Over the last few years, magnetic resonance image‐guided radiotherapy systems have been introduced into the clinic, allowing for daily online plan adaption. While quality assurance (QA) is similar to conventional radiotherapy systems, there is a need to introduce or modify measurement techniques. As yet, there is no consensus guidance on the QA equipment and test requirements for such systems. Therefore, this report provides an overview of QA equipment and techniques for mechanical, dosimetric, and imaging performance of such systems and recommendation of the QA procedures, particularly for a 1.5T MR‐linac device. An overview of the system design and considerations for QA measurements, particularly the effect of the machine geometry and magnetic field on the radiation beam measurements is given. The effect of the magnetic field on measurement equipment and methods is reviewed to provide a foundation for interpreting measurement results and devising appropriate methods. And lastly, a consensus overview of recommended QA, appropriate methods, and tolerances is provided based on conventional QA protocols. The aim of this consensus work was to provide a foundation for QA protocols, comparative studies of system performance, and for future development of QA protocols and measurement methods. Measurements conducted during commissioningWhile many of the comments and recommendations will be applicable to other configurations of MR‐linac devices, the authors are working with Elekta Unity systems, and thus the tests are designed around this system.The recommendations within this report are intended to give an overview of suitable tests and methods and do not replace the manufacturer instructions for use or other labeling provided with the device. Individual institutions should assess their QA needs based on their own regulations and risk analysis as appropriate.1.12O2S:Tb) scintillator‐based amorphous silicon megavoltage (MV) imaging panel is integrated into the system enabling machine setup, calibration, and quality assurance procedures. The imaging panel, located at 265.3 cm from the source has a pixel size of 0.04 cm and a field of view of 21.0 cm × 8.5 cm, limited by the cryostat gap separating the main MR coils that generate the static magnetic field (B0).Elekta Unity  shown in Fig. ® and Elekta Monaco®, a Monte Carlo‐based treatment planning system that optimizes and calculates the dose distribution in the presence of a magnetic field.The system is used in combination with Elekta MOSAIQ1.2® . Second, nonstandard source axis distances result in different beam characteristics, such as profile shape and certain penetrative quality measures, for example, dose maximum, than would be obtained at the conventional 100 cm SAD. Lastly, the beam lines of MR‐linacs may be different from conventional machines. On Elekta Unity, the beam passes through the MR cryostat, gradient coil support structure, quadrature body coil (QBC), anterior and posterior receive coils, and the patient support system. The cryostat is specifically designed to have a low attenuating, homogenous region between the superconducting coils while maximizing the B‐field homogeneity to enable diagnostic quality MR images. In this homogenous region, small variations in material thicknesses (0.01 cm) and helium level result in small variations in attenuation of the radiation beam with gantry rotation. Manufacturing tolerances of the cryostat body and QBC result in a change in beam output at isocenter for a fixed dose at the monitor chamber. The helium level also affects the attenuation. To optimize imaging performance, the superconducting magnet was designed as one system (non‐split magnet) and thus helium is present in a small annulus connecting the cryostat in the longitudinal axis through which the beam passes. The system incorporates a zero boil off MR system and as such the helium level is fixed. It is only expected to change on exceptional events, for example, a cryostat cooler fault or some service operations, including ramp up and down of the magnet. The attenuation of the liquid helium is <0.9% and thus on a change in helium level, the output of the system may change with gantry rotation when the Helium annulus contains both gaseous and liquid helium. This occurs at helium fill levels of less than 93%. During installation, the system is characterized and the attenuation of the components and helium in the beam line is determined as a function of gantry angle. This information is then used in the treatment planning system to account for the attenuation. In the case where the helium level changes after initial system characterization, the cumulative worst‐case effect for a treatment plan with multiple beam angles is less than 0.5%.Both the geometry and environment of MR‐linac systems such as Elekta Unity are different to conventional radiotherapy devices. First, systems are bore type machines where the linear accelerator rotates around the MRI system. As a result, the radiation collimator is not visible in the treatment room. Unlike conventional systems, no light field or side lasers are present. This requires alternative methods of equipment alignment at the MV isocenter, for example, remote isocenter lasers, positioning of equipment using table indexing Fig. , or use Given the above, QA tests should take the cryostat effect into account by use of a gantry‐dependent output baseline or via comparison to the treatment planning system output.1.33) and superimposed on this is the gradient fields, which vary during MRI. In terms of dosimetry, the magnetic field can be considered uniform and the gradient field strengths neglected as these are on the order of 10’s of mT/m. Dosimetry measurements during MRI show no discernible difference to those conducted in a static magnetic field.Due to the presence of the magnetic field, the dose distribution characteristics are different to conventional radiotherapy systems. Note that MR systems have two main magnetic field generation systems. The main static magnetic field (B0), which is always present and highly uniform  but in the presence of the B‐field the dose distribution will be shifted.Symmetry — Due to the profile shift the symmetry of the beam along the IECX axis, which is evaluated around the beam central axis, can be approximately 1% higher (for a 22 cm × 30 cm field at a depth of 10 cm) than if the magnetic field was not present. Measurement of symmetry about the field edge midpoint would yield a symmetry comparable to the case if the magnetic field was not present.Asymmetric penumbra — The geometric size of the penumbra on the opposing field edges perpendicular to the B‐field can vary by 1 mm for a 10 cm × 10 cm field.In addition to the above other effects can be seen at interfaces of different densities. If the electrons exit a higher density material into a lower density material, they can “return” to the higher density material, which increases the dose at the interface Fig. . This efIn all the above cases, the TPS takes the magnetic field into account during plan optimization and dose calculation.22.1Measurement equipment used on conventional RT systems must be assessed for compatibility with an MR‐linac. MR compatibility falls into three categories:MR safe: These devices pose no known hazard in an MR environment.MR conditional: These items pose no known issues in a specified magnetic field under specified conditions of use.MR unsafe: Devices that are known to pose a hazard within the MR environmentIn many cases, existing QA vendors sell specific MR/RT equipment with the intended environmental conditions of use noted. Users should however ensure their suitability for the field strength of their system and ensure that “similar” equipment, not compatible with an MR‐linac device is not used on the system. TableThe main considerations for use of equipment on an MR‐linac are discussed in sections 2.2 to 2.4.2.2Devices may contain ferrous material that could make the device a projectile hazard — this would make it “MR unsafe.” Appropriate testing should be conducted by the user or certification should be sought from the QA device manufacturer as some ferrous content may not pose a hazard if this forms a small proportion of the overall mass of the device. Equipment packaging also needs consideration given these may not be MR safe even if the device is. In addition, “MR conditional” devices may require special considerations. This might include slow positioning of the device into the bore due to eddy currents or limitations on where and when the device should be used. In the latter case, an object placed in the bore may distort the image due to its susceptibility properties and may cause heating. These effects can be minor in plastics or can cause severe imaging artifacts in the case of metallic equipment. For electronic devices that contain metal, use of the device during MRI will either lead to image artifacts or to erroneous measurement due to electrical noise induced in the measurement device by the MR system. Some devices may be safely used for radiation measurements in a 1.5T field but may be damaged if MR imaged due to the induced currents in the device from the gradient coils or RF pulses from the system. The use of a passive detector, for example, gafchromic film2.3Detectors may not operate or may have their characteristics modified by the presence of a high magnetic field. In terms of active (electronic) QA devices, modifications may be required to allow them to operate correctly. In addition, radiation detectors contain materials of different densities so their measurement can be affected in the presence of a magnetic field. Specifically, the following effects can occur and need to be considered when choosing the method and detector type:Dose response of ionization chamber — It is well documented that the response of typical ionization chambers depends on the B‐field strength and the orientation of the chamber to the B‐field and photon beam. Correction factors need to be determined per detector type, orientation, and magnetic field strength.Response of ionization chamber in plastic water — Air gaps between the ionization chamber and water equivalent plastic cause under or over response. This effect can be controlled by the addition of water (thus removing the air gap) around the chamber but modifications must be made to current methods and the practicality of the technique assessed.cav) for PTW 30013 Farmer chamber to 1.01 mm for a PTW Diode P for a 10 cm x 10 cm field, indicating that large ionization chambers overmeasure the lateral offset and diodes undermeasure without any corrections applied. Thus, in the case of a beam from a gantry angle of 0 degrees, the Farmer chamber would require an offset of −0.61 mm in the IECX axis. Variations of the lateral EPOM with depth were less than 0.2 mm for a 10 cm x 10 cm field. Note that the lateral offset occurs in the direction perpendicular to the beam and magnetic field. No shift is present in the direction parallel to the magnetic field. The vertical EPOM for ionization chambers is also affected, halving for the larger ionization chambers such as PTW 30013 (B0T = 2 mm (~0.5Rcav), B1.5T = 1 mm(~0.4Rcav)) and reducing to almost zero for the PTW Pinpoint 3D. Given the authors' findings, an assessment of the effect on the chosen detector should be made by the user prior to their use.Additional dose shift — Depending on the ionization chamber volume, the dose distribution measured by the device might be shifted compared to the actual dose distribution, effectively resulting in a different effective point of measurement (EPOM). The EPOM shifts both laterally  and vertically. The vertical movement of the EPOM is surmised to be due to electrons moving in a more lateral direction compared to the case without a B‐field and the fact that the electrons do not directly enter the detector from the top due to their modified trajectory.Profile