The aim of this work is to study a new analytical model which describes the dose–response curve in megavoltage photon beams of the radiochromic EBT3 film measured with two commercially available flatbed scanners. This model takes into account the different increase of the number of two types of absorbents in the film with absorbed dose and it allows to identify parameters that depend on the flatbed scanner and the film model, and parameters that exclusively depend on the production lot. In addition, the new model is also compared with other models commonly used in the literature in terms of its performance in reducing systematic calibration uncertainties.The new analytical model consists on a linear combination of two saturating exponential functions for every color channel. The exponents modeling the growing of each kind of absorbent are film model and scanner model‐dependent, but they do not depend on the manufacturing lot. The proposed model considers the different dose kinetics of each absorbent and the apparent effective behavior of one of the absorbents in the red color channel of the scanner.The dose–response curve has been measured using EBT3 films, a percentage depth dose (PDD) calibration method in a dose range between 0.5 and 25 Gy, and two flatbed scanners: a Microtek 1000 XL and an EPSON 11000 XL. The PDD calibration method allows to obtain a dense collection of calibration points which have been fitted to the proposed response curve model and to other published models. The fit residuals were used to evaluate the performance of each model compared with the new analytical model.The model presented here does not introduce any systematic deviations up to the degree of accuracy reached in this work. The residual distribution is normally shaped and with lower variance than the distributions of the other published models. The model separates the parameters reflecting specific characteristics of the dosimetry system from the linear parameters which depend only on the production lot and are related to the relative abundance of each type of absorbent. The calibration uncertainty is reduced by a mean factor of two by using this model compared with the other studied models.The proposed model reduces the calibration uncertainty related to systematic deviations introduced by the response curve. In addition, it separates parameters depending on the flatbed scanner and the film model from those depending on the production lot exclusively and therefore provides a better characterization of the dosimetry system and increases its reliability. Their composition based on light elements, low energy dependence, near biological tissue equivalence, and high spatial resolution make them suitable detectors for advanced techniques in radiotherapy. Their measurement principle is based on the optical density change measured by a flatbed scanner.1, 2 rational functions, a linear relation combined with a power function to account for the nonlinear response, exponentials, or more complex parameterizations.The curve relating the digital signal produced by the scanner, as a measurement of the light attenuation through the film, and the absorbed dose is commonly referred as the dose–response curve. This relationship is not linear and several phenomenological models have been proposed to describe it. Various analytical models have been used: polynomials,5 rati Alternatively, a PDD calibration method can be used.To measure the dose–response curve, multiple film pieces are usually irradiated inside a reference field with the film positioned perpendicular to the beam central axis.3 Alte have determined the attenuation due to the film active layer as a spectral property. Devic et al. first showed that it was possible to describe the spectral attenuation combining several Lorentzian functions. Callens et al. established a model linking polymer conjugation length, energies of electronic and vibronic transitions, and polymer color phases. The absorption spectroscopy measurements are adequately described by this model.The absorbed dose by a radiochromic film, induces a polymerization reaction in the monomer crystals forming the emulsion layer of the film, which alters their spectral light attenuation properties. The dose–response curve, as measured by the scanner, condenses this spectral information in one or three color bands. Several authors15 hav and León‐Marroquín et al. have measured spectral absorptions under broadband conditions.A limitation of the previous studies is that the measurement conditions of spectral absorption could be different from those used to measure light attenuation in flatbed scanners. Spectral absorption is usually determined employing monochromatic light while the flatbed scanner uses broadband emission and detection, including some light scattering and wavelength shifting phenomena. Callens et al.17 andThe aim of this work is to obtain, using a PDD calibration method, dose–response curve data that allow us to study the behavior of an analytical model which describes the dose–response curve of two commercially available flatbed scanners in a dose range between 0.5 and 25 Gy. The model is based on the increase with dose of the number of two types of absorbents, each one with its own kinetics. This model enables to separate the parameters depending on the flatbed scanner and the film model from those depending on the production lot exclusively. In addition, the new model is compared with other models commonly used in the literature in terms of its performance in reducing systematic calibration uncertainties.22.1 giving rise to two detectable polymer color phases, each one with a characteristic torsion angle between two consecutive repeated linked units.The emulsion layer of the EBT2 and EBT3 radiochromic film models is composed of LiPCDA microcrystals dispersed in a water soluble material. The microcrystal constituents are diacetylenes monomers which under exposure to an external agent, such as heat, UV rays or ionizing radiation, undergo chain polymerization. The backbone of the polymer can be arranged according to different geometrical conformations,20 giv The absorbents are named after their predominant absorption color band. They are commonly referred as the red and blue phases. The total light absorption in the film results from the combined action of the two absorbents. The absorbent density increases with absorbed dose, but the dose dependence is different for each color phase which results in a complex relationship between dose and light absorption.As the two polymer color phases exhibit different optical absorption properties, ionizing radiation induces the growth of two types of absorbents in the emulsion layer.17 Theρr and ρb be the red and blue absorbent density, number of absorption centers per unit of volume, for a given dose D. Let σr and σb be the red and blue effective light extinction cross sections per absorbent for a broadband light intensity I. Then, the relative differential reduction in light intensity −dI/I of a light beam crossing a differential volume of emulsion layer with unit area and thickness dx is equal to the total cross section of centers.Let  centers.17  3) resulp, the number of centers per unit area ρp·t is a function of absorbed dose. Let us consider there is a number np of lithium pentacosa‐10,12‐diynoate (LiPCDA) monomers per unit of volume laid out in such a way that they generate polymers of the phase p. After being irradiated by an absorbed dose dD, the number of polymers formed will be proportional to np and an interaction probability κp. On average, every polyPCDA macromolecule will be a chain of nm LiPCDA monomers, being nm the mean conjugation length. The decrease in the number of monomers that are not yet polymerized will beFor the color phase  phase p.15 Aftp color phase in the emulsion layern0,p is the initial number of monomers per unit of volume arranged to become polymers of the phase p. The sum of free and conjugated monomers have to be equal to n0,p. Taking into account ρp(D), the density of formed polymers, thenp per unit volume as function of the dose will beSolving this differential equation results in the remaining number of not yet polymerized monomers for the Combining Equations  and 8)  8) resulnm, κr, and κb are characteristics of the emulsion layer material and they do not depend on the production lot. Following the determinations from Callens et al.,nm is not dependent on the polymer color phase either. The other constants t, n0,r, n0,b can change with the production lot. Since σr and σb are broadband effective cross sections, they depend on the spectrum of the light used to measure d. Furthermore, the measurements from Callens et al. and León‐Marroquín et al. have shown that the absorbance curves of the active layer are different under monochromatic and broadband conditions, and that the difference between them increases with absorbed dose. In that sense σr and σb should be expressed as dose functions. Unfortunately, the determination of the dose dependence of each σp is a complex task, even analyzing the absorbance curves determined under broadband conditions. The specific characteristics of the device used to measure broadband absorbance curves affect their value. Besides that the effect of the σp dose dependence always appears mixed in with the dose variation of the number of polymers in these curves.The constants s et al.,15nm  we proposed that the optical density as measured by a flatbed scanner can be expressed askr and kb, were characteristics of the emulsion layer and were considered not dependent on the scanner color channel nor the production lot. The parameters ϕr and ϕb were related to the relative abundance of each polymer color phase and proved to be color channel and production lot dependent in the dose range considered. Identifying terms in Equations  we pr and by León‐Marroquín et al., both covering a dose range up to 50 Gy, seem to indicate that the values of the absorbance corresponding to wavelengths in the red band behaves differently than the values in the green and blue bands. The red band is dominated by the main absorption peak of the red phase of the polymer. In that sense, Equation  were retrospectively analyzed. In this case, calibration irradiations were made using square pieces of 3 × 3 cmFor every production lot, the film pieces were scanned all at the same time, using the Microtek ScanMaker 1000 XL, 24 h after irradiation. The scanning protocol was the same that the one previously explained. In this case, the average digital signal of every calibration piece was converted to optical density and related to the absorbed dose.kr,R, kr,GB, and kb determined using the production lot 06191802 and the PDD calibration procedure.The response curve was fitted using a least square method allowing all the ϕ parameters to vary, but using the 3d values depending on the scanner and on the color channel. The figure also shows the fit of these measured values to the Equations (In Figure quations .Figure 2 normalized by the number of degrees of freedom of each fit for every response curve model, color channel, and scanner model.In Figure As the exponential terms in Equations  are cons4As shown in Figure  and originate in the design of the optical elements of the scanner.Figure 24 and developed a model, based on theoretical and experimental basis, to determine the contribution of each color phase to the optical density, and the mean conjugation length of the polymer chains in the films. This study showed that the polymer phases have different dose‐dependent reaction kinetics. Using the spectral absorbance model proposed by Callens et al., a sensitometry model for EBT film models and flatbed scanners was developed based on the assumption that the color phase optical density dose dependence is related to the variation with dose of its integral absorbance. This study assumed as well that the variation of the optical density with absorbed dose for each color phase is color channel‐independent, that is, this variation is proportional to the actual number of polymers. The dose range considered in the study was up to 5 Gy. According to the spectral absorbance measurements of the emulsion layer of EBT3 films under broadband conditions, similar to those of flatbed scanners, the absorbance curves present broader absorption peaks compared with the monochromatic absorbance. The relative intensities of the peaks are different as well and it is no longer acceptable to assume that each color phase optical density is proportional to the polymer number. The higher the dose, the more marked the difference between monochromatic and broadband absorbance curves. The difference is notable for the main absorption peak corresponding to the transition from the ground level to the first excited electronic level of the blue phase, λ= 635 nm. As this peak is in the red color channel of the scanner, the optical density dose dependence in this channel is different to those in the green and blue channels. In spite of the absorption peak broadening being due to optical effects, it can be interpreted by the scanner as an effective absorption center number variation, which allows us to formulate Equations (kr the specific characteristics of the digital scanner optical systems considered in this study.Equation  allows u15 devquations . These eThe residuals shown in Figure 2 listed in Table Figures It is interesting to note that some of these analytical models, despite having different functional forms, produce quite similar results in the end. It is the case of Tamponi and SHGD models, specially for the EPSON scanner. For the Microtek scanner, there are substantial differences only for doses higher than 15 Gy.One of the characteristics of Equations  is that 5A response curve model has been proposed which is based on the increase with dose of the number of two types of absorbents, each one with its own kinetics. The model takes into account as well the apparent different effective behavior of one of the absorbents when it is measured under broadband conditions.To study the dose–response curve, a PDD calibration method has been used. By means of this method is possible to efficiently produce a dense collection of pair calibrations points, relating absorbed dose and optical density which allows us to determine the parameters of the dose–response model and to identify systematic deviations introduced by the analytical functional form.The proposed response curve model does not introduce systematic deviations up to the degree of accuracy reached in this work. One additional strength of the model is that it enables to separate the parameters depending on the flatbed scanner and the film model from those depending on the production lot exclusively. The first ones are related to the specific characteristics of the dosimetry system. The lot dependent ones, which reflect the relative abundance of each type of absorbent, are linear parameters so they are easily determinable facilitating the calibration process. Using this model the uncertainties related to the fitting process in the calibration are, on average, halved compared with other published models.The authors have no conflict of interest to disclose.All listed authors contributed substantially to the study design and execution, and the drafting and reviewing of the manuscript. Each author approved the final submitted version of the manuscript.


