S‐index”) is defined as the standard deviation of the normalized differential DVH curve. In a study of 16 patients with brain tumors at our institution, the S‐index was found to vary from 0.80 to 3.15. Our results showed that the S‐index provides a more reliable and accurate measure of dose homogeneity than that given by conventional methods. A guideline for evaluating the dose homogeneity of treatment plans based on the S‐index and its relation to equivalent uniform dose is discussed.The goal of the present study was to develop a new dose–volume histogram (DVH)– based homogeneity index for effectively evaluating the dose homogeneity of intensity‐modulated radiotherapy plans. The new index, called the sigma‐index (“PACS numbers: 87.53.Xd, 87.53.Tf Expressing the dose distribution in the form of isodose curves or surfaces is useful, because the curves or surfaces graphically show the anatomic locations of regions of uniform dose, high dose, and low dose.dDVH) and the integral DVH (iDVH) are both very useful for assessing tumor volume coverage and the dose delivered either to healthy tissue surrounding the target or to specific structures in the vicinity of the target. The DVH is thus a powerful tool for summarizing and quantifying complex dose distributions.Like many physical quantities, dose is a scalar field, where “field” refers both to the region and to the value of the physical quantity in the region. Other types of quantitative evaluation of dose distribution are based on the dose–volume histogram (DVH) concept in either differential or integral form. Entities like the DVH (which are not used in other areas of physics) have proven to be invaluable for summarizing a three‐dimensional dose distribution in a two‐dimensional graph. The differential DVH  is defined as the ratio of the maximum dose (Dmax) in the PTV to the prescription dose (Dp),(‐) with a value closer to 1 indicating better homogeneity. The H‐index generally varies from 1 to 1.5 in real‐world patient treatment plans. The index's simplicity has led to its being extensively used for quantifying dose homogeneity in tumor volumes. However, although calculating this index is easy, the value obtained sometimes fails to accurately represent the homogeneity of the treatment plan, and obtaining a complete understanding of the dose homogeneity of the entire target requires consideration of the entire DVH curve.The conventionally used homogeneity index (ose (Dp),‐9 wiH‐index also exhibits another problem. Fig. H‐indices are 1.06 (Patient 1) and 1.10 (Patient 12). The H‐indices indicate that the homogeneity of the plan for patient 1 is better than that for patient 12. However, the degree of this difference is unclear, because the index itself has no physical or clinical meaning, indicating the need for a new type of quantitative homogeneity index to be developed.The dDVHs, and we developed a new index that provides objective scores of the dose homogeneity for the PTV. The intensity‐modulated radiotherapy (IMRT) plans for the brain tumors of 16 patients, as obtained using BrainSCAN version 5.2.1 software , were used to explore the effectiveness of the new homogeneity index. We also compared the new index with currently used homogeneity indices based on biologic effects on tumors.In the present study, we examined the dose homogeneity of treatment plans for brain tumors based on statistical analyses of H‐index, another homogeneity index called HI has been definedIn addition to the D2 and D98 represent the doses to 2% and 98% of the PTV, respectively. For example, D98 indicates that at least 98% of the target volume receives this dose, and hence D2 and D98 are considered to be the maximum and minimum doses, respectively. Equation HI values indicate a more homogeneous target dose. Together with the H‐index, the effectiveness of the HI value in evaluating dose homogeneity is discussed and compared with our proposed new index later in this article.where iDVH is a plot of the volume of a given structure that receives at least a certain dose, and the dDVH is a plot of the volume receiving a dose within a specified dose range. Because of its simplicity, the iDVH has been found to be more useful, and it is more commonly used than the dDVH is. However, the dDVH is also useful in that it can provide unique information regarding the extent of dose variation within a structure. The iDVH can be defined asThe vi is the ith volume element receiving a dose of at least (Di), and V is the total volume. In this study, the x‐axis and the number of points  in the dDVH curve are, respectively, the percentage dose normalized by the prescription dose and 1 point for each percentage dose. Fig. iDVH and dDVH for a real patient at our institution, and Fig. iDVH and dDVH. The V(D=100)=0.95 in iDVH indicates that 95% of the total volume receives at least the prescription dose, and the V(D=100)=0.02 in dDVH indicates that only 2% of the total volume receives exactly the prescription dose.where f(x)] of an independent variable x (the dose). The mean value of a random variable x is also called its expectation value and is defined asAs can be seen in Fig. “S‐index”) and is defined asBased on the statistical analysis, we can use the standard deviation of the dDVH curve to quantify the spread or dispersion of the average dose within a structure—that is, the degree of inhomogeneity. This number—that is, the standard deviation—is called the sigma‐index (DSD represents the standard deviation of the dose.where (Dmean) and the corresponding S‐index. These values indicate that the PTV receives, on average, 105.96% of the prescription dose, with a variation of 2.59%. In other words, most of the PTV will receive from 103.37% to 108.55% of the prescription dose.As an example, Fig. S‐index to reinvestigate the two DVHs shown in Fig. S‐indices were 0.80 and 2.57 for the plans of patients 1 and 12 respectively. The difference of 0.04 in the H‐index between patients 1 and 12 does not clarify the distinction between the two DVHs in Fig. S‐index clearly indicates that the homogeneity of the PTV dose is about three times better for patient 1 than for patient 12. In other words, the dose in the PTV ranges from 99.2% to 100.8% of the mean dose for patient 1, but from 97.43% to 102.57% for patient 12. These results show that the S‐index provides a better quantitative measure of the dose homogeneity than the H‐index does.To investigate how the new homogeneity index quantifies the dose homogeneity, we used the H‐index and HI) are related to the S‐index, we analyzed data from 16 patients at our institution. Table S‐index corresponds to a lower HI, as expected. However, some exceptions are seen, in that the HI can provide incorrect information on the dose homogeneity. Unlike the HI‐based system, the H‐index is not well matched with the S‐index, which suggests either that the H‐index should be avoided when attempting to quantify the degree of homogeneity accurately or that the definition of the H‐index should be modified.To further investigate how the conventionally used homogeneity indices . The dose homogeneity of the DVH is clearly better for patient 5 than for patient 11. This mismatch between the real dose homogeneity in a PTV and its index is attributable to the H‐index being defined based on the point doses of the DVH rather than on the entire DVH curve. In contrast, Fig. S‐indices for the two DVHs in Fig. To illustrate the inadequate representation of dose homogeneity within the tumor volume by the currently used homogeneity indices, consider the two pairs of patient DVHs for brain tumors shown in Fig. HI system and the S‐index reveals similar problems. Fig. HI values for the DVHs of patients 6 and 9 do not accurately represent the dose homogeneity. The HI system indicates that the homogeneity of the two DVHs is the same, which contradicts the better homogeneity of the DVH of patient 6 as compared with that of patient 9. This result shows that, like the H‐index, the HI method can give incorrect information about dose homogeneity. Fig. S‐index, the dose homogeneity of the DVH is 17% better for patient 6 than for patient 9, which indicates that the S‐index uniquely provides quantitatively accurate information on dose homogeneity.A comparison between the S‐index is recommended for evaluating dose homogeneity, the accuracies of the conventional homogeneity indices can be improved by modifying their definitions. For instance, the H‐index can be made more robust by using D5 instead of Dmax (modified H‐index), and HI can be improved by using D5 and D95 instead of D2 and D98 (modified HI). Fig. H‐index and the modified HI for the DVHs shown in Figs. S‐index does, the modified H‐index and the modified HI both appear to provide appropriate values for the dose homogeneity of the DVHs.Although the H‐index. Moreover, Fig. HI system also gives incorrect information. Although the modified DVH of patient 6 is more homogeneous than that of patient 9, the modified HI values incorrectly indicate that the dose homogeneities are the same.However, the modified indices still have problems. Fig. We believe that any homogeneity index based on the doses at only a limited number of points of the DVH may provide incorrect information about dose homogeneity in the PTV. Moreover, another point should be considered when evaluating the dose homogeneity of a treatment plan: namely, the radiobiologic impact of dose inhomogeneity on the PTV. The EUD is defined as the biologically equivalent dose that, if given uniformly, would lead to the same reduction in the tumor volume as the actual dose that has an inhomogeneous distribution. Therefore, if the homogeneity index is properly constructed, it should be closely related to the EUD.The concept of the equivalent uniform dose (EUD) for tumors as proposed by Niemierko is one of the methods used to show the relationship between the dose homogeneity and radiobiologic effects.,12 TnEUD) values for three patients, where the nEUD is defined as the ratio of the EUD to the prescription dose. The figure shows that nEUD increases with the overdose‐induced inhomogeneity. For example, the nEUD of the DVH of patient 11 is larger than that of patient 5 because the overdose‐induced inhomogeneity is larger in patient 11 [labeled “B” in Fig. nEUD in the PTV will both increase with the overdose‐induced inhomogeneity in the DVH, which suggests a proportionality relationship between the S‐index and the nEUD.Fig. nEUD data of 16 patients at our institution and their corresponding S‐indices. Note that the outlier for patient 6 in Fig. S‐index without a big change in nEUD value. Just as the same radiobiologic effect results if two nEUD values are the same, the equivalent value of the S‐index usually seems to provide the same radiobiologic effect. Here, we assumed that the two DVHs are normalized, so that 95% of the PTV receives the prescription dose. Although the S‐index closely reflects the nEUD value in general, two DVH curves can be seen to have the same standard deviation  and different nEUD values (different biologic effects) under certain conditions—as seen in the outlier for patient 6 in Fig. Fig. S‐index as indicated in Table S‐indices of less than 2 (meaning less than 2% standard dose deviation in the target volume) were classified as “outstanding” dose homogeneity. The nEUD values in this category typically show doses that differ by 1% from the prescription dose. The DVHs of the next grade (those for which the S‐index is greater than 2 but less than 2.5) were classified as “good.” The Radiation Therapy Oncology Group (RTOG) has provided a guideline for clinically acceptable DVHs.No more than 20% of any PTV will receive more than 110% of its prescribed dose.The prescription dose is the isodose that encompasses at least 95% of the PTV.No more than 1% of any PTV will receive less than 93% of its prescribed dose.For simple evaluation of the dose homogeneity of a plan, the degrees of dose homogeneity of the DVHs have been categorized arbitrarily based on the S‐index meet the criteria recommended by the RTOG for the no‐variation condition.It turns out that the treatment plans classified “good” or “outstanding” based on the H‐index and the HI because it provides complete information for the entire DVH curve in a treatment plan. The results presented here show that the S‐index provides better evaluation of the degree of dose homogeneity. It has been also shown that the S‐index is generally proportional to the nEUD, where nEUD is the quantity known to be related to the inhomogeneity‐induced radiobiologic impact to the tumor.We have used statistical analyses of DVHs to propose a simple index for the objective assessment of the dose homogeneity of radiotherapy in the PTV. We believe that this new index is superior to conventional methods such as the This investigation was supported by a research grant from the National Cancer Center, Korea (No. 0610060).


