The evaluation procedure is supplemented by a detailed measurement precision assessment by applying an error propagation through the non-linear model fitting approach. While the algorithms are tailored to SCLAS, they can be transferred to other spectroscopic methods, that similarly require an instrument function. The presented methods are validated using gas cell measurements of methane in the near infrared region at pressures up to 8.7 bar.A set of algorithms is presented that facilitates the evaluation of super continuum laser absorption spectroscopy (SCLAS) measurements with respect to temperature, pressure and species concentration without the need for simultaneous background intensity measurements. For this purpose a non-linear model fitting approach is employed. A detailed discussion of the influences on the instrument function of the spectrometer and a method for the  Tunable diode laser absorption spectroscopy (TDLAS) fullfills these requirements2. But unless a complex laser modification of the standard diode laser setup is used4, TDLAS is limited to a small spectral range of several cm−1 . Therefore the application of TDLAS in high pressure environments is difficult because of the broad spectral lineshapes due to collisional broadening. On the other hand there are spectrocopic techniques like classic fourier transform spectroscopy (FTS), that cover a broad spectral range with high spectral resolution but lack the time resolution to resolve the progress of chemical reactions or process fluctuations. These disadvantages have partially been reduced in new FTS methods like dual comb spectroscopy5 (DCS) which increases the time resolution while also avoiding moving parts in the setup. Nonetheless DCS still requires the sampling of several laser pulses for recovering one spectrum. This limits the theoretically achievable repetition rate. The recently introduced super continuum laser absorption spectroscopy (SCLAS) is theoretically able to measure a whole spectrum in a single laser pulse while still covering a broad spectrum of several 100 wavenumbers7. The broad spectral range that is covered facilitates the simultaneous detection of several species6 as well as the simultaneous determination of pressure and temperature from a single measurement8. This supercontinuum laser absorption spectroscopy is based on spectrally broad pulses with a duration in the order of 100 ps. First of all this pulse is filtered to limit its spectral bandwidth to the wavelength region on interest. These filtered pulses can then either be dispersed in time7 or in space8 to resolve their spectral content. This work focuses on the dispersion in time, but the presented method is applicable to the space dispersed variant of SCLAS as well.Strict regulations of pollutant emissions as well as the economic demand for efficiency necessitate the optimization of combustion systems and other high temperature and pressure processes. Measurement systems that allow for the To achieve the dispersion in time the initially short pulse is passed through a dispersion compensation module (DCM), which results in a temporally stretched pulse with a duration of several 100 ns. This temporal pulse profile reflects the spectral distribution of the initial SCL pulse. The dispersed pulse then passes through the measurement volume where it gets partially absorbed. This step can also be swapped with the DCM as the results are the same in stationary measurement volumes. In instationary volumes passing the pulse through the measurement volume before the DCM can have advantages in terms of a higher signal to noise ratio as a temporally short pulse avoids additional averaging over transients in the system. Nonetheless the evaluation procedure for both cases is the same.et al., Kaminski et al., Dupont et al. and Blume et al.10. For details we refer to Blume et al.11, as this work will give a general discussion of the analysis methods for SCLAS measurements.Afterwards the intensity profile is detected by a fast photo diode and recorded on a high bandwidth oscilloscope. This measurement principle is summarized in Fig. While a simultaneous reference measurement of the initial intensity distribution simplifies the evaluation procedure, it also introduces disadvantages regarding setup complexity and applicability to harsh industrial environments. E.g. in combustion environments broadband absorbing materials can easily agglomerate or condense on optical measurement accesses. The absorption bands of these solid or liquid materials are significantly broader than gas absorption features but are still covered in the broad spectral range of SCLAS. Thus applying the measured initial intensity distribution in the evaluation would introduce a low frequency distortion in the signal. Furthermore the short time scales of the laser pulses make an accurate tuning of the light path lengths of the reference beam path to the measurement beam path necessary. Otherwise the two measurements would be shifted on the time axis. To avoid these disadvantages the procedure presented here does not rely on a reference measurement, but uses a fitted background model for evaluation.12 in the order of 0.1 cm−1 the instrument function of a SCLAS spectrometer can not be neglected. An important influence on the instrument function is the photo diode whose influence can become visible as so called diode ringing11. Blume et al. neglect the instrument function and evaluate only the line areas. Therefore this approach does not allow for the evaluation of pressure values. Werblinski et al. determine the instrument function of the photo diode by a separate measurement, but the influence of the initial SCL pulse length and profile are neglected. To overcome these disadvantages a comprehensive approach has been developed for the in-situ determination of an instrument function covering all effects given by the SCL source, the photo diode and the data acquisition. Furthermore many of the current evaluation procedures evaluate the data in the absorbance domain and not in the intensity domain. Hence a discussion of the advantages of the evaluation in the intensity domain is given. These advantages mainly become visible in measurements with strong absorption, which are often unavoidable due to boundary conditions of the experiment. Regarding the unknown initial intensity distribution a simultaneous fitting approach13 is applied, while previous evaluation procedures used estimation algorithms for the background14. These components are combined to a comprehensive SCLAS data analysis method, that enables the evaluation of mole fraction, temperature and pressure from SCLAS measurements even for partial total absorption. To estimate the potential of SCLAS for measuring different quantities under various conditions a detailed uncertainty estimation regarding noise, uncertainty in wavenumber axis and uncertainty in the instrument function is given.Current challenges in the SCLAS data analysis are the high absorption line count that needs to be evaluated, the determination and application of the instrument function and the unknown initial intensity distribution prior to absorption. The high number of absorption lines of a species that is covered in a single measurement on the one hand improves measurement precision. On the other hand it makes the evaluation using a model fitting approach computationally very expensive. Regarding the achieved spectral resolutions15. Hybrid methods like indirect hard modeling (IHM) utilize physical knowledge, like line shape functions, to reduce the measurement data and calibrate a more robust statistical model16. An approach that does not depend on calibration data is the fitting of a comprehensive physical model. This model can be derived from a physical model and certain fixed parameters (e.g. line data). This approach has already been applied to SCLAS data14. The preference for the physical model approach is easily explained regarding the availability of line databases like HITRAN17 and the possibility to avoid calibration measurements. Therefore the method explained in this article will also follow this approach.For the processing of broadband absorption spectroscopic measurements different methods have been proposed. These methods can be distinguished regarding the amount of physical knowledge that is utilized in the data analysis. Statistical methods like partial least squares (PLS) do not require any knowledge of the physical system but rather calibrate a purely statistical model using several calibration measurementsIn the following sections a model for the measured voltage signal is developed, that incorporates the actual absorption effects as well as the behaviour of DCM, photo diode and oscilloscope.T, the pressure p and the absorber mole fraction q. The line data that is needed for such a simulation can be taken from databases like HITRAN201617, which is used in this work. According to Beer-Lambert’s law the absorbance of several absorption transitions can be described asThe first part of this physical model is a model for the absorbance depending on the temperature Iν is the transmitted spectral intensity at wavenumber ν, Iν,0 is the initial spectral intensity, q is the absorber mole fraction, L is the absorption path length, Si(T) is the line strength of the line i, Φ is the line profile function and νi0, is the center frequency of transition i . N describes the number density in the measured gas sample, which in the case of ideal gas can be calculated as N = qp/(kbT) with the Boltzmann constant kb. For many technical systems the line profile function Φ is sufficiently described by a Voigt profile, which is a convolution of a Gauss profile with a full-width-at-half-maximum (FWHM) ΔνiG, and a Lorentzian with a FWHM ΔνiL,. These profile widths are calculated using line data supplied by HITRAN and therefore are limited to self- and air-broadening.In equation , Iν is tSi(T0), the line strength at temperature T can be calculated asUsing HITRAN reference line strengths Z(T) is the total partition sum at temperature T, E″i is the energy of the lower state of transition i and h is Planck’s constant.In equation , Z(T) iset al.18 was chosen for its reduced computational effort.The calculation of absorbance spectra for the evaluation of SCLAS data can be computationally very expensive, because the number of lines to be simulated can easily reach several thousands . The calculation of the Voigt profile constitutes the largest part of the calculations during the absorbance simulation. Therefore the Voigt approximation approach of McLean I0 of the SCLAS measurement can either be measured simultaneously8 or be estimated from the actual measurement signal. In the SCLAS spectrometer regarded in this work, the incident background intensity I0 before absorption is not measured separately or simultaneously. For modeling this unknown background intensity, different approaches are possible. The first approach is to estimate the background using only the measured I with absorption lines11. This approach relies either on a certain shape of the absorption lines or on a difference