samples were the same size for accurate comparisons to be made. This enables accurate comparison between cropped areas of the photos taken of the fuel samples as seen in The blend ratios as seen in =D,B,EandxD+xB+xE=e blends . Centroi


An algorithm for the simulation of two-dimensional spectral domain optical coherence tomography images based on Maxwell’s equations is presented. A recently developed and modified time-harmonic numerical solution of Maxwell’s equations is used to obtain scattered far fields for many wave numbers contained in the calculated spectrum. The interferometer setup with its lenses is included rigorously with Fresnel integrals and the Debye-Wolf integral. The implemented model is validated with an existing FDTD algorithm by comparing simulated tomograms of single and multiple cylindrical scatterers for perpendicular and parallel polarisation of the incident light. Tomograms are presented for different realisations of multiple cylindrical scatterers. Furthermore, simulated tomograms of a ziggurat-shaped scatterer and of dentin slabs, with varying scatterer concentrations, are investigated. It is shown that the tomograms do not represent the physical structures present within the sample. The depth profiles are obtained using a broadband interferometer setup by interfering electromagnetic fields from a reference arm and a sample arm. The technique is used in many fields such as ophthalmology4, dermatology7, intravascular imaging9, and art conservation12. It is expected that the global OCT market will reach US$1.8 billion by 202413. On the experimental side, several innovations have led to improved imaging resolution14, imaging speed15 and working distance17. On the theoretical side, however, the link between microscopic properties of the scattering medium and the resulting tomograms is not yet fully understood21. It is desireable to link the microscopic properties of the sample to features of the recorded tomograms, to reduce interpretation errors due to artifacts and to improve existing imaging devices. Since calculating the light scattered by an arbitrary sample geometry is complex, computer simulations are employed in many cases. Many computer simulations of OCT images use a Monte Carlo model (see for example25) which neglects the wave nature of light, meaning that phenomena such as diffraction and coherence are also neglected. There are models which use the extended Huygens-Fresnel principle, meaning that the investigated sample is treated as a turbid medium where the refractive index exhibits random spatial fluctuations28. Full-wave approaches which rigorously consider Maxwell’s equations and include the interferometer configuration are often necessary in order to model image formation in OCT accurately. For example, the simulation of OCT imaging of blood cells requires the use of Maxwell’s equations29. There are simulations of OCT images of lung tissue31 where the interferometer setup has not been taken into account. Simulations of single microspheres have successfully been compared to experimental OCT data32. There are several proceedings dealing with OCT simulations based on Maxwell’s equations35. However, calculations including Maxwell’s equations for both the interferometer setup and the interaction of the electromagnetic field with the investigated media for large volumes are computationally expensive and only few works exist37.Optical coherence tomography (OCT) is an imaging modality that enables the recording of high-resolution depth profiles of semi-transparent scattering media38 is used to simulate two-dimensional OCT tomograms. Our previous work39 is extended to complex scatterers. The interferometer setup is included rigorously by using Maxwell’s equations. The structure of this study is as follows: First, the derivation of the OCT algorithm is presented and both the resulting incident beam and the achieved imaging depth are discussed. In the second section, tomograms of different scatterers are presented. Both single and multiple cylindrical scatterers, a ziggurat-shaped scatterer and dentin models are investigated. The results for single and multiple cylindrical scatterers are validated by comparing them to tomograms calculated with a finite-difference time-domain (FDTD) version of the algorithm described in37 and the features of the simulated OCT images are discussed. By going from single to multiple cylindrical scatterers and then onwards to a more realistic scatterer model for tubules in dentin, it is shown for both simple cases and more involved cases that the resulting tomograms do not represent the exact geometrical shapes.In this study, a modified version of the algorithm presented in37. Since the computational effort is considerable, we restrict ourselves to two-dimensional simulations. Figure L1 and L2 in the sample arm and L3 and L4 in the reference arm, respectively. For arbitrary two-dimensional scatterers, a numerical solution of Maxwell’s equation is employed in order to obtain the scattered fields in the sample arm. The scattered fields are propagated back through the lens system and coupled back into the detection fiber, for each wave number, interfering with the reference fields. This means that illumination and detection is modeled as a superposition of coherent modes which closely resembles a swept-source setup. However, since the light in the reference arm and the sample arm comes from the same source, and since only the correlation between the reference and the sample arm is measured in OCT, the calculation is mathematically equivalent to a spectral domain setup with a spectrally broad light source40. In the following, a description of each calculation step of the OCT model is given.Optical coherence tomography uses the interference between a reference