Current signal recording and processing systems have come a long way since their initial inception and use. There is, however, still ample scope for improvement, not only in the troubleshooting of their limitations, but also in the expansion of the boundaries in the recording of intracardiac signals. Here, we recount our experience with the use of the PURE EP™ signal recording system  in the animal laboratory. Developments in processing systems have strengthened the real-time display of information and allowed for the making of decisions on the go.10 Despite the technological advances, there have been notable limitations in the performance of current recording and mapping systems.12 Maximum procedural success has been achieved in the localization of the sources of focal arrhythmias, and in delineating re-entry circuits.15 Uncertainty in the “reality” of the signals obtained during various maneuvers, however, is still a challenge, and it is in this direction that most of the current efforts are being directed.16 In this paper, we describe the results of a series of experiments involving cardiac mapping and ablation in a canine model, comparing conventional electrophysiology (EP) signal processing equipment with an advanced signal processing platform called PURE EP™ .Electroanatomical mapping systems (EAMs) made their way into clinical practice in 1997.Figure 1A. In this architecture, various gain and filter settings are implemented in hardware. As a result, signals are recorded from a specific hardware configuration, and the original raw information generally cannot be recovered.The cardiac signal data-acquisition module of typical present-day EP recording and mapping systems is depicted in . In addition, there are means to provide defibrillator protection and radiofrequency (RF) noise suppression. In this architecture, there is no need for gain switching, and the full range of input signals (± 250 mV) is digitized with optimal resolution. Raw signals acquired by the acquisition module are filtered and processed in the accompanying software, using the digital processing module. In the real-time window, waveforms of interest can be displayed as raw signals or as any combination of raw and filtered signals to enable better visualization of signals in the presence of noise and artifacts. All displayed signals are time synchronized. On the review screen, the operator has the option of opening multiple review windows, with the ability to display the results of various signal-processing algorithms, independent of the real-time tracings.The PURE EP™  system has the ability to allow for the recording of raw  cardiac signals with multiple display options. This is achieved using a proprietary amplifier topology, which combines a very low-noise instrumentation amplifier  and a high-resolution 24-bit A/D converter ® SmartTouch® catheter , with a 3.5-mm tip and an interelectrode spacing of 1-6-2 mm.During all the experiments, we used standard electrophysiology catheters like the Blazer™ II (4-mm tip) or Blazer™ II XP (8-mm-tip) RF catheters , with 2.5-mm interelectrode spacing; or a ThermoCoolSaturation artifact. High gain settings in recording and mapping systems are designed to improve the visualization of very low-amplitude intracardiac signals. When presented with signals that have large amplitudes, the high gain settings may result in output signals larger than the supply voltage of the front-end amplifiers, causing saturation of the amplifiers. This shows up on the display as an artifact in the form of truncation of the signals. In addition, amplifier saturation is often followed by long recovery times before resumption of proper function, causing a further loss of signal recording during the intermediate aftermath of the saturation for a finite period. The only current solution to this is the reduction of system gain, which negatively affects the successful visualization of small signals.17Figure 2 shows an example of saturation adversely affecting displayed signals. The lead labeled “Pure EP Uni” shows a large unipolar signal where the entire signal has been captured by the PURE EP™ system . The lead labeled “Conv EP Uni” shows a truncated version of the same lead as captured by a conventional EP system with typical gain settings. The truncation is evident from the flat tops and bottoms of the signal. However, when filtering is applied (“Filtered Conv EP Uni” with 50-Hz low-pass filter plus power line notch), as is often done to remove power line and high-frequency noise, the truncation is not evident, and much of the true signal detail is also lost.In addition, the problem is magnified several-fold when pacing is attempted. Pacing signals are orders of magnitude larger than the EGMs, and are typically applied directly to the sensing electrodes, resulting in profound saturation. This, in turn, increases amplifier recovery time, compromising the ability to view or record signals for several seconds after the pacing artifact. Additionally, the pacing artifact may also be present on adjacent electrodes because of cross-talk, masking EGMs on those electrodes also. For maneuvers like entrainment and differential output pacing, this causes inaccuracies, since there is always obscuration of the local EGMs by the pacing artifact. 3B show examples of saturation. Pacing has overdriven both systems into saturation. In the first example, the conventional EP system takes almost 10 s to recover from saturation, while the PURE EP™ system , with its larger dynamic range, recovers in about 2 s.The clipping solution; or is it the clipping problem?. It is not uncommon for operators to clip signals to remove overlapping signals on the display screen. This involves clamping the display of signals above a set voltage limit to prevent them from obscuring small signals in adjacent leads. Since this is a display action only, it still allows large signals to be recorded without creating artifacts. This works well, as long as the signals are discrete and separated in time. However, if there are fused signals or signals of equal amplitude present near-simultaneously, there could be a smudged view of the entire complex signal owing to clipping.18 This may hide significant features of the signal and may, in extreme cases, change the way the signal is interpreted. These interactions occur randomly, and cannot be anticipated by the operator. One possible solution to this clipping problem is to decrease the gain of the system to minimize saturation effects at the expense of increased system noise. The PURE EP™ system  minimizes problems here by virtue of its very low hardware gain, high dynamic range, and good noise performance (1 μV root mean square over the full bandwidth). In the case of software clipping, the high bandwidth ensures visualization, and that any data below the clipping level are not compromised by saturation of the hardware.Noise due to ablation. Ablation involves the application of high-frequency electrical power from an RF ablation generator between the distal electrode of an ablation catheter and a reference patch (dispersive or return electrode) placed on the back of the patient. The electrical resistance to the passage of this current generates heat, which in turn causes localized necrosis by thermocoagulation. Ablation essentially creates an unbalanced energy input into the amplifier connected to the distal electrode of the ablation catheter, resulting in additive noise, baseline shift, and obscuration of the underlying signal data. Additional front-end filters designed to block signals at the ablation frequency are required. Specifically, it is important to use passive filters to reduce ablation noise before it enters the active circuitry to prevent nonlinear effects from propagating through the system.20Figure 4 shows an example of baseline drift at the onset of ablation. In many cases, the drift can push the signal outside the bounds of the display so that it cannot be viewed. It also shows a solution implemented by the PURE EP™ system  for extreme baseline wander via the opening of a new review window that “pins” each beat to the baseline so that the data can be viewed throughout the ablation procedure. The ultimate solution is improved front-end filtering so that acquired and displayed waveforms are visible and clean during the application of RF energy.Unipolar signals. Unipolar signals refer to signals recorded between an electrode placed in the tissue and a neutral reference electrode. In EP recording systems, the preferred reference is the Wilson central terminal (WCT), derived from the electrocardiogram (ECG) limb leads, since this does not require an extra catheter. The unipolar signals are distance-dependent but not direction dependent, making them valuable in mapping and ablating arrhythmias.23 Unfortunately, in conventional recording systems, unipolar signals referenced to WCT tend to be noisy. The low cut-off frequency for the highpass filter requires an expanded dynamic range to fully capture the recorded signals with baseline drift. Since the reference is not a dedicated one, the signal is susceptible to noise and surface ECG lead polarization artifact. Extra care thus needs to be taken to ensure the fidelity of the unipolar WCT signal by matching the path of the WCT and the active electrode, which is difficult to achieve in the EP laboratory environment. Other techniques are required to ensure the fidelity of the signal. Filtering of the power line frequency is generally necessary, and filtering of EP laboratory interference may also be required. The use of adaptive notch filters that do not add artifact to this unipolar signal offers the best solution to maintaining the fidelity and usefulness of the unipolar WCT signal.Figure 5 shows an example of a unipolar signal with fixed frequency noise labeled “Acquired EGM.” This noise can be removed by a conventional notch filter, but this leads to a loss of detail, the development of artifacts, and changes in the shape of the waveform features, as shown in the lead labeled “Notch Filter Applied.” The PURE EP™ system  uses a different approach with proprietary adaptive notch filtering that removes the noise without affecting the base waveform shape, as shown in the lower lead of the figure. This preserves the integrity of the WCT referenced lead.Contact force. Maintaininggood contact with tissue remains one of the greatest challenges in modern electrophysiology. It influences mapping and ablation and remains one of the most important factors in determining the success or failure of a procedure. Rightly so, there has been acceleration in innovation to assess contact with tissue.25 Current technology requires the use of relatively stiff catheters to transmit the force to the force sensor effectively. In addition, the angle of contact made by the distal electrode with the cardiac tissue and the effect of the intracardiac structures coming into contact with the catheter system may produce significant variability.26 The sampling rate of the contact force data and the stability of contact remain as parameters that affect the contact force readings. and 6B.However, with the expanded frequency and dynamic range of the PURE EP™ system , the current of injury (COI) caused by the