in the frequency content of absorption lines and the background. According to an approach originally developed by Kranendonk et al. the derivative of the intensity I08. While these two approaches either estimate the background or limit the influence of the background on the fitting result, it is also possible to create a background model that is fitted to the measurement data simultaneously13. This approach promises good accuracy and is already used commonly in TDLAS data analysis. In the context of SCLAS the fitting process grows in complexity due to the additional background variables that need to be optimized. Nonetheless the simultaneous fitting approach was chosen for the implementation in the current work as it promises the best results. Therefore the raw model of the intensity is described byβ0, …, βm are the background parameters, that define the shape of the background. Polynomials are often chosen as parametric background functions13 but due to the strong fluctuations in the SCL intensity distribution the polynomial approach is not expected to describe the profile sufficiently accurate. Therefore a cubic spline function with equidistant support points is chosen to describe the background. The number of support points of the spline has to be adapted to the problem and the regarded spectral window. It has to be high enough to describe the background sufficiently, but if the number of support points is chosen too high the background tends to partly approximate the absorption lines. This effect can influence the fitting results and has to be avoided. As of now the maximum allowable number of support points has to be estimated according to the maximum expected absorption line FWHM Δνmax. Under the assumption that collisional broadening is the dominant broadening mechanism the maximum line widths are reached at maximum expected pressure and lowest expected temperature. For these parameters the Lorentz FWHM of all absorption lines in the regarded range are calculated. As collisional broadening is also dependent on the collision partner, these widths have to be calculated twice: once for absorber concentrations approaching zero (e.g. q = 0.001) and once for a pure absorber (q = 1). The largest Lorentz FWHM of both cases is used as Δνmax. As a rule of thumb the distance between the equidistant spline support point should not be chosen smaller than Δνmax.The initial or background intensity 19. Nonetheless the SCL pulse profile is still unknown and the transfer of the manufacturer supplied photo diode response function to a different measurement system with even minimal deviations in the electrical characteristics is critical. Therefore we present an approach for the determination of the instrument function directly from a measurement with known absorber mole fraction, temperature and pressure. The three already mentioned influences on the instrument function will be discussed separately before finally describing the combined method. Note that in the following discussion transmission loses other than the absorption in the measurement volume will be neglected because they do not influence the spectral resolution or instrument function in the absorbance signal.The instrument function of the SCLAS spectrometer is influenced by different parts of the system: the finite length and temporal pulse profile of the undispersed SCL pulse, the dynamic response of the high bandwidth photo diode and the finite sampling rate and pulse response of the oscilloscope. The sampling rate is known and the pulse response function of the high bandwidth photo diode is either supplied by the manufacturer or can be measured using pulsed lasersIν,SCL . Due to dispersion and other effects during super continuum generation a significant wavelength dependent delay can be introduced. In other words, the peak wavelength is dependent on time during the pulse. To simplify the modelling of the SCL pulse this wavelength dependent delay is incorporated into the model of the DCM discussed below. Thus the model for Iν,SCL  discussed here neglects this time dependency of the spectral content of the pulse.The initial SCL pulse spectrally typically covers wavenumber regions that are not needed for the actual measurement. Therefore the pulse is bandpass filtered for the spectral region of interest in order to maximize the achievable repetition rate. This filtering does not influence the instrument function as it only limits the covered spectral region. The resultant SCL pulse can be described by a time dependent spectral intensity distribution Therefore the qualitative spectral intensity distribution of the pulse is assumed to stay constant over the whole pulse duration while only the intensity is dependent on time. Put differently the normalized spectral distribution Iν,SCL  to be described byThis assumption allows the spectral intensity distribution Note that this model of the initial SCL pulse is still a rough approximation of the SCL pulse, but allows for a simple derivation of the instrument function and gives good results in practical application as will be shown in the experimental results.D : t → ν can be defined, that assigns a wavenumber to every time step in the measurement. An example for this function D(t) is given in Fig. et al.6. But as D(t) needs to be known accurately, a method using the absorption lines as a reference is applied11. Using the DCM dependent function D(t) the time dependent spectral intensity distribution after the DCM is described byBy passing through the DCM the spectral components of the pulse are delayed according to their wavenumber. Hence a function After the DCM the dispersed pulse is passed through the measurement volume and partially absorbed:photo diode refers to the whole detection system including the measurement amplifier as it significantly influences the response of the photo diode. Furthermore only the dynamic response of the photo diode is regarded in the following section.The dispersed pulse, modulated with the absorption features (equation ) is thenIABS(t) at the photo diode instead of the spectral intensity distribution. The intensity IABS(t) is defined by the integralThe wavelength dependent sensitivity of the photo diode will be neglected at this point as it does not have any effect on the instrument function such that only the time dependent pulse response of the photo diode is regarded. Thus it is reasonable to only consider the time dependent intensity The normalized spectral intensity distribution equation  to −∞ wiequation  can be rD(t). Thus Equation (The relation between time delay and wavelength is given by Equation  can be tISCL(t) describes a short pulse in the vicinity of t = 0, it is valid to assume that ISCL(t) is only non-zero in a short time interval 0 ≤ t ≤ ΔtSCL. Thus for a constant time t the integral in equation (τ of length ΔtSCL. The pulse length ΔtSCL is in the order of 100 ps and thus too short for any significant change in t in equation (As equation  is only entclass1pt{minimaEquation  can thusEquation  describet ≥ 0 equation (IABS(t) at the photo diode decreases. This behaviour is intuitively explained as the SCL pulse only consists of a limited amount of energy. Thus an elongation of the pulse in time results in a smaller intensity.With these integration limits the integral describes a convolution. Therefore for all equation  can be w−1 to 6145 cm−1). Thus the derivative t1 and t2 describe the regarded measurement window. This approximation is illustrated in Fig. Furthermore the factor entclass1pt{minimahPD(t)20. The resulting voltage at the photo diode is given as the convolutionThe photo diode itself is modeled as a linear time invariant system and is therefore fully described by its pulse response hDAQ(t):hSCLAS(t) is the pulse response of the whole spectrometer.Similar to the photo diode the finite sampling rate and pulse response of the oscilloscope is modelled as another pulse response I0(ν) is highly unsteady. Thus several pulses have to be averaged in accordance with the trigger of the pump pulse:Due to the stochastic process involved in the creation of a SCL pulse, the spectral intensity distribution of a single pulse and therefore t = 0 always refers to the trigger of the corresponding pulse. Furthermore equation (A(D(t)) is constant and thus that the gas state does not change during the measurement interval. The averaging results in a less noisy averaged measurement signal In equation , the meaequation  assumes equation  to equatequation  shows thhSCLAS in the time domain. Due to the discrete sampling of the oscilloscope, measurements are only available at the time points tl = lΔtsampling with According to equation  all insthSCLAS [l] from measurements, UDAQ[l] and I0[l] eA(D[l])− have to be known. UDAQ[l] is easily attained as it is a SCLAS measurement, whereas I0[l] eA(D[l])− can not be measured directly. Therefore I0[l] eA(D[l])− needs to be simulated according to theory. Thus SCLAS measurements with known temperature, pressure and mole fraction are used, which allow the simulation of A(ν) from HITRAN data. The background intensity distribution model I0[l] (a spline function) is then fitted to the measurement using an estimated instrument function. The result is a discrete deconvolution problem with the measured UDAQ[l] and the simulated Ith = I0[l] eA(D[l])−. The discrete convolution can be described as a linear equation systemTo determine hSCLAS are limited to a small range −μ ≤ l ≤ μ with t = l = 0. The elements outside of this range are neglected. With n × (2μ + 1) matrix A describes an overdetermined linear equation system for which the least square solution hSCLAS can be found. This yields a valid estimate for the instrument function that can be used for different measurements with the same setup. The best value for μ has to be determined empirically. Too low values will cut away parts of the pulse response function and therefore distort the profile. Too high values increase the number of variables and thus increase the instability of the equation system and the influence of noise on the result.It is valid to assume that the SCLAS pulse response function is narrow compared to the actual measurement duration. Thus non-zero elements in the vector UOSC and measured voltage i:The cost function is chosen to compare modelled voltage σi at the time points ti are assumed to be equal σI = σi. As the standard deviation σI can not be estimated beforehand, it is set to σI = 1, because the result of the fitting is not dependent on a constant factor in the cost function. Whereas the commonly used cost function compares absorbances13, equation  uncertainty σI is given by equation (The measurement uncertainties equation  directlyHence the absorbance uncertainty increases inversely to the measured intensity (or voltage). Thus strong absorption lines with a small measured intensity lead to big uncertainties in the absorbance, that are not considered in the cost function as they are not known beforehand. Therefore the direct evaluation of the measured voltage promises better evaluation results in measurements with strong absorption features.UDAQ is calculated directly and supplied to the optimization algorithm.The cost function is minimized using the non-linear least-square optimizer function lsqnonlin in MATLAB. To increase the numerical efficiency of the optimization the Jacobian of the signal model T′, p′, q′ and βj for the parameters T, p, q and βj. These values minimize the cost function and therefore represent the evaluation result of the measurements in temperature, pressure and mole fraction. This evaluation procedure is summarized in Fig. The optimization process yields the optimum values To evaluate the capabilities of the SCLAS measurement and analysis method, the resulting uncertainty of the model fitting approach is calculated. Uncertainties of the HITRAN parameters are not an essential part of the SCLAS measurement and analysis method described here and can be replaced with higher quality line data. Therefore the uncertainties of the HITRAN parameters are neglected in the following uncertainty estimation to allow for an unbiased estimation of the potential of SCLAS. Put differently, the uncertainty estimation presented in this paper sets a maximum limit for the achievable precision in the regarded SCLAS setup by assuming perfect line data. The practically achievable precision and accuracy of course still depend on the quality of the line data.The uncertainty of the SCLAS measurement and analysis can be divided into the uncertainty induced by measurement noise, by the determination of the wavenumber axis and by the instrument function. These factors will be discussed separately and are combined in the end.p′, T′ and q′, the variance of the measurement noise is estimated. With k sampled points and m fitted variables aj  the standard deviation of the signal noise is estimated to be:ri is the final fit residuumTo determine the influence of the noise of the intensity measurement on the fitted parameters m. The determination of fitting parameter uncertainties from measurement data uncertainties is well known for linear fits. But the nonlinear model demands the consideration of inter-parameter dependencies. According to Burell21 the uncertainties aj for this non-linear model can be calculated asεjl is the inverse of the matrix αjl:Equation  considerThe model derivatives of first and second order are estimated using finite differences.D(t) is also subject to measurement uncertainty. This uncertainty is similar to an uncertainty in the measurement times ti. Solutions to this uncertainty problem for nonlinear models are discussed by Cecchi22 under the assumption of uncorrelated uncertainties. But the assumption of uncorrelated errors in the measurement time ti is not valid for this case since the deviations are correlated by the function D(t). Hence the methods applied by Burrell and Cecchi need to be adopted to account for this correlation. First an assumption concerning the correlation has to be made. One systematic source of such a correlated error could be introduced by the determination of D(t). But the results for D(t) lead to no visible deviations between measured absorption lines and HITRAN line positions. Even if a small deviation was present, the method used for the determination of the instrument function is able to compensate small and constant deviations in the wavenumber axis. A possible source for a stochastic error in the wavenumber axis is given by jittering of the trigger signal for the data acquisition. A deviation in the trigger time would lead to a constant offset in D(t) as all measurement times ti would be shifted equally. This error source is assumed to be dominant. Therefore the dispersion function is decomposed toThe experimentally determined time-wavelength-relation D0 is assumed to have an uncertainty, which leads to a constant and therefore directly correlated offset in the wavenumber axis. With this definition D0 becomes an additional parameter of the simulated signal UDAQ.Only the constant offset D0 is limited by the time step size respectively the sampling frequency. The function D(t) can be approximated as linear in the measurement window and the values of D0 are assumed to be uniformly distributed in an interval of the width gΔtsampling with g defined according to equation . Thus the uncertainty in equation . Hence tσD0 on the resulting optimum parameters the implicit description of the fitting optimum22D0 and yields (see22)To determine the influence of the uncertainty σi = σ, equation  yields the equation systemSimilarly to the application of Burrell’s approach in the wavenumber uncertainty assessment, the differentiation of the implicit equation  with resaj/∂UDAQ(ti) for 1 ≤ j ≤ m and 1 ≤ i ≤ k, which allow for the calculation of the instrument function induced uncertainty aj:This equation system is solved for the derivatives ∂aj results from the noise induced uncertainty Now all three uncertainty components are known. The final uncertainty All experimental datasets and evaluation results of this contribution are available on request from the authors.The theoretical derivation of the presented methods need to be applied to real measurement data demonstration and validation. The first step is therefore the determination of an instrument function from 20 separate measurements of methane. These measurements are carried out using the same setup as for the latter actual measurements, without the need for additional equipment. After that, ten measurements of methane at ambient pressure are evaluated. The low pressure ensures that the line shapes stay narrow. This way diode ringing and instrumental broadening have an significant effect on the measured line shapes and thus need to be reproduced by the instrument function. The ambient pressure measurements are therefore mainly used as a test bench for the instrument function. After that ten measurements at a higher pressure of 8.732 bar are evaluated. These measurements were conducted with a methane mole fraction of 0.223. In combination with the elevated pressure this leads to some wavenumber intervals being dominated by total absorption. This way two of the claims made for the presented methods can be tested: evaluation at higher pressures and evaluation in case of partial total absorption. All measurements were conducted at room temperatures as this is the simplest way to achieve homogeneous temperatures in the absorption path.et al.11 and therefore only the most relevant parts are discussed here. The photo diode is a Newport Newfocus 1544-B-50 with a bandwidth of 12 GHz. The pulse response function of the photo diode is supplied by the manufacturer. For the data acquisition a LeCroy SDA 820Zi-A oscilloscope with a bandwidth of 20 GHz at a sampling rate of 40 GSps is used. The oscilloscope is also employed to directly average over 9849 consecutive SCL pulses. An example for the captured measurement is depicted in Fig. The spectrometer setup is equal to the one described and employed by Blume The measured methane is contained in a high pressure double-pass absorption cell with a total absorption path length of 367 mm. The reference temperature is measured with a thermocouple of type E in the cell. For room temperature measurements as they are conducted in this paper the combined reference temperature uncertainty of thermocouple and measurement system is ±2.9 K. The reference pressure is measured with a Keller PAA-33X 0–10 bar absolute pressure transducer with a measurement uncertainty of 5 mbar. To achieve well-defined methane-air mixtures the gases were premixed in a larger mixing vessel, before filling the evacuated measurement cell. This way the impact of insufficient mixing of the gases in the tubing is minimized. The reference values for the mole fraction were determined according to the pressure values in the mixing vessel. Therefore the reference uncertainty of the mole fraction is dependent on the absolute pressure and is given separately for each measurement.cm−1 11. Thus Dicke-Narrowing can be neglected in this investigation. The noise induced uncertainty of the determined instrument function decreases with the number of used measurements. For this evaluation a number of 10 measurements per pressure level in the wavelength region from 5995 cm−1 to 6145 cm−1 are used. For this wavenumber region the time-wavenumber-correlation D(t) is determined by applying the method described by Blume et al.11. As temperature, pressure and mole fraction are known, only the background I0 has to be determined, to be able to simulate a corresponding theoretical spectrum. This background is determined by fitting the whole model UDAQ  and 0.497 bar (abs.) at a temperature of 295.5 K are used. Please note that the spectral resolution of the instrument is in the order of 0.1 Equation ) with vaI0 the simulated signal I0(t)eA(D(t),T,p,q)− is known and the linear equation system . For an easier comparison the normalized diode pulse response is shown in the same figure.If this fit was processed without assuming any instrument function the background could easily get distorted in the neighborhood of narrow absorption lines. Hence the yet unknown instrument function is roughly estimated as a Gauss pulse with a FWHM of 170 ps during the fit. With the resulting background n system  can be sAs expected the determined instrument function shows a broadened pulse with a diode ringing artefact. The diode ringing does not exactly match that of the manufacturer supplied diode transfer function. This can be explained by the different input capacitances and impedances of the oscilloscopes used by the manufacturer and in this work. The high speed photo diode amplifier system is very sensitive to these changes. Another important difference between the diode response itself and the determined instrument function is the width of the actual pulse. The increased width, compared to the diode response, is introduced by the duration of the SCL pulse. Accordingly the small dip on top of the pulse represents another diode ringing artefact similar to the response on a step function.Thus the effects observable in the instrument function can qualitatively be explained. The suitability of the instrument function for measurement evaluation is discussed in section 4.As already mentioned the current work aims to develop and validate the evaluation algorithms while the problem of accurate line data will be approached in future studies. In order to be able to differentiate between effects induced by the evaluation algorithms and effects induced by the line data, simulated measurement signals are evaluated. These simulated measurement signals build on the same absorption line data as the evaluation. Thus the corresponding evaluation results can be regarded as unbiased by inaccurate line data. The simulation procedure is similar to the signal model described in the methods section. The only difference lies in the simulation of the initial intensity distribution. The real initial intensity distribution does not necessarily follow the restrictions of the spline curve assumed in the model. Hence for the simulated signals differently scaled and filtered random signals are summed in order to