IntroductionThe Golden Hour is a term used in the trauma setting to refer to the first 60 minutes after injury. Traditionally, definitive care within this period was believed to dramatically increase a patient’s survival. Though the period of 60 minutes is unlikely to represent a point of distinct inflection in survival, the effect of time to definitive care on survival remains incompletely understood. This study aims to measure the association of time to definitive hemostasis with mortality in patients with solid organ injuries as well as the effect of survival bias and a form of selection bias known as indication by severity on the relationship between time to treatment and survival.MethodologyThis is a retrospective cohort study using data obtained from the American College of Surgeons National Trauma Data Bank (NTDB) from the years 2017 through 2019 selecting patients treated for blunt liver, spleen, or kidney injury who required angioembolization or surgical hemostasis within six hours. A Cox proportional hazards regression was used to analyze time to death. The association of probability of death with time was examined with a multivariate logistic regression initially treating the relationship as linear and subsequently transforming time to hemostasis with restricted cubic splines to model a non-linear association with the outcome. To model survival and indication by severity bias, we created a computer-generated data set and used LOESS regressions to display curves of the simulated data.ResultsThe multivariate Cox proportional hazards analysis shows a coefficient of negative 0.004 for minutes to hemostasis with an adjusted hazard ratio of 0.9959 showing the adjusted hazard of death slightly diminishes with each increasing minute to hemostasis. The likelihood ratio chi-square difference between the model with time to hemostasis included as a linear term versus the model with the restricted cubic spline transformation is 97.46 (p<0.0001) showing the model with restricted cubic splines is a better fit for the data. The computer-generated data simulating treatment of solid organ injury with no programmed bias displays an almost linear association of mortality with increased treatment delay. When indications by severity bias and survival bias are introduced, the risk of death decreases with time to hemostasis as in the real-world data.ConclusionDecreasing mortality with increasing delay to hemostasis in trauma patients with solid organ injury is likely due to confounding due to indication by severity and survival bias. After taking these biases into account, the association of delayed hemostasis with better survival is not likely due to the benefit of delay but rather the delay sorts patients by severity of injury with those more likely to die being treated first. These biases are extremely difficult to eliminate which limits the ability to measure the true effect of delay with retrospective data. The findings may however be of value as a predictive model to anticipate the acuity of a patient after an interval of unavoidable delay such as with a long transfer time. Blunt abdominal trauma is a major cause of morbidity and mortality in the United States. The most commonly injured solid organs are the liver and spleen, followed by the kidney and pancreas . Solid oIntuitively one would expect prompt interventions to be more beneficial. This is a familiar concept in trauma known colloquially as the Golden Hour, referring to the first 60 minutes after injury in which achieving homeostasis has been thought to dramatically increase survival. Recent evidence has exposed the golden period of precisely 60 minutes as largely a myth but the genuine effect of time to definitive care on survival remains incompletely understood. Some studies have shown a direct correlation between delayed time to hemostasis and increased morbidity while others have aroused skepticism showing minimal correlation between survival and treatment delay .This study has two aims: first to measure the association of time to definitive hemostasis with mortality in patients with solid organ injuries. Measuring the probability of survival continuously over time will allow us to identify any inflection of the survival curve at one hour or any other time. Second, we will examine the effect of survival bias and a form of selection bias known as indication by severity bias on the relationship between time to treatment and survival. Our hypothesis is that shorter time to hemostasis is associated with improved survival in blunt solid organ injury patients who require procedural hemostasis.Study designThe first part of the analysis is a retrospective cohort study measuring the association of mortality with time to hemostasis for solid organ injuries. The next part of the analysis is a computer simulation created to analyze the shape of the survival curve.Data collectionData for this analysis were obtained from the American College of Surgeons National Trauma Data Bank (NTDB), which compiles patient information from over 900 trauma centers across the United States. The years 2017 through 2019 were chosen.Inclusion and exclusion criteriaWe selected patients with blunt liver, spleen, or kidney injury who required angioembolization or surgical hemostasis within six hours. We selected the six-hour time frame because early trauma deaths are generally due to bleeding; hence, the mortality for these patients should be mostly influenced by the timing of hemostasis. Limiting the analysis to the early period also makes the data less skewed and eliminates outliers. We use the time of arrival at the hospital as time zero. The time of trauma is not available in the NTDB, and transport times are recorded less consistently than hospital arrival. Statistical analysisDescriptive data are displayed for baseline characteristics and the distribution of time to hemostasis. To analyze time to death, we used a Cox proportional hazards regression with the outcome of mortality. Time to hemostasis was entered as a time-dependent variable. Fixed covariates include age, sex, injury severity score (ISS), initial systolic blood pressure, liver, spleen, and kidney severity scores, and units of packed red blood cells and plasma transfused within the first four hours.Examination of the non-linear association of the probability of death with time to hemostasis was performed with multivariate logistic regression. We used a simplified model with death as the outcome and time to hemostasis as a predictor of interest, controlling for covariates of sex, age, and ISS. Covariates were chosen based on a priori importance and on the Wald test for the contribution of individual variables to the model.The logistic regression was performed initially treating the probability of death versus time to hemostasis as linear with no transformation. Restricted cubic splines with five knots were then used to transform the predictor of interest (time to hemostasis) to a model involving a non-linear association with the outcome. We used this regression to generate a line graph with confidence intervals displaying the relationship between time to hemostasis and the probability of survival. To compare the goodness of fit of the models with time to hemostasis as linear versus non-linear terms, we calculated the likelihood ratio chi-square difference between the model with time to hemostasis included as a linear term versus the model with the restricted cubic spline transformation. Statistics were performed with R and reviewed with the department statistician.Computer simulationPotential biases arise when the severity of the injury influences the time to hemostasis. To model these biases, we next created a computer-generated dataset. A Python script was used to simulate a cohort of 12000 patients of whom 9000 are actively bleeding. Time of observation is divided into 36 intervals. We specify a mortality per interval between 1 and 5 percent with random variation. All patients underwent a hemostasis procedure. The time to procedure is set to random times across the range of intervals. The procedure reduces the mortality per interval by 95%. All patients are included in the dataset even if they die before the procedure. We then used the computer-simulated data to demonstrate the effects of survival bias and of confounding by indication by severity of injury. First, we introduced the indication by severity bias. We simulated this by dividing the time to hemostasis by the percent mortality per interval. Thus, the time to hemostasis is proportionately shorter for injuries with a greater risk of death. This simulates providers acting faster for more severe injuries and selecting sicker patients for more prompt treatment.Next, survival bias was introduced by eliminating all the patients who died before scheduled hemostasis. Patients with more lethal injuries die sooner, thus those still alive and present at later intervals tend to have less serious injuries. Finally, we combined the simulated indication by severity and survival biases into a single analysis.To display simulated data, we use LOESS regressions to smooth the curves of the generated plots. We do not require a mathematical model such as logistic regression with splines for this purpose because we are not interested in making inferences about the simulated data but only in showing the visual changes in the curves.Nine thousand sixteen patients with a solid organ injury required a hemostatic procedure (HP). Tables Figure The multivariate Cox proportional hazard analysis with admission to hemostasis as a time-dependent variable with the additional covariates listed in the methods shows a coefficient of negative 0.004 for minutes to hemostasis. The adjusted hazard ratio is 0.9959. 95% confidence interval is . The likelihood ratio test is 1947 on 10 degrees of freedom (p=<0.001). The coefficient represents the expected change in log hazard of death associated with a one-minute increase in time to HP. The adjusted hazard ratio is the adjusted hazard of death with each increasing minute of time to HP. The negative coefficient shows that the adjusted hazard of death slightly diminishes with each increasing minute to HP.The multivariate logistic regression that treats the probability of death versus time to hemostasis as linear with no transformation shows a coefficient of negative 0.0054 for time to hemostasis. Standard error of the coefficient is 0.0004. The odds ratio of death for each increasing minute to HP is 0.995. Comparing the odds of death at each interquartile boundary shows the adjusted odds of death at 211 minutes versus 104.4 minutes is 0.56 . The model that uses restricted cubic spline transformation of time to hemostasis to account for non-linear probability of outcome over time shows a coefficient of negative 0.0185 for minutes to hemostasis. Standard error of the coefficient is 0.0025. The odds ratio of death for each increasing minute to HP is 0.982. Comparing the odds of death at each interquartile boundary shows that the adjusted odds of death at 211 minutes versus 104.4 minutes is 0.53 . As in the Cox analysis, the odds of death diminishes with time to HP for the multivariate logistic regressions.The likelihood ratio chi-square difference between the model with time to hemostasis included as a linear term and the model with the restricted cubic spline transformation is 1367.53-1270.07 = 97.46. This chi value corresponds to a p-value of <0.0001. This shows that the model with restricted cubic splines is a better fit for the data.Figure Figure Figure Figure Figure The term Golden Hour was coined in the 1970s by military physician Robert A. Cowley after his experience treating injured soldiers in a field trauma setting. Dr. Cowley used his experience in combat to pilot numerous trauma centers around the United States, ultimately founding the Shock Trauma Institute in Baltimore. It was his expertise in prioritizing transport and triage that helped give rise to the trauma care system we know today. His “Golden Hour” concept was originally described as a golden period which referred to the first six hours after injury; however, now it is commonly thought of as the first 60 minutes after injury. Cowley argued that achieving hemodynamic stability within the golden period gave trauma patients a dramatically higher chance of survival . Dr. CowWhile it is logical that prompt treatment should benefit trauma patients, the