Wood ants were trained indoors to follow a magnetically specified route that went from the centre of an arena to a drop of sucrose at the edge. The arena, placed in a white cylinder, was in the centre of a 3D coil system generating an inclined Earth-strength magnetic field in any horizontal direction. The specified direction was rotated between each trial. The ants’ knowledge of the route was tested in trials without food. Tests given early in the day, before any training, show that ants remember the magnetic route direction overnight. During the first 2 s of a test, ants mostly faced in the specified direction, but thereafter were often misdirected, with a tendency to face briefly in the opposite direction. Uncertainty about the correct path to take may stem in part from competing directional cues linked to the room. In addition to facing along the route, there is evidence that ants develop magnetically directed home and food vectors dependent upon path integration. A second experiment asked whether ants can use magnetic information contextually. In contrast to honeybees given a similar task, ants failed this test. Overall, we conclude that magnetic directional cues can be sufficient for route learning. Summary: Wood ants can learn and remember overnight the direction of a short foraging route that is specified magnetically. Cataglyphis noda . During the walks, ants loop around the nest, periodically turning toward it, giving themselves the opportunity to memorise views that can guide their later returns to the nest. Manipulation of the direction of the magnetic field during normal outdoor learning walks reveals that turns to face the nest are directed by path integration, with the Earth's magnetic field providing compass information. Desert ants do not seem to rely on magnetic information in later life and switch to a time-compensated sun compass that may give more precise directional information  and about a third of this group completed training.Experiments were performed on three colonies of laboratory-maintained wood ants, Experiments were performed in an artificially lit laboratory with some natural light from a window distant from the arena. The arena was lit directly by a ring of white LEDs fixed above a 25 cm diffuser  and placed centrally above the magnetic coils.info@claricent.de), generated a uniform Earth-strength inclined magnetic field  within a volume of about 50 cm3  with its bottom removed was placed in the centre of the platform in the centre of the coil system B. Ants tTraining was in two stages. In the first stage, individually marked ants taken from the nest were trained in groups of 5–6 individuals. A group of ants was put in a small release cylinder in the centre of the arena. The cylinder was 7 cm wide and 5 cm high with a 30 deg exit in the wall that pointed in the direction of the sucrose. At the start of training and periodically through training, but not in tests, the magnetic direction to the sucrose was reinforced by a black vertical bar (6 cm high by 0.5 cm wide) held by Blu Tack to the inner wall of the cylinder just above the sucrose B.In the second stage of training, marked ants were released individually in the centre of the arena from a portable release compartment B. The deAs training progressed, the bar was removed on many of the trials. If an ant failed to reach the sucrose when there was no bar, the bar was replaced to ensure that the ant was rewarded. Because the arena is so small, it is difficult to exclude the possibility that, during training, ants are attracted directly to the drop of sucrose by cues emanating from it. Consequently, training trials were only analysed to examine paths in which the ants travelled directly away from the goal.−1 from the moment the cylinder was raised above the base plate. Paths were stored on computer as MP4 files that were later re-coded as AVIs and then processed with custom-written code in MATLAB to extract both the position and facing direction of the ant. These data were checked and corrected by hand using methods described in An HD webcam , which surveyed the whole arena from a position just above the coils, recorded the ant's path at 30 frames sOnce ants were trained, nine test trials were given with no bar and no sucrose. To see whether ants have a ‘longer’ term memory of the magnetic direction, four of these tests were given at the start of the experimental day, before there had been any training (referred to as ‘early’ tests). Tests were separated by a varying number of training trials see , with noBecause it became clear that ants do not sustain their paths, we analysed the facing direction of the ant's body relative to the line from the centre of the arena to the goal (body angle relative to 0 deg). We examined body angle over time in successive intervals of 2 s from the start of a test. Data extraction and selection used MATLAB scripts developed by A.O.P.V-test with a predicted value of 0 deg body angle to see whether distributions have a mean direction. We also obtained the circular mean, R, the resultant vector length, and the circular standard deviation.The figures show the data accumulated in various ways in 10 deg bins. Statistics were performed on the values of the modal bins of single test trials. Circular statistics are calculated in MATLAB . We usedIn order to determine whether ants have food or home vectors, we extracted episodes in which ants fixated the food or the start point in the centre of the arena for at least 7 frames within ±10 deg. Fixation is defined as the facing direction being within ±10 deg of the goal position (food or start). As the heading direction is a bit variable, we allowed a single frame within the 7 frame episode to be outside the ±10 deg limit. We then examined the body angles of these fixations relative to 0 deg.A second experiment was conducted with newly trained groups of ants. Ants were released in the centre of the arena as already described. They chose between approaching an upright or an inverted triangle on the cylindrical arena wall to find sucrose at the bottom of one, depending on the orientation of the magnetic field. The sucrose was beneath the upright triangle, when the direction from the central start point to the upright triangle was West. The inverted triangle was rewarding when the magnetic direction to that triangle was North. The triangles  were cut from black card. They were fixed in position, 90 deg apart with their tip or base on the arena floor. The ants’ paths were recorded during training and test trials following the same procedure as in the route-learning experiment.We show below that ants learn the magnetic direction of a route from a start point to food and that they are also likely to compute food and home vectors. Their behaviour is somewhat complicated. The likely reason is that other directional cues interfere with the expression of the ants’ magnetic knowledge. We start by investigating the ants’ uncertainty about using magnetic information during training trials with food. We then analyse test trials in which food is missing.Fig. S1). This behaviour cannot be attributed to some quirk in the applied magnetic field (coil West) as it also occurred in three trials at the start of the day  when the coils were inactive, and the ants were guided by the Earth's natural magnetic field .During training trials in which the ant can be guided to the sucrose by the black bar and by magnetic cues, the ant usually reached the food e.g. A. We fouTo assess when during the start of the trajectory ants most frequently switched to facing toward a goal in the opposite direction, we analysed the 49 training trials with a bar that were performed by the 9 focal ants. The orientation of the ants’ body was accumulated over 1 s intervals for the first 4 s of their paths. In all plots C, the maEight of the tests were given with coil West pointing in a variety of directions relative to local West .Test 26 differed in that it was guided by local West. During these tests, ants also faced toward the real and fictive goals.We first illustrate this behaviour through plots of body angle during the initial stretch of two of the nine tests taken by ant E . Goal faV-test, V=13.08, P=0.013, n=69; R=0.19). For other 2 s intervals up to 8–10 s , indicating that the ants’ memory of the magnetic direction from the start to the food endured overnight. But there was a difference in the ants’ behaviour over later 2 s intervals: the peaks first rotated anti-clockwise followed by intervals with three clear peaks. The distribution of all the peaks had three rough clusters: toward the food, in nearly the opposite direction and perpendicular to the food-nest direction . To examine how performance varied between ants and across tests, the frequency of body angles during the 0–2 s interval is shown separately for each of the nine test trials and for each of the focal ants in Fig. S3. Tests appear to become better over the course of the experiment and some ants  performed more precisely than others.The subset of 32 tests given early before any training were initially similar to the full set. From 0 to 2 s, ants faced close to coil West . But, at a low level of significance, they were directed toward a room cue . The vector lengths were R=0.16 for magnetic cues and R=0.19 for room cues. Circular mean and s.d. were −17.1 and 74.4 deg (n=53) for magnetic cues, and −28.0 and 72.8 deg (n=53) for room cues.Lastly, we asked whether room cues play a role over the 0–2 s interval. To do so, we excluded tests in which coil West was the same as local West. Body angles of this depleted sample are plotted relative to coil West and local West in Although initially ants tended to face along the magnetically specified direction, thereafter their direction often wandered, with occasional episodes in which they faced in the direction of the food. Do these episodes indicate that the ants are guided by a food vector through path integration ? A food Two criteria must be met to infer that facing the food when distant from the line between the start and the goal is a reliable sign of the use of a food vector or that facing the nest indicates the use of a home vector. The first is that facing the food or nest occurs over a considerable period. We analysed instances of food and nest facing (±10 deg) that persisted for at least 7 consecutive frames (230 ms). The second is that the body angle when facing these points differs substantially from 0 deg, indicating that the ant is relatively distant from the line connecting the start to the food and so is not merely facing in a remembered direction.Cataglyphis . These fixations can occur throughout the trajectory as might be expected for an expression of path integration and were often much longer than 7 frames . The antaglyphis .Fig. 5.Fig. S3]. Before the test, ants had three consecutive training trials in which the magnetic field was in the same direction relative to the room and to any cues associated with it. In the test that followed, the magnetic direction was rotated 90 deg clockwise from the training direction. At the start of the test (0–2 s), ants as a group faced toward coil West . Later on (14–20 s), their body angle had rotated by 90 deg in the direction of the three prior training trials  are also included in g trials A. IntereThe group's behaviour was reflected in the ants’ idiosyncratic paths B. PointsIn this experiment, different groups of ants were again released at the centre of the arena. They were confronted with an upright and an inverted triangle in fixed positions, 90 deg apart, on the cylindrical arena wall with the base or apex close to the floor A. When tSucrose was below the inverted triangle when the magnetic direction to that triangle was West A. Thus, Despite many training trials, ants did not learn to approach the correct triangle. During the course of training and testing, individual ants were recorded in 10 trials between trials 27 and 60. For each recorded trial, we noted which triangle the ants reached first, ignoring the two times in which an ant failed to reach either triangle. We then plotted the proportion of ants approaching the upright triangle B, left. The performance of individual ants was assessed using 14 individuals that were recorded on five or more of the 10 trials. For each ant, we plotted the proportion of trials in which the ant first reached the upright triangle against the proportion of trials in which the ant first reached the correct triangle: 9 out of 14 ants chose incorrectly on half or more of their recorded trials; 11 out of 14 ants preferred to approach the upright triangle.The ants’ behaviour is illustrated through individual paths taken from the last training trial (60) in which the upright triangle was rewarded C. WhicheP=0.014). Again, there was no sign that ants learnt to choose the correct triangle.In another experiment of the same kind with a different set of ants, the triangles were placed 180 deg apart. In this case there was a weaker but just significant preference for the upright triangle . The confusing effects of daytime training may have dissipated overnight, allowing the ants’ memories to emerge more clearly at the start of the day.A comparison between the ants’ behaviour during early morning tests before training and that in later tests supports the suggestion that following magnetic cues can be disturbed by competing cues . In the Ants in these experiments often moved toward a fictive goal in the opposite direction from the real goal. The ants’ uncertainty about the correct path to follow could be why they often faced or moved in precisely the opposite direction to the magnetically signalled route. Both directions frequently occurred in the same path . While sFig. S1). In these training trials, a vertical bar was fixed above the food. Ants tended to move directly away from the bar toward the start of their path. The bar, as it was not always present during training, may have made them hesitant that it was correct to move in the food direction.In sum, while we are confident that ants were unsure of the correct direction to take, we have little direct evidence of a link between their uncertainty and their tendency to face in both directions. One indication is facing toward the fictive goal when room cues are made prominent . AnotherDrosophila. The flies, when confined to a small arena with two inaccessible visual targets, 180 deg apart, oscillated between them for several hours, like Buridan's fictional ass trying vainly to choose between two identical piles of hay (Ants were also unsure what to do in the second set of experiments. We suggest that turning away from the current direction may occur because ants are uncertain about the correct destination. The undecided ant may not know the location of the triangle that it is not currently approaching. By turning away, it unlocks its attention and frees itself to switch its path directly towards the other triangle C. This ks of hay . Were thIn sum, these experiments demonstrate that ants can learn to face in a magnetically defined direction and that their memory of the direction persists overnight. There is suggestive evidence that path integration can work with magnetic cues. The ants also showed uncertainty in their behaviour. Their indecision could be a consequence of the small area in which the experiments were conducted. It could also arise from competition between magnetic and other directional cues.


Real haptics is a technology that reproduces the sense of force and touch by transmitting contact information with real objects by converting human movements and the feel of the objects into data. In recent years, real haptics technology has been installed in several surgical devices. A custom-made surgical drill was used to drill into the posterior lamina to verify the time required for penetration detection and the distance the drill advanced after penetration. A surgeon operated with the drill and the same aspects were measured and verified. All experiments were performed on female miniature pigs at 9 months of age with a mean body weight of 23.6 kg . There were statistically significant differences in the average reaction time and the distance travelled after penetration between a handheld drill and the drill with the penetration detection function (p < 0.001). The reaction time to detect penetration and the distance after penetration were both significantly improved when compared with those of the handheld surgical drill without the penetration detection function, with mean differences of 0.049 ± 0.019 s  and 2.511 ± 0.537 mm . In this study, we successfully conducted a performance evaluation test of a custom-made haptic interface surgical drill. A prototype high-speed drill with a haptic interface accurately detected the penetration of the porcine posterior lamina. As a treatment for various musculoskeletal diseases, spine surgery involves drilling into the bone near the spinal nerves and vascular organs. This is an extremely demanding procedure because it involves the manipulation of hard tissue in the vicinity of highly vulnerable soft tissue. As a result, intraoperative complications of spine surgery occur at a certain rate. Bone drilling is usually performed with a high-speed drill, and severe complications, such as dural lacerations and spinal cord injury, may occur during the operation. Neurological complications are devastating and can severely diminish a patient's quality of life due to loss of motor function, limiting the patient’s functional ability and causing excruciating neurological pain; every effort should be made to prevent the occurrence of such complications.6. Among these surgical complications, dural lacerations have been reported to affect 2–13% of cases. Most dural lacerations are caused by bone drilling during decompression of the spinal cord7. It has also been reported that the incidence of intraoperative complications, such as nerve injury, is inversely related to the number of years of surgical experience of the surgeon. Imajo et al. found that spine surgeons with less than 5 years of surgical experience had the highest frequency of complications1. It is widely accepted that in spine surgery, when drilling bone using a high-speed drill, the surgeon relies on the feel of the drill to determine whether the bone has been penetrated. Therefore, it is thought that the number of years of surgical experience affects the surgeon’s ability to detect bone penetration, and this directly affects the incidence of complications.Previous reports have shown that the complication rate of spine surgery is relatively high, ranging from 1 to 17%1. In other words, although the number of years of experience in spine surgery influences the complication rate to some extent due to the nature of the procedure, dural lacerations and neurological injuries still occur regardless of the number of years of experience.In contrast, Imajo et al. described the results of a survey of spine surgery complications and concluded that the incidence of intraoperative dural injuries was similar between spine surgeons with less than 10 years of experience and those with more than 10 years of experience9. Real haptics is a technology that reproduces the sense of force and touch by transmitting contact information with real objects and the surrounding environment in both directions by converting human movements and the feel of the objects into data. By converting movements and textures into data, it is possible to program force tactile sensation data into machines, adjust the degree of force, and transmit force tactile sensation over a long distance9. Specifically, by monitoring the drilling force in real time when using the drill, it will be possible to detect penetration, automatically stop the drill, and proceed with the surgery while obtaining detailed information, such as the bone hardness and the identity of the drilling layer. In addition, by extracting intraoperative drilling sensation and movement data, we believe that such data can be applied to develop a training simulator that reproduces a realistic sensation of force and touch and adds an assistance function based on the movements of a skilled surgeon.In recent years, real haptics technology, which has been utilized in various fields, has been installed in high-speed drills to establish a surgical method with greater safetyWe hypothesized that a prototype high-speed drill with a haptic interface could detect the penetration of the porcine posterior lamina more accurately and more reproducibly than experienced spine surgeons could recognize Fig. A,B. In t10. The custom-made drill has a master–slave integrated structure and is driven by a linear motor with two optical encoders and a rotary motor with a rotary encoder10. A schematic drawing of the prototype is shown in Fig. 10. Moreover, when the custom-made surgical drill detects penetration, position control is applied to the linear motor on the slave side to pull in the drill bit while the rotary motor is stopped automatically . The block diagram of the DOB is shown in Fig. The rotary motor is controlled by velocity control. Figure mentclass2pt{minimm, s, Cp(s), and Cf(s) are the master part, slave