generate a background distribution similar to a real signal. After applying the instrument function on the simulated signals, Gaussian white noise is applied to them in order to achieve different signal to noise ratios (SNR). Here SNR is defined as the maximum difference between a signal and the corresponding background without absorption divided by the standard deviation of the noise:SNRs of 20, 100 and infinity, which corresponds to zero noise, were simulated for both gas parameter sets (ambient and high pressure). For each SNR and pressure level 100 simulated signals were evaluated. The mean and standard deviation of each evaluation series are summarized in Table νmax = 2.96 cm−1. Thus with the spectral range spanning 150 cm−1 the maximum allowable number of equidistant background spline support points is 51. As this estimation is just a rule of thumb and does not regard effects like superimposing absorption lines, the number of spline support points is chosen conservatively to be 45. Thus temperature, pressure and mole fraction as well as 45 equidistant background spline support points are fitted to the measurement data during the evaluation. The fitting result of one of the ten measurements is shown in Fig. The experimentally determined instrument function is now applied in the evaluation of methane-air-mixtures. To validate the quality of the instrument function ten ambient pressure measurements at 0.980 bar, 295.6 K and a methane mole fraction of 18.06% are evaluated. The ambient pressure ensures that the diode-ringing effects are still visible and therefore have to be described by the instrument function in order to produce reasonable results. As the actual pressure is usually not known beforehand, a maximum pressure of 10 bar is assumed in order to determine the number of background spline support points. For the methane absorption lines in the regarded spectral region this leads to a maximum FWHM of ΔThe resulting values for temperature, pressure and mole fraction for the ten measurements are given in Table The low pressure SCLAS evaluations show maximum temperature deviations of 10.2 K (measurement 1 & 2) from the reference measurement of 295.6 ± 2.9 K. Nonetheless the temperatures determined by SCLAS only fluctuate in a range of 3.7 K with a mean value of 304.4 K. The calculated precision uncertainties give values between 2.91 K and 3.60 K. These uncertainties are mainly induced by wavenumber and instrument function uncertainty. As the temperature was constant during the ten low pressure measurements, the determined temperature values can be used to empirically estimate the precision uncertainty of the SCLAS evaluations using the empirical standard deviations over the ten measurements. Hereafter standard uncertainties estimated by the standard deviation of the measurement series are called empirically estimated uncertainties while the standard uncertainties calculated using the approach presented in this paper are referred to as directly calculated. For the temperature measurements this leads to an empirically determined standard deviation of 1.16 K. Therefore the directly calculated precision uncertainty gives greater values but in the same order of magnitude as the empirically estimated ones.The pressure results in the ambient pressure measurements 1 to 10 show a maximum deviation of 0.030 bar from the reference measurement of 0.980 ± 0.005 bar. The directly calculated standard uncertainties overestimate the empirically estimated standard uncertainty of 0.017 bar but are still in the same order of magnitude. Regarding the mean value of 0.982 bar and the standard uncertainty no relevant inaccuracy can be identified.The mole fraction results in the ambient pressure measurements show a maximum deviation of 2.3% from the reference measurement of 18.06 ± 0.5%. The directly calculated standard uncertainties overestimate the empirically estimated value of 0.35% by a factor of four. But similar to the derived temperature, the mean value of the mole fraction 19.42% deviates from the reference value of 18.06%. This behaviour is discussed in section 4. Regarding the direct uncertainty estimation the uncertainties in all parameters are overestimated to a reasonable degree, which can mainly be attributed to the wavenumber induced uncertainty.Additionally to the ten ambient pressure measurements, ten high pressure measurements at 8.732 bar and 295.6 K of a methane-air-mixture are evaluated. The methane mole fraction of 22.3% in combination with the elevated pressure leads to spectral regions with total absorption. As the pressure is not assumed to be known beforehand, a maximum pressure of 10 bar is assumed. Thus similar to the low pressure case temperature, pressure, mole fraction and 45 background spline support points are fitted to the measurement data.The fitting result of one of these high pressure measurements is shown in Fig. A) approach zero, (1) does not allow the determination of a definite I0. As a result the fitted background in regions with strong absorption is not trustworthy. Nonetheless, the important evaluation variables temperature, pressure and mole fraction are not affected by this. The derivatives of the signal model UDAQ with respect to these variables in regions of total absorption, are small compared to regions with weaker absorption. Hence the influence of these variables on the cost function is very low in strong absorption regions. Thus the determined optimum of the cost function in these variables is not influenced by the total absorption region but by the wings of the absorption lines, that still show moderate absorption.As visible in the detail view in Fig. The resulting values for temperature, pressure and mole fraction are given in Table The mean value of the pressure measurement in the high pressure case matches the reference pressure value to within 216 mbar. The direct calculation of the uncertainty overestimates the standard deviation of the measurement series, but still shares the same order of magnitude.Similar to the temperature the mean value of the mole fraction deviates by 2.24% from the reference value and the uncertainties are overestimated but in the same order of magnitude.Summarized the high pressure case shows an overestimation of temperature and mole fraction similar to the ambient pressure case. Additionally the uncertainties in all parameters are overestimated to a reasonable degree, which again can be attributed to the wavenumber induced uncertainty. The reasons for this are discussed in the next section.Ambient pressure measurements as well as high pressure measurements show reasonably good agreement in the spectral line shape between model and measurement. Especially in the ambient pressure measurements the distortion of the line shapes and the diode-ringing is clearly visible and correctly depicted by the model.Furthermore the pressures derived from the spectroscopic measurements match the reference pressure measurements very well. These observations indicate a good quality of the estimated instrument function.As already mentioned the temperature as well as the mole fraction value show a deviation in their mean values from the reference values for both pressure levels. As it was shown in the evaluation of simulated signals the evaluation procedure gives accurate results if the line data and the instrument function match the real values. Because the derived temperature is mainly dependent on the ratio between the line strengths and the instrument function affects all absorption lines similarly, the more probable cause for the deviation in temperature is inaccurate line data. Additionally this deviation in the temperature can also explain the deviation observed for mole fractions, as the methane absorption line strengths in the regarded region decrease with increasing temperature. Therefore an underestimated temperature in the fitting optimum leads to the compensation of the reduced line strength by an increased value for the mole fraction. Thus the main reason for both deviations is most probably inaccurate line data. The pressure was determined accurately in both pressure levels. This could only be achieved by the novel determination method for the instrument function.The results of the uncertainty estimation show for both pressure levels a reasonable overestimation of the empirical estimated standard deviations. This overestimation is mainly caused by the wavenumber axis uncertainty while the noise and instrument function induced uncertainty alone would give a tight estimate of the standard deviation. This can be explained as equation  is only in-situ determination of the spectrometer instrument function from low pressure measurements, without additional hardware. Only with this fully specified instrument function the exact evaluation of pressures becomes feasible. In addition a detailed statistical uncertainty propagation for several uncertainty sources and all evaluation parameters is proposed. This enables the estimation of precision uncertainties of all parameters from a single measurement.The current work provides a set of algorithms that facilitates the simultaneous evaluation of SCLAS measurements in temperature, pressure and absorber mole fraction without recording an additional background measurement, also at elevated pressures. An important part of this evaluation is the These algorithms are applied to methane measurements with an absorption path length of 367 mm at ambient pressure (0.980 bar) and elevated pressures (8.732 bar). Especially the ambient measurements depict a very good agreement with the determined instrument function. The resulting temperature, pressure and mole fraction values match the reference measurements reasonably well for both cases. Furthermore the precision uncertainty calculation results give a conservative estimate of the fluctuations observed in both measurement cases. Thus the uncertainty propagation can be used to estimate the achievable precision of SCLAS measurements. The results promise precision uncertainties of below 3 K in temperature, below 50 mbar in pressure and below 0.4 % in methane mole fraction for the regarded experimental setup. This shows the high potential of SCLAS as a multi parameter line of sight measurement method.Of course this potential can only be fully exploited with accurate line data. Therefore an important step to the maturity of SCLAS is the derivation of more accurate line data or other spectrum models. Additionally the simulation of absorbance spectra has to be optimized as the currently achievable evaluation time is limited by the high computational effort of simulating several thousand absorption transitions.The presented uncertainty estimation is limited to the measurement uncertainty without line data uncertainty as it was only used as a measure of the achievable precision. By applying an analogous approach for the error propagation of the line data, the total uncertainty of SCLAS measurements can be directly estimated. With these enhancements SCLAS can be used as a valuable tool in research and development without the need for additional reference sensors.


