The advent of novel technologies revealed that other geophysical signals than those directly related to fault motion could be used to probe the state of deformation of the Earth's crust. Electromagnetic signals belonging to this category have been increasingly investigated in the last decade in association to natural earthquakes and laboratory rock fractures. These studies are hampered by the lack of continuous recordings and a systematic mathematical processing of large data sets. Indeed, electromagnetic signals exhibit characteristic patterns on a specific frequency band  that correlate uniquely with the paroxistic rupture of rocks specimens under uniaxial laboratory tests and were also detected in the atmosphere, in association to moderate magnitude earthquakes. The similarity of laboratory and atmospheric VLF offers an unique opportunity to study the relation between VLF and rock deformation on at least two different scales and to enlarge the dataset by combining laboratory and atmospheric data. In this paper we show that the enlarged VLF dataset can be successfully used, with a neural network approach based on LSTM neural networks to investigate the potential of the VLF spectrum in classifying rock rupture precursors both in nature and in the laboratory. The proposed approach lays foundation to the automatic detection of interesting VLF patterns for monitoring deformations in the seismically active Earth’s crust. The very low frequency (VLF) range of the electromagnetic (EM) spectrum represents a very strict and promising example of these types of geophysical signals that could extend our knowledge about transient phenomena associated with the deformation of the Earth’s crust and the seismic cycle. Anomalies in the propagation of the signals of VLF radio stations have already been studied in the past as a precursor to the earthquake occurrence4. Although the source of these signals and their propagation through the atmosphere is not conclusive, the use of VLF signals as possible earthquake precursors has been widely investigated also in the laboratory with experiments on rock samples (e.g.7). Possible mechanisms of generation and propagation are summarized and discussed in previous literature (e.g.9). Limitations to the study of the natural VLF signals have been: 1. the non-systematic data acquisition due to a limited data storage and/or data transmission through the network; 2. processing of large data set, due to the high acquisition rates which are of the order of tens of kHz; 3. the fact that the study of the electromagnetic signal was mainly focused on the observations of narrow EM bands. These three limitations hampered the research of EM signals as possible detectors of transient geophysical signals. However, EM signals are actually sensitive to transient phenomena: the sensitivity of the LF signals becomes apparent for seismic events with magnitude M ≥ 5.510, whereas in the case of VLF signal this threshold lowers to M ≥ 4.56. Nardi et al.6 suggested that three VLF signals were correlated with a magnitude M ≥ 4.5 earthquake, within an average of 3.6 days after the appearance of characteristic signals  in the VLF spectrum and within 270 km maximum distance from the survey site, during an observation period where no other relevant earthquakes have occurred.Seismic sequences are characterized by a seemingly irregular occurrence of earthquakes along main faults. This irregularity challenges our understanding of the physical processes governing the seismic cycle and represents a serious limitation in the modeling strategies for predictive purposes. More information can be achieved by observing continuously the rock volume which contains seismogenic faults as rutinariely done by monitoring deformation at surface or ground (and ocean) motion. Along with these signals other geophysical observables may contain relevant constraints on transient phenomena coheval to the deformation such as geochemical signatures of permeability and temperature changes  and very low frequency (VLF) ranges, precisely in the band 0.8–12 kHz. In the laboratory, the antenna was positioned horizontally at a distance of 10 cm from the sample. The direct transformation of the EM signal into electrical signal in the acoustic frequency band also offers the advantage to allow memorizing and analyzing the data using a common sound card, the ‘‘wav’’ file format and existing software in the field of acoustics. Still in the laboratory, to record the reference acoustic signal of the cracks, we used an omnidirectional preamplified microphone Accord PA196, with a sensitive band 0.05–16 kHz5 who also conducted experiments on rocks to verify the correlation of EM emission with rock deformation. Experimental evidence7 documented that VLF signals associated with rock fracturing occur in the form of mainly two impulsive event types occurring before, during, and immediately after the rupture episode. The observed signals, which are visible both in the amplitude and frequency domain, have been divided into two groups . However, the recognition of interesting patterns in the atmosphere and thus their associations with transient natural events is largely hindered by the lack of a systematic approach to the measuring systems, the lack of continuous data over a statistically consistent temporal and spatial interval and in general, by the approach to data management. Moreover, although these patterns appear clear in the spectrogram from visual inspection, designing a specific mathematical algorithm to find them automatically is not trivial. For the aforementioned reasons in this study we have overcome these limitations using:VLF signals have been studied discontinuously and non-systematically using different measuring systems and investigating different frequency bands. Indeed, previous studies were focusing on specific frequencies whereas the evolution of the EM signals during the rupture sequence is only visible using the entire spectra of the ELF-VLF band Fig. , from7. 7.A VLF EM monitoring network termed as “Cassandra” which is a systematic instrumented network for VLF detection in the atmosphere. The type of antenna and the network design deployed on the national territory is described in Nardi and CaputoA machine learning algorithm for automatic signal detection by scanning VLF spectral patterns over the huge amount of signals typically acquired at 44.1 kHz in a frequency band 20 Hz—22 kHz.Versatility, because by only changing input data they can solve many problems.Automation, in fact once the algorithm is designed, the only requirement is to provide enough data to let the machine learn how to solve the problem and then the system will be able to autonomously and automatically solve the same problem in similar different situations.Objectivity, as machines are not “biased by emotions”. For example, medical research procedures require double blinded protocols to avoid biases. When using machine learning techniques there are no specific protocols to be fit.The use of machine learning (ML) techniques have been tremendously increased in the last few years with the increasingly large amount of data available. More specifically, the neural networks are used for a wide range of different purposes such as image and text recognition, speech interpretation and many other uses. The main benefits in using ML are:12. The main advantage in using neural networks as compared to other types of machine learning techniques is that features relevant for the prediction do not need to be selected in advance as they are automatically found by network training. In this work, we designed the NN to detect the most typical signal patterns of the natural atmospheric noise as well as OIS patterns in the VLF band. NN was trained on VLF signals which preceded rock failure in the laboratory under uniaxial tests as well as on natural VLF signals which occurred a few days (< 5 days) before a moderate magnitude earthquake (M > 4.5) exploiting the apparent scale invariance of the VLF OIS. Laboratory OIS can be used effectively to enlarge the data set and to train the neural network for the detection of OIS signals in the atmosphere.A particular case of machine learning techniques is represented by neural networks (NN). The neural network “architecture” is inspired by the structure of the primate brain cerebral cortex, in order to learn increasingly abstract features of the input and best support the desired output. The basic idea of RNN shown in Fig. 16. Basically, LSTM blocks are an update of RNN ones to keep more sequence memory. An example of a LSTM is explained in Yildirim17. The LSTM blocks replace the neurons in Fig. 17. The main limitation of LSTM networks is that only “past” input in the sequence can affect the “future” neurons. That’s why a new version of LSTM called “bidirectional LSTM” (BI-LSTM) has been designed to put together a forward and a backward LSTM. Both networks are connected to inputs and to outputs. A general classification shallow LSTM neural network consists of an input layer with a number of neurons as the input sequence, a LSTM blocks layer, a fully connected layer (with a number of neurons as the output classes) a softmax layer and a classification layer. As described by Sagheer and Kotb18, it is possible to create a deep LSTM neural network architecture, adding more hidden LSTM layers after the first one.We have implemented an Artificial Neural Network (ANN) architecture whose structure is designed to deal with time series. More specifically, such architectures are based on recurrent neural networks  and some of them not (“QUIET”). Signals composing the background noise, including the anthropic signals and natural signals such as sferics, are all belonging to the category QUIET.It is worth noting that the LABset OIS data labeled as rock rupture precursors showed spectral patterns similar to the AIRset OIS events collected a few days before the Italian earthquakes listed in Fig. A deep BI-LSTM neural network with 1000 hidden units has been trained in order to fit the known classification so implicitly understanding the most important features and cut offs to split the “potential events” to “not events”.Our case study dataset is composed of signals acquired using both laboratory experiments and an experimental monitoring network. The first dataset, hereafter “LABset”, are VLF signals generated by 36 rock samples ruptured under uniaxial compression. The 36 samples belong to 14 different lithotypes including porphyry, granites, basalts, peridotites, limestones and concrete. To detect VLF signals during laboratory tests we have used the ELF-VLF spectrum sampled at the frequency of 44.1 kHz. Detecting and discriminating DIS and OIS from the background noise, consisting of natural (e.g. atmospheric) and artificial  signals was a complex procedure requiring a different type of spectrum analysis for OIS and DIS. Therefore, in this first monitoring phase, it was chosen to only focus on the OIS type for reasons explained below. OIS emissions are regular  and sampled at 44,100 Hz from a number of stations deployed in Italy along the central Apennines and belonging to the Cassandra monitoring network. The OIS signal appeared to occur a few days before (seemingly up to 5 days) a moderate magnitude seismic event (M 4.5). A very important point is that OIS are not common signals of the VLF spectrum, they do not belong to any of those signals observed even rarely in the background and were only recorded when followed by a short term (days) significant earthquake  provide a powerful source to verify the model accuracy. In fact, such data include the true labels but the user can use them as “new data” to predict the label by the trained model. The comparison between the real and the predicted labels allows the accuracy percentage computation Accuracy = matched results/total test data. Moreover, when dealing with neural networks training, a common practice is to divide data into three groups: 1. training data; 2. test data and 3. validation data. Training and test data are used as described before, validation data are used during the training process to check  that there is no overfitting. In fact, such data, as test data, is not used for the learning process but sometimes, during the training, the predicted classification of such data are compared to the real output. If the training accuracy is too much higher than the validation one, then an overfitting problem is detected.A machine learning method tries to predict a variable (answer or label) when some others (predictors) are known. In order to make the algorithm “learn”, the first step is to collect data including both the predictors and the answer (labeled data). Such data will feed the algorithm that will “learn by itself” if predictors show patterns (said features) able to predict the answer. This machine learning stage is named training. After the training, if successful, the algorithm is able to automatically predict the answer for any other datasets including all the provided predictors. Such a “trained algorithm” able to predict is named “model”. An example of a machine learning task may be recognizing a person's gender by using his/her bibliometric and anagraphic data. In this case, height, weight, age and other data are the predictors and “male” and “female” are the labels. In order to realize a machine learning model working we need to provide the system with already labeled data. In order to check how the model is able to “generalize” its validity for data not included into the training and avoid overfitting problem20.We are here using a Neural Network Architecture (ANN) consisting of a set of layers of neurons Fig. . Input d21 which include recurrent neural networks (RNN) (e.g.13) and Long short term memory (LSTM) networks (e.g.17).A big drawback of the backward propagation process is that input data are considered independent and weights networks updates calculation is not affected at all by input sequences. This makes backward propagation of neural networks not particularly suitable for time series analysis, speech recognition or any other application where input sequence sorting does matter. That’s why some network structures dealing with time and using previous states memory have been introduced19. A confusion matrix is a table counting where the predicted classes agree or disagree with the true values. Conventionally the rows represent the true classes and the columns represent the predicted ones. The number in the intersection between rows and columns represent how many records characterized by the row true classes have been classified with the predicted one specified by the column. So the numbers along the diagonal show all the records correctly classified and the other numbers the wrong ones. In addition to general accuracy, confusion matrices indicate the accuracy of the method for each single class.When dealing with machine learning, the best way to show methods accuracy is using the “confusion matrix”After selecting randomly 281,949 blocks , they have been used to feed and check the network and use their label to match at best the results. More specifically the training and the check has been performed as described in “The neural network approach and definitions” and 73,649  has been used for the test; 146,072 data (85% of the remaining data) have been used for the training and 25,777 data (15% of the remaining) have been used for validation. Each block is an ensemble of 1323 recorded samples. The number 1323 has been evaluated as the correct one to be sure that the signal contains the anomalous pattern. During the process we have checked that validation accuracy was similar to the training one (in other words that overfitting did not occur). Once the training has been completed, the trained network has been used to predict the classification of the 73,649 test sequences initially excluded from training . The capability of the NN to distinguish signals of relevance from the background which are possibly related to transient geomechanical properties is very satisfactory as revealed by the confusion matrix in Fig. The VLF signals have been proved in the laboratory to be a reliable rock rupture precursor. Indeed, recurrent and regular pulse trains defined as OIS from previous literature seem to systematically anticipate the paroxistic rupture event under uniaxial compressive tests. At the same time, OIS with similar characteristics (period and band width) have been detected in the atmosphere. The similarity of VLF-OIS in the atmosphere and in the laboratory offers the opportunity to investigate the potential of VLF signals in the study of transient phenomena on scalable temporal scales. However, the recognition of characteristic patterns in the atmosphere and thus their association to transient natural events is largely hindered by 1. the lack of a systematic approach to the measuring systems, 2. the lack of continuous data over a statistically consistent temporal and spatial interval and 3. because, even if these patterns appear clear in the spectrogram from visual inspection, designing a specific mathematical algorithm to find them automatically is not trivial. That’s the context where neural networks found applications as they are designed to detect features that can be used to classify signals as “human” would do. In this paper, we challenged such limitations to show that artificial intelligence and, more specifically LSTM neural networks, are effective “automatic detectors” for OIS. Such characteristic spectral patterns have been correctly classified by our neural network within the defined categories: precursors to rock “rupture” and “quite”. The classification has been proven effective not only for laboratory data but also for atmospheric signals recorded in association with transient natural events involving fracturing of rock volumes (e.g. earthquakes).A couple of methodology limits have to be mentioned, mostly correlated to events magnitude and distance between events and recording stations. In fact, at the state of the art, VLF signals are not able to detect earthquakes having magnitude below 4.5 and the maximum distance range where events are detectable is not yet defined.Despite these limitations, such results look very promising. Indeed, a precursory event can be studied and recognized when the dataset of events is statistically consistent. VLF signals cannot be studied over a statistically consistent data set because the actual dataset is typically large and relevant signals cannot be easily extracted. The proposed approach has the potential to provide a systematic study of OIS patterns because we demonstrated that the use of NN allows the automatic detection of characteristic signals in the air. The automatic detection would allow real-time monitoring, detection and extraction of characteristic patterns which would remain otherwise hidden. Moreover, we demonstrated that these OIS signals can be studied on at least two different temporal and spatial scales, at the controlled conditions of the laboratory and on natural recordings in the atmosphere, providing the opportunity to study their potential to probe the state of deformation of rock samples as well as rock volumes of the seismically active Earth’s crust.


Removable partial dentures (RPDs) are inserted with the aim to restore masticatory function. There is however inconsistent evidence supporting the alleged improvements, posterior occlusal contacts being one of the decisive factors. We hypothesized that the distribution of abutment teeth in RPDs influences masticatory performance and functional parameters. To evaluate the masticatory performance and functional parameters in patients with a RPD using a single mathematical parameter (tilting index [TI]) for both jaws that predicts biomechanical behavior on the basis of the distribution of abutment teeth.X50). Mastication on the preferred and nonpreferred chewing sides was analyzed. Total muscle work (TMW) was calculated using bipolar electromyographic recordings of the masseter and anterior temporalis muscle. Functional parameters were subjected to multiple linear regression analysis including X50 as a dependent variable and functional units (FU), the number of teeth, bite forces, and sagittal and frontal components of TI (TI α and TI β) as independent variables.Masticatory performance was measured in patients wearing long‐time adapted RPDs using the standardized test food optocal, yielding the mean particle size . This means that increased dental support improved chewing performance with RPDs under these conditions.When the preferred chewing side was tested, none of the investigated parameters correlated significantly with In well‐adapted RPDs, the distribution of abutment teeth as expressed by the tilting index seems to be of subordinate importance for masticatory performance. In particular, caries and periodontal diseases lead to tooth loss and/or shortened dental arches. Depending on the number and location of lost teeth, removable partial denture (RPD) is used to restore masticatory function, esthetics, and phonetics. Previous studies have shown that the masticatory performance of patients with RPD is less than for fully dentate subjects  reveals that symmetrical activation of the masticatory muscles improves chewing performance  with telescopic RPD participated in the study. The prostheses were incorporated for at least 3 years. Participants underwent a conventional clinical examination used for prosthodontic cases.The study included 58 jaws and 366 telescopes (166 in the upper and 200 in the lower jaw) distributed with no side preferences. In Table The study was approved by the Ethics Committee of the University Medical Center, Heidelberg (S‐570/2014) and all patients gave their written consent to the experiments.2.2i was used for all parameters associated with the maxilla whereas index j indicated an association with the mandible) at their respective positions (including missing teeth). For mathematical analysis, a Cartesian coordinate system is chosen, which lies in the OP with x and z pointing in the anterior and vertical directions, respectively  is modeled by a rigid plate. The teeth are modeled by springs that are attached perpendicular to the OP . In our biomechanical model, we included mechanical support up to the positions of the second molars (each 14 teeth tooth positions for maxilla and mandible). The model can contain any number of theoretical support positions, for example, z toward the OP, thus introducing prestress into the system. When the plate is released, it will move to a new position of equilibrium, depending on the support. This position will usually be displaced and twisted compared with the original position. The mathematical basics of the model are summarized in the appendix.To evaluate the equivalence of the support present in the maxilla and mandible, the position of the rigid plate is fixed and all spring ends are moved by a distance Δα: tilt in the x‐direction, β: tilt in the y‐direction) is appropriate for evaluation of the equivalence of the spatial distribution of the support elements for both jaws. Under the action of spatially equivalent support in the maxilla and mandible, the plate remains in the horizontal position. The more the situations in the maxilla and mandible differ from this state, the larger becomes the tilting, that is, more unfavorable is the static situation of the system. The vertical displacement at the position of the resultant force in each jaw (zrf) is a measure of total stiffness differences of the supports in the maxilla and mandible and will always be oriented toward the more