arm and a sample arm signal in order to reconstruct the relative phase of the light. In order to develop a realistic algorithm for an OCT setup based on Maxwell’s equations, the electromagnetic fields need to be propagated through the optical system. Note that the model for the interferometer setup presented here uses a similar approach as that derived iny-direction or in z-direction, while propagation is in x-direction as indicated in Fig. w0 is half of the fiber’s mode field diameter, taken to be w0 = 4.6 μm in the given setup. E0 is the maximum magnitude of the field. In the following, the vectorial character L1 can be calculated ask is the wave number and ky is its component in y-direction. Setting L1 (or L3) and using the paraxial approximation to write L1 is given byf1 is the focal length of the first lens L1. For the reference arm, the same expression can be used, especially when L1 and L2 (or L3 and L4), a two-dimensional Fresnel integral is used together with the paraxial transmission function of the lens 41 where nL1 is the refractive index of the lens L1 and DL1 is its maximum thickness at the center. The two-dimensional Huygens-Fresnel integral is given as42θ is the angle belonging to a point between the x-axis and the y-axis, Uinc is the incident field at the input plane. λ is the wavelength. Similarly to the three-dimensional approximations for propagation through lenses in41, the distance L1 and L2 (L3 and L4) in paraxial approximation at the focus becomesAssuming a weakly guiding fiber that emits a wave field, and omitting a time dependence of mentclass2pt{minimDL can be neglected. The paraxial approximation is valid here since OCT uses a small numerical aperture. In many imaging systems such as OCT, an afocal and telecentric lens setup with the aperture stop between L1 and L2 (L3 and L4) is used. The aperture stop appears then at infinity from both the sample space and the detector space. For this reason, the Debye-Wolf formalism can be used45 which represents a rigorous solution of Maxwell’s equations. As described above, the scalar Fresnel integral is sufficient to calculate the propagation of the fields through a lens system with a low NA as is used in OCT systems. However, a rigorous solution of Maxwell’s equations is needed in order to couple the fields into the simulation grid of the numerical solver38. For this reason, the Debye-Wolf integral is employed in order to calculate the electromagnetic field in the sample space. In three dimensions, the Debye-Wolf integral is given by47The thickness of the lens z is the distance from the focal position. L2 (and L4)49. Note that many authors omit the global phase eikf. In three dimensions, the rotation matrix L which is given by θ and θ stemming from the Jacobian element from the conversion from Cartesian to spherical coordinates disappears in cylindrical coordinates. In order to determine the prefactors for the cylindrical Debye-Wolf integral, we follow the derivations in46: The two-dimensional Green’s function is given asR is the radius of the reference circle, Here, 46 with the developed two-dimensional approach, the Debye-Wolf integral is obtained as50. By using the one-dimensional Debye-Wolf integral, the aperture is assumed to be a slit and the lenses are approximated as cylindrical lenses. Sheppard50 gives comparisons of the two-dimensional  and the three-dimensional case  and states that “The general features of both the cylindrical and spherical cases are similar …” which gives an idea of what the approximation of a three-dimensional interferometric system as two-dimensional encompasses. Continuing therefore with the one-dimensional Debye-Wolf integral with lenses fulfilling the sine condition in a coordinate system as shown in Fig. 45. Identifying Both the Green’s function and its gradient can be used in the vectorial Kirchoff-Fresnel diffraction integral. Following the derivation inmentclass2pt{minimL1 and L2) or on a reference mirror (L3 and L4). For simplicity and because a good optical system is well-aligned, we assume that the reference and the sample arm have the same fields in their focal regions, stemming from similar lenses. However, different lenses for the reference and for the sample arm can easily be considered and the derivation leading to equation  is fulfilled. First, the plane wave in equation  to the middle plane of the lens. Subsequently, the fields in front of the detection fiber are calculated with the Debye-Wolf integral. Since the fields emitted by the fiber have been described by a Gaussian shape, the amount of light coupled back into the fiber isThis removes the autocorrelation and the DC-artifact.39, the calculation time necessary for a simulation, and the maximum distance ctmax from the focus that will be visible in the tomogram. If the number of sampled wave numbers, N, in a given interval between 55δk is the spacing between the sampled wave numbers. The factor of 2 in the denominator indicates that the light undergoes a roundtrip through the considered medium. Equation  and at 59, we consider an imaging system with λ0 = 1.3 μm and we use a FWHM of Δλ = 170 nm (related to the maximum amplitude) in all presented simulations. The OCT signals are calculated with equation 38 in the simulation grid to reduce artifacts caused by the finite size of the grid cells on the scatterer boundary. Compared to previous work39, the new simulations include rigorously the interferometer setup and the phase shift due to the detection fiber being at another position in lateral direction when the beam is scanned over the sample. In OCT, scanning over the surface of a sample is done by moving the position of the illumination and detection fiber or by deflecting mirrors. This means that the scanned beam would impinge on the lenses