Multiple Input and Multiple Output (MIMO) is a promising technology to enable spatial multiplexing and improve throughput in wireless communication networks. To obtain the full benefits of MIMO systems, the Channel State Information (CSI) should be acquired correctly at the transmitter side for optimal beamforming design. The analytical centre-cutting plane method (ACCPM) has shown to be an appealing way to obtain the CSI at the transmitter side. This paper adopts ACCPM to learn down-link CSI in both single-user and multi-user scenarios. In particular, during the learning phase, it uses the null space beamforming vector of the estimated CSI to reduce the power usage, which approaches zero when the learned CSI approaches the optimal solution. Simulation results show our proposed method converges and outperforms previous studies. The effectiveness of the proposed method was corroborated by applying it to the scattering channel and winner II channel models. In Multiple Input and Multiple Output (MIMO) wireless communication systems, with proper transmitter precoding and receiver signal processing, high diversity and multiplexing gains can be achieved to meet the increasing demands of high data rates and low latency applications . To realThe era of 5G and 6G, where the CSI is not required to improve the performance of these systems in some cases , but is In downlink beamforming, the benefits offered by MIMO rely on the degree of availability of CSIT ,20,21,22In MIMO systems, the CSI is estimated at the receiver and fed back to the transmitter for optimal precoding design. The feedback accuracy depends on the degree of CSI quantization and the condition of the feedback channel. In the case of limited feedback, it has been shown that the Analytic Centre Cutting Plane Method (ACCPM) provides better CSI estimation accuracy with fewer feedback values ,31,32. MA novel method for MIMO CSI estimation in single and multiple users is proposed through the use of null space during the learning phase, so the power consumption is significantly reduced during the estimation period.A new mathematical concept is used to determine the common null space of multiple users for their CSI estimation. Thus, the number of feedback bits in multiple user systems is only one, as compared with other research, where the number of bits is equal to the number of users.Multiple orthogonal beamforming vectors are applied to reduce the CSI estimation time. This will reduce the number of feedback bits and hence reduce the error in estimation that leads to lower CSI uncertainty.The methods and algorithms are not verified by only using a randomly generated channel matrix; two standard channel matrices were used—the scatterer channel for a single user and the winner channel for multiple users. The convergence for these two channels was approved.Numerical results show that the proposed method outperforms all existing research.The main contributions of this paper can be summarized as follows:The rest of this paper is organized as follows. Throughout the paper, lightface letters represent scalars and boldface letters represent vectors (lower case) and matrices (upper case). Logdet(X) means the logarithmic determinant of matrix X. m transmitting antennas and n receiving antennas, the channel can be characterized by the following matrix:i and receiving antenna j.In any wireless communication system, the channel plays a key role in determining communication performance. In a MIMO system with Due to its importance for various applications, the MIMO channel has been extensively studied and various MIMO channel models have been proposed, most of which are based on measurements. There are many ways to classify channel models ,34,35,36The physical channel model characterizes the multipath bidirectional wireless environment where the electromagnetic wave propagates between the transmitter and receivers. The model depends on the direction of departure (DoA), the direction of arrival (DoA), the signal amplitude, and the delay of the multipath component (MPC). The physical model is independent of the system bandwidth and antenna configurations, making it suitable for signal reproduction.The analytical model represents the channel impulse response or the transfer function between a pair of transmitter and receiver antennas without considering wave propagation. The combination of these impulse responses can be represented by the channel gain matrix. The main advantage of the analytical model is that it is independent of technology and hardware and can be easily generated and synthesized.The standard model focuses on the development of new radio systems. Various techniques, such as signal processing and multiple access, are incorporated into such models to improve the system’s performance. Different organizations propose several reference models on MIMO channels. The standard model depends on physical and analytical models. Most of the preliminary results in this paper depend on the randomly generated channel matrix. To approve the method, we used two types of standard channels—the scatter channel model for single users and winner II models for multiusers.This model is used for both single and multiusers, but the approach used in this paper assumes a single user. In the MIMO system, the signal follows a multipath where multiple copies of the signal propagate between transmitter and receiver at a different propagation delay and different angle and strength due to the existence of the scattering object . The received rays are added destructively or constructively at the receiver side which gives rise to the channel fluctuation. In this model, the received rays depend on the antenna configuration , the number and location of scattering object(s), the angle of departure, and the angle of arrival of the signals.This model used a multiuser MIMO system where multiple base stations and multiple mobile stations can be assumed with their geometrical location to simulate a spatially defined mMIMO system.The model is a stochastic base approach for channel modeling, with a randomly generated bidirectional radio channel model. This model is independent of the antenna since a different pattern setting and different antenna configuration can be assumed. The statistical distributions determined from the channel measurement can be used to determine the channel parameters. The channel investigation can be obtained by adding the effect of the rays using different parameters such as power, time delay, angle of arrival, angle of departure, etc.ACCPM is used to solve semi-convex or general convex optimization problems  with theAt every iteration, the algorithm calculates the analytical center of the new working set defined and generated in the previous iteration. If this center of analysis is a a wailed solution, the algorithm is terminated, otherwise, the new plane is returned and added to the system. The algorithm finds a solution to the problem as the number of iterations increases, while the working set is shrinking.N-antenna transmitter broadcasts confidential signals K receivers each equipped with kth receiver. Without the loss of generality, an assumption that considers the transmitted signal to be random with unit variance, and zero means that lead to kth receiver becomes:kth receiver.A broadcast MIMO system with multiusers as shown in K = 1) is assumed which results in omitting the second term and reduces PtThe value of With the Note that P-1 is a convex optimization problem and can be solved using the interior point method or other convex optimization tools such as CVX . AdditioThe initial beamforming vector is generated randomly from the complex Gaussian distribution with zero mean and unit variance. The beamforming vector must be updated after each iteration using the following equation:To satisfy the above equation, it is clear that The above method is summarized in the following algorithm. This algorithm was implemented using MATLAB 2021 with a Razer Blade laptop  with 8 GB RAM and a Core i7 processor . It can implemented by any PC that can run MATLAB 2014, such as a PC with 4 GB RAM and a Core 2 Duo processor.Algorithm 1 CSI estimation for single user single beamforming vectorInitialization: Set the maximum allowed error between the estimated and actual CSI βC. using  3.Evaluate H^ using .  4.find the null space vector of   5.Update beamforming vector using .Until According to Algorithm 1, the simulation is terminated when the difference between the estimated covariance channel matrix and the actual covariance channel matrix is small enough or the number of time intervals exceeds the predefined value.To speed up the covariance channel matrix estimation, the receiver also compares the estimated power with the actual power at the same time interval and sends back an additional Combining  and 10)10), the rding to .Algorithm 1 can be slightly adjusted to estimate Note that Equation  generateZ represents the number of the orthogonal vector needed) from a non-orthogonal vector set These beamforming vectors should be orthogonal to generate orthogonal cutting planes so that they can be easily separated by the receiver. Vectors are orthogonal if the dot product between any two vectors is equal to zero. The Gram-Schmidt process ,40 can bOn the receiver side, a comparison is performed to extract The transmitter utilizes these values to estimate the channel covariance matrix. Accordingly, the following convex optimization problem is formulated:The beamforming vectors are updated using the null space of the estimated covariance channel matrix. Since each vector uses different null spaces, the number of vectors used to estimate the matrix is equal to the number of null spaces, which is determined and restricted by the number of antennas at the transmitter and receiver sides.Algorithm 2 CSI estimation using orthogonal beamforming vectorInitialization: Set the maximum allowed error between estimated and actual CSI (Evaluate H^ using .Update beamforming vector using .Repeat  1.  2.Determine the new C. using   3.Evaluate H^ using .  4.Find the null space vectors of   5.Update beamforming vectors using .Until The following algorithm summarizes the procedure to estimate M signals and a comparison is performed at each receiving antenna. The comparison results In this scenario, a single user can be considered as multiple receivers, where each antenna is considered as an independent receiver, and a signal from the transmitter is sent to each antenna. As a result, each column of the channel covariance matrix According to the results from , the traIn this case, the beamforming vector is updated by adding the principal eigenvector of Algorithm 3 CSI estimation using multiple non-orthogonal vectorsInitialization: Set the maximum allowed error between the estimated and actual CSI (Update beamforming vectors by adding the previous vector to the corresponding principle in vector.Repeat  1.  2.Determine the new C. using .  