resilient side.Tilting of the rigid plate  was not taken into account.Clinically, the algebraic signs of the three variables are possibly misleading; absolute values of |2.3After completing a test cycle, the patients performed three different masticatory performance tests. Each test entailed 15 chewing strokes for habitual and unilateral chewing on the right and left sides of the jaw. Habitual chewing was repeated two times. Standardized artificial test food  was used  ratios were used. The side that showed greater muscle activity (TMW) was determined to be the preferred chewing side. Similar methods were applied by Yamasaki et al.  and RatnIt is known that EMG activity is greater on the preferred chewing side during deliberate unilateral chewing Stohler, . Thus, mFor silicone test food, the standard approach is to determine the weight of the particles retained by sieves and to curve‐fit the cumulative weight by the use of the Rosin–Rammler equation. In the present study, in contrast to classical sieving methods, optical scanning analysis methods for measuring masticatory performance were used . After cleaning the skin with alcohol (70%), electrodes were placed on both sides of the middle part of the masseter and on the anterior temporalis. The reference electrode was positioned on the neck over the seventh vertebra. The EMG signals were differentially amplified , recorded at a sampling rate of 1500 Hz, saved on a personal computer, and band‐pass‐filtered (10–700 Hz) off‐line.2.5Bite force was measured with a recently available bitefork . The two functionally separated sensors of the instrument were placed between prefabricated bite blocks individualized with silicone impression material and positioned between the second premolars and first molars Figure . This co2.6After the installation of the EMG recording device, the three masticatory performance tests were conducted in one session. First, the subjects performed three maximum bites in maximum intercuspation. After a test cycle for habitual chewing to familiarize them with the procedure, subjects were asked to chew habitually, followed by chewing on the right side and, finally, on the left side. Habitual chewing was used to identify the preferred chewing side and was repeated two times. Under each condition, 15 chewing cycles were performed. The minced food was spat out and the mouth was rinsed with water to collect all the particles in a filter bag.The bitefork was adjusted with silicone impression material. Initially, while holding the bitefork in the mouth and stabilizing it by hand, the patient was asked to bite on the bite blocks and to perform some bites to become familiar with the feedback device. If the subject was able to control the bite force by watching a bar on the feedback screen, he/she was asked to bite three times with 50, 100, and 150 N on the device.2.7X50 values were calculated by use of the Rosin–Rammler algorithm. The EMG data were analyzed by use of AcqKnowledge 3.9.1 software and a semiautomatic Matlab program. Root mean square (RMS) normalized recordings adjusted to maximum biting EMG were used to compute the area under the curve  of the EMG bursts. TMW for all bilaterally measured muscles was summed for the 15 chewing cycles and analyzed. The ratios of side‐specific TMW (SMW) for the preferred and nonpreferred chewing sides were also analyzed. Bite force recordings were evaluated separately for each side of the jaw.The collected and dried chewed artificial test food was scanned and analyzed by use of a validated procedure  for each person were matched with the chewing sides.The tilting indices TI To achieve a realistic comparison of the functional data with the remaining dentition, all analyzed recordings were adjusted in relation to the preferred or nonpreferred chewing side . Means and standard deviations (SD) were calculated for all results.2.8X50, TI, TMW, FU, bite forces, and the number of teeth. Multiple linear regression analysis was also used. Preferred and nonpreferred chewing sides were analyzed separately. Chewing side differences for TMW, SMW ratios, and bite forces were tested by repeated‐measures analysis of variance. The level of significance was set at p < .05.Spearman correlation was used to analyze correlations between the variables 2.9All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional review board and with the 1964 Helsinki declaration and its later amendments.3α and TI β are listed in Table A broad distribution of the total number of remaining teeth was observed for the maxillae and mandibles; the minimum was three and the maximum 20 Figure . The finX50 values presented in Table X50 values for the nonpreferred chewing side were smaller than those for the preferred side but did not differ significantly (p = .68). Values of X50 and the tilting index β  were significantly positively correlated  for chewing on the nonpreferred side  with TI β was observed for the preferred chewing side, however. Correlations between X50 and TI α values of both preferred and nonpreferred chewing sides were not significant  .On the preferred chewing side, the mean number of teeth was 6.52 (SD = 2.6); on the nonpreferred side the mean was 6.10 (SD = 2.7). A significant negative correlation was observed for the total number of teeth and the X50 for the preferred chewing side , but not for the nonpreferred chewing side . The SMW ratios for the preferred  and nonpreferred  chewing sides differed significantly (p < .0001).A significant negative correlation was observed between TMW and X50 values (p > .05), and no significant differences between bite forces for either side were found for any force level   Table .4The objective of this study was to quantify the functional effects of different abutment teeth distributions for a sample of patients restored with RPDs that had been worn for at least 3 years. The performance of the nonpreferred chewing side was assumed to correspond to a nonadapted situation, for instance, a condition immediately after tooth loss or a short‐term response after modifying static tooth distribution in the context of various prosthetic reconstruction techniques. In the context of this study, we refer to adaptation in a general sense as the reaction of the organism to an alteration of internal or environmental conditions. Several physiological variables were included in the statistical analysis. For the preferred chewing side, the main result of this investigation is that—on the basis of the rather small data set of this pilot study—masticatory performance correlates neither with a static biomechanical balance of the restored chewing system, characterized by the TI nor with the number of FU, the number of teeth, TMW, or bite force. A small, nonsignificant influence on the number of functional units can be seen on the preferred chewing side as well. We assume that in a larger patient collective, this effect might be more pronounced. Even so, for long‐term adapted RPD and the preferred chewing side static biomechanical variables seem to be of subordinate significance for performance.X50 values. The TI α, representing the tilting of the denture in the sagittal plane, is of greater importance than the TI β, referring to the frontal plane. Summed up, the distribution of abutment teeth has a significant effect on the nonpreferred side. We would attribute this effect to the unfamiliarity of this condition to the patient. The lower TMW indicates that patients may have exerted greater caution when chewing on the nonprefered side. As their masticatory performance tends to increase over an adaptation period  differ significantly under both chewing conditions; this was not observed for the TMW values. The ratios for the preferred side correspond well to those for natural dentition during unilateral chewing . The broad range of abutment teeth distribution may, however, have balanced this possible limitation. Second, all patients were restored with telescopic RPD, which usually ensures a better balance of the denture than the use of clasps. This has consequences for the generalizability of our results. One significant open question also remains unanswered—the overwhelming dominance of preferred right side chewing. The finding is in agreement with population‐based studies . The clinical correlate to this observation would be, for example, the common case of the unfavorable dynamics of free‐end saddles (Preston, It might be argued that the TI, based on the static behavior of the combined jaws, may not be of substantial benefit for estimating the real dynamic conditions in function. However, the results for chewing on the nonpreferred side show that the parameter is sensitive enough to model an effect. Under these conditions the Preston, . The adaAs outlined above, our new biomechanical model describes the static equivalence of both jaw situations in maximum intercuspation and uses the tilt after loading of the combined system as a measure of inequality. A biomechanical description with the help of a continuous mathematical parameter, comprising both jaws, has not previously been available. It is the first attempt to use the distribution of teeth in both jaws as a predictor of the biomechanical balance of the compromised masticatory system. The fact that there are no significant results when the patients were chewing on their habitual chewing side suggests that static considerations that are based on the Eichner‐ or Kennedy‐Classification should be re‐evaluated. When the chewing process is viewed as an asymmetrical movement, it is obvious that a complex procedure such as stabilizing a denture cannot be described with simple static designs. Many treatment plans still follow the idea that only a symmetrical chewing pattern results in good chewing performance. This concept has to be refuted: The chewing process can be regarded as an interaction between acquired neuromuscular patterns, the innate preference for one chewing side, and external conditions. For the first phase of usage after the incorporation of RPD the model might enable the prediction of the quality of performance of the masticatory system and assist decision‐making, regarding the use of additional tooth and/or implant support for a prosthesis. Future investigations must, however, replicate the results of this investigation and validate this new model for cases analyzed before and immediately after the prosthetic restoration of patients. When planning a new restoration, the patient's tooth status could be entered into a software mask based on our mathematical model. The resulting tilting indices show what kind of tilt and torsion can be expected for the planned denture or prosthesis immediately after incorporation. Based on these results, the dentist could decide to add another implant or abutment tooth to achieve a smaller tilting index and greater stability. A major advantage of the developed model is that the tilting index can be calculated individually for each patient time‐saving and efficient.On the basis of this pilot study, and despite its inherent limitations, it might also be of future interest to evaluate the previously preferred chewing side of patients, to enable comparison of this preference with the preferred chewing side in long‐term use, in particular, because for the human population and all age groups chewing side preference seems to be normal behavior (Barcellos et al., Lydia Eberhard, Stefan Rues, Jürgen Lenz and Hans J. Schindler contributed to study design. Lea Bach and Lydia Eberhard contributed to data acquisition. Lydia Eberhard, Lea Bach, Stefan Rues and Hans J. Schindler contributed to data analysis. All authors contributed to the drafting of the manuscript and approved the final version.The authors declare no conflicts of interest.Informed consent was obtained from all individual participants included in the study.