anomalies — In the case of closely packed ion chambers within an array, a difference in average density around a chamber can cause a variable chamber response.Intentional modification of dose response of a detector — The use of a high‐density material above a detector such as film has been exploited to remove the B‐field effect on the dose distribution. One such example is to sandwich film between copper plates which has the effect of nullifying the dose shift. This technique works as the higher density copper sheets result in a reduction in electron path length which in turn results in a reduction in the effect of the Lorentz force on the electrons. Effectively this results in dose being deposited more locally to the photon interaction and without the asymmetric dose shift due to the Lorentz force. This technique has been used to measure profiles and star shot films allowing for common QA techniques to be used from conventional radiotherapy systems with only minor modifications to the methods.2.4The Elekta Unity system does not require the use of conventional three axis lasers due to daily adaptive planning. Clinically a single sagittal laser is used to check patient yaw rotation relative to the table. Additionally, indexed patient positioning devices are used to approximately align the patient. Subsequent MRI and plan adaptation align the beam to the patient geometry of the day.As most electronic QA equipment cannot be MR imaged, it is not possible to use the same approach as for a patient. On Elekta Unity, the provided solution is a table‐indexable QA Alignment Platform which is used in conjunction with the on‐gantry MV imager and a phantom containing radio‐opaque markers for calibration and alignment checks. The QA platform in Fig. During machine setup, the location of the MV imager relative to the megavoltage isocenter is determined. This procedure, provided by the device manufacturer, determines the source detector distance (and hence isocenter pixel size), panel yaw rotation, and the offset of the center of the panel‐sensitive area relative to the beam central axis. Note that the offset is determined as a function of gantry angle to correct for any gravity‐related changes in panel‐source geometry. This enables the position of phantoms/platforms to be determined and for the position of system components, for example, MLC position, to be measured relative to the reference MV coordinate system.3Compared to a state‐of‐the‐art CBCT image‐guided radiotherapy system, machine QA for an MR‐linac device can be summarized as:QA requirements of a Linac, modified as required due to the machine geometry or magnetic field.Imaging QA of the CBCT system is replaced by comparable MR diagnostic QA with additional checks on geometric distortion.CBCT to MV coordinate system alignment tests are replaced by comparable tests for MR to MV.These recommendations draw on the previous QA protocols such as AAPM TG‐142,The frequency of the tests in this section is based on the expert opinion of clinical users and forms the basis for QA on MR‐linac systems. Across the authors institutions, daily/weekly QA (machine and MR) ranges from 45 minutes to 1 hour per day on average. Monthly QA ranges from 3 to 4.5 hours noting that some sites spread the QA over the month rather than conducting in one session. As with any new technology or treatment technique introduction, a cautious approach should be considered. As longitudinal QA results are published the frequency of the QA tests are expected to reduce leading to reduction in the cost and resource required. At the authors institutions, this is an ongoing assessment based on their experience with the system and their growing QA results database. Automation of QA tests, especially utilizing the MV imager benefit from being cost‐effective and quick to perform. As with conventional machine QA, the frequency of the tests and the action levels are to be assessed by individual institutions based on recommendations, such as this report, experience, and the clinical use of their machine. QA baselines are expected to be acquired at first commissioning of the machine, when the nominal beam parameters are changed, when the measurement equipment is calibrated or if a drift occurs between a “gold standard” reference and the measurement device. QA programs should be designed such that test equipment is regularly compared against other devices/methods. The guidance provided in the following sections highlights the use of different equipment at the various frequency levels, not only for allowing more in‐depth testing but also to allow comparison between measurement devices and techniques.3.1Table Specific recommendations on the methods are described below with a consensus view on the optimal method and frequency. Note that while some of these tests and methods are applicable to other MR‐linac systems, these tests are described for Elekta Unity where some solutions are integrated into the system, for instance, the MV imager. Furthermore, note that other systems may require additional tests, for example, lasers and lateral table checks. In all cases, limits are based on AAPM TG‐142 or manufacturers’ guidelines. Note that the user shall assess these tests in line with their clinical practice and adjust the limits as appropriate for the technique being used. Note that the QA acceptance level is not to be interpreted as the device performance specifications. Radiotherapy systems have a tiered level of performance. Typical performance is the performance measured on a sample of machines and the device performance is that performance specification declared by the manufacturer either in the product data or performance specifications, for example, IEC60976/977. Lastly, the QA acceptance level defines when actions need to be taken by the user. These acceptance levels may be set below the device performance specifications and may be chosen by the user based on the clinical use and age of the machine. As an example, the manufacturers' device specification for dose output through the week is <1% (IEC 60976/977) whereas the QA acceptance level is 3% for daily and 2% for weekly. The latter difference allowing for different test method precision.3.23.2.1The rigid gantry‐mounted MV imaging device can be easily used for a daily output check.Weekly or monthly absolute dose checks in the presence of magnetic field require consideration of the experimental setup and the application of specific magnetic field correction factors. Calibration of the monitor chamber should be made in water to avoid air gaps and it is recommended to calibrate 1 MU to 1 cGy at user calibration depth, at a gantry angle of 90 degrees to ensure liquid Helium is always present in the beam path. Weekly or monthly absolute checks should follow the same setup, although the use of plastic water can be considered for constancy checks. Note that the users of solid water must ensure air gaps are removed around the chamber  and measurements made in water must guard against air bubbles.20,10TPR) as the energy specifier as this is less affected by the magnetic field and is practically easier to measure than similar parameters in TG‐51.In addition, publications recommend the use of the tissue phantom ratio 20, 10 (3.2.2For high‐frequency tests (weekly), the use of an integrated MV imager for profile deviation checks over the visible area (21.0 cm x 8.5 cm) can be used. Given that daily output checks make use of the MV imager, the beam profile shape constancy can be measured daily but must be measured at least monthly. Note that in the case of the MV imager tests, the profile measurements are relative. Profiles are measured across the images acquired and compared to a baseline image. Given that MV imagers have a flood calibration, knowledge of the actual profile shape is lost. However, for QA, the relative change from the baseline image is important. Appropriate specifications, taking into account that profile test parameters will be dependent on field size should be determined if utilizing a test field smaller than the largest.For less frequent testing (monthly or greater), covering the clinically used field size area, tests can be conducted with gafchromic film or the use of a multi‐axis linear array placed in the bore of the magnet.Tests at multiple gantry angles are less impactful for the MR‐linac due to a vertically mounted waveguide and thus no bending or steering magnets are present that may cause deviations with gantry rotation. Thus, measurements at cardinal angles on an annual basis are appropriate. Annual tests are expected to be conducted with film, multi‐axis ion chambers, or with 3D water tanks.3.2.320,10, can be measured using either a 1D water tank or water equivalent plastic, noting the additional measures required to ensure air gaps are removed. The energy measurement should be consistent with that used in the national protocol used for absolute dosimetry. More frequent energy checks can utilize the beam profile shape as a surrogate.Beam quality, such as the TPR3.2.4The output constancy with gantry angle and the field size output factors dependency (incl. off axis) should be checked. Both should be compared to the cryostat characterization and output factors present in the TPS beam model. Measurements can be collected with a detector in a buildup cap (for gantry angle dependency), with use of a 3D/1D water tank or water equivalent plastic for the field size output factors which can acquired at a fixed gantry angle.3.2.5Measurements should use the same technique at the time of TPS commissioning and compare to the baseline taken during the commissioning phase. For the Elekta Unity system, treatment planning data collection is carried out using a 3D scanning water tank with PTW microdiamond and 3D Semiflex detectors. Use of alternative techniques for the annual QA (film or linear/2D detector arrays) are envisaged as per conventional radiotherapy devices as familiarity with the MR‐linac technology increases.3.33.3.1Frequent checks of the leaves and diaphragm positions for accuracy and repeatability can be made quantitatively using the integrated MV imaging device or qualitatively using film. The weekly test may cover a reduced area and may be qualitative only, for example, using a picket fence test. Monthly tests should consider whether quantitative tests over a larger area and with gantry angle changes should be included.Semi‐automated routines using the on‐gantry MV imager provide the quickest check of the beam limiting device performance. It is recommended a series of rectangular fields with varying central axis offsets is irradiated to check the position of the leaves within the available field area (21.0 cm × 8.5 cm). An example test field is shown in Fig. For frequencies of greater than a month and at least annually, a wider field area or changes in gantry angle should be considered. Methods including 2D arrays (subject to appropriate resolution) or film (with or without copper plates) in the bore of the machine can be used. Film tests, either using a picket fence consisting of abutting fields or a series of rectangular fields with varying offsets as in the MV imager test can be used. Note that dependent on the method, the baseline of the expected collimator positions may vary. As described previously, the B‐field results in the field edges perpendicular to the B‐field and beam direction to be shifted. Hence, the expected leaf position will be different to that displayed by the machine. Use of a technique to remove the effect, for example, copper plates, use of device not within the magnetic field, for example, on gantry MV imager, or comparison to the treatment planning system dose distribution should be considered.For repeatability measurements, a series of repeat exposures of the same field with the collimators approaching from different directions will verify the mechanical stability of the collimator