catheter contact with tissue provides a good estimate of contact force without the need for stiff catheters. This reflects the local injury to the tissue caused by contact of the catheter. The degree of local injury is proportional to the amount of contact force applied. This COI is reversible, and correlates well with standard contact force-sensing catheters Poor catheter visualization. Withthe increase in the ablation of arrhythmias from the sinuses of Valsalve, there are significant concerns about injuring the conduction system and the coronary arteries at this location. It is imperative to be able to distinguish catheter position above and below the aortic valve in order to avoid injury to the coronaries. Currently, no reproducible way to do that exists.27 However, respiratory and cardiac motion, coupled with catheter artifact, contribute to poor images in this location.16 Imaging is also relatively insensitive to small changes in the position of the catheter tip during ablation. It is extremely important to be able to visualize the catheter tip in these locations, owing to the clustering of important structures in the vicinity.Regions at the base of the heart remain largely dependent on accurate positioning of an intracardiac echocardiography probe for their visualization.. While recording these signals, we confirmed the catheter position and orientation using intracardiac echocardiography and fluoroscopy. We also used contrast to confirm the position of the catheter within the coronary artery. We believe that the unipolar recordings made were suggestive of COI, and given appropriate filtering, frame rate, and sampling rate, they could be reproduced in any system.Using the PURE EP™ system , distinctive changes in unipolar signal EGM morphology were noted when ablating above the valve compared with ablating below the valve or within the coronary artery The PURE EP™ system  allows for the display of signals with more than one filter at the same time, a feature not found in current systems. This allows for the ability to look at signals filtered in multiple ways for specific reasons. Further, the finding that we may be able to make a distinction between ventricularized signals within the coronary artery vs. contact EGMs above and below the valve is novel. This is being further evaluated for consistency. The ability to predict the location of the catheter can be used as an extra checkpoint along with intracardiac echocardiography and fluoroscopy prior to ablation.Our hypothesis for the change in the EGMs is that, given that COI emanates from myocardial injury with contact, no COI changes were noted above the valve owing to the dominant muscle mass being present below the valve directing the resultant COI away from the point of contact. This causes a negative COI at this location. In contrast, below the valve and from within the coronary arteries, the COI originates from the local contact within or on the myocardium, giving rise to a typical positive COI from that location.Conduction tissue signal recording. Conduction tissue within the distal ventricles remains intimately associated with the muscle. Mapping these signals with a bipolar electrode is difficult because there is fusion and overlapping of signals originating from the muscle and the conduction system. Arrhythmias originating from the conduction system, therefore, are more difficult to map, since successful annotation of the smaller signal within the bigger myocardial signal needs considerable experience, and may not always be possible. The assessment of ablation lesions targeting any of these two components remains problematic.28 Large local sharp spikes produced during depolarization will cause the filter to “ring” at its resonant frequency and produce false positives. Additional processing is required to suppress these unwanted transient responses  and 8B.A distinguishing feature of signals from the conduction system is their high-frequency content. This difference allows for these signals to be selectively displayed by using a high-pass filter. In the PURE EP™ system , this can be done as a separately displayed item to show the presence of signals from the conduction system. However, one of the problems in using a simple linear filter is the transient response to impulse-like signals produced by the myocardium.Reliable marker for transmurality of lesion. We do not have any reliable measures of lesion transmurality. The use of surrogate measures like drop in impedance, changes in the COI, and change in EGM need to be integrated into a definitive variable for its determination. In addition, the effects of ablation upon the local unfiltered unipolar EGM, to help discern the effects of ablation in creating a transmural lesion, remains to be evaluated.EGM-based contact force estimation. There has been a correlation noted between the amount of contact force and the rise in the COI. However, changes in the COI have not been standardized for radius of the catheter tip, the difference between the COI in the atrium and ventricle, the effect of intracavitary structures on the COI, or the standardization of COI at the annular region.Discerning the underlying substrate based upon the recorded EGM. Currently, there are only approximations made to evaluate the underlying substrate when mapping cardiac tissue. There are no EGM characteristics that are able to recognize and differentiate between scar tissue, fat, or myocardium consistently in all circumstances. Using the better fidelity in bipolar and unipolar characteristics of the PURE EP™ system  we hope to be able to create areas of scar and map them to identify unique characteristics of local EGMs.Evaluate the performance of current automated mapping algorithms. The PURE EP™ system  can reproduce recorded cardiac signals with greater fidelity than existing commercial systems. Current algorithms are not very efficient in discerning between components of an EGM, especially in multicomponent recordings. With the PURE EP™ system , we plan to reevaluate the current automated algorithms for accuracy in the annotation of EGMs, and perhaps clarify the complex process of mapping and tagging points correctly.Application of the system in other areas of signal recording and processing. Signal recording and processing is an integral part of various mapping techniques in fields other than cardiology. We hope to apply the expanded hardware configurations of the system for mapping and recording signals in fields like neurology and ophthalmology, in order to better be able to characterize regional electrical activity.We thank Jennifer Mears for her help in preparing and proofreading this manuscript.


When jammers move rapidly or an antenna platform travels at high speed, interference signals may move out of the null width in the array beampattern. Consequently, the interference suppression performance can be significantly degraded. To solve this problem, both the null broadening technique and robust adaptive beamforming are considered in this paper. A novel null broadening beamforming method based on reconstruction of the interference-plus-noise covariance (INC) matrix is proposed, in order to broaden the null width and offset the motion of the interfering signals. In the moving case, a single interference signal can have multiple directions of arrival, which is equivalent to the existence of multiple interference sources. In the reconstruction of the INC matrix, several virtual interference sources are set up around each of the actual jammers, such that the nulls can be broadened. Based on the reconstructed INC and signal-plus-noise covariance (SNC) matrices, the steering vector of the desired signal can be obtained by solving a new convex optimization problem. Simulation results show that the proposed beamformer can effectively broaden the null width and deepen the null depth, and its performance in interference cancellation is robust against fast-moving jammers or array platform motion. Furthermore, the null depth can be controlled by adjusting the power parameters in the reconstruction process and, if the direction of interference motion is known, the virtual interference sources can be set to achieve better performance. As antenna arrays have been applied to radar, sonar, radio astronomy, medical imaging, and in other areas, beamforming is widely used for interference suppression, target detection, and direction of arrival (DOA) estimation ,2. The pTherefore, null broadening techniques can provide effective and robust approaches to overcoming this problem. In References ,12,13,14In this paper, we combine the null broadening technique with robust adaptive beamforming and propose a novel adaptive null broadening beamformer based on covariance matrix reconstruction. First, a new interference-plus-noise covariance (INC) matrix is reconstructed by placing several virtual interference sources around each of the actual jammers. Then, the signal-plus-noise covariance (SNC) matrix is reconstructed using the Capon spatial spectrum . Next, tThe main contributions of this paper can be summarized as follows:1) Compared with other existing null broadening beamforming techniques, we propose an algorithm based on INC matrix reconstruction by setting up several virtual interference sources around the actual jammers directly in the Capon spatial spectrum, which can simultaneously broaden the nulls and eliminate the self-null phenomenon of the desired signal. Furthermore, the null depth and width can be controlled by setting the parameters of the virtual interference sources.2) Based on the reconstructed INC and SNC matrices, we design a convex optimization problem to obtain the steering vector of the desired signal, in order to further improve the robustness of algorithm.The rest of this paper is organized as follows. In N omni-directional sensors receives M interfering signals come from directions d is the sensor interval, K is the number of snapshots. The output of the adaptive beamformer is expressed asAssume that a uniform linear array (ULA) comprised of However, in practice, the INC matrix cannot be obtained directly, and the sample covariance matrix high SNR .In this section, a new adaptive null broadening beamforming method is proposed. The main idea is to reconstruct a new INC matrix by utilizing virtual interference sources and estimate the steering vector of the desired signal by solving a new optimization problem. As the sample covariance matrix is usually inevitably mixed into the desired signal, which can cause the self-null phenomenon with a high signal-to-noise ratio (SNR) and result in significant performance deterioration, it is necessary to remove the desired signal component from the sample covariance matrix and reconstruct the INC matrix. On the other hand, to avoid convergence to the DOA of the interfering signal, the SNC matrix is also reconstructed to effectively improve the pointing accuracy of the estimated steering vector. Moreover, the method used to reconstruct the INC matrix is also used for SNC matrix reconstruction. Therefore, this section is divided into three parts: The first part discusses the INC matrix reconstruction, the second reconstructs the SNC matrix, and the last estimates the steering vector of the desired