Since the first identified case of COVID-19 in Wuhan, China, the disease has developed into a pandemic, imposing a major challenge for health authorities and hospitals worldwide. Mathematical transmission models can help hospitals to anticipate and prepare for an upcoming wave of patients by forecasting the time and severity of infections. Taking the city of Heidelberg as an example, we predict the ongoing spread of the disease for the next months including hospital and ventilator capacity and consider the possible impact of currently imposed countermeasures. Since the first appearance of the novel coronavirus disease (COVID-19) in Wuhan, China, in December 2019, the disease has spread worldwide to be classified as a pandemic by the WHO on 11 March 2020.1. The exponential increase in new confirmed cases in Germany has reached 16,290 positively tested cases by 20 March 2020, with many more cases presumed undetected due to a possible transmission from minimally symptomatic or asymptomatic persons carrying the virus3.In Germany the first confirmed COVID-19 case was recorded on 28 January 2020. Contact tracing and isolating of affected individuals interrupted the chains of transmission. However, with tourists returning from holidays in high-risk regions of northern Italy combined with a carnival celebration near Cologne, the virus spread to 13 out of 16 federal states within only a month4.The spread of COVID-19 in other countries, such as China and Italy, indicate the major challenges German hospitals, especially intensive care facilities, will have to face in the weeks to come. On 17 March 2020, the Robert Koch Institute declared the population risk as “high”. For the elderly and individuals with pre-existing health conditions, there is an increased probability of severe disease requiring hospitalization and critical care treatment, as well as higher mortality ratesThe spread of the disease varies locally and depends on several factors, such as the temporal progression of the infection in the specified region, the size and the age distribution of the population, the available capacities and the implemented countermeasures. Thus, to prepare hospitals and assist health authorities in the most efficient way, the resulting challenge and burden on the health care system should be modelled locally. As an exemplary case, in this manuscript we are concerned with the spread of the COVID-19 epidemic for the city of Heidelberg in Germany.5. The vast majority were home-comers from vacations in a high-risk region in Italy and their immediate contacts. All positive tested individuals were isolated and several hundred contacts were traced and quarantined to prevent further spread. Only thereafter, the total numbers of positively tested COVID-19 cases rose to 20 on March 15, and 34 on March 16.The first COVID-19 positive case in Heidelberg was confirmed on 28 February 2020, with eight more cases sporadically appearing in the following two weeksThe capabilities to cope with the increasing burden on the health system on a local scale depend on the preparedness of available local resources in the hospitals and the implemented countermeasures  to slow down the disease spread. In such an extraordinary situation, it is crucial for hospitals to rely on future predictions about the upcoming need of beds as well as ventilators. Predictions from mathematical modelling can assist authorities in their decision making on suitable countermeasures to slow down the viral transmission and prepare the health systems.We model the transmission dynamics of the COVID-19 outbreak by an individual level susceptible (S) – exposed (E) – infected (I) – removed  compartmental model. The transitions between the compartments are described through the following set of differential equationsγ is the inverse of the infectious period andκ is the inverse of the incubation period.Extended data, Table 16 lists all parameter choices for our model. For the main part of this manuscript we assume an incubation period of 5 days7 and an infectious period each of 2.9 days8. The infectiousness of the disease is given byR0 and is defined as the mean number of people that get infected by a single infected individual over the whole infectious period. Due to the limited data about the COVID-19 pandemic, there is much uncertainty in the estimation ofR0. As baseline we follow the value of 2.2 obtained from the COVID-19 outbreak on the Diamond Princess ship9. For a sensitivity analysis, we also consider a lowerR0 of 1.5 with respect to the estimation of Duet al. (2020)10 and a more severe outbreak progression with 3 in accordance to the estimation ofR0 ranging between 2.2 and 3.6 for the SARS epidemic in 200311.Where12, for our model we assume a closed population of 150,000, without individual entering or leaving the system. We consider this as a realistic scenario because we do not expect major demographic changes to occur within the six months of the projections.According to the office for statistics and urban development HeidelbergThe true number of initial infected individuals remains unknown due to the asymptomatic progression in some cases. Furthermore, due to limited testing capacities, only high-risk individuals  were tested, resulting in missing positive cases in the population that resulted from possible community transmission. To capture this uncertainty, we assume that the outbreak starts on 12 March 2020 with 10 initial infectious individuals who have not been isolated and are able to transmit the disease.R0. In the baseline scenario (R0 = 2.2) the peak of the infection will be 10196 individuals infected on 26 May and a total outbreak size of 126,574 (84%) of the population infected after six months. This aligns with estimates for the proportion of British people get infected if no countermeasures are put in place13.For now, we assume that no countermeasures have been set in place and predict the future dynamics of the outbreak for the upcoming period of 6 months.In the mild outbreak scenario, there will be 77952 (52%) total infections by 12 September 2020, whereas in the severe scenario we will have a rapid growth and a shorter doubling of infections time with 141,084 (94%) infections in total.Extended data, Figure A16). The model showed more robustness to this kind of variation on the overall outbreak curve with a shift in max 42 days for the peak of the outbreak curve and difference of 8856 infections during peak time .For sensitivity analysis, we also investigated, with respect to expert opinion, shortening and prolonging the incubation period [3.5–5.5 days] as well as the infectious period [1.5–4.5 days] .When varying the number of first infected individuals, the dynamics of the outbreak stay the same. The more infected individuals we start the simulation with, the further we assume the outbreak has progressed already  this is a decrease of 62%  in the total number of people infected.However, such a permanent lock-down of the population will not be feasible due to economical and psychological health reasons. Thus, in6).If after three weeks contact rates will revert to their original state, the outbreak curve will only be shifted by weeks with no effect on the height nor on the outbreak size 14, the mortality probabilities for each age category are 0.1% (<55), 1.5% (55–65), 7% (65–75), and 20% (>75 years of age). By means of these percentages we calculated the cumulative number of deaths in Heidelberg as proportions of the people in the “Removed (R)” compartment over time 15, who assume that the duration from hospital admittance to death is 11.2 days.As the severity of the disease and thus the number of expected deaths from the COVID-19 outbreak depends on the age of the individual, we obtained the specific age distribution for the city of Heidelberg: <55 (72.4%), 55–65 (11.06%), 65–75 (7.77%), and >75 years of age (8.77%)er time  for diffAs preparation for the health system to plan the bed and ventilator capacity that needs to be enabled, it is important to predict the number of patients who require hospitalization and ventilation.16. In the beginning of an epidemic, people are hospitalized for isolation purposes but not because of their severe symptoms. Besides, due to asymptomatic or minimally symptomatic infections that do not present for testing, the true number of infections remains unknown and thus estimates of hospitalization might be overly pessimistic. For this simulation study, we nevertheless assume, that 5% of all infected individuals need hospitalization. Because the duration of hospitalization varies in the literature, we take the smaller and more conservative estimate of 11.5 days18 19, the average duration of ventilation is 3-20 days. Supposing a conservative estimate of 7 days, we can model the capacity in the Intensive Care Unit given the calculated death numbers.For calculating the number of ventilators needed, we assume that 90% of all people who died from the disease were ventilated before. The remaining 10% possess a patient do not resuscitate/do not intubate (DNR/DNI) status. We further assume that 60% of ventilated die. This leads to 1.5 (0.9/0.6) people ventilated per death count. According to Chen20. Even if we unrealistically assume that all spaces can be used for COVID-19 patients, the capacity will not be enough within the following six months, unless we have a permanent reduction in the disease transmission by 50% .Extended are available under the terms of the et al., the authors propose a SEIR  differential equations model estimating the spread of COVID-19 in the city of Heidelberg.In Koppel The proposed modelling approach is performed on a closed population with no new entry/leave. As declared by the authors, a more realistic approach could extend the current model with taking into account the scenario where people with different underlying health conditions/immune systems would have an unexpectedly higher rate of susceptibility/infection, or a second scenario where a "recovered" case does not necessarily mean "not susceptible" anymore. However, for the sake of this paper, the deployed model benefits from high robustness which makes it a reliable prediction algorithm. A sensitivity analysis of the model has been performed via applying on a range of assumptions for countermeasures and infection rates. As a result, changing the number of initially infected individuals at the beginning of the pandemic seems to only shift the distribution linearly with no effect on the peak. Considering the limitations in the intensive care unit, a set of policies have been introduced to help the authorities in Germany.  While using a well-known robust algorithm, and bearing in mind that this is not supposed to be a methodological research, the paper still lacks some validation tests. A validation test on the existing benchmarking experiment can be carried out in several ways. At the time of this review, there might be access to the real data of COVID-19 spread in Heidelberg in order to compare with the estimated results. Alternatively, the same approach (of course with a different rate of daily infection) could be applied on a data of SARS to test the performance of the model.Is the work clearly and accurately presented and does it cite the current literature?YesIf applicable, is the statistical analysis and its interpretation appropriate?YesAre all the source data underlying the results available to ensure full reproducibility?PartlyIs the study design appropriate and is the work technically sound?YesAre the conclusions drawn adequately supported by the results?YesAre sufficient details of methods and analysis provided to allow replication by others?YesReviewer Expertise:Biostatistics, Machine LearningI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. The authors used a compartmental model that they referred to as a SEIR model where they divided the population into four compartments: Susceptible (S), Exposed (E), Infected (I), and Removed .  Now the model could be specified differently as their scheme of the equations was not clear to me. The susceptibles pool depleted at a constant rate of exposure , but they seem to have substituted and mixed up with an SIR model .  S -> E -> I -> R should be the correct order of their model specification, so one way to formulate would be dS/dt = - Rate of Exposed dE/dt = Rate of Exposed - Rate of Infected dI/dt = Rate of Infected - Rate of removal dR/dt = Rate of accumulation of removed persons from the pool But from reading their formulae it does not appear that they went with these equations, nor did they explain the basis or provide a reproducible code. Hence, although the epidemic curves look plausible, it is difficult to interpret how they were specified or their true significance. The authors may want to clarify this aspect of the methods section.Is the work clearly and accurately presented and does it cite the current literature?PartlyIf applicable, is the statistical analysis and its interpretation appropriate?PartlyAre all the source data underlying the results available to ensure full reproducibility?NoIs the study design appropriate and is the work technically sound?PartlyAre the conclusions drawn adequately supported by the results?NoAre sufficient details of methods and analysis provided to allow replication by others?NoReviewer Expertise:EpidemiologyI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. In this work, the authors predict the ongoing spread of the disease including hospital and ventilator capacity for the next months in the city of Heidelberg and consider the possible impact of currently imposed countermeasures.The paper should be checked by a native.A Conclusion section should be added.The novelty and contribution should be clearly bolded.The introduction should be updated by recent researches. The authors should consider some works about statistical methods (including time series models or regression models) that can be applied to model and predict medical datasets.Comparative Analysis should be given in more details.It’s better to suggest some subjects for future works. This is an interesting contribution to the existing literature, but the paper suffers from several shortcomings listed in the following comments:Is the work clearly and accurately presented and does it cite the current literature?YesIf applicable, is the statistical analysis and its interpretation appropriate?YesAre all the source data underlying the results available to ensure full reproducibility?YesIs the study design appropriate and is the work technically sound?YesAre the conclusions drawn adequately supported by the results?PartlyAre sufficient details of methods and analysis provided to allow replication by others?YesReviewer Expertise:Data Analysis, Statistics, Data ModellingI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.