literature shows an inconsistent relationship between time to definitive treatment and survival. Some studies show that shorter delay is associated with better survival ,7-11. OtOur findings show a pattern that is almost the inverse of a golden period. The odds of death decrease with increased delay to hemostasis with an infection at about two hours. These results appear counterintuitive. A causal inference would be that delaying hemostasis improves survival, which is not medically plausible. One explanation for such heterogeneity and paradoxical results in the literature is that trauma is heterogeneous. Disparate results are expected because survival depends on many factors including the type of injury, severity of injury, duration of elapsed time, effectiveness of definitive therapy, and effectiveness of field and emergency department stabilization. In addition to complex predictive factors, biases play a role in the interpretation of survival data.The second part of our study focuses on two biases that strongly influence outcomes in trauma: survival bias and confounding by indication by severity. Survival bias occurs when patients who are less severely injured are more likely to survive to receive treatment at a later time. This idea is explained by Anderson et al. who describe “immortal time bias” whereby all patients within a cohort who are selected for an exposure are essentially immortal until they receive that exposure. In this case, the immortal time period is the time between the trauma and the hemostatic intervention . PatientConfounding by indication occurs when a covariate predicts both treatment and outcome . When thThese biases are known to affect the results of epidemiologic trauma research . To furtWith awareness of these biases, we can use statistical methods to reduce their effect. Stratifying comparisons by injury and employing multivariate analysis can be used to control confounding by severity. Previous studies have shown that some highly lethal injuries are time-sensitive when analyzed this way ,9,17. LiOur analysis as well as the analyses seen in other studies shows that we can control for some elements of selection and survival bias, however we cannot completely eliminate them. In an epidemiologic study designed to test methods of controlling indication bias, Bosco et al. analyzed the effect of adjuvant therapy on breast cancer. In describing what they termed a “most stubborn bias”, they employed multiple statistical methods to control confounding by indication but found that they could not completely eradicate imbalance between treatment groups .Recognition that these biases are universal and incompletely controlled will help us interpret the measured association between time to treatment and survival. One implication is relevant to field stabilization versus rapid transport. Better survival after longer delay could be attributed to the benefit of more time devoted to field stabilization, but this conclusion may not be justified. If no treatment whatsoever were rendered in the field, the above-described biases would still produce an association between increased transport time and lower in-hospital mortality. Our results can be used as a predictive rather than explanatory model. Patients received after long prehospital time or transfer are likely to be more stable because the sicker patients have already been eliminated. This expectation could factor into triage and activation criteria. Limitations We used time of arrival rather than time of trauma as the beginning of the time to hemostasis interval. Time of arrival is a less accurate measure of total time to hemostasis, but it was more consistently recorded, and it is relevant for delays within the hospital. We did not stratify patients by the type of solid organ injury or method of hemostasis. Delays might have had a different effect on a subset of our population. The protective association of delay to HP with death is probably still influenced by selection bias with unknown/unmeasured risk factors. Time to hemostasis may be based on radiologic findings, hemodynamic trends, initial lab values, response to volume resuscitation, pressor requirements, and clinical indices of perfusion. There may be some residual survival bias. When the delay to procedure is not random, Cox analysis cannot control for timing of treatment due to underlying risk; thus, survival bias can still exist .Mortality decreased with increasing delay to the HP in trauma patients with solid organ injury. The association persisted after controlling for multiple confounders including time-dependent covariates. This counterintuitive result is likely due to confounding by severity and survival bias. Our computer simulation shows how severity and survival bias can produce a survival curve that shows a beneficial association of survival with increasing time to treatment, even when delay actually diminishes survival. Our interpretation after taking these biases into account is that delay to hemostasis does not improve survival but rather sorts the patients by severity of injury with those more likely to die being treated first.These biases are difficult to eradicate, even with multivariate statistical models controlling for time-dependent covariates and multiple injury characteristics. More sophisticated models will help reduce these biases but not eliminate them completely. Our results may be of value as a predictive model to anticipate the acuity of a patient after a given interval of unavoidable delay such as a long transfer time. We recommend caution in the interpretation of delay to treatment as an explanation of outcomes in studies based on retrospective data.


Digital pathology offers the potential for significant benefits in diagnostic pathology, but currently the efficiency of slide viewing is a barrier to adoption. We hypothesised that presenting digital slides for simultaneous viewing of multiple sections of tissue for comparison, as in those with immunohistochemical panels, would allow pathologists to review cases more quickly.Novel software was developed to view synchronised parallel tissue sections on a digital pathology workstation. Sixteen histopathologists reviewed three liver biopsy cases including an immunohistochemical panel using the digital microscope, and three different liver biopsy cases including an immunohistochemical panel using the light microscope. The order of cases and interface was fully counterbalanced. Time to diagnosis was recorded and mean times are presented as data approximated to a normalised distribution.Mean time to diagnosis was 4 min 3 s using the digital microscope and 5 min 24 s using the light microscope, saving 1 min 21 s , using the digital microscope. Overall normalised mean time to diagnosis was 85% on the digital pathology workstation compared with 115% on the microscope, a relative reduction of 26%.With appropriate interface design, it is quicker to review immunohistochemical slides using a digital microscope than the conventional light microscope, without incurring any major diagnostic errors. As digital pathology becomes more integrated with routine clinical workflow and pathologists increase their experience of the technology, it is anticipated that other tasks will also become more time-efficient. Over the past few years, digital pathology has been deployed for primary diagnosis in a few flagship projects around the world. Our institution now routinely digitally scans 100%Of late, much of the research has been concerned with diagnostic accuracy as the introduction of a new technology must not compromise patient safety. Many single papers and a systematic reviewPreliminary work from our group showed that early whole slide imaging (WSI) viewers were 60% slower than the microscope,A more recent development of this software though allows for simultaneous viewing of multiple sections of tissue for comparison, something that is simply not possible with light microscopy. With as little as 61.9% of a pathologist’s time spent viewing an image when at the microscope,We therefore designed an experiment to compare the time with diagnosis using a digital system with the microscope. The digital system minimises the effort to manually switch between separate slides, instead offering the user a one-touch method of reviewing the case. It is hypothesised that the use of an appropriately designed digital pathology workstation can offer a reduction in time to reach a diagnosis without compromising diagnostic confidence.A purposive sampling strategy was employed to recruit 16 participants from within our institution: 8 senior trainee histopathologists and 8 consultant histopathologists from a range of subspecialist fields (not including liver pathology). Pathologists were recruited in person and those included were those who were approached and were willing to be involved. This study took place prior to the digitisation of our department resulting in a wide range of experience of using a digital microscope, from very little to many days’ cumulative experience.All participants were asked to view three cases using the digital microscope, and three different cases using the light microscope. The order of cases and interface was fully counterbalanced.All cases were liver needle core biopsies of tumours, with clinical details available in 10.1136/jclinpath-2021-207961.supp2Supplementary dataEach case contained one or more H&E-stained slides, as well as multiple slides stained with immunohistochemical stains that were relevant to the case. An example of the slides for a case can be seen in Before the experiment was undertaken, each participant was given a 15-minute training session using the digital microscope. A standard training session was divided into three sections: section one in which the researcher would show the participant how to use the software, section two where the participant would use the software themselves and the researcher would evaluate their use of it, and section three where they were asked to perform a diagnostic task.All trials took place in a quiet, windowless room in the histopathology department of our institution usually used for teaching purposes. The digital microscope was placed at one end of the room with a light microscope set up on an adjacent table. The only light source was a standalone lamp placed next to the door behind the participant, to standardise the effect of ambient lighting on the display as far as possible at 10 lux.A Dell Precision T5500 with AMD W5000 graphics card was used for this experiment. A Barco Coronis Fusion (6 MP) display  was used in conjunction with a Barco Nio (2 MP) display . The larger 30-inch screen was a split screen setup, the left side displaying the H&E slide (which was in constant display) and the right displaying the immunohistochemical slide. The smaller screen 21-inch on the right side of the participant displayed thumbnail images of all slides and highlighted which slide was currently in view. Viewing software was the Leeds Virtual Microscope.The light microscope was a Leica DMR microscope  with ×2.5, ×5, ×10, ×20, ×40 and ×100 objectives and ×10 eyepiece. The microscope lens was reset by the researcher to the lowest magnification at the start of each trial (×2.5 magnification). The participants were provided with a practice slide in order to familiarise themselves with the microscope prior to the experiment.The experiment was recorded using a three-video camera setup: one captured a ‘down-the-microscope’ view from the microscope camera mount, one captured the microscope work area (placed in the corner of the room furthest from the microscope) and the other directly captured the participants’ face and body from in front while at the digital workstation . Timing began when they picked up the first glass slide and finished when the set was completed. The video data were analysed to capture each instance of the participant interacting with the slide, panning, zooming, using the microscope condenser and writing notes. A screenshot of the participant video recording with the synchronised view down the microscope can be found in 10.1136/jclinpath-2021-207961.supp1Supplementary dataStatistical analysis was performed in Stata V.16. Data approximated a normal distribution and therefore are summarised by the mean and SD.As analyses indicated a wide variation in time to diagnosis according to the case, a normalised time to diagnosis  is