In the past years, a significant amount of effort has been directed at the observation and characterisation of caries using experimental techniques. Nevertheless, relatively little progress has been made in numerical modelling of the underlying demineralisation process.The present study is the first attempt to provide a simplified calculation framework for the numerical simulation of the demineralisation process at the length scale of enamel rods and its validation by comparing the data with statistical analysis of experimental results.FEM model was employed to simulate a time-dependent reaction-diffusion equation process in which H ions diffuse and cause demineralisation of the enamel. The local orientation of the hydroxyapatite crystals was taken into account. Experimental analysis of the demineralising front was performed using advanced high-resolution synchrotron X-ray micro-Computed Tomography. Further experimental investigations were conducted by means of SEM and STEM imaging techniques.Besides establishing and validating the new modelling framework, insights into the role of the etchant solution pH level were obtained. Additionally, some light was shed on the origin of different types of etching patterns by simulating the demineralisation process at different etching angles of attack.The implications of this study pave the way for simulations of enamel demineralisation within different complex scenarios and across the range of length scales. Indeed, the framework proposed can incorporate the presence of chemical species other than H ions and their diffusion and reaction leading to dissolution and re-precipitation of hydroxyapatite. It is the authors’ hope and aspiration that ultimately this work will help identify new ways of controlling and preventing caries. The continued demand for dental treatment and disease prevention has led to a great need for a deeper understanding of the underlying mechanism causing caries, namely, the mineral dissolution of the complex hierarchical structure of human teeth interrod enamel. Therefore, the remaining region is named as rod head. These rods are primarily composed of Hydroxyapatite (HAp) crystals (approximately 25–30 nm thick and 200 nm long), which are covered by a nanometre-thin layer of enamelin. It is important to note that there exists no discontinuity between rod head and interrod regions as the HAp crystals gradually change their orientation from one region to the other. This interrod enamel shows a more prominent gradient in HAp crystal organisation, particularly its orientation. An interface between the rod enamel can also be identified, which is termed the rod sheath. The rod sheath is a highly organic domain, containing substances like enamel protein. It is thought to be acid-resistant and in some cases can be absent. Its thickness may vary from 0 to hundreds of nanometre nm, depending on the size of HAp crystal at the border between two adjacent rods . In general, enamel shows 3%–4% content of water by weight, a high fraction of other inorganic matter (92%–96%) and its remaining composition is 1%–2% organic material.Within each rod, it is possible to identify a region occupying a minor area that is generally called ameloblasts during tooth development, well in advance of tooth eruption. These cells are generally of columnar shape, with approximately hexagonal cross section and their size directly reflects the final size of the enamel rods aprismatic layer is present on the outermost surface of the enamel characterised by a laminate-like structure. The aprismatic layer is deposited by the ameloblasts each day of life, ant it is thought to occur predominantly during the diurnal hours, and for this reason the aprismatic layer appears to be periodically repeating forming this laminate look The crystallisation of the HAp is carried out by Overall, the enamel shows a hierarchical structure, starting at the nano-scale with HAp crystallites being the main constituents. At a higher hierarchical level  the length of reference is the enamel rod. While at the macroscopic level , bundles of enamel rods are considered to build up the actual geometrical shape of the tooth that is visible with bare eyes.Structurally, the early enamel caries process is thought of as enamel density reduction and removal by demineralisation in the presence of acids. In natural caries, demineralisation is primarily driven by the products of the anaerobic metabolism of cariogenic bacteria within dental plaque producing acid. In general, the enamel demineralisation progression is mediated by the acid diffusion into enamel causing the dissolution of HAp crystals. The most prominent properties of the acidic solution in terms of affecting the demineralisation progress are: the acid type, the pH, the acid concentration, the acid titration, the extent of enamel surface exposure to the acid and the exposure time The rate and directionality of enamel demineralisation at the millimetre scale is thought to be mainly driven by the rod orientation with respect to the acid wet surface. To understand the influence of the preferential demineralising direction it is of paramount importance to consider the anisotropic diffusivity properties of a given etching agent within the HAp crystals This heterogeneous structural organisation of the HAp crystals gives rise to different patterns of etching at the rod enamel scale. As described by Silverstone et al. The purpose of this work is to create a simple and reliable mathematical descriptive model for the prediction of the enamel demineralisation process at a length scale comparable with that of the enamel rods.Cast within the framework of the Finite Element Method (FEM), the model consists of an implementation of the reaction-diffusion differential equations within a Representative Volume Element (RVE) that reflects the periodic structure of rod and sheath domains within human tooth enamel. The first attempt at modelling the physics of this problem was proposed in 1966 by Zimmerman Additionally, the effect of the etching attack acting with respect to the rod longitudinal axis on the etched surface morphology needs to be investigated. To shed some light onto this aspect of the process and to demonstrate the power of the newly developed analysis tool, a simplified simulation was carried out with etching surface being normal to the rods’ longitudinal direction. On a qualitative basis, the impact on the acidity of the etchant was also evaluated, using the model through parametrical analyses at several levels of pH, and discussed below.In this study, an intact human third molar was used that was extracted for therapeutic reasons other than caries . The root tips of the analysed tooth were removed using a rotating diamond saw at very low speed to avoid microstructural modifications. After that, the molar was fixed in 10% formalin buffered solution  for four days. Subsequently, the sample was rinsed with water and sectioned to obtain 500 μm-thick slices.The tooth slice was embedded in a bespoke envelope made of Kapton sheet and nail varnish, in order to allow only for the natural tooth free surface to be exposed to the etching agent. The varnished tooth slice was incubated at 37 °C in lactic acid  to simulate the carious process ex situ exploiting the X-ray beams generated at the UK synchrotron facility, Diamond Light Source . The experiment was conducted on the I13-2 beamline dedicated to imaging and tomography studies. Synchrotron X-rays facilities provide unrivalled compromise between x-ray flux and spatial resolution achievable The sample was examined X-ray micro-computed tomography (X-ray μCT) was performed using a pink X-ray beam with moderate energy spread. The average energy of the X-ray beam employed was 21.8 keV. To accomplish this task, a set of absorption contrast images were taken in transmission mode at different sample rotation angles with respect to the incident X-ray beam, ranging from 0 to 180°, with an angular step of 0.05°. Each image was acquired at room temperature using an exposure time of 0.3 s. Full data acquisition for the entire set of sample rotations took less than 30 min. To acquire the transmitted radiographs, a PCO Edge 5.5 camera  was positioned at a distance 940 mm downstream from the sample. The imaging detector is equipped with a CMOS sensor with a 10x objective that provided the pixel size of 0.325 × 0.325 μm (so that the reconstructed data had an isotropic voxel size of 0.325 μm), and the field of view of 0.83 × 0.70 mm. The distortion correction of the acquired images was performed prior to the reconstruction using an array of points with known position. Dark and flat field images were also captured prior to the acquisition of a sample to eliminate noise due to pixel sensitivity variation and other background artefacts. Subsequent processing for 3D reconstruction was performed using SAVU processing pipeline .The analysis of reconstructed 3D data from X-ray μCT imaging was performed using Avizo v2020.1 software (Thermo Fisher Scientific).The full dataset contained 2510 × 2510 × 2110 voxels. To reduce the computational cost, detailed analysis of the features of interest was conducted on sub-volume of 1380 × 850 × 861 pixels (448.175 × 275.925 × 279.500 µm). De-noising processing step was applied using a median filter and a non-local mean filter in Avizo software. For image processing, the background was removed via watershed thresholding and manual segmentation. For the purpose of tracking the eroded rods and therefore the lesion advance, segmentation was performed on the filtered data with the segmentation editor. The statistical information was extracted for the evolution of volume of segmented areas as a function of the distance from the enamel surface.Morphological investigation of the etched surface was conducted by means of electron microscopy using FIB-SEM-STEM technique at the Multi-Beam Laboratory for Engineering Microscopy (MBLEM), Department of Engineering Science, University of Oxford . Some examples of previous applications of these techniques in biomaterials are given in Due to the low sample conductivity, the surface was coated using a magnetron sputtering technique to produce a thin Au-Pd coating layer of 5 nm thickness, thus improving the imaging resolution achievable to a few nm. Due to the local roughness of the sample it was expected that the coating layer thickness could experience some deviation from the nominal value without affecting the imaging results. In addition, STEM imaging was carried out at a position close to the fully demineralised region of the enamel. A STEM lamella sample was extracted from the tooth by exploiting the capabilities of Tescan LYRA instrument . Focused Ion Beam (FIB) was used together with the Gas Injections System (GIS) and nanomanipulator for sample lift out. The lamella preparation procedure produced lamella of ~100 nm thickness thin enough to resolve individual HAp crystallites.The result of the segmentation of 3D tomographic reconstruction of the artificial caries lesion is shown in By performing the segmentation of the eroded enamel rods, in a selected sub-volume of the whole probed sample, it was possible to extract the statistical data for the demineralised area fraction as a function of the depth from the sample free surface subjected to acid treatment. The numerical values obtained from this post-processing analysis are reported in the section of this paper below that is dedicated to the comparison with numerical predictions.In order to obtain