The Stoner–Wohlfarth model predicts the crossing of the ascending and descending branches of the hysteretic magnetization curve. This crossing behavior has widely been dismissed, with the claim that it violates the laws of thermodynamics. Experimental verification of hysteresis branch crossing has not been acknowledged in the literature. Here we show, both theoretically and experimentally, that the crossing of the ascending and descending branches of the magnetization curve is a robust, reproducible phenomenon which does not violate any fundamental law. In particular, the Stoner–Wohlfarth model applies to non-interacting, magnetic particles which change their state of magnetization by coherent rotation, rather than domain wall motion. This widely used model predicts the crossing of the ascending and descending branches of the magnetization curve when the applied field is close to the magnetic hard-axis. However, experimental observation of this hysteresis branch crossing has not been acknowledged in the literature. In fact, the crossing of hysteresis branches has been labelled as being nonphysical2 and a problem3 with the Stoner–Wohlfarth model. Here we show that the crossing of the ascending and descending branches of the magnetization curve does occur and is consistent with the Stoner–Wohlfarth model. We found that the crossing of hysteresis branches in the Ni:LiNbO3 system (thin film nickel on lithium niobate) is a robust and reproducible phenomenon. Contrary to previous reports, we show that the crossing of hysteresis branches is physically realistic4 and violates no fundamental law. Additionally, we note that several previously published reports7 appear to show the crossing of hysteresis branches in their data under similar conditions without providing an explanation of the phenomenon. Our results demonstrate that the crossing of hysteresis branches must be considered when analyzing or modelling the magnetic behavior of anisotropic, ferromagnetic materials that undergo coherent rotation. Given the recent interest in anisotropic magnetic materials in the fields of spintronics8 and nanomagnetic logic9, we anticipate that the crossing of hysteresis branches may influence the understanding and development of magnetic devices for sensing, data storage, and nanomagnetic computing.The Stoner–Wohlfarth modelThe Stoner–Wohlfarth (SW) model calculates the magnetic free energy for a single, uniaxial particle, based on the anisotropy energy and the Zeeman energy. Using a zero temperature approximation, the model allows the calculation of the magnetization as a function of the magnitude and direction of the applied field.s) and the anisotropy field (Hk) of the simulated particle, and the angle of the applied field with respect to the hard axis (α). Supplemental Section k = 500 Oe at three different angles of applied field with respect to the hard axis. It is clear from Fig. 1, however, very little discussion was provided. Hysteresis curves for both components  of the magnetization can be generated numerically for a SW-particle. We have implemented an algorithm to generate numerical solutions to the SW-model, based on the values of the saturation magnetization , the finite temperature algorithm requires a parameter representing the effective particle volume (Veff), or the volume over which the exchange interaction acts. The height of the energy barrier between the two states is calculated from the energy landscape and the approach-to-equilibrium equation is solved for an appropriate measurement time to yield the fractional population of magnetization in the two states (n1 and n2). The net magnetization is calculated as the weighted sum of the two states. Supplemental Section S2 gives the details of the numeric simulation of the finite temperature SW-model.In order to more accurately simulate a real particle, we have implemented an algorithm to numerically generate solutions to a temperature dependent SW-model. To account for thermal fluctuations, the procedure outlined by Lanci and KentHk. Measurements of the transverse remanence of the samples in this study indicated that angular distribution of the anisotropy was less than 0.5° FWHM. As a result, subsequent simulations ignored the angular (orientation) distribution and modeled only a distributed magnitude of anisotropy field. Supplemental Section The algorithm implementing the finite temperature SW-model can then be extended to simulate a distributed anisotropy. Whereas the original SW-model assumes a unique orientation and magnitude of the anisotropy field, a more realistic simulation must account for a finite distribution of anisotropy orientation and magnitude. Any physical sample, comprising an ensemble of particles, must have some finite variation in anisotropy from particle to particle. In order to better model such a physical sample, the finite temperature SW-model algorithm was used to simulate the magnetization of a SW-particle over a range of anisotropy magnitudes, and the behavior of the ensemble was calculated as the weighted sum of the individual simulations, assuming a normal (Gaussian) distribution with a standard deviation of σSimulations were performed for a SW-particle with a unique anisotropy field at several different temperatures. The hysteresis branch crossing (HBC) is still present in the simulated magnetization curves, provided the temperature is sufficiently low. The effect of increased temperature is to reduce the area of the HBC and to move the HBC to lower values of applied field. This effect is demonstrated in Fig. Hk). The effect of a distributed anisotropy magnitude is to smooth out the sharp feature of the HBC. While the area of the HBC is found to remain unchanged, the fact that the crossed region is “smeared out” makes it less obvious. This effect is demonstrated in Fig. Additional simulations were performed for a SW-particle at fixed temperature and a variable anisotropy distribution width . Details are provided in the “11. All samples in this study demonstrated HBC consistent with the finite temperature SW-model when the angle of the applied field was in the range 0.5° <|α|< 5° from the hard axis. Figure Nickel thin films on LiNbOIn order to elucidate the physics behind the HBC, we examine the energy landscape (magnetic free energy as a function of theta) of a Stoner–Wolhfarth particle at four specific values of applied field: below the crossing, at the crossing point, in the crossed region, and above the crossing. These four energy landscapes are shown in Fig. When the reduced applied field, h = 0.6, the energy landscape has two minima. This curve corresponds to a point on the hysteresis curve below the crossing on the positive side. On the ascending branch of the hysteresis curve, the rightmost minimum (at θ ≈ 142°) is occupied, while the leftmost minimum (at θ ≈ 36°) is occupied on the descending branch. From Fig. It is clear from this analysis that when the applied field is just below the hard axis , the energy minima of both the ascending and descending branches move toward the applied field as the field is increased. However, at an applied field greater than h = 0.70, the minimum occupied on the ascending branch moves toward the applied field more rapidly than the descending branch, resulting in the crossing of the branches. If the field were applied at θ = 92°  sample can cause the appearance of an HBC when none actually exists or cause the disappearance of the HBC when it does exist. Similarly, a poorly saddled sample can increase or decrease the area of the eye in a measured hysteresis loop, thereby creating the appearance of a violation of conservation of energy. All samples in this study, when properly saddled, showed measured hysteresis loops with a net energy loss. Because the HBC only occurs when a major loop is traversed, repeatedly traversing the region of the HBC does not result in a net energy gain. This fact is demonstrated experimentally in Fig. It is important to note that close examination of Fig. 4. In fact, the crossing of hysteresis branches has been referred to as being energetically incorrect and nonphysical2, and a problem3 with the SW-model. A number of authors have proposed other models or modifications to the SW-model in order to eliminate the “problem” of crossover15. It is unclear why many researchers have assumed that HBC is non-physical. It appears that some researchers have rejected the idea because of the erroneous assumption that it violates the laws of thermodynamics (1st or 2nd law). The data presented in this paper clearly indicate that HBC is a real physical phenomenon, is well described by the finite temperature SW-model, and does not violate the laws of thermodynamics.It is widely assumed that hysteresis branches cannot or do not cross16 in their data. While many of these previously published works do not call attention to the HBC or provide an explanation for the phenomenon, the figures in these reports show qualitatively similar results to those presented above. As predicted by the SW-model, the HBC’s observed in these reports occur in magnetization measurements performed at or near the hard axis16.In addition to the data demonstrating HBC presented in this work, a number of reports in the literature clearly show HBC3 substrates. The nickel layers were between 90 and 100 nm thick, as measured by stylus profilometery. A 2 nm tantalum capping layer was deposited by DC magnetron sputtering, without breaking vacuum, in order to prevent the formation of nickel oxide on the surface. To avoid the effects of in-plane shape