signal based on the INC and SNC matrices. Furthermore, the complete algorithmic flow is detailed in the third part.The Capon spatial spectrum  estimatoUsing the Capon spatial spectrum , an INC M regions of virtual interference sources are located in In the INC matrix reconstruction, the range of virtual interference sources can be determined according to the motion of the antenna platform or jammers. Assume that In order to accurately estimate the steering vector of the desired signal, the SNC matrix can be reconstructed by the Capon spectrum. In the same manner as , we can Due to many mismatches, such as gain and phase perturbations, incoherent local scattering, signal look direction error, array geometry errors, the steering vector of the desired signal is often inaccurate. To obtain the optimal weight vector, we introduce the convex theory and obtain the accurate steering vector by solving the optimization problem. In the presence of steering vector mismatches, the mismatch vector put SINR . TherefoTherefore, the optimization problem can be written asBy solving the optimization problem, the desired signal steering vector can be obtained. Set The first constraint can be expressed asAccording to –18), th, th18), With Karush–Kuhn–Tucker (KKT) conditions , the LagHence, we can obtainBy solving , we can  problem  can be e problem , 11) an anx and  problem , the weiAs shown in Step 1. Calculate the Capon spatial spectrum  and searStep 2. Reconstruct the new INC matrix  based onStep 3. Estimate the steering vector  problem ; andStep 4. Calculate the weight vector opt−null .In the proposed algorithm, the computational complexity is dominated by solving the optimization problem, which can be efficiently solved using an interior point method with a computational complexity cost of ON3.. The robON3.. The robON3.. The robeference .(22)wopb of Rec-MVDR is set to 1.2. The optimal weight vector In our simulations, a ULA consisting of eference , (ii) theference , (iv) theference , and (v)eference . Among tained by , which ial power ,35. For In the first example, the beampatterns of null broadening were examined. Assumed that the desired signal with fixed 5 dB SNR comes from the direction In the second example, we examined the performance of the considered methods in suppressing fast-moving interfering signals. The directions of the two interfering signals were assumed to be random and uniformly distributed in In the third example, the impact of the signal look-direction error on the output SINR was examined. We assumed that the random DOA mismatches of the desired signal were uniformly distributed in In the fourth example, the effects of array geometry mismatches on the tested beamformers was investigated. Mismatches in antenna geometry are usually modelled by sensor position errors, which can be assumed to be subject to a Gaussian distribution. In the simulations, the mismatches were modelled as a Gaussian process with a mean of 0 and a variance of 0.002. At the same time, the DOAs of the interferers changed from run to run, but were kept fixed from snapshot to snapshot. In addition, the other parameters were the same as those in the third example.From In the fifth example, the impact of the steering vector mismatch due to incoherent scattering was considered. Assuming that the steering vector of the desired signal changes over time, the desired signal can be modeled as (23)x˜s(itten as (24)SINRmized by (25)woptIt can be seen, from In the last example, the performance against multiple interferers is examined. Assumed that five interfering sources with random waveforms were uniformly distributed in The output SINR of the tested beamformers versus input SNR for the number of snapshots The paper proposes a novel robust beamforming technique using null broadening for mitigating intereference caused by fast-moving jammers or antenna platform motion. The weight vector can be obtained by introducing the reconstructed INC matrix and the estimated the steering vector of the desired signal into the traditional MVDR beamformer. The DOAs of interfering signals can be estimated with the Capon spectrum estimator, and a new INC matrix can be reconstructed by placing several virtual interference sources around each of the actual jammers. The null depth can be controlled by the spatial power parameter in the process of reconstruction. Using convex optimization software, the steering vector of the desired signal can be estimated by solving a new optimization problem based on the reconstructed INC and SNC matrices. The simulation results demonstrated that the performance of the proposed null broadening beamformer dramatically outperforms other methods including classic robust adaptive beamforming and null broadening techniques, and can achieve good robustness against fast-moving jammers or antenna platform motion. Moreover, if the DOAs of the interferers or the direction of antenna platform motion is known, more virtual interference sources can be set along the actual interference DOAs, in order to achieve better performance.


In this article, using numerical simulations we investigate the self-assembly of rod-like particles in suspension due to depletion forces which naturally emerge due to the presence of smaller spherical depletant particles. We characterize the type of clusters that are formed and the evolution of aggregation departing from a random initial configuration. We show that eventually the system reaches a thermodynamic equilibrium state in which the aggregates break and reform dynamically. We investigate the equilibrium state of aggregation, which exhibits a strong dependence on depletant concentration. In addition, we provide a simple thermodynamic model inspired on the theory of self-assembly of amphiphilic molecules which allows us to understand qualitatively the equilibrium aggregate size distributions that we obtain in simulation. Due to their entropic origin, depletion forces are ubiquitous in complex fluids. They originate in solutions with high concentrations of microscopic particles of at least two different sizes. Above certain threshold concentrations of the different species, the larger particles tend to aggregate to increase the entropy of the system and thus minimize its free energy. This phenomenon can be seen as an effective interaction between the larger particles mediated by the smaller ones, as formulated by Asakura and Oosawa in their seminal work . DepletiSince the first theoretical account of depletion forces was given by Asakura and Oosawa , much efThe interaction due to depletion can be markedly anisotropic depending on the shape of the interacting colloids. A number of different naturally occurring and custom-made shapes have been investigated that self-assemble due to depletion forces into a rich variety of complex structures ,18,20,21In this article, we investigate using numerical simulations the aggregation of rod-like particles due to depletion forces. In contrast to most simulation studies where the depletion interaction is introduced via effective potentials between colloidal particles, in our approach we simulate the dynamics of both the rod-like particles and smaller spherical depletant particles, which naturally induce the emergence of depletion forces. Departing from a random configuration, we solve the evolution of the system until the average aggregate size  reach a stationary state which we identify with thermodynamic equilibrium. We characterize the equilibrium size distribution of the aggregates and the dependence of aggregation on the concentration of depletant. In addition, we propose a new theoretical approach in the discussion of depletion forces inspired on the theory of self-assembly of amphiphilic molecules, which allows us to understand qualitatively the equilibrium distributions that we obtain in simulation.L and square section of side a, are modeled with the help of a group of spherical beads which collectively behave as a rigid body [We performed computer simulations of a system of long rods in a solution of smaller depletant spherical particles in contact with a thermal bath. The rods, of length gid body   of length =80σ see . We consTo understand the results from numerical simulations we built a simple thermodynamical model which allows us to predict the type of aggregate distribution that we might expect in equilibrium. The model is based on self-assembly theory ,27 and aT the temperature, D, which is much smaller than the length and width of the aggregating rods.The effective depletion interaction potential between two colloidal particles is, to a good approximation, given by ,7,28(1)L and radius R μn,m=ggregate ,29,30. THere, Equation ) does noWe identify the formation of aggregates as the simulations advance. Aggregation of rods is observed to increase as the depletant concentration increases, in agreement with Equation . A signiIn lane see . The thilane see . The lasTo characterize the formation of aggregates, we calculate the average number of discs per aggregate as a function of simulation time, In In Equation , inducinn  for systems with different depletant concentration. For the lowest value of From our simulations we also have access to the size distribution of clusters. We analyzed the average size distribution of the different systems considered once they reach thermodynamic equilibrium. In ue of cD a, the diThe predominance of side-to-side aggregation and the type of size distribution of clusters should be independent of the cross section of the rods as long as their aspect ratio is small enough R/L≪. In contWe investigated the self-assembly of slender rods in solution due to the effective depletion interaction that emerges due to the presence of depletant particles. We performed computer simulations at fixed temperature and volume of a simple model composed of rod-like particles and smaller spherical depletant particles which interact solely through repulsive interactions. In this system, depletion forces between rods emerge naturally for certain values of the concentration of rods and depletant, inducing the aggregation of rods into clusters.We verified that, as expected, clusters form preferentially due to side-to-side interactions, growing in the plane perpendicular to the axes of the rods. Aggregation due to tip-to-tip interactions is anecdotal. We characterized the growth of these clusters over time, observing a monotonical increase in the average size of aggregates up to a stable plateau when the system reaches the equilibrium state.The equilibrium state was analyzed for systems with different depletant concentration. We observe that the aggregates’ average size increases rapidly with depletant concentration, as expected. Indeed, mean-field thermodynamic models of aggregation anticipate an exponential increase of the average aggregate size with the strength of the interaction governing self-assembly. In our case, the formation of clusters is led by the depletion interaction, which increases linearly with depletant concentration. We also obtained the size distributions obtained in equilibrium from our numerical simulations. We show that the numerical results are consistent with a geometrical distribution predicted by basic thermodynamic models of self-assembly.