at another position in y. However, as long as this deviation is small, especially since a good interferometer setup is designed to reduce such abberations, it is reasonable to assume that the beam still propagates through the center of the lenses in the simulations, which proves to be a reasonable approximation. For this reason, scanning over the sample consists of either moving the sample relative to the imaging system or moving the imaging system relative to the scatterer, yielding equivalent results.In the following, selected tomograms are presented in order to give an overview of the effects that appear in OCT imaging simulated with Maxwell’s equations in two dimensions. Similar toequation . It is n39. In order to validate the developed algorithm, the tomograms calculated using a modified version of the MAXWELL-solver and the interferometer setup described in the theory section are compared with the two-dimensional version of the FDTD-based code presented in37. Figure a = 5 μm and a refractive index of ncyl = 1.42 in a medium of nmed = 1.35, which is similar to biological tissue in an organic solution61. The light is incident from the negative x-axis. Figure 37. Figure 37. Since the tomograms are symmetric, only half of the scanning positions are shown. Furthermore, the agreement between the simulations is quantified by calculating an integrated error over all image points asIMAXWELL and IFDTD are the absolute values of the OCT signal amplitude for MAXWELL- and FDTD-based model, respectively, and the sum extends over rows and columns of the image. In order to do this, the result obtained with FDTD is interpolated to match the number of image points obtained with the MAXWELL-solver. It can be seen that the tomograms show good qualitative agreement. As described in39, the distance between the cylinder front and back side is d = 2aμm in Fig. 39, refractive indices closer to the case of biological tissue have been used and the interferometer setup has been included rigorously according to Maxwell’s equations. With the obtained results, it is possible to explain the structure of experimental tomogram images presented elsewhere62.We extend the investigation of simple cylindrical scatterers that has been described in previous worka = 20 μm, and with the same refractive indices as in Fig. d = 2aμm. In the tomograms, the surface waves appear when the incident beam interacts with the cylinder at the lateral boundary (|y| ≈ a). Due to the low contrast between cylindrical scatterer and medium, the surface waves appear only at one depth position. Simulations with a higher refractive index contrast lead to more surface waves in the performed computations. As explained in the theory section, compared to our previous work39, different refractive indices have been used and the interferometer setup has been included rigorously. As a consequence, the fine substructure of the surface waves can be observed in the computed tomograms. To the best of our knowledge, this is the first time such a structure has been observed in computed tomograms of cylindrical scatterers. Whilst performing these simulations, we found that the structure of the surface waves is sensitive to the source condition that is used to generate the incident field. In particular, in the FDTD code, the full time-harmonic incident field should be calculated for every wave number k and then Fourier-transformed into time domain. Approximations to this approach that allow for an analytical expression of the incident field in time-domain in the FDTD code63, however, change the structure of the surface waves observed in the simulated tomograms. This is the first time, to our knowledge, that such an effect has been observed. At greater depths, signals from reflections inside the cylinder can be seen. Again, the comparison between MAXWELL-solver and the FDTD-based model show good agreement. However, the rippled structure that seems to trace the lateral side of the cylinder surface with an intensity that is orders of magnitude lower than the signals from the front and back of the cylinder differs between the simulations. Increasing the length of the absorbing boundaries in the MAXWELL-solver or changing the type of solver from the convergent Born series to a GMRES64 or a CGNR solver65 approach does not remove these structures. When the grid resolution is increased so that the size of the individual grid cells is smaller (and the curvature of the round structure is approximated better), it can be shown that the intensity of the rippled lateral structures tracing the cylinder curvature is reduced. In other words, it can be shown that these signals are artifacts from the numerical grid in both the time-harmonic MAXWELL-solver and the FDTD-based model. Another way to show this is to replace the numerical calculation with the analytical solution, by Bohren and Huffman54: Every plane wave incident at a certain angle gives rise to a scattered far field which can be weighted and integrated. Figure y-values for clarity. The simulation shows that when there are no artifacts from a numerical grid, the rippled structures at the lateral cylinder sides disappear and the cylinder curvature cannot be seen. The black line shows where the cylinder surface facing the incident beam is located. The dotted black line shows where the other half of the cylindrical scatterer appears in the tomogram when the delay due to ncyl along the path is taken into account. It can be seen that the scatterer appears distorted. Experiments with bubble raft phantoms and recorded OCT images67 show similar distortions. At a greater depth position behind the cylinder, at x = 63.75 μm, the signal from multiple reflections inside the cylinder  can be seen clearly.The structure of the tomograms naturally changes when