Following the publication of this article , the autFigure Figure After checking the original data, the authors realized that there were errors with the image selection during manuscript preparation. The corrected figures Figs. , 8 alongIn addition, in Section “Mouse skin wound model and treatments”, the statement “Thirty male C57BL/6 mice  were used in this study, and all procedures were approved by the Animal Research Committee of FMMU” at the beginning of this Section was wrong. In fact, the mice were purchased from the Laboratory Animal Center of FMMU instead of the Animal Research Committee of FMMU.The correction will not affect the finding and conclusion of the manuscript. The authors regret for the inadvertent errors during manuscript preparation and apologize for any inconvenience caused.


During the COVID-19 pandemic, the bidirectional relationship between policy and data reliability has been a challenge for researchers of the local municipal health services. Policy decisions on population specific test locations and selective registration of negative test results led to population differences in data quality. This hampered the calculation of reliable population specific infection rates needed to develop proper data driven public health policy. Over the past decade, Dutch municipalities and their public health services  have been experimenting with data-driven policy development and decision-making. During the COVID-19 pandemic, the growing role of data in governmental decision-making became visible to the general public as well. The numbers of positive test results, hospital admissions and deceased individuals were reported by the national news media on a daily basis. In addition, the use of more complex concepts, such as the reproduction number, were no longer exclusively used by scientists but became a regular part of the national press conferences during the pandemic. The use of terms such as ‘road maps’ and ‘signal values’ implies data reliability is high. However, in practice, data reliability is not always self-evident. The Dutch public debate focused on relevance of public health data for decision-making and policy development. In this article, we illustrate the bidirectionality of this relationship by showing how political decision-making and policy influenced data quality of the Dutch COVID-19 registries. We use data from a case study of three municipal health services in the southwest of the Netherlands. This study was initially designed to determine population differences in SARS-CoV2 infection rates [Initially, variation in data completeness and quality mainly impacted the work of epidemiologists and other public health researchers serving local governments, as they were responsible for providing reliable information to policymakers. On the one hand, the political pressure to quickly provide reliable information on, for example, compliance with the test policy or infection risk among specific population groups was high. On the other hand, researchers could not influence the data collection process and the corresponding reliability of these data. For public health researchers, this created some tension between their data responsibilities and their ability to influence data.The primary aim of this article is to illustrate how the interaction between decisions of governmental institutions and decisions of individual citizens affected the quality of COVID-19 registry data. By data quality, we refer to a combination of completeness of the source data and reliability and relevance of the information derived from these data. Based on our findings, we will formulate a few recommendations that could contribute to improved quality of registry data in a potential future health crisis.Centraal Bureau voor de Statistiek) [The initial research goal of our case study was to determine population differences in the prevalence of diagnosed COVID-19 in the southwest of the Netherlands, with a focus on living and labour conditions. Therefore, COVID-19 registry data of three municipal health services were merged with microdata of Statistics Netherlands (tistiek)  at an inDatasets merged at the individual level can reveal different details of information than datasets merged at an aggregated level, such as district level. This method contributes to the prevention of incorrect extrapolation of associations present at the aggregated level to the level of the individual. In epidemiology, this phenomenon is referred to as ‘ecological bias’ . In our Ecological bias could have occurred when residents of one district were at a higher risk of SARS-CoV2 infection than residents of another district due other factors than the determinants studied. Determinants studied in our case study were SEP, demographics, and living and labour conditions. Statistics Netherlands is highly motivated to preventing this type of bias. Therefore CBS offers scientists the opportunity to reuse its microdata at the individual level and pseudonymise merged datasets. The study period ranged from June 2020 to April 2021.The data quality of the COVID-19 registries was influenced by two aspects of public health policy. The first was the registration policy that stipulated which test results ended up in the official registries. The second aspect was the testing policy that stipulated who was eligible for a test at either the governmental test facilities or any of the alternative test locations. Test compliance of individuals was an additional complicating factor (a schematic representation is shown in Fig. The registration policy was unaltered during the study period. Citizens tested at the governmental test facilities were all registered in the CoronIT registration system, regardless of their test results. Those with a positive test result were also registered in the Osiris registration system, regardless of the test facility. As a result, negative test results of all citizens tested at alternative test facilities did not end up in the official registries. In addition, data from Dutch citizens tested across the border in Belgium were missing as there was no policy on the exchange of test results between Belgium and the Netherlands. In epidemiology, this type of missing values is referred to as ‘selective missing values’.The testing policy was more complex and was changed over time. For example, the Dutch government strongly discouraged children to visit a test location at the start of our study period, but they were encouraged to do so in a later phase of the pandemic. Large-scale testing in nursing and care homes was part of the national policy advice at the start of the pandemic, when elderly were assumed to be the most vulnerable to SARS-CoV2 infection. In addition, citizens of specific districts or municipalities were actively approached for testing by their municipal health services, for example in response to a local outbreak or when low compliance with the test policy was suspected. Moreover, employers were made responsible for their own testing policy in the workspace. Therefore, testing policy is assumed to have varied largely between sectors and individual employers. As these local policies have not been documented in a structured way, their influence on the data quality for specific groups of employees is largely unknown.In summary, the data quality for a population group is high if many citizens in this group were allowed to test at the governmental test facilities of their municipal health service and they were highly compliant with this testing policy. Our case study data showed an overrepresentation of members of households with children and with a high to very high household income derived from salary or shares at the governmental test facilities. If many citizens in a specific population group were not allowed to test at the governmental test facilities but only at the alternative test facilities according to the test policy, the data quality for this group is low. More specifically, the registry of positive test results in these groups is relatively complete, whereas the registry of negative test results is not. One method to identify population groups with a large proportion of missing negative test results is to study the proportion of positive test results detected at the alternative test facilities in each group. Our data showed that this proportion was the highest amongst citizens aged 65 years and over, those living in nursing homes or other care facilities and those with pension benefits as the main source of their household incomes. For population groups in which many citizens were not allowed to test at the governmental test facilities, were tested abroad or were not compliant with the test policy, the data quality is very poor. These groups can be considered as ‘data invisible’ as they could not be identified from the data of our case study. These examples illustrate how the complex interaction between test policy, registration policy and compliance with the test policy could have led to population differences in COVID-19 registry data quality.Clearly, population differences in data quality lead to population differences in the reliability of calculated infection rates. The infection rate in each population group is often calculated according to the formula in fig. For population groups with good data quality, the calculated infection rate is quite reliable. For population groups with complete data for positive test results but not for negative test results, the calculated infection rate will always be an overestimation of the true infection rate. After all, the calculated denumerator will always be lower than the true denumerator. For population groups with very poor data quality, the infection rate cannot be calculated in a reliable manner. Only infection rates with similar reliability can be compared fairly. For example, in our case study, the infection rate could not be fairly compared between specific labour sectors nor between residents in private households and those in nursing homes due to these differences in data quality.However, these limitations do not imply that data-driven working is not possible at all. They rather substantiate the importance of comparable data quality. For example, differences in infection rates between different types of households within care institutions can be studied. After all, similar testing and registration policies applied to both types of households. As a result, the calculated denominator reflects a comparable underestimation of reality in both groups. By making proper comparisons, more reliable management information can be extracted. This reduces the risk of misinformation and can therefore eventually contribute to reduced stigmatisation of population groups. Decision-making based on simple geographical statistics, such as the infection rate at district level, is not recommended because of the abovementioned bias induced by testing and registration policies. Our case study shows that data quality can vary greatly within but also between districts.We believe our case study provides a number of valuable insights. Existing legal frameworks, procedures and registration systems that worked well in regular practice of infectious disease control did not match the needs of a crisis as large as the COVID-19 pandemic. To better facilitate public health researchers to unravel indicators of true relevance for the third sustainable development goal of the United Nations—that is to ensure healthy lives and promote well-being for all at all ages—in the future, the following two steps are recommended.Our advice is to invest in a single reliable registration system of positive and negative test results that is suitable for all possible test locations and for all possible infectious diseases. The use of such a system could avoid many of the population differences in data quality illustrated in our case study in the future. Besides more reliable estimates of population differences in infection rates, it would also provide valuable information about testing behaviour. Instead of mapping testing rate data obtained at test facilities of the Dutch Municipal Health Services only, testing rate data could be obtained from all inhabitants, regardless of their test location. That would provide an opportunity to calculate and visualise meaningful population differences in testing rates, estimate population differences in compliance with the test policy and develop population-specific public health interventions, in close collaboration with other countries.Compliance with the test policy is an important determinant of data completeness and quality. However, it is difficult to properly measure compliance on a full population level. In a future health crisis, we would not only benefit from more complete data but also from more varied data relevant for public health policy development. That is, data that can be directly used to substantiate the development of interventions, such as data on citizens’ motivations for either accepting or refusing a test or vaccination.Our second recommendation is therefore to involve citizens more intensively in both data collection and public health policy development. Ideally, the future public health data architecture is flexible enough to ensure that citizens gain ownership of their public health data. It would be transparent and suitable for adding new data, and it would include an option for citizens to enrol in or withdraw from scientific studies at any moment. Having this kind of system would be an important step towards a more efficient approach to reduce health inequalities based on true citizen science.