reported as the primary outcome measure. We also report actual time to diagnosis for ease of understanding.Multiple linear regression was used to estimate the time to diagnosis in minutes adjusting for the binary fixed effects of experience level (trainee vs consultant) and interface . CIs were generated to the 95% level. A sensitivity analysis of variance (ANOVA) was also performed on normalised time with the within-subject factor being light microscope versus digital microscope and between-subject variable being experience level (trainee vs consultant).A summary of the main results can be seen in In terms of overall results, the mean time to diagnosis was 4 min 3 s using the digital microscope and 5 min 24 s using the light microscope, as shown in Overall, normalised mean time to diagnosis was 85% on the digital pathology workstation compared with 115% on the microscope; a relative reduction of 26% , as can be seen in When subcategorising the results by experience, the mean time to diagnosis for trainees using the digital microscope was 3 min 31 s, and 5 min 25 s using the light microscope. This equates to a time-saving using the digital microscope of 1 min 54 s . The mean time to diagnosis for consultants using the digital microscope was 4 min 35 s, as compared with 5 min 24 s using the light microscope. This results in a non-statistically significant time-saving using the digital microscope of 0 min 48 s . The normalised mean time to diagnosis for trainees was 74% and 116% compared with consultant times of 96% on the digital microscope and 114% on the light microscope, respectively. This equates to a reduction of 42%  for trainees, and again a non-statistically significant reduction of 18%  for consultants.When evaluating the results by set, the mean time to diagnosis for set A across all participants using the digital microscope was 3 min 13 s, and 5 min 11 s using the light microscope. This equates to a time-saving using the digital microscope of 1 min 58 s . The mean time to diagnosis for set B across all participants using the digital microscope was 4 min 53 s, and 5 min 38 s using the light microscope, a difference which was not statistically significant . The mean normalised time to diagnosis for set A was 77% compared with 123%, set B 93% compared with 107% on the digital microscope and the light microscope, respectively. This equates to a statistically significant reduction in normalised time to diagnosis per case for set A of 47% , but again a non-statistically significant difference of 14% .When combining the data across both interfaces, the data are summarised in The mean time to diagnosis by trainees was 4 min 29 s, as compared with consultants with a mean time of 5 min 0 s. Therefore, trainees were faster across both modalities by a mean time of 31 s, but this difference was not significant . Similarly, when combining data across both interfaces, the mean time to diagnosis for set A cases was 4 min 13 s, and set B was 5 min 16 s, which represents a non-significant mean difference of 1 min 6 s .Results of multivariable linear regression adjusting for experience level were largely the same as the unadjusted results presented above. Time to diagnosis on the digital microscope was 1 min 21 s faster than on the light microscope . Consultants took 31 s longer to reach diagnoses than trainees, but this was not statistically significant . There was an adjusted reduction in normalised time to diagnosis of 30% on the digital microscope as compared with the light microscope . There was an adjusted reduction in normalised time to diagnosis by trainees of 10% as compared with consultants but this was, again, non-significant . Similarly, there were no notable differences in the results of the sensitivity ANOVA performed on actual and normalised time, the details of which can be found in 10.1136/jclinpath-2021-207961.supp3Supplementary dataThere were no major diagnostic errors made on either interface. Two participants gave a discordant diagnosis of hepatocellular carcinoma in a case of probable metastatic carcinoma—in set B. Review of the case revealed some cytological features that might support such a diagnosis and further immunohistochemistry (IHC) might be required to entirely rule it out, making the true diagnosis ambiguous. Therefore, either diagnosis was acceptable for the purposes of this study.The significant difference in time to diagnosis demonstrates that for this set of diagnostic tasks, the digital microscope was quicker with a time-saving of 1 min 21 s per case, or a 26% relative reduction in time to diagnosis. As far as we are aware, this is the first study evaluating the impact of the digital microscope on time to diagnosis for cases involving immunohistochemically stained slides.A mean reduction of 1 min 21 s per case using the digital microscope is a considerable time-saving in a health service that is struggling under ever-increasing demand for services. Previous work from our group has estimated the number of cases in our institution requiring extra stains to be approximately 5%. Given that our institution handles approximately 60 000 surgical cases per year, the use of the digital microscope for these cases alone may result in a time-saving of 67.5 histopathologist hours over the course of a year within our institution. This saving is likely to be higher in institutions where immunohistochemical or special stains are used more frequently.The effect of increased experience of many of the specialist trainee participants with digital system was possibly reflected in the results, with the trainee cohort being just over 1 min quicker than the consultants on the digital pathology workstation per case, but this difference was not statistically significant . Alternatively, this may reflect the fact that trainee histopathologists were more familiar with liver specimens, as no consultant liver histopathologists were included in this study. However, no major diagnostic errors were made, thus suggesting unfamiliarity had no overall impact. Moreover, the use of only liver biopsies across the two interfaces will have prevented unfamiliarity with the type of case from biasing the results.The time to diagnosis for set B was longer than the time to diagnosis for set A, although not statistically significant . This may have been due to some diagnostic difficulty surrounding one case; a diagnosis of hepatocellular carcinoma was made twice during the trial and given as a differential diagnosis once. If this was given only on the digital microscope, this may pose questions regarding image quality and the regulations regarding the use of digital slides for diagnostic work. However, these diagnoses were made on both interfaces: once on the digital microscope and the second on the light microscope. This would lead us to believe that these mistakes were due to the diagnostic difficulty surrounding of that case, as opposed to issues regarding image fidelity on the digital microscope.Participants largely reported positive experiences using the digital microscope. A large proportion of participants used the system as intended and as was shown in the training session. However, some opted for a different technique, and rather than using the digital microscope at low power to identify an area of interest and then zoom, many participants used the digital microscope at very high magnification and scrolled the whole length of the image. This technique is inevitably time-consuming and may be due to unfamiliarity of the digital microscope for some participants; time reduction per case will likely increase with continued use and increased experience.Fourteen participants reported that they found the two-screen digital microscope very useful for comparing areas of tissue side by side and aiding a diagnosis. Two participants commented on the controls and the ergonomics of the mouse and keyboard design and felt that a less cumbersome method of panning and switching between slides needed to be implemented. One participant commented that this implementation of a digital microscope to look at immunohistochemical cases was the best they had used to date. It is unsurprising that comments were not completely unanimous regarding the digital microscope workstation; it is well known that there is not a ‘one-size-fits-all’ workstation for digital radiologists.The design features of the digital microscope are a major strength to this study. The use of medical grade displays with high technical specifications enables more tissue to be viewed on the screen at one time reducing the need for interaction by the histopathologist (pan/zoom) for this low power assessment task; an 8-megapixel display shows approximately the same amount of tissue as a ×10 light microscope eyepiece. A high luminance and contrast ratio increases, the ability of the pathologist to make a confident diagnosis at low magnification. This, in combination with a viewer that facilities fast viewing, is likely responsible for the considerable time-saving afforded by the digital microscope in this study as this reduces refreshing time. Additionally, side-by-side viewing reduces cognitive load and streamlines the process of referencing the H&E WSI to check an area for relevance; light microscopy demands that pathologists remove the IHC glass slide, load the H&E glass slide and navigate to the area of interest, which is challenging and time-consuming.Previous work from our group did not demonstrate a time-saving from increased screen resolution,et alet alet alAnother study by Hanna et al.The effect of working digitally does not just result in potential time-savings in time to diagnosis, but instead impacts the entire laboratory workflow. Although it is outside of the scope of this work to discuss the impact of a whole system evaluation of digitised pathology services, this has recently been addressed by the work of Baidoshvili Inevitably, there were several limitations to our work. First, this was a small study with known considerable user variation. Second, the time difference to diagnosis will be affected by the participant’s familiarity with the case type. Although this should not impact the primary outcome in this study (difference between the two interfaces), having participants diagnosing case types that they are familiar with may be more reflective of the time-savings observed in routine clinical practice. It should also be noted that many routine cases do not require immunohistochemical slides to review and therefore the observed time-saving may not be applicable to these cases. Third, it would be advantageous to spend longer familiarising participants with the digital microscope; familiarity with one interface and not the other will inevitably bias the results in favour of the familiar interface. Lastly, due to the prototype nature of the digital microscope, the usability could be improved. There were some issues regarding slide registration , which proved particularly problematic for participants who were less adept with the digital microscope. Further, time-saving will be likely observed as the digital microscope becomes more user-friendly.To conclude, the digital microscope reduced time per case by 1 min 21 s per case and a relative reduction of 26%, without any major diagnostic errors as compared with the light microscope. This is likely due to the ability to view multiple slides simultaneously, which is not possible using analogue systems. We anticipate that these time-savings will have a major improvement on pathologist productivity at a time where pathology services are strained, and serve as a point from which to build other user interfaces to enhance pathologist productivity.Digital pathology may offer benefits over microscope in viewing whole slide images—one unique capability is the ability to review side by side, with synchronised pan and zoom.We designed a pragmatic study looking at evaluation of liver biopsy cases including an immunohistochemical panel, where serial comparisons are needed.We used a custom viewer that allowed side-by-side viewing, and rapid review of immunohistochemical images in the sequence.We found that mean time was 5 min 24 s on the light microscope and 4 min 3 s on digital microscope, a reduction of 1 min 21 s  and a relative reduction of 26%. These benefits were seen with relatively little training and exposure to the system, and further work is needed to evaluate the real-world impact.