system. The use of Linac log files can also be considered to map the performance changes, taking into account that the logs are not independent of the machine.3.3.2Differences between the machine reading and measured gantry angle using a spirit level on the gantry should be recorded to stay within specifications. Alternatively the use of a phantom that can be imaged using the MV detector can be considered.3.3.3Table accuracy and repeatability can be verified by attaching radioopaque markers to a table index location and using the MV detector to measure their position relative to the MV panel reference pixel or previous images. Movement of the table to different locations within the imager field of view and repeated extraction to table position 0 and repositioning to isocenter will allow assessment of the accuracy and repeatability of the table. Unlike conventional kV‐IGRT, table moves based on the fusion of the daily MR image to the planning reference image are not supported on Elekta Unity. This is because of the ability of the system to move the beam rather than the table to setup the patient in the correct location.3.3.4The radiation isocenter can be measured using either a Winston‐LutzThe frequency of the radiation isocenter test is considered monthly because of the use of SBRT and additionally because the Unity systems include a unique beam line service system for which the isocenter size is advised to be checked after servicing. This system allows the movement of the whole beam line from the gantry structure using a swing in swing out (SISO) winch system for easy servicing and hence it is advised to check the radiation source position after the beam line is restored in addition to the other tests recommended by the manufacturer.3.43.4.1The sections below detail the suggested minimum QA for the MRI device. The reader is also referred to Tijssen et al.The following checks should be completed: element‐wise signal to noise ratio for the anterior and posterior coils, scaling test of the transverse and coronal planes, coils/accessory condition checks, and call bulb function check. Standard sequences Table , phantom3.4.2These tests include flood field uniformity, spatial linearity, slice profile, and spatial resolution. The criteria for QA passing depend on the sequence used. The tests, imaging sequences, and pass criteria for the Elekta Unity, Philips periodic image quality tests (PIQT) are given in Table 3.4.3The MR periodic image quality checks (PIQT), MR scaling tests, and MR to MV procedure test the small field MR geometric accuracy. In the monthly test, a phantom that measures the large field MR geometric fidelity is used. This phantom, shown in Fig. 3.4.4Analogous to the kV/MV alignment on mainstream IGRT systems, this test measures the alignment (transform) between the MR and MV coordinate systems. The MV coordinate system axes are defined by the projection of the leaf sides, the IECY axis origin by the longitudinal midpoint of the field defining x‐diaphragms, and the IECX origin as the average projection of the radiation central axis with gantry rotation. The MR system imaging center is determined by the gradient fields. Fig. The purpose of the QA test is to verify that this transform has not changed due to either a MR or MV coordinate system shift or MR/MV geometric distortion. In the case of Elekta Unity, the transform is measured on machine install and applied in the treatment planning software. Specifically, all MR images are maintained within the MR coordinate system and an offset to the planning isocenter is applied prior to optimization and dose calculation. Note that only translations are applied. Rotations are not taken into account and thus limits are placed on the allowable rotation between coordinate systems.To determine the MR to MV transform, a phantom with markers visible in both MR and MV is used. The Elekta Unity system is provided with a phantom and software to QA the transform. Fig. The test frequency is noted as weekly, consistent with current manufacturer recommendations. QA data from the systemTreatment centers have also developed other phantoms that can be imaged by the two modalities3.5Table For the MV imager this is a critical piece of equipment as many of the QA tests utilize this for efficient QA. QA recommendations are made for this device below, noting that whenever it is being used for measurements should a QA failure be detected then alternative devices should be used to determine if the machine or the measurement device is in error. MV imager‐based devices have been shown to have a very stable response over time (>6 months) but may require periodic recalibration if artifacts become visible in the images.Miscellaneous equipment includes ferromagnetic sensors and oxygen sensors which form part of the ancillary equipment for the treatment room. The manufacturers’ guidance should be followed for the methods and frequency for those devices.3.5.1An appropriate MV phantom, for example, Las Vegas, should be used to assess the contrast, uniformity, noise, and spatial resolution of the MV imager. Evaluation of the image with comparison to a reference data set to observe any failing sections of the panel.3.5.2An image of a phantom of known size and position, for example, Las Vegas, should be used to measure its dimensions against a baseline.3.5.3The location of the isocenter projection on the panel is recommended to be checked monthly to ensure that QA measurements do not provide false negative results. The stability of the panel position can be checked by imaging fixed components of the machine, for instance, by imaging the cryostat cross over pipe and cryostat coil gap. Measurement and trending of the component edges or features of these fixed items will give an indication of imaging panel movement.4This report has provided an overview of the QA equipment and techniques required for measurements on MR‐linac systems particularly with a focus on the Elekta Unity system. QA on other MR‐linac systems will be comparable but will depend on machine design. A literature overview of relevant research has been reviewed and a consensus overview of typical QA measurements has been presented, in line with conventional Linac guidelines. Furthermore, an introduction to the tests and methods required on MR‐linac systems has been provided. The presented work describes a framework for comparative studies of system performance, enabling future development of QA protocols and measurement methods.David Roberts and Carlos Sandin are employees of Elekta Limited. Panu Vesanen is an employee of Philips Healthcare. All the other authors are from institutions that are part of Elekta’s MR‐linac consortiumData sharing not applicable to this article as no datasets were generated or analyzed during the current study.


In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity. In  BedrossiThese results were recently extended to the case of compactly supported Gevrey regular perturbations to Taylor–Couette flow in .do not capture the full echo chains  and hence do not rule out subsequent asymptotic stability. Furthermore, that work imposes a logarithmic smallness restriction  inviscid damping then follows as a corollary. However, this is a strictly stronger condition than the physically observed phenomenon of inviscid damping, which is the convergence of the velocity field. Indeed, we show that the linearized problem exhibits solutions which exhibit norm inflation due to complete echo chains. Here, the velocity asymptotically converges despite the divergence of the vorticity as time tends to infinity.secondary linear mechanism, where we linearize around an arbitrarily small low-frequency perturbation around Couette flow. More precisely, we show that in arbitrarily small neighborhoods of Couette flow  there are traveling wave-like solutions of the form c small. However, we think that an extension to more general low-frequency perturbations, while technically tedious, should follow by similar arguments. We show that this (simplified) linearized problem exhibits the same echo chains and norm-inflation results as for the nonlinear problem, globally in time. Thus, echo chains are a linear mechanism. Furthermore, we identify a critical regularity threshold at which stability of the vorticity in Gevrey regularity fails, but where damping of the velocity nevertheless persists.In the study of nonlinear inviscid damping, the main challenge is given by cascades of resonances, also known as echo chains. This mechanism is, however, not present in the linearized problem around shear flows due to the decoupling structure in Fourier space of these equations. In this article, we identify echo chains as a In order to introduce our model consider the Euler equations near Couette flow  linearized problem for a perturbation, which, with slight abuse of notation, we again denote by traveling wave-like solution. We remark that The main source of possible growth and instability in the nonlinear problem is thus given by the interaction of the mentclass2pt{minimecho chains as the full nonlinear Euler equations near Couette flow , butdamping, that is asymptotic convergence of the velocity field, holds, while asymptotic stability in Gevrey regularity fails.(Summary) Let tions of  with \doC is  optimal in the sense that there exists a (larger) constant u(t) converges in Furthermore, the constant As we discuss in the ensuing subsection, a key challenge in this result is that the (nonlinear) growth mechanism of echo chains stops after a finite time for any given chain. Hence, in order to obtain non-trivial asymptotic behavior like blow-up it is necessary to work with countably many echo chains of increasing length. In particular, one has to allow for arbitrarily long chains, which had been ruled out by logarithmic smallness constraints in prior works. A key effort of this article in Sections inviscid damping in analogy to Landau damping in plasma physics. Following this result, there has been much renewed interest in inviscid damping, both linear and nonlinear, also around other flows. In particular, it turns out that the linearized problem  is stable already in relatively low Sobolev regularity and that, unless one assumes compact support of perturbations, geometries with boundary do not allow for (linear) stability in high Sobolev regularity.In their seminal work Bedrossian and Masmoudi  establisfluid echoes . Here, as a first easy, non-optimal estimate, we may roughly control the multiplier byX be as in Proposition X as Let k dependence. The result then follows by Gronwall’s lemma. By Proposition plier in  by \docuWe stress that these estimates are far from optimal. However, they allow us to control the evolution until a small positive time Indeed, in Section c is very small compared to In order to introduce ideas, in the following Section Theorem sumption  as in [3k for In order to develop a heuristic for the growth mechanism, formally consider the equation with udied in , 5, the terature , 13 we sDue to the difficulties involved in controlling a nonlinear evolution around growing perturbations,  establisequation  instead c, i.e. the size of the low-frequency part. Furthermore, as we discuss in Section c is then comparable to multiplication by a constant.The more explicit structure of the associated Duhamel iteration scheme allows us to establish more precise upper and lower bounds and in particular quantify the dependence on the parameter Identifying the sharp growth behavior and removing this constraint allows us to construct families of initial data, for which the velocity field converges while the vorticity diverges  arbitrarily large We stress that for any fixed (Resonance chain) Let }ω(τ) of  with\docWe remark that such a result also yields bounds for frequency-localized initial data globally in time by using Theorem We note that in equation  modes onpathk is a finite sequence integral associated to