anisotropy, samples were patterned into 7 mm disks using conventional photolithography and wet-etch. Samples were thermally annealed under vacuum (< 10–6 Torr) at 325 °C for 2 h.Nickel films were deposited by DC magnetron sputtering on polished 128° Y-cut LiNbO3 (as opposed to 128° Y-cut) and samples annealed at 250 °C showed the HBC, but with significantly different anisotropy fields.While all data in this report were obtained from a sample processed as stated above, it is worth noting that a large number of samples were produced which showed the same behavior. The HBC was repeatedly observed in samples deposited by e-beam evaporation, as well as sputtering. Samples with and without capping layers and adhesion layers all showed qualitatively the same behavior. Samples deposited on Y-cut LiNbOIn plane MH-curves were measured at room temperature using two different commercial vibrating sample magnetometers (VSM). The first, a so-called vector VSM, was equipped with both X and Y-axis pick-up coils to allow the simultaneous measurement of the x- and y-components of the magnetization, while the second VSM only allowed the measurement of the x-component of the magnetization. Again, it is worth noting that the phenomenon of HBC was observed in multiple measurement systems, on multiple samples. Removal and reinsertion of the sample had no effect on the HBC.3 samples to be robust and reproducible, having been repeatedly observed in more than 20 individual samples, using multiple measurement systems. Although the crossing of magnetic hysteresis branches may be difficult to observe in experiment, we note that a number of previously published reports demonstrate the phenomenon. We conclude that under the appropriate conditions, magnetic hysteresis branches can and do cross. Our results demonstrate that the crossing of hysteresis branches must be considered when analyzing or modelling the magnetic behavior of anisotropic, ferromagnetic materials that undergo coherent rotation.We have demonstrated that the crossing of hysteresis branches in highly anisotropic ferromagnetic materials is physical and is well-described by the finite temperature Stoner–Wohlfarth model. This phenomenon, which has been widely ignored for more than 70 years, violates no fundamental law. We find the hysteresis branch crossing in annealed nickel-LiNbOSupplementary Information


This study aimed to investigate this effect by comparing the prescribed treatment planning dose with the actual dose in vivo to the peristomal clinical target region. A retrospective, dosimetric study was performed with approval of the institutional research ethics board.The tracheostomy site and adjacent skin is at risk for recurrence in head/neck squamous cell cancer patients. The tracheostomy tube is an in vivo dose measurements were included. The radiation dose at the skin surface underneath the tracheostomy device was measured using an optically stimulated luminescent dosimeter (OSLD) and was compared with the prescribed dose from the radiation planning system. The effect of the tracheostomy flange and/or soft tissue equivalent bolus on the peristomal dose was calculated.Fifteen patients who had received high‐dose radiotherapy to the tracheostomy region with in situ were found to have a 3.7% difference between their prescribed and actual dose. With a tissue equivalent bolus there was a 2.0% difference between predicted and actual. The mean prescribed single fraction dose  and OSLD measured dose  were found to have no significant difference. However, with the flange excluded from the planning simulation (density = air) target skin dose deviated from predicted by an average of 55.3%  and volume coverage was not achieved.Patients with tracheostomy equipment In summary, the tracheostomy flange acts like bolus with a twofold increase in the skin surface dose. Changes in the peristomal apparatus from simulation to treatment needs to be considered to ensure that the simulated dose and coverage is achieved. When locally advanced tumors cause dyspnea, orthopnea, and stridor, patients may undergo an emergency tracheostomy procedure to protect the airway. In addition, tracheostomy is required following total laryngectomy and other radical surgeries to help manage secretions. This is clinically important as it can affect the ability to effectively deliver radiation therapy to this region. Rates of peristomal recurrence have been described between 1 and 11% and are associated with significant morbidity and mortality.Although modern treatment planning systems (TPS) are reliably accurate for regions located beyond the depth of maximum dose, there remains an element of dosimetric uncertainty in the surface and build‐up regions.The issues around dose uncertainty in superficial regions is of particular relevance for head and neck cancer, where the planning target volume (PTV) often encroaches upon the patient surface.in vivo dosimetry is occasionally required. This technique allows for direct measurements of the dose to ensure that the patient is exposed by the appropriate amount for the region of interest. Traditionally, TLDs, diodes, or metal oxide semiconductor field effect transistors (MOSFETs) have been used for in vivo dosimetry. Recently, dosimeters based upon optically stimulated luminescence (OSL) have been proven to be useful and increasingly popular.3 TPS.Due to the uncertainties in TPS predictions for dose in the surface/build‐up regions, in situ has not been previously described. Therefore, a dosimetric study was performed to evaluate the impact of the tracheostomy hardware on the measured dose delivered to patients and the predicted dose calculated by the TPS.To our knowledge the dosimetric effect of the actual tracheostomy tube and flange 2A retrospective, dosimetric study to assess the impact of tracheostomy hardware was performed with approval of the institutional research ethics board. All head and neck cancer patients were identified from a retrospective database and included in the study if they met several criteria. These criteria included patients who: had tracheostomy, received radiotherapy and had a physical OSL dosimeter (OSLD) measurement of the dose at the stoma site between 2013 and 2017. The dosimeter location was known to be a predetermined region associated with the highest prescribed dose from the planning distribution. This is an institutional policy that is followed for all patients.2.A3 TPS version 9.2  for all patients. Plans typically employed a six or seven co‐planar beam arrangement with additional noncoplanar beams if required. Patients were aligned in the supine position and immobilized with a thermoplastic mask. No treatment plans using electrons, orthovoltage tomotherapy, or VOLUMETRIC arc therapy (VMAT) were included in this study. The IMRT plans were optimized so that at least ninety five percent of tracheostomy site PTV received the prescribed dose, for example, V56 Gy > 95%. The radiation treatment plans for all patients were copied to a research database for review and dosimetric analysis.Head and neck contouring was completed by the attending radiation oncologist based on the institutional standard agreed upon for contouring of organs at risk and target volumes.2.Bin vivo dosimetry, a commercial OSL system was used consisting of the InLight microStar reader  with Landauer nanoDot dosimeters. These devices were prescreened by the manufacturer for accuracy. This system can be used to measure dose at or near the skin surface.2O3:C) encapsulated in 0.2 mm thick, 5 mm diameter discs. This sensitive material is enveloped by a light‐tight plastic casing that measures 10 × 10 × 2 mm. In the first year of using OSLDs at our institution, monthly quality control tests determined that the OSL system is accurate within ±3% for therapeutic doses (approximately 10–300 cGy/fraction) These results are in line with the manufacturer's specifications and prior data.For 2.CFor any high dose head and neck radiotherapy plan, the treating radiation oncologist may request OSLD measurement for verification of the delivered dose relative to the planned dose. The institutional policy was that the measurement would not result in any treatment changes unless a discrepancy of >5% was detected and felt to be clinically significant. An OSLD measurement was performed for one of the fractions during the treatment course for each individual patient involved in this study. Once the patient was set up on the treatment couch, an OSLD was placed on the skin directly adjacent to the stoma. This was a predetermined region associated with the highest prescribed dose from the planning distribution. The tracheostomy flange was located directly over top of the OSLD and centered over the stoma as per the usual clinical practice. The OSLDs were left on during cone‐beam CT imaging for positioning and verification. This was done for practical purposes, since patients were treated in an immobilizing thermoplastic mask. This made it unfeasible to position the patient with CBCT and then place the OSLD under the mask and other apparatus. It was determined that the CBCT dose only added between 0.2 and 0.6 cGy of dose reading (uncorrected for the kV response of the OSLDs) to the OSLD during a phantom study. This was determined with OSLDs that were taped to an anthropomorphic head‐and‐neck phantom . These phantoms were then CBCT‐scanned using the clinical H&N settings. The imaging dose was a small, negligible percentage as compared to the typical 200 cGy dose per fraction delivered in a head and neck radiation treatment plan.After