3.Evaluate G^ using .  4.Eigendecomposition   5.Update beamforming vectors by adding the previous vector to the corresponding principle in the vector.Until The simulation of P-4 can be performed with the assistance of Algorithm 3.When there are multiple users (i.e., In this case, the received power at each user is determined by . At eachThe receivers feed back the values of K convex optimization problems in (All users’ CSI is determined separately by solving blems in .K beamforming vectors K null space Then, the Algorithm 1 with multiple optimization problems is applied to the above problem.In this scenario, the transmitter sends a single beamforming vector to all users by finding the common null space of these receivers ,42. Let itten as:y=[h1Hw,hThen, according to the values of Finally, the beamforming vector is updated using the null space of M right singular vectors and The null space of Algorithm 4 CSI estimation for multi-user single beamforming vectorInitialization: Set the maximum allowed error between the estimated and actual CSI (Repeat  1.  2.Determine the new t. using .  3.Evalute H^ using .  4.Eigendecomposition   5.Update beamforming vectors by adding the previous vector to the corresponding principle eigen vector.Until The method can be implemented following the procedure in Algorithm 4.This section provides the simulation results to illustrate the performance of the proposed ACCPM methods. The initial beamforming vectors for all cases are generated randomly from the complex Gaussian distribution with zero mean and unit covariance. Each simulation result is averaged over 500 trials.The actual correlation matrix is determined by When When When When For the purpose of verification of the method, the method is tested using one of the standard channel models: the scattering channel  that is Finally, for a multiple user case with a standard channel, the winner channel model is tested  using P-N represents the number of transmitting antenna in the base station, Where refer to .The paper demonstrates different methods of using ACCPM for channel covariance matrix estimation in various MIMO systems including single user, multiuser, and multi beamforming vectors. The simulation results show that the proposed methods not only converge but also outperform existing benchmark methods. In particular, the use of null space during the learning phase is a better choice to reduce the power consumption. Using ACCPM, the learning is achieved at the transmitter side and requires little feedback from the receiver. Additionally, the transmitter starts learning without any information about the estimated CSI. The effectiveness of the method was corroborated by applying it to standard channel models. The method can be considered in terms of artificial intelligent as a simple classification problem or can be solved using prediction methods. This can be considered for future studies, where it can be improved further by reducing the feedback time intervals. Additionally, it could be tested for all channel models and the probabilistic mathematical concepts could be considered for future work.


The problem of the interaction of waves with a serpent-type wave energy converter was investigated, and a novel 3D analytical solution was derived. The optimal parameters of the energy converter were derived for the first time from wave energy principles. The results show that the wave power captured by the device increases with increasing wave lengths until a maximum and then decreases. The efficiency increases with decreasing stiffness of the device in shallow and intermediate waters. In deep water, the efficiency increases with increasing stiffness until a local maximum and then decreases. Moreover, the efficiency increases with the increasing mass of the device in shallow and intermediate waters. In deep water, the efficiency of a converter decreases with the increasing mass of the device. An original approach to determine the optimal parameters of a device for given wave conditions was derived. The derived analytical formula shows that the top efficiency level of power capturing cannot exceed 50%. This is the theoretical maximum for this type of converter. The power take-off optimization analysis also identifies the spectrum of wave conditions for which the efficiency of the generator is close to the maximum. A series of laboratory experiments were conducted in a hydraulic laboratory to verify the model. The comparisons of the recorded data with the analytical solution show a very good agreement. It is a substantial resource that is not being utilized into electrical energy. The perspective of rising global energy consumption and the unwanted side-effects of fossil fuels energy production brings even more importance to the issue of acquiring wave energy. Despite the current low level of conversion, the studies regarding this topic and propositions of mechanisms for capturing this resource are quite extensive. This suggests that the optimal technology is yet to be determined and potential for advancements and innovations in this field exists.Potential energy of ocean waves is estimated to be about 2.96 TW2 more than a century ago. A serious revoked interest in the subject came in the seventies3 in the midst of the international oil crisis when a promising wave oscillating mechanism was presented by Salter in 19744. These circumstances brought more funding and scientific interest to the topic of wave energy utilization. After not identifying a viable, affordable solution for wave energy conversion, the interest and investments declined around 1984. Only in the last twenty years, with the growing concern of climate change the technology have again gained more attention. Correspondingly, a surge in research and development of new devices has ensued, especially after the commercial success of other renewable energy technologies, e.g. wind turbines.First investigations into the possibility of utilizing wave energy were conducted by StahlA variety of new proposed designs and prototypes led to attempts to categorize and organize wave energy converters (WECs). Different types of classifications emerged. Oscillating water column (OWC), overtopping devices, diaphragm pressure differential devices and oscillating body devices are groups that sort converters depending on their principle of capturing wave energy.5. Overtopping devices collect water from overtopping waves to a tank from which the potential energy of that water is utilized. Please see the example of a working device presented by Margheritini et al.6. Pressure differential devices are submerged constructions that use the changing pressure, due to the wave’s motion, above them to pump fluid inside the devices. A large class of devices, the oscillating body converters, extract energy from waves induced oscillations of parts of their bodies. Pelamis7 and Oyster8 are more commonly known oscillating body converters.Oscillating water columns work by having an air chamber above water from which wave oscillations are pushing air in and out through a turbine. An example of an OWC device is described by Gish9, the horizontal surge motion10, the tilting pitch motion11, or some combination of those.Another main classification separates converters by the type of mode of motion of the working surface of devices relative to the water surface. For the most efficient wave energy extraction, the working surface of a wave energy converter moves in the vertical heave motion12, and for a general review of various working wave energy converters, see Joubert et al.13.Devices can still be more precisely categorized by specifying the type of location where they operate: offshore, near-shore and shoreline; or their orientation regarding the incoming waves: terminators, attenuators and point absorbers. A more complete description of current technological classifications is provided by Drew et al.14. This is most commonly attributed to the high cost of maintenance and the cost of the device itself15. One solution is to develop an optimal WEC design for effective and reliable energy production, so it can be easily reproduced and scaled. Another arising idea regarding solving this issue is to justify the cost by adding additional functions to devices or using them in highly specific settings: combining breakwater and coastal protection with wave energy conversion17, combining wave energy devices with wind energy devices18, powering desalination plants19, fishing farms or offshore rigs20.Despite various proposed devices, the main problem concerning the implementation of this technology remains the same. The Levelized Cost of Energy (LCOE) from the wave energy conversion exceeds several times that of other, more developed renewable technologies21) or partially-submerged plates installed on a floating tube perpendicular to the shoreline .In this study, a new concept of an efficient 3D WEC that is capable of subtracting energy from the entire water column is introduced and investigated. The device is a terminator, an oscillating body-type WEC, which captures energy from the wave motion. The converter is planned as a row of independently moving plates that oscillate in the horizontal direction. It is intended to operate near-shore with plates standing from the sea bottom to the sea surface. Wave energy is retrieved by the mechanism of capturing and converting the mechanical energy of the oscillating plates. The proposed concept is original and substantially differs from a single porous-plate WEC installed at a seawall  is the plate displacement.The situation considered in the study is the wave interaction with a device constructed as a row of plates, which can move horizontally in piston-type oscillations. The plates are connected by flexible-type joinings. The situation is shown schematically in Fig. A and frequency ω. These are reasonable and widely recognized approximations applied in the analyses of real working WECs.The solution for the wave-device interactions is derived on the basis of the potential wave theory, which is a standard theory used in the modelling of wave energy converters. This approach has some limitations and requires the following assumptions. The fluid is inviscid and incompressible, and the fluid motion is irrotational. Moreover, it is assumed that the sea bottom is impervious, and that the excitation is provided by incident waves of small amplitude V , may be computed from the velocity potential Φ .) is the two-dimensional vector differential operatorAccording to these assumptions, the velocity vector, The fluid motion is governed by the continuity equationThe kinematic boundary condition,The dynamic boundary condition,x, y, t) is the free surface oscillation, g = 9.81 m/s2 is the acceleration due to gravity, and h is the water depth.The kinematic boundary at the wave energy converter,28.Furthermore, the velocity potential must satisfy the boundary conditions at infinity and