length of Let integral  the inteWe remark that equation  introduci. Indeed, if l.Given a time interval k. We call resonant if non-resonant else. A path non-resonant if i.Let k and l. This agrees with the absolute value if Let ution of  with \doLet ution of  with \do(Single resonance bound) Let k. Here, we first note that if a path is non-resonant, we may estimate its contribution by k and i).We argue by a Duhamel-iteration approach and summing over all paths starting in Ad 1 Let k and denote i and let l starting from c and bounded one of the integrals by In order to bound Repeating this argument for all c and thus Expanding the product in the numerator and looking at each summand separately we need to consider monomials in the numerator. If a factor Ad 2 We proceed similarly as in case 1, but note thatl and ending in We remark that by using the linearity of the problem this further yields a convolution-type bound for entclass1pt{minimaUsing the single-resonance results of Proposition We use local well-posedness to control the evolution from time l is maximal with this property and prescribe We next choose entclass1pt{minimaWe then iterate this control another By our construction of the coordinate mentclass2pt{minimWe proceed in multiple steps: We note that it suffices to establish upper and lower bounds on mentclass2pt{minimLet thus entclass1pt{minimaentclass1pt{minimaWe may thus repeat our application of Proposition While this result is quite useful and shows the growth mechanism, the logarithmic boundWe remark that also , 5 contac, in the following we consider an abridged model which considers the evolution for l and its neighbor In order to better study the effect of resonances and, in particular, the behavior for large The present model is very similar to the one considered in , except udied in  both anaAs we will see in the following this yields important cancellation properties and further exposes a connection of this problem to the 1D Schrödinger problem with scaling critical potential.two-mode system:As a simplification of calculations, we use that In order to introduce ideas, we first consider the case where , v(0). While the solution follows immediately by computation of the matrix exponential, for later reference we note an alternative proof arguing at the level of second derivatives. Formally differentiating the equation for u is given by a stationary Schrödinger equation with potential Having discussed the case of small v instead is a degenerate elliptic problem. However, we note that . Then oWe prescribe initial data nd solve  to obtaiWe then solve  to obtaiFinally, we solve  on \docuCombining these three maps, we show in Proposition In the following Section l to obtain sharp Gevrey 2-type norm inflation results for this model problem.In order to understand the mapping properties of Theorem We remark that  is the sLet We observe that a general solution of  is givenOn the exterior intervals we similarly obtain an explicit solution, but with very different dependence on Let We note that a general solution of  is givenLet ution of  and 24)\documentCombining Lemma We thus see that, while the evolution on the small interval Having establish a precise description of the evolution for times close to a single resonant time In Section In what follows we instead consider an iterated version of the model of Section c by c/2 for brevity. This further highlights resonant times, where v  does converge in Consider the chained two mode model with In particular, choosing For any given Let now Then by construction, the Building on the insights obtained in this model problem in the next Section we consider the full problem.In the previous section we have shown that for large ll model 32\documeConsidering thatnd hence  reads asWe approximated In Section a(k) in the following Section three-mode model. In contrast to the two-mode model of Section In view of the sizes of We remark that in Section In this section we introduce the solution operator of the homogeneous three-mode model:By symmetry it holds that We first establish a rough power law upper bound on the growth of solutions on Subsequently, we iteratively improve this bound to We then show that the resonant mode is decreasing as Combining these results, we construct solution operators on the intervals Concatenating the solution operators we obtain the solution operator from time The main results of this section are summarized in the following theorem:We recall that, by the results of Section  Let E system  for \docSuppose that at time u satisfies the assumptions of this theorem with We note, in particular, that the exponent here is different from 1 (which was not visible in prior works due to the logarithmic constraints) and that at time  Let nd solve  with\docin solve  with newWe remark that the precise choice of Let Our proof proceeds by considering the three intervals As a first step in our proof of Theorem  Let u(t) satisfiesWe consider the problem  on the iIf in addition We may write the homogeneous three-mode model  as42\doc(Reduction 1) Consider the problem , then itWe observe thatequation . \documeIn order to solve  we firsttions of . In Lemmt| becomes small.As a first heuristic, note that on In order to establish these asymptotics we make use of a self-improving estimate. That is we will assume for the moment that there exists some (Improvement) Consider the problem  on the ier bound .c), such thatThen there exists a constant Furthermore, if We note that A toy model In order to introduce our method of proof let us first consider the approximated problem where we replace a(t) by exactly sumption it holds In particular, we observe thatent over  if \docuent with  with thic is small, With this improved upper bound at hand, we now can establish the lower bounds . For thiProof in the actual model This proof is analogous with the only difference being that the solution matrix is now not anymore given by power laws but rather by explicit hypergeometric/Legendre functions, which are bounded above and below by such power laws.t, which has the general solutionP is the Legendre function of the first kind and Q is the Legendre function of the second kind . We here write t where https://dlmf.nist.gov/14.8)We note that the problemunctions  https://https://dlmf.nist.gov/14.10)The corresponding values of S(t) are comparable to the power laws obtained above and M vanishes. Given these homogeneous solutions, we may thus again make the ansatz thatGiven the solution operator of problem , we retu(Particular Solution) Consider the problem  with iniS(t) of the exact problem . Here, we introduceSimilarly to the proof of Lemma S(t) satisfies the power law bounds discussed in the proof of Lemma We hence consider the ansatzThus, it follows thatFinally, we return to the proof of the a priori bound by a power law .C and c but are uniform in Consider the problem  on the iAnalogously, for the interval Our proof is based on concatenating estimates on a large number of very small intervals, where we can use perturbative arguments.S(t) is the solution operator to the problem with b replaced by Thus, consider an interval We further recall that equation  and withPlugging in this ansatz, we obtain thatu(t)| also only grows by a constant factor bounded by 100 on the interval We may hence controlIn order to study the evolution of  on the m In the homogeneous case on the middle interval In particular, at time An explicit solution can be given in terms of hypergeometric functions, as we explore in the “Appendix It remains to discuss the evolution on the interval  Let  problem  on the iWe note that This proof is largely analogous to the one of Proposition c is sufficiently small.As a first heuristic let us again consider the approximated two-mode model, whereb(t) is replaced by It remains to discuss the extension to the full problem. We note that under the transformation S asS. As we know that We may then explicitly compute the inverse of In the following section we use multiple bootstrap arguments to show that this behavior persists in the full problem.diverge in (any) Sobolev regularity but their force field nevertheless converges. Damping persists despite the failure of scattering. The estimates for a single resonance in the full problem are summarized in the following theorem:Building on the results for the three-mode model of Section u be the solution of the full model (k andu at time (Summary) Let ll model :\documenAs in Section As in Section p ansatz  to 63)..\documenu be as in Theorem  Let By linearity without loss of generality we may set Then at least for small positive time it holds thatIn what follows we argue by bootstrap that the maximal time satisfying these estimates is given by T and that hence T could be chosen larger by continuity, which contradicts the maximality.Indeed, suppose there were nditions  to 62)  \documenAd  of the full model is comparable, and hence the exact same proof immediately applies:Ad  and 60)Ad (and (stimates  and 61)Ad  Let We here use the Duhamel iteration, which we phrase as properties of the solution map for single mode initial data. Arbitrary initial data can then be realized as a linear combination. We emphasize that on this short time interval the action u satisfies where Suppose that at time Ad (i) Let Ad (ii) Let us first consider the special paths Ad (iii) Let u satisfiesa and b, i.e. Let Let k from Next consider a path with at least one resonance. We group all paths that share the segment from first to last resonance and observe that at least Using the result of the preceding lemmas, we can now prove Proposition By linearity we may decompose our initial data mentclasspt{minimau(k) for Concerning the contributions by mentclasspt{minimak) to construct chains of resonance and associated solutions exhibiting norm inflation. Furthermore, we combine countably many chains to construct solutions with non-trivial asymptotic behavior.It remains to control the evolution of the full equation on the right interval Similarly to Section u be as in Theorem  Let We again make a bootstrap ansatz similar to the one in Proposition  Ad  and 65)65): We mWe first compute f and integrating it follows thatIt hence remains again to study the perturbations due to the inhomogeneity. As in the proof of Proposition r bounds  and 67)\documentThis concludes the proof of Theorem not compactly supported in Fourier space.In the preceding sections we have shown that the linearized Euler equations  exhibit diverges to infinity as time tends to infinity but despite this the associated velocity perturbation converges. Linear inviscid damping persists despite the failure of scattering and nearby traveling wave-like solutions are the underlying cause of Gevrey blow-up! We stress that one of the main challenges we had to overcome in this work is that one has to be able to consider infinitely many echo chains of increasing length. In particular, one has to allow for arbitrarily large frequencies, while prior works had imposed a logarithmic smallness constraint.As our main result we thus show that the linearized Euler equations not only exhibit full echo chains and the associated Gevrey norm inflation but that for data in a critical Gevrey regularity class we obtain blow-up and damping at the same time. That is, we construct smooth initial data u(t) converges in (Modified Scattering) Let On the other hand, for every Furthermore, for We proceed similarly as in the case of Theorem entclass1pt{minimaLet now On the one hand the norm inflation mechanism of Section R as a frequency cut-off On the other hand resonant times are well-separated and for any given Any instability will thus have to sustain a sequence of infinitely many separate echo chains for a sequence of times tending to infinity to ensure that the flow is not asymptotically stable after all stabilizing effects, whose interaction makes this a very challenging problem:mentclasspt{minima Sobolev , 15 and  Sobolev  is fundaIn the present work we have, for simplicity of calculation and presentation, considered a single-mode perturbation