irradiation, the OSLDs were read out by the microStar reader after at least 10 min had elapsed.2.DThe density of the tracheostomy hardware was calculated using CT images. This result was then compared to physical measurements of the device’s density. This was done in order to ensure that the tracheostomy tube and flange were not composed of any high atomic number elements that would erroneously increase the calculated density on CT images,3 TPS. The physical measurements of the density involved careful determination of flange volume using the Archimedes' principle of displacement.CT images of patients with a tracheostomy tube and flange were acquired. These scans were obtained using a Philips Brilliance Big Bore CT scanner . The images were acquired at 1.2 mm × 1.2 mm axial pixel pitch with a 1.5 mm voxel thickness using 120 kVp. The CT images were used to delineate the tracheostomy tube and the flange. The mean density of the flange volume was then calculated using Pinnacle2.Eperistomal volume. The tracheostomy tube and flange were then contoured using a fine 1 mm brush on lung density CT window  and is referred to as the trach contour. The mean dose to this OSLD volume was measured from the published patient treatment plan.In order to evaluate the tracheostomy tube and flange’s impact on the dose to the adjacent target region, a retrospective dosimetric analysis was performed. The OSLD location was contoured using a 5 × 5 × 1 mm volume at the skin surface directly adjacent to the stoma site. This was done to estimate a CTV and this region was defined as the peristomal volume without the tracheostomy equipment, the density of the trach contour was set to zero and the dose was recomputed.The number of monitor units was held constant for each patient’s treatment plan and then a smaller dose grid with a resolution of 0.1 mm was used for further analysis. To estimate the dose to the For cases where additional bolus material was applied overtop of the tracheostomy flange, this additional material was contoured. The treatment plans were then evaluated again with normal density and then a second time with the bolus density set to zero using fixed monitor units for each plan. A screenshot of the tracheostomy tube, flange, 95% isodose line, and the effect that zeroing the trach contour has on the isodose line can be seen in Fig. 2.FThe patient, tumor, and treatment characteristics (n = 15) were analyzed and summarized using descriptive statistics. The difference between the prescribed dose from the radiation plan and the measured OSLD dose was calculated as an absolute value (cGy) and as a percentage difference to normalize for the variation in the absolute prescribed doses. The mean of the differences was calculated along with the standard deviation of the differences. For each patient, a paired t‐test was carried out to assess the mean difference between the OSLD measurement and the radiation dose predicted by the planning system at the peristomal region.33 which was comparable to the predicted value from CT planning datasets, measured as 1.168 ± 0.2 g/cm3. Radiation plans consisted of IMRT technique with the prescribed dose to the tracheostomy site ranging from 50 to 70 Gy (median 70 Gy). Tables This single institution study identified 15 patients with biopsy proven head and neck cancer. These patients had tracheostomy prior to radiation and at least one measurement of the dose received using an OSLD. A complete set of patient, tumor, and characteristic information can be seen in Table P = 0.34.In Fig. trach contour was excluded from the planning CT scan  the target coverage and dose to the peristomal volume decreased significantly. The predicted mean dose was reduced by an average of 53.5% (SD 22.5) and therefore coverage of the peristomal target volume was inadequate.When the For patients with bolus, the target dose and coverage also decreased when the bolus and applicator were excluded from the planning CT . This was seen to be an average of 62.7% (SD 3.50) for patients with a bolus and an applicator and 46.8% (SD = 1.22%) for patients with a bolus but no applicator.4in vivo surface measurements to validate the predicted doses.This study demonstrates that the tracheostomy tube and flange have a significant impact on head and neck radiation target coverage in the peristomal region. This was achieved through analysis of 15 separate patients. Patients 1–10 only had a tracheostomy device and did not have any bolus material. With the tracheostomy device density set to air equivalent a significant difference between the predicted plans can be seen. For patients 11–13 a bolus material was present in addition to the tracheostomy device. By comparing the dose changes with the tracheostomy device and bolus set to air equivalent we can directly observe the similar effect that these two materials have on the predicted dosage. Additionally, patients 14 and 15 further reinforce this conclusion, as even without a tracheostomy device the significant dose effect of the bolus can be seen. This dose effect is caused by how the radiotherapy beams interact with materials. Radiotherapy penetrates the surface layer it encounters and then irradiates deeper layers. Therefore, to treat superficial lesions a bolus material is commonly used to act as a layer of scattering material to replicate the skin surface. By setting the density of our tracheostomy applicator to zero we see a significant drop in the radiation dose delivered to the peristomal region. This directly demonstrates how the tracheostomy device is acting as a bolus material to cause an increase in the dose that the skin surface and subcutaneous tissues receive. To our knowledge, this is the first study to demonstrate the effect of tracheostomy equipment on radiation dose, using An important consideration for these findings is the accuracy of the OSLD measurements, and the differences between the measured and planned dosages. There are a couple of factors which can cause deviation between these values. The first major cause is related to the location of the dosimetry measurements. The dosimeters were placed on the skin directly adjacent to the stoma, within a region of high‐dose gradient. In this region there is the largest potential for deviation between the OSLD measurement and predicted dose value. Since there is a large gradient in this area, minor changes in positional accuracy will have large effects on the OSLD's measurement accuracy as compared to the predicted value. Second, the calculated TPS density for the applicator flap could potentially cause a deviation from the actual radiation dose delivered. However, the calculated density from the CT images of the applicator flap correlated with the measured physical density. This indicates that the applicator is constructed out of a polymer material with a low average atomic number; as such, the electron density used for the dose computation is accurate. Despite these sources of potential error, the difference between the planned and measured doses were on average <4%. This is within acceptable limits as described by evidence‐based treatment guidelines.As seen with the tracheostomy devices in this study, medical devices can have a significant effect on radiation treatment planning. The dosage effects of medical devices must be carefully accounted for to ensure target coverage and to avoid excessive toxicity.This study has several important limitations. This was a retrospective study with variation in the clinical presentation, primary disease sites, and total prescribed dose, which reflects clinical practice. All patients included in the study had IMRT and therefore it is not possible to extrapolate these results to patients who may undergo other treatment techniques/modalities such as conventional radiotherapy, VMAT, electrons, or proton therapy. The OSLD measurements were taken on average once or twice per patient and it is possible that there may be differences in setup due to air gaps, device placement or inter/intra‐treatment motion through the full course of radiotherapy. The number of patients included in this study is low and reflects an uncommon but important clinical situation. Future studies are needed with a larger cohort of patients to allow for further statistical validation of these findings for specific clinical presentations and disease sites.5in vivo OSLD measurements at the peristomal region. The tracheostomy flange applicator was found to have a density similar to water equivalent bolus. Using OSLD measurements during treatments, the actual measured dose in the peristomal tissues was, on average, within 4% of the predicted dose from the radiation treatment plan. This was deemed to be clinically acceptable. Overall the tracheostomy flange causes a bolus effect, with a twofold increase in the skin surface dose. If there is no tracheostomy tube or flange in place, then a customized ring of bolus 1 cm thick, may be used to achieve similar dose coverage at the skin surface. Dose coverage of the tracheostomy site should be carefully evaluated for in each patient depending on the specific in situ device and institutional practice. Any changes that may occur in the peristomal device placement from the time of simulation to treatment completion could have a clinically significant impact on radiation dose and may require adaptive planning or other modifications.This is a retrospective, dosimetric study of 15 head and neck cancer patients who underwent high‐dose radiotherapy and had The authors do not have any conflicts of interest to declare.