In this study, we analyze the DSA images of a subject-specific left posterior AVM case and customize a generic electric analog model for cerebral circulation accordingly. The generic model consists of electronic components representing 49 major cerebral arteries and veins, and yields their blood pressure and flow rate profiles. The model was adapted by incorporating the supplying and draining patterns of the AVM to simulate some typical AVM features such as the blood “steal” syndrome, where the flow rate in the left posterior artery increases by almost three times (∼300 ml/min vs 100 ml/min) compared with the healthy case. Meanwhile, the flow rate to the right posterior artery is reduced to ∼30 ml/min from 100 ml/min despite the presence of an autoregulation mechanism in the model. In addition, the blood pressure in the draining veins is increased from 9 to 22 mmHg, and the blood pressure in the feeding arteries is reduced from 85 to 30 mmHg due to the fistula effects of the AVM. In summary, a first DSA-based AVM model has been developed. More subject-specific AVM cases are required to apply the presented  These asyndrome . Still, syndrome . For exasyndrome .A number of AVM models have been developed over the past two decades, in particular by Young and his co-workers . Among tin silico simulations, which are all lumped parameter models where the mass of vessels within an AVM are lumped into one or several electronic components to simulate the blood pressure gradient and flow through it , and Rn is the resistance under a normal pressure range (80–120 mmHg).where The autoregulation mechanism of Eq. (1) was tested using an artificial pressure waveform, which contained five phases with 10 cardiac cycles for each phase . A normaThe Endovascular Intervention Unit of the Neurosurgery Department of the West China Hospital (WCH) treats approximately 70 AVM patients annually. Of these patients, about 50% receive surgeries following endovascular treatments. We randomly selected an AVM case for flow simulations.An 18-year-old female patient was referred to the WCH with symptoms of eyesight loss of her right eye and headache. CT scans revealed left cortex deformation and a suspected AVM growth at the left occipital lobe. The AVM was confirmed by a DSA procedure  . The lefAn endovascular embolism procedure was performed for the patient . Intra-pThe supplying and draining vessels revealed from DSA images were critical for the diagnosis and treatment of the AVM. The small vessels and the AVM nidus can be lumped into several electronic elements. In our implementation, the AVM was modeled by two parallel resistors with equal resistance to simulate the AVM plexiform. Another parallel branch consisting of a resistor with much lower resistance was used to mimic the fistula . This arSince the feeding arteries arise from the left PCA and PICA, the left PCA is rerouted onto the nidus network, bypassing the posterior capillary bed, and adjoining to the AVM nidus .In the current implementation, the AVM does not have an intranidal adaptation mechanism, following what is suggested in literature . In otheThe baseline model shown in With the introduction of a left AVM in the model, the flow rates at the left PCA, PICA, and the right PCA are shown in Compared to the left PCA (diameter ∼2 mm), the diameter of the left PICA (diameter ∼1 mm) is smaller, and its contribution to the AVM flow is not as significant as the PCA. However, the flow rate (∼35 ml/min) in the left PICA is much higher than the normal value of 10 ml/min , leadingA question raised is whether the circulation “steal” effects are located in the vicinity of the AVM  or they We also compared the blood pressures proximal and distal to the AVM, i.e., in the feeding artery, the left PCA, and the draining vein SSS. The results are shown in Advanced imaging technologies nowadays have revealed more AVM cases, including many that are asymptomatic . Clinicain silico models can provide useful insights in certain aspects of AVM, such as the extranidal adaption to hypotension (It has been shown that otension  and the otension . The latotension .However, it is still controversial whether regional regulation functions may be deprived due to the presence of an AVM, or just their lower limits are displaced . An updaOur model provides certain support to the hyperemia theory. We have shown in the transnidal pressure simulation  that theThe novelty of the work is that it is the first AVM model that incorporates subject-specific information from DSA image. Since the morphological arrangement of the AVM in the cerebral circulation model is subject-specific, a different AVM patient requires having an AVM model customized and re-adapted in the generic 0D model to account for the specificity.in vivo data. This is another important limitation of the current study.There are some limitations pertaining to the current model. Firstly, the implementation of the autoregulation mechanism is rather simple. The model assumes that the CBF responds to blood pressure variations only, yet CBF autoregulation is a complex process that involves neurogenic, metabolic, and myogenic mechanisms . Secondlin vivo measurements such as CBF velocities measured using transcranial ultrasound (It is also worth noting that while our model results were compared to DSA data, the calibration was rather phenomenological than quantitative. In trasound  or phasein silico platform for further AVM investigations. An immediate future work would be to numerically investigate the embolization of AVMs and those AVM cases located at different regions of the brain, e.g., in the anterior circulation region. Indeed, AVM occurs at the anterior circulation in 70–80% of the time (Nevertheless, the presented model provides an the time . Anotherin silico model be used as the basis for further AVM investigations.In this study, we used the information from DSA images of an AVM patient to alter an electrical analog model for cerebral circulation. Our simulations reproduced multiple flow phenomena that can be observed from the DSA images. We suggest the current All datasets generated for this study are included in the article/Supplementary Material.The studies involving human participants were reviewed and approved by The Ethics Committee of the West China Hospital, Sichuan University. The patients/participants provided their written informed consent to participate in this study.CZ performed the endovascular procedure of the AVM case and provided clinical insight. NC made the numerical simulation. HH drafted the manuscript. All authors reviewed and agreed the current form of the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The difficulty in combining high fidelity with fast operation times and robustness against sources of noise is the central challenge of most quantum control problems, with immediate implications for the realization of quantum devices. We theoretically propose a protocol, based on the widespread stimulated Raman adiabatic passage technique, which achieves these objectives for quantum state transfers in generic three-level systems. Our protocol realizes accelerated adiabatic following through the application of additional control fields on the optical excitations. These act along frequency sidebands of the principal adiabatic pulses, dynamically counteracting undesired transitions. The scheme facilitates experimental control, not requiring new hardly-accessible resources. We show numerically that the method is efficient in a very wide set of control parameters, bringing the timescales closer to the quantum speed limit, also in the presence of environmental disturbance. These results hold for complete population transfers and for many applications, e.g., for realizing quantum gates, both for optical and microwave implementations. Furthermore, extensions to adiabatic passage problems in more-level systems are straightforward. It follows that, if the evolving eigenvector connects the initial state to a certain target final state, adiabatic tracking produces the desired quantum state transfer.The near-term development of quantum technologies is fundamentally linked to the challenge of controlling quantum systems with high precision. Ideally, precision should be further flanked by flexibility and resilience against experimental imperfections, and by the capability of realizing many control operations on the system within its natural lifetime. Within the field of quantum control, a class of methodologies which satisfies many of these requirements – especially precision, flexibility and robustness – exploits the concept of adiabaticity4, have been explored in great detail both from the theoretical point of view and from the experimental community. Indeed, they have been implemented in two and three-level systems in different experimental contexts such as cold atoms7, trapped ions8, N-V centers12, superconducting qubits16, and micromechanical oscillators18. While most experiments rely on the system driven by optical excitations, superconducting qubits can exploit the additional flexibility offered by microwave excitations. The aim of these protocols is to produce the same population transfer as that given by a truly adiabatic evolution but in a shorter time. The price to pay for the speedup is typically an increase of the needed control resources20, especially in terms of tuneable Hamiltonian terms. This is in most cases too demanding from the experimental point of view: complete time-dependent control of all the couplings may be needed, for few-level systems, or even of interactions between distant subsystems for more complex systems. In order to overcome the experimental difficulty, it was soon realized that, via appropriate time-dependent unitary transformations, the scheme can in general be modified in such a way that the correcting Hamiltonian contains only realizable terms25. Physically, this typically means that the concept of adiabaticity is abandoned, and even if initial and final states are the desired instantaneous eigenstates of the original Hamiltonian, transitions are allowed throughout the intermediate path.Despite the many advantages offered by the adiabatic methods, they intrinsically suffer from the requirement of slow evolutions, which drastically limits the number of operations to be performed on the system before environmental noise spoils its quantum features. Recently, so-called counterdiabatic (cd) or shortcut-to-adiabaticity or transitionless methods26. This is based on the application of radiofrequency sidebands to each optical excitation of the three-level system and it will be denoted as frequency-modulated STIRAP (fmod-STIRAP). For a given modulation frequency, the only control parameter is the time-dependent amplitude of the sidebands. The key advantage of the method is that of joining very high fidelities with fast timescales, close to the quantum speed limit without needing ad hoc adaptations of the standard STIRAP experimental control setup. Given the wide-spread use of STIRAP protocols, the present technique is applicable to a broad range of experimental systems. Since it does not require the system to completely abandon the adiabatic path, the method maintains the robustness of STIRAP while dramatically enlarging the parameter regions in which STIRAP is efficient. Moreover, the shortcut can be straightforwardly adapted to the implementation of quantum gates via fractional STIRAP27, and it is not limited to a specific temporal regime: it can be beneficially used at all timescales from a quasi-adiabatic to a fully nonadiabatic regime. Finally, as compared to similar methods