part, position controller, and force controller, respectively. Bilateral control is applied to the two linear motors to realize transmission of the haptic sensation. The block diagram of the bilateral control is shown in Fig. These equations represent the law of action and reaction. Thus, the haptic sensation between the two linear motors is realized by bilateral control.11. In the drilling state, the reaction force from the cutting object is applied to the slave part. In contrast, when the drill penetrates the object, the reaction force is decreased. Additionally, the velocity of the slave part is increased due to the decrease in the reaction force. Therefore, large displacements of position and reaction force are generated by penetration. Thus, the drill detects the penetration when Eqs.   11. In t by Eqs. .6\documeWhen the custom-made surgical drill detects penetration, position control is applied to the linear motor on the slave side to pull in the drill bit. Additionally, the rotary motor is stopped automatically.After review and approval by the Judging Committee of Experimental Animal Ethics of Keio University School of Medicine, all experiments were performed on female miniature pigs at approximately 9 months old with a mean body weight of 23.6 kg . All animals were purchased from the laboratory  and housed and treated in accordance with rules approved by the Ethics Committee (no. 18047). All experiments were performed in accordance with any relevant guidelines and regulations. This study follows the recommendations in the ARRIVE guidelines.The miniature pigs undergoing surgery were premedicated with midazolam  administered subdermally. Subsequently, inhalant anaesthetics and isoflurane  were used with maintenance at a 2% flow rate after orotracheal intubation. All pigs were killed immediately after surgery by an overdose of isoflurane. After establishing anaesthesia, a standard midline dissection was performed by 1 of 3 board-certified spine surgeons using electrocautery, and the laminae of the thoracic vertebrae and lumbar vertebrae were carefully exposed.The experimental setup is shown in Fig. First, the custom-made surgical drill was used to drill the posterior lamina without the penetration detection function. After penetration was detected, the threshold was set as a control, where − 7 N/s was The time required for penetration detection was defined as the time from the start of the decrease in the reaction force of the slave until The time required for penetration detection.The distance advanced by the drill after penetration.After that, the penetration detection function was turned on, and the measurement results of the drilling of the posterior laminae were used to verify:The distance the drill advanced after penetration was defined as the distance that the slave advanced before the device detected the penetration and started retracting the drill Fig. A,B.FigurThe time required for penetration detection.The distance advanced by the drill after penetration.Next, the surgeon operated the drill without the penetration detection function and verified:The time until the surgeon recognized the penetration detection was set as the reaction time, which was defined as "the time when the drill position started to move forwards rapidly" to "the time when the drill stopped moving forwards (the time when the speed was displaced negatively)", that is, the time taken by the surgeon to detect the penetration and start retracting the drill body. The participants were instructed to move the master in the opposite direction of penetration (retracting the drill).The distance travelled after penetration was defined as the distance travelled from "the position where the drill started to move forwards rapidly" to "the peak of the drill (slave) position", i.e., the distance travelled until the surgeon detected the penetration and started to retract the drill body Fig. C,D.Finally, the haptic drill with the penetration detection function was used by the surgeon, and the same items were measured and verified.Next, to assess if the haptic drill is able to detect the penetration of lamina in an actual surgery setting, the surgeon operated the haptic drill with horizontal movement similar to the actual surgical procedure . A p value less than 0.05 with a CI of 95% was considered statistically significant. A p value less than 0.05 was considered statistically significant. Data were analysed with the Statistical Package for the Social Sciences .The average time to detect the penetration of lamina by the drill with the penetration detection function was 0.015 ± 0.005 s (range 0.01–0.02 s), and the travel distance after penetration was 0.11 ± 0.063 mm , and the distance travelled after penetration was 3.52 ± 1.770 mm . These results clearly showed that even in an experienced surgeon’s hand, it is difficult to recognize the penetration of the lamina early and to stop the drill immediately after detecting the penetration of the lamina during surgery.There were statistically significant differences for both the average reaction time and the distance travelled after penetration between the handheld drill and the drill with the penetration detection function (p < 0.001). The penetration detection time of the device was approximately one-tenth as fast as that of the handheld drill, and the mean difference was 0.154 ± 0.024 s  and 2.862 ± 0.741 mm .Similarly, three spine surgeons drilled the posterior laminae 3 times each until they detected the penetration by the custom-made haptic drill with the penetration detection function. The average reaction time was 0.020 ± 0.012 s (range 0.01–0.04 s), and the distance travelled after penetration was 0.541 ± 0.351 mm . The average reaction time was 0.169 ± 0.041 s, and the distance travelled after penetration was 2.982 ± 1.242 mm when using a handheld drill without the penetration detection function. Additionally, the lack of correlation between the detection time and distance indicated that bone drilling using a handheld drill was less reproducible. However, when using the handheld surgical drill with the penetration detection function, the reaction time and the distance after penetration were both improved. The average reaction time to detect penetration and the distance after penetration were both significantly shorter than those of the handheld surgical drill without the penetration detection function, with mean differences of 0.049 ± 0.019 s and 2.511 ± 0.537 mm, respectively. This result clearly showed that even when the drill was used as a handheld device, the reaction time and the distance after penetration can be significantly shortened if it is equipped with real haptics.Interestingly, analysis of the reaction time and the travel distance after penetration of the handheld surgical drill with the penetration detection function showed no statistically significant differences among the surgeons for their average reaction time and the distance travelled after penetration. According to this result, surgeons can recognize penetration extremely quickly with haptic drills regardless of their years of experience. As we have described in the introduction section, gaining surgical experience has thus far been considered very important in reducing the possibility of neurological complications caused by surgical drills. However, it is difficult to gain experience in actual spine surgery without years of experience performing spinal surgery. Therefore, the establishment of a surgical simulator using real haptics technology will be important for young spine surgeons to gain experience in order to avoid neurological complications caused by surgical drilling.14. Meyer et al. compared the drilling performance of a bone simulator with a haptic system between the trainee resident group and the expert staff surgeon group. They concluded that there were no significant differences in surgical time or the accuracy of the surgical technique13. Thus, the use of a spine surgery simulator with a haptic system may be useful for spine surgery education, which has a steep learning curve. Previous reports showed the utility of real haptics technology only for surgical simulators. This study is the first report indicating the accuracy of the penetration detection function and automatic stop function of haptic drills when performing real surgery.Several previous reports described the availability of haptic technology for surgical simulatorsIn this study, we also proved the accuracy of penetration detection when the surgeon operated the drill in the horizontal direction, which is similar to the actual surgical procedure. It was possible to detect penetration according to the shape of the lamina by recognizing the position by dividing it into 2 mm increments. On the other hand, the position information of the lamina can be moved by the deviation of the spine during the procedure, so overdetection may occur; however, overdetection never causes damage to the spinal cord. Currently, we are planning to add alert systems to the penetration detection system. Further refinement of the penetration detection system will include minimization of the overdetection of penetration.15. These systems are controlled with a haptic device that feeds the exerted drilling force into the hand of the surgeon in real time. Thus, surgeons can detect the differences in tissues and then perform surgeries more safely and accurately. Real haptic systems are also considered to be an essential technology for the development of robotic telesurgery16.Furthermore, the concept of using real haptic technology and virtual reality (VR) for robotic surgery has also been developedWe acknowledge the limitation that we were unable to use human cadavers due to resource limitations. In this study, the drill was used only in a vertical direction against the vertebral lamina of young miniature pigs. However, in actual surgery, the surgeon moves the drill in various directions against the vertebral lamina in various age populations while drilling. Further study for the improvement of this evolutional instrument may require the evaluation of the safety and utility of the haptic interface surgical drill in various bone conditions, including osteoporotic bone.In the present study, a prototype high-speed drill with a haptic interface successfully detected the penetration of the porcine posterior lamina more accurately and more reproducibly than experienced spine surgeons. The integration of haptic technology into spinal drills may be expected to have future applications in accurate, safe robotic surgery and telesurgery.Supplementary Video 1.