We calculate the change of the properties of a resonator, when coupled to a semiclassical spin by means of the magnetic field. Starting with the Lagrangian of the complete system, we provide an analytical expression for the linear response function for the motion in the case of a mechanical resonator and the current for the case of an electromagnetic resonator, thereby considering the influence of the resonator on the spin and vice versa. This analysis shows that the resonance frequency and effective dissipation factor can change significantly due to the relaxation times of the spin. We first derive this for a system consisting of a spin and mechanical resonator and thereafter apply the same calculations to an electromagnetic resonator. Moreover, the applicability of the method is generalized to a resonator coupled to two-level systems and more, providing a key to understand some of the problems of two-level systems in quantum devices. Resonators and spins are ubiquitous in physics, especially in quantum technology, where they can be considered as the basic building blocks, as they can collect, store and process energy and information13568910and the spin, to find the coupled equations of motion (EOMs) that describe the resonator displacement and the spin magnetic moment, finding that this magnetic moment depends on the path the resonator takes. This is fundamentally different from conventional magnetic force microscopy (MFM)1314We start with a Lagrangian description, that includes the degrees of freedom of the resonator spin bath - resonator coupling, similar to the curves found in experiments where the tails have heuristically been fitted with power laws5et al.Furthermore, we find in our analytical results that the interaction amplitude as function of temperature is a curve that for certain conditions shows an optimum, see Sec. spin - electromagnetic resonator), thereby making the results suitable for direct use by other fields in physics. Moreover, we will show the applicability of the theory to the case of the resonator coupled to two-level systems (2LSs) and higher level quantum systems.Although we start calculating the susceptibility of the more intuitive mechanical resonator, we will as well derive explicitly the  impedance for electromagnetic resonators  that is produced by a magnet. The magnet is attached to a mechanical resonator that has spring constant k and (effective) mass m. The origin of the laboratory frame is chosen to be the equilibrium position of the magnet’s center. The displacement of the magnet from this equilibrium position is denoted by q(t). See The configuration of our theoretical analysis is given in IS stands for an expression with the internal spin degrees of freedom that needs to be included to derive the spin EOM. A more detailed account is left in App. A.The resonator-spin system does not live in an isolated world. Therefore we include dissipation and decay to the environment into the EOMs. The first differential equation, derived with respect to the resonator displacement, includes the Raleigh dissipation Fext(t), is an external force that is exerted on the resonator.where the last term, μ ⋅ B), assuming that μ does not depend on the position of the resonatorμ follows the classical path, we will show by solving the spin EOM that μ is influenced by the resonator and it is therefore a priori not at all obvious that Starting with the Lagrangian, which contains the degrees of freedom for the resonator and the spin, leads to the force interaction term T1 and T2; the time constants associated with the decay of the semiclassical magnetic moment longitudinal and perpendicular to the magnetic field, respectivelyμ∞, according to the master equationThe other set of differential equations can be found by deriving the EOM with respect to the spin degrees of freedom. Since the spin interacts with the environment, we can expect an effectively decaying amplitude that is often described by γs is the gyromagnetic ratio and Here μ∞(t) is the vector to which the spin magnetic moment would decay to if given the time. As the resonator moves, the magnetic field changes, and so does μ∞. We will assume that the environment of the spin is a heat bath, connected to the spin by means of the relaxation timeset al.n for particles in a two-level system isThe spin equilibrium magnetic moment W is the probability rate that the particle changes energy level due to an applied field and n0 is the population difference between the energy levels when the ensemble has the temperature of the heat bath. In other words −2Wn is proportional to the incoming energy and where n∞ is the steady state solution. Thus when 2WT1 ≪ 1 the spin ensemble, and hence our semiclassical spin, is connected well enough to the heat bath to assume that our spin has a well defined temperature. For spin-where g(ω) the spin’s normalized absorption line that is usually described by a Lorentzian or Gaussian that peaks around the Larmor frequency. This makes this condition hard to satisfy when the resonator has a resonance frequency around the Larmor frequency, and one should minimize the resonator’s movement q. When this condition is not met, the spin saturates and the temperature increases or might be undefinedwhere μ∞ from the canonical ensemble and find for spin-Assuming the condition is satisfied we can now derive μs ≡ Sħγs is the magnitude of the non-averaged spin magnetic moment with spin number where q. Higher order terms will give rise to nonlinear effects. Interaction terms with even powers in q are usually experimentally uninteresting since they will produce even multiples of the fundamental resonance frequency. These multiples are not measured or can easily be filtered. Uneven powers of q can, however, lead to disturbing nonlinear effects like Duffingq to suppress higher order terms and therefore the nonlinear effects, but in experiments this is usually limited by the signal-to-noise ratio.To find the resonance frequency and quality factor of the resonator, we will need to calculate the interaction term up to linear order in The zeroth order term does not contribute to the dynamics of the system, however it does give rise to a constant deflection of the resonator. This can be solved by shifting the origin of the laboratory frame by the amount of the deflection; this causes, however, a  change of the coordinates of the spin. We will provide an estimate of the deflection in App. C and leave it further out of account.q, we need to solve q-dependent parts. By substituting q → λq we use perturbation theory to findTo find the interaction term μ is perturbed into a q-independent part μ0 and a linear term μ1. The higher order termswhere At first we are mostly interested in solutions that do not decay over time and do not depend on initial conditions because then the linear response function can conveniently be given in the Fourier domain which makes it easy to compare with experiments. The Fourier Transform κ = κ1 + κ2 + κ3, with κ1 ≡ −μ0 ⋅ B′′ and κ-terms, which turn out to be:where B0 ≡ B(q = 0) and the notation v parallel and perpendicular to v. κ2 and κ3 are derived from where κ1κ1 is real and therefore it cannot describe the extra dissipation channel that has been seen in experimentsκ-terms is shown separately in If we compare this result with the conventional approach that neglects the effect of the resonator on the spin, we see that in that approach we have only the term κ-terms for each spin to find the susceptibility of the resonator connected to a whole ensemble of spins, i.e. κ = ∑sκ1(rs) + κ2(rs) + κ3(rs). Moreover, if the spins in the sample have an average nearest neighbor distance smaller than the typical spatial scale of the applied magnetic field, we can see the sample as a spin continuum and hence, instead of summing, integrate over the sample with spin density ρ(r).We assume that all spins in the system act individually and do not influence each other, except through the relaxation times. We can then sum over the If we calculate the result for a volume with constant spin density, it is found in App. E by partial integration of the volume in the direction of the movement of the resonator thatwithq-direction are large. The The boundary term vanishes when the volume boundaries in the κ we can calculate the frequency and Q-factor shifts as seen in experiments by den Haan et al.From κ causes the change in Q-factor. The new Q-factor is given byThe imaginary part of T1. In this simulation we have set T2 to zero only after we checked that the T2 = T1 gives an additional frequency shift of about 1 nHz and a five orders of magnitude lower shift in Q-factor compared to the results shown in In In this section we calculate the complex impedance coming from a spin interacting with an electromagnetic resonator. The derivation is very similar to the mechanical resonator and hence we will largely copy the results. We will assume that the system can be described by a lumped element model, which is a valid approximation when the typical size of the system is much smaller than the wavelength. The results might be generalized to work for other resonators by using the distributed element model1As there is a direct analogy with the mechanical resonator, it is straightforward to write down the complete Lagrangian and derive the EOM. From this we calculate something similar to the susceptibility, but more commonly used in electromagnetism, the impedance q is the charge Qe. However, instead of writing down Qe and ‘momentum variable’ Qe then becomes The electromagnetic analog of the displacement q → I. This results in an extra impedance z = z1 + z2 + z3, equivalent to the κ-terms, whereThe resulting interaction term is slightly different compared to that of the mechanical resonator. The zeroth order term vanishes conveniently due to the time derivative, leading to the impedance interaction term The resonators complex impedance then becomesz-terms as done in Sec. spin bath - resonator coupling when partially integrating over a whole sample because I is, unlike q, not a Cartesian direction. However, one thing simplifies the z term reasonably: the law of Biot-Savart shows a linear dependence on I implying that z1 vanishes. Note that it is very well possible that