initial conditions, the Taylor series expansion technique and the perturbation method are applied. The free-surface kinematic and dynamic boundary conditions and the kinematic boundary condition at the wave energy converter were expanded into Taylor series about a mean position29.To obtain an analytical solution of the boundary-value problem and to avoid problems arising from an unknown location of the free surface η The application of Eq.  to 1) r r1) resuBy assuming perturbation proceduren, n = 0, 1,… is of order of (Ak)n in which A is the characteristic wave amplitude, k is the characteristic wave number and Ak = ε ≤ 1.All the following analysis will regard the first order solution of the perturbation method and for the purpose of clarity, the left subscript will be left out.x,y,z) is the spatial velocity potential.To further break down the solution, the spatial velocity potential is introduced, such thatI, reflected ϕ1 and transmitted ϕ2 spatial velocity potentials may be written in the following formThe incident ϕ.j = {– ik, α2, α3, …}, where k is the wavenumber.Rj can be derived by multiplying the wave energy converter’s kinematic boundary condition 35\documeC as added dampingFD10 as the force due to diffracted wavesThe Eq.  may be wCp. The selection of an optimal Cp is a complex problem. For small values of Cp, the wave energy radiates from a conversion system in the form of reflected and transmitted waves, and the efficiency of a wave energy converter is low. When the values of Cp are large, the wave energy radiate from a system in the form of reflected waves, and the efficiency of a wave energy converter is again low. These indicate that there is an optimal value of Cp. The optimal value of Cp may be derived analytically by determining the minimum of wave energy radiated from a wave energy converterThe motion of a wave energy converter depends on the value of the coefficient CpAfter some algebra one obtainsPI is calculated fromPc by the converter isFor further evaluation of a wave energy converter efficiency, the incident wave power Cp and the energy radiated from a converter, E =|R1|2 +|T1|2, are the significant factors in the modelling of a wave energy converter. The magnitude of the wave energy radiated by the converter is calculated for a wide range of the parameters of the model. The results are plotted versus the dimensionless value of Cp in Fig. kh = 0.5, 1.0, 1.5 and 2.0 which corresponds with wave periods T = 13.20, 7.27, 5.44 and 4.60 s.The value of Cp for a small range of lower Cp values and then increases. This is due to the fact that both very low and very high values of power take-off coefficient would stop the energy absorption. The extreme values would either lead to a standstill of the device or a stop in capturing energy. The results clearly show a distinctive minimum in radiated wave energy in a range of low Cp values. For this value, wave reflection and transmission are the lowest. This means that the energy captured by the device is the highest possible, and the value of Cp is optimal for the purpose of energy conversion. To the best of the authors’ knowledge, the optimal value of Cp from a radiated wave energy has been derived with this approach for the first time.The results in Fig. Cp were applied to determine the power efficiency of the wave energy converter operating with such optimal power take-off settings. Figure kh. The results are presented for representative values of kh ∈ (0, 3 > , corresponding to the spectrum of wave periods T < 3.67 s, to cover a broad range of wave and water conditions, from shallow to deep waters. The values of stiffness and mass parameters were chosen to comprise typical material parameters as well as extreme values of real engineering material that could be used in the construction processes.The derived analytical solution and the formula for the optimal The results in Figs. The derived analytical solution also showed that the top efficiency level of the power capturing mechanism cannot exceed 50%. It is the theoretical maximum for this type of converter for any condition and parameters.10|/A, and are plotted against the dimensionless wave number, kh ∈ (0, 3 > , corresponding to the spectrum of wave periods T < 3.67 s, to cover a wide range of wave conditions.The model was then applied to determine the displacement of the converter plate. The displacement of the plate was also calculated for the wave energy converter operating with optimal settings for the value of the power take-off coefficient applied in the calculations. The results are shown in Fig. The results in Fig. F10|/ρgAh and |F50|/ρgAh2, and are plotted versus the dimensionless wave number kh = (0, 3 > and spectrum of wave periods T < 3.67 s.The derived analytical solution was applied to determine the hydrodynamic wave loads on the wave energy converter. The loads were calculated by integrating the dynamic pressure acting on the working plate of the converter. The calculations were conducted for the wave energy converter operating with the optimal value of the power take-off coefficient. The results are shown in Fig. The results in Fig. 30. A wave gauge was installed 4 m from the plate system to measure free-surface oscillations. The measurement sampling frequency was 50 Hz. The setup is presented in Fig. h = 0.6 m and h = 0.4 m.Laboratory experiments were conducted in the wave flume of the Institute of Hydro-Engineering of the Polish Academy of Sciences in Gdańsk. The wave flume had 64 m in length, 0.6 m in width and 1.4 m in height. A horizontal steel plate of size 0.6 × 1.4 m was modelling the converter’s oscillations and was installed at one end of the flume. A porous wave absorber of a 1:7 slope was located at the other end of the flume. Wave reflection from the absorber is below 5% for waves considered in typical laboratory experimentsFirst, preliminary experiments were conducted to establish the accuracy of the measurement system. Then, a set of 5 regular waves of lengths from 1.6 to 7.2 m and wave periods from 2.15 to 5.03 s were generated for each water depth setting. For these parameters, the plate displacement and free-surface oscillations were recorded. Wave parameters and recorded data for each experiment are presented in Table To obtain wave parameters, the Fourier analysis was applied. Then, the transmitted wave amplitudes were calculated in the model for the input data from the experiments. The outcome was compared with the experimental results. The transmitted wave amplitudes are presented alongside the regular wave amplitude from the wave flume in Figs. h = 0.6 m and does not recur in the second set of results. It can be concluded that the theoretical model allows to predict the laboratory data with satisfying accuracy.The plots illustrate that the theoretical results calculated by the model are in good agreement with the experimental data. The difference between the theoretical results and experimental data is usually below 5%. The discrepancy occurs only for the shortest wave and water depth 31. These relations provide forces from transmitted wave properties. Although it seems to be partial, this type of validation of the model is widely recognized and sufficient from a scientific and an engineering point of view. In fact, the relationship between wave kinematics and wave forces has already been successfully validated in previous works32.The radiated waves predicted by the analytical model were compared with experimental results and match waves recorded in laboratory experiments with good accuracies. The validation process chosen in the study relies on the Haskind relationsThe problem of wave interaction with a serpent-type wave energy converter was investigated. An original 3D analytical model for wave interaction with a wave energy converter was derived, including the derivation of equations governing the motion of the energy device and the efficiency of a power take-off system. On this theoretical basis, among others, the displacement of the plate of the converter, wave loads on the converter, and the optimal wave energy conversion model were analytically derived.The analysis shows a strict relation between the basic parameters of the model and the wave energy recoverability. An original analytical formula for the conditions of the optimal operation of the proposed wave energy converter was derived, and a novel analytical formula was obtained to determine the efficiency of a power take-off system. The effect of the values of mass and stiffness of the converter on hydrodynamic forces and converter kinematics were presented.The results obtained from the analytical formula derived for the efficiency of a wave energy converter are consistent with the original analysis conducted by applying a formula obtained for wave energy. The results show that the wave power captured by the converter increases with increasing wave length until a maximum and then decreases. This is observed basically for the whole spectrum of wave conditions. A more complex effect on the efficiency of a wave energy converter possesses the parameters of a wave energy converter. The results show that the efficiency of a wave energy converter increases with decreasing the stiffness of a system in shallow and intermediate waters. In deep water, the efficiency of a wave energy converter increases with increasing stiffness until a local maximum and then decreases.Moreover, the results show that the efficiency of a wave energy converter increases with increasing the mass of a system in shallow and intermediate waters. In deep water, the efficiency of a wave energy converter decreases with the increasing mass of a system. The values of the optimal stiffness and mass of the device for given wave conditions were determined. The derived analytical formula shows that the top efficiency level of power capturing cannot exceed 50%. This is the theoretical maximum for this type of converter for any condition and parameters. The power take-off optimization analysis also identifies the spectrum of wave conditions for which the efficiency of the generator is close to the maximum.Laboratory experiments were conducted in a wave flume to verify the derived analytical model. Radiated waves obtained for different wave lengths and different water depths were compared with theoretical results. A fairly good agreement between the theoretical results and experimental data is observed. According to widely validated and recognized Haskind relations, this outcome verify also forces obtained by the derived model.The studies indicate that the proposed converter may also be applied for coastal protection by capturing part of wave energy and reducing wave intensity in the coastal zone. Moreover, adding additional functions to the device or using them in highly settings: combining the device with breakwater functions, with wind energy devices or powering desalination plants, fishing farms, or offshore rigs may justify its construction.