In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations  is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains. Many problems in applied sciences can be modelled by a parametric polynomial system, and therefore to solve such problems we must be able to explore the properties of these systems. In particular, we often seek to identify areas of the parameter space where a property holds. The contribution of this paper is a new methodology for exploring these.We are motivated by the problem of understanding the multistationarity behavior of a Chemical Reaction Network (CRN). In a CRN, variables represent the concentrations of the species. These change as time passes and are studied as part of the field of dynamical systems. This is of polynomial type when the kinetics is assumed to follow the mass action rules. The equilibria of such a dynamical system are therefore the solutions to a system of polynomial equations. However, the coefficients of the terms on the polynomials may involve some parameters. These parameters are usually the rates under which a reaction occurs and the total amounts .Both the variables and parameters can only attain non-negative real values. A network is called multistationary if there exists a choice of parameters for which the network has more than one equilibrium. There are already many algorithms developed for answering the binary question of whether a system can exhibit multistationarity –9. The iIn the case where multistationarity can exist it then becomes important to determine the parameters where the network has this behavior. There has been less progress in this direction in the literature to date: the present paper offers a promising new development for this problem.Reviewing the state of the art in the literature, we see one vein of work focused on specific reaction networks, with success following heuristic or manual calculations to find a suitable parameter which may not work in generality , 11. TheRecently in  a new apIn this work, we propose using polynomial superlevel sets to approximate the union of the hyperrectangles from  as a setThe organization of this paper is as follows. The mathematical framework of reaction networks and the definition of the multistationarity region is given first followed by a section containing the notations regarding parametric functions and definitions of the sampling and the rectangular representations of the multistationarity region from  is used to show the hyperractangle S of a hyperrectangle S with respect to B, i.e.X is distributed by a normal distribution with mean g(X) when X is distributed by a probability distribution q is denoted by The cardinality of a set All computations for the examples of this paper were done on a computer with the following information. Processor: Intel(R) Core(TM) i7-10850H CPU @2.70GHz 2.71 GHz. Installed memory (RAM): 64.0 GB (63.6 GB usable). System type: 64-bit Operating System, x64-based processor.The software and programming languages used for the computations reported in this paper had the following version numbers: Maple 2021, Matlab R2021a, YALMIP, SeDuMi 1.3, Julia 1.6.2, MCKR 1.0.In this section, we introduce the concepts of reaction network theory that are needed throughout the rest of the paper, with the help of a simple gene regulatory network example.One can think of a gene as a unit encoding information for the synthesis of a product such as a protein. First, a group of DNA binding proteins called transcription factors bind a region of the gene called promoter. Now an enzyme called RNA polymerase starts reading the gene and produces an RNA until it arrives in the terminator region of the gene. The process until here is called the transcription step. After transcription is completed, the resulting RNA leaves the nucleus (in eukaryotes) and reaches ribosomes. In ribosomes, the second step, called translation, gets started. Ribosomes assemble a protein from amino acids using the manual guide written in the RNA. A gene encoding of a protein recipe is said to be expressed when it gets transcribed to an RNA, and the RNA translated to the protein. A gene is not always expressed in a constant rate. There might be proteins that bind the transcription factors or the promoter region and, as a result, inhibit the RNA polymerase starting the transcription process. On the other hand, there might be other proteins in which their binding to the transcription factors or the promoter region enhances the transcription.A, B, and C as their final products. Denote their concentrations at time t by [A](t), [B](t) and [C](t) respectively. The concentration of these proteins will not remain constant all the time, and we have an Ordinary Differential Equation (ODE) system describing the variation of the concentrations as time passes, see Fig. A activates the expression of the second gene with Michaelis-Menten kinetics with the maximum rate A and B together with the product of two Michaelis-Menten kinetics, with maximum rate C,  and protein C has an inhibitory effect on the expression of the first gene, captured by the denominator Consider a simple example from  then a grid on B is achieved by dividing B along each axis to equal parts. Then for each sub-hyperrectangle of B in this rectangular division we assign the average of the number of solutions of k coming from the sub-hyperrectangle. This approach hereafter is referred as the rectangular representation approach. See Fig. A different discretization can be done using a rectangular division of Consider the gene regulatory network in  to 100 equal sub-rectangles and then solving the system for 10 points superlevel set of f is the set of the formU(f). Naturally, a polynomial superlevel set is a superlevel set of a polynomial.Consider d. A sum of squares (SOS) polynomial of degree 2d is a polynomial d by Polynomial sublevel sets are defined similarly as in Definition K a closed subset of B. For  where U(p)  of degree d can be written as p(x) which are the variables of the optimization problem.Let p(x) has to be nonnegative on B. This can be enforced by lettingK or in other words K. This holds if and only if Now let us look at the constraints. First of all d YALMIP , 32 whicd YALMIP .K be the union of rectangles B, 9 of them satisfy this condition. These sub-rectangles are colored orange in Fig. p of the PSS representation of degree 2. Figure U(p).We continue with the example from Fig. $B= and then transforming the PSS polynomial back to the original B allows a better PSS approximations via YALMIP. For degrees 2 and 6 the numerical problem message is avoided but for degree 10 it remains. The results are shown in Fig. Let us illustrate this with our ongoing example. Consider the sampling representation of the multistationarity region of the network of Example 3.4 given at Fig. B takes the same amount of time as solving the system at 1000 random points with only 2 of their coordinates varying took time in the previous case. It took about 1.5 s to compute the 45 coefficients of the polynomial of the PSS representation of degree 2. The polynomial found is the following.To demonstrate that the number of free parameters need not be 2 to be able to compute the PSS representation, we repeated the above process with all 8 parameters of the system being free in the following hyperrectangle:p is a polynomial of degree d. K is impossible. In order to save or show the sampling representation one needs to use a matrix of the size When K using the sampling representation is not a straightforward task. However, using the PSS representation one needs to verify only one condition of the evaluation form, To test if a point K, using the sampling representation, if K and then take the minimum. Using the PSS representation, whether To compute the distance of a point In a typical example from CRN theory, r is usually much higher than 2, and therefore item 1 is really important. When rm, one can use less memory by saving the PSS representation instead of keeping all the points of the sampling representation in the memory. Further, as items 2 and 3 show, this will not cause a loss of information about the multistationarity region.Let us address why one should find a PSS representation of the multistationarity region using the sampling representation if one already has a sampling representation. Let We showed earlier  that the PSS representation can be generated for examples with a higher number of parameters than two. There, we let all 8 parameters of the network in Fig. Consider a gene expression system with 4 species me as in :5\documeIn  it is sh,,5,14]$$\,,5,14]$$\As shown in the previous section, it is possible to get the PSS representation both from the sampling and the rectangular representations. If one needs to solve the system at random points and take an average in order to get the rectangular representation, then using the rectangular representation does not have much advantage over using the sampling representation for the purpose of finding the PSS representation.m equal parts, then the number of sub-hyperrectangles in the rectangular representation becomes B to equal sub-hyperrectangles. When using a bisection algorithm, the rectangular decomposition usually contains fewer number of sub-hyperrectangles .But in some cases it is possible to compute the average number of the solutions without solving the system. One such case is introduced in . InsteadB. The goal is to express B. One of the shapes of multistationary networks, observed often for realistic models, are bistable networks with a folding type of bifurcation, such as in dual phosphorylation cycle [Let us simplify the question. There is a hyperrectangle on cycle .(i)This can happen if and only if for almost every (ii)This can happen if and only if for almost every (iii)This can happen if and only if The proof is straightforward by noting thatB is a subset of B is a subset of B along only one axis into two equal sub-hyperrectangles. We continue in this fashion until each sub-hyperrectangle is inside two-value bisection search hereafter. This algorithm is formally written in Algorithm 1.In our settings these networks have one steady state for some choices of parameters and threeA similar algorithm was presented in  is not nB are of different scales, then it is better to replace the termination condition of Algorithm 1 with the following:If the length of the edges of q is a distribution on S and then find a rectangular representation using two-valued bisection search and afterwards a PSS representation.We are not going to explain how to use the Kac-Rice integral in the CRN framework as it is not the topic of this paper. All algorithms in this paper are independent from the choice of the algorithm to compute the expected number of solutions of a parametric system on a given parameter region. A simple algorithm to achieve this is to solve the system for several random points from the given hyperrectangle according to a random distribution of interest and take the average. Thus we may assume the existence of a method capable of computing To illustrate this method we use Example 2.1 of , shown halues in ), the gogiven in  it takesgiven in  as the fS in the process of this algorithm. Then the total number of expectations that are needed to be computed until the termination of this algorithm is equal to B along each axis to Note that having fewer hyperrectangles in the rectangular representation obtained by the two-valued bisection search does not guarantee a faster speed than the simple approach for finding a rectangular representation discussed in the previous section. Consider the setting in Algorithm 1. Assume length of all edges of S is an almost subset of S to Now consider a more general case where S which has the same zero measure sets as Lebesgue measure’s we have Note thatq on S we have q one can attain any point in q, we will not get Returning to our goal assume B is almost subset of q on B with the same zero measure sets as Lebesgue measure’s.Let two-step bisection search.Note that in  it is meConsider the following univariate polynomial of degree 5 with two parameters from [lication  which co