Squall lines are known to be the consequence of the interaction of low‐level shear with cold pools associated with convective downdrafts. Also, as the magnitude of the shear increases beyond a critical shear, squall lines tend to orient themselves. The existing literature suggests that this orientation reduces incoming wind shear to the squall line, and maintains equilibrium between wind shear and cold pool spreading. Although this theory is widely accepted, very few quantitative studies have been conducted on supercritical regime especially. Here, we test this hypothesis with tropical squall lines obtained by imposing a vertical wind shear in cloud resolving simulations in radiative convective equilibrium. In the sub‐critical regime, squall lines are perpendicular to the shear. In the super‐critical regime, their orientation maintain the equilibrium, supporting existing theories. We also find that as shear increases, cold pools become more intense. However, this intensification has little impact on squall line orientation. In the subcritical regime, squall lines are perpendicular to the wind shear, until equilibrium between wind shear and cold pool spreadingIn the supercritical regime, the orientation of the squall lines reduces the incoming wind shear, and maintains the equilibriumChanging properties of cold pools with background shear, notably their deepening, has little effect on the squall line orientation One key ingredient in the organization of squall lines is the presence of cold pools below precipitating clouds. These are areas of cold air with negative buoyancy anomaly, driven by the partial evaporation of rain and concomitant latent cooling, and observed to span 10–200 km in diameter  has a sponge layer to absorb gravity waves. We neglect the Earth rotation (a reasonable approximation in the tropics). All simulations are run to radiative‐convective equilibrium (reached in about 30 days), after which we start our analysis, from day 30–35, with hourly outputs.The simulations use the cloud‐resolving model SAM  to 20 m s−1, with 2.5 m s−1 increments. To avoid an impact of this imposed surface wind on surface fluxes, the domain‐mean surface wind is removed before computing surface fluxes . With shear, convection triggering is favored in the downshear (decreasing x) direction, as described in more detail in Section Figure 2.2a of the squall line orientation, as the angle between the squall line and the y direction. This definition allows to have a reference angle of 0 when the line is perpendicular to the background wind. To measure this angle a, we investigate the spatial autocorrelation of precipitable water  we can see a deep convective line, with a length of 75 km, up to 10 km of altitude. The shear seems to constrain the convective cells at the edge of the cold pool. This is consistent with the aforementioned RKW theory of equilibrium between the incoming wind shear and that associated with spreading cold pools.Figure Usfc = 10 m s−1, the line is perpendicular to the imposed wind, suggesting that it is a critical case  or to the left (decreasing y direction) of the cold pool, which sometimes gives rise to a V‐shaped pattern, with two lines, one to the left and one to the right of the cold pool.Consistent with RKW and Robe and Emanuel , the cony‐axis. This orientation is believed to keep the projection of the background wind perpendicular to the squall line direction, close to the propagation speed of cold pools. In the following, we propose to quantify this with the angle detection described in Section In short, in the presence of shear, the triggering position of the convective cells is always at the downshear edge of the cold pool, and as the shear rate increases, so does the angle to the 3.2−1. To quantitatively test the theory of squall line orientation, we plot the projection of the basal shear velocity perpendicular to the squall line Usfc  cos(a) in Figure Based on the angles estimated from the different simulations, several conclusions of RKW theory have already been validated: the existence of two regimes, subcritical and supercritical, and the emergence of a critical shear near 12.5 m sUsfc = 12.5 m s−1, and at this moment, the wind speed and the cold pool spreading speed are equal. We then enter the super‐critical regime, and we expect saturation, that is, the angle of the line maintains the cold pool/background wind equilibrium. In orange, we have plotted the results of our simulations. We observe a very good agreement in the sub‐critical regime. In the supercritical regime, we also have a reasonable agreement with the theory, roughly within error bars. At this stage, we can conclude that the theory of the conservation of the shear projection on the direction perpendicular to the squall line allows to predict the orientation angle of the convective bands. We note in passing that the results are robust to reasonable changes in the relaxation timescale used to impose wind shear and to a prescribed large‐scale upward vertical velocity, following , we observe a cold pool nearly symmetrical to the maximum precipitation and not very intense. In the case −1  the composite value of buoyancy in  cross section around the location of maximum precipitation  higher in altitude. To determine why rain evaporation increases with shear, we show the evolution of relative humidity vertical profiles (estimated as averages within 10 km of the maximum precipitation location) for all shear cases −vp′ (instead of Usfc  cos(a)). For the subcritical regime, we still have the same agreement. For the supercritical case, we observe that the correction brings the theoretical expectation in closer agreement with the CRM simulations. Since this correction is small, we conclude that the intensification of the cold pools is not a determining factor for the orientation of squall lines.Since the potential energy increases with the imposed shear (see Figure ibrium,Usfccos(5While the RKW angle theory was so far qualitatively supported, for the first time we support it with an objective and reproducible quantification of squall line orientation. Using autocorrelation image analysis of the integrated humidity profiles, we measure the orientation from CRM simulation of the squall lines and identify the subcritical and supercritical regimes, depending on the ratio between the basal shear velocity, and the spreading velocity of cold pools. The hypothesis of the conservation of the projected basal wind perpendicular to the squall line, that predicts the orientation of squall lines, is here validated.We further investigate the properties of cold pools, and notably their evolution with shear. We measure an intensification and a deepening of the cold pools due to the drying of the lower atmosphere with increasing shear. However, the resulting intensification of the cold pools has a secondary impact on the orientation of the line.Although these results are within an idealized framework, they draw our attention to the sensitivity of convective organization to wind shear. Squall lines are associated with extreme rainfall, and an interesting question could be the impact of squall line orientation on extreme precipitation rates. In the longer term, more work is desirable to extend these results beyond the CRM framework, for example, by using satellite data (Fiolleau & Roca, Supporting Information S1Click here for additional data file.Movie S1Click here for additional data file.


Primary malignancies of the nasal cavity and paranasal sinuses account for 0.3% of all head and neck cancers with a proportion of tumor in sphenoid and frontal sinuses estimated to 0.001%.A 67 year-old man was referred to the Department of Otolaryngology for the evaluation of a few days history of monocular diplopia, left facial and frontal pain. Patient did not report fever or continuous rhinorrheas. The general medical history revealed that the patient had previously been treated for a prostatic carcinoma by surgery, radiotherapy and hormonal therapy. The ENT clinical examination has just reported the presence of a divergent strabismus of the left eye and frontal pain increased on palpation. Concerning the nasofibroscopic examination, we did not found rhinorrhea, a nasal extension of the tumor and signs suggesting a complication of a rhinosinusitis. The frontal mass seemed not be complicated by a rhinosinusitis. A first physician suspected the diagnosis of a rhinosinusitis with a possible sinus cavernous thrombosis. Therefore, he prescribed a Magnetic Resonance Imaging (MRI) and a Computed Tomography (CT) that showed two focal lesions, the first in the left cavernous sinus and the second in the left frontal sinus . The lesa ≥ 2 ≥ mm dural thickening, the loss of hypointense zone  and nodular dural enhancement could be highly predictive for a dural invasion6The mechanism of prostatic metastases to the paranasal sinuses involves a spread cells that probably comes from the vertebral venous plexus.Concomitant metastases of the frontal and cavernous sinuses are a rare condition that can lead to a misdiagnosis of rhinosinusitis. In the context of a history of cancer, even if it remains exceptional, an imaging must be made to precise the etiology of the symptoms. In the case of frontal malignancy, dural thickening and nodular enhancement combined with the loss of hypointense zone could be highly predictive for a dural invasion implying a poor prognosis.The authors declare no conflicts of interest.