the cylinder size changes. In Fig. μm and a refractive index of ncyl = 1.42 are distributed within a circular area in the numerical grid of the MAXWELL-solver. Note that we do not expect to be able to resolve the individual cylindrical scatterers due to their size, but the total size of the circular area is larger than the resolution limit of the optical setup. The refractive index of the outer medium is again nmed = 1.35. In Fig. 36. Furthermore, as expected, the individual cylinders cannot be resolved by the given OCT system, however, the general outer contour of the round shape is visible. The shape of the simulated tomograms is complex due to the many interactions of the light with the cylindrical scatterers. There is good qualitative agreement between the results obtained with the MAXWELL-solver and the results obtained with the FDTD-based model. The influence of the individual scatterers is investigated by randomly shifting their positions from the original positions shown in Fig. x- and y-directions, and for each cylinder, resulting in the cylinders being shifted by a minimum of 0.06 nm and a maximum of 230 nm in x-direction and by a minimum of 14 nm and a maximum of 192 nm in y-direction. The maximum overall shift is 253 nm and the average overall shift is 88 nm. Note that cylinders were not prevented from overlapping, however, this does not occur frequently. Figure It has been shown that the calculated cylinder tomograms do not reveal the physical surface of the scatterers. In order to investigate whether the general shape of a distribution of multiple cylindrical scatterers can be imaged, many cylinders with a radius of 0.5 IMAXWELL − IFDTD for all points in Fig. Figure x = 0 μm as one might expect  is smaller than the lateral width of the grid (69.3 μm) so that edge effects are reduced. Figure y ≈ 8 μm cannot be distinguished due to the lateral resolution. The tubules at greater depths cannot be identified in the tomogram. Assuming that the tubules can be approximated as cylindrical scatterers, the mean free path length in the slab model with dimensions Lx and Ly is calculated as Csca is the scattering coefficent for infinitely long cylinders as described by Bohren and Huffman54. For Lx = 40.4 μm, Ly = 69.3 μm, and for an incident wavelength of μm when cylindrical scatterers with a radius of a = 1 μm are assumed. This shows that most of the light undergoes multiple scattering, causing signals to appear at positions that are not easily linked to the positions of the tubules.Having presented several basic scatterer configurations in order to investigate OCT imaging with Maxwell’s equations, more realistic scatterer configurations for water-filled tubules in a dentin slab are now shown. Since the previous sections have shown that OCT does not always reveal the true physical surfaces of individual scatterers, it is interesting to see whether OCT images of dentin slabs show the actual position of the tubules. The dentin slab with the tubules varies less along the equation  are diviLx × Ly = 162.5 × 63.7 μm2. Over the first half of the depth coordinate it can be seen that most signals in the tomograms can be linked to the positions of the tubules. With increasing depth, the correlation with the tubule positions decreases. It can also be seen that the intensity of the tomogram decreases with depth. The mean free path length is L = 24.8 μm for the perpendicular polarisation shown in Fig. In order to alleviate this effect, the concentration of tubules is decreased. Figure μm. It can be seen that more correlation between the actual positions of the tubules is achieved compared to Fig. Figure μm and all tubules can be identified. However, at greater depths, some artifacts are visible where no scatterers are located. Therefore, the scatterer concentration is reduced further in Fig. μm. In Fig. Reducing the scatterer concentration further shows the location of all tubules in the tomogram. This is shown in Fig. A two-dimensional algorithm for the simulation of OCT images was implemented and validated with an independent FDTD-based model. An expression for the maximum visible depth was derived and simulated tomograms of several scatterer geometries were investigated. Tomograms of single cylindrical scatterers were calculated and the origin of the individual signals was discussed. Tomograms of multiple cylindrical scatterers, of plane boundaries and of dentin models were presented as well. The simulated tomograms showed that care must be taken when interpreting OCT images. The smooth round surface of the considered single cylindrical scatterers was not visible in the tomograms. Furthermore, it was shown that in dentin slabs, artifacts can be mistaken for the position of scatterers, depending on the concentration of the water-filled tubules inside the dentin. Tomograms of many cylindrical scatterers yielded complex images with patterns that were sensitive to small shifts in the cylinder positions which showed that in order to understand the formed tomogram, the precise location of the scatterers must be known. The individual cylinder positions could not be detected, but the general outline of the outer contour of the scatterers could be traced. In tomograms of a ziggurat-shaped scatterer, the boundaries were much more readily identified. In short, by computing tomograms for various scattering geometries, the correlation between OCT tomogram and sample structure has been investigated successfully for several cases, showing that the tomograms do not accurately represent the surface of the scatterers. The simulations suggest that in order to understand image formation in OCT, both the wave properties of light and the properties and positions of the scattering structures must be taken into account.Supplementary information