The artificial neural networks (ANNs) have been often used for thin-film thickness measurement, whose performance evaluations were only conducted at the level of simple comparisons with the existing analysis methods. However, it is not an easy and simple way to verify the reliability of an ANN based on international length standards. In this article, we propose for the first time a method by which to design and evaluate an ANN for determining the thickness of the thin film with international standards. The original achievements of this work are to choose parameters of the ANN reasonably and to evaluate the training instead of a simple comparison with conventional methods. To do this, ANNs were built in 12 different cases, and then trained using theoretical spectra. The experimental spectra of the certified reference materials (CRMs) used here served as the validation data of each trained ANN, with the output then compared with a certified value. When both values agree with each other within an expanded uncertainty of the CRMs, the ANN is considered to be reliable. We expect that the proposed method can be useful for evaluating the reliability of ANN in the future. Therefore, the ANN can provide optimized solutions for any type of data. With such advantages, ANNs have been widely used in numerous fields, such as gaming, medicine, image processing, and voice recognition2. One of the main characteristics of an ANN is that it can extract meaningful data from complex information that contains noise. Moreover, unlike traditional algorithms that require initial values and multiple iterations, an ANN algorithm provides output in a very short time when trained once in advance.An artificial neural network (ANN) is a well-known machine learning algorithm used to execute unspecified operations to achieve a certain goal without the operator’s intervention. The ANN is a method that comprehensively utilizes a variety of information by imitating the synapse structure of neurons making up the nervous system14. A typical example is to determine the thickness of a thin film using the spectral reflectance spectrum. The reflectance spectrum includes not only information related to the thickness of the thin film but also various effects such as the bandwidth and distribution of the light source, the optical properties of the optical components in use, the uniformity of the material, the linearity of the detectors, environmental fluctuations and external noise. It is practically impossible to extract information of only the thin film thickness from the reflectance spectrum obtained experimentally, which increases measurement uncertainty. In this respect, given the advantages of an ANN, several studies have been reported to measure the thickness of transparent thin films or multiple thin films with a broad spectral light source and X-rays19. However, in all of these earlier works, the performance evaluation was limited to the level of a simple comparison with the results of existing analysis methods or other measurement techniques. As a result, it was concluded that an ANN works well if the difference between the results from the comparisons is relatively small. Even if the difference is small enough, it doesn’t mean that the ANN algorithm works accurately because both analysis methods can be incorrect. Therefore, because of insufficient verification on the effectiveness of the ANN algorithms, the results obtained with an ANN algorithm should only be used as initial values in a model-based analysis to reduce the difference with less iteration18. Despite of the practical difficulty regarding analysis reliability, no methods beyond the simple comparison have been proposed and adopted for verifying the ANN algorithms. Besides thickness measurement applications, several studies of ANN algorithms have been also reported in real-life applications related to weather forecasting, yield improvement, and so on24. Similarly, the performance of the models used in these studies were evaluated in terms of various types of errors like root mean square error between model output and label. Unfortunately, label values based on actual measurements were also basically lack of measurement reliability. The original idea of this work is to exploit the traceability chain of the length standard of SI unit for evaluating the ANN algorithm for the first time. The reasonable determination of the ANN parameters and the reliable evaluation of the ANN algorithm can be achieved based on this concept, not using simple comparison with conventional methods.Metrology throughout semiconductor and display manufacturing processes involves significant amounts of complex dataIn this article, an ANN algorithm for thin film thickness measurements was designed and verified using four certified reference materials (CRMs). Certification was done using a standard instrument based on spectral ellipsometry at the Korea Research Institute of Standards and Science. The ultimate goals of this study are (1) reasonable selection of the parameters of ANN algorithms and (2) a performance evaluation based on an international standard instead of a simple comparison with current analysis methods or measurement techniques. To accomplish this, ANNs for the thin film thickness analysis were built in 12 conditions with different combinations of the number of hidden layers  and the number of nodes in each layer . These 12 ANN algorithms were trained in a thickness range of 1–110 nm. The training spectra were created by a numerical simulation based on a multiple interference within the thin film. To evaluate the reliability of each ANN algorithm, the measured spectra of CRMs with nominal thicknesses of 10 nm (CRM-10), 30 nm (CRM-30), 50 nm (CRM-50), and 100 nm (CRM-100) were used as the validation data sets. When a trained ANN algorithm provides outputs that are in good agreement with the certified values from the CRMs, it is considered to function properly. To the best of our knowledge, this study is the first to evaluate ANN algorithms intended to measure thin film thickness value based on the traceability chain of a length standard. It is very difficult to conclude that the simple comparison methods in previous works provide measurement reliability applicable to metrology.14. The lights reflected from the top and bottom surfaces interfere with each other, as shown in Fig. d) as well as the Fresnel reflection coefficients at the interfaces (r12 and r23), the complex refractive index of medium (k) and the refracted angle (θ) according to the Fresnel equation14.For thin film thickness measurements, the spectral reflectometer has been widely employed due to high measurement speed and simple configurationWhen the reflectance spectrum at an unknown thickness is obtained experimentally, the thickness cannot be easily determined by analyzing the spectrum itself. In such a case, each measured spectrum is compared individually with many spectra generated by Eq. . The ref19. A multilayer perceptron (MLP) type ANN algorithm was constructed and trained using Python, similarly to the previous work19. In the wavelength range of the spectrometer to be used for the CRM measurement, a wavelength range in which the intensity of the measured light is sufficiently greater than noise was selected, and the number of samples for that range was established as the number of input nodes. Therefore, reflectance spectra are received from 881 input nodes and thin film thickness analysis values are output from one output node through a hidden layer. A sigmoid function was applied as the activation function like the previous works19, and the loss was calculated according to the average of the mean squared error between the outputs and the ideal value as determined in the simulation. After the error estimation is completed, a basic backpropagation algorithm is utilized to update the weights connecting each layer. Batch gradient descent was used with a learning rate of 0.000001. For weight initialization, Xavier initialization was utilized. Because the purpose of this study is not to improve the performance of ANN algorithms, of which basic form is only exploited without any additional techniques among other advanced algorithms. In our work, in the thickness range of 1–110 nm, 110 reflectance spectra were numerically generated with equal steps of 1 nm based on a mathematical model with the Fresnel equation, as expressed by Eq.  thin film on a thick silicon substrate. Table 14. The Deuterium lamp used as a light source emits white light in a wide wavelength range of 112–900 nm. The light was normally incident with regard to the thin film surface with a beam diameter of 2.6 mm in the form of plane wave created using a collimating lens, as shown in the spectral reflectometry part in Fig. Figure δ) in cases were 1.2 nm for 100 nodes with one hidden layer (N = 100 with L = 1) and 0.54 nm for 150 nodes with one hidden layer (N = 150 with L = 1) and 0.81 nm for 100 nodes with two hidden layers (N = 100 with L = 2) and 0.74 nm for 150 nodes with two hidden layers (N = 150 with L = 2). Thus, we finally selected the ANN algorithm with the lowest value of the averaged offset, which in this case was 150 nodes with one hidden layer (N = 150 with L = 1). Unexpectedly, the ANN algorithms with more hidden layers, i.e., L = 3, and more nodes, i.e., N = 200, did not always result in better reliability. In the authors’ view, the results showed that the ANN algorithms with more nodes and hidden layers were over-trained by only the training data set of ideally generated reflectance spectra, which may result in overfitting. Hence, the proposed method can be beneficial for evaluating accuracy of the ANN algorithms with the help of a traceability chain of a length standard.For clarity, Table 18. This allows the iteration steps of the model-based algorithm to be reduced. In such case, the output of the model-based algorithm was used as a reference value or as a true value regardless of its reliability. The smallest deviation between two outputs cannot always mean that the ANN algorithm is designed well and works properly.More importantly, in some previous works, when the difference between the ANN algorithm and the model-based algorithm was found to be relatively large, the output of the ANN algorithm was used as the initial value of the model-based algorithm to reduce the differenceIn this article, a novel method to design and evaluate an ANN algorithm used to determine the thickness of thin films was proposed and demonstrated. As a reference value, a CRM certified value directly traceable to a length standard was utilized. Twelve ANN algorithms with different conditions  were developed in-house and then trained using 110 numerically created reflectance spectra in a thickness range of 1–110 nm. With reflectance spectra of the 4 different CRMs obtained by experiments, the thickness values were determined by 12 well-trained ANN algorithms and then compared with the corresponding certified values of the CRMs. As a result, based on a traceability chain to the length standard, each ANN algorithm was evaluated. Finally, in this work, an ANN algorithm with 150 nodes with one hidden layer was chosen as the best case with an average offset of 0.54 nm, as derived from the differences between the outputs and the certified values of the CRMs. The practical applications of this study can be limited to only cases providing certified values for the present. It is expected that the proposed approach will be beneficial for those involved in developing and verifying machine-learning algorithms for rigorous metrology. In the future, for completeness of this study in a metrological view, uncertainty evaluations need to be performed according to Guide to the Expression of Uncertainty in Measurement.