Internal jugular vein catheterization is widely used in clinical practice, and there are many related studies on internal jugular vein catheterization. However, the omohyoid muscle, which is adjacent to the internal jugular vein, is a rarely mentioned muscle of the infrahyoid muscles group. The purpose of this study is to explore the anatomical relationship between the omohyoid muscle and the internal jugular vein on ultrasound guidance and provide a theoretical reference for jugular puncture and catheterization.The study included 30 volunteers. The volunteer’s head lay in the neutral position and was then turned to the left at an angle of 30°, 45° and 60° with the bed surface, as verified using an adjustable protractor. A high-frequency ultrasound probe (6–14 Hz) was used to examine the plane of the apex of sternocleidomastoid triangle (PAST), the triangle consists of anatomical landmarks: a base was clavicle, its sides – heads of sternocleidomastoid muscle. And the plane of the middle of sternocleidomastoid triangle(PMST) which was a horizontal line, connecting midpoints of both sides. The right omohyoid muscle (OM) and the right internal jugular vein (IJV) were observed and recorded for statistical analysis.There were statistically significant differences in the number of overlapping cases of OM and IJV at each head rotation angle between the PAST and PMST groups. There were statistically significant differences between the angles which OM and IJV centre point line and the left horizontal position of the PAST and PMST at different body angles.The traditional middle route puncture point is the apex of the sternocleidomastoid triangle, which can effectively avoid injury to the omohyoid muscle, to an extent.ChiCTR2000034233, Registered 29/06/2020. www. Chinese Clinical Trial Registry.gov. The placement of a central venous access into the internal jugular vein(IJV) is a commonly practiced clinical technique that allows fluid and blood transfusion and other rescue treatments. It is widely used to treat critically ill patients with severe trauma, shock, and acute circulatory failure. It can also be used for total parenteral nutrition therapy, long-term intravenous fluid and antibiotic therapy, and temporary cardiac pacemaker transcatheter installation. The rephttp://www.chictr.org.cn . Thirty volunteers (15 women and 15 men), who were healthy, were included from September 2020 to June 2021. Participant age ranged from 18 to 80 years. In addition, included patients must have the ability to cooperate with the observation throughout the process. Volunteers were excluded if they had thyroid disease, abnormal neck anatomy, history of neck puncture or surgery. Volunteers who could not correctly communicate were also excluded.This study was approved by the Ethics Committee of Medical Research and Clinical Trials of Huzhou First People's Hospital. All methods were carried out in accordance with relevant guidelines and regulations and Declaration of Helsinki. This clinical trial was registered at Each volunteer’s head lay in the neutral position and was then turned to the left at an anlge of 30°, 45° and 60° with the bed surface, as verified using an adjustable protractor Fig. .An ultraThe data collected included the following: width of the omohyoid muscle , the width of the internal jugular vein  Fig. , the angAll of the data are presented as the mean ± standard deviation (SD), median (interquartile range), or number (%), as appropriate. The measurement data were analysed by Shapiro–Wilk. Normally distributed data that satisfied the homogeneity test of variance were tested by ANOVA, if not by Kruskal–Wallis. The abnormally distributed data were analysed by Kruskal–Wallis to compare among groups. Pearson’s chi-squared test was used to assess the comparison among groups in count data. Data were analysed using SPSS software . A p value of less than 0.05 was considered to be a significant difference.P < 0.05) (see Table p < 0.05), and there is no significant difference in the number of cases where OM and IJV overlap at different angles in the same plane (see Table p < 0.05)   was significantly lower than that of the PMST plane (1.56 ± 0.36 cm) at 30 degree angles (Our study mainly observed the anatomical relationship between the OM and the IJV in two planes under ultrasound guidance. The selection of the two planes corresponds to the punctured plane of the traditional middle method and posterior method for IJV puncture. The main results included the overlap of the IJV and OM at the corresponding head angle for each plane and the anatomical angle α of the IJV under ultrasound with OM. We found that the lower the punctured plane was, the more significant the overlap and α of the IJV and OM.The overlapping relationship between OM and IJV has not been previously documented. Our results showed that there was no significant difference in the overlap of the IJV and OM at different head angles on the same plane. Then, the number of overlapping cases, OWOI, and the OR of the PAST planes with the same angle are smaller than those of the PMST plane. This also indicates a greater likelihood of injury to the OM by a puncture at a relatively low level.The results show that for the α angle in the PAST plane, the median values of the three individual bit angles were 14.17°, 14.97°, and 12.89°, indicating that the OM is mainly located on the upper medial side of the IJV in the PAST plane. The α angle in the PMST plane was larger, and the median values of the three individual bit angles were 62.75°, 45.68°, and 107.33°, indicating that the right OM was mainly located on the medial and lateral surfaces of the right IJV. The PAST plane was the insertion point for our middle puncture. Generally, the puncture needle was oriented towards the posterolateral during middle puncture, which also suggested that the probability of injury to OM during middle puncture was lower. However, the PMST plane corresponds to the posterior puncture point, and the OM below the PMST plane is mainly located on the upper and lateral sides of the IJV, and the needle insertion direction of posterior puncture is generally towards the posterior and medial sides, which also indicates that the probability of passing through the OM during puncture is higher.We find anaesthesiologists rarely focus on the relationship between the OM and the IJV. But we can still find a close relationship between the two in clinical work see Fig. . In 1988The present study had some limitations. We only evaluated the anatomic data of healthy volunteers who had never undergone IJV cannulation. And the number of subjects in our study was small. Moreover, we thought that the discomfort in the throat and scapular region might be caused by the injury of the OM during puncture, but no study has confirmed the inevitable relationship between the two. Therefore, further research is needed to confirm our findings. More other studies are also possible, including data from healthy volunteers and patients with different conditions.The levels of PAST and PMST correspond to the puncture planes of the traditional middle method and posterior method for IJV puncture. Based on the results of this study, it can be seen that the traditional middle approach, if the insertion point plane is too low or the traditional posterior approach is chosen, is more likely to damage the OM and cause discomfort to the patients. At present, in many hospitals, it is impossible to perform IJV catheterization under ultrasound guidance. The traditional middle route puncture point was selected at the top of the triangle, where the sternal head at the lower end of the sternocleidomastoid muscle and the clavicle meet, which could effectively avoid injury to the OM, to an extent. Even ultrasound-guided catheterization of the IJV can cause damage to the OM if it is not well recognised. Therefore, it is crucial for clinicians to popularize the anatomical relationship between the OM and the IJV.In conclusion, the traditional middle route puncture point is selected at the top of the triangle where the sternal head at the lower end of the sternocleidomastoid muscle and the clavicle meet, which can effectively avoid injury to the OM, to an extent. Different head rotation angles did not effectively prevent damage to the OM during puncture.


Porous materials are of great interest in multiple applications due to their usefulness in energy conversion devices and their ability to modify structural and diffusive properties. Geometric tortuosity plays an important role in characterizing the complexity of a porous medium. The literature on several occasions has related it as a parameter dependent on porosity only. However, due to its direct relationship with the morphology of the medium, a deeper analysis is necessary. For this reason, in the present study, the analysis of the geometric tortuosity is proposed considering the porosity and the pore size distribution. Geometric tortuosity in artificially generated digital porous media is estimated using the A-star algorithm and the Pore Centroid method. By performing changes in the size of the medium and the distribution of the pore size, results are obtained that indicate that the geometric tortuosity does not only depend on the porosity. By maintaining the same porosity, the geometric tortuosity increases if the pore size is reduced. Similarly, these pore size effects are greater if the size of the medium is reduced. The A-star algorithm was found to be more suitable to characterize the majority of paths within the half-pore. On the other hand, to increase the size, the Pore Centroid method is the most appropriate. Finally, three types of correlations were generated relating tortuosity with porosity and pore size. All the correlations were determined with 95% of interval confidence. Porous media are constituted by a solid skeleton (matrix) and the pore spaces (pore network)2. Following the International Union of Pure and Applied Chemistry (IUPAC), porous materials can be classified into three categories according to their pore diameter: microporous (< 2 nm), mesoporous (2—50 nm), and macroporous (> 50 nm) materials3.Porous materials are composed of two phases, i.e., solid and pore volumes, distributed disorderly5. Tortuosity is a singular parameter used to characterize the flow paths inside a porous medium6, giving an idea of the medium's complexity and quantifying the flux resistance of the structure. Tortuosity is commonly correlated with porosity6, but it is also related to other parameters depending on the described phenomena6. There are several types of tortuosity: electrical, hydraulic, diffusive, and geometrical6. Geometric tortuosity emphasizes the microstructure configuration, focusing on the geometric flow paths inside the media6, in contrast with the other types of flux-based tortuosity5. Geometric tortuosity is valuable for different areas, considering that changes in the microstructure of electrochemical devices can affect their performance7. In electrochemistry, it has been useful for the microstructural characterization of lithium-ion batteries8 as well as porous support layers9. Even a correlation between geometric and flux-based tortuosity for LiFePO4 electrodes10 has been developed. For earth sciences, it provides a reliable estimation of pathways for 3D rock porous media samples as well as can be used to estimate other parameters such as permeability and effective diffusion coefficient helping in applications such as gas storage and reservoir production11. Additionally, for durability evaluations of porous composites such as fuel cell electrodes, filtration membranes, polymer foams, ceramics, and powder beds, geometric tortuosity is important12.Some microstructural descriptors let to characterize the porous media. Porosity, specific surface area, correlation functions, pore size distribution, Minkowski functionals, constrictivity, and tortuosity are essential parameters to describe porous materials6:L is just the length of the straight line between the inlet and outlet surface of the analyzed medium.The mathematical definition of geometric tortuosity is as follows13. Moreover, it lets a better analysis of its geometrical properties.Modeling allows the generation of porous media with a great variety of microstructures and different complexities. Voxel-based models are particularly interesting because they are directly linked with the experimental data collected from volumetric imagesThis study focusses on the analysis of digitally generated artificial porous media rather than real porous media obtained by tomography. This methodology allows the reduction of expenses by not needing to acquire expensive equipment for the