We consider the Fröhlich model of a polaron, and show that its effective mass diverges in the strong coupling limit. The polaron model introduced by Fröhlich  represensee Eqs.  and , and quroved in  using opm of the polaron is defined viaThe effective mass The effective mass of the polaron satisfiesm that diverges as According to  =1. In  it was srinciple  it is noV, we cannot directly apply  ar ar\documIt remains to derive the claimed lower bound  on \docuV. Similarly to (To derive a lower bound on ed as in . The firlarly to  above, wIn particular, if we choose q. By assumption the functions k. We plug these bounds into (R. By translation invariance, we may assume without loss of generality that this ball is centered at the origin. The relative error in the energy coming from the additional terms in (Next we shall localize the electron. With The remainder of the proof is now identical to , and we


The design of loading systems to test biologic samples is often challenging, due to shape variability and non-conventional loading set-ups. In addition to this, large economic investments would not be justified since the loading set up is usually designed for one single or for a limited range of applications.•Applying cyclic tensile loads on the suture wire, mimicking the physiologic condition where both suture anchorage points have a certain compliance;•Designing a loading system as versatile as possible, in order to be able to accommodate organs with different geometries and sizes;•Keeping low both the complexity and costs of realization.All these considerations and the design calculi are here reported in detail, discussing the novelty of the system, and its main advantages.The object of this work is the development of a loading set-up finalised to on-site testing of sutures whose main function is applying a localised tensile load. The main challenges of this design process can be so summarized:  Specifications TableIn this methodological work the design of a non-standard loading system is described. The authors have designed a loading system which was intended to perform cyclic tests on sutured organs .Fig. 1Eq•Interfaceability/Compatibility with the testing machine: The loading system must be interfaceable with a standard testing machine with reference to both its base and its actuator. In this case the testing machine was an Instron Electropulse E3000.•Loading the suture along its axis: The suture had to be loaded along the direction joining its two anchorage points on the organ.•Testing the suture on-site: This requirement came from the necessity of taking into account the compliance of suture anchorage points at both ends.•Versatility: The sutured organ does not have a standard shape  and it might show significant anatomic variability among subjects.•Stability and stiffness: The design had to provide an excellent stability and stiffness in order to guarantee a good repeatability of results and to minimize the bias produced by the adopted loading system.•Feasibility: The components should be immediately available or they should be simple to be manufactured.First of all, its main requirements have been identified, as described in the following.The first step of the design required the identification of the best way to properly load the suture wire once it has been implanted into the organ.Loading configurations used in previous works to stress sutures on-site 1T1 and T2 applied to the two branches of the suture changes during the test because of the variation of α angle (A or FB) and T1 and T2 changes during the test, as resulting from equilibrium equations:The direction of forces This approach exhibits some important issues: 2a and b in T1 and T2 forces . In addition, T1 and T2 forces are always horizontal, along the suture wire, independently from anchorage point migration. The above-mentioned conditions apply for a specific design where the diameter of the upper pulley is equal to the sum of the radii of the smaller pulleys (r1 and r2) and the distance between their centres (i):This means that this loading system allows calculating in every instant the exact value of the tension which acts on the suture once the force applied by the testing machine is known and controlled: when the loading machine applies a force Having established the working principle, that is the ‘conceptual design’, the next step was the detailed design of each component.(1)The Support Base with perforated plates: holes have been made in order to allow constraining the organs through one or more bolts. The support base's weight is over 20 kg in order to provide a good stability: for most applications, it can be simply overlaid on the loading machine base without the need of additional constrains(2)The Support Plate: this plate has been soldered on the support base and it allows rising the height of the lower pulleys with respect to the support base(3)The Support Bar with slot: this bar allows regulating the horizontal position of the lower pulleys(4)The Lower Pulleys: the lower pulleys function is to load the thread along its axis; in order to achieve this aim, their height must be regulated, as visible in Fig. 5(5)The Upper Pulley: the upper pulley is constrained to the actuator of the loading machine.The whole loading system is shown in All these components will be analysed in detail in the following.Lower pulleys are bolted to the support bar n.3 in ; this coMore in detail, the support bar bears a longitudinal slot where the lower pulleys are fixed; this slot allows a fine and continuous tuning of the horizontal position of the pulleys. In addition, the bar bearing the lower pulleys can rotate around a vertical axis. These two regulations guarantee that the line connecting pulleys’ centres and the line passing through the two anchoring points belong to the same vertical plane.The support bar is connected to the loading system via the support plate; this one is fixed to the support base with three screws, while the support bar is connected to this plate by two bolts. The plate has seven holes which allow adjusting the vertical position of the support bar in relation to the size of the sutured organ.δ) at the point of load application.The support bar must guarantee a sufficient stiffness when its end is loaded with the peak load, which has been set equal to 100 N. It was calculated admitting 1.5 mm displacement , the following equations hold, respectively:With reference to overturning moments .The cost of all components has reached about 660 Euros.After all the components have been designed, their assembly and their interaction with the larynx have been evaluated numerically. The geometry of single components has been replicated with 3D CAD and the whole system has been assembled.The three-dimensional model of the organ can be obtained from CT data.Technical drawings complete with dimensions are provided for each designed component, as supplementary material.A virtual loading test has been performed, considering all elements as solid bodies .Fig. 9Mu3); they have been fully constrained, with the only exception of the support base, whose contact with the base of the loading machine has been simulated making use of eight spheres, put at the eight vertices. The simulation has been performed applying the gravitational force and a rising external force from 0 N to 200 N , the system needs to be clamped to the base of the machine.This loading system has been employed to perform experimental tests on an equine laryngoplasty. In relation to this application, cyclic loads have been applied with displacement control 1 mm/s), between two loads limits (30 N and 50 N), corresponding to physiologic swallowing loads. In the supplementary material, a video is provided where the application of the system to a laryngoplasty procedure is shown. Results obtained on 8 samples are reported extensively in other works by the same authors  mm/s, be: the sysThis set up has been designed to test sutured organs behaviour during follow-up, focusing on a relevant complication that is suture loosening The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


The peak particle velocity (PPV) is an important indicator for predicting blasting excavation disturbances. However, the PPV distribution in the deep underground space is significantly different from that on the outdoor ground. Therefore, it is difficult to predict the underground PPV by Sadovsky’s vibration formula. The PPV sidewall distribution characteristics were studied during site blasting in an underground cavern in the Taohuazui mine in China, and a similar numerical model was used to verify the site test data. We derived a PPV prediction formula for the underground cavern sidewall surrounding rock using a mechanical analysis model of a simply supported plate and beam in combination with dimensional analysis. The model considered derived boundary constraints, comparison with site measured data, the value predicted by Sadovsky’s vibration formula, and numerical simulation results. The results showed that the PPV distribution on the middle 1/3 section of the underground cavern sidewall showed a “platform” or “bulge” different from the curve from Sadovsky’s vibration formula. The PPV amplification coefficient in this section was distributed in a drum shape. The PPV prediction formula for the middle section of the sidewall derived in this paper was highly consistent with the data measured on-site and the numerical simulation results. The mechanical analysis model with a simply supported plate and beam included an underground cavern sidewall length–height ratio of 5 and effectively supplemented the PPV prediction formula for the middle section of the traditional underground cavern sidewall. With rapid developments in the scale and buried depth of underground excavation5, the construction of a large number of underground caverns has been commenced successively, and the associated difficulties in blasting excavation construction and vibration prediction control of upright high sidewalls have been increasing9. During blasting excavation, energy propagates along the sidewall in the form of vibration waves, inducing damage to the surrounding rock and lining and leading to sidewall instability, closed function failure, and other problems. This greatly increases the construction safety risk and operation maintenance costs. Engineering practices have shown that there is a good correlation between the PPV and structural failure10. A large number of studies on the attenuation law of PPV and its fitting formulas have emerged, such as empirical prediction formulas represented by Sadovsky’s cube-root formulas11 and USBM square-root formulas12. Therefore, the PPV of underground cavern sidewalls should be strictly and accurately tested and predicted13.To date, drilling-blasting is still the main excavation method for constructing tunnels, mines, underground hydropower stations, and other underground spaces. Researchers continue to study blasting safety during construction17. Many researchers have studied the slope elevation effect under blasting vibration22 and proposed modified Sadovsky’s formulas. These modifications use a power function or an approximate curve to predict the PPV monotonicity attenuation with the distance from the blasting centre or survey point elevation23. Besides, many scholars have developed a series of soft computing methods based on modern computer technology for PPV prediction model, such as artificial neural network24, gray relational analysis25 and genetic programming26. Compared with the traditional empirical prediction formulas, through the soft science method, generally more input parameters can be considered, and the predicted value and measured value can have a higher degree of fitting through intelligent algorithm based learning and training of the model27. Table 32.The theoretical and empirical formulas derived by Sadovsky from spherical charge initiation conditions in an infinite free field  have significant limitations when used to predict the PPV distribution of blasting vibrations in underground jointed rock masses42. Therefore, describing underground sidewall PPV simply by the amplification effect has limitations due to the lack of a reasonable mechanical model for dynamic analysis and a formula suitable for underground engineering practices. As a result, the industry requires a PPV prediction formula suitable for underground high sidewalls. Some scholars believed that, different from blasting vibration propagation in continuous ground media, blasting vibration attenuation in underground projects is affected by special spatial geometry and constraints, and attempted to consider the high sidewall as a simply supported beam for vibration modal and mode analysis, but no in-depth study or discussion was carried out44.The boundary constraints of