proposed theoretically29 and also realized in the lab15, the present sideband protocol does not produce diagonal terms in the system Hamiltonian, and hence it does not require to compensate variations of dynamical phase introduced by the control-induced ac-Stark shifts.In this contribution, we introduce a cd scheme for producing state transfers in three-level systems which consists in an accelerated STImulated Raman Adiabatic Passage (STIRAP) built upon a general framework recently proposed30. The driving scheme involves a first “Stokes” pulse on the The starting point of our analysis is the three-level STIRAP protocol in ladder or mentclass2pt{minim28. The present analysis is restricted to Gaussian pulses, typically used for STIRAP30, of the form3133, which evolves only in the 30, the system initially prepared in the ground state The quantities t}κ Fig. . In the cd-STIRAP. Among possible shortcuts to adiabaticity, much attention has been given to the idea of cd driving19, also known as transitionless quantum driving20. This protocol consists in applying additional control fields which compensate for nonadiabatic transitions exactly by instantaneously decoupling the adiabatic states. As a result, the latter are followed with unit fidelity at all times. The price to pay for this precision is the necessity to realize new time-dependent couplings in the Hamiltonian: indeed, counterdiabatic STIRAP (cd-STIRAP) requires a direct control on the 34. Specifically, for resonant STIRAP  is computed by performing a sigmoid fit to the temporal evolution of the populations, in order to remove the oscillatory pattern, as detailed in Methods.An important target for the applications of protocols such as cd-STIRAP and fmod-STIRAP described in this work is the possibility of speeding up, using limited resources, the quantum transfer process between initial and final states. For these protocols, we compare the transfer time from being in the state 38, since the norm of the Hamiltonian is proportional to the Rabi frequency of the corresponding pulses for the present protocols. In realistic experimental implementations, upper bounds may be imposed on the strength of the correcting Hamiltonian, or more precisely on the Rabi frequency ), obtained for ) in Fig. 15, a single sideband was produced by a separated microwave source with When an increase of the driving frequency t}Ωd(t)  which it}Ωd(t)  which iing Eqs.  translat30. In Fig. 39, and it is related to the construction of the fmod Hamiltonian: while this is engineered so to cancel nonadiabatic transitions to the leading order in The infidelity minima (fidelity maxima) in Fig. 43. A fractional STIRAP27 (f-STIRAP) prepares the system in an arbitrary superposition state41Modified STIRAP protocols have been studied for the realization of superposition states and quantum gates41 separated in time by The corresponding counterdiabatic correction is similar to the one of Eq.  depictedmentclass2pt{minim34, which should be realized independently. By computing the 44. The other parameters are Two f-STIRAPs are needed since a single one introduces a deleterious phase factor which is then compensated by the second f-STIRAP. The combined pulses, together with the corresponding These results highlight the potential of the protocol for applications of interest in quantum computation and quantum simulation, where the possibility of increasing the number of gate operations of even a small fraction, or the precision of single-qubit gates, can represent a substantial gain.The fmod-STIRAP protocol produces high fidelities systematically, opening a new route for the applicability of the STIRAP methodology (and modified STIRAPs) beyond the strictly adiabatic parameter regime. Crucially, the method can be easily realized experimentally, since it does not demand additional excitation resources, and it relies on the production of symmetric low-frequency sidebands. Indeed, the sideband generation requires easily accessible radiofrequency sources only, and it may take place at the expense of the laser intensity of the underlying STIRAP process. The potential production of undesired symmetric sidebands at larger frequencies is not a limitation: in the case in which their Rabi frequency can be controlled independently from that of the fundamental harmonic, they are a precious resource as discussed in the following. If they are not controllable in an independent manner, one can still adapt the construction of the effective Hamiltonian described in Methods in order to harness their effect: this would lead in general to a modified choice of the Rabi frequency inter alia for its simple implementability, producing competitive results with a modest increase of the STIRAP complexity. Nonetheless, the fmod strategy can be adapted and extended ad hoc in order to focus and optimize different aspects of the controlled dynamics. This can be done by controlling symmetric sidebands at higher frequency harmonics. The increase of control parameters can then be exploited for fine-tuned shaping of the effective Hamiltonian. Possible objectives of this procedure may be the minimization of the oscillations around the target adiabatic path45, or the robustness against selected types of external noise47. In this direction, the method may be combined with successful techniques from optimal control for carrying-out the optimization task, such as stochastic gradient (learning) algorithms47. All the mentioned features of fmod-STIRAP make it particularly fascinating from the experimental point of view, showing its potential for generalizations to speed up adiabatic passages in more-level configurations, and hence for successful applications in many branches of quantum science.As benchmark figure of merit, we concentrated especially on the final fidelity, timing and stability with respect to variations in the choice of parameters. In this respect, the use of (two) sidebands at the same frequency is interesting 48, whose use for the derivation of approximate counterdiabatic fields has been recently proposed49. We consider a general form of the correcting Hamiltonian in which the 48. Assuming to use only terms oscillating at the fundamental harmonic The shortcut Hamiltonian is constructed such that it reproduces The diagonal elements in Eq.  represenWe consider a possible experimental realization where an input laser intensity 50, using the parameters5115, where Energy relaxation is taken into account in the numerical simulations by using the following Lindblad master equation for the density matrix 36For a constant Parametrizing the states like  and Eq.  can be rSince, under cd-STIRAP, the system instantaneously follows the dark state with unit fidelity, the occupation probabilities of states entclass1pt{minima28.The difference For determining the transfer time of the fmod-STIRAP protocol, the dynamics of the populations entclass1pt{minima52. In producing Fig. Finally, the transfer time is computed as entclass1pt{minima


Social trust has been an important mechanism in overcoming crises throughout history. Several societies are now emphasizing its role in combating the COVID-19 pandemic. This study aims to investigate how variations in social trust across 68 countries are related to the transmission speed of COVID-19. Specifically, using cross-national index data from the World Value Survey, the study tests how variations in social trust across countries generate different time durations at which each country reaches the peak in terms of increases in new infections of COVID-19. Using data drawn between December 31, 2019 and July 31, 2020, this study found that in countries with a high level of social trust, particularly trust among ingroup members, or with a narrower or wider range than the intermediate range of trustees, the number of new infections tended to reach the first peak within a shorter time duration than in other countries. These results imply that in such societies, on the one hand, high cooperation among people to achieve common goals and strong compliance to social norms may allow them to begin neutralizing COVID-19 faster. On the other hand, however, the low risk perception and prevalence of cohesive relationships among people may lead to speedier transmission of COVID-19 before neutralization takes place. Trust is widely considered an important type of social capital , 2. A nuOwing to these benefits, throughout history, societies have often encouraged people to trust each other and be trustworthy when faced with global adversities such as economic or institutional crises . The COVWill societies with strong social trust among people decelerate the transmission of COVID-19? Although some high-trust societies may have successfully curbed the spread of COVID-19, it is still too early to conclude that social trust will control the spread COVID-19 for three reasons. First, to date, there is little empirical evidence of the significant relationship between the degree of social trust and epidemics. Second, as prior observations have mainly focused on trust in local governments , it stiThis study explores the relationship between social trust and the transmission speed of COVID-19 worldwide. Using cross-national index data from the World Value Survey (WVS) 2017–2020 , the stuIn addition, given that social distance contradicts social trust at a certain level, an emphasis on social distance in high-trust societies generates an unexpected paradox for the social actors involved. The results will show how social actors behave under this paradox in high-trust societies and extend the literature on social trust.The rest of this paper is organized as follows. The second section introduces the concept and types of social trust based on the previous literature and predicts how social trust influences the speed of the spread of COVID-19. The third section describes the methods used to test the predictions, and the fourth section reports the results. The final section discusses the implications of the findings for both practice and the literature.Social trust has been defined as the belief that other people will not cause any harm to us but will rather look after our interests . It is aSocial trust is based on interactions between the actor placing the trust, namely, the trustor, and the target, namely, the trustee . PreviouPrevious studies have also suggested another concept that determines trust in society: the trust radius, which is defined as a circle of people among whom trust is operative . This coThis study investigated how the different levels and width of social trust in each country affect the fast peaking time for COVID-19, measured by the time taken for the number of new COVID-19 infections to reach the first peak. Here, fast peaking has two different meanings. First, it refers to the steep slope of increase in the new infections before the peak—that is, the transmission speed of COVID-19 is fast after the country finds its first case. Second, it refers to the faster control of such transmission because the peak indicates a starting point for the decrease in the number of new infections. However, fast peaking does not measure a steep or flat slope after the peak to approach the normal state. The neutralization speed of COVID-19 is less testable at the current moment because COVID-19 is still in progress.In countries with high levels of particular and general social trust, the number of new infections will reach the peak faster. In other words, these countries will experience faster tramsmission and neutralization of COVID-19. The following two reasons support faster transmissions. First, social