This paper introduces an interactive framework to guide decision-makers in a multi-criteria supplier selection process. State-of-the-art multi-criteria methods for supplier selection elicit the decision-maker’s preferences among the criteria by processing pre-collected data from different stakeholders. We propose a different approach where the preferences are elicited through an active learning loop. At each step, the framework optimally solves a combinatorial problem multiple times with different weights assigned to the objectives. Afterwards, a pair of solutions among those computed is selected using a particular query selection strategy, and the decision-maker expresses a preference between them. These two steps are repeated until a specific stopping criterion is satisfied. We also introduce two novel fast query selection strategies, and we compare them with a myopically optimal query selection strategy. Computational experiments on a large set of randomly generated instances are used to examine the performance of our query selection strategies, showing a better computation time and similar performance in terms of the number of queries taken to achieve convergence. Our experimental results also show the usability of the framework for real-world problems with respect to the execution time and the number of loops needed to achieve convergence. Supplier selection is the process of determining the best suppliers for acquiring the necessary materials for the production activities of a firm. This is a key aspect of Operations Management (OM) for a firm of any size. Although decision-makers (DMs) still proceed manually in some contexts, many automated methods and tools have been adopted to solve the problem. The main benefits of using these instruments include the reduction of the decision process time and the capability to take into account complex aspects arising when the business grows. These frameworks do not merely select the least expensive suppliers. They can also consider multiple criteria  to sharpen the company’s competitiveness. The task of quantifying the relative importance of such criteria in a specific decision process can be tricky, partly because it involves a variety of factors and business goals. This is usually achieved by conducting long interviews with multiple experts and stakeholders.Determining the weights by eliciting the DM’s preferences on the criteria;Solving the problem given some fixed weights of the criteria.Recent works on multi-criteria supplier selection follow this general structure. This eases the development of hybrid approaches based on a pair of techniques, one for each of the two tasks. The first task is generally covered by Multicriteria Decision Making (MCDM)  methods, such as the Analytic Hierarchy Process (AHP), the Analytic Network Process (ANP) or fuzzy-based extensions taking into account incomplete data  technique where the learning algorithm is allowed to choose the data from which it learns (Settles ). We adodiscrepancy.We also considered a second research question related to the query selection process of our active learning loop. Our framework evaluated the quality of a solution by considering the max regret of the utility function with respect to compatible preference models. A related myopically optimal query selection strategy in terms of the value of information is the setwise minimax regret criterion  in Kaur and Singh . A stochExact approaches based on mathematical programming have a tendency not to be scalable when the problem modelled is NP-hard. This often happens in supplier selection, and the best supplier configuration is computed using means of metaheuristics approaches. Genetic algorithms for supplier selection have been, for example, designed by Taleizadeh et al.  and He eMany recent supplier selection approaches are based on hybridising two or more techniques. As an example, Shaw et al.  hybridizcontent-based, collaborative filtering and knowledge-based , or if there is a purely qualitative  representation of the uncertainty. Bayesian approaches include, for example, that shown in the work by Chajewska et al. , BoutiliDifferent approaches have been explored to interact with the DM  weights define the DM’s preferences. Our goal was to define a procedure to find a suitable solution with a low cognitive effort for the DM. Instead of precisely computing the DM’s weights through elaborated interviews as in standard MCDM techniques, we adopted an active learning loop to reduce the uncertainty of the DM’s preferences by asking comparison queries until the max regret of a potential recommendation is below a fixed threshold.n is the number of criteria, so that there is a weight i. In our supplier selection framework, we consider four criteria, so n functions, This section presents some of the key notations used in this paper. Let A weighted vector is preferred toLet is a set of optimal solutions with respect to Let The main novelty of our supplier selection framework is the fact that the importance of each criterion is defined using a series of interactions with the user, with an interleaving of the elicitation and optimization. In this way, the user drives the solution process in order to reduce the uncertainty with respect to the DM’s trade-offs among the objectives. We define the combinatorial optimization problem ith function. The first three signs are negative because the first three functions have to be minimized, whereas The weighted sum used as the objective function of a priori, but more typically, the trade-offs among the objectives are harder to define. In general, the precise definition of the preference vector The aim of the learning loop described in Fig. Let us consider, as a query The following lines describe how the framework works in practice, referring to the block diagram in Fig. SelectRecommendedSolutionStopCriterionComputeQueryThe next step is to initialize the set of constraints SelectRecommendedSolutionComputeQueryStopCriterionAs shown in Fig. Let us consider a set of suppliers The parameters Note that the variables Our MILP model is based on the following integer decision variables:A feasible solution M”) constants and the other variables/parameters are defined previously. The objective function : select a query Max min discrepancy (MMD): select a query Max discrepancy sum (MDS): select a query Each of these methods can be used to implement ComputeQueryOnce we make O(n) linear programming problems (b)(c)Because of Proposition StopCriterionOur iterative procedure is possible to repeat until we find a necessarily optimal solution in The aim of this section was to assess the computational effectiveness of the framework by considering the three different preference elicitation strategies described in Sect. The computational experiments were performed on randomly generated instances that represent realistic scenarios as described in Sect. i, j), , dividing the result by the corresponding best result among the three methods in that row. For example, the RWB value for the SMMR query time for the first row is equal to 1.12/0.81. We then considered the mean of the values over all 48 rows. These values were recorded in the last row of the table.We needed a common measure to compare the rows of Table We solved the MILP problem using the extreme points We computed the value We set the weighted vector to The idea is to try to define an approximation of the range of each objective in order to re-scale a random vector with respect to the ranges of the objective functions.The 20 instances generated for each triple The bar chart in Fig. As we can see from Fig. Figures Generally speaking, the results show that the strategies SMMR and MMD look better than MDS in terms of the number of queries generated. A possible explanation is that the discrepancy sum computed in MDS, which drives the query generation process, can be high even if one of the two solutions in the selected pair It has been proven that the SMMR method generates the most informative query , and to select two solutions that are maximally different, i.e., that maximizes the mutual discrepancies with respect to the corresponding associate weighted vectors. From our experimental results, it seems that MMD performs better on average than the setwise minimax regret in terms of execution time. Furthermore, we also got a lower average number of queries to achieve convergence. MMD seems to be a good alternative to the setwise max regret, especially when the computational time to generate a query significantly affects the overall execution time.Some of the recent developments in MCDM the purpose of supplier selection are related to introducing fuzzy theory in order to manage data incompleteness/uncertainty with respect to the DM’s response. This feature is not currently included in the proposed framework. However, the possibility of allowing fuzzy answers to the queries may be an interesting future research direction. Furthermore, the type of queries included in the framework may be extended to allow the user to express preferences among a certain set of solutions. In this case, the queries are less intuitive but could lead to a reduction in the overall number of queries required.Future research may also involve an extension of the combinatorial problem to a stochastic case where aspects like the stochastic demands of the components are included in the problem definition. This can be easily achieved by replacing the MILP model considered in Sect. It would be interesting to extend our framework in a multi-agent context where the purpose is to find a common solution between two conflicting agents. In this case, the weighted vector corresponding to an optimal recommendation needs to consider the tradeoffs of the different DMs, which may be conflicting.Finally, another future research direction is related to adapting the framework to a case where the combinatorial problem is solved using heuristic algorithms with no optimality or quality guarantee. If we increase the size of the instances considered, the current MILP model would not scale well, and the high computing times would make the interaction with the DM impractical.