Competing claims exist about how the geographic distance between parents and their adult children has changed historically. A classic modernisation hypothesis is that people currently live further away from their parents than in the past. Others have argued for stability and the remaining importance of local family ties, in spite of a long‐term decline in co‐residence of adult children and parents. The current paper uses a novel design that relies on reports by grandchildren to study long‐term changes in intergenerational proximity in the Netherlands. The analyses show that there has been a clear and continuous decline in intergenerational proximity between the 1940s and the 1990s. Mediation analyses show that educational expansion and urbanisation are the main reasons why proximity declined. No evidence is found for the role of secularisation and increasing international migration. Proximity to parents declined somewhat more strongly for women than for men. In family studies, the geographic distance between parents and children has been considered a dimension of family solidarity in an ageing society . The implications of this trend are not straightforward. On the one hand, one would expect that migration is positively associated with the geographic distance between parents and children because many immigrants have parents who are living in the country of origin .2N = 7,051).I used data from the first wave of the Netherlands Kinship Panel Study (NKPS)  and father–grandparent dyads when the parents divorced before the respondent was 7 years of age (n = 227).n = 27). The remaining number of dyads was 13,720. Dyads involving deceased grandparents were included in the descriptive part . In the remainder of this paper, the respondents' father and mother will be called (adult) children, and the respondents' grandparents will be called parents. The ‘grandchild design’ of the study is illustrated in Figure Respondents were asked where their two sets of grandparents lived when they were growing up. The parent–grandparent dyads constituted the units of analysis in this paper. For each respondent, there was information on two such dyads: between the mother and her parents and between the father and his parents. If a set of grandparents was no longer together, the grandparent who lived closest had to be selected.The main time variable used in the design was the period in which the adult children were raising the respondent. During the interview, childhood was defined as the time from birth to age 15. To define the historical observation period, I used the midpoint of the child‐rearing period (7 years). Respondents (grandchildren) were born between 1936 and 1985 with a median birth year of 1960. Using this variation, the period that I analysed began in 1943 and ended in 1992  of all ages at the time of the survey, but especially at the higher ages, a large part of the birth cohorts will already have died. For example, a 90‐year‐old respondent could provide information on proximity about 80 years ago—an attractive piece of information from an historical perspective—but few 90‐year old people were still alive in 2004, the year when the survey was held. Because life expectancy was highly stratified by education  yes, during the entire (respondent's) childhood; (2) yes, during part of (respondent's) childhood; (3) no; or (4) (mother's/fathers's) parents no longer alive. The option ‘during part of childhood’ was uncommon (4.7%), pointing to the high degree of residential stability typically observed in the child‐rearing period. The variable was coded into a dichotomy, contrasting 1/2 to 3.Because a retrospective design was used, the outcome variable was somewhat crude. No precise distance was measured, but the focus was on living in the same town or city. Although this variable had face validity and a good empirical distribution, it also had drawbacks. Living in the same town or city will mean something different in a city than in small town or rural area. Most cities in the Netherlands are small, however. The average population size of the 20 largest cities in the Netherlands was 254,000 in 2018, and the largest city (Amsterdam) had 854,000 inhabitants.2.32), strongly urbanised (1500–2500 addresses/km2), moderately urbanised (1000–1500 addresses/km2), hardly urbanised (500–1000 addresses/km2) and not urbanised or rural (<500 addresses/km2). Immigrant status was based on whether or not the child was born abroad. With the data, it was not possible to define the second generation. A possible second‐generation effect was partly covered by the non‐Western religious background variable.To statistically explain the trend, several independent variables were used. All measures applied to adult children and to the time they were raising their children. No information was available for the characteristics of the parents. Education was scaled in a linear fashion using the ISLED coding .Two control variables were used: the age of the adult child when his/her child was 7 years old. With this variable, I was able to adjust for possible changes in the timing of parenthood. I also controlled for the gender of the adult child. Previous studies examined if couples lived closer to the husband's or the wife's parents, but there was little consistent evidence in this direction . The current sample was limited to respondents aged 18–65 to avoid possible bias due to (selective) mortality. I checked whether a more stringent age cut‐off would affect the trend parameter. Limiting the sample to ages 18–55 yielded the same trend (b = −.043). Broadening the sample, however, to ages 18–75 yielded a somewhat smaller trend parameter (b = −.033). This would be in line with selective mortality by education. In the oldest cohorts, the lower educated may have been under‐represented, leading to an underestimate of intergenerational proximity in these cohorts and hence a somewhat flatter trend line.Model 1 shows the effect of time and control variables. This trend effect was −.043 showing a 4 percentage‐point decline in proximity per decade. A test for non‐linearity showed that the quadratic effect was not significant . The interaction of gender and partner's education was not significant.To examine the effects of education in more detail, I plotted the effects in Figure In Model 3, I added effects of urbanisation. The role of urbanisation was clear in Table Migration played a role as well, as Model 4 shows. Adult children who were foreign born were 5.2 percentage points less likely to live close their parents. The effect of time was hardly reduced when adding this variable. A possible explanation was that the share of immigrants was still rather small, which mitigated the impact on the overall trend.Religion had small associations with proximity and in the expected direction (Model 5). Adult children who were Catholic were more likely to live close to their parents than secular children. An even larger effect was observed for persons who had a non‐Western religious denomination (5.9 percentage points), but this was not significant, probably due to the low number of Muslims and Hindus in the data. Interesting was that Protestants were not more likely to live close to their parents compared with secular persons; the religion effect was limited to Catholics and persons with a non‐Western religion. Because of these small effects and the opposing compositional changes Table , there wIn Model 6, all independent variables were included simultaneously. The effects of independent variables that were significant in the previous models remained significant in the full model, and some effects increased in size. Moreover, the effect of time was no longer significant. All the independent variables together explained 81% of the trend, a sizeable and significant portion. When looking at the separate models, it was clear, however, that education and to a lesser extent urbanisation were the prime components of this explanation.In the last step of the analyses, I assessed if there were significant interactions of key independent variables with time Table . Period There were no significant interactions of time with education, migration and religion. There was a significant interaction between time and gender, however. The unadjusted graph in Figure The interaction with urbanisation was also significant and in the negative direction, suggesting that urban–rural differences in proximity grew stronger over time. Figure To obtain more insight in the role of urbanisation, I compared the places where respondents lived at the time of the survey and the places where they lived when they were 15 years of age. For three levels of urbanisation , I calculated the percentage of respondents who grew up in a similar type of place. Figure 4Using retrospective data and a novel design, this paper demonstrated a long‐term decline in proximity between parents and their adult children, in line with classic ideas in modernisation theory. The decline was clear but not spectacular in magnitude: About 50%–55% of the adult children lived in the same town or city as their parents after World War II; this declined to about 35% in the early 1990s. Proximity to parents declined for both genders but more strongly for women.A large part of the trend was due to educational expansion, with a more modest role for urbanisation. There was no clear evidence that the role of education itself changed over time. The association between education and proximity to parents was strong and remained so over time. Because there was also a substantial expansion of education, this turned out to be the most important cause of the trend.Secularisation did not contribute to the trend, in part because religious differences were small and inconsistent. The impact of secularisation was counteracted by a growth of the non‐Western religious population. Muslims, Hindus and persons with other non‐Western religions tend to live closer to their parents than secular persons. Migration also played a role, with first‐generation immigrants living further away from their parents, but the share of immigrants in the most recent period was too small to explain the trend.My findings are consistent with an earlier Swedish study that analysed a shorter and more recent period (Chudnovskaya & Kolk, My study also added a comparison of competing explanatory variables, which turned out to have effects on proximity, but effects not always large enough to explain the trend. My conclusion is in line with a more general idea in demography that structural change underlies family change (Blossfeld et al., The role of education was clearly linear, suggesting that the location of universities was not the only reason for the association between education and intergenerational proximity. Labour market considerations will have played a role as well. Previous studies have demonstrated that education has a linear and positive causal effect on migration during the life course (Malamud & Wozniak, Gender differences were observed as well. Husbands' education had a stronger effect on where couples lived than wives' education, a finding that is in line with an economic interpretation of the educational effect because in these cohorts, few mothers were employed and many mothers were employed part‐time (De Graaf & Vermeulen, The role of urbanisation requires more discussion. My analyses confirmed earlier studies by showing that people in urban areas live further away from their parents compared with people in rural areas (Michielin et al., Although the grandchild design made it possible to look at long‐term changes, something that had not been possible with repeated cross‐sections in the past, the design also had limitations. First, the data only provided characteristics of adult children and not of parents. It is known where parents lived and if they were alive, but there was no other information on parents. It would have been ideal to analyse intergenerational proximity with characteristics from both sides of the dyad. Still, it is plausible that most of the ‘action’ will come from the adult child generation, as this is the more dynamic, more mobile member of the dyad. Previous studies confirmed that for proximity between adult children and parents, child characteristics were more influential than parent characteristics (Michielin & Mulder, Second, the data can be affected by mortality. I used a cut‐off of age 65 to avoid this as much as possible, but even for this age range, more of the older respondents will already have died, and this will be more common for the lower educated than for the higher educated. This remains a disadvantage, but if this was a bias, it will probably work against my conclusions because the increase in educational attainment will have been underestimated rather than overestimated. Given these caveats, it remains important to also study changes in proximity using register data. Such data are available and important, but they are more limited in terms of their explanatory variables. In European countries where register data are available, information on education is often incomplete for older cohorts.The author declares that he has no conflict of interest.