Time series within fields such as finance and economics are often modelled using long memory processes. Alternative studies on the same data can suggest that series may actually contain a ‘changepoint’ (a point within the time series where the data generating process has changed). These models have been shown to have elements of similarity, such as within their spectrum. Without prior knowledge this leads to an ambiguity between these two models, meaning it is difficult to assess which model is most appropriate. We demonstrate that considering this problem in a time-varying environment using the time-varying spectrum removes this ambiguity. Using the wavelet spectrum, we then use a classification approach to determine the most appropriate model (long memory or changepoint). Simulation results are presented across a number of models followed by an application to stock cross-correlations and US inflation. The results indicate that the proposed classification outperforms an existing hypothesis testing approach on a number of models and performs comparatively across others. It is not often the case that a given data set has a known explicit model from which it is generated. Analysts will look to fit an appropriate model to such a series in the hopes of understanding the underlying mechanisms or to make predictions into the future. The models proposed are expected to be distinct in their properties such that there is a clear prevalence of a suitable model for the data. However, models with certain structural features have been known to have similar properties to other models  model, characterised by its autoregressive (AR) parameters For the purpose of this paper, we define the changepoint and long memory models as:The ambiguity present between diagnostics of the competing models given in Eqs.  and 2)  2) can cBefore discussing the wavelet spectrum, we provide a brief background to wavelets and the specific spectrum we propose to use.Wavelets capture properties of the data through a location–scale decomposition using compactly supported oscillating functions. Through dilation and translation, a wavelet is applied across a number of a scales and locations to capture behaviour occurring over different parts of a series. Further information on them and their application can be found in Daubechies  and NasoWe follow the definition in Fryzlewicz and Nason  for a loz.Define the triangular stochastic array j and location (time) k.As in the traditional Fourier setting, the spectrum is the square of the amplitudes and as such the evolutionary wavelet spectrum can be defined asConsidering both scale and location, the two dimensions allow the differences between the proposed models to be seen. Examples of the differences in these spectra are given in Fig. Due to the fact that the wavelet spectrum gives a distinction between the two models, we propose to use this as the basis for our inference regarding the most appropriate model. Whilst the Fourier spectrum could be used here as in Janacek et al. , we chooyzlewicz .In the next section, we detail how to use the wavelet spectrum of the two models in a classification procedure.Testing whether a long memory or changepoint model is more appropriate whilst under model uncertainty comes with the hazard of mis-specification. A formal hypothesis test places assumptions on the underlying model in both the null and alternative, but the allocation of the null is hazardous—should the changepoint model be the null or alternative? It would be entirely feasible to choose the changepoint model as the null model, not reject In the previous subsection, it was demonstrated that the wavelet spectrum can be used to distinguish the changepoint model from the long memory model, and the classifier proposed here builds on this. However, to begin a classification method must first ‘teach’ itself on the structure of the classes through sets of training data. These are data sets already determined to be in each category and are the basis for calculating the distances from each group. This previous knowledge allows for determination of patterns and features of each category (that are unique from other categories) for comparison to the candidate data set. A common example is the spam filter on mailboxes, which is trained on previous spam emails so that it can classify if a new email that arrives is spam or not. The decision is made by comparing it to a number of patterns already determined to be features in spam email for example, short messages or hidden sender identities. Further information on classification methods and training them can be found within Michie et al. .n, the classifier has no previous information to train on. To remedy this we create training data through simulation. Given a candidate series we first fit the competing models in Eqs.   n, the c in Eqs. . When co in Eqs.  and sele in Eqs. ].M is the number of simulated series and n is the length of the original series. Note that we are not sampling from the original series, we are generating realizations from the fitted models.Following the identification of the best changepoint and long memory models, the training data are then simulated as (Monte Carlo) realisations from these, denoted byj by concatenating scales, hence k is larger than CN) is bounded by Now we have the training data and the observed data, denoted a et al. ,3\documea et al.  to allowa et al.  where thnd Ombao  for compAssumption 2.1 from Fryzlewicz and Ombao (z where (possibly infinitely many) functions (nd Ombao ) The setAssumption 2.2 from Fryzlewicz and Ombao ((nd Ombao ) There eSuppose that assumptions mentclass2pt{minimThe proof is given in Appendix A summary of the proposed procedure is given in Algorithm 1.To test the empirical accuracy of our proposed approach, simulations were conducted over a number of models. Here, these models are chosen over a number of parameter magnitudes and combinations to show the effectiveness of the approach outlined in Sect.  For each model given in the tables below, 500 realisations of each model were generated and classified, using We have used re Nason  requiresre Nason  that addFor the changepoint models, we used the simulations given in Yau and Davis . Table 1It should be noted that as the Yau and Davis method is a hypothesis test we would expect results around 0.95 for a 5% type I error.d parameter, values close to 0 are closer to short memory models and values close to 0.5 are stronger long memory models .In contrast to the changepoint models, the classification of a long memory model is expected to be less clear. This is due to the variation within the wavelet spectrum of long memory series that could be interpreted as different levels and hence a changepoint model would be more appropriate. To demonstrate the effect of the classifier on long memory observations, a larger number of models were considered. We simulated long memory models with differing levels of long memory as measured by the d from 0.1 to 0.4 we improve the classification rate.The results in Table p, q was 2.Some interesting things to note include, when there are strong AR parameters . This coupled with autocorrelation and partial autocorrelation function plots given in Fig. These data have been analysed previously by Bertram et al.  to deterThe wavelet classification process presented within this paper provides the user a distinct choice over a number of proposed models, and when explicitly applied to an ambiguity such as long memory or a changepoint as in Sect. The evolutionary wavelet spectrum provides a representation of non-stationarity which is lacking in the commonly used (averaged over time) spectrum. This gives an advantage when drawing comparisons between non-stationary and stationary series, since the wavelet spectrum may appear substantially different. Quantifying this visual difference allows for a direct comparison between the series and each proposed model.The variance-corrected squared distance metric used in the proposed classifier has been demonstrated to be quite accurate under the ambiguity of long memory and changepoint models. It is particularly effective at identifying changepoint models correctly, as the results in Table As mentioned in Sect. An aspect not covered in this paper is the precise form of ARMA and long memory models in the LSW paradigm, i.e. how the model coefficients relate to the An R package (LSWclassify) is available from the authors that implements the method from the paper.


La Gioconda. The effect occurs reliably but was not tested with Mona Lisa herself. Remarkably, viewers judged Mona Lisa’s gaze as directed to their right-hand side irrespectively of the image zoom, its horizontal position on screen, and the distance of the ruler that was used for measuring the gaze direction.If the person depicted in an image gazes at the camera or painter, a viewer perceives this as being gazed at. The viewers’ perception holds irrespectively of their position relative to image. This is the Mona Lisa effect named after the subject of Leonardo’s famous painting  Mona Lisa effect named after the subject of the famous painting La Gioconda  bitmap of the painting were presented on a 35 cm by 26 cm computer screen at a viewing distance of 66 cm. To test whether some aspects of Mona Lisa’s face may lead to different results, we manipulated whether only the eyes and nose or the entire head were visible to the participants. The image zoom ranged from 30% to 70% in steps of 10%. All images were centred to the same pixel on the bridge of the nose and cropped to match the screen resolution , see centre column of Instead of asking participants for the binary judgement of whether they feel being looked at, we obtained a metric of their perceived gaze direction on a 2 -m carpenter’s rule. The responses ranged from 0 to 200 avoiding negative numbers. The ruler was positioned between the participant and the painting  and sligLa Gioconda. Thus, participants could be tempted to repeatedly enter the same number without assessing the stimulus. We addressed this by displacing the entire picture by 100 px (3.4 cm) to the left or right , The significant increase in the judgements from 14.0 cm to 19.5 cm defines a line of gaze that is not directed at the viewer. These measures and the geometry in The statistical evidence against an interaction of distance of the ruler and lateral displacement also contradicts the idea that Mona Lisa’s line of gaze was constantly directed at the viewer. Rather, the line of gaze and all its intersection points shifted with the image to the left or right, respectively.A Bayesian linear regression revealed evidence against an effect of image zoom  on the participants’ judgements .Viewers move in real space and look into a fictional pictorial space through the window of a painting (cf.