For reliability, the experiment (transwell migration assay) was performed again, and the description of the experimental method was revised. The corrected In the original article, there was a mistake in The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.5 cells in 100 μl serum-free medium) were added to the collagen pre-coated transwell insert , and the culture well was filled with 600 μl of the complete medium containing 10% FBS. After 18 h, cells were fixed with 70% ethanol for 10 min and stained with crystal violet solution. Then, non-migrated cells in the upper side of transwell insert were removed using a cotton swab. Three random fields of each insert were counted and photographed under a light microscope .Serum-starved B16F10 cells (1 × 10The 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.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Outdoor breathing trials with simulated avalanche snow are fundamental for the research of the gas exchange under avalanche snow, which supports the development of the international resuscitation guidelines. However, these studies have to face numerous problems, including unstable weather and variable snow properties. This pilot study examines a mineral material perlite as a potential snow model for studies of ventilation and gas exchange parameters. Thirteen male subjects underwent three breathing phases—into snow, wet perlite and dry perlite. The resulting trends of gas exchange parameters in all tested materials were similar and when there was a significant difference observed, the trends in the parameters for high density snow used in the study lay in between the trends in dry and wet perlite. These findings, together with its stability and accessibility year-round, make perlite a potential avalanche snow model material. Perlite seems suitable especially for simulation and preparation of breathing trials assessing gas exchange under avalanche snow, and potentially for testing of new avalanche safety equipment before their validation in real snow.The study was registered in ClinicalTrials.gov on January 22, 2018; the registration number is NCT03413878. The main research has been focused on the gas exchange occurring in the air contained within the snow6 and the role of an ‘air pocket’ in gas exchange7. This research is fundamental for development of the international resuscitation guidelines9.A number of outdoor breathing trials with healthy volunteers has been conducted in order to investigate factors affecting the survival of victims covered with avalanche snowUsually, breathing experiments with volunteers are organized in the mountain terrain with simulated avalanche snow. Even in this environment it is complicated to include all factors affecting the survival . The design and preparation of these experiments are very demanding also due to a limited accessibility of snow and due to unpredictable weather conditions.10. However, the snow properties differ among the published studies. The average avalanche snow density ranges from 200 kg m−3 (dry snow) to 550 kg m−3 (wet or dry)11. In their studies, Brugger et al. used snow with median density of 376 kg m−3 (range 144–546 kg m−3)1, Roubik et al. used snow of a similar density with much smaller variance (380 ± 14 kg m−3)3. Strapazzon et al.4 even conducted three series of experiments, time separated, with the median snow density of 364 kg m−3 (range 155–481 kg m−3).It is a well-accepted presumption that the physical properties of snow can contribute to the rate of avalanche victim survival12. Some of the limitations can be overcome in a special laboratory capable of simulating various aspects of the mountain environment13.The field environment poses additional challenges to the experiments. For example, standard medical devices used outdoors can be a source of erroneous readings of the physiological parametersAnother approach could be finding a stable material that can simulate avalanche snow in a laboratory environment. This snow modeling material could facilitate research focused on an isolated factor of gas exchange of avalanche buried victims and aid testing of protocols for breathing experiments before their implementation in real snow..14 in order to select a material which could serve as a possible model of avalanche snow for studies of ventilatory and gas exchange parameters. From three loose porous materials , perlite was found as the most suitable one to simulate avalanche snow because the resulting concentrations of breathing gasses copied concentrations measured during breathing into real snow. Furthermore, perlite has favorable properties, mainly the homogeneity and it is easy to manipulate with. By changing the water content in perlite, they were able to modify its physical properties and thus affect the gas exchange when simulating a burial in this material.A small prospective laboratory study was conducted by Roubik et al15, or buried in sand16.To our best knowledge, there has not been published other study dealing with breathing under different materials simulating the avalanche snow. There have been only reported cases of victims covered with rubble, mainly following an earthquakeThe aim of the present study was to find a suitable material as a surrogate of avalanche snow for breathing experiments aimed at studying ventilation physiology and factors affecting survival of victims covered with avalanche snow.The study protocol was approved by the Institutional Ethical and Review Board of the Faculty of Biomedical Engineering, Czech Technical University in Prague , and written informed consent was obtained from the volunteers before enrollment in the study. All methods were performed in accordance with the relevant guidelines and regulations. The study was registered in the ClinicalTrials.gov register (NCT03413878) on January 22, 2018, before enrollment of the first volunteer.The prospective interventional randomized double-blind crossover pilot study was conducted in Spindleruv Mlyn, Krkonose Mountains, Czech Republic, at the altitude of 762 m above sea level on January 29, 2018 – February 1, 2018.19. All eligible volunteers were members of the Czech Armed Forces and students at the Military Department of the Faculty of Physical Education and Sport, Charles University in Prague.Thirteen male volunteers participated in the study, eleven were included in the final analysis. The number of participants was selected as a standard number of subjects included in published studies with a similar scope and design20. They underwent an initial examination performed by an experienced physician, including assessment of the past medical history, smoking history, physical examination and spirometry . The exclusion criteria were the Tiffeneau Index less than 0.70, any acute respiratory infection, and a history of cardiovascular or respiratory disease.All study subjects were fit and well, classified according to the American Society of Anesthesiologists as ASA IEach volunteer underwent three phases of the experiment: phase “S”—breathing into snow, phase “PD”—breathing into dry perlite, and phase “PW”—breathing into wet perlite , non-invasive blood pressure , and peripheral blood oxygen saturation (SpO2). Using a respiratory sensor D-Lite , connected between the mouthpiece and the tubing leading to the cone , tidal volume (VT), airway pressure (Paw) and airflow (Qaw) measured at the airway opening. The spirometry sensor D-Lite also provided a gas sampling port for the Datex-Ohmeda S/5 monitor measuring inspiratory and end-tidal fractions of oxygen , carbon dioxide , and nitrous oxide , using an E-CAiOVX  anesthesia and spirometry module. The analyzer was calibrated before each experimental day using a standard calibration gas. Apart from the monitor Datex-Ohmeda S/5, there was another vital sign monitor, CareScape B650 , used as a backup SpO2 and HR monitor of the subjects.Continuous monitoring of all study subjects was performed throughout the whole experiment. Before initiation of each breathing experiment, the subject was attached to vital sign monitors. Datex-Ohmeda S/5  served as a primary monitor, which monitored and recorded the following physiological parameters: electrocardiography (ECG), heart rate . All the devices were placed in a tent equipped with an electric heater and the gas sampling lines were wrapped in a polyurethane foil for insulation and supplemented with a heated wire in order to prevent condensation and freezing of water in the tubing.The experiments were recorded on three independent camcorders; the first one continuously recorded the vital sign monitor screens, the second one recorded the study subject and the third one the overall situation on the site. The video and audio recording was intended as a backup of the data recorded by the laptop computer and written protocols. This enabled reviewing the situation on the site later during the data processing and evaluation. The investigators communicated via PMR (private mobile radio) strictly adhering to the protocol protecting the double-blind design of the study.21.The subject was connected to vital sign monitors and laid down in a prone position on an insulated mat in front of a custom-made apparatus described below. The prone position was chosen as it is the most common pose of buried avalanche victimsAt the initiation of each breathing phase, the subject had a nose clip properly placed and was attached to a mouthpiece so that he was able to breathe only through the specially designed breathing circuit. The mouthpiece was at this point connected only to the flow sensor, not to the cone yet, allowing measurement of respiratory gasses and baseline ventilation parameters. After reaching stable ventilation parameters, especially a steady tidal volume and breathing frequency (after approx. 