Previous work has shown asymptotically tight results in two dimensions with N is the d dimensions must have weak constructions in two dimensions . This not only opens the possibility for having constructions with bichromatic discrepancy: a natural extension of the concept of discrepancy of a set of points. In this paper, we define this concept and extend known results to the chromatic setting.We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in  For example, it is well known that the intersection of two segments is always a segment (that could possibly degenerate to a point or even become empty). The definition of other mathematical concepts heavily depends on the definition of segments .Euclidean line segments are one of the most fundamental objects of geometry. Although often loosely referred to as round the Euclidean segment into the digital space. The digital segments will look very similar to the Euclidean counterparts . However, we cannot guarantee the useful properties and concepts that follow from the axiomatic definition of Euclidean segment  introduces imprecision. The most common approach used in practice is to somehow see Fig. .In the aspect of the consistency of digital segments, we look for a deterministic method to construct digital segments in a way that (i) the analogous of Euclidean axioms are satisfied and (ii) the digital segments resemble the Euclidean ones as much as possible.Our aim is to construct a digital path partial set of consistent digital segments on S  if for every pair p and q under the 2d-neighbor topology.Grid path property: mentclass2pt{minimSymmetry property: if Subsegment property: for any Prolongation property: Monotonicity property: for all For any These axioms give nice properties of digital segments analogous to Euclidean line segments. For example, (S1) and (S3) imply that the intersection of two digital segments is another segment (that could degenerate to a single point or an empty set). (S5) implies that the intersection of a segment with an axis-aligned halfspace is a segment [and connected by (S1)], and so on.consistent digital segments (CDS for short). Although our final goal is to have such a construction that works for the case in which o is the origin or any fixed point in consistent digital ray system (CDR for short), as it contains all segments (or rays) from o to o to any other point of A partial CDS for entclass1pt{minimaentclass1pt{minimaHausdorff distance. The Hausdorff distance H of two objects A and B is defined by L-infinity norm.Another property that we want from partial CDS is that they visually resemble the Euclidean segments. The resemblance between the digital segment entclass1pt{minimaS is defined aserror of the partial CDS construction. We are interested to see how the error grows as we enlarge our focus of interest. Thus, we limit the domain to the case in which both points are in the N centered at the origin . When the weak CDR has zero inner leaves (and thus is a proper CDR) our bound matches the lower bound of Håstad. As the number of inner leaves increases, the lower bound decreases. In Sect. For any This shows an important relationship between (iv) Intuitively speaking Theorem Any CDR defined on d dimensions  Our lower bound argument shows that for a CDR in For any O(c) error and Although the number of inner leaves is large , this construction has other interesting properties that would make it suit for practical purposes. Moreover, we extend this construction to have a tradeoff construction with O(c) error and For any Discussion of this construction and other variations is given in Sect. (vi) In order to reduce the number of inner leaves we instead look at how the mapping of any weak CDR with constant error to points would look. Discrepancy results give us a condition on any such CDR with B and R be two sets of points whose discrepancy satisfies Let Note that the constant 1 is crucial for our proof: although we would like the statement to hold when Further discussion on the implication of these results is given in Sect. T of x, y) (except the root) is either T, where u is the parent of v, we define T(e) as the subtree of T that is rooted at the child node v of e. We slightly abuse the notation and use T(v) to denote the subtree that is emanating from v towards the leaves . The proof now considers a few cases depending on whether k is small or large  and if Q contains too few or too many points.Let k  be the subtree rooted at v.We do a slight abuse of notation and use the same terms as in two dimensions. For simplicity of the notation, we assume that d dimensions. In other words, the subtrees rooted at the vertices in From Theorem see Fig. .Given any CDR in The proof follows from a packing argument. Consider the set N choices for Let v is the root of the problem. We conclude with the following statement.The existence of this vertex Any CDR defined on v is on the Apply Lemma d dimensions.Combining with Theorem Any CDR in By Theorem In this section we describe how to construct a weak CDR in two dimensions with constant error. Specifically, we show that the weak CDR has at most 5/2 error and at most N is a power of 2. We partition ith slice between greedy digital path from each point p and 2p is defined by picking a point in each y-coordinate). Lemma Assume that The last step of the construction is as follows: for those points The following lemma shows that every two consecutive points we picked in xy-monotone.For u and v are not connected in the 4-neighbor topology, i.e., u and v are grid points, this implies that u and v respectively, i.e., xy-monotone because we only pick one grid point per each It is trivial for Next, we show that any two greedy digital paths in the same slice For L-infinity distance to the corresponding line segment. Given any two consecutive line segments v to By the way we picked the grid points on the greedy digital paths, we can see that any grid point on the greedy digital path has at most 0.5 Then, we show how the greedy digital paths in For When p will connect horizontally to p horizontally until we hit some greedy path. Since p is between When By repeatedly applying Lemma For Now we have all the tools to show that GREEDY is a rooted tree at the origin so that GREEDY is a weak CDR.xy-monotone paths to all the vertices.GREEDY is a rooted tree at the origin with xy-monotone paths connecting to all the grid points from the origin because the graph is connected in xy-monotonicity comes from the fact that all the grid points are connected towards the origin. Clearly, this holds in If we can show that every grid point xy-monotone and disjoint. Hence, for any point By Lemmas Furthermore, the last step of GREEDY only applies to the grid points Then, we can talk about the quality of GREEDY in term of the number of inner leaves and the Hausdorff error in the next two lemmas.There are at most By Corollary The Hausdorff error in GREEDY is at most 5/2.n line segments such that each line segment lies in some slice H is the Hausdorff distance between A and B under Given any point v is between v is also between v is not in v to We first show how to partition p  error and From the above theorem, we can also extend it to have a tradeoff construction with O(c) error and For any O(1) error and c so that the original grid edges have length c. Therefore, the error of the tree in O(c), but the tree does not cover all the refined grid vertices. Then, we draw some vertical or horizontal line segments that branch from the tree as shown in Fig. c, i.e., c steps. Hence, their errors are still O(c). We first apply the GREEDY construction in p in p to some vertex q in the tree with distance at most 1. Then, O(1). Let In addition to the GREEDY construction for a weak CDR, we can also observe some nice properties in GREEDY. If we remove all the branches which end up at some inner leaves, we have an infinite tree which covers 2/3 of the grid points of V be the vertices in O(1) error. Moreover, for any vertex Let xy-monotone. And by Corollary O(1) error.By Lemma By definition of greedy digital path, in each R and blue B point sets such that the absolute value of their discrepancy is 1. Let In this section we construct red x-coordinate and the same y-coordinate. The blue point x-coordinate and smaller y-coordinate.A staircase is a sequence of alternating blue and red points Given a staircase, we can define a curve by connecting consecutive points on the staircase. Additionally we add a vertical segment at the beginning and a horizontal segment at the end, in order to connect the curve to the boundary of the unit square, see Fig. The transformation in Sect. Below every red point there is a corresponding blue point. Moreover, by Lemma x, y) to the bottom-left of the staircase satisfies x, y).Assume that a set of blue and red points forms one staircase. Then the curve induced by the staircase splits the unit square into two parts. The set of points :mxy represents the expected difference between the numbers of blue and red points in the axis-aligned rectangle with corner points  and . Every point  along the curves i many blue points more than red points. Figure Recall the definition of discrepancy of mentclasspt{minimam staircases such that the staircase approximating The idea of our construction is to approximate the level curves k such that the blue point  between the ith and Consider any point .Theorem Discussion. As mentioned in the introduction, this result could be the first step towards showing that GREEDY construction is tight : we want to create a weak CDR with O(1) error and few inner leaves. Theorem This non-crossing requirement is key and follows from the fact that we require o(N) error in arbitrary dimensions].Common intuition would say that the O(e) error (for some We also find that weak CDRs are an interesting research topic on their own. In particular, we would like to find the relationship between the number of inner leaves and the error of the construction. That is, say that we want a CDR with