further insights and confirm the nature and orientation of the HAp crystallites, SEM and STEM investigations were carried out within the sample.A STEM image of the partially demineralised enamel is shown in FEM methods are nowadays employed in a number of numerical analyses and simulations to solve differential equations in complex shaped domains. Due to its great versatility, FEM fields of application span from structural to biological, from thermal to optical and so on In this work, the dissolution of HAp crystals was considered to be a first-order kinetic process driven by the action of a single component in solution, i.e. the concentration of hydrogen ions, The nature of the hierarchically structured material studied here was such that at the small scale of consideration involved the mineralised dental tissue could not be considered homogeneous, and its diffusivity response was not isotropic. For the correct calculations to be performed, it was necessary to implement the rotation of the diffusivity matrix from the orthotropic form at the micron-scale aligned with the local rod orientation to the global coordinate system. Further correction to the diffusivity tensor was required to account for the evolving porosity that affected material permeability. The former issue will be addressed in the section devoted to this issue below, while the considerations required to incorporate the porosity effect are presented here. In mass transport problems, material porosity is known to affect the overall diffusivity of a component within a structured host material It is important to mention that diffusivity is a temperature-dependent parameter whose variation can be expressed by the Arrhenius equation:Here The second term on the right-hand side of Eq. As concerns the discrimination of the demineralised regions over the time, the approach adopted here makes the use of the level-set phase-change model. This approach is preferred over the method that considers element deletion because it avoids deactivation of elements over the time, therefore the mesh remains the same during the time-dependent simulation.At the length scale considered, a HAp crystallite can be considered to be either pristine or completely dissolved. To achieve numerical stability, the phase parameter Here the terms on the left-hand side describe the motion of the interface in terms of the partial time derivative. The terms appearing on the right hand side are necessary to ensure the stability of the numerical solution. Parameter l × l, extruded normally to its own plane by the length L. In the model described here l was set equal to 7.1 µm, and L was chosen to be equal to 250 µm, so that with the dimensions given in micrometres, the RVE domain can be written mathematically as:The above equations were solved within the finite element framework over a three-dimensional domain representing the periodical structure found in the tooth enamel. The software package used in this work was COMSOL. As illustrated in L/l ~ 35 was dictated by the experimentally observed high diffusivity of H ions in the z, direction  compared with the corresponding values in the x or y direction, µCT observation. The well-known keyhole-like geometry of the enamel rod was reproduced in the RVE model as shown in The choice of the large ratio xy with a mapped method and then sweeping the elements along the longitudinal direction z. This permitted a very low degree of element distortion while maintaining a great mesh uniformity, ideal to minimise numerical errors.In the context of the FEM analysis, the discretization of the RVE was performed by firstly meshing the 2D plane x and y, namely, periodic boundary conditionsy as the normal axis, and x as normal axis.Eq. In widespread practical use, pH is used as the parameter to describe hydrogen ion concentration according to the following relationship:Mol/litre was set within the domain V, corresponding to pH = 14. Same value of concentration was kept fixed throughout the time-dependent simulation at the flat surface having z-axis as the normal axis , the etchant acidity was varied from pH 2.0 to 5.5. The plot in Aimed at finding relationships between the pH and the front of the lesion, the depth at which the lesion had reached 80% of the mineral content was recorded for each pH and plotted, A simplified model was employed in this simulation to improve the understanding the origins of different types of etching patterns. The underlying HAp crystal orientation is thought to affect the direction of the demineralisation front and therefore contribute in determining the type of etching pattern. In order to enhance the visualisation of this phenomenon, this section of the paper considers a rotated etching attack angle (90°) with respect to the Type 1 scenario.yz and xy. Whilst, on the surface opposite to the exposed one, the boundary condition applied was only the absence of a flux of hydrogen ions. Parameters for coefficients of diffusivity and reaction remained the same as the simulation described in the previous section of this paper concerning the simulation of the experiment, while the etchant pH was set to 5.To perform this task, the same model as portrayed before was utilised with some modifications in both the geometry and boundary conditions. Firstly, the surface that was the actual source of hydrogen ions was no longer perpendicular to the longitudinal direction of the rod but, normal to that as depicted in y-direction, which can help demonstrate the region where more material had undergone phase change, in other words where it had demineralised. It was revealed that the core of the rod head did not experience deep demineralisation, as seen before in the scenario studied experimentally. Instead, the outer part of the core demineralised more, giving rise to a different etching mode, which was clearly associated with Type 2. Additionally, in the early stage of the demineralisation the interrod regions dissolved first. This simulation is not to say that the etching mode was solely and undoubtedly induced by the acid attack orientation, but that this aspect played an important role In In this work, the demineralisation process induced by an etchant agent at the micron-scale of human enamel was successfully simulated. For the first time, it was possible to understand and model the dissolution process occurring at this scale and the role played by the HAp crystallites. By accounting for the internal enamel HAp organization and for the disordered and hypo-mineralised content in the sheath regions, an exhaustive agreement was found with respect to the experimental evidence. Only thanks to the micron resolution provided by micro CT using a synchrotron X-ray facility, detailed information about the dissolution progression was observed and therefore directly compared with the simulation at the same length scale. Although the validation was only done at one reference time (after 3 weeks) of the etching process, and although the model only considered the effect of hydrogen ions diffusing and therefore demineralizing the HAp, the model presented here accurately captured the mechanisms involved and resembled the actual morphology observed in both the micro-CT and direct SEM studies.The model gives some insights into the nature of Type 1 and Type 2 etching patterns. If the enamel rods were exposed to demineralizing agent on a surface parallel to the longitudinal extent of enamel prisms, Type 2 etching pattern was observed. Particularly, the head of the rod appeared to be less prone to hydrogen ion diffusion because of the crystal orientations of HAp crystallites, while the interrod demineralised rapidly. Without claiming this aspect to be fully responsible for determining the etching pattern type, this analysis appears to provide some interesting and thought-stimulating insights.The novel simulation framework presented hereby constitutes a solid starting point towards the final aim which is to simulate typical tooth demineralization, especially when the tooth is infected by natural caries. To accomplish this task, the capabilities offered by the modelling framework presented here may be fully exploited, for instance, to implement numerous substances that diffuse and react within the enamel, other than simply H ions, e.g. F, P and Ca ions. Other future works may be addressed at transferring the findings of this analysis to a “larger” scale model that would make use of the micro scale diffusion-reaction properties, opportunely homogenised. This would enable the simulation of an entire three-dimensional tooth, with nearly infinite degrees of freedom to analyse a near limitless number of scenarios.Eventually, it is very important to mention that, this numerical method can also be utilised for the reverse action of the tooth erosion, i.e. re-mineralisation, that general occurs at the enamel top layer thanks to the “healing” action of the saliva.In this study, an intact human third molar was used that was extracted for therapeutic reasons other than caries .The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


We present a new optomechanical probe for mechanical testing of soft matter. The probe consists of a micromachined cantilever equipped with an indenting sphere, and an array of 16 single‐mode optical fibres, which are connected to an optical coherence tomography (OCT) system that allows subsurface analysis of the sample during the indentation stroke. To test our device and its capability, we performed indentation on a PDMS‐based phantom. Our findings demonstrate that Common Path (CP)‐OCT via lensed optical fibres can be successfully combined with a microindentation sensor to visualise the phantom's deformation profile at different indentation depths and locations in real time.This work presents a new approach to simultaneously perform micro‐indentation experiments and OCT imaging. An optical fiber array‐based sensor is used to develop a new hybrid tool where micro‐indentation is combined with optical coherence tomography. The sensor is therefore capable of compressing a sample with a small force and simultaneously collecting OCT depth profiles underneath and around the indentation point. This method offers the opportunity to characterize the mechanical properties of soft materials and simultaneously visualize their deformation profile. The ability to integrate OCT imaging with indentation technology is promising for the non‐invasive and precise characterization of different soft materials. Therefore, the investigation of mechanical properties is crucial in several research fields and at all length scales, from cellular up to tissue and organ scale.The local mechanical properties of soft biological materials are typically assessed via nano and microindentation by means of Atomic Force Microscopy (AFM).To contribute to this field, in 2010, our group proposed a new approach to precisely and locally map the tissue's viscoelastic properties at microscale by means of ferrule‐top technology. The technique relies on a ferrule‐top probe, which is obtained by assembling a rather macroscopic (compared to AFM) cantilever on a millimetre‐sized glass ferrule. The cantilever is equipped with a sphere at its free‐hanging end, which is brought in contact with the sample's surface to allow the user to apply the mechanical stimulus. An optical fibre is then used to measure the deflection of the cantilever providing the information needed to assess the mechanical properties of the sample, as in AFM nano‐indentation.Although indentation techniques