underground upright sidewall planes or cylindrical sidewalls are significantly different from those of the infinite space considered by Sadovsky’s vibration formula or the semi-infinite space on open ground. Many studies have proven that the vibration attenuation of underground structure blasting is different from that of surface blastingBased on the blasting excavation project of the underground space in Taohuazui mine of China, the mechanical calculation model of PPV prediction and amplification effect distribution characteristics of underground engineering blasting is established through the combination of in-situ test, theoretical analysis and numerical simulation. Combined with dimensional analysis method, the PPV prediction formula of surrounding rocks induced by underground engineering blasting excavation is obtained and the application scope of the simplified end constraint method is determined.The underground cavern in the Taohuazui mine located in the middle and lower reaches of the Yangtze River in Daye City, Hubei Province, China was chosen as the site for this test in consideration of its spatial structure and surrounding rock characteristics representative of underground projects. Geological conditions of surrounding rocks in the underground cavern: the orebody is simple in shape and lenticular as a whole. The ore is a skarn bearing copper–gold ore, with the main minerals of pyrite and chalcopyrite, followed by magnetite and siderite, and a small amount of bornite. Generally, the ore is compact and massive. The upper and lower walls of the ore block are skarn or skarn with quartz syenite diorite porphyritic. The surrounding rock of this cavern was highly intact and unsupported, which reflected the vibration performance of the original cavern sidewall surrounding rock. The overall dimensions of the underground cavern were 32 m × 10 m × 12 m (L × W × H). The section of the underground cavern tested is shown in Fig. We conducted two blasts in this test and recorded the blasting vibration information on the two survey lines during each blast. We modelled the underground cavern sidewall surface as a simplified rectangle, and the intersection of the sidewall bottom plate and the left boundary was set to the origin. The bottom plate and the left boundary were defined as the X-axis and Y-axis, respectively, to establish a rectangular coordinate system as shown in Fig. The two survey lines arranged for the survey points in this test were named Survey Line I and Survey Line II as shown in Fig. Blasthole I was initiated first to obtain the three-dimensional PPV of survey points I1-I8 is shown in Fig. . Therefore, the study was carried out on the PPV in the Y-axis direction perpendicular to the sidewall. Comparing the PPV of survey points on Survey Lines I and II under the initiation conditions of Blastholes I and II showed that the smaller the horizontal distance from the blasting centre, the higher the PPV. Although the PPV variation in the three directions did not completely monotonically decrease with increasing elevation difference, it showed an overall decreasing trend. The separate analysis of each set of data showed that the PPV in the upper section of the underground cavern sidewall varied with the elevation difference. The PPV in the middle section of the sidewall showed a “platform” or “bulge” generally distributed in the elevation range of 5–8 m. The data on the two blast test survey lines were compared with Sadovsky’s vibration formula . The dimensions of this cavern were 32 m × 10 m × 12 m (L × W × H). The whole model included 535,028 units and 556,632 nodes, as shown in Fig. The PPV at the survey points of the same positions as the on-site test was extracted and compared with the site survey points, as shown in Fig. To study the distribution characteristics of the amplification effect on the sidewall, we extracted the PPV at all nodes on the sidewalls in the whole model cavern over time The PPV was extracted and divided by the corresponding value predicted by Sadovsky’s vibration formula to obtain the sidewall PPV amplification coefficient, as shown in Fig. Since the PPV prediction in the middle section of underground cavern high sidewalls by Sadovsky’s vibration formula are consistent with actual conditions, in this section we describe a mechanical analysis model more in line with actual conditions. The model considered structural dynamics in combination with high sidewall constraint conditions during underground blasting excavation. In addition, the vibration characteristics and dimensional methods were analysed to obtain a prediction formula suitable for blasting vibrations in the middle sections of high sidewalls.Under actual engineering conditions, the upper and lower ends of the surface rock surrounding underground cavern sidewalls are subject to constraints from the roof and the bottom plate, while the left and right ends are subject to constraints from the boundaries. Therefore, the surrounding rock typically has simply supported plates on four sides, but are different from those of ordinary four-sided simply supported plates. Simplified conditions were used in this study to analyse such characteristics, as shown in Fig. We simplified the four-sided constraints and the lateral forces on the surface surrounding rock. Since the explosion source was near the bottom of the sidewall, the blasting load was equivalent to a simple harmonic concentrated load F acting on any point  on the simply supported plate on four sides. The forced vibration equation for the simply supported plate on four sides is:For the blast impact loads, when not considering the system damping effect, it can be assumed that:Then, the oscillatory differential equation changes to:46:When the dynamic basic solution of the deflection surface equation of the simply supported rectangular plate on four sides acts as a simple harmonic transverse unit concentrated load on any point  on the plateIt is not difficult to generalize from the dynamic basic solution that when the force F(t) at any point  on the simply supported rectangular plate on four sides is not a unit harmonic force, the deflection surface equation is:The partial differential of the time t was calculated by Eq. . The PPVGenerally, for rock and soil structures with higher stiffness, such as underground caverns, only the first-order vibration mode is considered. Moreover, for each blast, the load acting on point  is a known point; namely, for each blast, We analysed the section of the underground cavern sidewalls with a greater length–height ratio, as shown in Fig. t. The differential equation for the bending vibration of a simply supported beam can be expressed by the D’Alembert principle:When only the deflection of the beam caused by the bending moment is considered and the thickness of the selected surrounding rock is small, the plane cross-section assumption of the Euler–Bernoulli beam is satisfied. The initial conditions of beam vibration are:The boundary conditions of beam vibration are:The displacement equation solution form of Eq.  is obtaiThe influence of the high-order vibration mode is ignored. Only the first-order vibration mode is considered.When The equation above is used to obtain the partial differential of time From the above equation, we can obtain the maximum vibration velocity of the simply supported beam appears in the middle of beam Combined with the dynamic analysis results in “According to the number of parameters, there is a total of 12 physical quantities analysed. The independent variable is  according to the π theorem, and there are 9 π components. Equation  is substIn addition, the following dimensionless quantity can be obtained:Under the same site conditions, ρ and c are approximated as constants. Therefore, from Eq.  there isIf the influence of the four-side constraints of the sidewall on PPV is considered, the following can be obtained by solving Eq. :22\documWhen Similarly, considering the influence of the constraints at the upper and lower ends of the cavern sidewall on the PPV, the vibration velocity formula of the sidewall can be derived as follows:Unitary regression and binary regression analyses were performed on Sadovsky’s vibration formula and prediction Eqs.  and 24)24) accorThe variation in the measured value in the Y-axis direction at survey points on Survey Lines I and II, the value predicted by Sadovsky’s vibration formula, and the values predicted by the simply supported plate formula and simply supported beam formula with the elevation difference were drawn into curves on the same diagram for comparison Fig. .Figure 1According to Fig. Since the dimensions of the selected underground cavern in the Taohuazui mine were limited to a length–height ratio less than 3, it was consistent with the mechanical model of the simply supported plate and not as consistent with the mechanical model of the simply supported beam. When the length–height ratio of the underground cavern sidewalls is great, the left and right constraint effects on the sidewall are insignificant. When the whole sidewall is simplified to the simply supported plate model, it is no longer applicable. Instead, the sidewall section should be simplified to a simply supported beam for analysis. The distribution characteristics of the PPV and amplification effect under different length–height ratios will be further studied.We established a group of similar numerical models according to the parameters of the underground cavern in “The blasting zone of the four models was designed in the middle of the sidewall. The blasting PPV was selected as the survey line directly above the blast zones which were located at 6 m, 18 m, 30 m, and 42 m. A schematic of the blast zones and survey points is shown in Fig. We see from Fig. By comparing the closeness of the two absolute value curves of the difference between the predicted values and simulated values to the curve y = 0 we observed that the closer to y = 0, the better the prediction effect of the formula. In addition, when the length–height ratio of the sidewall was 1, 3 and 5, the closeness of the red curve to the curve y = 0 was basically the same and better than the blue curve. When the length–height ratio of the sidewall was 7, the red curve started to deviate from the curve y = 0. However, the blue curve became increasingly closer to the curve y = 0 with the increase in the length–height ratio and was very close to the red curve when the length–height ratio was 5. The blue curve was even closer to the curve y = 0 than the red curve when the length–height ratio was 7. This indicated that when the length–height ratio of the sidewall was greater than 7, the values predicted by the two-end constraint model were more accurate than those predicted by the four-side constraint model. Therefore, to ensure more accurate prediction results, we suggest using the mechanical model of a simply supported plate or simply supported beam and its prediction formula in the underground space with a sidewall length–height ratio of 5.The PPV on the underground cavern sidewall showed a “platform” or “bulge” at the sidewall middle elevation. That is, an elevation amplification effect appeared in the middle 1/3 section of the sidewall. The maximum amplification coefficient in the middle of the sidewall reached 1.9. We found by comparing the prediction ability of Sadovsky’s vibration formula and the PPV prediction formula herein that the prediction formula considering “boundary constraints” could more accurately predict the PPV drum distribution characteristics in an underground cavern sidewall middle section.When the length–height ratio of an underground cavern sidewall is smaller than 5, the mechanical model of the simply supported plate and its prediction formula are recommended.In this paper, the four-side constraint and two-end constraint PPV prediction formulas were derived using an underground cavern blasting test in the Taohuazui mine in China with the mechanical model of a simply supported plate and simply supported beam. The following conclusions were drawn by comparing the measured sidewall PPV of the underground cavern with the PPV predicted by the formula:The boundary conditions of an underground cavern sidewall surrounding rock are complicated. Therefore, we simplified the boundary constraints when establishing a mechanical analysis model, leading to some limitations in the use of the model and prediction formula. However, the end constraints on the surrounding rock are actually a kind of force constraint between simple and clamped supports. Further study on this problem should be done. Since it is difficult to obtain complete and accurate data on the blasting PPV of underground cavern high sidewalls, the mechanical models and prediction formulas proposed in this paper need to be tested in more projects.