processing of porous materials. Instead, only computational power would be required for analysis using simulation techniques. Furthermore, the simulation methods are non-destructive, unlike other processes that require the application of forces or pressure for analysis, such as mercury porosimetry or gas absorption processes.15. This reduces the differences between digitally generated media and real media, allowing similar results in both methodologies. As a result, generating artificial media digitally allows correlations to be obtained, since it is easier to experiment with a large number of samples that would be difficult to achieve in reality depending on the type of media to be studied.There are well-known programs that allow the generation of artificial porous media with specific properties allowing the control of multiple geometric parameters of the medium16, Voronoi tessellations1, random packing of simple solids4, or imitating physical process methods17. Considering the several methodology generations, different software has been employed to analyze the generated porous media. PuMA18, Tort3D19, OpenPNM15, and Porespy14 are some of the computational tools to generate and study porous materials from different approaches.There are numerous ways to generate explicit microstructure models: overlapping of spheres14, has been used to carry out the current research. The porous media are generated using the Gaussian blur method. Similar studies have been performed with some modifications related to processing and generating methods22. One of the main analyzed parameters in the current study is the geometric tortuosity based on Eq. .As part of the study, pore size distribution (PSD) is also computed. The pore size distribution can be defined as a measure of abundance for each existing pore size in a volume of each porous medium. It can be represented as a function of the pore radius, whose values vary in every part of the mentioned volume42. Since the current research considers digitally created porous media, the bulk porosity of media is known from the beginning. Therefore, it is intended to define specific sizes and porosities, assuming that these values will not change over time. The volume from which the REV starts will be obtained from an artificially generated medium of larger dimensions and intermediate microstructural characteristics than our study.In studies where porous media are reconstructed, the representative elementary volume (REV) is initially definedOn the other hand, porous media samples are generated based on random images, assuming they represent porous media, and tortuosity is estimated. This study seeks to test the A-star algorithm as an alternative to tortuosity evaluation as well as to review the implications of its use. As mentioned, and cited earlier in this section, tortuosity evaluation has been carried out with path-finding algorithms in various situations, even with different particle shapes.The rest of the paper is divided as follows: “This section explains the methodology for solving paths through the porous media with the A-star search algorithm. Before the A-star application, the variables included in the study are defined, and the generation process of the porous media are explained.The porous media creation starts with a matrix image of random noise generated with zeros and ones. Then, a multidimensional Gaussian filter is applied. It is implemented as a sequence of 1-D Gaussian filters as follows:14. The blobiness is the variable that controls the morphology of the porous media and falls in the 0 to 1 range. By reducing its value, the standard deviation of the Gaussian Filter increases, and larger pore diameters on the final structure are expected.The sigma formulation is intended to allow images with visually pleasing solid and porous space distribution at low blobiness valuesψ0, which falls in the range from zero to one. All pixel values in the last resulting image greater than the threshold are rounded to the value one and defined as the solid phase. In contrast, a value of 0 is defined as the pore phase, resulting in a set of binary values 42. Therefore, the pore indicator function is written as:42:A Gaussian convolution is applied to the defined kernel and previously generated random image. The result is a normally distributed image converted to a uniform (flat) distribution. Therefore, the resulting image can be defined as a grey-scale matrix An image of random noise is generated. Its size is defined by the shape parameter. It is a Python matrix filled with numbers between 0 and 1.14. It controls the size and aspect ratio of the solid particles.A Gaussian kernel of sigma determined by the mean of shape parameter dividing by a factor is generated. The mentioned factor is a number chosen such that a blobiness of 1 gave reasonable-sized blobsGaussian blur convolution is applied over the image of step 1. This process is performed by using the defined kernel.The resulting image is normalized and rescaled with values between 0 and 1.Thresholding is applied over the image until the value of specified bulk porosity is reached.The following steps describe, in summary, the algorithm to generate the porous media:A simplified description of the porous media generation process is presented in Fig. To analyze the effect of the blobiness over the generated porous media, Fig. 5. The size domain of the porous media is given in voxels, which result in the tridimensional representation of pixels. Considering an image resolution of 1 µm per pixel as well as other studies that also use Porespy as the generator of porous media46, it can be concluded that every voxel represents 1 µm3 in size. The more voxels the porous media has, the more resolution of the geometrical properties can be achieved. However, an analysis of porous media with high dimensions requires high computational power. Therefore, the current study's size domain is a cubic volume with 40 × 40 × 40 pixels or 403 voxels, representing a porous media of 403 µm3.A binary 3D matrix represents porous media, so each voxel is characterized by either a zero or one. The solid material is represented by one, while the void spaces are considered zeros. The mentioned approach is commonly used in artificial porous media computation47, metal foams (0.75–0.95)48, gas diffusion layers (0.60–0.90)31, Lithium-Ion battery electrodes (0.30–0.60)49, catalyst layers (0.40–0.60)31 and porous media in other fuel cells (0.30–0.55)50. Figure Six different values of blobiness are evaluated, from 0.5 to 1.0, with a leap of 0.10. For each blobiness, eleven different porosities are analyzed, i.e., from 0.45 to 0.95 in steps of 0.05. In the end, a total of 3300 samples were generated. The number of samples is determined considering statistical approaches to guarantee the appropriateness of the results (see Online Appendix A). On the other hand, the porosity values were selected based on practical applications in several fields: biomaterials (0.70–0.95)To generate a medium, it is necessary to establish the mold media represented by the volume. It determines how many nodes occupy the three-dimensional space. Considering the computational cost, size-independent analysis (see Online Appendix B), and Representative Elementary Volume analysis (see Online Appendix C) a cubic volume with forty elements by side has been selected to represent the porous media.Then, once the shape is defined, it is required to establish a porosity range to study. In this step, it is important to consider the more common porosities in porous media applications. The porosity range is determined considering the previously mentioned materials, so the porosity range is from 0.45 to 0.95. The blobiness value is modified and varied from 0.5 to 1.0 to represent a wide variety of materials in which the morphological structure of the solid-pore interaction has effects. With the variables established, a medium is generated for the Porespy library. All the generated domains are just candidates because it is needed to ensure at least one possible effective path between the inlet and outlet surfaces. If no connection is possible, the geometric tortuosity cannot be computed. Porous media with at least an effective route are stored as stereolithography CAD in a NumPy array. Paraview is used to open the stereolithography CAD and report a better image of the generated porous media.Slice the porous media layer into 100 partsApply the pore centroid approachApply A-star algorithmSince the generated media are 40 × 40 × 40 shape, to estimate the number of paths the number of inlet The first approach is implemented to reduce the computational cost of considering all the initial and final nodes that form the porous media paths. The process takes the matrix representation of the porous media previously saved in the porous media generation process. The first layer of the matrix is taken, considering that the z-axis indicates the depth of the 3D porous media. Then, the layer represented by z = 1 is selected and separated into smaller slices. Figure 7. The current study only uses the pore centroid approach to determine the pore centroid nodes. Following the mentioned procedure, 100 paths at maximum are guaranteed to reduce the computational cost. Figures The pore centroid of every sub-region in layers z = 1 and z = L are determined once the Matrix Division procedure is carried out. The objective of the pore centroid calculations is to get the possible initial and final nodes that will form the path that allows the computation of the geometric tortuosityMatrix division and the selection of the midpoint processes are resumed in the following flowchart in Fig. 46. The algorithm uses a heuristic search and search-based methods on the shortest route. The main objective of the algorithm is to find a path from a selected starting point to an endpoint. Both points or nodes are obtained with the Pore Centroid approximation considering the lowest cost, i.e., the shortest route traveled. Analyzed nodes can be freely located in the space or arranged in the form of a structured grid. In the current study, given the binary geometry, the void nodes serve as a computational space for the A-Star algorithm. In such a case, the path can pass only through the nodes with a value representing the pore space. One limitation is that a route cannot pass twice through the same node on the network. The A-Star algorithm is defined as a best-first algorithm because the value evaluates each cell in the configuration space as follows:h is the heuristic distance from the cell to the desired state, g is the path length from the initial state to the desired state according to the selected series of cells. This list ends with the scheduled cell. The value f evaluates each cell adjacent to the target cell. The cell with the lowest value, f, is selected as the next cell in the sequence. The Euclidean distance is used to determine the distance between two nodes. It is used to value the nodes in the heuristic formula, and the following equation gives it:The A-Star algorithm belongs to the best-known path search algorithms widely applied in a space with a metric or topological configurationSeveral paths are generated from the inlet to the outlet surfaces of the porous domain. In Fig. 5. Online Appendix D shows a pseudocode of the A-star algorithm.A-Star algorithm finds the shortest route from one node to another, but the number of paths is related to the number of starting and ending nodes. To understand this, call N the number of start nodes and M the number of end nodes, both found considering the Pore Centroid Approach. There will be N times M paths that should be analyzed. Due to the high number of possible paths, only the shortest paths are considered to achieve a maximum of one-hundred paths. Numerically talking, the path with the minimum value of nodes to pass through the medium is selected from the set of paths respective to the starting node. The minimum value of the paths is used on algorithms like the Direct Shortest Path Search method and the Skeleton Shortest Path Search methodi represents the index for the starting nodes, i. In addition, m is the amount of starting nodes, and L measures the shortest length in the domain, i.e., the side length of the domain, which is equal to 40 in this study. Figure To compute the geometric tortuosity, Eq.  is applif(r), which is proportional to the combined volume of all pores whose effective radius is within an infinitesimal range centered on r. The algorithm provided by the Porespy library from Python computes the density function of the pore size distribution by analyzing the histogram of voxel values in the distance calculated over the