can accurately probe the local mechanical properties of biological tissue, and simultaneously provide subcellular imaging they are less suitable for in‐depth optical imaging of tissues. Microindentation sensors can, in fact, provide data on the bulk mechanical response of the sample (at tissue level), without the possibility to discriminate subsurface deformations due to the application of the external load. A visualisation of the inner structures of the sample  would provide additional information for a better interpretation of the mechanical properties of the indented material.Triggered by this series of considerations, in 2013 we demonstrated that the ferrule‐top probes used for indentation can be modified to hold an additional optical fibre that, when connected to an optical coherence tomography (OCT) system, enables one to visualise how the subsurface layers of a sample deform under the application of an external load.Both the designs incorporate only a single OCT‐fibre, which acquires a one‐dimensional OCT depth profile of the sample at the indenter contact point. However, to visualise the sample's deformation during the indentation stroke, one needs to transversally scan the region of interest and reconstruct a cross‐sectional image of the tissue around the indentation site. To this end, in this paper, we propose a new experimental sensor that combines an array of multi‐OCT fibres with microindentation capability. The instrument is based on a miniaturised cantilever probe that compresses the sample with a small force and simultaneously collects OCT depth profiles around the indentation point. The integration of microindentation and OCT allows one to investigate, in principle, the mechanical properties of any material while acquiring 2D‐profiles that show the tissue deformation during a single indentation stroke in real time. Before going into the details, we believe it is important to clarify that the goal of this paper is to test the working principle of the probe and not to validate its performance in identifying specific mechanical properties of the material. The latter would require an experimental effort that goes beyond the resources currently available. It is, however, fair to remind the reader that the use of ferrule‐top technology for material characterisation has already been discussed in numerous other papers.22.1a ≈ 65 µm diameter barium titanate sphereThe building block of the sensor is a 5 mm × 5 mm × 10 mm 3D‐printed ferrule, which has a 300 μm ridge, 1 hole in the centre, and 16 microgrooves on the front side of the rectangular ferrule (pitch ≈ 300 μm), as illustrated in Figure 2.2The optical fibre mounted in the central groove and aligned with the reflective part of the cantilever is connected to a commercial interferometer  to measure, via Fabry‐Perot interferometry, the deflection of the cantilever. The 16 OCT fibres are connected to a spectral‐domain OCT system  implemented in Common Path‐OCT mode (CP‐OCT). The layout of the system is shown in Figure Finally, the sensor is mounted on a piezoelectric stage operating in closed‐loop mode, which is then used to drive the indentation stroke. Briefly, a cantilever with a spring constant in the range of 3.4±0.2 N/m, calibrated according to Ref.  and a sp2.3Before each experiment, two types of calibration are performed: background subtraction and axial offset alignment. The background image is obtained when no scattering material is present in front of the sensor and is then subtracted from all subsequent scans. As far as the axial offset alignment calibration is concerned, one should consider that the facets of the OCT fibres are not perfectly aligned. This misalignment introduces relative axial shifts in the recorded A‐scans. To overcome this issue, the probe is mounted in front of a flat surface, which allows us to measure the phase delay due to the misalignment. The misalignment is then compensated for during the actual indentation measurements. Both calibration procedures are directly implemented in Labview; therefore, no further postprocessing of OCT data is performed. After the OCT calibration, the actual mechanical measurement starts. As the tip is pushed into the sample, the deflection of the cantilever, the indentation depth , and the 16 OCT depth profiles are simultaneously acquired.2.42 particles suspended in OH‐terminated silicone oil, following the fabrication steps indicated in Bartolini et al.To test our approach, we used the sensor described above to indent and simultaneously image a PDMS‐based phantom. The sample consists of a ∼300 µm thin membrane, made of Sylgard 184 polydimethylsiloxane  poured on a plastic ring. To improve the scattering properties of the material, the phantom was prepared with the addition of TiO3The indenter is based on ferrule‐top technology,Following the production protocol described in the method section, we produced several sensors. For a more systematic use of this probe, it is therefore mandatory to find more suitable (possibly automated) assembly techniques. It is worth stressing that the lensed fibres proved to be prone to breaking during the fabrication phase; furthermore, the  gluing process is rather complex and often leads to nonfully functioning probes. The ultrathin fibres are glued in the u‐groove of a 3D printed fibre array and mounted on a motorised Z‐stage, while the sample is positioned perpendicularly to the fibre axis. However, the resolution of the groove of the 3D printed ferrule does not ensure a straight alignment of the fibres toward the sample, making the coupling back of the light very difficult. This is a severe limitation of this probe. As a matter of fact, the best probe we were able to fabricate could only count 11 working OCT fibres out of the 16 expected.In our study, we refer to A‐scan as the signal recorded by one single fibre. A set of consecutive A‐scan acquired from several imaging fibres produces a cross‐sectional OCT image called B‐scan. Figure y‐scale).As expected, the data acquired before the calibration, reported in Figures In Figure n = 1.00); therefore, to obtain the correct geometrical distances, one has to consider the effect of the refractive index of the material's light travel through. Considering that the fibre lens is made of barium titanate with a refractive index n = 1.95, the peak‐to‐peak geometrical distances (gd – peak 1 vs peak 2 and peak 3 vs peak 4) correspond indeed to the dimension of the lens (∼ 65 µm in diameter). The effect of the two reference surfaces (front and rear lens facet) adds two additional peaks in the OCT scans. Those kinds of reflections are unavoidable in this type of sensor, as they arise from the reflections due to the several interfaces the light has to encounter before reaching the sample. The gd for peak 1 to peak 3 and peak 2 to peak 4 has been measured to be ∼0.8 mm. Those distances correspond to the microscope glass slide . This set of data, together with the graphs showed in Figure The distances between the small peaks (peak 1 vs peak 2) are ∼125 µm for fibre 12 and ∼118 µm for fibre 14. However, the scale bar refers to the image depth that one would observe in vacuum  is ∼126.8 µm as for Figures gd = 65 µm). On the contrary, the distance between peak 1 and 3 (black arrow) is, on average, ∼305 µm for both the figures and represents the PDMS membrane's thickness as measured with a commercial OCT system .To further test the ability of the sensor to distinguish different types of deformations due to the indentation stroke, we performed two indentation experiments in two different locations of the PDMS‐based phantom. The first location was selected close to the rim of the ring over which the PDMS membrane is suspended Figure , while f4Our work demonstrates that the general concept of a fibre‐array optomechanical sensor able to simultaneously indent and collect depth information of a phantom has the potential of assessing the morphological properties of materials while performing mechanical characterisation. Our findings prove that CP‐OCT systems based on lensed optical fibres can be successfully combined with a microindentation sensor.z‐axis is suboptimal. Thus, more often than not, the OCT fibres are not pointing straight toward the sample, and the back coupling of the light from the sample back to the OCT fibres is usually weak, with significant deterioration of the OCT signal to the point that, for some fibres, the losses are so high that no A‐scan can be collected.However, it is fair to mention that the sensor still has several limitations. The assembling of the 16 OCT fibres on the 3D‐printed ferrule is tedious work, since the alignment, the mounting and the gluing is performed manually and under an optical microscope. Furthermore, the printing resolution of the grooves, created to precisely mount the OCT fibres, is smooth; therefore, the alignment in the Although we recognise that several improvements in the fabrication steps need to be implemented, it is also important to highlight some strengths of the method. To the best of our knowledge, this is the first fibre array sensor able to collect depth profile and indentation simultaneously, without limitations on the sample thickness or preparation. The use of a high refractive index lens allows the imaging fibre array to be held far from the sample due to the depth‐of‐field offered by the barium titanate lenses. This feature leaves sufficient space for the cantilever to bend and probe the material without touching the OCT fibre lenses. Moreover, the integration of OCT imaging with indentation techniques could be further exploited by introducing ad hoc mathematical models able to extrapolate multiple information from single OCT depth profiles; one could think of providing a more extensive evaluation of deformation profiles in relation to the local material composition or to apply specific OCT algorithm for a more accurate extrapolation of peak profiles or other information hidden in the sample. Thus, the combination of OCT with micro indentation opens new ways to explore a whole range of information  that are neglected while choosing either microindentation or optical imaging separately.This work has been financially supported by the European Research Council under the European Union's Seventh Framework Programme (FP/20072013)/ERC grant agreement no. [615170].D. Iannuzzi declares a potential conflict of interest as cofounder, shareholder and advisor of Optics11.FIGURE S1 3D‐printed ferrule‐top probe assembly: The 3D‐printed ferrule is mechanically embedded on a rigid arm used to mount the sensor on the Z‐piezoelectric actuatorClick here for additional data file.FIGURE S2 Sketches of the setup: A ferrule‐top probe is equipped with an optical fibre for interferometric readout of the cantilever and with a spherical tip to indent the sample. The 16‐OCT fibres are mounted on the top facet of the 3D printed ferrule (left). The sensor is then mounted on the Z‐piezoelectric actuator, which is solidly attached to an XYZ manipulator (right)Click here for additional data file.FIGURE S3 Image of the setup: OCT imaging is obtained from the 16‐fibres connected to the Optical switch while the sensor hovers above the sample. Indentation measurements make use of the interferometric readout and are driven by the piezoelectric elementClick here for additional data file.FIGURE S4 PDMS membrane B‐scan: Commercial OCT B‐scan image of the PDMS sample employed for indentation. The thickness of the membrane is also shownClick here for additional data file.