the frequencies of interest are comparable to ωs. In this case one should calculate the whole term. Moreover one should be careful with the implied condition of It is much harder to simplify the In et al.T1 and T2 dependency.So far we have done nothing more than rigorous math to calculate the susceptibility of a system were the physical process is precisely known. However, the physical nature of the interaction between a resonator and a general two-level systems (2LSs) can be different from the simple magnetic field interaction and will often even be unknown. This subject has been studied in glassy systems long before it found its application in quantum technologyet al.B-field dependent density of states and heat capacity. Indeed, expanding the average energy term, as we did in App. B, leads to the heat capacity which resulted in x the Zeeman energy – temperature ratio. These similar results in combination with the results obtained in this paper imply two things: First the B does not have to be the physical magnetic field. It is always possible to rewrite the two state Hamiltonian toMeanwhile the field of glassy dynamics revived when it was found in experiments that the electric permittivity and loss factor of a nonmagnetic glass do actually depend on the magnetic fieldB can be any field that splits the energy levels, leading to an energy difference ε when q = 0. Here E0 is an uninteresting energy-offset and σ is a vector containing the Pauli matrices. The interaction strength is determined by B depend on q, which is the generalized coordinate of the mechanical resonator, or generalized velocity of the electromagnetic resonator. Because the expectation values of the Pauli matrices σ are described by the Bloch equations, the derivations in this paper apply to any resonator-2LS system. Just substitute κ and z terms.where S + 1 energy levels  by expanding the Brillouin function from App. B and substitutingSecondly, this result can be easily generalized to a system with 2κ and z-terms. This 2S + 1-state quantum system must be isomorphic to a spin-S particle and hence meet two conditions: 1) the energy levels are equally spaced and 2) transitions are only possible to adjacent energy levels.into the B and finding the energy gap ε. This way, we hope the formulism can be of use in many systems.The general formulism presented in this section can thus be made very practical again by substituting the actual physical field responsible for the dynamics into μ1, it might also be important to consider the transient solutions.We have calculated the linear response function of a mechanical and electromagnetic resonator coupled to a spin. The linear response function of the resonator shows extra terms that result in a shift of the resonance frequency and a drop of the Q-factor of the resonator, compared to the bare resonator characteristics. Moreover, we have generalized these results to the coupling with an energy level system with an arbitrary amount of equally spaced energy levels. In practice this means that despite having nonmagnetic samples and frequencies that are not even close to the Larmor frequency, one encounters dissipation of the resonator due to the inhomogeneous field it creates. Eventually this might not be a surprise since the resonator alters the heat capacity of the spin’s equivalent spin ensemble. Although this is closely related to the magnetic loss enhancement in nonmagnetic glassy systems56et al.T1, are not taken into account in our description. Furthermore, when pulses are applied, for example in spin resonance techniques, a precise time evolution of the system is needed. Moreover, sending hard pulses might violate the condition for the temperature and linear response of the spin that we have encountered in Sec. basic principles. In this case one might move to a calculation involving the spin-operators. The theory presented here would still give a fair indication about the enhancement of dissipation, which is of importance in the field of hybrid quantum systems that are pushing the limit of macroscopic superpositions22We have chosen to do the calculations completely in the (semi)classical regime as we are especially interested in the expectation value of spin and resonator. Moreover this leads to an intuitive description and fairly simple calculations. The classical treatment has it limitations though: Berman How to cite this article: de Voogd, J. M. et al. Dissipation and resonance frequency shift of a resonator magnetically coupled to a semiclassical spin. Sci. Rep.7, 42239; doi: 10.1038/srep42239 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


There were several minor typos present in the Methods section of the original article.C and B as torque vectors, i.e., already multiplied by In the paper we defined variables I1 and I2 were accidentally switched with the values for center of mass distances s1 and s2. The corrected table is:When reporting the model parameters in Table 4 in the paper, the values for inertia parameters fmax and vector ũ, and not × operator. The corrected equation in the text is: Since we previously defined operator × as element-wise multiplication, the joint torque calculation equation (page 12) should have a regular multiplication between matrix The authors apologize for these typos and state that they do not change the scientific conclusions in any way.The original article has been updated.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The Checkpoint Ordering Problem (COP) is both of theoretical and practical interest. It has several applications and is conceptually related to some well-studied combinatorial optimization problems, namely the Single-Row Facility Layout Problem, the Linear Ordering Problem and a variant of parallel machine scheduling. In this paper we study the complexity of the (COP) and its special cases. The general version of the (COP) with an arbitrary but fixed number of checkpoints is NP-hard in the weak sense. We propose both a dynamic programming algorithm and an integer linear programming approach for the (COP) . Our computational experiments indicate that the (COP) is hard to solve in practice. While the run time of the dynamic programming algorithm strongly depends on the length of the departments, the integer linear programming approach is able to solve instances with up to 25 departments to optimality.We suggest a new variant of a row layout problem: Find an ordering of  An instance of the Checkpoint Ordering Problem (COP) consists of i and the checkpoint with respect to a particular permutation where Let us start with elaborating on the connections of the (COP) to the Linear Ordering Problem (LOP), the Single-Row Facility Layout Problem (SRFLP) and scheduling on identical parallel machines with the objective of minimizing the sum of weighted completion times.1.1.n one-dimensional departments, with given positive lengths The simplest known layout type is single-row layout. It arises as the problem of ordering stations on a production line, where the material flow is handled by an automated guided vehicle (AGV) in both directions on a straight-line path . An instn] and i and j with respect to a particular permutation where Several practical applications of the (SRFLP) have been identified in the literature, such as the arrangement of rooms on a corridor in hospitals, supermarkets, or offices , the assn weights.Similar applications are conceivable for the (COP), e.g. the rooms on a corridor could be arranged such that the weighted sum of their distances with the office of the head is minimized or planes could be assigned to gates such that the weighted sum of their distances from the entrance of the airport terminal is minimized. When comparing the (SRFLP) with the (COP), we observe that the problems are quite similar. One difference is that an (SRFLP) instance has 1.2.n] a profit and the goal is to find an ordering of maximum profit. In the simplest case of the Linear Ordering Problem (LOP) , this profit is determined by those pairs u comes before v in the ordering. Thus in its matrix version the (LOP) can be defined as follows. Given an A such thatOrdering problems associate to each ordering (or permutation) of the set [G with vertex set G containing for every pair of nodes i and j either arc  or arc , but not both. Then the (LOP) consists of finding an acyclic tournament, i.e. a tournament without directed cycles, of G of maximum total edge weight.is maximized. Equivalently, we can interpret Although the (LOP) and the (COP) have apparently a similar structure, it is harder to directly relate these two problems. We will show in Section 1.3.m identical parallel machines. Each job Furthermore we can relate the (COP) to the NP-hard  schedulij is denoted by m part of the input, and by m.Every machine can processat most one job at a time, and every job has to be processed on one machine in an uninterrupted fashion. The completion time of job cheme of , this scm part of the input, the problem is NP-hard in the strong sense by transformation from 3-PARTITION [For the special case of only one machine the problem can be solved in polynomial time by Smith’s Rule  that sugproaches –10. For ARTITION . For furARTITION . For a mARTITION .From a computational point of view  methods –17, for  methods .(1)The checkpoint must not lie exactly at a splitting point of two departments but it can also be covered by a department. I.e. the checkpoint does not necessarily define a partition of the departments. When considering a scheduling set-up the (COP) can be described as follows: It is allowed to split one arbitrary job into two parts at any point and then the two parts have to be scheduled first on the two machines.