Background Children are still prescribed age contraindicated drugs, but information about the number and type of these drugs dispensed for children in the Netherlands is limited. Objective To determine the incidence and prevalence of contraindicated drugs that were dispensed for the use by children. Setting The study was conducted in the Netherlands with routinely collected data from 95% of all community pharmacies. Method We performed a one-year nationwide observational study where all patients aged 17 years or younger who have received at least one prescription in 2016 were included. Contraindicated drugs were selected, according to the 5th level of ATC code, using different information sources. Main outcome measure The proportion of (newly) contraindicated drugs that were dispensed to children. Results In total, 3.9% of all children received at least one drug that was contraindicated for their age. The highest percentage of contraindicated drugs that was dispensed, was observed in patients aged 1–2 years and 13–17 years  and the percentage of contraindicated drugs that were dispensed was higher in female than in male patients . Conclusion The results of this study show that a substantial percentage of children received a drug that was conta-indicated for their age, and it happes more in female than in male patients. Furthermore, the information about this type of contraindications is limited and inconsistent. Information about contraindications or warnings should be improved to support prescribers and other healthcare professionals when prescribing contraindicated drugs which are deemed necessary.It is not clear why contraindicated drugs are more often prescribed for female than male children. Further research in this field should be carried out to address these gender differences and to improve our knowledge about the safety of drugs  in thse patient groups.Many drugs are prescribed off-label because they are not approved for use in children , 2. GPs One of the recent examples is the label revision for products containing codeine and tramadol which states that these drugs are contraindicated for the treatment of pain in children younger than age 12. Adverse event reports showed that the use of codeine was associated with respiratory depression and death, with the majority of cases involving children under the age of 12 . This coNumerous studies have described the contraindication of medication use in relation to specific diseases or patient characteristics, but the focus on age and gender in children in these studies is very limited –9. MoreoTo determine the incidence and prevalence of contraindicated drugs dispensed to children, with routinely collected data from community pharmacies in the Netherlands.Informed consent was not required, since this was an observational, non-interventional cohort study.Data were obtained from the Dutch Foundation for Pharmaceutical Statistics . This database contains dispensing data from 95% of all community pharmacies in the Netherlands since 1990 . In the All patients aged 17 years or younger who have received at least one prescription during the study period between 1 January 2016 and 31 December 2016 were included. Patients with an unknown gender were excluded.th level ATC code of the World Health Organisation [Drugs that are contraindicated in children were selected according to the 5nisation . The prenisation . The desnisation . The folFor all patients, we determined the number of drugs dispensed between 1 January 2016 and 31 December 2016. In our study, a person was identified as user of a contraindicated drug when having at least one contraindicated drug dispensed during the study period. In the analyses, we have calculated the cumulative 1-year incidence as the proportion of the total cohort to which a contraindicated drug was dispensed in the year 2016 for the first time since birth. In addition, the cumulative 1-year prevalence was determined by the proportion of children with a contraindicated drug dispensed in the study period, including those who had started before 1 January 2016. This was also stratified according to age, which was classified into five groups: < 1 year, 1–2 years, 3–6 years, 7–12 years and 13–17 years. In addition, we investigated the gender differences and we calculated the number of contraindicated drugs that were dispensed per region and per type of prescriber. The analyses included almost the entire Dutch population and therefore the calculation of confidence intervals for the incidence and prevalence were not deemed necessary. We also investigated the information available in the Farmacotherapeutisch Kompas, SmPC, Kinderformularium about the reasons for a drug to be contraindicated, which are classified into the active ingredient, high/unknown dosage, formulation, lack of information about efficacy or safety and the presence of excipients. Finally, we have calculated the number of patients, the number of drugs that were dispensed and mean age for the most commonly dispensed contraindicated drugs.p value < 0.001) and highest in patients aged 1–2 years (7.0%). The majority of patients received only one type of drug that was contraindicated for age in the study period ; 2 or more: 1,496 (0.3%)).Patients with an unknown gender n = 122,144) were excluded from the analysis. The remaining 1,527,021 patients received at least one drug in 2016 where 3.9% of these patients received a drug that was contraindicated for age . The cumulative incidence was highest in the age categories 1–2 years (5.5%) and 13–17 years (4.2%), followed by the age categories 0–1 (2.3%), 7–12 years (1.8%) and 3–6 years (0.6%). The postal code information available for each patient indicated that the number of patients who received a contraindicated drug in the Netherlands is fairly equally distributed . The majority of patients received the contraindicated drugs from their GP , followed by medical specialists  and other healthcare professionals  but the contraindicated drugs were nevertheless relatively more often prescribed by medical specialists than by GPs . The most common drugs that were relatively more often prescribed by specialist than GPs or other healthcare professionals are hydrocortisone/neomycin/polymyxin B , atropine  and dimetindene .The number of new users of a contraindicated drug in 2016 was 43,206 (2.8%), which was significantly higher in females (3.1%) than in males (2.6%)   Table . Higher One of the most common contraindications is due to the active pharmaceutical ingredient, which is seen in 49 types of drugs and in 25,875 patients Table . Drugs tFigure Table It is not always clear why a drug is contraindicated and how many of these drugs are prescribed to children despite their contraindication for age. In this nationwide study, we have calculated the incidence and prevalence of drugs with a contraindication for the age groups to which they were nevertheless dispensed. In our study we observed that 3.9% of all children received at least one drug that was contraindicated for age, which are mainly newly prescribed drugs (2.8%). These results are almost similar to the results observed in the study by Bensouda-Grimaldi et al. (3.2%) . Our resAlternatives are available for many contraindicated drugs, but in some circumstances the contraindicated drug is needed. Prescribing an on-label drug does not always have to be accurate, whereas prescribing a contraindicated drug does not have to be considered bad practice either. One of the examples is the oral suspension cholecalciferol which is contraindicated because of the high dose of vitamin D. These drugs are necessary for children with an impaired vitamin D absorption such as cystic fibrosis or serious vitamin D deficiency in rachitis. However, it is not clear if they were all prescribed for these type of indications, as we observed a large number of patients who received the high dose of Vitamin D in our study. The increase of vitamin D use since the global recognition of vitamin D deficiency in the general population has also led to an increase in vitamin D intoxications, which were partly due to inappropriate prescribing . This maMetoclopramide is still prescribed for chronic use for the indication symptomatic gastroesophageal reflux as the substantial need is not met by other medications . HoweverOne of the strengths of our study is that we used a population-based dispensing database which registers drugs that were dispensed in more than 95% of all community pharmacies in the Netherlands. Pharmacists play an essential role in drug safety and it is likely that the number of contraindicated drugs is higher in prescription data than in our study as we have used dispensing data. Detailed information was available on all drugs that were dispensed to children. The relatively long history available also enabled us to study any drugs that were dispensed since birth to determine the 1-year cumulative incidence and prevalence.Our study also has several limitations. Our data do not include the drugs that were dispensed by hospital pharmacies where the numbers of contraindicated drugs that were dispensed may be higher. No information about the month of birth was available. Therefore, we only included drugs in our analysis where the contraindication for age was based on the number of years and not months. Also, no information was available with regard to the indication of the drugs prescribed.The results of this study showed that a substantial percentage of children received a dispensed drug which is contraindicated for that age. This is more common in females than in males. Gender differences in disease prevalence or healthcare may explain this observation. We also observed that the information about the contraindication (contraindication or warning and the reason) is limited and not consistent between the different information sources. Since prescribing an on-label drug can be inaccurate, as much as prescribing an contraindication drug can be good practice, it is important that all prescribers and healthcare professionals are informed sufficiently about the contraindications. Further research is needed to investigate the reasons for prescribing particular drugs despite their contraindication for age, and the gender differences that were found in this study.


In this work, we have developed a compact and efficient microdevice for on-chip trapping of micro-sized particles. This hydrodynamics-based trapping system allows the isolation of polystyrene micro-particles with a shorter time while inducing a less hydrodynamic deformation and stress on the particles or cells both after and before trapping. A numerical simulation was carried out to design a hydrodynamic trapping mechanism and optimize the geometric and fluidic parameters affecting the trapping efficiency of the microfluidic network. By using the finite element analysis, the velocity field, pressure field, and hydrodynamic force on the micro particles were studied. Finally, a PDMS microfluidic device was fabricated to test the device’s ability to trap polystyrene microspheres. Computational fluid analysis and experimental testing showed a high trapping efficiency that is more than 90%. This microdevice can be used for single cell studies including their biological, physical and chemical characterization.Capturing cells or biological entities is an important and challenging step toward  One of the most exciting advancements in the application of microfluidics has happened in the biological and biomedical areas. Reduced sample volume, short time analysis, control of spatio-temporal chemical composition, precise and predictable fluid flow regime (laminar flow), portability and integration with sensors actuators, controllers and automation systems have made microfluidic devices an attractive miniaturized platform for biological and biomedical applications4.In recent years, the use of microfluidics, and microfabrication has been adopted in the various areas of engineering and science. Microfluidics is an important technology, suitable for numerous biomedical application fields such as drug delivery, cellular analysis, diagnosis, cancer detection, and