The results are often comparatively elementary and seem to be isolated. On the other hand, the intuitive character of the statements and the lack of a general method of approach make the field attractive...In this paper we focus on closed planar curves but the consideration above identifies pretty well the context of our contribution. In this area interesting questions keep on coming up, as in the recent [Closed curves are natural mathematical objects which have been studied since a long time. In his well-known paper  Fenchel e recent , where acurvature. If arc-length parametrized planar curve, i.e., a twice differentiable function from the interval turning angle function curvaturek of The natural and complete geometric descriptor we associate to a curve is its entclass1pt{minimak and f for What are the conditions on Here and in the following with closed we just mean that starting and end point of the curve coincide . Figure Given another function mentclass2pt{minimInterpolation of curvature functions is a tool used in computer graphics to gradually transform one curve into another, while mantaining the length of the curve . This mek and f are shown. On the existence side, Sect. k, f) that satisfy our constraints. In Sect. As for an outline of the contents, Sect. f for F :Let Let n-th derivative of F and evaluating it in An analytic function is everywhere 0 if and only if all of its derivatives vanish in at least one point. We conclude by computing the We want now to better understand this condition, by discussing some of its implications. Our main tool will be an approximation argument based on the observation that, if g bounded and integrable.mentclass2pt{minimmentclass2pt{minimg. More explicitely, for any The ‘if’ part is trivial. For the ‘only if’ we use a density property of polynomials in our class of functions to approximate g to be f to By Lemma mentclass2pt{minimmentclass2pt{minimI, level sets of such regular values are therefore finite.Before moving to the next lemma, which provides a much more local characterization of our constraint, it is convenient to recall that a level set If mentclass2pt{minimIn Lemma The reason we excluded the level sets In order to have the equivalence  \documenentclass1pt{minimaWe collect in one theorem all the conditions we have proven equivalent.0.The curve with curvature 1.2.Moreover, they implyLet We also point out the following corollary, which rules out the possibility of vector spaces of curvatures of closed curves.For n even can only be satisfied by Setting In this section we discuss some periodicity properties that If entclass1pt{minimaj-th component, numbered from the left  such that the curve obtained by integrating the curvature In Sects. If The first existence statement implies the second just by taking j is an integer multiple of j going to infinity determines periodicity and differentiability of the function . From nmentclass2pt{minimverticescurvatureturning angleN vertices, which is closed  entclass1pt{minimabalanced, meaning In order to find non-trivial pairs such that the polyline associated to n copies of the pair of unit vectors summing up to Note that it is easy to construct polylines with no balanced proper subsets of edges. Consider for example k of a closed curve, when is it possible to find f such that the curve associated to Given the curvature function


We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems. Fractional calculus has a long and rich history. Its wide range of applications made this field an active area of mathematical research. Frequently investigated properties of FDEs include existence and uniqueness problems, multiplicity of solutions, stability of solutions and analytical study of the analytical properties of solution. Parallel to these area of research, one of the most explored and interested areas of research in this field is the design of new numerical methods for finding the approximate solutions to problems of this category. Many scientists and mathematicians are trying to design efficient and reliable techniques to find possible estimates to solutions of FDEs or their coupled systems.There are many analytical, semi-analytical, numerical, and spectral approximations of solution to FDEs and their coupled systems. Among others, one of the easiest method is the differential transformation method (DTM). In , DTM is Nonlocal FDEs arise in mathematical modeling of various problems in physics, engineering, ecology, and biological sciences ,29,30. Sf and g are nonlinear functions of Keeping in view the increasing interest of mathematicians in fractional calculus, we are strongly motivated to design a new spectral approximation technique for complicated problems such asWe start our discussion by introducing some definitions and preliminary results.The following definitions and notations are important for our further analysis. More details and theoretical understanding of these results, see ,43,44,45Definition 1.The Riemann–Liouville Definition 2.For where From the above definition, it is easy to extract These polynomials play a central role in approximation theory. Generally, these polynomials are defined on the domain cally as∫0τρiτ(t)The above equation has an equivalent vector representation given asfound in .Lemma 1.The integral of product of any three LP is a constant, represented by, wherefined in  andThe constant defined above is important in the solution of FDEs. We recall one more important result of the Legendre polynomials, which is their application in the study of convergence and developing of error bounds.Theorem 1.Let there exists a constant C such thatandwhere.Let ∏MC is constant and can be chosen in such a way that Z is storm livoliel operator of legendre polynomials with where In the next section, we first recall some of our previously designed operational matrices and then develop new operational matrices.OP acts as a basic block in developing approximation techniques. The purpose of operational matrices is to replace a given derivative term with its matrix notation. The following matrices are important in our further investigation.Lemma 2.Let where wherewhere.Let ΛMCorollary 1.The error Proof. ConsiderLemma 3.Let fined in ; then thwhere wherewith .Let ΛMFurthermore, fined in  andCorollary 2.The error Proof. The proof of this corollary is similar as Corollary 1. □Lemma 4.Let where fined in  andwhereThe matrix and .Let u under the effect of initial conditions. However these matrices do not have the ability to solve FDEs with integral types of boundary conditions. Therefore, the invention of new matrices that can easily handle integral types of boundary conditions are of basic importance. In the forthcoming discussion, we will design two new operational matrices having the ability to deal with integral type boundary conditions.Lemma 5.Let fined in , the folwhere the matrix wherefined in .Proof. Let Lemma 6.Let fined in ; the folThe matrix Furthermore, the entries are defined asProof. The general term of the equality can be written aswhich completes the proof of the lemma. □The operational matrix method for the solution of fractional differential equations is, in fact, a spectral method. The main aim of the spectral method is to convert a typical differential equation to system of easily solvable algebraic equations, which can be solved to obtain the solution in the series form of some orthogonal polynomials. The application of these methods to nonlinear differential equations results in a nonlinear system of algebraic equations, which can be solved using some iterative algorithms (the Newton–Raphson method is a frequently used method), see for example ,51,52,53f and g to convert the nonlinear fractional differential equation into a recurrence relation of linear fractional differential equations with variable coefficients.In this paper, we implement a different approach. We first use the Taylor series method to linearize the nonlinear part We first consider the following coupled system of linear fractional differential equations with variable coefficientsAssume the solution of  in termscDσ1u(t)=Now using the values of nd e1 in , we can (u(t) asu(t)=KMHMation asv(t)=LME(tions asu(t)=KMEKM and LMApproximate the initial solution, the solution of the linear part, by the method presented in previous section and name it Linearize the nonlinear part at Repeat step 1.Nonlinear FDEs cannot be directly solved using the OP method; however, combining it with the quasilinearization method makes it easy to recursively solve nonlinear FDEs. The procedure of this technique is as given as follows.Consider the following nonlinear FDEs.It can be easily noted that  is fractIn this section, we calculate a upper bound for error of approximation of solution with the proposed method.Consider the following fractional differential equation.The solution of the above problem can be written in terms of shifted Legendre series such thatnow from  and the an writecl(t)u(t)nd using  we getKIn view of Theorem (1), it is evident that if the function Equation  also estConsider the following system of FDEs.now from  thatu(tnd using  in 25)  (32)cDσuWe solve one single equation, three systems of linear FDEs with variable equations, and two systems of nonlinear problems, and analyze the convergence of the approximate solution by measuring the following error norms.Test Problem 1.Test Problem 2.Test Problem 3.Test Problem 4.Test Problem 5.Test Problem 6.The first test problem is solved using the proposed method. The problem is linear and is relatively easy to solve. We compare the approximate solution with the exact solution of the problem. We observe that by increasing the scale level of approximation, the approximate solution draws closer to the exact solution as expected; see for example The analysis of the second problem is given in The same analysis is performed for Test Problem 3 and Test Problem 4. The results are shown in The nonlinear Test Problem 5 is solved with the proposed iterative scheme. We use three different choices of the parameters This article presents a new algorithm for the solution of fractional order differential equations. The Newton Raphson method is combined with the operational matrix method for the solution of these problems. The convergence of the proposed method is checked analytically and is confirmed by solving several test problems. It is found that the approximate solution is highly accurate, and one can get high accuracy by using high scale levels. The mathematical proof of convergence and error analysis is our future plan of research.