(2)For the (COP) the sum of the lengths of the departments that are placed to the left and to the right of the checkpoint are predetermined through the position of the checkpoint. E.g. for a centered checkpoint the sum of the lengths of the departments to the left and to the right of the checkpoint has to be equal. For Due to these differences it is not possible to directly carry over polyhedral results, dynamic programming algorithms and (mixed) integer linear programming (ILP) models and their corresponding approximation results [When comparing  results  from sch1.4.Now let us further clarify the similarities and differences of the (SRFLP) and the (COP) with the help of a toy example: We consider 4 departments with lengths Finally we also want to clarify the workings of the (LOP) with the help of a toy example. We consider 4 objects and the weights 1.5.We propose a new combinatorial optimization problem that is both of theoretical and practical interest.We study the complexity of the (COP) and its special cases, pointing out several connections to related problems.We propose two exact approaches for the (COP), namely a dynamic programming algorithm and an ILP approach.We demonstrate the practical difficulty of the (COP) in a computational study. In this context let us also refer to our companion paper  for a coThe paper is structured as follows. In Section The main contributions of this paper are the following:2.M we ask whether there exists a permutation of the departments such that the obtained costs are at most M:Consider the decision variant of the (COP) : given some value Decision Checkpoint Ordering (DCO):Instance: n departments with given lengths Question: Is there an ordering In the following proof we assume w.l.o.g. that the checkpoint is located at the center in order to simplify the presentation.Theorem 1:DCO is NP-complete.Proof 1It is clear that DCO To prove that DCO is NP-complete, we give an NP-complete problem and a polynomial-time transformation to DCO. The following problem is NP-complete  by a department a with length t with length Now we transform an instance of PARTITION to an instance of DCO as follows. We replace each element t is located exactly above the checkpoint because of the large weight of department t. Due to the definition of the weights of the departments t. Hence the only way to influence the objective value is to decide whether the departments should be located left or right of t. If we can find a subset t and all other departments right of t, then the corresponding ordering t in the optimal ordering B, then there exists no subset If the ordering In summary we have shown that there exists an ordering In the layout context the above result can be interpreted as follows: The minimization of the inter-row costs that occur in all multi-row layout problems is NP-hard, even in its simplest version. This is not only an interesting theoretical insight on its own but also supports our definition of the Single-Row bound in our companion paper . Next leFirst we assume that all departments have the same length. In this case the optimal permutation of the departments can be obtained by a simple greedy selection. We choose the permutation Fact 2:A permutation i from the checkpoint.where Proof 2The change of the objective function caused by swapping two departments i and j is i and j. Now swapping the two departments improves the objective value. As all other departments are not affected by pairwise swaps, it is not possible to improve the objective value by an arbitrary number of pairwise changes if  where all department lengths are equal and the weights are binary is still NP-hard . This pripartite . (LA) waipartite ,24 to deipartite , single ipartite ,27 and cipartite ,29. Therartite [ ,31 and  tite [ [ ,33. For tite [ [ .i, the smaller is its distance from the checkpoint. Next we give a short formal argument for the above claim and refer to Figure (1)the checkpoint is left-aligned and(2)the relative weights The following result is in fact known as Smith’s Rule [Next we assume that the checkpoint is left-aligned or right-aligned. Also in this case the optimal permutation of the departments can be obtained by a simple greedy selection. We choose the permutation h’s Rule  in the sFact 3: The permProof 3Assume that there exists an optimal permutation that does not satisfy condition  with left-aligned checkpoint also as a (LOP) with special structure. We collect the lengths of the departments in a column vector W are not independent but determined by an outer product of two vectors. We have just seen in Fact 3 that the (COP) with left-aligned checkpoint can be solved by a simple greedy heuristic. Hence this special structure of the (LOP) cost matrix W as an outer product of two vectors is the reason why this (LOP) version can be solved in polynomial time.is minimized. Contrary to the general (LOP), which is NP-hard , the matWe summarize the above results as follows: The (COP) is NP-hard and its ‘hard’ part is to determine where the centers of the departments are located with respect to the checkpoint in the optimal solution. In the following section we show that the (COP) with one checkpoint that is neither left- nor right-aligned can be solved by a dynamic programming algorithm and hence is NP-hard in the weak sense. If the number of checkpoints is not part of the input, then the (COP) is NP-hard in the strong sense.3.In every (COP) layout there is at most one department covering the checkpoint. All other departments lie completely to the left or to the right of the checkpoint.In an optimal (COP) layout all departments, except for the department covering the checkpoint, have to be arranged in non-increasing order from the checkpoint to the border of the layout with respect to their relative weights property ,36.In order to simplify the description of our dynamic programming algorithm we assume w.l.o.g. that the checkpoint is centered and hence that the sum of the lengths of the departments to the left and to right of the checkpoints has to be equal. Now let us outline the workings of our dynamic programming algorithm in detail.We exploit the following two simple properties of (COP) layouts for designing a dynamic programming algorithm:The input for our algorithm is a (COP) instance, where the list of departments is given in non-decreasing order with respect to their relative weights Next we choose an alignment of the center department above the checkpoint: We start with the right end of the center department above the checkpoint and then shift the center department 0.5 to the right in each iteration until the center of the center department is placed directly above the checkpoint. Clearly it suffices to consider 0.5 shifts of the center department as the checkpoint is centered and s indicates the remaining free space to the left of the checkpoint and M gives the overall remaining free space either to the left or to the right of the checkpoint, which is equal to the sum of the lengths of the departments not yet assigned. As we arrange the departments in non-increasing order with respect to their relative weights where We refer to Algorithm 1 for a detailed description of our dynamic programming algorithm that we implemented in C. In Section Hence the (COP) with one checkpoint is NP-hard in the weak sense. If the number of checkpoints is not part of the input, then the (COP) is NP-hard in the strong sense, which can be proven by a deduction from 3-PARTITION.To further clarify the workings of our dynamic programming algorithm let us consider a toy example with centered checkpoint and the following input data: re 1, 3, , 2,  . To simplify the notation, we consider the (COP) on n that is again w.l.o.g. assumed to be centered. To ensure transitivity on the ordering variables we use the 3-cycle inequalitiesin order to relate the positions of the that rule out the existence of directed 3-cycles and are sufficient for guaranteeing that there is no directed cycle.i plus S / 2, i.e. the total length of the departments divided by 2. Then we subtract i from the checkpoint, if department i is located to the left of the checkpoint. If department i is located to the right of the checkpoint, this difference is minus the distance of the center of department i from the checkpoint. Therefore we multiply this difference by the term i lies to the left of the checkpoint and i lies to the right of the checkpoint:Now the distances of the departments from the checkpoint can be expressed as quadratic terms in ordering variables: For department withThe additional multiplication with Expanding and simplifying  yields(1To model the (COP) as an ILP we apply standard linearization and introduce new variables for all products of ordering variables in :(12)i appears twice in the indices. Now (Note that we have to introduce ces. Now  can be fMoreover we use the following standard constraints to relate the orderings variables and their products:In summary we obtain the following ILP model for the (COP) :In the following section we computationally compare an ILP approach based on this model with our dynamic programming algorithm introduced in the previous section.5.i as checkpoint and deducing the (COP) weights from the pairwise weights as follows: L to the instance name for indicating the instances with the new random department lengths. All instances considered can be downloaded from http://tinyurl.com/layoutlib.We report the results of computational experiments with our dynamic programming algorithm and ILP approach respectively. All computations were conducted on an Intel Xeon E5160  with 2 GB RAM, running Debian 5.0 in 64-bit mode. The algorithms were implemented in C  and Gurobi 6.5 (ILP) respectively. To generate (COP) instances we use layout benchmark instances from the literature by simply choosing one department In Tables In Tables Comparing our two approaches we observe that the dynamic programming algorithm is clearly superior for all benchmark instances from the facility layout literature that only contain departments with short lengths 6.In this paper we proposed a new variant of a row facility layout problem and two exact algorithms for solving it. The Checkpoint Ordering Problem (COP) is weakly NP-hard. It is both of theoretical and practical interest and has several important relations to other well-studied combinatorial optimization problems. In our computational study we showed that the (COP) is hard to solve in practice for both dynamic programming and ILP approaches.It would be interesting to examine if the models, results and algorithms for scheduling on two parallel machines, which is a very well-studied problem, can be used to obtain stronger approximation results and/or to design stronger exact approaches for the (COP) in particular and layout problems in general.