We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink. For instance, in quantum mechanics the Wigner transform takes an arbitrary probability amplitude x-space  to which we make correspond subsets in phase space; these are obtained using the notion of polar dually from convex geometry. Our constructions are somewhat related to the so-called “Pauli reconstruction problem”, which can be formulated in the Wigner formalism as follows: knowing the probability densities uniquelyare waves! Quantum mechanics is a probabilistic theory from which we can extract consequences on the statistical behavior of many measurements performed on a great number of equally prepared systems. The real issue is the calculation of probabilities, and the objects used to study them  The uncertainty inequalities are statements about the variances and covariances of the random variables corresponding to independently measured position and momentum in an ensemble of equally prepared particles. We emphasize that they are neither a statement about the measure of both momentum and position of a single particle nor an effect of the interaction with a measurement device as early interpretations of the Heisenberg inequalities seemed to suggest  have in common that they associate to a ve shown , 20 thatpointillisme, that is the view that mathematical points make sense in physics. We totally agree with Butterfield’s views and assume in this paper that the basic elements of configuration space  are infinitesimal regions with non-zero volume. Indeed, in practice we can never experimentally determine a point in physical space with absolute precision; as Gazeau [Nothing is mathematically exact from the phsical point of view.In fact the notion of point-like particle is a mathematical abstraction, which we can (in principle) approximate with arbitrary accuracy. However, these regions cannot be made arbitrarily small, because the uncertainty principle would then lead to violations of special relativity (at least for massive particles) since in the limit quantum phase space itself is ambiguous, to say the least; the aim of this paper is to propose a substitute, which is a collection of fiber bundles. The simplest of these is the “canonical bundle”X centered at In two brilliant publications , 9 Jerems Gazeau  humorousoyal (in ), even tmentclass2pt{minimn of degrees of freedom, the properties of such “quantum state” and to relate them to the theory of Gaussian wavepackets; our study will unveil unexpected and beautiful geometric properties of quantum mechanics.The aim of the present paper is to study, for an arbitrary number X cannot be localized in the momentum representation in a set smaller than its polar dual x, y, z) and by the vector of conjugate momenta t. This extends to many particle systems by introducing the generalized position and momentum vectors x, p). This way of writing things explicitly singles out the two subspaces n-dimensional linear subspaces of Lagrangian planes which have the property that the canonical symplectic form on pure quantum state. Admittedly, this definition of a quantum state is rather abstract. The reason will become clear to the reader in the course of this article, but there is a rather immediate  motivation. It turns out that the Cartesian product quantum blob, that is the image of a phase space ball with radius We introduced in  the geommentclass2pt{minimlsewhere , 17 a qumentclasspt{minimax axis, in which case the classical phase space is just the x, p plane. The pseudo quantum phase space consists of parallelograms x, p plane, x-axis and p-axis this condition becomes Here is a basic example. Suppose that the configuration space is the geometry . ChoosinIn Sect. n degrees of freedom will in general be written The configuration space of a system with J is the standard symplectic matrix:S of A, B, C, D satisfy the sets of equivalent conditionsThe standard symplectic form ns imply  that a rRecall  that them is defined modulo 4 and corresponds to a choice of the argument of the determinant In the last line of this table the integer refer to . The nonz and n: a linear subspace Lagrangian plane if When n-vector n non-conjugate coordinates. For instance the set of all The most typical  example of Lagrangian planes is given by the “coordinate Lagrangian planes”. They are obtained by picking out in the 2A is a Lagrangian plane: it has dimension n andA is symmetric. More generally, a subspace A and B are real Lagrangian Grassmannian and is denoted by The subspaces consisting of all n-dimensional subspaces for which There is an alternative way of interpreting Lagrangian planes as subspaces of n. A symplectic automorphism U is called a symplecticrotation if U(n) the group of all symplectic rotations; one shows [U(n) is the image in A and B satisfy the conditionsIn the phase plane ne shows  that U..\documenLet U(n) of The subgroup U be a linear mapping taking This is proven as follows : let \doO(n) of U(n) consisting of all symplectic matricesU(n)/O(n) with the Lagrangian Grassmannian  in the result above. For instance, in the case Notice that we cannot replace It follows from Proposition The following property is useful when considering phase space shifts of the origin:Every phase space point The case ess, see  for an eX by Let us identify the position space ellipsoidn-dimensional ellipsoid centered at Let A is a symmetric positive definite matrix. Clearly, Q, and R(Q) is also a quantum blob. It is sufficient to assume entclass1pt{minimaThe quantum blob described in the result above is not unique. For instance there exist infinitely many quantum blobs X be a convex body in configuration space X contains 0 in its interior. This is the case if, for instance, X is symmetric: polar dual of X is the subsetWe begin by briefly recalling the usual notion of polar duality from convex geometry (we are following our presentation in ); for thIn  we prove(i) Let resp. resp. (ii) Let A be a real invertible and symmetric X is “small” since X and Blaschke–Santaló inequality: assume that X is a symmetric body; then there exists [c is  related ; these se exists  \documen/n! (see ) for a dLet now Let Let The following table summarizes the main properties of Lagrangian polar duality:The following characteristic property of quantum blobs is also useful:Let n-dimensional ellipsoids Q is represented by the inequality G in block matrix form G:We have to show that nditions , taking  by (see )\documenThe definition of quantum states we are giving here generalizes the Definition 3 in .Centered case)Let Let  The symplectic actionfined by  is transfined by  is also (i) it is sufficient to show that there exists A such that ellipsoid X is (ii) is proven in a similar way. To prove part uch that  holds. LO(n) of U(n) consisting of all matrices Proposition The quotient O(n) is the largest subgroup of O(n) is the stabilizer (or isotropy subgroup) of the action of O(n) is a closed subgroup it follows from the theory of homogeneous spaces that we have the identificationLet us state things in a more precise way. We first note that the “orthogonal symplectic group” r bundle \documentO(n). On the analytical level J plays the role of a Fourier transform.The complex structure rotation In this section we identify a subset of X be a convex body in any Euclidean space Xis the unique ellipsoid inwith minimum volume containingX and the John ellipsoid is the unique ellipsoid inwith maximum volume contained inX. If A is an invertible linear mapping thenX is a centrally symmetric convex body, then There is a vast literature on the Löwner and John ellipsoids of a convex body; a classical reference is . Let X bng sense :37\documThe John ellipsoid of B are real The inclusionRecall  that theS. This property is instrumental in proving the symplectic covariance of Weyl quantization, and implies that the metaplectic group acts transitively on the Gaussian wavepackets we define below.A crucial fact is that the Wigner transform enjoys the property of symplectic covariance , 14, thaPhys. Reps. paper [A and B are real symmetric A positive definite and Following our work in  we intros. paper  with F. ions see ). It turt}T^(z0) , 21. In G is the positive definite symmetric and symplectic We will denote by ne shows , 11, usi(i) There is a bijective correspondence fined by  and 49)\documentQ is the quantum blob Q we associate the function S. It follows that we haveA and B. (ii). see [(i) In view of the discussion above the Wigner transform associates to  formula ,\documenii). see . \documeFor a detailed study of the correspondence B(n) see , 17.X is the ball U be a symplectic rotation and define a Lagrangian frame U takes this set to same John ellipsoid, and to both states thus corresponds the fiducial Gaussian wavepacket X is the ellipsoidU(n) is a group the relation Consider first the very simple case where property . The stang again  the John formula  that we There is a canonical identificationQ is also the John ellipsoid of the stateG, hence Let nsion of  to formunsion of  is straiX or X is an ellipsoid, then the centroid coincides with its center in the usual sense. To define the polar dual of unique interior point X  such that the polar dual minimal for all possible interior points X is an ellipsoid. We note that the practical determination of the Santaló point is in general difficult and one has to use ad hoc methods in each particular case. See [So far we have been considering Lagrangian products of ellipsoids. The next— and fundamental!—step would be to generalize our constructions to products entclass1pt{minimabe the canonical Lagrangian frame and Having in mind that the polar dual is calculated with respect to the Santaló point (not the centroid!) we can define the associated canonical Lagrangian quantum state exactly as follows let ussed in  followinentclass1pt{minimaussed in  All thes