Back pain may be related to an improper sitting position. The aim of the study was to assess the sagittal curvatures of the spine in a sitting position in children with generalized joint hypermobility (GJH). The study included 302 children aged 8–14 years. The sagittal curvatures of the spine  were assessed using the Saunders digital inclinometer. In order to assess GJH a 9-point Beighton scale was used. The study revealed no significant differences (p > 0.05) in sagittal curvatures of the spine in a relaxed sitting position between children with and without GJH. Regardless of the occurrence of GJH, kyphotic alignment of the spine was noted in a relaxed sitting. GJH does not affect the position of the trunk in a sagittal plane in a relaxed sitting position in children aged 8–14 years. A relaxed sitting position in children with and without GJH is characterized by a kyphotic position of the spine caused by an improper position of pelvis and lumbar segment of the spine. Kędra et al.6 revealed on a group of over 11,000 children and adolescents aged 10–19 years that BP occurred in nearly 75% of the respondents. In this study children aged 10–13 years the pain was most often located in the cervical region, while in adolescents aged 14–19 years—in the lower back7.Back pain (BP) typical of middle-aged individuals, occurs in young persons more often nowadays7. Therefore, it is important to identify and modify the risk factors of BP that could reduce its occurrence or recurrence8.The prevalence of BP leads to increase in socio-economic costs connected with its diagnostics and therapy, absence from school/work or social reimbursements because of chronic disability8. Kędra et al.7 revealed that students who experienced BP spent 5 or more hours a day in a sitting position more often than students without pain . This study also revealed that the majority of students (67.4%) were not familiar with the rules of ergonomics used to modify a sitting position in order to reduce the intensity of BP or to prevent it7.Improper sitting position for a prolonged time is perceived as one of the factors leading to the development of BP10. Generalized joint hypermobility may significantly affect body posture and the pattern of sitting due to decreased proprioception which leads to adoptions extreme positions in joints15. These factors together with increased activity of nociceptors may cause BP in children with GJH15. However, to date, no analysis of sagittal spinal curves in a relaxed sitting position in children and adolescents with GJH has been published. Therefore, because of the relation between a sitting position and BP as well as between BP and GJH, the aim of the study was to assess the sagittal curvatures of the spine in a relaxed sitting position in children with GJH.Generalized joint hypermobility (GJH) is diagnosed when the mobility of small and big joints is higher than normal for age, gender and race, when systemic diseases are excludedThe study included 302 children  aged 8–14 years. The inclusion criteria were: age between 8 and 14 years, no injuries or pain in the musculoskeletal system within three months prior to the study and no other disorders that could affect the results.The study was approved by the local ethical committee and a written consent was signed by a legal guardian of each child included in the study.17.In all the participants sagittal curvatures of the spine were assessed using the Saunders digital inclinometer 19. The measurements of the sagittal curvatures of the spine were performed in a standard relaxed sitting position. The child was sitting on a therapeutic table with adjustable height. This height of the table was set to keep the 90° angle in the hip and knee joints with feet fully and freely supported on the floor. Upper limbs were kept relaxed on both sides of the body. The participant was asked to “keep a natural relaxed sitting position”. Five seconds after that, sagittal curvatures of the spine were measured.Prior to the measurements a waterproof pen was used to mark the following anatomical landmarks on the back of each patient: the lumbosacral junction (L5–S1); the thoracolumbar junction (Th12–L1); the apex of thoracic kyphosis Th6–Th7; the cervicothoracic junction (C7–Th1)16. To measure sacral slope the inclinometer was calibrated to 0° in the horizontal plane and after that the angle at L5–S1 level was noted. In order to determine the angle of lumbar lordosis, the inclinometer was calibrated to 0° at L5–S1 level, and then the angle at Th12–L1 was assessed. Thoracic kyphosis was determined after calibrating the inclinometer to 0° at Th12–L1 level and then the angulation at C7–Th1 was measured17. Moreover, thoracic kyphosis was divided into its lower and upper part. The angle of the lower part of thoracic kyphosis was measured between the thoracolumbar junction (Th12–L1), where the device was calibrated to 0° and the Th6–Th7 junction. An upper part of thoracic kyphosis was measured from the Th6–T7 junction, where the device was calibrated to 0° up to the cervicothoracic junction (C7–Th1). According to the methodology all of the parameters were measured three times and the mean values of these three measurements were used in further statistical analysis19. The lordotic position of the examined segment was marked with “−” (minus).The measurements of the sagittal curvatures of the spine were performed according to the guidelines of the American Medical Association18.Previous study revealed good repeatability and reliability of the measurements of sagittal curvatures of the spine with the Saunders inclinometerAll the measurements were taken by one researcher (Researcher #1) who was blinded to the diagnosis of GJH.10.Researcher #2 assessed the GJH in all of the patients by use of a 9-point Beighton scale including: (1) passive hyperextension of the 5th finger of the hand over 90°; (2) passive apposition of the thumb to the forearm, (3) hyperextension of the elbow over 10°; (4) extension of the knee over 10°; (5) active forward bending of the trunk with the knees extended so that the palms of the hands rest flat on the floor10.The tests regarding limbs were performed symmetrically on both sides. One point was given for the positive result of each part of the Beighton scale. Five points or more for girls and four points or more for boys were the cut-off values for the diagnosis of GJHThe study group was divided into two subgroups of girls and boys. In each of these subgroups the sagittal curvatures of the spine were compared between the individuals with and without GJH.All the participants gave their written informed consent. The study was conducted in accordance with the Declaration of Helsinki, and the research was accepted by the Senate Scientific Research Ethics Commission of Olsztyn University, Poland (3/2020).20. Data distribution was verified by use of the Shapiro–Wilk test. The means were compared with Mann–Whitney U test and Student’s t-test. The level of significance was set at p < 0.05.Statistical analysis was performed by use of Statistica 7.1 software (StatSoft Poland) In both subgroups (girls and boys) no significant differences were found between the participants with and without GJH in the age, height, weight and body mass index (BMI) (p > 0.05) Tables  and 2.TaNo statistically significant differences were found in all of the sagittal curvatures of the spine measured between girls with and without GJH (p > 0.05) Table .Table 3CSimilarly, no significant differences in all the sagittal curvatures of the spine measured were noted between boys with and without GJH Table .Table 4CThe aim of the study was to assess the influence of GJH on sagittal curvatures of the spine in children in a relaxed sitting position. To date, no such analysis has been published.19. The referential value for sacral slope and lumbar lordosis in a standing position is (−) 15°–30° and (−) 30°–40°, respectively19. The study revealed that regardless of the GJH children aged 8–14 years adopted an improper kyphotic alignment of sacral slope and lumbar lordosis.The study revealed no significant differences in all of the sagittal curvatures of the spine measured in a relaxed sitting position between girls with and without GJH as well as between boys with and without GJH. What we find interesting is the kyphotic position of the entire spine in all the subgroups (regardless gender and GJH). Thoracic kyphosis was within the normal values for a standing position (30°–40°) in all of the subgroups21. Thus, the observed manner of sitting may significantly determine the occurrence of BP. It seems particularly significant in the case of children with GJH, since they present decreased ability to recognise the position of particular body segments with regard to one another. What is more, children with GJH present limited passive stabilisation of the spine connected with higher flexibility of connective tissues15. Therefore, appropriate educational programmes should be implemented in order to explain optimal manners of sitting aimed at reducing the risk of musculoskeletal system overloads and pain.Children and adolescents spend significant amount of time in a sitting position, and this time increases with ageThe authors evaluated the individuals in a relaxed sitting position without back support. Therefore, a relaxed sitting position adopted by children during activities of daily living may differ from the position assessed in this study. However, this sitting position could be standardised and what is also important it enabled to obtain access to the examined curvatures of the spine.The evaluation of sagittal alignment of the spine was performed shortly after a command “keep a natural relaxed sitting position” (5 s). We decided for such solution as we did not verify factors which could potentially influence the sagittal profile of the spine after longer period of time . However, it would be interesting for future studies to analyse the influence of time on the sitting position and subsequently on the changes of the sagittal spinal alignment.18.The study was conducted by use on an objective reliable measurement tool—the Saunders digital inclinometer with previously verified technique23. On the other hand, the aim of the exercises addressed to children with GJH is improving the stability of lumbo–pelvic–hip complex and proprioception as well as on education in order to avoiding the adoption of extreme position in the joints during activities of daily living24.The analysis revealed that, regardless of GJH, children aged 8–14 years present a kyphotic position of the spine in a relaxed sitting position. It was a result of an improper position of the pelvis and the lumbar spine. Therefore, educational, prophylactic and therapeutic activities should be aimed mainly at improving the position of these two segments in sitting position. Moreover, the shape of the spine in the sagittal plane in a relaxed sitting position did not differentiate children with and without GJH. As GJH may be one of the factors leading to BP, it is important to apply specific tests for GJH which will help to diagnose it in clinical practice. It is important as the rules of physiotherapeutic procedures regarding individuals with GJH differ from the standard exercises. Firstly, children with GJH should not perform stretching exercises since they may increase lumbosacral painThe evaluation of lower and upper part of the thoracic kyphosis is an additional advantage of the study. It revealed that both the lower and the upper part of the thoracic spine were in a normal kyphotic position. Regardless of the occurrence of GJH, higher values of kyphosis were noted for an upper part of the thoracic segment, what indicates the tendency for bigger inclination of this part of the thoracic spine, which may subsequently lead to a head protraction. However, this observation needs further research that would include the assessment of the position of head and cervical segment of the spine.Generalized joint hypermobility does not affect the position of the trunk in a sagittal plane in a relaxed sitting position in children aged 8–14 years.A relaxed sitting position in children with and without generalized joint hypermobility is characterized by a kyphotic alignment of the whole spine caused by an improper position of pelvis and lumbar segment of the spine.The sagittal curves of the spine in sitting position should not be used to diagnose generalized joint hypermobility. Thus the tests specific for generalized joint hypermobility should be applied in clinical practice.