Arabidopsis thaliana (Columbia accession) using the chemical ethylmethane sulfonate. We measured the effects of new mutations relative to a premutation founder for fitness components under both natural (field) and artificial (growth room) conditions. Additionally, we measured three other quantitative traits, not expected to contribute directly to fitness, under artificial conditions. We found that induced mutations were equally as likely to increase as decrease a trait when that trait was not closely related to fitness (traits that were neither survivorship nor reproduction). We also found that new mutations were more likely to decrease fitness or fitness‐related traits under more stressful field conditions than under relatively benign artificial conditions. In the benign condition, the effect of new mutations on fitness components was similar to traits not as closely related to fitness. These results highlight the importance of measuring the effects of new mutations on fitness and other traits under a range of conditions.Mutations are the ultimate source of all genetic variations. New mutations are expected to affect quantitative traits differently depending on the extent to which traits contribute to fitness and the environment in which they are tested. The dogma is that the preponderance of mutations affecting fitness will be skewed toward deleterious while their effects on nonfitness traits will be bidirectionally distributed. There are mixed views on the role of stress in modulating these effects. We quantify mutation effects by inducing mutations in  BecauseTwo major experimental approaches have been used to study the effects of mutations on fitness  What is the distribution of fitness effects of mutations derived from mutagenesis? (2) Does the distribution of fitness effects differ between field and relatively benign laboratory environment conditions? (3) Does the distribution of mutation effects for fitness proxies differ from the distribution for traits that are less likely to be related to fitness? To date, three laboratories have used 22.1A. thaliana using ethylmethane sulfonate (EMS). EMS is an alkylating agent that most commonly results in G:C to A:T substitutions (transitions) . We therefore used dry mass of survivors multiplied by proportion of plants surviving to harvest as our measure of fitness for each mutant line and premutant founder. This is a reasonable proxy of fitness for a selfing annual plant  in Beltsville MD (N 39.05378 W −76.95387). Weather data records for the Beltsville Experimental Agriculture Station can be found here: 2.3Plants from the same lines were grown in a walk‐in growth room at the University of Maryland. The same experimental design was sowed into plug trays in Spring 2014. Seeds were cold treated for 2 weeks at 2°C and allowed to germinate on benches at 20°C with 24 hr of fluorescent light and 8 hr of incandescent light for 2 weeks, as above. These plants were cultivated in the growth room at 20°C with 24 hr of fluorescent light. Seedlings were then transplanted in random order into plug trays (give size of the plug) and continued to grow at 20°C with 24 hr of fluorescent light on a bench and 8 hr of incandescent light. A total of 15–25 plants from each of the two sublines and the founder sublines were grown in three blocks and were measured for four traits: silique number (=number of fruit), Julian day of first flower, number of trichomes per mm of midrib length, and the ratio of side branch mass to main branch mass. They were allowed to grow from April 2014 to December 2014. All plants were harvested above ground when they went to seed unless they had not bolted by December 2014, at which point all remaining plants were harvested. Because we were not certain whether these plants would have bolted or not, they were left out of measurements that relied on bolting . These plants represented about 6% of the overall plants measured. Julian day of first flower was assessed daily. The largest leaf at first flower was collected and used to determine trichome number per midrib length. Aboveground biomass was collected at the time plants went to seed. Silique number and the mass of the main branch and side branches were measured on the dried samples. We consider silique number to be a fitness proxy.2.42.4.1Statistical analyses were conducted in R , indicating that induced mutations contributed to among line variance. The Columbia founder line was found to have the second highest fitness (and the highest mass when survivorship was not included in the data) .Under field conditions, there was a significant mutation line effect on fitness     . A sign test only indicated a significant deviation from the median value of the mutant lines for the founder for number of trichomes per midrib length , the founder had more trichomes than 15 of the 20 mutant lines, but this was not significant after sequential Bonferroni correction. The rank of all other traits was not significant. In other words, there is little evidence of the mean character state of the mutant lines differing from the founder for the four traits measured in the growth room.There was a significant mutant line effect for day of first flowering  Figure , silique) Figure , and tri) Figure , indicatp > .29, n = 21).There were no significant correlations between pairs of traits that demonstrated significant line effects  suggested that beneficial mutations occurred, but that the magnitude of deleterious mutations was greater  × survivorship) to that under growth room conditions (silique number), we see that there is a significant line effect for both. However, the rank of the founder differs in both. Under field conditions, all but one line performed worse than the founder Figure . Under a4.2Traits that are not as closely related to fitness are expected to be bidirectionally affected by new mutations. Three other traits were investigated under artificial conditions . These traits are likely less closely tied to fitness with the exception of branching (Lortie & Aarssen, A. thaliana independent of the alleles at the major flowering time loci (Samis et al., While the traits flowering time and the presence or absence of trichomes have been shown to be under major gene control (Johanson et al., 5A. thaliana, but strays from those results in an explicable way. Increasing the number of mutations by three‐ to fourfold increases the likelihood that a strongly deleterious mutation will occur and counter the effects of slightly beneficial mutations. Fisher used the analogy of focusing a microscope. If a specimen is reasonably near focus, then large movements with the course adjustment are more likely to reduce focus than small changes with the fine focus (Fisher, The results of this study are congruent with the mainstream view of the effect of mutations on fitness in that most of the lines decreased fitness relative to a premutation founder and that effect was more pronounced under field conditions. This is contrasted to previous recent results from  Fisher, . While oNone declared.


This study aims to investigate the effect of natural sounds on the anxiety of patients undergoing coronary artery bypass graft surgery (CABG).In this clinical trial, 90 patients, who were candidates for CABG in an urban area of Iran, were selected and randomly assigned to intervention and control groups by the minimization method. In the intervention group, natural sounds were broadcast through headphones for 30 min. In the control group, headphones connected to a silent device were used. The research instruments were a demographic questionnaire and the Spielberger State-Trait Anxiety Inventory (STAI). These were used before the intervention, 30 min after the music, and before the surgery in the waiting room for both groups. Data was analyzed using SPSS software.p = 0.001). Moreover, the mean anxiety of the intervention group decreases, while it increases for the control group over time (p < 0.001).The mean anxiety level of the intervention group has been found to be significantly lower than that of the control group half an hour after the intervention as well as in the waiting room in the preoperative period (Natural sounds can be used as a non-pharmacological way to reduce the anxiety of patients undergoing CABG.RCT2017011723190N3, Registered 1 March 2017.I Coronary artery bypass graft surgery (CABG) is one of the most valuable and effective methods for treating coronary artery diseases, but it is also one of the most stressful events that can occur during a person’s lifespan  and the Spielberger State-Trait Anxiety Inventory (STAI). The reliability of the demographic questionnaire was determined by the content validity method as well as the opinion of professors and experts. The STAI is a commonly used measure of trait and state anxiety. Form Y, its most popular version, has 20 items for assessing trait anxiety and 20 for state anxiety. Internal consistency coefficients for the scale have ranged from 0.86 to 0.95, while test-retest reliability coefficients have ranged from 0.65 to 0.75 over a 2-month interval  was referred to the hospital according to the cardiac surgery program in the night shift before the operation and following the selection of samples according to the inclusion and exclusion criteria. Consent was obtained from the participants, and the demographic data form and the Spielberger STAI were completed to assess the level of state anxiety through face-to-face interviews. The patients were then randomly assigned to the intervention and control groups. For the intervention group, nature sounds were played for 30 min via headphones in order to remove additional sounds. In the control group, headphones were connected to a silent device and were placed on the ears of the patients. Thirty minutes after the completion of the music playing and on the morning of the operation day in the waiting room, the anxiety questionnaires were completed by the members of the two groups. Nature sounds used in this study were the artwork “Sounds of Nature” recorded by Dr. Arnd Stein in t-test and repeated measures ANOVA with SPSS software version 18. The significance level was less than 0.05.Finally, the data was analyzed using descriptive statistics, such as mean and standard deviation, and analytical statistics including the independent The demographic results indicate that in the intervention group, the mean age of patients was 58.61 ± 9.55 years and the mean duration of the disease was 9.51 ± 5.02 years. In the control group, the mean age of patients was 57.71 ± 9.88 years and the mean duration of their occurrence was 11.66 ± 3.99 years; they were homogeneous in terms of demographic characteristics . However, there was a significant difference between the two groups after the intervention and the mean anxiety level in the intervention group was lower . An intra-group analysis of variance with repeated measures was used: this resulted in no significant effect on the time (p = 0.46). The interactive effects of time and group were also evaluated using the repeated measures analysis of variance. While the result was significant and showed a decrease in the mean anxiety of the intervention group, the mean anxiety of the control group increased (p < 0.001) (Table p > 0.05).Based on the results, no significant difference was found between the two groups before the intervention (independent 1) Table . Normal Based on the results of this study, there was a significant statistical difference between the two groups in terms of playing natural sounds half an hour after and before surgery in the waiting room: the mean of anxiety in the intervention group was also lower. In the field of natural sounds, the results of the study by Aghaie et al.  were conAlthough a similar study was not found in terms of the use of natural sounds before CABG, music intervention has been studied in several studies and various results have been reported. For example, the results of McClurkin et al.  are consDespite the results of this study, the findings from Ebneshahidi and Mohseni , SendelbBased on the results of the present study, the mean anxiety of the intervention group decreased but the mean anxiety of the control group increased; in the preoperative waiting room, the intervention group had lesser anxiety than the control group. In this regard, the results of Kipnis et al.  are consQuran and cultural books like Divan Hafez, thinking of nature and the consequences of thinking about it have been emphasized. These can help reduce this challenge.Music can facilitate anxiety reduction due to its physiological and neurological effects. After studying 23 studies on ischemic heart disease, Bardt and Dileo  stated tSince the special environment of intensive care units increases the anxiety of patients due to factors such as sensory deprivation, severe medical noise, sleep deprivation, fear of diagnostic procedures, and separation from family members and friends (Rezaie et al. Physiological indices such as heart rates as well as blood pressure of these patients and the cortisol level taken at the time of the start of the surgery could have provided further evidence of the value of the administration of music to cardiac patients. These were not assessed in this study and are lost opportunities. Therefore, it is recommended that future studies be conducted as to assess these variables. This study was limited to patients with CABG. Lack of similar research on natural sound music limited the possibility of comparing the results of this study. So, doing similar studies in people with a different age range and other illnesses, comparing this kind of music with other types of music, and comparing the duration of the music playback are recommended for future studies.Natural sounds can be used as a way to reduce the anxiety of patients before surgery. Nurses can use this non-pharmacological approach along with other types of care to reduce the anxiety of patients before CABG.