A variety of robot-assisted surgical systems have been proposed to improve the precision of eye surgery. Evaluation of these systems has typically relied on benchtop experiments with artificial or enucleated eyes. However, this does not properly account for the types of head motion that are common among patients undergoing eye surgery, which a clinical robotic system will encounter. In vivo experiments are clinically realistic, but they are risky and thus require the robotic system to be at a sufficiently mature state of development. In this paper, we describe a low-cost device that enables an artificial or enucleated eye to be mounted to standard swim goggles worn by a human volunteer to enable more realistic evaluation of eye-surgery robots after benchtop studies and prior to in vivo studies. The mounted eye can rotate about its center, with a rotational stiffness matching that of an anesthetized patient’s eye. We describe surgeon feedback and technical analyses to verify that various aspects of the design are sufficient for simulating a patient’s eye during surgery. However, to date, little attention has been paid to patient head motion, which is frequent among patients undergoing eye surgery under monitored anesthesia3, also known as conscious sedation, which makes a patient calm and somewhat sleepy but still often awake. Head motion in this state is due to factors such as breathing, talking, snoring, and other voluntary and involuntary motions of the patient. Brogan et al.4 measured the motion of 12 patients’ heads during cataract surgery and found that, over the course of a procedure, head drift was 2–7 mm medially, 2–4 mm laterally, 1–5 mm superiorly, and 1–4 mm inferiorly. In one robot-assisted in vivo study in humans, there was difficulty in initiating subretinal injection in a patient due to head drift5. Additionally, 16% of patients snore under monitored anesthesia, and half of those have sudden head movement during surgery6. Sudden movements can be unpredictable in general. Movement must be compensated for by the surgeon, to the best of their ability, to avoid complications.A variety of robot-assisted surgical systems have been proposed to improve the precision of eye surgery7, using visual-servoing techniques8, or moving the headrest to counteract patient movement9. Others have pursued passive approaches to motion compensation, which are not mutually exclusive with active approaches. Some passive approaches include immobilizing the patient’s head12, mounting the robot to the patient’s head13, and forming a mechanical connection between the robot and the eye15.Any clinically deployed robotic system for eye surgery will have to cope with patient motion. Active compensation  can involve sensing the force between the surgical instrument and the eye7. The robotic device proposed by Natalius et al.9 could potentially simulate patient motion , since it was designed to have the range of motion to accommodate for the patient head motion measured by Brogan et al.4. Any method of simulating patient motion will provide a more challenging evaluation of a robotic system than using a stationary eye, although artificially generating truly accurate physiological motion is challenging. In addition, the performance of head-mounted-robot systems and head-immobilization systems is dependent on the geometry of the patient’s head and the compliance of the soft tissues, which will not be captured by a human-head replica  or a live animal . In vivo studies are more clinically realistic, but require the robotic system to be at a mature state of development, as evidenced by the relatively small number of groups conducting in vivo studies17.Benchtop experiments with artificial or enucleated  eyes, which are typical in the development of robotic systems, rarely capture the effect of patient motion. Patient motion was simulated using a linear piezoelectric actuator that generated one-dimensional step motions by Ebrahimi et al.Another aspect of eye surgery that is often neglected when robotic systems are evaluated using artificial or enucleated eyes is the tendency of eyes to rotate due to applied forces. During vitreoretinal surgery, trocar cannulas are embedded into the sclera, which act as entryways for surgical instruments. Forces exerted on the cannulas by surgical instruments may cause the eye to rotate in its orbit. Intentional rotation of the eye enables the surgeon to view different parts of the retina under the surgical microscope, but unintentional rotation of the eye due to inadvertent forces may negatively impact surgical precision. By neglecting to incorporate a way for an eye to rotate during the evaluation of a surgical robot, the eye will remain in the same orientation, whether or not there are inadvertent forces exerted on the cannulas by the surgical instruments. This may result in unrealistically high performance measures on evaluation tasks.18, may be particularly important for robotic systems using force-sensing instruments; in a clinically realistic experiment, the forces between the trocar cannulas and the surgical instruments vary based on the orientation of the eye.Furthermore, the extraocular muscles and other tissues surrounding the human eye result in elastic behavior when the eye is rotated. During eye surgery, if the eye is rotated away from its primary  orientation, it experiences a restoring torque that tends to return it back to its primary orientation. An eye model that incorporates this elastic behavior will more realistically simulate eye surgery. The relationship between restoring torque and orientation, which can be approximated as a constant rotational stiffness21. These artificial eyes can be rotated, but they do not simulate the rotational stiffness of the eye. Two groups have created gimbal mechanisms to simulate eye rotation, but there is no mention of rotational stiffness23. Mohammadi et al.24 developed an eye-fixation device for surgical practice on enucleated eyes that enables the eye to rotate about its center. This system uses a vacuum pump to hold the eye in place inside of a cup. The cup rests on a low-friction surface, which enables it to be rotated easily. A weight is suspended from the cup, creating a pendulum, which simulates the rotational stiffness of an eye.Several groups have developed devices that enable artificial or enucleated eyes to be rotated. Some of these devices involve an artificial eye that rests in a bowl-shaped depression or socketIn this paper, we propose a device that mounts an artificial or enucleated eye to a set of standard swim goggles, which can be worn by a human volunteer during practice with, and evaluation of, eye-surgery robots Fig. . Swim go24 in that it uses a pendulum design to simulate the rotational stiffness of an eye. However, our device does not require a mass to hang below the bowl, which makes our device compact enough to be mounted to a set of goggles. In Mohammadi et al.24, neither the magnitude of the device’s rotational stiffness nor its relationship to that of a human eye is discussed. We review literature on the rotational stiffness of an anesthetized human’s eye, and present the design process for our device to achieve this stiffness. Finally, using a vacuum pump to hold the eye in place, which could potentially distort the shape of the eye, is not necessary with our device. We use adhesive methods to attach the posterior surface of the eye to our device. The features and limitations of existing eye-mounting devices, as well as those of our proposed device, are provided in Table Our device