Clinical Neuroradiology also slightly dropped with the impact factor from 3.156 to 2.8  but many other factors.Clinical Neuroradiology to establish this journal as one of the leading journals in the field.We would like to thank all authors and reviewers for their continuous support of Martin BendszusEditor-in-Chief


Microcantilevers are at the heart of atomic force microscopy (AFM) and play a significant role in AFM‐based techniques. Recent advancements in multifrequency AFM require the simultaneous excitation and detection of multiple eigenfrequencies of microcantilevers to assess more data channels to quantify the material properties. However, to achieve higher spatiotemporal resolution there is a need to optimize the structure of microcantilevers. In this study, the architecture of the cantilever with gold nanoparticles using a dip‐coating method is modified, aiming to tune the higher eigenmodes of the microcantilever as integer multiples of its fundamental frequency. Through the theoretical methodology and simulative model, that integer harmonics improve the coupling in multifrequency AFM measurements is demonstrated, leading to enhanced image quality and resolution. Furthermore, via the combined theoretical‐experimental approach, the interplay between induced mass and stiffness change of the modified cantilever depending on the attached particle location, size, mass, and geometry is found. To validate the results of this predictive model, tapping‐mode AFM is utilized and bimodal Amplitude Modulation AFM techniques to examine and quantify the impact of tuning higher‐order eigenmodes on the imaging quality of a polystyrene‐polymethylmethacrylate (PS‐PMMA) block co‐polymer assembly deposited on a glass slide and Highly Ordered Pyrolytic Graphite (HOPG). Gold nanoparticles are precisely deposited on a cantilever's surface through integrated drop casting dip‐coating approach, adjusting higher eigenmodes to align with integer multiples of the first eigenmode. Through statistical and image analysis methods, it is shown this customized cantilever enhances image quality and resolution during the imaging of block co‐polymer PS‐PMMA surfaces using AFM tapping mode and bimodal amplitude modulation techniques. Among microscopy techniques including optical, confocal, electron, and near‐field scanning optical and atomic force microscopies, due to its applicability in ambient conditions, ease of use, high resolution, and ability to perform both imaging and characterization, atomic force microscopy (AFM) and its modalities have become the unique remarkable technique.Imaging and nanoscale characterization of the organization, structure, interfaces, and properties of the emerging engineering, functional and biological materials such as 2D, topological, quantum, hyperbolic, and carbon‐based materials4, 5Microcantilever is a critical element of AFM whose properties determine the speed and resolution of images as well as the identification of materials properties.14 The primary principle of AFM operation is classified into two modes of operation: contact or static and dynamic mode. In most modern studies, the dynamics AFM is preferred, as the microcantilever is externally vibrated to enhance sensing accuracy while better preserving the tip and the sample. More recently, based on the principle of dynamics mode AFM, through simultaneous excitation and detection of eigenfrequencies of microcantilevers several advanced techniques called multifrequency AFM are developed to explore more information from the materials during imaging. Due to the dynamic nature of advanced AFM techniques, during the nonlinear interaction between the cantilever‐tip ensemble and specimen, higher harmonics of the fundamental frequency of the vibrating cantilever emerge. The main source of these harmonics’ responses is short‐range attractive and repulsive forces as well as contact time. As a consequence, these harmonics carry plentiful information on the mechanical, physical, and chemical properties of the specimen. Recent studies demonstrated the harmonics of microcantilever signal are highly sensitive to the local elasticity of the materials, and higher contrast with detailed imaging were achieved from the higher harmonics of the microcantilever signal rather than its fundamental frequency. Hence, using harmonic amplitude images with higher spatial resolution can open a path for fast, robust, and non‐invasive mapping of mechanical, physical, and chemical properties of advanced materials at the nanoscale.The principle of AFM performance is based on the deflection and/or vibration of the microcantilever utilized to probe the tip‐surface interaction. Depending on the type and range of interaction as well as the time scale of measurements, AFM can provide a diverse range of information such as atomistic and molecular structure of the surface, and physical, chemical, thermal, mechanical, electronic, and magnetic properties of specimen.15 The To enhance the higher harmonics, several efforts have been performed either by exciting the cantilever at a submultiple of its natural resonance frequency or by driving its torsional mode as well as using multifrequency AFM via coupling the eigenmodes to increase the harmonics of first eigenmode. Although these methods could enhance the specific harmonics of the microcantilever signal, the way the cantilever is excited affects the quality of the captured topography images which is based on the excited frequency to which the feedback controller is applied. So, rather than an excitation strategy, it may be better to optimize the microcantilever structure as a force sensor. Recent studies demonstrated that changing the mass distributions of the microcantilever via either adding or removing local masses is the most convenient way to optimize the performance of the harmonics.However, the main challenge in the detection of higher harmonics signals is obtaining a reasonable signal‐to‐noise ratio (SNR).27 To  tuned the third flexural eigenmode to the 16th harmonic of the natural resonance frequency by removing the mass from the cantilever.Via this approach, the higher eigenmodes of the cantilever can be tuned to be an integer multiple of its fundamental frequency which leads to the enhancement of the corresponding harmonics response. Based on this methodology, Sahin et al,34 tun However, in addition to design and fabrication complexity, there is a significant change in the effective stiffness of the modified cantilever especially at a small frequency ratio. Cai et al, modified the standard cantilevers by varying the width of the cantilever as well as adding a step cross‐section to the cantilever. The fabrication process of these designs is rather complex and results in difficulty in the geometrical control in manufacturing.Felts et al. introduced a paddle structure on the cantilever to tune the ratio of the first two eigenfrequencies in a diverse integer range.35 How Zhang et al. tuned the frequency properties by altering the mass distribution of the cantilever by cutting a rectangular slot. However, this approach makes a hole in the cantilever which affects its dynamic response in ambient conditions and liquids especially when the cantilever is squeezed by interfacial liquid at the solid surface or when the environment is humid.To enhance the harmonic response of microcantilevers, following Sahin's approach,34 Zha and Xiang et al. attached a concentrated particle to the cantilever to simultaneously tailor the eigenfrequencies. However, to optimize tuning, the heavy attached particle is required. Furthermore, in their modelling approach, they just consider the changes of eigenfrequencies attributed to the added mass of the particle which leads to the decrease of eigenfrequencies while the primary mechanism of the resonant frequency change occurs through both induced mass‐change and spring constant‐change (stiffness‐change) and can lead to either increase or decrease of eigenfrequencies. Despite the mentioned valuable efforts, there is a lack of comprehensive understanding of the effect of the attached particle geometry and its mass, the cantilever structure and the interplay between the mass distribution and the stiffness of the cantilever in decreasing/increasing eigenfrequencies and the simultaneous tuning of the higher eigenmodes to the fundamental frequency ratio to an integer value.Li et al.38 andIn this paper, we detail our straightforward and comprehensive theoretical, numerical, and experimental methodology to model, analyse and optimize the tailored cantilevers.Using our presented approach, we study the effect of the interplay between the induced stiffness and mass changes of the attached particle on the frequency shift and tuning the eigenfrequencies of the microcantilever. Our simulation predicts that the nature of the frequency shift (positive and negative shift) and stiffness of the cantilever depend on the location, shape, size and geometry of the attached foreign particles. Based on our theoretical and numerical findings, we modify the cantilever with gold nanoparticles using a dip‐coating method to adjust the third and fourth eigenmode to be an integer harmonic of the modified fundamental resonant frequency. Via dynamic AFM and bi‐modal AM AFM measurements, we demonstrate that the excitation of the third and fourth eigenmode of the modified cantilever can significantly enhance the spatial resolution thus providing an opportunity for a robust and non‐destructive mechanical mapping of polymer and carbon‐based materials.2Figure For our presented analysis, the cantilever geometry as presented in cL, width cb and thickness ch oriented with the x‐axis with flexural displacement along the z‐axis. The attached particle length is aL, width ab and thickness is ah.The cantilever is singly clamped at one end and free at the other end. The length of the cantilever is cE and aE are Young's modulus of cantilever and attached mass, respectively. The cI denotes the cantilever moment of inertia while aI is the position‐dependent added mass moment of inertia. The density of the cantilever and added mass are denoted by ρc and ρa, respectively. To solve Equation  is the nth eigenmode shape of cantilever and ωn is the nth eigenfrequency of the cantilever, A is the arbitrary value of the amplitude.By neglecting rotatory inertia and shear deformation, the flexural displacement of the cantilever with added particle obeys the following differential equation:∂2∂x2EcIn for the first fourth eigenmode are 1.8751, 4.6940, 7.8547, and 10.9955.In this method, the eigenmode shape of the cantilever to satisfy the boundary conditions is expressed as:40 by altering the length and 2) by adjusting the height.Figure For the variation of mass by changing the length of the added mass while keeping the height fixed, we plotted the first eigenfrequency shift. It was evident that as the length of the added mass increased, the frequency shift became more negative, indicating the dominance of the mass effect when length is varied Figure 3a. Then,Subsequently, we plotted another simulation graph, where we depicted the variation of frequency shift under three conditions: 1) when the length and height were kept constant, 2) when the mass was increased threefold by triplicating the length of the added particle while keeping the height constant, and 3) when the mass was increased threefold by triplicating the height of the added particle while keeping the length constant Figure . This scFurthermore, the frequency ratio of the 2nd and 1st eigenmodes for these two cases Figure  are plotAs the moment of inertia is proportional to the cube of the height of the cantilever, according to Equation , the incIt means the changes in stiffness and subsequently the resonance frequencies of the modified cantilever are a function of the geometry of the added particle. This finding can explain why the resonance frequencies of the cantilever show both positive and negative shifts when particles with the same mass are added to it.32The simulation results are further compared with the finite element model (FEM) using three cantilever materials: (a) a silicon nitride‐based cantilever (E = 310 GPa), (b) a pure gold‐based cantilever (E = 79 GPa) and (c) a silicon cantilever (E = 179 GPa). The comparative plots for all the eigenfrequencies using these cantilevers are represented in Figures 3.2To examine the effect of added particles on the dynamics and frequency response of the microcantilever as well as the spatial resolution of multifrequency AFM, we modified the cantilever by adding Au‐nanoparticles on the cantilever surface through a combination of micro drop assisted dip‐coating approach.The SEM images of the as‐synthesized Au‐nanoparticle are represented in Figure To demonstrate the generality of our method, we have used two different cantilevers in our experiments: NuNano Scout 70 and FMV‐A (Bruker) cantilevers. The experimental results and details of analysis for the FMV‐A cantilever are given in supporting information . To quantify the comparative resolutions of tapping mode images obtained for PMMA‐PS and HOPG using the pristine and Au‐modified cantilevers, comparative image analysis algorithms using entropy and EAV methods are employed and the normalized values  are tabulated in  As explained in the mentioned references, depending on the sample and cantilever stiffness, at the specific effective stiffness, the specific eigenmode can provide higher sensitivity in imaging. So, as HOPG is stiffer than PS‐PMMA, using a higher eigenmode can provide higher resolution as for 4th eigenmode as seen in our analysis. It is important to mention that PS‐PMMA images obtained for certain eigenmodes, such as for the 4th eigenmode of the pristine cantilever and 1st eigenmode of the Au‐modified cantilever lack of details and show very poor features that the image analysis algorithm is unable to be applied to these images. The reason for better contrast for HOPG rather than PS‐PMMA using 1st eigenmode of the modified cantilever in comparison with the 1st of the pristine one can be explained by the reduced frequency and spring constant of the modified cantilever.For HOPG, the values of both the parameters are found to be below the unity for the 4th eigenmode imaging of the pristine cantilever, indicating a poorer image quality. For the 1st eigenmode of the modified cantilever, the values are almost unity, indicating the Au‐modified cantilever can obtain a similar resolution as the 1st mode of pristine one for HOPG. However, the values are greater than the unity for the Au‐coated cantilever at 4th eigenmodes indicating a much‐improved resolution as compared to even the natural frequency of the pristine cantilever. This finding is particularly important, and it can be noted that by exciting the cantilever with its harmonic frequency through tuning the higher eigenmodes, a higher‐resolution image can be obtained. A similar statistical analysis is carried out for the images obtained for the PS‐PMMA sample. However, all the images obtained for the Au‐modified cantilever exhibit values less than unity. Thus, the comparison between the Entropy and Sharpness values of the modified cantilever for PS‐PMMA and HOPG shows better performance of the 4th mode of the modified cantilever for HOPG. It can be explained by the higher stiffness of HOPG which enhances the harmonics and provides more sensitivity by using higher‐order eigenmodes of cantilevers.46 As Figure  is also matches with the images obtained from the 1st‐4th eigenmode of the modified cantilever. The detailed cross‐sectional analysis of second phase, imaged with the 1st‐ 2nd and 1st‐ 4th eigenfrequencies are represented in Figure The bimodal amplitude modulation (AM‐AM) AFM technique is then introduced to explore the effect of higher‐order eigenmode tuning on the imaging quality of the polystyrene‐polymethylmethacrylate (PS‐PMMA) block co‐polymer assembly deposited on the glass slide. The imaging parameters are set at a setpoint of 0.55 V, a scan rate of 1 Hz, integral gain of 50, with a scan size of 10 µm x 10 µm. The AM‐AM images for the pristine and Au‐modified cantilevers for the 1st‐2nd and 1st −4th eigenfrequencies are collected. The first and the second phase mapping in the air for the pristine and Au‐modified cantilevers are represented in Figure 47 is As a summary, we have developed a theoretical and simulative model to analyze the effect of added mass on the frequency response of cantilevers and subsequently through the positioning of added practice, tune the ratio of higher eigenmode to the first resonance of cantilever to an integer value. In our study, we investigated how the cantilever frequency response is affected by the mass, length, and height of the attached particle, as well as its position on the cantilever. Our results indicate that increasing the height of the attached particle, rather than its length, leads to higher stiffness of the modified cantilever, even with the same attached mass.To achieve specific changes in the cantilever's frequency response, we employed simulations based on our analytical model. By carefully selecting the appropriate mass and geometry for the attached particle, we were able to tune the cantilever's frequency response. This tuning allowed us to modify the eigenfrequencies to become integer multiples of the fundamental frequency, as detailed in Table 44,3H2O), citric acid, and sodium hydroxide (NaOH) of AR grade were purchased from Sigma–Aldrich . All the aqueous solutions were prepared using ultrapure deionized water (Millipore Milli‐Q system) having a resistivity of 15 MΩ. HOPG was purchased from MikroMasch .Gold (III) chloride trihydrate  added. The solution was then placed on a vibrating plate in a stoppered flask for 24 h to ensure full PMMA dissolution and mixing.Polystyrene  and polymethyl methacrylate   were used to create the necessary 50%PS/50%PMMA weight‐to‐weight ratio (referred to herein as PS‐PMMA). To make a 3% casting solution of this demixed system 1.5 g of PS was mixed with 1.5 g of PMMA and 97 g (63.7 mL) of chloroform (119.38 g molGlass coverslips were placed in a beaker containing 99% isopropanol (IPA)  and sonicated in an ultrasonic bath  for 15 min to remove any surface contamination before being rinsed with deionized water and placed into an oven operating at 70 °C to dry. Clean, dry substrates were placed in the vacuum chuck of an SCS G3P‐12 spin coating device  and the PS/PMMA casting solution was applied dropwise onto the surface to create a fully coherent liquid layer. The spin program was then initiated which comprised ramping the chuck rotation up to 6000 rpm for three minutes. The film was then dried at room temperature before being stored in a desiccator. with an average particle size of 0.432 µm .Particle size distribution was determined by using scanning electron microscopy (SEM). SEM was carried out using a Hitachi SU5000  microscope and images were acquired at 5 keV with a nominal spot size of 30 nm. Figure 49 witAFM Probe: The cantilever properties and the AFM imaging in the air were carried out using an Asylum research Oxford Jupiter AFM system. An uncoated Nu‐Nano Scout‐70 silicon cantilever  and an FMV‐A silicon cantilever  were used for the analysis and imaging. In our experiment, we first utilized the micro drop technique, applying a specific quantity of sub‐micron‐sized droplets with a diameter of roughly 40 nm and a known concentration in water onto a hydrophobic silicon surface. To regulate the added nanoparticle's mass, adjustments were made to the concentration and droplet volume. The mass of the desired Au‐nanoparticle was initially calculated from the simulation. In order to deposit this desired mass, initially a stock solution of higher concentration of Au‐nano dispersion was prepared and to obtain the specific concentration, the stock solution was serially diluted. For example, to deposit the Au nanoparticle on the NuNano cantilever, initially 20 ml of a 10 mM aqueous dispersion of Au nanoparticle was prepared. Then this stock solution was serially diluted to obtain an Au nano dispersion of 10 µM (diluted from 10 mM to 1 mM to 100 µM to 10 µM). Then using a precise micropipette 0.2 µl of this 10 µM dispersion was deposited on the hydrophobic silicon surface. Subsequently, the tiny droplet was transferred to the desired location on the cantilever by bringing the cantilever closer to the droplet and gently dipping the specific part of the cantilever. The use of a camera‐enabled programmable micro‐dip coater can make the deposition precise, controllable, and repeatable. Due to the hydrophobic nature of the cantilever surface, the gold nano‐dispersion assembled in a concentric area on top of the cantilever's body after a successful transfer. The cantilever was then dried at room temperature, and as the water molecules evaporated, the gold nanoparticles became attached to the cantilever.To deposit the Au‐nano dispersion at a specific location on the cantilever, we employed a combination of drop casting, dip coating, and transfer protocols. This methodology drew inspiration from the particle‐imprint method elucidated in the literature.51 In The relative position of the mass deposition can be controlled by varying the angle of the cantilever concerning the droplet during the transfer process. Moreover, the length of the added mass can be altered by adjusting the volume of the nano‐dispersion, while the height can be changed by repeating the procedure multiple times.To verify the accuracy of the deposition method in specific locations, it was performed ten repetitions of the deposition at targeted positions. The quantitative error analysis was conducted, and the average percentage error was calculated . The obtained results revealed an average error of ≈1.3% of the total cantilever length. The error in the mass deposition is further quantified and the average percentage error is further calculated  can be represented as:P(iX) is defined as the probability of a pixel having a grayscale value iX . I is the maximum grayscale value of the pixels.In the entropy method,54 theThe entropy of all the images was calculated using MATLAB and all the values are normalized and tabulated in Table  was used to calculate the sharpness of all the images by evaluating the change of rate in a specific direction of the grayscale image. The sharpness for all the pixels was expressed as:dfdx is the gradient between adjacent pixels.The improved EAV algorithm55 wasThe algorithm was applied to all the images and the normalized values are tabulated in Table All the data presented in this study were obtained through at least three measurements. To determine the error in the position of the added mass, the percentage error (δi) was first calculated as:c is the center of the targeted deposition of the added particle and xm is the measured center of the deposited nanoparticles.To calculate the average percentage error ten depositions were conducted at the same locations. The average percentage error is then measured as:m:dm (desired mass) is the desired deposited mass calculated from the volume and concentration and the deposited mass (depm) is the mass of the added particle calculated from the effective mass (em) of the cantilever.The error in the mass deposition was calculated as the percentage error as Δamm and apm are the actual mass of the modified and pristine cantilever respectively, calculated from the frequency and stiffness of the cantilevers.To calculate the actual deposited mass in our experiment we calculated the actual mass of the pristine and modified cantilever and then the actual deposited mass was calculated using the following relation:Five different added masses were analyzed and the average percentage error was further calculated as:The AFM data was analyzed using the in‐built software provided by Asylum Research. MATLAB was utilized for simulations and quantitative image analysis. FEM analysis was performed using SOLIDWORKS and ANSYS software. The size of the nanoparticles, as well as the location and positions of the added mass, were measured with the ImageJ software. All graphs were plotted, and data analysis was carried out using the OriginPro graphing software.The authors declare no conflict of interest.Supporting InformationClick here for additional data file.