trust is not created in a single moment but rather builds up gradually through historical interactions among people . TherefoSecond, previous studies have shown that people in high-trust societies tend to take high risks. For instance, Siegrist et al.  used surOn the other hand, in high-trust societies, once people recognize the risks in society, they will immediately control their behaviors that may amplify such risks further. The two following reasons support this idea. First, high-trust societies are characterized by cohesive relationships among people. In these societies, people tend to be oriented toward a common goal and perceive strong responsibility to contribute to that goal . This orSecond, once a common goal is established in high-trust societies, compliance to social rules and norms should follow suit as a medium of achieving the common goal. Several previous studies have pointed out the importance of compliance to social rules in managing COVID-19 transmissions. For instance, scholars have shown that the higher infection ratios and morality of males, in comparison to females, are relevant regarding gender differences in compliance to health rules imposed by societies. That is, females are more risk averse, concerned about health consequences, and in favor of activity-restraining public policy measures than males. These differences allow females to be more compliant with the social rules for health-promoting behaviors during the COVID-19 pandemic such as washing one’s hands, maintaining distance from others, and staying at home , 31. SocFinally, as assumed, if both particular and general trust accelerate the fast peaking time for the number of new infections, the trust radius will form a U-shaped relationship with the acceleration. In countries with a narrower range of trustees, people have intimate interactions within ingroups and rely heavily on ingroup members. This can facilitate both frequent contact among the ingroup members and sensitive reactions to risks posed to them. Where there is a wider range of trustees, people are generally embedded in high trust-based environments. In these societies, people are likely to have cohesive forms of social contact, low risk perception, and sensitive reactions to the common risks for society. The intermediate range of trustees cannot take either of these effects, so the fast peaking time will be less likely in countries with an intermediate trust radius.Following these arguments, this study predicts that (1) in countries with a high level of particular or general social trust, the number of new infections of COVID-19 reaches its peak faster, and (2) the relationship between the trust radius and the fast peaking time for the number of new infections is U-shaped. Specifically, in countries with a narrower or wider trust radius rather than an intermediate one, the number of new COVID-19 infections reaches its peak faster. The analysis results provide general support for these predictions and show a significant relationship between social trust in the countries and the fast peaking time for COVID-19.To test the influence of social trust, this study used data from the World Values Survey (WVS), which is the largest cross-national survey investigating different attitudes, values, and beliefs across countries and has been carried out since 1981. The most recent investigation is the WVS 7th wave, which took place between 2017 and 2020, in which about 104,000 respondents across 80 countries participated . This suThe data on COVID-19 infections around the world were collected from the European Centre for Disease Prevention and Control (ECDC) . These d0(t) is the baseline hazard function, and β1 to βk are the vectors of parameters for k explanatory variables. According to this formula, the different hazard rates across countries rely on the explanatory variables and not on the baseline hazard. The sample period commenced on December 31, 2019 and ended on July 31, 2020. Each country was at risk when it had COVID-19 infections for the first time. The dataset entered the number of new infections every day for sample countries since the countries acquired the first case until either the number reached its peak or the end of the study period.The dependent variable in this study is the time duration each country takes to reach the peak of new infections. This study uses the number of new infections per million per day in each country as its measure. This measure is more effective than the use of the direct number of new infections in order to control the inherent differences in population sizes across countries. To estimate the association between social trust and the fast peaking time of new infections, this study applied the Cox proportional hazards model, following other similar studies such as those on the outbreak of influenza . This moTo define the peak, this study developed a dummy variable coded as 1 when the given day met two conditions and otherwise coded as 0. The two conditions are as follows. First, the given day has the largest number of new infections between 14 days prior to and 14 days after that day. The 14 day-interval is appropriate because the maximum incubation period for the disease is 14 days. People would need around 14 days to recognize the risk of the spread of COVID-19. Second, the cumulative number of cases on that day should be larger than one for every million. With the application of the second condition, the peak cannot be determined until each country reaches a certain level of total cases.In the sample periods, some countries experienced multiple peaks by experiencing multiple cycles of drastic increases and decreases in new infections after they neutralized the first wave successfully. As these additional waves may have arisen from more complicated sources across countries, this study only focuses on the first peak and excludes data after it.The first explanatory variable in this study comprises both types of social trust, namely, particular and general, in each country. Previous studies have defined these trust levels based on the following questions , 0.33, 0.66, and 1 (completely trust). Then, the averages of the rescaled answers from Q1 to Q3 and those from Q4 to Q6 per country were calculated, and each was defined as trust radius, the measurements suggested by Delhey et al. [general trust−βparticular trust. In line with this definition, the trust radius indicated the extent to which respondents viewed their outgroup people as generally trustworthy. Finally, by adding one to all values, the negative values were fixed. The final measure of the trust radius ranged from zero (the narrowest) to one (the widest). Figs To define the y et al.  were apppopulation by dividing the number of people by the land area of the country. The unemployment rate and log-transformed GDP were controlled to reflect the different economic conditions across countries. As countries with high rates of annual overseas movement may have been more vulnerable to the faster transmission of COVID-19, this study used the international movement expenditure  and receipt  to control for influences from general international movements. These variables refer to the expenditure of international outbound and inbound visitors, respectively. The death rate (per thousand) was also included to control the different health conditions of people across countries. Additionally, the number of hospital beds (per thousands) is included. This variable controls the differences in the general state of medical treatment across countries because such differences can influence the risk perceptions of people toward a health emergency. All of these data were obtained from the annual World Development Indicators that the World Bank provides in the most recent year [internet penetration rate from the Internet World State 2020 by using the percentage of the total population of each country that uses the internet [To disqualify alternative explanations, this study entered multiple control variables as follows. Heterogeneous demographic or economic conditions across countries may create differences in the transmission of COVID-19. Hence, this study controlled the ent year . In sociinternet .lockdown period on the focal day, and otherwise coded as 0. The data on the lockdown period for each country were obtained from “National responses to the COVID-19 pandemic” in Wikipedia [The faster or slower transmission of COVID-19 can be deeply related to different government interventions. Indeed, as the cases of COVID-19 continue to rise, several countries have entered lockdowns to restrict movement and thus slow down the spread of the virus. These governments’ interventions and the corresponding timing of their implementation may create significant differences in the peaking time for new infections across countries. To control this non-pharmaceutical intervention of each country, this study entered a dummy variable coded as 1 when the focal country was under the ikipedia . Some conew tests (per thousand) is an influential element in determining the identification of new infections. However, as several countries (24 of the sample) did not provide this information, missing values were frequently generated. Thus, this study provided results based on models with and without this variable. According to the ECDC, there may be a time gap between the new tests and the announcements of the results owing to unforeseen delays in reporting systems in each country and the time it takes to get the results [Finally, the number of  results . To cons results , 40.Models 3 to 6 represent the results for the influence of particular and general trust. As predicted, in countries with a high level of particular trust, the number of new infections reached a peak faster than other countries . The influence of general trust has been shown to be weak. It presents a positive and significant coefficient in Model 5  but an insignificant coefficient in Model 6 .To visualize the results, this study separated the sample countries into groups of high and low particular or general trust based on their median values and plotted their survival functions using the Kaplan-Meier estimator. As shown in Models 7 and 8 test the curvilinear pattern of the trust radius to explain the time duration that each country would take to reach the peak by including the squared term of the trust radius. The addition of the quadratic terms in Models 7 and 8 leads to significantly better models than do Models 1  and 2 , respectively. The squared terms of the trust radius show a significantly positive impact on the fast peaking time , suggesting that the fast transmission and control of COVID-19 will be more likely in countries with narrower or wider ranges of trustees than in those with a medium range of trustees. Using the appropriateness test for a U-shaped relationship , this stTo check the robustness of the results, this study redefined the peak only if the given day had the largest number of new infections between 14 days before and after that day, although the total number of cases on that day was less than one for every million. Then, the study reran the aforementioned regressions based on the redefined peak. The results replicated those shown in To gain additional insight, this study conducted two further examinations. First, it tested how the transmission speed of COVID-19 differed based on the trust level in the public sector, namely, political trust across countries. To define political trust, following previous studies , this stThe political trust may rely heavily on the county’s political regime, such as the