is similar to that of Mohammadi et al.22 and Mohammadi et al.24), but they do not simulate patient motion. Lack of eye motion might make it more difficult to translate skills obtained during training to performance in clinical practice.Although our primary motivation is improved evaluation of robotic eye-surgery systems, many of the benefits of our proposed device also apply to manual practice in wet labs. Many eye-fixation devices have been developed for this application , will fit inside of the bowl.Based on the above values, we selected an inner bowl diameter of 32 mm; this will accommodate the 99.85th percentile of adult-pig eyes and also leave room for a layer of masking tape, which is approximately 0.15 mm thick. Because the measurements of pig eyes were substantially larger than the human-eye measurements provided by Augusteyn et al.18. Taking the radius of the eye to be 12 mm, this range of forces results in torques ranging from 35 to 47 mN mm per degree of rotation. Converting from degrees to radians, this corresponds to torsional spring constants in the range 2.0–2.7 N mm/rad.The extraocular muscles and other tissues surrounding the human eye cause viscoelastic behavior when the eye is rotated during surgery. The forces between the surgical instruments and the trocar cannulas are directly related to the orientation of the eye through the eye’s rotational stiffness, so an eye model that has a rotational stiffness similar to the eye of a patient undergoing eye surgery will serve as a more realistic tool for simulating eye surgery. Patients undergoing retinal surgery typically receive a nerve-block anesthetic. Although we do not have measurements for the rotational stiffness of an eye that is affected by a nerve block, we estimate that it will be similar to that of a patient under deep general anesthesia. The force required to rotate the eye of a patient under deep-surgical-plane anesthesia varies approximately linearly with the angle of rotation away from the eye’s primary orientation , and has been measured to be 0.3–0.4 gf  per degree of rotation24, we use a pendulum model to approximate the rotational stiffness of an anesthetized eye. Although the gravity-induced torque of a pendulum varies nonlinearly with angular displacement, the torque–displacement relationship approximates the constant stiffness of an anesthetized patient’s eye for the range of eye rotations typical of retinal surgery  of the bowl can be modeled as a point mass at the location of the center of mass (CoM), which is located at a distance K (units N m/rad), whereFigure entclass1pt{minimaK of the pendulum with the constants reported in the literature. To compare the true torque–rotation relationships of the pendulum and the eye, which are slightly nonlinear, we directly compare the mean torques measured by Rosenbaum and Myer18 and the torques that result from the pendulum model . This comparison is illustrated in Fig. Using the small-angle approximation enables us to compare the approximated spring stiffness K (units N m/rad) using the bowl, its outer radius R (units m) must ber (units m) is the inner radius of the bowl. The details of this derivation are provided in “After a preliminary analysis, we determined that the brim around the edge of the bowl has a negligible effect on the bowl’s rotational stiffness, so we initially design a bowl without a brim, and later verify that the addition of the brim does not have a considerable effect. To approximate a rotational stiffness K, we considered the possibility of placing lightweight inserts  into the bowl to accommodate eyes with diameters smaller than 32 mm. Because the density of such inserts could be much lower than the density of the solid bowl, the addition of an insert would not significantly increase the rotational stiffness of the device. We chose the rotational stiffness of the device to be 2.0 N mm/rad, which is the lower limit of the range of stiffness values described by Rosenbaum and Myer18 and leaves open the possibility of adding a lightweight insert without causing the device to exceed the upper limit of clinically realistic stiffness values . For the selected value of When choosing the rotational stiffness of the device, In this paper, we use 3D-printed PLA due to its low cost and ease of manufacturing. Although a bowl with sing Eq. , we findEquipped with the final bowl design (other than the brim), we are able to finalize the locations of the BTUs, and the design of the BTU holder.a from the center of the triangle  base possible, then the inequality  should ba for which the lowest point on the bowl is at the same height as the lowest points of the BTUs. This configuration is shown in Fig. To limit the height of the device, we consider the value of BTU Fig. c, we finReferencing Fig. into Eq.  and 7),,\documena numerically, choosing the solution in the open interval For our BTUs (Bosch Rexroth R053010810), we measured The prototype BTU holder, which places the BTUs in the poses described above, is shown in Fig. To prevent the bowl from losing contact with the BTUs, we add a brim around the edge of the bowl to act as a rotational hard stop. Without the brim, the bowl may lose contact with a BTU if it is rotated by more than .2∘ Fig. b. We desher Fig. c. The brTo verify that the rotational stiffness and the range of possible eye orientations are realistic, and that the bowl is not inadvertently lifted off of the BTUs when a mounted eye is rotated, three ophthalmic surgeons used surgical instruments to rotate an enucleated pig eye that was mounted in the final prototype. Trocar cannulas were placed in the eye approximately 3 mm from the limbus at the 4 o’clock and 8 o’clock positions (from the surgeon’s perspective). Instruments were then inserted into the cannulas and used to rotate the eye. The bowl prototype used for initial evaluations had a rough lower surface and did not include the brim. One surgeon indicated that they disliked the feeling due to the rough surface of the bowl, which prompted us to fabricate the bowl as described in “We assume that the volunteer lies supine, with their head nominally oriented such that the bottom surface of the BTU holder is horizontal. For the BTU poses determined above, we can calculate the smallest angle of rotation of the volunteer’s head, The details of this derivation are provided in “The offset between the mounted eye and the eye of the volunteer wearing the goggles is expected to result in exaggerated movement  of the mounted eye. Fortunately, this will lead to conservative experiments when evaluating robotic systems, but we would ultimately like the movement of the mounted eye to be as realistic as possible. We can quantify the effect of the offset using a simple model for the motion of the volunteer’s head Fig. . We assuWe estimate To estimate 28; these ranges are 29, which combine measurements from males and females, and we found that the measurements are in the approximate range of 25. Using digital calipers, we measured To quantify For each sex, we numerically search the parameter space of 13. To measure such benefits, our future plans involve performing surgical tasks  on enucleated eyes while the robot is mounted to a volunteer’s head. To effectively evaluate the effectiveness of head-mounting, the eye must also be mounted to the volunteer’s head. Our group’s eye-surgery robot30 is far too weak to penetrate the goggles in the event of a system failure, and we have already obtained IRB approval for the use of the head-mounted robot and the eye-mounting goggles.Our research group is evaluating the benefits of compensating for patient motion by head-mounting surgical robotsUnder some circumstances, it may be necessary to prevent the eye from rotating. For example, if the anterior section of the eye is removed for some experimental reason, then it may not be possible to insert trocar cannulas into the eye, and thus not possible for an instrument to rotate and/or stabilize the eye. For such cases, the bowl can be directly attached to the swim goggles , since the BTUs and BTU holder are not necessary. For a nonrotating bowl, the brim is not necessary, and the bowl can also have thinner walls to reduce the size of the device.R is located at a distanceR with another hemisphere of a smaller radius r removed. The result is a bowl of massThe CoM of a solid hemisphere with radius R, which is given by Eq. (Combining Eq.  and Eqs.n by Eq. .For our final prototype, depicted in Figs. 31 to derive We use the product of exponentials convention for forward kinematics as described by Murray et al.For our device, we havex-component of x-component of The configuration shown in Fig. sing Eq. , we findSupplementary Information.