Reversible and programmable shaping of surfaces promises wide-ranging applications in tunable optics and acoustic metasurfaces. Based on acoustic radiation pressure, contactless and real-time deformation of fluid interface can be achieved. This paper presents an experimental and numerical study to characterize the spatiotemporal properties of the deformation induced by acoustic radiation pressure. Using localized ultrasonic excitation, we report the possibility of on-demand tailoring of the induced protrusion at water–air interface in space and time, depending on the shape of the input pressure field. The experimental method used to measure the deformation of the water surface in space and time shows close agreement with simulations. We demonstrate that acoustic radiation pressure allows shaping protrusion at fluid interfaces, which could be changed into a various set of spatiotemporal distributions, considering simple parameters of the ultrasonic excitation. This paves the way for novel approach to design programmable space and time-dependent gratings at fluid interfaces. In the aim of designing a reversible and programmable patterning process, several strategies have been explored using an external energy such as mechanical static energy8, thermal expansion9, microfluidics11, chemical13 or electrical tuning14. Among them, the use of fluids reveals to be particularly suitable to achieve dynamically tunable devices19. Recent advances carried out by Melde et al., showed the possibility of assembling particles at the water surface into a variety of 2D shapes using acoustic holograms and radiation pressure20. However, most of these concepts suffer from the use of discrete lattice structure to approximate smooth surfaces, or are constrained by mechanical actuators, resulting in a non-real-time of surface shaping and limiting the range of applications in a time-varying environment.Contactless deformation of surfaces is a topic of growing interest in the field of tunable optic and acoustic metasurfaces25, a promising approach is to remotely control the shape of plane fluid interfaces into 3D out of plane deformation, using acoustic radiation pressure (ARP). ARP has attracted numerous research in past years29 due to the variety of applications in biomedical33, for non-contact surface tension measurement36, or to realize functions including acoustic tweezer37, enhanced electrospinning38 and acoustic diode16. Recent study has highlighted the possibility of controlling the height of the interface deformation induced by the ARP as a function of the transient excitation parameters39. Nevertheless, determining the spatiotemporal displacement of an interface exposed to ARP remains challenging due to strong hypothesis set by the analytical model. Numerical works have been carried out to tackle these limitations and have conducted to better understanding of the dynamic of the protrusion induced by ARP40. However, to our knowledge, no studies have been carried out on the characteristics of the 3D surface deformation in space and time.Among the extensive work on non-contact deformation of soft interfacesThe present study investigates both spatial and temporal characteristics of the surface deformation induced by a transient ARP, as a function of the spatiotemporal characteristics of the incident pressure field. It provides new insights for novel application of the ARP for on-demand deformation of fluid interfaces.Based on theoretical formulation of the ARP induced by a transient ultrasonic excitation, we model the induced surface displacement using Finite Element Method (FEM). Measurements using a polychromatic confocal displacement sensor are performed to measure the time evolution of the interface displacement induced by the transient ARP. The global shape of the deformation is investigated performing a scan over the surface. Several spatiotemporal distributions of the incident pressure field are investigated.41:a space, g the gravity, In a homogeneous medium, considering an acoustic radiation pressure exerted on an interface, as well as an isentropic change of state, the Lagrangian form of the equation of motion is given by42:42.Given the space laterally unconfined, it is assumed that the Lagrangian pressure mentclass2pt{minim44. Camera based devices are exposed to several limitations, such as their field of view, which prevents access to the negative amplitudes of the surface displacement and to the 3D shape of the interface deformation. Also, the lighting condition and the tilt imposed to the camera reduces the accuracy of the measurements39. Other experimental strategies have been explored, such as the use of a laser probe47.Experimentally, the main challenge lies in measuring the entirety of the displacement of an optically transparent interface such as air–water. Classic experiments to observe the surface displacement induced by ARP are based on the use of a High-Speed Camera49 (see Methods). A white light is transmitted from the controller to the sensor head through an optical fiber. The light is then focused by a chromatic dispersive lens in the displacement probe, at different focal lengths depending on the wavelength. The position of the interface is determined by the analysis of the spectrum of the light backscattered by the surface to the spectrometer in the probe controller. More specifications about the chromatic confocal displacement sensor are given in Supplementary Table These methods are still restricted to study the time evolution of the interface deformation characterization on a few places. In the present work, tracking the surface displacement of a transparent interface has been possible through the use of a polychromatic confocal sensorz- direction. The aim is to demonstrate that different pressure fields can determine dynamics of a liquid surface under the effect of ARP. Transducers were selected to generate specific deformation for different spatiotemporal shaping and applications.The experimental setup used in this work is described in Fig. entclass1pt{minimaThe influence of the incident transient excitation parameters on the shape of the deformation is investigated experimentally and numerically. The model allows focusing on both spatial and temporal evolution of the deformation considering the formulation of the input excitation. Three different time steps can be distinguished as shown on the pictures taken with a Nikon D5600 camera Fig. a–c: riset = 8 ms, both experiments and simulations exhibit a deformation height of 0.13 mm and a deformation width of 4 mm radius for a 2.1 MPa incident pressure at r = 0. It highlights that for a water–air interface, modelling the surface displacement in space and time caused by an incident ultrasonic beam can be accurately performed using Eq.  induced by the ARP is presented in Fig. ) is the same as the pressure field distribution as highlighted in Fig. ), followed by a further widening during the decrease (6 mm at -6 dB for t = 16 ms). Finally, deformation splits into a quasi-axisymmetric circular capillary wave, centred around  as denoted by the 2D representation (black solid line) on Fig. For the spherical transducer, it can be observed that the shape of the deformation during the rise  of diameter 40 mm are used successively: a spherical one focusing at 40 mm, and a planar one with a Rayleigh distance at 120 mm. More specifications on the transducers are given in Supplementary Table The Finite Element Analysis is performed using the Comsol Multiphysics software. The module “Laminar Two-Phase Flow, Moving Mesh” is used to solve the Navier–Stokes equation for incompressible flow. This method enables to study the time evolution of the displacement of the interface between two fluids. The model consists in a 2D geometry and is bounded by two types of boundaries. The top boundary is modelled with a free surface condition, including the surface tension between the two fluid σ and two no-slip wall boundaries specifies the stationary solid walls on the bottom and right sides of the fluid domain. The geometry of the sample is large enough so the surface waves resulting from the interface deformation do not interfere with the walls during the simulation time (150 µs), with the left and right-sided wall located at ± 50 mm from the centre of the domain, and the bottom at a depth of 5 mm. The gravity is considered as a force directed downward (with ers Fig. c,d.Supplementary Tables.Supplementary Video 1a.Supplementary Video 1b.Supplementary Video 2.