Elicitation is a commonly used tool to extract viable information from experts. The information that is held by the expert is extracted and a probabilistic representation of this knowledge is constructed. A promising avenue in psychological research is to incorporated experts’ prior knowledge in the statistical analysis. Systematic reviews on elicitation literature however suggest that it might be inappropriate to directly obtain distributional representations from experts. The literature qualifies experts’ performance on estimating elements of a distribution as unsatisfactory, thus reliably specifying the essential elements of the parameters of interest in one elicitation step seems implausible. Providing feedback within the elicitation process can enhance the quality of the elicitation and interactive software can be used to facilitate the feedback. Therefore, we propose to decompose the elicitation procedure into smaller steps with adjustable outcomes. We represent the tacit knowledge of experts as a location parameter and their uncertainty concerning this knowledge by a scale and shape parameter. Using a feedback procedure, experts can accept the representation of their beliefs or adjust their input. We propose a Five-Step Method which consists of (1) Eliciting the location parameter using the trial roulette method. (2) Provide feedback on the location parameter and ask for confirmation or adjustment. (3) Elicit the scale and shape parameter. (4) Provide feedback on the scale and shape parameter and ask for confirmation or adjustment. (5) Use the elicited and calibrated probability distribution in a statistical analysis and update it with data or to compute a prior-data conflict within a Bayesian framework. User feasibility and internal validity for the Five-Step Method are investigated using three elicitation studies. The knowledge held by expert practitioners is too valuable to be ignored. But only when thorough methods are applied, can the application of expert knowledge be as valid as the use of empirical data. The responsibility for the effective and rigorous use of expert knowledge lies with the researchers.” certainty is the key feature of Bayesian statistics, uncertainty reveals the extent of our knowledge and ignorance Elicit the location parameter of the SN using the trial roulette method.(2)Use software to provide instant feedback on the interpretation of the expert’s beliefs by the researcher so the expert can accept this representation or adjust their input.(3)Elicit the (un)certainty of the expert by determining the scale and shape parameters of the SN using expert’s statements on the lower and upper bounds for a plausible range of the parameter values.(4)Use software to provide instant feedback on the interpretation of the expert’s (un)certainty about the location parameter by the researcher so expert can accept this representation or adjust their input.(5)Use the elicited and calibrated probability distribution in a Bayesian analysis to update it with data or to compute a prior-data conflict.https://osf.io/wvujz.The remainder of the paper is ordered as follows. We first provide details on the Five-Step Method. Thereafter we present a user feasibility study in which we elicited beliefs regarding a trivial sports related question from respondents to investigate visual and procedural preferences of users for the digitized version of the trial roulette method. A second study was carried out by asking experts working at a staffing company about certain key performance indicators which we used to validate the internal validity of steps 1 and 2 of the elicitation procedure. A final study was done with regional directors working at a large financial institution. They provided actual forecasts concerning average turnover per professional in the first quarter of the year 2016 with the Five-Step Method. The participating companies already make predictions concerning the parameters we elicit, yet they do this in the form of point estimates. The experts are thus already used to thinking about these data and predicting these data which makes them highly suitable to include as experts in an elicitation exercise. Yet, it is an extension for them to actively specify and separate knowledge and uncertainty. Because the companies also provided us with data on the predicted parameters we were able to compare the forecasts of the experts with data and thereby get an indication of the internal validity of the elicitation procedure. The proposition to split the elicitation process results in a procedure differing from the existing elicitation procedures as, for example, proposed by In this section we describe the technical details of the Five-Step Method which has been programmed in R  using thFigure 1. Instead of vertical “bins” a grid is used and the digital “chips” can be placed on the grid. Experts provide estimates for the expected minimum and maximum value of the parameter of interest, represented by the left and rightmost digital “chips” in the grid, based on which the range of the grid is determined. Thereafter they place additional “chips” in the grid. In specific, the input grid, denoted by G, is a matrix size 600 (columns) × 300 (rows) and cells are activated by the placement of a digital “chips” in the grid. The cells where a sticker is placed obtain a value of one, all other cells are set to non-available. A second matrix, denoted by R of the same dimensions is created in which all rows are equal and the columns are a sequence of numbers with equal intervals running from the reasonable lower to upper bound provided as input. We then create output matrix O which contains values from R activated by the placement of dots in G and after the deletion of all non-available values in O, the remaining values are stored in a vector.The first step of the Five-Step Method consists of a digitized version of the trial roulette, which can be seen in The vector of values that is elicited in step 1 are used to fit a SN distribution. The SN distribution is defined in this paper as a normal distribution with the additional shape parameter γ. The shape parameter is based upon a general method for the transformation of symmetric distributions into skewed distributions as described in Figure 2. Note that the distributions would be exactly mirrored with respect to the mode if the γ values would be The effect of the shape parameter on the allocation of mass can be seen in To fit the SN distribution we make use of the snormFit function from the fGarch package . This fuFigure 1. The visual feedback indicates how we interpret the information that is provided by the expert. The expert can accept the representation of their beliefs or adjust input until the representation matches their beliefs. Once the expert approves the representation of their beliefs, the mean value is extracted from the distribution which is to be used in step 3.The SN distribution that is fitted based upon the expert’s input is provided as visual feedback to the expert, see Figure 3. We restricted the priors that represent the experts’ beliefs to be SN distributions so πd(𝜃) ∼ SN, where subscript d denotes expert d = 1,…,D, μ0 denotes the prior mean, 0 denotes the prior skewness. The value for μ0 is assumed to be known, either obtained through steps 1 and 2 or stated directly. In step 3 the expert is required to provide values for the reasonable lower and upper bounds they perceive as likely for their estimate of μ0 The value for μ0 is repeated 100 times, the values for the reasonable lower and upper bounds for the estimate are both repeated 10 times.Step 3 of the Five-Step Method is used to derive the distributional representation of the expert’s prior beliefs concerning the parameter of interest and can be seen in 0. The 120 values, μ0 repeated 100 times and the values for the reasonable lower and upper bounds both repeated 10 times, are provided to the snormFit function by means of which a SN distribution is fitted. The estimates for 0 are obtained and μ0 is constrained to the input value. Visual feedback is provided to the expert of the resulting SN distribution, which can be seen in Figure 3. The expert can accept the representation of their beliefs or adjust input until the representation matches their beliefs.Based on the input provided in step 3 we will obtain estimates for the scale parameter Use the elicited distribution that represents the expert’s beliefs.In this section we describe the three studies we conducted. During the user feasibility study R version 3.1.2 was used and R version 3.2.3 was used during the elicitations done with the staffing company and the large financial institution. We conducted the elicitations in a semi-structured face-to-face setting so that the researcher could provide interpretations accompanying the visual feedback. An advantage of a face-to-face setting is that it allows clarification of procedural and elicitation related questions thereby improving the validity of the responses .D = 9), obtained through convenience sampling from a population of university trained adults, were randomly allocated to two out of the four possible variations of the software.With the user feasibility study we evaluated the usability of the first two steps of the Five-Step Method. Procedural and visual preferences were investigated. Four variations of the shiny application were tested. The respondents  providing predictions about five sales results concerning the first quarter of 2016: contract hours, hourly cost buying and selling, turnover and hourly sales margin. A staffing company is a link between companies that want to hire staff and staff looking to work at companies. They buy work from individuals and thereafter place them to work at other companies. The amount of hours they place an individual at another company in the quarter are the contract hours. The hourly cost buying is what it will cost them per hour to buy the work from the