Irreversible electroporation (IRE) is an emerging technique that has drawn attention in the field of cancer treatment. IRE uses non-thermal electric pulses to induce death of cancerous cells. However, recent studies have shown that the application of this technique may result in heating of the tissue. There is still room for improving its efficiency and defining better treatment protocols. This study investigates the optimal IRE protocols that avoiding the thermal damage during the IRE treatment.Electrode and pulse parameter are investigated. Finite element models are created to evaluate the ablation area and the temperature changes in the tissue. The model is validated experimentally in bovine liver tissue, while the parameters were optimized using response surface method (RSM).P = 0.020 and P = 0.003 respectively). Meanwhile, only the input voltage significantly affects the ablation area (P < 0.001). The optimal result from RSM showed that for maximum ablation area 250.82mm2 with no thermal damage, the IRE protocol consisted of an active electrode length of 10 mm, a distance between electrodes of 10 mm, and the delivery of 50 pulses of 41.21 µs and 3000 V.From analysis of variance, the parameter of electrode distance and input voltage has significant effect to the temperature rise in the IRE treatment of bovine liver (The approach presented in this study allows the optimization of the IRE protocols. An optimal IRE protocol that maximizes the ablation area was successfully calculated which can be applied with no risk of thermal damage to the tissue. Pancreatic adenocarcinoma is the most common malignancy of the pancreas and it is the fourth leading cause of cancer death in men and women , 2. ThisIrreversible Electroporation (IRE) is a minimally invasive surgical procedure that has drawn interest in the field of cancer ablation over the last decade. The electroporation technique consists in the exposure of cells to strong electric fields delivered by electrodes inserted in the soft tissue. If the applied electric field is strong enough, electroporation can be irreversible, which is characterized by the irreversible generation of nanopores in the plasma membrane leading to eventual cell death , 7. TheoSeveral parameters can have an influence on the outcomes of an IRE treatment. These parameters establish the IRE protocols of the treatment, and they are mostly related to the pulses that are delivered to the cells (pulse parameters) and to the configuration of the inserted electrodes (electrode parameters). The electric field generated in the tissue and the consequent temperature results change depending on the applied configuration. As a result, by adjusting some of these parameters, the thermal effect caused by heating can be reduced without compromising the ablation process.what can we do to optimize the outcomes of an IRE procedure?Optimization of the IRE protocols is therefore essential to avoid the thermal damage that may occur on vital structures near to the liver, such as vasculature, gall bladder, or bile duct . NeverthTo the best of our knowledge, there are still undefined guidelines and uncertainty about the ideal IRE protocols that should be applied to a specific treatment, particularly related to combination of pulse and electrode parameters configuration. As a result, this paper presents an approach that allows the calculation of optimal IRE protocols by developing an IRE model with experimental validation. The final goal is to present the optimal IRE protocols that maximizing the ablation area and avoiding the thermal damage during the IRE treatment.This section presents published studies related to IRE and the corresponding IRE parameters in relation to the electric distribution and the thermal development during the IRE treatment. At the end of this section, the approach proposed in this study is described.During IRE treatment planning, multiple parameters are considered to determine the ablation zone, including the electrode and pulse parameter. In terms of the electrode parameters, several studies have been done to investigate their effect on the IRE treatment. Davalos et al.  demonstrRegarding the pulse parameter, Garcia et al.  investigAlthough several investigations have been done related to the electrode and pulse parameter in IRE, much of the research only considered them separately, despite the fact that both parameters need to be planned together. It is still not known the optimal configuration for both electrode parameter and pulse parameter in order to obtain the maximum ablation area with no thermal damage.The present research contributes to the development of an experimentally validated model to adjust the IRE protocols that related to electrode and pulse parameter, and thus give guidance to the operator regarding the optimal configuration needed. An experimental apparatus is developed to validate finite element models for IRE simulation. A set of experiments was designed and performed in ex vivo bovine liver tissue to validate the models. In these experiments, temperature measurements were taken and then compared to the results obtained from simulations. If the temperature values measured experimentally are similar to the values calculated from the models, then they can be assumed as a reliable representation of the real phenomena. After validation, the models will be used to find the optimal IRE parameter configuration using response surface methodology (RSM).This section presents the developed models to simulate IRE and the experiment setup and procedure that were performed to validate the models.A schematic representation of the experimental setup is presented in Fig. Fiber-optic temperature measurement probes “T1 Fiber Optic Temperature Probe”  were inserted in the liver tissue to measure the temperature between the two electrodes. Another fiber-optic probe was inserted in the bovine liver tissue, 3 mm away from the right electrode and at the same depth of the tip of the electrodes, in order to monitor the temperature outside the electrodes. Fiber-optic probe was chosen to measure the temperature due to its invulnerability to the electromagnetic interference that may occur around the electrode. The actual experiment setup is shown in Fig. Five IRE parameters were tested in different levels. Two cylindrical electrodes were inserted into biological tissue. The distance between electrodes, the applied voltage, the number of pulse repetitions and the pulse width were evaluated in three levels . Meanwhile, the active length of the electrodes was tested in two different levels (low and high). The different levels of the parameters in this study present in Table Finite element models were created in order to analyze the effect of the IRE parameters in the electric field and temperature responses. The computation of the finite element models was performed using COMSOL Multiphysics v.5.4. The geometry consists of a 2D longitudinal cut perpendicular to the two electrodes inserted into bovine liver tissue. Finite element method (FEM) was used to solve the mathematical models. The mesh was built automatically by COMSOL with a custom size for each domain. For liver and electrodes domain, maximum element size was set to 3.3 mm and 1 mm, while minimum element size was 3.1 mm and 0.8 mm, respectively. The elements had a triangular shape, where the number of elements for each configuration of electrodes is presented in Table The electric field distribution in the liver was determined by the Laplace equation:To model heat transfer within the biological tissue, the Bioheat Transfer Equation was considered:terature . \documeTo fully define the model, boundary conditions such as electrical insulation and thermal insulation of the electrodes were established. The liver tissue was also electrically insulated from the external environment. The initial temperature of the tissue was set at 37 °C. Table The electrical and thermophysical properties of the bovine liver tissue were included into the model and they are summarized in Table In this section, the results of the experiments performed to validate the models are presented. After model validation, the optimization process to obtain the optimal IRE parameter configuration for bovine liver is also described.Finite element model was empirically validated by comparing the model result with the temperature measured in the bovine liver tissue during the IRE experiments. The experimental results are presented in Fig. Figure The result from the experiment with the pulse width is shown in Fig. 1 34) Taguchi design was applied using Minitab 18, resulting 18 combinations of parameters.According to Table After validating the models, the IRE experiment using 18 combination of five IRE parameters was performed by using the FEM analysis. The response of these IRE parameters to the maximum temperature and ablation coverage area (EF > 800 V/m) are presented in Table RSM was used in Minitab to analyze the result in Table Based on the result in Fig. P = 0.020 and P = 0.003 respectively). Meanwhile, only the parameter of input voltage significantly affects the ablation area (P < 0.001).From analysis of variance, the parameter of electrode distance and input voltage has significant effect to the temperature rise in the IRE treatment of bovine liver . Therefore, the error is not significant, since it is less likely that thermal damage occurs in the tissue.Once verified the reliability of the optimization process, the electric field and temperature distributions on liver tissue were calculated using simulations. The graphical representation of the simulation result for the electric field and temperature distribution is displayed in Fig. The electric field distribution between the electrodes seems to be quite homogeneous. However, in Fig. The temperature increase is more substantial in the area between the electrodes, and it follows a pattern similar to the electric field distribution as shown in Fig. This study presented a statistically based  approach to calculate optimal IRE parameter configuration taking into account the thermal effects. Although IRE generally considered as a non-thermal treatment, our result showed that the temperature in bovine liver tissue increased mainly in the range of 37–50 °C and can reach up to 72 °C in the high voltage (3000 V) parameter. Similar to the result from Agnass et al. , IRE canTwo-dimensional models were built instead of 3D models for the sake computational speed and based on the assumption that the electric and temperature distributions are symmetric. The models simulate two needle electrodes in the real IRE setup and still considered a good approximation for the planning of IRE treatments . In addiThe models were validated based on experimental measurements of temperature. Validation was successfully achieved, where a great part of the experiments presented percentage errors lower than the assumed threshold of 10%. However, some measurements presented relatively high errors. A possible source of error might be related to the insertion of the temperature probes in a wrong place. The bovine liver tissue has the disadvantage of not being a transparent material. This lack of visual information makes it difficult to make sure that the temperature probes were positioned at the correct locations.y-coordinate of 0 mm represents the point of the vertical line that is aligned with the tip of the electrodes. y-coordinate values between 0 and 10 mm correspond to the portion of the vertical line that is aligned with the active part of the electrodes. As a result, the center measurement point is located at the y-coordinate of 5 mm. This figure shows that the temperature distribution along the vertical line is not constant. By having a small error in the probe location, the temperature measurement can have a different result. Therefore, it is suggested to have a visual feedback for the probe position, such as live image from ultrasound, to ensure that the probe is inserted at the right place.Temperature measurement error due to a wrong probe placement can be illustrated using temperature distribution in Fig. 2 and a maximum temperature of 45.36 °C were the measured responses by inserting two needle electrodes with an active length of 10 mm, separated by 10 mm, and the delivery of 50 pulses with a width of 41.21 µs and amplitude of 3000 V. No thermal damage in bovine liver tissue is expected to occur after using this IRE configuration and, consequently, it could be applied in a clinical context.The approach presented in this study enables the optimization of the IRE parameter using validated models. An optimized combination of IRE parameters for bovine liver was achieved by maximizing the electric field between the electrodes while limiting the maximum temperature in the tissue at 50 °C. As a result, an ablation area of 250.82 mmApart from five different parameters that were studied, there are other parameters that can also be investigated. One of them is the implementation of breaks between pulses. There is evidence that the rest periods between sequences of pulses can contribute to a decrease of the resultant temperature in the tissue , 28. ThiUsing another algorithm, such as nondominated sorting genetic algorithms II (NSGA-II), to calculate the optimal IRE protocol would add valuable information to the optimization process . Conside