A. Davis et al. Nature Communications 10.1038/s41467-022-28836-1 (2022)1 argue that Rossby wave energy reflection in the stratosphere contributed to the historic United States (US) cold air outbreak of February 2021. In their later article, Davis et al.2 present global climate model (GCM) sensitivity experiments where they conclude that the disruptions of the polar vortex and related reflection of planetary wave activity in the stratosphere during January and February had no discernible contribution to the February 2021 North American, including the US, cold air outbreak3. Though we acknowledge the importance of these and similar experiments, we also note concerns regarding the methodology in the experiments as well as the ability of the model to reproduce the February 2021 US cold wave. This debate has important implications for the accurate prediction of extreme weather and the application of numerical models to predict severe winter weather related to polar vortex variability at longer leads.Cohen et al.2 argue that downward reflected energy from the stratosphere did not contribute to the US cold wave of February 2021. We agree with Davis et al.2 that the source of the energy originated in the troposphere and not the stratosphere. We argue that, as shown in Figs. 5 and 7 from Davis et al.2 and in Fig. Davis et al.2 and Fig. 4 Fig. 3; Matthias and Kretschmer5 Figs. 4 and 5) and not downward as assumed by Davis et al.2. This is likely important as it facilitates convergence of WAF near the tropopause, which is crucial in amplifying the North American standing wave supportive of cold air outbreaks. As noted by Davis et al.2, WAF passing through a region is not sufficient to amplify or dampen a wave pattern, rather a convergence or divergence of WAF is required. Incorrect simulation of this convergence and/or divergence would most likely result in a model prediction that is too mild compared to the observations, as obtained in Davis et al.2 and discussed in the next subsection.The wave activity flux (WAF) vectors as shown in Figs. 6 and 7 of Davis et al.2 and Fig. 6 that WAF convergence between 180°W and 90°W is important for wave amplification over North America. However, the assertion that convergence to the east in the stratosphere  is not important is not supported and contradicts a previous study that argued it contributes to wave amplification and cold air outbreaks across North America7. During wave reflection, planetary waves tilt eastward with height over North America so the tropospheric low pressure trough (roughly between 120°W and 90°W) that delivers the cold to the US is connected to the stratospheric low pressure trough (between 90°W and 30°W) which is the region where WAF convergence occurs in the stratosphere .As seen in Figs. 6 and 7 from Davis et al.2 initialized the model with reanalysis data fully at 9 km and then nudge to a different state tapering off up to 12 km in their scrambled stratospheric initial conditions (ICs) experiments. Due to the varying degree of nudging, the model atmosphere is constrained by the observations either fully or partially throughout this layer at initialization of the experiment (but not in the forecasts); in Fig. Furthermore, Davis et al.6 treats the polar vortex and associated wave reflection as occurring in the mid to upper stratosphere. However, the polar vortex extends throughout the stratosphere starting at the tropopause8. Moreover, there is no physical reason to expect reflection to occur only in the upper stratosphere. In fact, Matthias and Kretschmer4 defined their reflective index based on eddy poleward heat flux at 100 hPa in the lower stratosphere. Furthermore, a recent study found that the static stability near the tropopause and in the lowermost stratosphere is sufficient to initiate wave reflection9, which is in the region of the initialization in the troposphere only (scrambled stratosphere ICs) experiments performed by Davis et al.2. Therefore, we argue that the methodological choice of transitioning between observed and scrambled conditions in the 9–12 km layer cannot fully remove the influence of the lower stratosphere.The methodological approach of Davis et al.6 provides new analysis arguing that the contribution from above the lower stratosphere was trivial to the critical WAF convergence resulting in the amplified wave pattern across North America during the period of February 1–6, 2021. From our own WAF diagnostics preceding the US cold wave of February 12–18, 20213, a more detailed analysis is required from January 26 through February 12, 2021 to better understand the role of WAF propagation and WAF convergence in the troposphere and stratosphere. But regardless, the new analysis presented by Davis et al.6 prima facie corroborates a previous study that demonstrated that the influence of wave reflection in the stratosphere is to trap wave numbers 2 and 3 in the lower stratosphere and troposphere. The suppressed vertical wave propagation then contributes to North Pacific blocking in the troposphere crucial for North American cold waves7.Davis et al.6 interpret their model forecasts initialized on February 1 and 8, both with the standard forecast and the scrambled forecasts, as predicting a cold air outbreak comparable to the coldest event in the MERRA2 reanalysis. The ability of their model to simulate the observed event is shown in their Fig. S3 in Davis et al.2 and Fig. 3 in Davis et al.6. We repeat their figure here in our Fig. 3. There is a small area of slightly colder-than-average temperatures in the central US but also some warmer-than-average areas as well. Thus, the North American average that their analysis focuses on, and that their conclusions are drawn from, is representative only of western Canada. That one ensemble member can predict cold approaching the magnitude of the observed anomaly6 suggests that the model can predict the cold attributable to natural variability but not the cold that is attributable to external forcings such as wave reflection.Davis et al.2, that the model did not simulate the wave reflection as observed, likely contributing to the model not reproducing the extremity of the cold across the US. This could be the explanation for the discrepancy between model temperature forecasts and observations.It appears, based on Figs. 5 and 7 in Davis et al.2 modeling experiments to assessing the role of lower stratospheric wave reflection in the February 2021 cold wave, especially over the US. As a way forward, we propose three modeling experiments , with the initial conditions based on observations as in Davis et al.2, but better suited to diagnose the involvement of wave reflection in this event. (1) To more cleanly assess the role of the troposphere, the upper bounds of the “scrambled stratosphere” runs should be changed from 9 km (full nudging)/12 km (no nudging) to something lower in altitude, for example 4 km (full nudging)/7 km (no nudging). This run would be considered the new “scrambled tropopause and stratosphere”. (2) To assess the role of the mid-stratosphere and upper stratosphere, Davis et al.’s2 “scrambled troposphere” can be retained as is; however it should be reconsidered as the “scrambled troposphere and lower stratosphere”, as the nudging in the initialization is full only until 20 km , while the polar winter tropopause is usually much lower. (3) To assess the role of the tropopause region and lower stratosphere, a possible third experiment in which the troposphere is scrambled  and also the mid- and upper- stratosphere is scrambled (full scrambling above 20 km transitioning to observations by ~15 km) may elucidate the role of the region near the tropopause for wave reflection. By combining these three experiments, perhaps further light could be shed on which regions matter most for the surface impacts. Furthermore, the underestimation of the extent of the surface temperature response in North America, and especially the lack of a response over the United States, may be ameliorated if nudging was applied over the duration of the forecast and not just at the initialization. Nudging experiments are planned as part of the Stratospheric Nudging And Predictable Surface Impacts10 experiments and are likely to advance our understanding of the role of the polar vortex on the weather. It would be worth systematically exploring which regions need to be nudged , for future work. An examination of the circulation differences associated with the model’s failure to extend the severe cold into the US would also improve our understanding of the tropospheric dynamics of the event.In summary, we question the applicability of the Davis et al.6 have performed interesting modeling experiments that question the role of the stratosphere and the polar vortex on the North American cold wave of 2021. While we agree that modeling experiments are important to testing the role of the stratosphere in this event, the difficulties of the model in reproducing the observed structure of wave reflection and convergence in the control run, as well as the relationship of the nudging approach to the lower stratosphere, raise concerns for us over the interpretation of the model results. Davis et al.6 also present additional observational diagnostics showing that upward WAF in the troposphere and the lower stratosphere is the primary contributor to WAF convergence in that region; we agree with this result but question the assertion that convergence in the stratosphere to the east played no role. In fact, the observational analysis of WAF propagation and convergence presented in Davis et al.6 provides strong empirical underpinnings for the theoretical construct on how wave reflection in the stratosphere contributes to North American cold waves7. We suggest that additional modeling experiments and observational analysis are necessary to determine the causal role of the lower stratosphere in determining the occurrence of this convergence.Davis et al.


This article has been retracted at the request of the authors due to a miscommunication resulting in the failure to obtain proper ethics review from Shiga University of Medical Science. The authors did obtain ethical review and approval from Seijo University and were under the impression that ethical review from Shiga University of Medical Science was not required. Once informed of their error, the authors contacted the Shiga University of Medical Science ethics committee in an attempt to receive retrospective ethical review and approval, however this request was denied. As a result, the authors have regretfully requested this retraction. After a thorough review, the journal has agreed and the article has been retracted.


Chaotic systems are hard to synchronize, and no general solution exists. The presence of hidden attractors makes finding a solution particularly elusive. Successful synchronization critically depends on the control strategy, which must be carefully chosen considering system features such as the presence of hidden attractors. We studied the feasibility of fuzzy control for synchronizing chaotic systems with hidden attractors and employed a special numerical integration method that takes advantage of the oscillatory characteristic of chaotic systems. We hypothesized that fuzzy synchronization and the chosen numerical integration method can successfully deal with this case of synchronization. We tested two synchronization schemes: complete synchronization, which leverages linearization, and projective synchronization, capitalizing on parallel distributed compensation (PDC). We applied the proposal to a set of known chaotic systems of integer order with hidden attractors. Our results indicated that fuzzy control strategies combined with the special numerical integration method are effective tools to synchronize chaotic systems with hidden attractors. In addition, for projective synchronization, we propose a new strategy to optimize error convergence. Furthermore, we tested and compared different Takagi–Sugeno (T–S) fuzzy models obtained by tensor product (TP) model transformation. We found an effect of the fuzzy model of the chaotic system on the synchronization performance. Synchronization of chaotic systems has received substantial attention theoretically and experimentally in recent years. Such attention is justified by the potential applications in secure communication ,2,3, theThe synchronization of dynamic systems is the problem of enforcing several systems to match and converge . ChaoticMoreover, the synchronization of chaotic systems with hidden attractors is even more challenging as they can lead to erroneous coupling behavior and instability. Besides, the system analysis can be complicated ,15,16; oIn order to carry out the synchronizations, it is necessary to leverage numerical systems so as to approximate the solutions of the systems involved. The global error up error ,18. The The goal of this research is to afford a new methodology to carry out the synchronization of two identical chaotic systems with hidden attractors by means of fuzzy control and a special integration method that exploits the oscillating property of the chaotic systems for numerical integration.Based on our previous expertise, we opted for complete and projective synchronization schemes. The complete synchronization employs the linear matrix inequality (LMI), whilst the projective synchronization employs parallel distributed compensation (PDC). The former proposal was applied to a set of chaotic systems with hidden attractors known in the literature.The main contribution of this research is to provide a new methodology to synchronize chaotic systems with hidden attractors through complete and projective fuzzy controls, evidencing that those fuzzy control strategies combined with such special integration method boost robustness over other popular approaches and, hence, are more likely successful in the task of synchronizing chaotic systems with hidden attractors.The convergence to zero of the error in the synchronization varies depending on the strategy used for the control. Projective