Peripersonal space (PPS), the interface between the self and the environment, is represented by a network of multisensory neurons with visual (or auditory) receptive fields anchored to specific body parts, and tactile receptive fields covering the same body parts. Neurophysiological and behavioural features of hand PPS representation have been previously modelled through a neural network constituted by one multisensory population integrating tactile inputs with visual/auditory external stimuli. Reference frame transformations were not explicitly modelled, as stimuli were encoded in pre‐computed hand‐centred coordinates. Here we present a novel model, aiming to overcome this limitation by including a proprioceptive population encoding hand position. We confirmed behaviourally the plausibility of the proposed architecture, showing that visuo‐proprioceptive information is integrated to enhance tactile processing on the hand. Moreover, the network's connectivity was spontaneously tuned through a Hebbian‐like mechanism, under two minimal assumptions. First, the plasticity rule was designed to learn the statistical regularities of visual, proprioceptive and tactile inputs. Second, such statistical regularities were simply those imposed by the body structure. The network learned to integrate proprioceptive and visual stimuli, and to compute their hand‐centred coordinates to predict tactile stimulation. Through the same mechanism, the network reproduced behavioural correlates of manipulations implicated in subjective body ownership: the invisible and the rubber hand illusion. We thus propose that PPS representation and body ownership may emerge through a unified neurocomputational process; the integration of multisensory information consistently with a model of the body in the environment, learned from the natural statistics of sensory inputs. Peripersonal space, the region of space immediately surrounding the body, is represented by multisensory neurons with visuo‐tactile responses in body‐part centred coordinates. The underlying reference frame transformations are crucial for body representation, and visuo‐tactile congruencies shape the sense of ownership for the body. We show how both can emerge in an artificial neural network model, based on Hebbian learning of the natural statistics of visual, proprioceptive and tactile inputs. HMDhead‐mounted displayIHIInvisible Hand IllusionPPSperipersonal spaceRBMrestricted Boltzmann machineRFreceptive fieldRHIRubber Hand IllusionRTreaction timeSDstandard deviationVRvirtual reality11.1Peripersonal space (PPS) is typically defined as the region of space immediately surrounding the body, or the space where we can physically interact with external objects, either actively, by reaching to touch them, or passively, when we enter in contact with an incoming object  or appetitive  purposes  subtending PPS representation.That the same learned associations that build PPS representation implicitly perform reference frame transformations in body‐part centred coordinates. Therefore, a key novel point of our study is to demonstrate that PPS representation and reference frame transformations can emerge spontaneously and simultaneously within a unified neurocomputational process, by learning the statistical associations in multisensory inputs that occur naturally when interacting through the body within the environment. As a key example of PPS representation, here we focused on visuotactile integration around the hand, in hand‐centred reference frames.The aim of the present study is therefore to extend the previously established model of PPS representation  position of the visual stimulus, and px the position of the hand in the same 2D plane, then the activity of the i‐th unisensory neuron iu is defined by:vu, pu, tu, respectively, denote neurons belonging to visual, proprioceptive and tactile populations, and x^ denotes the preferred position of a given neuron. The SD of the tuning curves (vσ and pσ) was set at three neurons for the visual population, and at one neuron for the proprioceptive population . 30 tactile units are used, and the preferred positions of the proprioceptive and visual neurons tile the space on a regular grid of 15 × 10 and 50 × 50 neurons, respectively. This includes the 1.2 × 0.6 and 1.2 × 1.2 meters of space represented by the neural populations, plus a safety margin  to avoid boundary effects. The width of the tuning curve was mainly determined during preliminary testing, on the basis of a set of heuristic criteria. We noticed that in order to allow efficient learning, the average learning signal from units from different populations needs to be approximately the same, hence the width of the tuning curve needs to be a fixed fraction of the total population range. Since the average firing rate of tactile units is fixed by the proportion of training inputs where touch is provided to approximately 5%, the width of visual and proprioceptive tuning curves was chosen to approximately match this value, while not requiring excessively large safety margins. In any case, the network's main predictions were robust with respect to the choice of such parameter, as shown in the In a probabilistic population code, such as the one used for the generation of stimuli in our network, the activity of neurons in the unisensory populations can be seen as a probability distribution conditioned on the position of the stimuli in the physical world, from which spike counts are drawn for each population. Let u is the vector of activity of all neurons in the lower layer (unisensory), m is the vector of activity of all neurons in the upper layer (multisensory), ijW is the synaptic weight connecting neuron mi to neuron ju, and ub and mb are biases for unisensory and multisensory neurons respectively. Note that the fact that the matrix used in the “down” pass is the transpose of the matrix used in the “up” pass implies that feedforward and feedback synapses are symmetric. In practice, the multisensory neurons' activity, given the unisensory neurons' activity, is a vector of samples of Bernoulli variables, whose mean is a sigmoidal function of the weight matrix acting on the unisensory neurons. Conversely, the unisensory neurons' activity, given the multisensory neurons' activity, is a vector of samples of Poisson variables, whose mean is the exponential function of the weight matrix acting on the multisensory neurons. In RBMs, the choice of sigmoidal and exponential “link” functions is the standard for Bernoulli and Poisson units, respectively  in its shared representation. Since it would be very difficult to decode such information directly from the multisensory layer, we proceeded as Makin et al. . In orde2.52.5.1The network uses a simplified set of sensory inputs, as visual information about the hand's position and appearance is not present. In literature, hand PPS representation in humans was typically assessed through a simple tactile detection task, in which in which reaction times to tactile stimuli on the hand are measured in the presence of task‐irrelevant auditory or visual stimuli, at various distances from the hand  coupled with the Steam VR software .Tactile stimulation was delivered through rotating mass vibrators (Precision Microdrives), driven by a dedicated microcontroller. A hand‐held button was attached to the same microcontroller, in order to collect reaction times to tactile stimulation on the same device and minimize unpredictable delays. Visual stimuli were delivered in a virtual reality scenario. A Head Mounted Display  was used, and rendering of the virtual environment was performed through a custom made software  were recruited for the study, and received monetary compensation for their time. Only right‐handed participants with normal or corrected to normal vision were recruited for the study. The study conforms with the World Medical Association Declaration of Helsinki, was approved by the ethical committee of the Vaud canton, Switzerland, and was performed with the understanding and written consent of each subject.Forty‐three healthy participants . Such responses were discarded. We then removed outlier responses by discarding, for each subject and experimental block, RTs falling more than 2 median absolute deviations away from the median RT. This cut‐off is a more robust equivalent of the standard cut‐off at 2 SDs, as suggested by theoretical and empirical justifications in methodological work  longer than 700 ms were automatically discarded by the microcontroller. This threshold can be considered safe as the average RT was 264.75 ms, with an average within‐subject RTs were measured in a 2 × 3 design with the factors Hand position , and Trajectory, indicating the direction of the ball, . Trajectory was recoded as Congruency: Congruent, when the ball was moving towards the tactilely stimulated hand, Incongruent, when it was moving towards the opposite side, and Receding. Reaction times were then analysed by means of linear mixed‐effects models. We fit a model on to multisensory facilitation (MF), including Congruency and Position as predictors. Different random structures were tested, assessing all the five possible combinations of Position and Congruency, including their interaction and just a random intercept, and the model giving the best fit was selected. Both in terms of Akaike and Bayesian Information Criterion, the best model was that which considered only the position as a random factor:Additionally, a model including a Position*Congruency interaction was tested, and statistical testing confirmed the selected random structure. Data preprocessing and further analysis was run in R . Linear mixed‐effects models were tested using the Satterthwaite approximation for the degrees of freedom from the lmerTest package.33.1The neural network was designed to learn a connectivity scheme that optimizes the reconstruction of the patterns of stimulation observed during the training. This led to a spontaneous diversification of the response of neurons in the multisensory layer to the different sensory modalities. Since at initialization all the multisensory neurons are connected with all the neurons in unisensory populations, after the training all neurons were to some extent multisensory, meaning that they received some input from all the unisensory populations. Nevertheless, a great variability in the modality tuning of different neurons and thus in the pattern of sensory responses emerged. In order to quantify the response of each neuron to a given modality, we computed the sum of the absolute value of the strength of synapses from a given unisensory population, normalized by the mean input for that modality. In particular, we focused on the response of multisensory neurons to tactile inputs. As appears evident from Figure In order to test whether and how the model might build up a PPS representation from capturing regularities in the environment, we tested how the spatial properties of the multisensory neurons depended on their tuning to the tactile modality. It is known from neurophysiological literature that the PPS is represented in the monkey cortex by a set of multisensory neurons that respond both to touch on a given body part, and to visual stimuli close to that body part. In the previous version of the model, this evidence was implemented by a hard‐wired connectivity whereby the projections to the multisensory neuron(s) were of fixed strength from