generated binary matrix that composes the porous media. On the other hand, the cumulative distribution function is also generated, representing the proportion of pore sizes that can fill the pore space.The PSD is the relative abundance of each pore size in a representative volume of a porous medium. It can be represented by a function porespy.metrics.pore_size_distribution function in Python code14. The mentioned function requires the input of four parameters. Given the characteristics of the current study, two of them are established, while the other two are left as default values. The two parameters are Im and Log, which are defined as follows:As mentioned, the PSD function of Porespy is used to describe the pore material. It is executed by the Im represents the array containing the sizes of the largest spheres that overlap each voxel. They are obtained from either porosimetry or the local thickness method. These two methods will be explained later in detail.Log is a boolean value; if true, the size data is converted to common logarithm values before processing. In the current study, this parameter is set as false.14. According to Jervis et al.51 the main difference between these two methods is that the simulated mercury intrusion porosimetry (MIP) incorporates a shielding effect. Whereby internal pores can be shielded by smaller pores nearer to the exterior of the sample, making them unreachable until higher pressures are applied. The local thickness method gives a more accurate account of the size of all the pores in the model. In contrast, the simulated MIP results provide a pore size distribution closer to the experimental procedure. A porous medium was analyzed with the two methods to show the differences, as presented in Fig. The Porespy library has two methods to extract the sizes of pores. These are the porosimetry and the local thickness method52. On the other hand, simulated MIP preserves the character of the Gaussian distribution used for the generation of the media since the largest pores are concentrated on the approximate value of 7 µm, unlike the Local Thickness method, where the majority are concentrated on the approximate value of 7 µm and 10 µm. Since this study aims to analyze porous media based on Gaussian generation, the simulated MIP method is preferred to estimate the pore size distribution.It can be corroborated in Fig. porespy.filters.porosimetry is employed. The input is a matrix of the porous media in 2-D or 3-D, setting the void phase as True. The invasion of the void space starts from the surface, giving a shielding effect for the greater internal pores, occupying them with smaller ones. This function computes the pore sizes by performing a distance transform to estimate the largest sphere that could be centered on each voxel14, and applying a second distance transform to obtain the invading fluid configuration in the trimmed mask. As a result, a copy of the initial porous media with the pore size values in each voxel is generated as a matrix.To implement the simulated porosimetry, the function Table Since the sigma value in the current research is a function of the blobiness value, it can be evaluated on Eq. . Table 2As shown in Table In Table Correlations from Table As the graphs in Fig. 5, for 3D porous media characterization it is common to use image processing techniques that allow reducing the dimensions of porous media at the cost of losing geometric characteristics.To discuss the implications of three-dimensional tortuosity computation, a recently done tortuosity study in 2D porous media is compared. It shows clear differences in both the methodology and the computational power required. First of all, despite being the same A-star algorithm, the method of obtaining paths consumes less computational power in two-dimensional porous media. This is due to the number of nodes that this algorithm must evaluate. While for a two-dimensional medium, considering samples of square sizes On the other hand, for 3D computation estimating a representative node for each layer or 2D slice using techniques such as pore centroid is considered a suitable methodology when performing downsizing. Although a factor to take into account with this methodology is that it must be guaranteed that each layer has a pore that allows connectivity with the previous pore, if this is not fulfilled, an adequate geometric tortuosity cannot be estimated. As mentioned before, the pore centroid technique was used to estimate the starting and arrival pores of the paths to be generated to guarantee the calculation of paths that our computational resources allow. Reducing the number of nodes by reducing the number of input nodes 43. This article worked with two-dimensional artificial porous media. The porosity that this study evaluated was in the range of 0.50–0.90. The sigma used was 2. Therefore, to perform an adequate comparison, the correlation obtained in Fig. By comparing with numerical data, the selected paper done by the co-authors of this study is presentedFigure According to Fig. On the other hand, something different can happen outside the established range, since it can be assumed that at values less than 0.45 the tortuosity of the porous medium in two dimensions will become greater than that obtained by the porous medium in three dimensions. This is very likely since even though there may be more nodes to analyze in the 3D porous medium than in the 2D one. As the porosity decreases, the 2D porous medium begins to run out of paths faster, so the resistance of the porous medium to the passage of species increases more. This occurs because the 2D porous medium does not have a transverse axis as occurs in the 3D porous medium, where in case there is no adjacent node in the direction of flow, the transverse nodes can be searched and an effective path can be found.55.This section shows a comparison between the obtained correlations and theoretical and empirical correlations found in the literature. Table Figure As shown in Fig. As mentioned, studies have demonstrated that tortuosity depends not only on the porosity, but another morphological characteristic has to be considered because it directly affects the paths of the media. Table According to the statistical parameters, the proposed surface equation can predict 98.6% of the tortuosity values as a function of the porosity and sigma with a 95% confidence interval.Pore size distribution (PSD) is computed and represented by the cumulative and probability density functions. The influence of blobiness on the PSD is analyzed by a graph comparison of different samples. For explanation purposes and considering the minor number of samples to use, there were considered porous media with larger sizes than the analyzed in the data collection process. Three-dimensional porous media of 100 × 100 × 100 voxels with 0.5 porosity were used.Two porous media samples were generated and presented in Fig. To determine the probability density functions, two different media are generated. Histograms of PDF are presented in Fig. Figure For the porous media that formed the pore size distribution in Fig. On the other hand, the average pore size was estimated by fitting the data of the probability density function presented in Fig. As Table Figure 5 where it is said that the size of voxels of the porous medium is reduced. The tortuosity tends to increase because the space through which species can pass is reduced, even if the same medium is maintained, the geometric information can be lost as the size of voxels decreases. Furthermore, this becomes more noticeable when the pore size is altered. Since even if the pore size is reduced in similar proportions, the tortuosity will increase more when the medium has a lower dimension in voxels.As shown in Fig. 5, the geometric tortuosity estimated with the pore centroid method can be preserved even when the resolution of the same medium is varied since it tends to consider the same path. This is an issue that can be evaluated in randomized means in future studies.It is worth mentioning that the A-star method can give a better idea of the complexity of the medium because it considers more paths than the Pore Centroid method which only considers one. The A-star algorithm becomes inefficient as the size of the medium increases. Of course, the pore centroid method is a good alternative for media of considerable size, as in the case seen in this subsection. Even, as mentioned by Fu et al.The cumulative distribution function (CDF) data are shown in Fig. Figure The implications of this study on the characterization of porous media are several. In the first instance, the effect of pore size on tortuosity could be corroborated. When comparing different media with the same porosity, the geometric tortuosity varied considerably as the pore size distribution changed. It can be established that porous media increase their capacity to allow the passage of species as the size of the pores increases. This implies that correlations of tortuosity with porosity alone must be neglected. Rather, the geometric tortuosity must begin to relate to other microstructural variables to achieve a better representation of reality. On the other hand, this tortuosity variation effect does not depend only on the pore size but also on the size of the medium. Since the approximate values of geometric tortuosity were observed to vary, they increased as the size of the medium was reduced. And this is somewhat feasible because as the medium shrinks, the number of nodes to analyze also shrinks, resulting in fewer possible paths to allow species to pass through. It should be noted that in our study, the sigma parameter can be used indistinctly from the size of the medium to relate a pore size distribution since this value alters the grouping of the grains within the medium.There is also the option of evaluating the average pore size, however, due to randomness, estimating this parameter directly from the generation parameters such as sigma will require taking a lot of data since it will depend on the size of the medium and it is not something that can be estimated. With complete certainty since there is still no correlation with the size of the medium or any other characteristic factor. Therefore, the adjustment technique to a Gaussian function for each porous medium was the best way to obtain the average pore size.The size domain is also another important factor that can alter the geometric tortuosity. As observed in this study, when comparing means of sizes 40 × 40 × 40 voxels with means of 100 × 100 × 100 voxels. It was observed that as the pore size increased, the tortuosity variation was greater in the smaller medium. This is due to the restriction of the paths being greater in the medium with fewer voxels, since it had fewer nodes through which to form a path, the passage of species within the medium would be more affected.The algorithm to estimate the geometric tortuosity plays an important role. For this study, the A-star algorithm was mostly used due to its characteristic of evaluating most of the porous medium. This covers larger nodes than other methods specifically designed to obtain a representative path such as the pore centroid method. It can be established that the use of the algorithm depends on the size of the porous medium to be analyzed, that is, the resolution of the medium in voxels. This is because the A-star algorithm analyzes more nodes, and requires greater use of computational resources, which implies that to perform calculations such as those shown in this study, good performance computers are required. Unlike the pore centroid method which estimates a single path based on the pore centroids of each 2D layer, saving computational resources. In this study it was possible to obtain adequate geometric tortuosity values for media of 100 × 100 × 100 using the pore centroid method, an issue that could not be achieved with the A-star algorithm, so the latter was used to estimate the geometric tortuosity in media porous 40 × 40 × 40.Finally, for better reproduction of the results obtained, correlations of the geometric tortuosity are established based on the porosity and the pore size distribution represented by the sigma generation parameter, in addition to leaving a correlation of the geometric tortuosity based on average pore size. These correlations were selected considering the Sum of squared errors and R-squared statistical parameters to guarantee the fitting. A possible advance in this study would be to calculate the geometric tortuosity using the pore network of the porous medium, also considering new approaches like artificial intelligence on 3D porous media generation.Supplementary Information.