biotechnology7. Before the advancement of microfluidics, conventional strategies such as sample tubes and pipettes were used to manipulate the biological particles at the microscale. The lack of sufficient precision, high-throughput, online monitoring, time and reagent consuming have limited their application in single cell analysis. Many microfluidic-based technologies have been developed to control the spatio-temporal of the environment of cells. Based on the types of forces used in the manipulation of the cells on the chip, they are classified into different categories: optical8, electrical9, magnetic10, acoustic11 and hydrodynamic manipulations12. Hydrodynamic trapping of cells offers a low-cost, simple, efficient and high-throughput arraying of the cell without controlling or using other equipment such as lasers, electrodes, magnets, and ultrasound transducers. In this technique, the sample and the medium are introduced by flow channels providing conditions for the observation of the responses of the cells to the environmental changes14. Depending on whether the target particles are in contact with a support surface, the hydrodynamic trapping systems are classified into two main methods: contact-based methods15 in which the fluid flow is implemented to physically confine the cells against microfabricated obstacles or walls, and contact-less methods16 which are based on micro eddies, micro-vortices and stagnation flow. Due to some limitations of contact-less trapping technique such as low efficiency and low-throughput, most of the efforts have been directed toward the implementation of the contact-based methods to capture and manipulate individual cells. Two concepts have been used for the contact-based trapping technique: (1) Path with the least flow resistance17 where single cells are directed into the trap sites possessing much smaller hydrodynamic resistance compared to the bypass or the main path, and after filling the trap site, the subsequent cell is redirected to another path having the least flow resistance, (2) Di Carlo18 method, in which micro-sieve arrays are designed to trap single cells from a suspension of flow. This method suffers from a relatively low-efficient single-cell trapping compared to the first concept and most of the samples are lost during the trapping process. In order to optimize the efficiency of hydrodynamic single-cell capturing systems, the geometric parameters, as well as the fluid flow factors, should be taken into account.Capturing and trapping of cells is the first critical step in single cell studies and can be defined as development of technologies for the immobilization of cells at a precise location for further analysis. At the same time, a fluid flow needs to be applied to provide the cells with nutrients and other media influencing the behavior of the cells. Isolation of single cells from the surrounding environment and their analysis have important implications for diagnosis and therapy. For example, capturing circulating cancer cells in peripheral blood is beneficial for detecting cancer at early stages and estimating the risk of metastatic relapse17 and other works, a channel with inclined walls is designed to increase the hydrodynamic resistance of the main channel, enabling to navigate single cells into the trapping sites. With the proposed design, the trapping device can be more compact as the main channel does not need to be extended to generate a higher flow resistance than trap sites. Compared with the previous studies using a similar concept, this design offers a fast loading time while less mechanical deformation and stress are introduced onto the target cells both before and after trapping. In this work, first, we used a computational fluid dynamic simulation analysis by COMSOL Multiphysics to optimize the geometric parameters of the channels and trap sites, and predict the fluid flow pattern inside the channels, considering the dimensions of the target biological particles. Finally, polystyrene beads were used in order to demonstrate the feasibility and the working mechanism of the fabricated device to simulate the capture of single cells.Here, we developed a lab-on-a-chip system using the contact-based hydrodynamic trapping method to immobilize single cells by positioning a series of micro-sieves on the main channel sidewalls. In this system, the combination of the two above-mentioned concepts is implemented to design an efficient trapping microfluidic device. In our design, similarly to Di Carlo design, micro-sieves are used, but on the sidewalls, to capture individual cells or particles while at the same time the concept of “path with the least resistance” is utilized to tune the geometric parameters of the main and the side channels in order to enhance capturing efficiency of the device. The network includes two side channels and a converging main channel designed to direct the streamlines toward the trapping sites. In contrast to Tan and Takeuchi19. When the particles’ Reynolds number is very low (<1), particles follow the flow streamlines. Microparticles/cells are in micro scale and the inertial effects which cause particles to travel across streamlines are very small compared to the viscous force, and they tend to follow streamlines20. The flow rate through each path can be determined by the corresponding hydrodynamic resistance. In fact, more flow can be passed through the path with a lower hydrodynamic resistance. The hydrodynamic resistance of the empty trap sites, main channel, and the flow parameters (input and output conditions) of the network need to be controlled to manage the trapping efficiency of the device and the fluid pressure difference between the main and side channels. In order to increase the efficiency of trapping, either the hydrodynamic resistance of the main channel has to be increased or the hydrodynamic resistance of the trapping path has to be reduced. The dimension of trap sites depends on the size of target cells, and the dimension of trap sites need to be small enough to not allow the target cells to pass through them. Due to the size constraints of the traps, only limited changes on the dimensions of the trap sites can be performed. As a result, to optimize the trapping efficiency of the system and to maximize the flow rate in the traps, one has to manipulate the geometrical parameters of the other paths such as the main and the side channels. The hydrodynamic resistance (R) of a channel with an arbitrary cross-section can be obtained from the following relationship21:1 and A2 are the microchannel cross-section area of the inlet and outlet of the channel respectively, L is the length of microchannel, Q is the flow rate inside the channel, ΔP is the pressure difference along the microchannel, µ is the dynamic viscosity of fluid, and Ip is the specific polar momentum of inertia. From this equation, it is obvious that the fluidic resistance of a certain channel is not only a function of its geometry but also is proportional to the flow rate inside the channel. For a constant cross-sectional area channel, the area of inlet and outlet are same, so that the last term of eq. 22. Increasing the fluidic resistance along the main channel is one of the simple ways to increase the trapping efficiency of the system23. In order to increase the hydrodynamic resistance of the system, the straight main channel can be replaced by a rectangular micro-channel cross-section with a linear wall profile, allowing to enhance the pressure gradient between the main and side channels. Using a converging micro-channel, the last term in eq. d/D). The length of a straight channel with the equivalent flow resistance is presented in Table d/D = 0.4, and L = 100 μm, the length has to be extended 5.2 times compared to the length of the converging channel (Leq = 5.2 × 100 = 520 μm).The illustration of the trapping system is shown in Fig. 24.25.ur = u − up is the relative particle velocity to the fluid velocity, up is the particle velocity, Fe is external force and Fb is the buoyancy force. The gravitational force (including the buoyancy) is considered to be negligible and no external force is applied on particles. The drag force acting on the particles can be obtained from the following equation26CD is the drag coefficient, Ap is the projected area of the particle in the plane perpendicular to the flow direction. The drag coefficient is a function of particle Reynolds number defined as:dp is the diameter of particles. For creeping motion (Stokes flow) when Rep < 1, the inertia force is negligible and the viscous force is the predominant force28, the particles follow the fluid streamlines19. Considering the maximum velocity of particles in trap sites, the particle’s Reynold number for the inlet velocity of 2.8 mm/s is less than 0.1, and remains less than unity, for the inlet velocity of 28 mm/s, confirming that for this range, particles indeed follow the streamlines with good accuracy.A 3D finite element analysis (FEA) was performed to estimate the velocity field and pressure gradient within the microfluidic device. Incompressible Naiver-Stokes and continuity equations were employed to characterize fluid flow parameters inside the micro channelsConventional soft lithography was used to fabricate the PDMS microfluidic chips. AutoCAD was used to draw the design layouts which were printed onto the masks . Photolithography was used to fabricate the master molds on silicon wafers. The negative photoresist (SU-8 2035) was spin coated at 1900 rpm for 30 s on a 4″ silicon wafer to obtain a layer with 70 μm thickness. Before exposing it to the UV, the photoresist was baked at 65 °C for 3 min and at 95 °C for 6 min. The transparent mask was then used to pattern the design using a mask aligner and UV light for 20 s and post-baked at 65 °C for 2 min. and at 95 °C for 7 min. In the final step, the SU-8 layer was developed for 5 min using SU-developer to obtain the mold and finally it was dried with an N2 gun. Before casting the PDMS, the surface of the mold was salinized in vapor phase for removing easily the cured PDMS. A mixture of pre-polymer and curing agent with a weight ratio of 10:1 was degasified in a vacuum desiccator to remove the bubbles. The PDMS was then poured onto the mold and placed on at 80 °C oven for 2 hours. Then, the PDMS layer was peeled off the mold, and the inlet/outlet were punched by a 1 mm punch. Finally, with the aid of oxygen plasma bonding, the PDMS chip was bonded to a glass slide to seal the channels.Polystyrene particles  with a diameter of 50 μm were dispersed in 0.05% Tween 20 in Phosphate buffered saline (PBS) in order to have a solution with a concentration of 50 particles/μl. The suspension was sonicated to ascertain a good dispersion.mm ID × 0.25 mm thickness. Before using the microfluidic chip for operation, the PDMS surfaces were incubated for 30 min with tween 20 (10% in PBS) in order to prevent the polystyrene particles from binding to the channel surfaces. Tween 20 can effectively disrupt the hydrophobic interaction between the particles and the PDMS surfaces29. The inlet tube then was connected to the syringe pump (KD Scientific Legate 110) for loading the suspended microparticles into the device at a specific flow. The trapping process was next observed under a microscope (Nikon Eclipse 80i). A digital camera (Infinity) coupled to the inverted microscope was used to capture photographs and videos.The microfluidic chip consists of one inlet for injecting the medium and the particles suspension, and three outlets, two from the side channels and one from the main channel. A 2 ml syringe with 23 G needle was connected to the inlet of the fabricated device through a polytetrafluoroethylene (PTFE) tube with size of 0.59 The fluid flow in the design shown in Fig. Pm and the side channels Ps for three different angles under a constant inlet velocity (2.8 mm/s). It is evident that when inlet and outlet cross-sections are similar, for a straight channel and Design 1 (θ = 0), the pressure difference between the main channel and side channels is small and may be even a negative value (for trap 6 and 7), meaning that