The aim of this study was to present a methodology for predicting changes in uric acid concentrations in the blood of chronically hemodialyzed patients based on an optical measurement of the intensity of selected wavelengths in the dialysate. Blood samples were taken from the arterial line every 30 min throughout the hemodialysis period, to measure uric acid levels. Simultaneously, optical measurements were made on dialysate flowing from the dialyzer. Uric acid concentration can be measured either directly from the blood or from dialyzer outflow with acceptable error. In addition, both methods reveal any increased dynamics in uric acid concentration in the initial phase of hemodialysis. The wavelength of the light was adjusted for optimal uric acid particle detection. Comparing the uric acid concentration measured in the blood of patients with the intensity of wave absorption in the dialysate, the functional relationship between the uric acid concentration levels was determined. Using the optical method for measuring uric acid concentration in the dialysate, the concentration of uric acid in the blood during hemodialysis can be non-invasively and accurately estimated. This method can be used to assess the adequacy of hemodialysis by computer acquisition of uric acid concentrations determined in on-line dialysate. The incidence of cardiovascular complications, and the life expectancy and quality of life of chronically hemodialysed patients all largely depend on the optimal treatment of hemodialysis5. To assess the adequacy of hemodialysis (HD) during clinical practice, both clinical and laboratory parameters are evaluated. The degree to which uremic toxins are removed is one of the important components in assessing the adequacy of hemodialysis6. Measuring and assessing uremic toxin excretion is usually done by testing the patient’s blood once in the month before and after hemodialysis.Renal failure requiring renal replacement therapy affects many people in the worldThis assessment enables clinicians to predict subsequent hemodialysis treatment procedures, however it is an inaccurate assessment method.7.Nowadays, the assessment of the adequacy of hemodialysis is based on, among others, the assessment of urea concentration in the blood serum before and after hemodialysis and the rate of its elimination from the body. It is calculated for this purpose Kt/V ratio determined in a one-compartment model8. In healthy kidneys, 70–75% of it is freely excreted in the urine9. In hemodialysis patients, hyperuricemia is found in 50% of patients on chronic hemodialysis10. Uric acid is a substance with a low molecular weight (168 Da), and is freely dialysed during the classical hemodialysis procedure, and removed with the used dialysate. Spectrophotometry is a method that has been described in medicine since the 1840s11.Uric acid is formed in the human body from the metabolism of purine bases13.The dynamics of changes in the concentrations of low-molecular-weight uremic toxins not bound to proteins in the blood is subject to kinetic modelling and can be approximated by exponential functions in single or multi-compartment models. Modelling of urea, which is a uremic toxin that is also an ineffective osmolyte , was developed and clinically confirmed many decades ago15.Optical monitoring of uremic marker molecules in the used dialysate for online estimation and removal of uremic toxins, while determination of Kt/V based on the UV absorption of the used dialysate has been clinically validated and is used in clinical practiceThe main of this study is to prove the hypothesis that the measurements of light absorption by a dialysate can be used to develop an a comfortable and no invasive method for estimating the uric acid concentration in patient's blood.16, conductometric systems17 or the most popular optical systems, see for example20. In this work, we focus on the optical system approach, which exploits the fact that electromagnetic waves of specific wavelengths are absorbed when passing through uric acid of different concentrations, see for example21Three main system-based approaches to on-line monitoring of HD can be distinguished, electrochemical systemsThe aim of the study was to assess the clinical usefulness of measuring uric acid concentration in dialysis fluid using a specially designed optic device, and then testing the validity of results by comparing with the levels of uric acid excretion determined in patients’ blood tests during hemodialysis.For our pilot study, we recruited 10 patients with end-stage renal disease undergoing hemodialysis. The reasons for kidney failure in the patients were: diabetic kidney disease , hypertensive kidney disease , chronic cardiorenal syndrome , amyloidosis secondary to seronegative arthritis , and autosomal dominant polycystic kidney disease (ADPKD) .2 were used. In 7 (70%) patients HD was performed via an arterio-venous fistula in the wrist or elbow, and in the remaining 3 patients HD was performed via a tunneled catheter in the internal jugular vein. Three hemodialysis patients were given drugs during HD. One patient was given an intravenous infusion of iron  at a dose of 100 mg; one patient was given intravenous paricalcitol at a dose of 5 µg, at the beginning of their HD; and one patient was given intravenous erythropoietin  at a dose of 2000 IU, at the end of their HD.All patients underwent hemodialysis (HD) on either the 1st or the 2nd of July 2019. Patients were treated with hemodialysis using a low-flux, single-use, hollow-fiber dialyzer with a POLYNEPHRON™ membrane. Dialyzers with an area of 1.7 or 1.9 or 2.1 mThis study was conducted in accordance with the principles of the Declaration of Helsinki and was approved by the Bioethics Committee of the University of Rzeszów (No. 9/05/2019).Selected clinical and laboratory data as well as the results of measurements made during a single hemodialysis procedure in the examined patients are presented in Table In the studied group, uric acid was measured from blood taken during a single hemodialysis. Blood for testing was drawn from the blood supply line to the dialyzer during hemodialysis procedures according to the scheme: within the first 3 min from the start of HD and then every 30 min thereafter. The last blood sample was drawn at the end of the HD, just prior to the conclusion of the procedure. The blood serum tests for uric acid were performed in a hospital laboratory by an automated method with uricase and peroxidase on an Atellica® Solution analyzer immediately following the taking of each blood sample. The results of clinical measurement of uric acid for patients presented in Table Due to the dynamics of changes in uric acid concentrations in the first phase of hemodialysis, the obtained results can be divided into two classes. The first class of results with lower dynamics corresponds to patients without diagnosed concomitant diseases, while the second class with noticeably higher dynamics of changes in uric acid concentrations corresponds to those diagnosed with diabetes. A graphic presentation of the results, divided into the two classes, is presented in the graphs in Fig. 25. The absorption depends on the wavelength, hence, to simplify the analysis of the process of the light transmission, a monohromatic electromagnetic wave was used. An accuracy of the proposed method as well as its correctness and reliability strongly depends on the wavelength of the light used in the measurements. A uric acid should absorb the chosen monochromatic light in a proper amount. It is obvious that too small as well as too high absorption makes our sensor not sensitivity enough or even blind. Therefore, it is crucial to find an proper for our measurements length of the light. After choosing an appropriate wavelength, the next step is expressing a uric acid concentration as a simple mathematical function of this transmission coefficient.For optical measurement, in our study we propose the use of a non-invasive dialysate drainage device  that measures uric acid concentration by spectrophotometry. For this purpose the absorption phenomenon of the light by a uric acid flowing through a transparent cable was employed. A photodetector placed on the opposite site of the cable can measure amount of transmitted light described by the transmission coefficient6 – 0.0003276103λ5 + 0.4928330339λ4 – 395.1139625939λ3 + 178,048.12236675λ2 – 42757228.1737871λ + 4,274,817,397.85099. The polynomial approximation indicates that the light absorption achieves its maximal value for λ = 573 nm. This wavelength value ensure that the light absorption is not too small which makes our sensor sensitive. Therefore, the proposed device uses a laser emitting the light of this wavelength and the photodetector sensitive to this wavelength.To better know the phenomenon of the light absorption by a uric acid, the influence of the wavelength on the light transmission was experimentally investigated. The light absorption by some uric acid probes of different concentrations was examined. Based on the experimental data, the transmitted coefficient was presented in Fig. 8 (system); 2000:1 for a single scan; siber optic connector: SMA 905 to single-strand optical fiber (0.22NA).Measurements were made with a high performance 0.065 nm HR2000 fiber optic spectrometer. The HR2000 is sensitive in the range of 200–1100 nm, but to be precise, the range and resolution depend on