the tactile area, whereas from the visual area they decreased as a function of the distance from the hand. We asked whether our model could simply learn a similar pattern of connectivity from the multisensory training, in which neurons that respond more strongly to touch code mostly for the close (trunk‐centred) visual space. Since our model uses several multisensory neurons, that spontaneously tune differently to each sensory modality, we tried to define a suitable approach to test this hypothesis. For each multisensory neuron, we defined its preferred visual distance as the preferred distance  of the visual unisensory neuron that projects the strongest excitatory synapse to that same multisensory neuron. Roughly, this corresponds to the peak of the visual RF of the multisensory neuron. This allowed us to explore how the properties of multisensory neurons vary depending on the region of the visual space that is stimulated. We found that, on average, the tactile input computed by multisensory neurons slightly decreases with their preferred visual distance, coded as described above Figure . Excitat3.2In an RBM, information from the different unisensory populations of the lower layer is encoded in the upper layer in a unified and compressed representation, embedding the statistical relations between the unisensory inputs. This allows the network to build a more compact and accurate representation of the input than each of its unisensory components  = 6.389, p = 0.0017) and a marginally significant effect of Position  = 3.59, p = 0.065). When looking at individual coefficients, using the Receding trajectory as a reference, we found Congruent trials to be significantly faster , and Incongruent trials to be not significantly different  from receding trials. In order to directly compare Congruent and Incongruent trials, and assess the role of proprioception in visuotactile integration, we fit the same model on the subset of Congruent and Incongruent trials. Again, the main effect of Congruency was significant  = 5.912, p = 0.015), with Congruent trials faster than Incongruent trials . This is in line with the model's qualitative predictions, shown in Figure F = 6.389, p = 0.0017), nor the comparison Congruent versus Incongruent = 5.90, p = 0.015), and the interaction was not significant (p = 0.76). Note that the facilitation when the visual stimulus is on the opposite side of the midline  is close to zero, similarly to when the visual stimulus is in the region outside the PPS , in line with the non‐significant difference found in our experiments. It is worth noting that the multisensory facilitation compared to unisensory trials was significantly below zero in all conditions, including Receding trials . This seems to contradict model predictions, as no significant tactile evoked activity is expected in the far space. We hypothesize that this may be due to overall stronger expectation effects in multisensory trials, compared to unisensory trials, due to the presence of the virtual ball providing a more precise cue about the likely time of stimulation. Our experimental design, unlike most previous studies, makes the delay of stimulation orthogonal to the three conditions of interest, which allows comparing them while controlling for expectation. We also investigated more in detail the possible interactions between PPS representation and expectation effects, by analysing trials separately by stimulation delay. We found an overall effect of stimulation delay reducing reaction times, compatibly with the presence of expectation effects. However, the decrease in reaction times with increasing delay (and decreasing distance from the hand) was significantly stronger in the Congruent condition, further confirming the presence of proximity effects in modulating reaction times  in virtual reality. RTs were compared for Hand position  and Congruency of the ball trajectory . A linear mixed‐effects model on the multisensory facilitation (MF), including Congruency and Position as predictors (see Methods for details), showed a significant main effect of Congruency  results show how visual and proprioceptive information combined can affect the encoding of tactile information to reproduce the associations learned during the training . Since the learned associations have no preferential direction, we expect the transfer of information between sensory modalities to take place also in the opposite direction: from the tactile to the visual and proprioceptive modalities. In particular, we focused on how visuotactile inputs affect the encoded proprioceptive information, as this link has been previously investigated in several behavioural works exploring the multisensory bases of body representation  position affects the response of some PPS neurons  between hand position and RF peak, as an index of hand position tuning. As shown in Figure In the previous paragraphs, we showed how the network can encode information in hand‐centred coordinates at the population level. This allowed to reproduce some important behavioural and neurophysiological aspects of PPS representation. However, while neurophysiological studies reported individual neurons with visual receptive fields spatially anchored to body parts in space . Then, feedforward kinematics were computed to determine hand position in Cartesian coordinates, and tactile units were activated if the distance between the visual stimulus and the hand was smaller than 15 cm.Figure 3.9θ represents the gaze angle, and zR the rotation matrix along the vertical axis. As usual, tactile stimulation was present if the body‐centred positions of the hand and the visual stimulus differed by less than 15 cm. As summarized in Figure Similarly, the network presented previously had to learn a simplified version of the actual reference frame transformations that are necessary to link retinotopic visual input and tactile input through proprioception, as we chose to ignore gaze angle. We therefore explored whether the network, in addition to encoding proprioceptive input in joint angles, could handle the additional degree of freedom of gaze direction in the horizontal plane. We trained another network that was identical to the one presented in the previous paragraph, except for a fourth population coding for gaze direction. This population consisted of 120 neurons, representing gaze angles from −π/4 to π/4, with 0 indicating looking straight ahead, and a tuning curve width of 1 neuron. The small width of the tuning curve, as explained in the methods, is motivated by the necessity to keep the average firing rate approximately constant across the different populations to allow efficient learning. Visual and proprioceptive inputs were encoded as in the previous section, with the difference that the coordinates of the visual input were now eye‐centred. Visual inputs and gaze angles were again uniformly distributed in the respective range, and the body‐centred coordinates of the visual stimulus were determined by rotating its eye‐centred position by the negative gaze angle around the body axis.44.1The multisensory bases of PPS representation have been studied first in animal neurophysiological studies  was measured. Such tactile induced activity can be interpreted as the network's prediction of tactile stimulation, based on the integration of visual and proprioceptive information. We found the network's tactile predictions to be based on the hand‐centred coordinates of the visual stimulus, with a maximal strength when visual stimuli are close to the hand and an activation profile depending on the distance from the hand, closely resembling what reported from single cell responses by neurophysiological studies in monkeys (as in Graziano et al., Importantly, the fact that the receptive fields are learned and not hard‐wired allows us to treat their properties as predictions generated by the model, and not assumptions that are set a priori. Specifically, the model predicts the existence of neurons responding to touch, with overlapping visual and proprioceptive RFs, and neurons not responding to touch with dissociated visual an proprioceptive RFs. The collective behaviour of such neurons leads to the encoding of tactile information being influenced by the hand‐centred coordinates of visual stimuli. Their receptive fields are broad and complex in shape, and neurons do not individually encode information in body‐part centred coordinates. This is consistent with what was found in literature in multisensory neurons, displaying broad RFs and only partially shifting reference frames (Avillac et al., 4.3Similarly to what we did with touch, we then tested the effect of visuotactile stimuli on proprioceptive encoding, by providing visual and proprioceptive inputs, and studying the effect of tactile input on the read‐out proprioceptive information. We found that, in the presence of touch, the encoded proprioceptive position got attracted towards the position of the visual stimulus, replicating the proprioceptive drift induced in the IHI. The maximal magnitude of the forecasted shift is around 40% of the visuo‐proprioceptive disparity, in line with behavioural data (Guterstam et al., 4.4As we introduced in the previous paragraph, it may be useful to approach the problem beyond the focus of Bayesian optimality, and under a more general perspective. A key function of the brain is to learn the regularities in the probability distribution of its sensory inputs. Those regularities are then exploited to compress inputs in a simpler, more compact representation, retaining the relevant information about their causes in the external world (Attneave, There are other important features that were not directly modelled here, but could be implemented in a model with an architecture similar to ours, designed to learn the probability distribution of its sensory inputs, in order to extend its level of compliance with such definition. For instance, the present model allows to accurately model only simultaneous stimuli. However, the combination between temporal and spatial processing is key for a dynamical model of the body in space, which is deeply linked to PPS representation, as well as for bodily self‐consciousness. Moreover, a perfect generative model, which does not include temporal features, should in principle not fully account for a PPS representation extending beyond the skin, as it would only learn associations between touch and stimuli currently causing it. Again, this could be instead achieved in the general framework of the learning of a complex probability distribution, extending not only in space, but also in time. Interestingly, the idea of PPS representation as a spatio‐temporal prediction system finds empirical support in the observation that it expands when faster stimuli approach (Noel, Blanke, et al., The authors declare no conflict of interest.TB, EM and AS developed the conceptual framework. TB designed and implemented the neural network, and performed the simulations, behavioural data collection and data analysis. EM and AS provided supervision in data analysis and interpretation of the results. TB drafted the manuscript, all the authors edited and contributed to the critical revisions of the manuscript and read and approved the final version for submission.https://publons.com/publon/10.1111/ejn.14981The peer review history for this article is available at Supplementary MaterialClick here for additional data file.