degree of democracy. To control the influence of individual countries’ political regimes, this study included the democracy scores in 2019 for each country as compiled by the Economist Intelligence Unit . This inSecond, this study argues that the fast peaking time for new infections in high-trust societies results from people’s perception of low risk towards an emergency and strong compliance with social rules in responding to common adversities in those society. To gain further insights concerning this argument, this study conducted an analysis to test how social trust affects the mobility of people in a pandemic. If the argument holds true, the rate of mobility should be generally high in high-trust countries; however, the high mobility in high-trust countries should be effectively controlled through stringent restrictions such as lockdowns.To test this prediction, this study utilized the data on Community Mobility Reports provided by Google . This daIn the analysis models, this study controlled the population and log-transformed GDP to consider the countries’ different conditions. The internet penetration rate was also controlled to reflect the availability of communication through the online space. Since mobility will be influenced by the severity of COVID-19 transmissions, this study entered the number of new infections per million people on the focal day in each country. Finally, the lockdown period is also controlled to reflect the influence of government interventions on mobility. For analysis, generalized estimating equations (GEE) with robust standard errors were applied. GEE models are appropriate for testing panel datasets since they are effective for controlling autocorrelation and heteroskedasticity. An identity link and a Gaussian distribution were applied for the tests.This study examined the influence of social trust on the spread of COVID-19 across 68 countries between December 31, 2019 and July 31, 2020. According to the results, in countries characterized by high levels of social trust  or a narrower or wider range of trustees, the number of new cases is likely to reach the peak faster than in other countries. These findings have the following implications for practice and the literature.First, social trust brings about the current good-news-bad-news conundrum. On the one hand, social trust may help speed up the neutralization of COVID-19 after the spread reaches a certain level because social trust facilitates cooperation among people in solving their common adversities as well as compliance to social rules for health-promoting behaviors. This observation is in parallel with other previous arguments that emphasize the benefits of social trust in managing COVID-19 transmissions , 13. NevSecond, the results of this study call for further studies on risks associated with social trust. Although it is apparent that social trust generates numerous forms of social capital, as several previous studies have shown , 5, 46, Addressing a balanced discussion on the effects of social trust will be critical for theoretical development and for comprehending social implications. As shown in this study, social trust is an intense and stable mechanism that explains actors’ social behaviors. According to the results, in high-trust societies, in-person contact, particularly contact among ingroup members, may be less controllable even in an emergency in which social distance must be maintained. This tendency does not change while controlling the internet penetration rate in society. This suggests that even in environments in which people can maintain contact online, routinized in-person or intimate contact within groups may be less controllable.In this line of arguments, other relevant factors such as group or national identity might also show similar effects as that of social trust. For instance, Van Bavel et al. , in theiLastly, according to the results of this study, societies may need to implement different strategies to respond to pandemics on the basis of their levels of social trust. For instance, in high-trust societies, the early phase of the pandemic will be a critical duration for controlling the crisis. With the strong tendency for compliance to social norms or rules in those societies, people will be significantly influenced by what others are doing and which rules are imposed upon them. Hence, strong restrictions against close contact and the dissemination of information through social media or networks about behavioral standards should be given at the early phase of the crisis, so that the societies can form the norms and rules for health behaviors immediately. In the low-trust societies, on the other hand, the policies and strategies should take longer-term perspectives because it may take more time to start neutralizing the pandemic.This study has several limitations. First, the varying duration of the peaking time for COVID-19 across countries will depend largely on restrictions imposed by government policies, and particularly on the strength and timing for the adoption of restrictions. For this reason, this study included a control variable that measures the lockdown periods of each country in the analysis. However, due to limited access to data, the study could not fully take into account many other potential influences from different government policies in the analysis. For more political implications, future studies will benefit from testing how the effects of government restrictions interact with the different levels or radius of social trust in countries when it comes to managing a global pandemic.Second, the discussion did not cover how long the countries take to revert to the normal state and how social trust may affect the second and third waves of infections following the first wave. As societies are still in the course of COVID-19 transmission, it is premature to examine these aspects at this point. Follow-up studies in the future should strive to deepen our understanding of the relationship between social trust and epidemic transmission.Third, this study uses each country as a unit of analysis, and the results thus did not reflect the influences of individual differences among the respondents. This is a critical limitation because compliance to social rules is significantly affected by individual aspects. Using alternative methods of analysis such as surveys and experiments at the individual level, future studies can provide more robust results.Finally, although this study, to a limited degree, focused on social trust as an antecedent of the fast peaking time for the transmission of COVID-19, many other socio-behavioral factors, such as the national culture, ethnic or religious diversities, and time perspective of the societies, may affect the peaking time through behavioral control and restriction among people. Future research that compares the influences of these diverse factors across countries will suggest further useful strategies to manage this global crisis.Despite these limitations, the theoretical arguments and empirical findings of this study offer important implications for the literature on social trust as well as the practical control of the current global crisis. It is hoped that the approach and findings of this study contribute to holistic safety in society.S1 File(DTA)Click here for additional data file.S2 File(DTA)Click here for additional data file.


A sustainable manufacturing company depends on the developments in three aspects in order to minimize harmful impacts on the environment, improve the social relations, and simultaneously maximize the economic benefits. Despite the increasing types of investigations that researchers have carried out in environmental and economic aspects, the minimum attention has been paid to social relations. In response to this deficiency, this paper proposes a new framework to obtain the overall sustainability index in manufacturing companies by encapsulating the sustainability criteria/sub-criteria. This article collected 33 sub-criteria for five pillars of sustainability as social, environment, economic, technological advancement, and performance management. The key contributions of this paper are highlighted as the hierarchical method that obtains the status of sustainability in uncertain conditions, the ability to identify the weak points, and a new framework for gathering the data about sustainability performance in manufacturing companies. The findings of this paper will aid both policymakers and decision-makers to assess the sustainability status of manufacturing systems and improve the performances of them. Sustainable development implies providing the way of meeting the needs by utilizing the maximum ability of current generation without compromising the abilities and technologies of future generation ,2. SustaIn the past decade, the increasing pressure of government regulations concerning manufacturing companies forced them to consider sustainability insights and think beyond only economic benefits ,6,7. TheSustainable manufacturing strategy involves three stages: (I) selection of appropriate criteria for measuring sustainability of manufacturing company, (II) a suitable assessment tool which determine the sustainability status and identify the fragile points, and (III) modifications of different parts of the system to enhance the sustainable manufacturing subsequently ,16. RegaThis study proposes a new benchmark approach which aims at measuring the current level of sustainability in manufacturing systems, identifying the weak points that have a negative impact on overall sustainability level and enhancing the efficiency of weak points to uplift the overall sustainability index subsequently. The fuzzy logic is utilized to consider the uncertainty and help decision-makers to express their subjective preferences by linguistic variables. Note that the fuzzy membership function is also utilized to transfer these linguistic terms. The application of manufacturing companies’ sustainability triple bottom line (TBL) criteria/sub-criteria, trapezoidal fuzzy set’s principles and fuzzy aggregation operator helped us in introducing a new as fuzzy sustainable manufacturing company index (FSMCI).➢The most appropriate manufacturing companies’ sustainability TBL criteria are collected from the extended literature to expose a holistic framework which adds to the state-of-the-art studies of sustainability in manufacturing sector by encapsulating a proposal and applications of a new and fuzzy sustainable manufacturing company index.➢The performance indicator abbreviated as FPII is utilized to identify the weak performances concerning sustainability TBL criteria which can help DMs immensely to improve the quality of weak points or so-called obstacles and consequently enhance the overall sustainability level.