synchronization depends on two parameters, Another contribution is to evidence that fuzzy controls used in this work are robust in handling multistable systems with hidden attractors, and the fuzzy controls used are also robust to changes in the initial conditions. Another contribution is that the tested chaotic systems were chosen arbitrarily from the literature. Therefore, these systems are interchangeable with any other continuous integer-order chaotic system with nonlinear terms with a state variable in common. Few works in the literature emphasize the numerical methods used to simulate chaotic systems since they can generate problems such as superstability. Our proposal is to use a special fixed-step numerical method that exploits the oscillatory characteristic of chaotic systems and where the integration step is easy to choose. Furthermore, an analysis of the performance of the synchronizations using different Takagi–Sugeno (T–S) fuzzy models was conducted. The results showed an effect of the fuzzy model of the chaotic system on the synchronization performance.There are two configurations in synchronization: bidirectional or mutual coupling systems, where both systems must adapt to each other, and master–slave systems or unidirectional coupling in which one system must adapt to another ,20. In bfixed-time synchronization observer, which emphasizes the time of convergence of the error in the synchronization under complete error [back-stepping control, synchronizes two different systems with complete error by setting an extra variable [The literature reports several efforts regarding the synchronization of chaotic systems in the presence of hidden attractors. One of the most-used controls is feedback control with complete error ,21,22,23te error . Finallyvariable .Most of the controls reviewed are specific regarding the set of chaotic systems that they synchronize. For example, SMC works with perturbed systems and with unknown parameters. Other controls are interested in the analysis of the convergence time of the synchronization. In this article, we addressed the problem of synchronizing two identical chaotic systems with hidden attractors considering the complete and projective errors. To solve this problem, it is necessary to have a control, but the choice of this control is not straightforward in the presence of hidden attractors . In addiOn the other hand, fuzzy control is an efficient and effective tool in nonlinear systems and has been successfully applied to chaotic synchronization before . Fuzzy cA known result in the literature is how the global error nditions ,18.h and the error in the representation are of utmost importance. If we choose an integration step that keeps the discretization error bounded, but which does not bound the growing of the round-off error, the later can alter the chaotic system; for example, the chaos may be lost (superstability) or make the system attracted to a different attractor, thus changing the system’s behavioral regime. Even the variable step numerical integration methods with error control exhibit difficulties when choosing the integration step to constrain the growth of the error computational chaos of systems that are not originally chaotic. For example, Skufca reported how the Runge–Kutta–Fehlberg method implemented with error control and a variable step (ode45 in Matlab) when applied to a system that models the coupling of two oscillators originates the computational chaos [The bounds of the global error are then given in terms of the error in the initial values al chaos .Whenever some characteristic or property of the dynamical systems is known, said property can be exploited by the numerical method. Special numerical methods are those that exploit some specific property present in the dynamical system. In such circumstances, the numerical approximation to the solution of the ordinary differential equations with some initial values can be enhanced capitalizing on such prior knowledge about the dynamical system. It follows that, since chaotic systems exhibit oscillations, it is convenient when integrating them to use a method that leverages such an oscillatory property. Therefore, in this research, we opted for using the numerical method developed by Gautschi . This ish constant, and the truncation error Gautschi’s method has been described and analyzed in detail in ,40, but imate it .Fuzzification translates crisp inputs into fuzzy values for the inference system. The fuzzy inference system applies the fuzzy rules and generates the system outputs in fuzzy form. The counterpart, defuzzification, proceeds to recover crisp values for the outputs.IF–THEN. These rules are capable of exactly describing nonlinear systems within some region of interest. Nonlinearities are expressed as inferred fuzzy outputs by indicating the fuzzy membership function in the premise and the associated coefficients in the consequence [Here, the fuzzy inference system employs the T–S fuzzy dynamic model . The T–Ssequence .I depends on the variables exhibiting the nonlinearities.Consider the continuous-time chaotic system given byUsing the singleton fuzzifier, fuzzy inference by the product, and defuzzification by the weighted average, the final outputs are calculated according toBy employing , System  is rewriWhen modeling a chaotic system using a T–S fuzzy surrogate, the goal is to build a T–S model following , represe)+bi} in . There eIn this paper, we considered the fuzzy modeling of chaotic systems of integer order with hidden attractors, as listed in The chaotic system given by Chua ,49 was oThe final output of the Chua fuzzy system is inferred byAnalogous to the Chua system, the Sprott A system assumes that The final output of the Sprott A fuzzy system is inferred by6 system, assume that For the NE6 fuzzy system is inferred byThe final output of the NE4 system can be fuzzy modeled as follows. Assume that The LE4 fuzzy system is inferred byThe final output of the LEThe general master–slave feedback control utilized to synchronize chaotic systems is summarized in d have the same values as the master system (Let us consider the fuzzy representation of the chaotic system  as the mDefuzzification is given by control:u(t)=−∑i=ted frome˙(t)=∑i=given in ,42, if t control:u(t)=−∑i=scaling factor.In the projective synchronization, the drive system is scaled to a scalar factor r system  and the In particular, for the case study at hand, our master system is given by its T–S fuzzy modeling  and the The final output of the fuzzy system  is infertions as .(11)RulThe final output of the slave fuzzy system  is givenLet the error  systems  and 12)e(t)=y(t)By means of PDC, which creates a controller fuzzy system with rule premises identical to those of the plant fuzzy system , the nexThe final output of the fuzzy control is given by:Substituting the fuzzy control  in the eP and constant A condition for the convergence of the error  is that,he error  is globaAnother convergence condition for the error  concernshe error  is globahe error  and 11)λj, j=1,…For projective synchronization, it is possible to find the value of the control parameter The aforementioned procedure was carried out using known strategies to determine the maxima and minima of a scalar function with a vector argument. However, when the expressions as a function of In this section, we present the results of the numerical simulations of the chaotic systems with hidden attractors in We further studied the rate of convergence by both control strategies applied to Chua’s system across a new set of initial conditions. The different initial conditions that we generated followed the methodology proposed by Danca . With prIn addition, we analyzed different T–S fuzzy models and we found an effect of the fuzzy model of the tested chaotic system on the synchronization performance.All the simulations were carried out on a personal computer with machine precision ode45 and requires fewer evaluations of the chaotic system than Runge–Kutta 4 (RK4) [In all numerical simulations, the differential equations were solved by using the Gautschi integration method  based on 4 (RK4) ,55.6, and LE4 and 250 time units the Sprott A system. Despite their popularity, time units may be ambiguous, and we, therefore, report our simulations in terms of iterations instead. The total number of iterations was calculated as /integration step. In our case, 150 time units correspond to 19,200 iterations and 250 time units correspond to 32,000 iterations.In order to employ the Gautschi integration method, we ought to establish the system frequency and the integration step. If the system frequency is unknown, it is possible and advisable to underestimate it . In the We synchronized each test system with another identical system, but with different initial conditions. First, we used the fuzzy models of each chaotic system with hidden attractors given in The gain matrices Upon every synchronization, the partial errors converged to zero as the simulation progressed, that is all synchronizations were successful. The proposed control is robust and capable of synchronizing chaotic systems with hidden attractors.Two Chua systems with hidden attractors were synchronized under error-complete and initial conditions e system . In ordee system .To carry out the synchronization the gain matrices for the Chua chaotic system of integer order with hidden attractors with the chosen initial conditions were calculated. The matrices  were obtSubstituting the gain matrices in the fuzzy control  and usinAnalogous to the previous synchronizations over the Chua system, we proceeded to synchronize the Sprott A system. The initial conditions used were r system  and y0=[In 6 system, we used as the initial conditions In the complete fuzzy synchronization of the NEr system  and y0=[The initial conditions used to synchronize this system were r system  and y0=[In this section, we present the results of the projective fuzzy synchronization for the test systems in 4.Fuzzy projective control  to analyze the effect of the scaling factor In the synchronizations presented so far, we used the T–S fuzzy models provided in All synchronizations with the different T–S fuzzy models were successful with the errors converging to zero. To objectively compare the simulations, we used the synchronization error and looked for the first iteration that satisfied that each error was less than 2% for all subsequent iterations.6 and LE4 systems, the performances of Base and the alternatives SNNN and CNO matched. In the case of projective fuzzy synchronization, this was only carried out with the Chua and LE4 systems due to the constraints of the fuzzy projective control, as explained in 4 system, 8.5% more iterations were required with the alternative SNNN, and the results of the alternative CNO and Base coincided.The results of these simulations are reported in 6), with a line of equilibrium points (LE4) and with a stable equilibrium point (Chua). All the synchronization efforts were successful, even in the case of the Chua system, where the synchronization through feedback control must take special care in the selection of its coupling parameter to avoid leaving the hidden attractor [In this research, multistable chaotic systems  with hidttractor .The tested chaotic systems were chosen arbitrarily from the literature. These systems are interchangeable with any other continuous integer-order chaotic system with nonlinear terms with a state variable in common. Chaotic maps and continuous systems of fractional order were excluded in this study. Chaotic systems with delay or with unknown parameters and/or initial conditions were not considered either. These latter would require a different design for the fuzzy controls used.Regarding the initial conditions, 30 new initial conditions for the Chua system were found and presented in In this work, the Gautschi special method of numerical integration was used. This method exploits the oscillatory characteristic of chaotic systems. This is a fixed-step method that can use integration steps up to  control ,55. Few  control .In traditional controls, pole assignment is a standard strategy to maximize the convergence rate (of the control). However, in fuzzy projective synchronization, this strategy is unfeasible because the equations of the fuzzy control  and the Every dynamical system has infinitely many fuzzy models that can approximate it. Considering this, we also showed that the synchronization performance was influenced by the choice of this model.We studied the feasibility of synchronizing chaotic dynamic systems with hidden attractors using fuzzy control. We exemplified the success of such an approach with a case study on the Chua system, a case where a previous attempt using a different control strategy was difficult. We did not intend to send the message that fuzzy control is somehow superior to other forms of controls, but instead, we just intended to highlight that, under some circumstances, this strategy might present specific advantages. For instance, to apply the fuzzy controls presented here, it suffices to fuzzify the system (analogously to We showed the flexibility of the approach by designing controls over two different synchronization errors; complete and projective. We stated the conditions that provide theoretical guarantees for the convergence to the solution. These conditions were previously reported in the literature. Here, we went one step further by empirically showing the practical consequences.For projective synchronization, we proposed a new strategy for error convergence optimization. This strategy capitalizes on the theoretical conditions of the control through the parameter We focused on four case studies. However, when reviewing the literature, we did not find theoretical limitations for this approach to be applicable to other cases of chaotic systems with hidden attractors. Moreover, for our case studies, the results behaved as predicted by theory. Hence, we believe it may be possible to extend our fuzzy control strategy to other chaotic systems.