individuals and the hourly cost selling is the price which they charge the companies where they stall the individuals. The turnover is equal to the contract hours multiplied by the hourly cost selling and the hourly sales margin is equal to the hourly cost selling minus the hourly cost buying.The goal of the second study was to test the internal validity of elicitations obtained with the first two steps of the Five-Step Method. We found a staffing company willing to participate with experts  were regional directors working at a large financial institution. They are considered experts in knowledge concerning market opportunities, market dynamics and estimating the capabilities of the professionals to seize opportunities. Based on these skills we expected that they could predict the average turnover per professional in the entire country in the first quarter of 2016. In this study the experts did not predict the distribution of the data y, but construct a prior for the mean denoted by πd(𝜃). As πd(𝜃) ∼ SN the elicitation results in the representation of each expert’s beliefs expressed in the hyper parameters μ00. We compare the predictions of the experts against actual results, expressed as the posterior distribution of the average turnover per professional, denoted by π(𝜃|y). π(𝜃|y) ∼ SN, μ1 denotes the posterior mean, 1 the posterior skewness. The prior for π(𝜃|y) is a N prior which is uninformative given the scale of the data.In the third elicitation study the experts  so that 𝜃∗ ∼ N.where 𝜃 represent the parameter of interest and 𝜃 ∼ N can be found in Figure 5, together with the posterior distribution π(𝜃|y). The values for the hyper parameters for πd(𝜃) and π(𝜃|y) can be found in Table 3. We can see, visually in Figure 5 and numerically in Table 3, that experts one and two provide very similar predictions, however expert 2 is less uncertain about the prediction. In the same manner we can see that expert four made a prediction that closely resembles the actual realization.All data were analyzed anonymously and were transformed to avoid revealing business-sensitive information. The elicited priors πThe Five-Step Method provides a first step for eliciting experts in a flexible manner such that no expert is forced to reason on a scale they are uncomfortable with, yet ending up with comparable priors for all experts.In essence the Five-Step Method resembles the structure for eliciting a distribution as is proposed by Figure 5. This provides an indication for the internal validity of the method, experts are able to accurately predict future data using the Five-Step Method. To see if this result holds in general or only in our sample, and to compare the results with other elicitation methods, we recommend a larger study in which experts use multiple elicitation methods including the Five-Step Method to predict future data.We acknowledge that asking experts for the reasonable lower and upper bound for their estimate in step 3 of the Five-Step Method could perhaps be an oversimplified procedure and other researchers might prefer to replace this step with eliciting quantiles. Besides providing graphical feedback it is desirable to stay as close as possible to the reasoning experts use on a daily basis. The method should be adjusted to fit the expert’s reasoning and not the other way around if we do not want to introduce unnecessary bias. As shown in the study with the large financial institution, the Five-Step Method allows for just that. We can help experts order their thoughts, whether they reason in terms of individuals, regions or totals. All these ways of reasoning can be used by simply altering the value for a in Eq. 2 and thereafter transforming the values back to be compared on the same scale.Using graphical feedback and flexible procedures remains a challenging task in an elicitation process. In the seminal work by This study was carried out in accordance with the recommendations of the internal Ethics Committee of the Faculty of Social and Behavioural Sciences of Utrecht University, with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the internal Ethics Committee of the Faculty of Social and Behavioural Sciences of Utrecht University.DV and RvdS mainly contributed to the study design. All authors have been involved in the design of (part) of the elicitation procedure. DV programmed the elicitation software. All elicitations have been facilitated by DV and DS. DV wrote and revised the paper with feedback and input of DS, MZ-Z, and RvdS. RvdS supervised the project.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Furthermore, we successfully sorted intrinsically magnetic magnetotactic bacteria with magnetic moments of 0.1 fAm2. In distinction to column-based magnetic sorting devices, microfluidic systems can prevent sample contact with superparamagnetic material. This ensures contamination-free separation of naturally occurring or bioengineered magnetic cells and is essential for downstream characterization of their properties.Magnetic cell sorting provides a valuable complementary mechanism to fluorescent techniques, especially if its parameters can be fine-tuned. In addition, there has recently been growing interest in studying naturally occurring magnetic cells and genetic engineering of cells to render them magnetic in order to control molecular processes via magnetic fields. For such approaches, contamination-free magnetic separation is an essential capability. We here present a robust and tunable microfluidic sorting system in which magnetic gradients of up to 1700 T/m can be applied to cells flowing through a sorting channel by reversible magnetization of ferrofluids. Visual control of the sorting process allowed us to optimize sorting efficiencies for a large range of sizes and magnetic moments of cells. Using automated quantification based on imaging of fluorescent markers, we showed that macrophages containing phagocytosed magnetic nanoparticles, with cellular magnetic dipole moments on the order of 10 fAm In recent years, there has been an increasing interest in adding a level of precision by genetically altering cells such that they would become specifically responsive to magnetic fields8. There is furthermore a long-standing hypothesis that certain animals may be able to sense Earth’s magnetic field via magnetomechanical signal transduction mediated by magnetic biomineralizations attached to mechanosensitive cellular structures12, analogous to what is observed in magnetotactic bacteria14. It is thus of interest to sort cells based on intrinsic magnetic properties in order to subject them to biophysical and chemical analyses for e.g. optimization of bioengineering applications.Targetable magnetic nanostructures have been valuable tools for studying biomechanical properties of cellular structures by exerting localized forces via magnetic field gradients15 have become a popular tool in biomedical research. However, this design introduces a risk for contamination with superparamagnetic material with which the samples come in close contact. In the case of cells dissociated from primary tissue, the large and not tunable magnetic fields can also cause magnetization and subsequent enrichment of undesired iron-rich cells, such as macrophages16. Furthermore, the tight spaces in the mesh, necessary to ensure short distances to the magnetized superparamagnetic material, can cause a considerable amount of unspecific mechanical entrapment of cells.For these downstream analyses it is crucial to avoid contamination with metallic materials during the enrichment step. It is furthermore desirable to have visual control of the sorting process and tunable separation parameters such that cells with a large range of sizes and magnetic moments can be sorted. Commercial magnetic sorting systems, in which a mesh of superparamagnetic particles creates strong magnetic field gradients in the presence of a permanent magnet18. In addition, these systems can be designed such that they prevent contact of the sample with magnetic materials in the sorting device. The general separation principle is based on transversal deflection of target cells from a longitudinal fluid flow. This is achieved through application of a magnetic field gradient transversal to the flow direction of the cells, which exerts a force on magnetized cells in the sorting channel  in a side channel can enhance the magnetic field gradient. The resulting magnetic force on the cells was 3.3 times greater than with a magnet alone23. We modified this principle, using a suspension of superparamagnetic nanoparticles (ferrofluid), which can be reversibly magnetized and retain phase stability27.Precise control of the magnetic field gradient can be achieved by introducing electromagnets in the systemM) of a suspension of superparamagnetic particles  in the presence of a magnetic field (H) is given by the Langevin function,H is the magnetizing field, n is the number of nanoparticles per unit volume in the ferrofluid, µ is the magnetic moment of each nanoparticle, µ0 is the permeability of free space, kB is the Boltzmann’s constant and T is the absolute temperature. For dense ferrofluids  the particles tend to align and form chains in the presence of the magnetic field, resulting in a higher initial susceptibility. This is due to a significant contribution to the magnetization from interparticle correlations, which can be taken into consideration by introducing into equation He) adding the contribution from the surrounding particles28. This effective magnetic field is given byML is the Langevin magnetization and H is the applied magnetizing field  in the calculation of the magnetization curves of the ferrofluid. Although a saturation magnetization of 60 Am2/kg was reported for 10 nm magnetite particles31, yielding a lower magnetic moment per particle, our calculation result was in good agreement with the reported saturation magnetization of the ferrofluid used in the macrophage