➢The trapezoidal fuzzy membership functions are employed which reflect more accurate results . A new t➢The following approach proposes a new method by which decision makers can decide about the performance of every sub-criteria based on the opinion of more than one expert. In other words, the mean of judgments will be set as the performance of sub-criteria which is obviously more reasonable than the opinion of one DM. However, the importance weights of criteria/sub-criteria will be determined based on the objectives of the organization.We highlight the contributions and innovations of this study as follows:The remainder of this study is organized as follows: in In this section, the up-to-date literature of sustainability in manufacturing companies and the application of fuzzy sets in sustainability context are reviewed which helps with illustrating the necessity of implementation and wide applications of the proposed approach in real-world manufacturing companies. Note that authors tried to sieve these reviewed studies despite the fact there are numerous masterpiece works in the context of sustainable manufacturing.This sub-section provides a comprehensive scheme of the methods and approaches which are developed for evaluating sustainability in a diverse industry by the usage of fuzzy theory. The fuzzy theory defined by Zadeh  opened aLin et al.  proposedPhillis et al.  modifiedAnother topic which is widely utilizing the application of fuzzy theory is the sustainable supplier selection problem. These past years, a huge number of studies have been carried out with the background of fuzzy sustainable supplier selection which mostly utilized fuzzy decision-making approaches ,42,43,44Due to the fact that manufacturing companies are playing an active role in the consumption of natural resources and generation of waste materials, they can significantly contribute to promoting sustainability development in developed/developing countries ,48. RegaMostafa and Dumrak  proposedMeanwhile, there are also a few studies which integrated sustainability pillars to propose assessment models for manufacturing companies by encapsulating related criteria, sub-criteria, and attributes. A selection of these studies is reviewed in In this section, first, the basic concepts that are utilized for elaborating this study are briefly denoted based on original references. After that, these concepts are merged to establish the proposed methodology.Regarding the type of fuzzy set which is incorporated for modeling the sustainability assessment in this study, some basic definitions are explained to provide a comprehensive illustration of trapezoidal fuzzy sets (TrFS). The TrFS are utilized instead of triangular fuzzy sets to cover more uncertainty and conclude more accurate results ,59,60. AAssume that n (1). A .(1)μA˜A˜+B˜A fuzzy number irements :A¯(r) isFor an arbitrary trapezoidal fuzzy number  follows :(6)Mag to obtain the Euclidean distances  between Step 4: use the fuzzy performance importance index (FPII) to determine the effect of every single sub-criteria on total sustainability. Identify the barriers through the manufacturing company using FPII and improve them in order to enhance the total sustainability index. For every single sub-criteria, the FPII will be obtained by using Equation (9) as follows :(9)FPIIiAfter obtaining the ranked FPIIs for every single sub-criterion, the ones that are lower than the managerial threshold will be identified as weak points. This managerial threshold also can be determined based on the standards of the company. However, in this study, we considered this threshold as the mean of the obtained FPIIs.A guideline illustration of the steps of this method is shown in In this section, an illustrative case is resolved which totally elucidates the extensive application of the proposed approach. Note that the only difference between this case and the real-world cases is in the collection of data which may differ in diverse cases. Given that, consider a real-world case is being resolved.The linguistic terms will be offered to DMs and experts, and they will be asked to assign the sub-criteria with a suitable abbreviation, e.g., “UI” for “Utmost importance”.Hence, consider that the criteria/sub-criteria shown in Once the sub-criteria and weights are determined, the DMs will cooperate with determining the performance status for each sub-criterion using linguistic terms. For this particular case, four decision-makers are selected to contribute to determine the performances. Once the weights and performance rates (mean of DMs’ judgements) obtained, the detailed data will establish The overall fuzzy sustainable manufacturing company index will be obtained by two levels of calculation using fuzzy aggregation operator Equation Using the same equation, the performance rates for criteria level are obtained and shown in Once the performances of criteria level are obtained, the overall FSMCI index will be calculated using the same equation and based on the results obtained in Once the FSMCI is obtained, the Euclidean approach will be utilized to match the FSMCI with sustainability status terms Using the Euclidean distance EquationUsing the same equation, the distances will be obtained as shown in Once the distances are all obtained, the minimum distance will be selected as the current status for sustainability in the associated manufacturing company. The results show that the FSMCI has the minimum distance with the status “Normally Sustainable” so this status will be announced for this particular manufacturing company.To improve the overall sustainability, there is no need for infrastructural modifications, while improving the weak points, they increase the overall sustainability significantly. These improvements firstly depend on the identification of weak points of sustainability.In this segment, a unified method is elaborated called fuzzy performance importance index (FPII) that help in identifying the obstacles. The FPII of all attributes is calculated and shown in To make these trapezoidal fuzzy sets prepared for the sieving process, the ranking score should be obtained for these FPIIs. A ranking method for TrFSs named magnitude of trapezoidal fuzzy set proposed by Abbasbandy and Hajjari  which isUsing the same equation, the ranking score for every single FPII obtained and shown in The weak points can be identified using Eight sub-criteria out of fifteen seem to be the obstacles of this case. These weak points are distinguished in Following the desire of firms and organizations in performing sustainability practices for remaining in the competitive marketplace, the first step is to determine the current sustainability status in the associated enterprise/system. This study originally adds to the state-of-the-art literature of sustainability in manufacturing systems by encapsulating a proposal and applications of a new and fuzzy sustainable manufacturing company index (FSMCI) which provides a benchmark for sustainability assessments in manufacturing companies.Meanwhile, this study proposes a conceptual model by accumulating 33 related sub-criteria and classifying them into five criteria of sustainability. Hence, the proposed framework can be considered as a benchmark approach for developing sustainability practices in varied systems. In other words, by collecting related sub-criteria and refreshing As a theoretical implication, the proposed model shown in Another worthwhile theoretical implication to be mentioned is that it motivates researchers to generalize the proposed method for other industries by following the steps shown in In light of the proposed methodology, the current status of sustainability for manufacturing companies will be determined straightforwardly. Given this, it can be implemented as an entrepreneurship project by involving several scholars and managers as a team for determining/improving the sustainability of manufacturing companies.Taking advantage of the proposed framework, policymakers can make policies that enhance the quality of sustainability in manufacturing companies and subsequently there will be quite a lot of advantages obtained including the prevention of waste, reducing the consumption of natural resources, promotion of reusing materials, etc.Due to hard computation processes, the previous MCDM approaches were ignored to be utilized in practice. This approach contributes in solving this problem by proposing a soft computing method whilst handling a good amount of uncertainty.The main implications on the theory and practice of this study can be highlighted as follows:According to the vast amount of literature which has been reviewed in the second section, most of the studies considered environmental and economic practices to maximize profit whilst minimizing the harmful impacts on the environment. The findings of this study motivate policymakers to consider social factors such as “work intensity”, “Health and safety”, and “Philanthropy” which have been taken into account less than other factors. In the meantime, this study contributes to consider “Environmental management” as a criterion for environmental aspect of sustainability. This will encourage managers to hire environmental managers and use the knowledge of them to enhance the quality of environmental-oriented practices.➢The model considered five dimensions for sustainability including social, environmental, economic, technological advancement, and performance management. In addition, thirty-three sub-criteria were collected from the literature and have been classified into those five criteria to obtain the fuzzy sustainable manufacturing company index (FSMCI).➢The trapezoidal fuzzy membership function was adopted to incorporate the impreciseness of data through sustainability assessment by which DMs are able to express their opinion based on subjective preferences instead of using risky deterministic numbers.➢DM. However, the importance weights of criteriaThe model led us to identify the weaker sub-criteria that are weakening the sustainability performance in manufacturing systems. In the particular case, eight obstacles were identified out of a total of fifteen sub-criteria. Enhancing the performance of these obstacles will affect the overall sustainability level and improves it.Despite the increasing amount of research and broad amount of literature with the background of sustainable manufacturing, the development of previous approaches is still a challenging concept. In this study, we proposed an integrated method to assess and obtain the sustainability status in manufacturing companies. In addition, the proposed model conducted a two-layered approach to amend the sustainability index by identifying the weak points in a manufacturing company. This paper developed a generic fuzzy-based assessment approach for measuring the current sustainability status in manufacturing companies by encapsulating thirty-three sub-criteria, collected from the literature and classified in five criteria. The major contributions of the proposed approach can be highlighted as follows:The summarized fuzzy sustainable manufacturing company index’s calculations includes: select the relevant sub-criteria based on the objectives of organization and determine fuzzy weights, assign linguistic performance based on expert’s judgement, obtain the overall FSMCI based on selected criteria/sub-criteria, match the FSMCI with sustainability status terms using Euclidean distance method to determine the current status of sustainability, calculate fuzzy performance importance index for every single sub-criterion to recognize weak points to improve the sustainability index, and finally modify the process to reach the maximum sustainability.There are also some limitations of this study that can be mentioned as follows: firstly, the linguistic approach which has been adopted for specifying the weights of criteria operates based on the subjective preference of experts which may cause inaccuracies in the final results. This is suggested to use one of weighting approaches such as AHP or BWM for obtaining the optimal weights. Another limitation to be mentioned is that this study proposes a generic framework for obtaining the status of sustainability. Given that, some criteria might be ignored in some cases. The authors suggest that practitioners first evaluate the criteria for the particular system and then apply the proposed calculations.