The ability to produce 3D maps with infrared radiometric information is of great interest for many applications, such as rover navigation, industrial plant monitoring, and rescue robotics. In this paper, we present a system for large-scale thermal mapping based on IR thermal images and 3D LiDAR point cloud data fusion. The alignment between the point clouds and the thermal images is carried out using the extrinsic camera-to-LiDAR parameters, obtained by means of a dedicated calibration process. Rover’s trajectory, which is necessary for point cloud registration, is obtained by means of a LiDAR Simultaneous Localization and Mapping (SLAM) algorithm. Finally, the registered and merged thermal point clouds are represented through an OcTree data structure, where each voxel is associated with the average temperature of the 3D points contained within. Furthermore, the paper presents in detail the method for determining extrinsic parameters, which is based on the identification of a hot cardboard box. Both methods were validated in a laboratory environment and outdoors. It is shown that the developed system is capable of locating a thermal object with an accuracy of up to 9 cm in a 45 m map size with a voxelization of 14 cm. It was chosen to represent the surface temperature distribution map using an OcTree data structure. The advantages of this approach are numerous: 1) it allows the filtering and discretization of the thermal point cloud, 2) it allows us to volumetrically represent the thermal environment and for the robot to interact directly with it, and (3) compared to a point cloud, an OcTree data structure allows for a more computationally efficient representation of space. Compared to [ it allow it allowApplication of an extrinsic calibration method, originally developed to calibrate one or more optical cameras with respect to a 3D LiDAR, to the calibration of an IR camera system.Development of an accurate large-scale long-range thermal mapping method, capable of working even in the presence of poor lighting conditions in both indoor and outdoor scenarios.Development of an occupancy grid mapping based on thermal voxels stored in an OcTree object. The method offers the possibility of producing several maps at different resolutions  from the same data structure.Error analysis and study of the influence of the voxel size on the accuracy with which a thermal cluster is recognized.The main contributions of this paper are as follows:The structure of the paper is divided as follows: in Thermography has a broad spectrum of applications ranging from rescue robotics to precision agriculture, industrial plant monitoring, and rover navigation. In the literature, many use cases can be found: Ulloa et al.  present Thermal mapping methods are mainly divided into four categories, as shown in This section describes how to calibrate the thermal camera with respect to the LiDAR, to compute the extrinsic parameters ested in  for the Perspective-n-Point (PnP) problem [The calibration algorithm analyzes a 3D point cloud and a 2D thermal image of the same object to determine the pose of the thermal camera in the reference frame defined by LiDAR. Here, the calibration target is assumed to be an ordinary cardboard box of known size, which must be oriented so that it has three walls in full view of the instruments at the same time. The camera pose in the LiDAR reference frame can be found by solving a  problem  that invThe first step of the proposed calibration process is to detect and cluster the walls of the box in the 3D point cloud. Each wall is represented by a subset The ground plane can be detected and removed because, unlike any of the box walls, it can be assumed to be perpendicular to the vertical axis of the world frame. The algorithm then searches the groundless point cloud for the plane fit with the highest number of inliers, sets those points apart, and repeats the process to find two other planes. At this point, the remaining outliers are simply discarded.M-estimator SAmple Consensus (MSAC) algorithm [The plane fitting function used here implemented a lgorithm  to identThe second step of the calibration process is to create and refine a suitable model of the target, from which to obtain the three-dimensional coordinates of its visible corners.The intersection of the three plane models shown in b uses algorithm [n reference points as the weighted sum of four virtual non-coplanar control points, whose coordinates in the camera frame are treated as the unknown terms of the problem. Some implementations of the EPnP algorithm include a subsequent phase of iterative Gauss-Newton optimization to further refine the results. In any case, the solution can be found if at least four corners of the target have been successfully identified in both the point cloud and the thermal image.Knowing both the world coordinates and the corresponding image coordinates of all the visible corners of the box allows us to set up a PnP problem with lgorithm . This mes is a skew coefficient. Here The extrinsic parameters of the thermal camera with respect to the LiDAR consist of a rotation matrix n valid points.If multiple point clouds and thermal images are available, the calibration algorithm automatically selects the set of extrinsic parameters which leads to the lowest average reprojection error over all the This section describes how a set of 3D LiDAR point clouds can be combined with a set of radiometric IR thermal images to create a three-dimensional representation of the temperature distribution in the environment the rover is traversing.This process occurs in three steps: first, the color map and the temperature map extracted from each thermal image must be projected onto the corresponding point cloud, producing a sequence of thermal point clouds; the thermal point clouds are then registered and merged together into a single scene, using the pose information obtained by reconstructing the trajectory of the rover with a suitable LiDAR SLAM algorithm; finally, the registered and merged thermal point clouds are processed by a spatial decomposition algorithm to produce a voxel map, in which each cell of the 3D grid can be associated with a specific temperature value.livox_horizon_loam  . An overview of the data fusion procedure is shown in Equation  can be uurned by  for a gihe pixelxi=ri,ci=Temperature maps can be extracted from each image file using dedicated software provided by the manufacturer of the thermal camera. Each radiometric thermal image includes among its metadata the coefficients required to convert the pixel intensity values into the apparent temperature values. The coefficients are automatically computed by the internal radiometric software of the thermal camera, taking into account a number of parameters set by the user: air temperature, relative humidity, target emissivity, and approximate observation distance must be specified manually before any data can be collected. At present, the user-defined parameters can be set only once per image and are applied to all pixels, so they must necessarily represent average values compatible with most of the observed scene.Before the thermal mapping process can begin, it is necessary to make sure that the point clouds and the thermal images acquired at the same time are correctly matched to each other. This means either associating each thermal image with a point cloud  or associating each point cloud with a thermal image . In both cases, the synchronization between the two datasets is performed by minimizing the absolute difference between their respective sets of timestamps.NaN) temperature value, to be easily identifiable by both the user and the algorithm.After data fusion, all elements of each thermal point cloud carry two additional pieces of information: the color of a thermal image pixel  and the corresponding value from the associated temperature map . Note that 3D points outside the area included in the thermal camera’s FOV correspond to 2D points outside the image, and therefore their coordinates in the image plane will be either negative or greater than the size of the image itself. Such points are represented with the same out-of-palette color and given an invalid  they contain. However, the merging cannot simply consist of grouping all the points into the same set: doing so would produce excessive concentrations of elements in the overlapping areas between two consecutive point clouds. Therefore, the algorithm processes the registered and merged thermal point clouds with a three-dimensional box filter. After the volume is divided into cubes of constant size, all elements within the same cube are replaced by a new point identified by averaging their coordinates. The purpose here is only to equalize the density of the resulting thermal scene, so the box filter can have a resolution  much higher than the grid which will later be used for the spatial decomposition algorithm. Notice that the box filter affects all the properties of each point, including its temperature : this means that the new point which replaces all the point cloud elements within a given cubic volume will have a temperature value equal to the average of the temperatures of the point cloud elements it replaces.Before proceeding with the spatial decomposition, the thermal scene must undergo some post-processing operations. First, to improve the efficiency of the spatial decomposition algorithm, the thermal scene is further downsampled by randomly selecting only a given fraction of its elements. Then the thermal scene could also be leveled by fitting a plane to the available points and determining the 3D rigid transformation that would make that plane normal to a certain reference vector, but only if the surface on which the rover is moving has a constant and known inclination. In all other cases, an “offset” described by the rotation matrix  derived  using thSIn the previous steps of the thermal mapping process, a set of point clouds and a set of radiometric thermal images were combined into a thermal scene S with a three-dimensional grid, grouping its points into objects that represent finite portions of space. Each of these regions will be associated with the average temperature of the points it contains, building a 3D temperature distribution map with the same resolution as the grid used.The final step of the process aims to discretize the scene OcTree model, a hierarchical data structure that recursively subdivides a volume into eight identical elements, also known as voxels [kd-tree data structures, because it guarantees a regular subdivision of space .The algorithm used to decompose the scene is based on the s voxels . This kibin of 3D points, represented by their coordinates and also by the index that identifies each of them in the original point cloud. The first bin encompasses the whole scene, and so contains all the points, but can be decomposed into eight children bins, which in turn could be decomposed further. Subdivision stops if an adequate condition is met, indicating that the required map resolution has been reached (or exceeded). Here, the stopping condition is expressed as a lower limit to the acceptable voxel edge length: the actual size of the voxels created, however, is determined only by the size of the volume to be decomposed and by the subdivisions required to reach that threshold.To process the scene, the algorithm attaches to each voxel a thermal voxels  must be processed back into xels see d: regionThe experimental setup for testing the proposed calibration and thermal mapping algorithms consists of a FLIR Vue Pro R 336 radiometric thermal camera and a Livox Horizon 3D LiDAR. One of the notable features of the LiDAR used is the adoption of a non-repeating scanning pattern, which gradually increases FOV coverage over time  was temporarily placed on a small cart and pushed down a flat straight corridor shown in . This arlow resolution map, with a voxel edge size of A shown in a.high resolution map, with a voxel edge size of A shown in b.The analysis was performed on two temperature distribution maps of the same environment, each of which was divided into two halves to simplify the representation of the results:The two maps were created from the same OcTree data structure, so they describe the exact same set of points. Specifically, the high resolution map represents the subdivision from the ninth level of the source structure, while the low resolution map represents the subdivision from the eighth level of the same structure.clusters, each made up of at least 3 voxels in close proximity to each other . The calibration method used to align the reference systems of the LiDAR and the thermal imaging camera is also presented. LiDAR point cloud registration with the UGV relative pose reconstruction was based on a LiDAR SLAM algorithm (livox_horizon_loam). For efficient representation of the 3D temperature distribution map, a discretization with a voxel grid by means of an OcTree-based spatial decomposition function was performed. The method was characterized in an indoor environment by tracing the presence of hot elements, such as radiators. The standard deviation of the localization error of a thermally significant element has been shown to be equal to This article describes a thermal mapping method based on data fusion between a 3D LiDAR and a thermal imaging camera. The target application sees thermal environment monitoring via UGVs (