experiments  similar to that of magnetite (480 kA/m). However, the non-Brownian nature of such particles (due to their larger size) would result in a smaller effect of interparticle interactions on the initial susceptibility32. Other metals with higher saturation magnetization such as iron (1711 kA/m) could be considered as material for a ferrofluid. On the other hand, lower magnetic core sizes than in the case of magnetite would be necessary for metallic iron to avoid phase separation so that the effect of high saturation magnetization at the material level would not translate into higher magnetic dipole moments of the magnetic cores27.Such an increase of the magnetic force is similar to what was obtained in previous implementations of particle-based microfluidic systems using microspheres of nickelηrv) on a sphere in laminar flow following Stokes’ law, the crossing velocity (v) and thus also the time needed for a cell to travel half of the channel width can be calculated. The passage time of a cell through the channel, and thereby the interaction time with the magnetic force, is determined by the dimensions of the channel and the flow rates. By setting the crossing time equal to the passage time, the minimum magnetic moment of the sorted cells is given bym is the magnetic moment of the cell, r is the radius of the cell, η is the viscosity of the buffer, h and l are the height and length of the sorting channel, respectively, f is the flow rate and , injected at 1 µl/min  in the presence of a 300 T/m gradient .In order to evaluate the performance of the particle-based microfluidic magnetic separation system, we first tested macrophages which were incubated with magnetite nanoparticles and labeled with a fluorescent nuclear marker for improved visualization Fig. . The macThe flow of cells was observed at the center of the sorting channel Fig.  and the 16. The analysis could further be extended to a quantitative comparison of two or more populations of magnetic cells simultaneously by using different fluorescent markers, similar to how we characterized the cell behavior in the absence and presence of external magnets . These bacteria are intrinsically magnetic due to their intracellular chain of magnetite nanocrystals (40 nm diameter), which however possess low magnetic moments in the range of 0.1·10−15 Am2 35. This is just above the theoretical limit for overcoming the thermal fluctuations (kBT) when interacting with the weak geomagnetic field (~50 µT). We tested the bacteria in a modified sorting chip that had reduced dimensions to decrease the perpendicular travel distance for the small cells . Specifically, as shown in the histograms in Fig. Next, in order to evaluate the performance of the microfluidic magnetic separation system also for naturally occurring magnetic cells, we tested magnetotactic bacteria (e.g. higher cell density in the medium), deviation from such assumption could be kept efficiently under control with small changes to the parameters of the analysis.Visual control of the sorting process in combination with the automated video analysis was useful in the evaluation of the system and could be employed in further optimization steps. We used a fluorescent stain for improved visualization of the cells with much better contrast than available through brightfield microscopy. The approach of reducing each frame into a vertical line through a mean intensity projection along the x-axis was chosen for two reasons. First, with this approach we could easily assess the width of each object passing in the channel, thus allowing us to filter out high intensity narrow signals which are caused by cell debris or free magnetic particles in the solution. Second, when cells were flowing outside the focal plane their signal appeared weaker. These cells could be recognized by the width of the signal. Thus, our analysis allowed us to more accurately detect the number of cells by choosing the appropriate thresholds for both width and intensity of the signal  ferritin is nominally an antiferromagnet with only a small fraction of uncompensated spins at the surface of the core. This results in a much weaker magnetization of 0.1 Am2/kg in a 1 T field40. In this context, the microfluidic magnetic sorting system could be valuable in a directed evolution approach7, where tunable enrichment of cells with a desired range of magnetic moments is important.We demonstrated that separation of intrinsically magnetic cells is feasible with a simple and robust microfluidic system. Despite the limitation of low cell throughput (1000 cells/min), such a system would offer reliable and tunable separation of magnetic cell populations without the risk of introducing contaminants compromising downstream analysis. Magnetic separation of red blood cells based on their intrinsic magnetic susceptibility was first demonstrated by Melville In summary, we here evaluated microfluidic magnetic sorting for contamination-free enrichment of cells with intrinsic magnetic properties. A ferrofluid in combination with permanent magnets could generate a strong magnetic field intensity gradient for a 10-fold increase in the separation efficiency of magnetic cells, compared to separation with a magnet alone. The system allowed for sorting of cells spanning a large range in size and magnetic moment, as demonstrated for macrophages with internalized magnetite particles as well as magnetotactic bacteria expressing magnetosome chains. Visual control and automated quantification were valuable for evaluating sorting efficiencies and fine tuning of the system parameters to achieve customized cell sorting.Microfluidic chips, compatible with microscopy, were produced by Mircronit Microtechnologies (Netherlands). The thickness of the glass above the channels was 1 mm, determining the minimum distance between the ferrofluid and external magnets. For the device used for macrophages, the dimensions of the channels were determined by the size of the mammalian cells. Thus the height of both channels was 50 µm while the width of the sorting channel was limited by production methods to minimum twice the height, and was thus 110 µm. The side channel was 150 µm wide and was filled with a ferrofluid containing 17% Vol magnetite particles with a saturation magnetization of 99 mT . In order to move the sample even closer to the ferrofluid , reduced dimensions were used for the experiment with magnetotactic bacteria. In this case, the height of the channels was 30 µm, the width of the sorting channel was 65 µm, the side channel was 100 µm wide, while all other dimensions remained the same , which was generating a magnetizing field H perpendicular to the flow direction in the channels . The largest possible dimensions of the microfluidic channels (for ferrofluid and sample) were used and these were separated by 10 µm. The channels were placed 1 mm away from a sintered Nd2. The macrophages were grown to confluence on 10 cm petri dishes and were incubated for 6 hours with 10 µl magnetite nanoparticles (nano-screenMAG/R Biotin 100 nm (30 nm magnetite core), Chemicell, Germany). The cells were washed twice in phosphate buffered saline (PBS) buffer to remove excess particles not phagocytosed by the cells. Macrophages were then removed from the plate using Accutase , spun down at 0.4 relative centrifugal force (rcf) for 3 min, and resuspended in 3 ml PBS. Cells were labeled with a fluorescent nuclear stain  for visualization and filtered through a cell strainer with mesh size 70 µm to avoid clogging of the microfluidic channels. The sample was diluted to ca. 1·106 cells/ml in PBS containing 20% glycerol to avoid settling of cells in the syringe.Cells of the murine macrophage cell line Ana-1 were grown in RPMI 1640 medium  containing 10% FBS and 1% PS, incubated at 37 °C with 5% COThe microfluidic device was placed on an inverted microscope  using a chip holder . The center of the sorting channel was imaged with fluorescence microscopy and recorded with a camera (Ximea MQ003MG-CM) at 20 fps and an exposure time of 10 ms. The cells were filled in a 1 ml syringe  and injected using a syringe pump , while the buffer (PBS) was injected using a 2 ml syringe , resulting in a constant 1:4.7 ratio of the sample to buffer flow rate. The sample flow rate was adjusted to the range of 0.5–2 µl/min. Assessment of the separation efficiency in the absence of ferrofluid was performed using the same sample preparation and experimental setup with a sample flow rate of 1 µl/min. Establishment of laminar flow conditions was visually confirmed by observation of two separate flows (sample and buffer) and documented for each experiment through video recording. Three permanent magnets  were placed on top of the microfluidic chip, approximately 1 mm away from the ferrofluid channel and videos were recorded. The performance of the system was evaluated for multiple placements of the external magnets to control for variability in position. For evaluation of the system and developing the quantification algorithm, we analyzed videos containing in total ~1000 cells passing the observation point, ~500 in the presence and absence of the magnet.Magnetospirillum gryphiswaldense (MSR-1) were grown following standard protocols41. The bacteria were immobilized by exposure to 75 °C for 15 minutes to avoid active swimming in the microfluidic channel, diluted in PBS buffer containing 10% glycerol and fluorescently labeled with DAPI . For this experiment a chip with reduced dimensions was used   and compared the resulting distribution histograms. First, a mean intensity projection across the x-axis was computed for each single frame to obtain an intensity profile that is a function only of the vertical position (y-axis) (Supplementary Fig. The datasets generated and analyzed during the current study, as well as the MATLAB code for the quantification software, are available from the corresponding author on request.Supplementary Information