5 min), the subject was connected to the cone and the breathing experiment was commenced. At the same time, nitrous oxide administration to the vicinity of the subject airways and the breathing circuit was initiated for potential leak detection and all the physiological and study parameters continued to be recorded and observed.Safety of the volunteers during the experiment was assured by advanced simultaneous monitoring and continuous evaluation of their physiological parameters by the attending physician with a specialty training in anesthesia and emergency medicine. Throughout the whole experiment, a clinical assessment of consciousness level of the volunteer was performed: the physician asked the subject to calculate simple mathematical operations and to show the result using his fingers which were not attached to the pulse oximeter probes. The physician could terminate the experiment at any time.EtCO2 of 62.5 mmHg), an accidental disconnection or a detection of the tracing gas (N2O) in the breathing circuit. The participant was disconnected from the test material and allowed to breathe ambient air through the mouthpiece with the respiratory sensor still attached and while the ventilatory and physiological parameters were still recorded. After further 2 min, when all parameters stabilized and returned close to the baseline, the subject was disconnected from the mouthpiece and the test phase was ceased.The criteria for termination of the breathing experiment were: a request of the subject, a supervising physician's command, reaching of the safety limit . The cone tip was connected to the rest of the breathing apparatus, which consisted of a dust filter, a 12 cm corrugated tube, a flow sensor with a gas sampling port, an antibacterial filter and a mouthpiece. There was also a water trap for draining off any excessive water.The apparatus, depicted in Fig. 2O) as a ‘tracing gas’. This technique has already been used during a similar study3. This easily detectable gas was administered via a 6 mm flexible tube into the vicinity of participant’s airways, breathing mouthpiece and along the breathing circuit at flow 20 L·min−1. The gas analyzer of the anesthetic monitor Datex-Ohmeda S/5 was sampling the gas from inside of the breathing circuit and thus was able to detect even a minimal concentration of nitrous oxide. The presence of the nitrous oxide in the system suggested either a leak in the tubing system, or a situation when the test subject inhaled gas not directly from the mouthpiece, but around the mouthpiece or through the nose, which was not properly sealed by the nose clip. The positive detection of N2O in the system was a reason to immediately stop the particular breathing phase.All parts of the breathing circuit were designed in an attempt to minimize the equipment dead space and resistance to breathing effort. The dead space of the whole breathing circuit was 220 mL and its airflow resistance was 475 Pa⋅s/L measured at the flow rate of 60 L/min according to EN ISO 8835–2. Owing to the fact that some components of standard anesthetic tubing were modified for this experiment, it posed a risk of leaks of the gas from the system. As it is a standard practice to check for leaks of breathing systems in anesthesiology and intensive care, also in this experiment a system for leak detection was used, utilizing nitrous oxide . It is a non-toxic material, perlite dust is in most countries listed as a “nuisance dust”22. However, as a special precaution, an extra High Efficiency Particulate Air (HEPA) filter  was inserted between the cone filled with the tested material and the rest of the breathing apparatus  full with the tested material and topping up with water and then the whole cylinder was weighted on the scales. Porosity was then calculated as:Filling of the cone with snow was performed gradually by adding and compacting small portions of homogeneous snow collected from a depth of 20–50 cm from the surface. The snow was identical throughout the depths. Both dry and wet perlite are non-clogging materials; thanks to this, they could be poured into the cone at once. The density of all materials was measured by weighing the whole cone filled with the tested material (54 L) before every experimental phase. The porosity of the snow was then derived from the knowledge of the ice density24 as wet snow without any impurities. Snow density was 542 ± 32 kg m−3, grain size was ‘very coarse’ (2–3 mm), snow hardness corresponded to ‘soft snow’ (4F), grain shape was classified as ‘melted forms’ (clustered rounded grains). The average ambient temperature during the breathing experiments varied from 0 to 3 °C. The minimum temperature during nights was − 5.4 °C. Atmospheric pressure was 91.8 ± 1.2 kPa (range from 90.5 to 93.0 kPa).Throughout the whole study, the snow temperature was between − 0.8 and 0 °C in the snow profile from the surface to the depth of 20 cm (not used for the experiments), and the temperature of snow was stable at 0 °C between 20 and 50 cm below the surface (this snow was used for the experiments). The snow used during the experiment was classified according to the International classification for seasonal snow on the groundN = 11) volunteers fulfilled the following criteria and were included in the final data analysis: sustained breathing during all three phases for at least 240 s and high-quality data recorded during all phases. Therefore, the number of participants included in the final statistical analysis was 11.The data from all 13 volunteers and tested materials were included in the analysis of the time to phase termination presented by the Kaplan–Meier plot. The Mantel–Haenszel test was used to statistically compare the cumulative termination rates between the individual phases . From the whole group of 13 volunteers, only 11  during S, PD and PW phases were calculated in 15 s intervals from the beginning of the experiment . The values were expressed as mean ± standard deviation and presented in graphs. The statistical significance of the difference was tested using the two-way ANOVA for repeated measures with Tukey HSD post-hoc tests  after the Shapiro–Wilk data normality test. P < 0.05 was considered as statistically significant.The statistical difference between the physiological parameters  in the breathing circuit in both S and PD phases and the breathing was also shorter than 240 s. There was no harm to any subject of the experiment.Eleven (85%) of thirteen volunteers were included in the data analysis. The characteristics of the subgroup of 11 analyzed volunteers are presented in the second data column in Table SpO2,  EtCO2 and  FiO2 of each of the volunteers involved in the study, color-coded for all three conditions , are presented in Fig. The individual curves of the physiological parameters SpO2 between the wet perlite and the dry perlite breathing phases appeared at 165 s and lasted until the end of the experiments, as depicted in Fig. SpO2 curve for the breathing phase into dry perlite overlaps the course of the SpO2 curve for snow. Moisturizing of perlite had a significant effect on the rate of desaturation development.The statistically significant difference in FiO2 were found between the dry perlite and the wet perlite breathing phases starting at 105 s, between the snow and the wet perlite at 195 s and between the snow and the dry perlite at 210 s. The statistically significant differences in EtO2 between the dry and the wet perlite occurred later, starting at 165 s.Inspiratory and end-tidal fractions of oxygen Fig.  were theFor inspiratory and end-tidal fractions of carbon dioxide, there was not found any difference between any of the tested materials in the whole study. In the graphs Fig. , all thrThe main finding of this study is that the characteristics of perlite in terms of gas exchange are similar to high density snow used in this study and furthermore, the trends of gas exchange parameters in snow lie in between the wet and dry perlite trends.FiCO2) and expiration (EtCO2) concentrations among tested materials. The trends of the curves representing EtCO2 during breathing into snow, wet and dry perlite overlapped each other and the same behavior was observed for FiCO2.Concerning the carbon dioxide, there were no differences in inspiration  and peripheral oxygen saturation (SpO2) during breathing experiments.Another advantage of perlite is the considerable capacity to hold water. The perlite water content can modify not only the properties of the material itself, but also its effect on respiratory gases  using N2O as a tracer gas. We believe that this care assured that the studied respiratory parameters are consistent within the studied groups  and show very small variances. This allowed to demonstrate the statistically significant differences in the properties of the studied materials (where they were present) even within a relatively short length of analyzed data.Another limitation is based on the published studies reporting survival time of real victims covered with avalanche snow. Approximately 10% of victims do not survive the first 15 min of the snow burialNext, the study was conducted in stable weather conditions with certain physical snow properties . Moreover, only a single type of snow was available at the site of the experiment.FiO2, EtO2 and SpO2, but the general trends of these parameters should not be affected significantly.Finally, the study was conducted at a low altitude site with higher oxygen partial pressure compared to common avalanche accident locations. This might affect the absolute values of 10.This study and the suggested avalanche snow model, i.e. perlite, is not intended for experiments modeling different snow properties. The snow properties themselves may have, according to recent research, an important effect on the survival time of avalanche victimsAnother obvious limitation of perlite is that there is no interaction between breathing air and snow. In real cases, the snow transforms when it comes into contact with the warm air exhaled by the buried subjects. This property of perlite represents an advantage for the reproducibility of the experiments, but it also represents a limitation of the perlite as a surrogate material of snow.The authors of the study see the possible application of this snow model material in preparation of breathing experiments, for testing of experimental protocols, technical equipment and specially designed breathing circuits preceding the field experiments in the mountain environment. It can also be used for studies on different population groups  and for initial testing of new avalanche safety equipment. Last but not least, it can help to train the subjects in laboratory environment for outdoor experiments if this training is a part of the protocol.This is the first study to document that it is possible to use a material which could simulate some of the properties of avalanche snow in laboratory conditions. The study suggests that perlite can be used as a surrogate of high density snow for studying gas exchange. Although these laboratory experiments cannot fully replace field studies in the real environment, they may be very beneficial in preparation and methodology validation of the outdoor experiments. Furthermore, due to the stability and homogeneity of perlite, the experiments yield consistent and repeatable results.Supplementary Information.