This essay applies Karl Polanyi’s concepts of embedding and countermovement to provincial Hungary during and after socialism. Comprehensive state socialist repression in the 1950s was a politics-led disembedding. An economy-led countermovement began in the 1960s, later augmented by elite discourses of civil society. The 1970s and 1980s were decades of socialist embeddedness. Neoliberal configurations after 1990 dislocated both economic and associational life. The illiberal democracy of Viktor Orbán is a more consequential countermovement than the earlier countermovements to state socialism. The argument is illustrated with data from long-term fieldwork in southern Hungary, in the region of the Danube-Tisza Interfluve. Many have viewed Hungary as the paradigmatic core of Central Europe, sometimes conceptualised as a distinctive ‘third region’ between East and West  continues to influence many fields of scholarship. Economic anthropologists hail him as the founder of the ‘substantivist’ school. Polanyi . The ‘do Polanyi . In thisThe global impact of neoliberalism in recent decades invites comparisons with the consequences for Britain and the world of an older liberal ideology in the nineteenth century. Polanyi taught that the very idea of the free market is an illusion, since markets always depend on a range of institutions, first and foremost the state. Subjecting such fundamental goods as land, labour and money to the logic of markets has far-reaching consequences because it disembeds economy from society. According to Polanyi, this was unprecedented in human history prior to the nineteenth century. The victims of the supposedly ‘free’ and ‘self-regulating’ markets respond with countermovements. These range from progressive initiatives to defend workers’ interests  to malignant forms of exclusionary politics.samizdat journal Beszélő. The buzzword that best encapsulated their aspirations was civil society.Polanyi’s conceptual apparatus can be applied outside the market-centric paradigm. He defined a countermovement in abstract terms as ‘a reaction against a dislocation which attacked the fabric of society’ Polanyi , p. 130.et al. Polanyi’s star rose with the advent of neoliberalism, and it was not long before scholars began to apply his insights to the postsocialist victims of ‘shock therapy’ marketisation and propertisation. Less attention was paid to the fate of civil society agendas after 1990, because the triumph of pluralist liberal democracy was long taken for granted. Three decades later, this is no longer the case. Viktor Orbán has had considerable success in imposing a new periodisation, according to which the decisive moment of discontinuity was not the change of system in 1989–1990 but accession to the EU and the formation of his second government in 2010. This representation has much to commend it. Orbán’s decade in power has been characterised in terms of ‘democratic backsliding’ , and despite the distortions of state-controlled media, my premise is that populist politicians such as Orbán possess democratic legitimacy. His success derives from the fact that, well before he began to tilt the playing field to favour his own party, the appeal and organisation of The first decade of socialist rule in Hungary was catastrophic. Political instability culminated in the revolution of 1956 and its brutal repression. Yet just a few years later, in a country in which agriculture was still the most significant branch of the economy, the regime of János Kádár pressed ahead with the mass collectivisation of this sector. In other sectors, the need to correct excessive centralisation was recognised. Economic reforms coincided with an easing of the political climate in the course of the 1960s. A comprehensive package known as the New Economic Mechanism (NEM) was formally introduced in January 1968.kolkhoz and sovkhoz respectively) enjoyed considerable freedom as market actors, including the possibility of establishing ancillary enterprises that did not have to be tightly connected with their core farming activities. Outside the new factory-type divisions of labour, members and employees were incentivised to produce commodities for sale on the plots allocated to them by the socialist institution. They did so by drawing on the traditional household divisions of labour. Continuity with presocialist smallholding agriculture was especially strong in regions where scattered vineyards and orchards made it costly to create large fields. In communities such as that of Tázlár, a mere 120 km southeast of Budapest, most farmers were members of a ‘specialist cooperative’ that supported them with supplies of inputs (fertiliser and fodder) as well as marketing services but did not interfere in the internal labour organisation of the household  needed to farm large fields remained in collective ownership, the new inequalities had little to do with ownership of the means of production. The transformation of rural Hungary in the last decades of socialism can nonetheless be viewed as a form of embourgeoisement, a continuation in socialist conditions of processes incubated in the first half of the century Szelenyi . In PolaMagyar Szocialista Munkáspárt—MSZMP), a new public library and a central Culture House that was accommodated in the town’s former bourgeois kaszinó. Yet Halas too experienced the impact of the New Economic Mechanism. Even before its official introduction in 1968, several large enterprises in the capital were instructed by the central planners to establish subsidiaries in Halas. A new industrial estate was created to make this possible, on the other side of the Budapest–Belgrade railway line. Within a few years, thousands of new jobs were created. Thanks to these central measures, by the mid-1980s over one-third of the town’s active labour force was employed in industry.Tázlár had a population of a little over 2,000 in 1976–1977. It is nowadays well below this figure . Since the mid-2010s, I have focused more on the nearby market town of Kiskunhalas, where a population in excess of 30,000 in presocialist years has declined considerably since 1990. Since Halas (as the name is commonly abbreviated) was the last town before the sensitive state frontier to Tito’s Yugoslavia, it was home to both Soviet and Hungarian garrisons. The main square in front of the secessionist town hall was renamed after Lenin. Halas had a standard range of socialist institutions, including a large state farm, a modernist office complex that served as the headquarters of the Hungarian Socialist Workers Party  basis. He then opened additional premises to take advantage of the construction boom of the late socialist years. His smooth transition to becoming a fully private entrepreneur after 1990 was facilitated not only by the networks he had established inside Hungary but also by high demand from across the border in Yugoslavia.Similar initiatives were taken by the Kiskunhalas State Farm, the town’s largest employer, which had long supported its workforce as petty producers and now afforded its managers generous scope for more autonomous economic action. During the last decade of socialism, numerous enterprises and cooperatives in the commercial and services sectors allowed their members to operate more independently as artisans or shopkeepers, incentivised to increase their own incomes by developing new products and improving customer service. Zoltán Krausz recalls that his path to prosperity as a leading retailer of household goods began in the 1970s as an apprentice with a state ironmonger. In the mid-1980s the company allowed him to open a new store on a semi-private . In the last decade of the Cold War, however, civil society was also cast as a singular unifying movement, to which an entire citizenry was supposed to subscribe. This second model was obviously incompatible with any version of the socialist state. Some activists did not distinguish these two senses and shifted the emphasis according to context. The collectivist sense triumphed in the emotional fervour of 1989. However, it was never clear how ‘we the people’ sentiments would lead to new forms of pluralist democracy rather than to postsocialist forms of populism.The traction of elitist intellectual countermovements is often exaggerated in Western accounts of the demise of Eastern European socialism. If civil society was a codeword for ‘liberal democracy’, populations assented readily on the understanding that the demise of socialism would enable their living standards as well as their freedoms to converge with those prevailing in the capitalist West. But even in the Polish case, where mass mobilisation was greatest, it became clear in the unravelling that took place after 1990 that class protest combined with strong sentiments pertaining to the nation and to religion was the most significant motivating force, not the agendas of intellectuals such as Michnik or Jadwiga Staniszkis Ost .et al. Although the dissidents of the 1980s were concentrated in the major cities , their critique was not altogether without echoes in smaller settlements. It may have gone unnoticed in villages such as Tázlár, but in Halas the reverberations were considerable. This was due to the fact that the town was selected in 1979 for a pioneering sociological investigation by a team of young social scientists. This research was itself a sign of political relaxation; in the 1950s sociology had been entirely suppressed. The Budapest-based researchers did not content themselves with deferential interviews with local power holders. In addition to such official encounters, they convened public meetings at which residents were encouraged to express their opinions openly. Some cadres became alarmed and accused the researchers of inciting young people to critique the entire system. The research was curtailed and Halas was not even named when the research was written up . The more prosperous strata were Protestant. The Calvinist pastor Áron Szilárdy (1837–1922) became a distinguished orientalist and academician in Budapest. The town’s grammar school (gimnázium) was renamed in his honour. In the course of the nineteenth century, however, Protestants were outnumbered by Catholic immigrants who colonised the town’s vast uninhabited hinterland, the puszta. Apart from church associations, many residents also belonged to guild-like bodies and clubs. These were independent of the state and supported by the contributions of their members. Almost all of these bodies were suppressed in the early years of socialism . After a few bureaucratic hurdles, permission was granted. The VIM was experienced by its enthusiastic members as the first stirrings of a free civil society. The association was nominally non-political. It organised summer camps with a focus on literature, including the work of Budapest dissidents. Culturally, László Lukács was a conservative, keen to revive the civis traditions of his town. In the free parliamentary elections of 1990, he campaigned on behalf of the Hungarian Democratic Forum (Magyar Demokrata Fórum—MDF), which won the elections nationwide. Under Lukács’s leadership, the VIM went on to spawn a number of specialised groupings to cultivate local and national heritage. He had the know-how to obtain grants  and to produce newsletters and magazines, in which the terms polgári (bourgeois) and civil were deployed as synonyms to propagate the concept of a civil society to wider audiences.No one was more excited by these developments than László Lukács, aged 16 and a Szabad Demokraták Szövetsége—SZDSZ). Zoltán Tóth had participated personally in their activities since his studies of horticulture in the capital in the 1970s. He was the prime mover in establishing the SZDSZ in Halas, managing to raise support from local entrepreneurs whose interests, he argued, would be promoted by the new party. Tóth and his allies launched a journal which they called Contrast (Ellentét). Articles in 1989–1990 invoked the concept of civil society in much the same way as the pamphlets and flyers of László Lukács. The main difference was