the flow may be reversed from the side channels into the main channel. Figure th and 7th traps, confirming that the flow is directed from side channels to the main channel. As two arrows show in Fig. θ = 4.4°), the pressure differences between the main and side channels are increased to remain positive for all trap sites, providing a flow direction toward the traps. For example, the pressure difference increases from 1.5 Pa to 2 Pa for the first trap . As can be seen in Fig. The set parameters were used to model the microfluidic in COMSOL Multiphysics and characterize the velocity contours and the pressure distribution within the microfluidic device. The pressure difference values between the main and side channels were computed for three different sidewall angles and plotted in Fig. kg/m3 as the working fluid. Table In order to characterize the effects of sidewall angle on the trapping efficiency of the system, flow passing through each trap site was calculated for three different designs using the geometric parameters shown in Table θ), the fluidic resistance of the main channel is increased and more stress will be applied on the trapped particles/cells due to the higher pressure in the main channel. Due to these reasons, we limited the minimum outlet width of the main channel such that only three particles could pass through the outer channel easily at the same time to prevent the accumulation of particles and imposing very high stress on the trapped particles/cells.The concept of using a converging channel to enhance the trapping capability of the microfluidic traps is confirmed with the derived flow rate ratios. The narrower outlet opening increases the hydrodynamic resistance of the microchannel; however, the choice of a very narrow outlet is limited because of the clogging possibility within the channel. In order to have a continuous flow inside the main channel without clogging, the dimensions of the outlet should be carefully determined. Experiencing a less stress by the trapped particle/cell due to the hydrodynamic forces is another determining factor for designing the outlet width. By increasing the angle  are not able to deliver cells into trap sites since theses streamlines do not pass through the center of cells. This concept and the various possible conditions for streamlines and the injecting particles are clearly shown in Fig. Te for each trap (i) can be defined as the fraction of flow at the effective area which can pass through the traps. Effective area is a portion of the inlet area (A) where the passing streamlines are able to deliver particles into traps (Ae = (D − 2r) (h − 2r)).As was mentioned earlier, the flow behavior inside the microchannel is laminar, so in order to trap particles into the designed sites, two conditions need be met. First, at least one of the streamlines has to pass through the center of mass of the entering particle which is supposed to be captured. Second, the streamline passing the center of the particle has to pass through one of the trap sites. Even though some streamlines very close to the sidewall of the inlet channel can pass through trap sites, they have no impact on particles since they cannot pass the particle center due to the physical limitation of the particleFlow rate at each trap is a mix of two flow rates coming from the effective and the ineffective areas. In order to estimate the efficiency of a specific trap, the portion of flow rate from the effective area which enters into the trap needs to be determined. Because of the laminar flow, the flow rate corresponding to a certain section can be predicted by counting the number of streamlines passing through the section of interest. Since the particles entering in a laminar flow follow the streamline paths, the probability of a trap site can be approximated by measuring the ratio of the passed particles at the trap site to the total number of arriving particles at the effective entrance of the main channel.Ae) of the main channel inlet and their trajectories were then simulated. It was assumed that particles do not affect the fluid flow in the channel. By monitoring the trajectory of each particle or cell, the efficiency of each site for trapping was evaluated by counting the particles flowing through it and comparing to the total number of the released particles. Figure For this purpose, with the aid of the particle tracing module of COMSOL Multiphysics, a uniformly distributed number of particles representing the streamlines were released across the effective region  on cells. When the cells or particles are far from the trap sites, they are subjected to a relatively low stress. After trapping, the particles/cells have to undergo much more mechanical forces due to the larger fluid pressure on their surfaces and their contacts with the traps. An additional FEA analysis was carried out considering a polystyrene single particle (microsphere with diameter 50 μm) sitting in the trapping site in order to characterize the mechanical stresses experiencing at the trap position. A fluid-structure interaction (FSI) analysis (details described in32) was created using the COMSOL Multiphysics to estimate the hydrodynamic stress acting at the fluid-particle interface and characterize deformation of the trapped single particles/cells due to the pressure difference across the cell. By assuming an incompressible and Newtonian fluid, the stress tensor acting at the interface of particle and fluid can be determined by the following formula34:σij is stress, δij is the Kronecker delta, vi is the fluid velocity in “i” direction, and p is the fluid pressure. Equivalent elastic strain (EQS) can also be used to quantify the deformation of cells/particles under hydrodynamic force32. In order to measure the mechanical stress and the hydrodynamic deformation, it was assumed that all traps are closed by particles, and the pressure and deformation on a cell trapped in the second site , Elasticity (3 MPa) and Poisson ratio (0.33) and water as the working fluid. For FSI simulation, the contact points are also assumed to be fixed and no displacement happens in contact points of particles and trap walls, and on the boundary of particle and fluid, the velocity of fluid equals the rate of change for displacement of the solid domain (particle). Similar to the computational section, for FSI simulation, the inlet velocity and outlet pressure were set 2.8 mm/s and zero respectively and no-slip conditions were applied for all walls.The fluid-induced stress can alter the functional behavior of both the healthy and cancerous cells such as proliferation, transport, differentiation or even cell death36, and the results are shown in Fig. 34 were also calculated and plotted in Fig. The FSI results Fig.  show thamax: 0.95 Pa) is applied to the trapped particle while in model II and III, the trapped particles experience 2.6 Pa and 9.2 Pa normal pressure respectively. In the current model, the maximum shear stress on the trapped particle is slightly smaller than the second model while is much smaller than the third model. The total force applied by only fluid flow on the trapped particles is reported in Fig. Two types of fluid force are acting on the particle wall at the boundary of the fluid domain: normal forces due to hydrodynamic pressure and tangential forces due to shearing effects. Under the same boundary conditions, the shear stress and normal pressure applying on the wall of trapped particles for our model and two other models are plotted in Fig. So far, the design parameters of the microfluidic device were optimized, and the trapping performance of the microsystem was analyzed using the simulation. The optimal parameters were used for following experiments and further analysis.e = 0.006Re.Dh, where Re is Reynolds number and Dh is the hydraulic diameter of the channel37. By considering the hydraulic diameter of the entrance channel and calculating the Reynolds number, the entrance length is very small. The distance of the channel before the inlet section is much longer than Le, confirming that in our experiment, at the entrance section always the fluid flow is fully developed. Three different designs were tested to observe their ability to trap particles and characterize their efficiency. It was observed that the main channel convergence affects significantly the efficiency of single particle trapping. Figure After optimizing the geometrical parameters of the microfluidic circuit, the optimized microfluidic device with different sidewall angles was used to test the trapping performance of the microsystem using microbeads with a diameter of 50 μm. Under the flow with a very low Reynolds number (Re ≤ 1), it is expected that particles move on the streamlines in agreement with the simulation results. Although trapping efficiency is not influenced by the inlet flow rate for a very low Reynolds number , the microparticles were loaded with a relatively low flow rate into the channel to observe the motion of the particles (<10 μl/s). In order to avoid clogging inside the microfluidic channel and monitor trajectories of particles, they were injected at a low concentration, and it was found that microbeads are distributed non-uniformly at the entrance of the main channel and non-sequential trapping was observed during micro particles loading. In the simulation section, the fully developed fluid flow at the entrance section was used to estimate the efficiency of the system. The entrance length (the distance before the fluid flow becomes fully developed) can be obtained from LFigure T6 and T7 are almost low, confirming their low flow rate ratio values, and the efficiency is decreasing from T1 to T7. However, for Design 3 the average trapping is higher compared to the other designs and more uniform. The average trapping for first, second and third trap locations are almost similar for Design 1 and Design 2, and it reaches to maximum for Design 3. For the fifths, sixth and seventh trap locations, the average trapping is significantly improved from Design 1 to Design 3. For 7th trap location, the average trapping is increased by almost 50% which is in good agreement with numerical simulation indicated in Fig. To assess and quantify the trapping efficiency of each microfluidic system, the ratio of the number of trapped particles to the total number of sites was calculated. For each device, the experiment was done at least six times, and the average value was calculated and reported as the trapping efficiency. In order to ascertain that similar conditions exist for all experiments, a new device, free of particles, was used. To characterize the trapping efficiency of each device, spherical 50 μm polystyrene micro particles were injected through the syringe pump at the velocity of 2 mm/s then the highest number of occupied traps was recorded. Figure In this study, a microfluidic platform was developed to capture single particles using a hydrodynamic-based approach and the fluidic resistance concept. A series of trap sites were located on both sides of the main channel to accommodate single particles by controlling hydrodynamic forces. In our design in order to increase the hydrodynamic resistance of the main channel and direct suspended particles or cells into the trap sites, inclined sidewalls were designed instead of extending the main channel length, resulting in a more compact configuration. Numerical simulations were performed to tune geometric parameters and optimize the trapping ability of the microsystem through estimating the trapping efficiency of the device. With the aid of this microsystem, single particles can be trapped rapidly, with the efficiency of 90%. Moreover, the numerical simulation revealed that single particles can be trapped at the trapping positions without experiencing a high level of stress in comparison to similar devices, suggesting a proper configuration for capturing delicate single cells. This system can be further developed for particle sorting by accurately designing and controlling the width of three outlets to allow all the small particles to be directed into two side outlets after passing through trap sites.