that the SZDSZ activists did not attach the same weight to local traditions and culture. In Halas, the SZDSZ out-performed the MDF, both in the parliamentary elections and in the local elections a few months later. Zoltán Tóth was elected as the first postsocialist mayor of the town, at the head of an SZDSZ–Fidesz coalition. It seemed that the prospect of a future-oriented liberal reconstruction of economy and society had triumphed over a conservative appeal to reconstitute the nation on the basis of its cultural heritage. The local elections of October 1990 represented the zenith of the liberal intellectual countermovement in Halas.But the MDF did not triumph in Halas. In the late 1980s other local activists were already in touch with well-known national dissidents, by now organised as the Alliance of Free Democrats  as well as the state farm. The latter enterprise had been struggling for years due to difficulties in marketing its showcase products, following the impact of Mikhail Gorbachev’s anti-alcohol policy. The unskilled were invariably the first to be made redundant. Virtually the entire Roma community, after decades of struggling to adjust to the socialist compulsion to work, now found itself unemployed and dependent on state assistance.vad kapitalizmus).The economic problems of Halas were accentuated by the closure of the Soviet barracks and by the instability created by civil war in nearby Yugoslavia. At the same time, each of these factors opened up plentiful black market economic opportunities. Unemployment, previously non-existent, rose to 12.5% by 1992 and the industrial workforce halved Hann , p. 553.Jakab’s views are shared by other ‘winners’ of the transformation, namely, former socialist managers who managed to build up private businesses after 1990. János Gidai was a young engineer at the Halas subsidiary of Ganz, a Budapest enterprise best known internationally for manufacturing trains. When this enterprise ran into financial difficulties in the late 1980s, the ministerial solution was to privatise and dismember the conglomerate to attract foreign capital. After a short period in British ownership, the core business was sold to an Austrian firm, which (according to János) like most other foreign investors was only interested in mopping up competition. The Halas subsidiary was divided into four units for privatisation. János and 26 colleagues took over compressor production, and soon succeeded in developing new products and opening new markets. But after buying out his partners and almost three decades of expansion, the workforce failed to reach three figures, compared with the 800 who had worked at this enterprise in socialist days. What riled János most when he looked back was that the lack of state support at the time of transition had led to the liquidation of Hungarian firms whose products were superior in quality to those of the foreign competitors who proceeded to take over domestic markets. He deplored the fact that the old system of industrial apprenticeships disappeared with the factories, such that, despite unemployment, it has become difficult to recruit skilled workers in Halas. Nowadays, János said plaintively at the end of our conversation, even college graduates opt for dishwasher jobs in London: they may earn more money that way, but what about their futures, he asked, and the future of their town?Független Kisgazda, Földmunkás-és Polgári Párt, generally abbreviated to Független Kisgazdapárt). In 1994 it returned László Varnai, a socialist (and ex-communist) who had formerly worked as a legal counsel at the state farm. In 1998 it elected a lawyer who had defected from the SZDSZ to Fidesz. Zoltán Tóth served as mayor throughout these years, but he was repeatedly obliged to change his coalition partners. Management of the municipality was hindered by the fact that the local coalitions never coincided with those governing in the capital. The mayor and most of his supporters quit the SZDSZ when they judged that the intellectuals who had founded this party had no interest in the issues that mattered in the provinces, in particular assuring jobs and livelihoods (Hann Városi Polgári Kör—VPK). According to its constitution, VPK members were required to distance themselves from all political parties. Although it never numbered more than a few dozen members, this association remained a significant force in local politics in Halas until 2002.The uncertainties of the first postsocialist decade were reflected in the results of parliamentary elections. In 1990 the Kiskunhalas constituency elected László Horváth, an elderly veterinarian, the candidate of the Independent Smallholders’ Party . During this period, the socialist-led government of Gyula Horn introduced new regulations for civil associations and their funding.civil társadalom) back to the level of a club movement, and the middle classes to the status of mass-consumption citizens with no means to realise their interests!  in 2002 and again in 2006—on the latter occasion by a young politician with a comparable lust for power, Ferenc Gyurcsány. But the welfare promises made by the socialists could not be kept.società civile. Ethnographers have shown that these grassroots associations contributed significantly to the consolidation of Fidesz in these years. Unlike the civic circle established by Zoltán Tóth in Kiskunhalas, whose members were precluded from party political activities, the activists of Orbán’s circles campaigned to present one political party as the true representative of the people, namely, of the entire Hungarian nation , the political temperature in this provincial town in the 2010s remained low. It did not rise significantly during local election campaigns. Even in more keenly fought national campaigns, turnout never rose above two-thirds. The young mayor and the member of parliament are not considered ideologues. They are perceived by my fieldwork interlocutors to be working quietly and pragmatically on behalf of their constituents. Apart from poster campaigns ahead of a referendum, the new one-party system is not omnipresent in everyday life. Those who suggest comparisons with Stalinism in the early 1950s are wide of the mark. While the present system undoubtedly has authoritarian aspects  and has no business subverting the edifice with irresponsible populist rhetoric. It is commonly suggested that one needs to investigate long-term history, not just the socialist experience but also the preceding centuries of economic and socio-cultural backwardness on the margins of Europe, to understand why the Magyars are incapable of implementing what enlightened Westerners have been implementing since at least the eighteenth century.The countermovement to neoliberal marketisation that currently appears so strong throughout provincial Hungary has itself generated powerful reactions. Civil society remains a prominent concept. In the new century it has become ever more important to take account of new forms of transnational connectivity. Since 1990, postsocialist investors driven by calculations of profit have been joined by legions of private philanthropists, the representatives of international aid agencies and civil servants of Western states disbursing know-how to the civilisational newcomers. In these circumstances, especially in the capital cities, novel forms of ‘global civil society’ Kaldor  have emeGlobal civil society for Kaldor  is a mulFidesz representative in the European Parliament. He had a ‘ringside seat’ (Schöpflin realpolitik within the EU that has brought systemic conflict between patriotic Christian conservatives in the East and increasingly punitive cosmopolitan liberals in the West (Schöpflin George Schöpflin  has arguIn Hungary, the coarsening of the political culture sank to new depths following the migration crisis of 2015, when the fence constructed by Orbán along the Serbian border became the prime symbol of national and even civilisational pride. Muslims were now the ‘other’ to be excluded from Christian Europe. The European Commission in Brussels, in the person of Jean-Claude Juncker, and the German government in Berlin, in the person of Angela Merkel, were accused in a daily barrage by government-controlled media of abandoning their duties. The media attacks claimed that these European leaders purported to represent Christian parties, yet in reality, aided and abetted by the Jewish financier George Soros, were pursuing policies that were endangering the civilisational integrity of the continent Bayer .41Focusing on the market town of Kiskunhalas, in this essay I have explored the socialist market economy, liberal civil society and illiberal democracy as three complementary kinds of countermovement. This involved applying Karl Polanyi’s heuristic to the dislocation caused by high socialism. The socialist market economy was, to a large extent, shaped from below, in both agricultural and industrial sectors. Civil society was a reaction to monolithic socialist rule on the part of an oppositional cultural intelligentsia. Illiberal democracy is a more orthodox Polanyian countermovement, a genuinely popular reaction to the ‘market society’ that replaced socialism after 1989. I have argued that the intellectual ideals of a free and democratic civil society as articulated before 1989 foundered thereafter in the face of a political economy that precluded such ideals. Instead, the impact of neoliberalism engendered illiberalism.Fidesz is so deeply entrenched in the Hungarian provinces that it is reasonable to diagnose a new one-party system. The question is how long leaders such as Orbán can maintain their image as populist anti-establishment rebels as they consolidate their hold on power. The political opposition has been hitherto fragmented and ineffectual. Criticism comes, as it did under socialism, primarily from intellectuals, who are nowadays likely to work with urban NGOs. But the new incarnations of an earlier liberal countermovement are stigmatised for undermining the national cause, and it is therefore hard to challenge today’s rulers. Unlike their socialist predecessors, they have electoral legitimacy.The populist politicians who came to exercise a near monopoly of power in the V4 in the 2010s can no longer be classified as a countermovement. They have become the new establishment. The power of urbanus and ruralis. ‘Talking civil society’ (Jensen & Miszlivetz Civil society promotion cannot compensate citizens for the massive disruption of livelihoods and security. This conclusion is valid for many other postsocialist states—certainly for all members of the Visegrád Group. The extreme nature of the Hungarian case derives from the fact that the sense of precarity and relative deprivation is greater in a country where so many households, especially in the countryside and small towns, participated in dynamic processes of accumulation in the last decades of socialism. These citizens have experienced an overwhelming sense of loss of control, for which the governments of the first postsocialist decades are held responsible. Those elites failed to take care of their constituencies, the people. Efforts to promote a civil society have instead functioned to widen social differences, especially the old gulf between Despite the scorn expressed towards Brussels, there is no question of withdrawal from the union that has provided substantial material aid to the region in the last 15 years and done much to buttress the emergence of a domestic capitalist class. But transfers received from Brussels through so-called cohesion policies are insufficient to close the historic gap. Provincial Hungarians do not perceive or experience any convergence with the ‘old EU’. Populist politicians pursue policies that address the claims and complaints of these citizens, emotional as well as material. In the relative economic stability of the 2010s, they were perceived to be telling the truth about power, in particular, its abuse by cosmopolitan elites. Policies that temper the impact of hyper-globalisation by a return to work-based redistribution in the interests of maintaining equality, well-being and the cohesion of the nation as a solidary community, have electoral appeal. But the low turnout at elections suggests that the noisy illiberalism that dominates the public sphere may not in fact have such deep roots in society. Support for illiberal populists is hardest to explain where it can be demonstrated that the commitment to redistribution grounded in need is purely cosmetic, since their economic policies in reality serve the interests of a new class and widen inequalities. But this too is a familiar act of deception in the double movement of Karl Polanyi’s ‘market society’.