the gratings and input slots selected. In Table The diode that was used for the device is a diode emitting in the spectrum of 465–640 nm controlled with a voltage in the range from 3.4 to 3.7 V. Power 3 W, current 700 mA and angle 120°.27).Applying the “brute force” method we empirically chosen the wavelength of the light equal to 573 nm as optimal for uric acid detection  of light absorbed by the detector in the spectrometer for a specific wave length λ, we have made a polynomial approximation of the intensities of peak positions  between uric acid concentrations as measured by dialysate spectrophotometry and uric acid concentrations as measured in a certified blood laboratory. In addition, by measuring both blood and dialysate, we observed a significant increase in the rate of uric acid excretion at the initial stage of hemodialysis in patients with underlying diabetes.33. de los Reyes et al.34 published a study describing the fluid dynamics of patients undergoing hemodialysis with ultrafiltration for one hour after the start of hemodialysis using a two-compartment model. In our study the steeper decline blood uric acid concentration in the first 30 min of hemodialysis diabetic patients is due to efficacious removal by dialysis, which is not matched by the effective refiling water-soluble toxins such uric acid in the extravascular compartment. Less efficient refilling in patients with diabetes mellitus may result from changes of a micro- and macrovascular nature accompanying dialysis patients with diabetes. This hypothesis of refiling disorders in the group of diabetic patients is confirmed by the known complication of more frequent intradialytic hypotension in diabetic patients compared to non-diabetic patients35. In our opinion the different dynamics of uric acid concentration in diabetic patients  is an interesting and original research result requiring validation in a larger population of patients.To the best of our knowledge the anomalous dynamics of uric acid concentration in diabetes patients in Fig. The appropriate wavelength for measurements of a light absorption by a dialysate was determined. The prototype device was built and the method for predictions uric acid in blood based on optical measurements was proposed. The performed investigations show the linear dependence between uric acid concentration in blood and the absorption coefficient. This fact was used to not invasively estimate uric acid concentration in blood. The relative error of the predicted value uric acid in blood was investigated. The relative error can achieve some great values, especially when the concentration is low, which is due to measurements errors as well as imperfection of the prototype device. The hypothesis that measurements of light absorption by dialysate can be used to not invasively estimated uric acid concentration in blood was confirmed.The use of photospectrometry as a method for assessing uric acid concentrations in dialysate during hemodialysis correlates with the uric acid concentrations found in patients’ blood sampled during hemodialysis, therefore it may be clinically useful as a non-invasive assessment of hemodialysis. For optical measurements, it appears that doubling the wavelength increases the efficiency of uric acid concentration estimation during hemodialysis.


In Persistent Spinal Pain Syndrome (PSPS), Pulsed Radiofrequency (PRF) is a used procedure. The results of PRF in PSPS performed with an electrode placed through the sacral hiatus were reported to be successful on pain in only 32% of patients. We have tried to improve the results by applying a new system to PRF.Ten patients were treated with PRF application through a steerable epidural catheter with a reference electrode outside the foramen. This method was named “Optimization Current Flow” (OCF). The duty cycle of PRF was 2 × 10 msec and total exposure time was 150 s. Follow up was planned for 1, 3, and 6 months. The NRS score was considered to be the primary outcome.In the first 10 patients treatment was successful (69% of the patients) at 6 months follow-up.This new modality of PRF in patients with PSPS seems to be superior to procedures done with a needle. Further prospective double-blind randomized long-term studies with a significant number of patients are required to validate this technique as there is a need to improve PRF results in PSPS. Failed Back Surgery Syndrome (FBSS) is a distinct medical condition that affects 15% to 30% of patients undergoing lumbar spine surgery . RecentlScar tissue is formed following surgery, involving the dorsal root ganglion and nerve roots, but epidural fibrosis is considered to be a radiological entity not correlating with the patient’s complaints  and the Treatment varies from psychological support on one side of the scale to re-operation on the other. Minimally invasive procedures such as spinal steroids injections, lysis of adhesions, epiduroscopy and both pulsed and continuous radiofrequency have a certain popularity but the results can be modest and sometimes temporary , 7.Pulsed Radiofrequency (PRF) close to the dorsal root ganglion is one of the therapeutic options, used mainly in patients with limb pain. The analgesic effect of PRF is not related to permanent physical neural damage  as in raThe mechanism of action could be a neuromodulatory-type process, which alters the synaptic transmission or the excitability of C-fibers . AlthougTwo approaches are possible. An electrode may be inserted by the percutaneous transforaminal route, or a catheter may be used, entering through the sacral hiatus in the epidural space. Both methods have an acceptable rate of success in non-operated patients , 12 and There are currently few studies on the use of PRF in patients with PSPS type 2, and almost always the series almost always include several etiologies other than PSPS type 2. Most of these reports use needle electrodes for the transforaminal route , 18–20. Three of these studies reported few cases of PSPS type 2 , 17, 20.In three studies  patientsCompared with monopolar PRF, it has been suggested that bipolar PRF would produce a denser and larger electrical field , 24 withThe setting of the treatments in the various series is usually 120 s with 45 volts at temperature of 42° using the cannula needle, while in cases treated with the epidural electrode , 22 the When the conventional monopolar technique is used such fields are found in a spherical zone around the active tip of the electrode, and a distance of approximately 2–3 cm is reached when 45 V are applied. Control of this distribution of electricity on the DRG is not always possible: in fact the distance D1 is not sufficient to cover the area of the entire ganglion .We have tried to improve this procedure by using bipolar application of PRF with a relatively large distance between the electrodes. This changes the geometry of the zone where the relevant fields are found into a prolate spheroid with its long axis along the line that connects the electrodes . Contrar®, Milan, Italy) with a 10 mm steerable tip, was then introduced and directed towards the relevant ganglion recess. An 18 G, 100 mm needle with an atraumatic active tip 10 mm long, was placed close to the foramen from a posterolateral approach  and normal saline solution to rule out incorrect placement of the catheter. Sensory stimulation at 50 Hz was performed at each level to check the correct positioning. PRF was then applied at 2 Hz × 10 ms and 25 V for a total exposure time of 150 s for every level treated. The average impedance was 275.8 ohms (190–350 ohms). We used 150 s because the coverage area of the ganglion is total. For the same reason it is pointless to use electrodes with a 15-mm tip.Inclusion criteria were persistent lower back pain and radiating pain of the leg with an NRS-score of at least 5 on a 0–10 scale, and duration of at least 6 months, following single or multiple surgical interventions on the lumbar spine. At clinical examination none of the patients had evidence of neurological deficit. The first 10 patients were considered: eight had lumbar surgery for removal of a lumbar discal hernia at a single level (two patients at two levels), and two patients for lumbar stenosis had bilateral pain.p < 0.05 was considered to be significant.The NRS score was used to monitor the clinical effect and was considered as the primary outcome. Success was defined as a fall of 3 points or more. All demographic data are described in The Medical Management Direction of ASST Settelaghi  hospitals requested only an informed consent read and signed by each patient as this was a no-experimental procedure with EEC marked material, since the applied medical treatment did not exceed medical standards.The PRF was administered bilaterally at the L3-L4 level in patients with bilateral pain, and in other cases at the level of the previous lumbar operation.p < 0.0001), demonstrating that the goal of reducing pain was achieved with statistical significance during follow-up.The treatment was successful at 1, 3 and 6 months’ follow-up in 69% of the patients. The improvement from preoperative median score of 8 the score improved to 3.9. There were no complications. The ANOVA analysis  showed aCurrently the treatment of drug-resistant PSPS type 2 in patients with radicular pain is based on epidural injections, percutaneous adhesiolysis and spinal cord stimulation ; howeverThe Spinal Cord Stimulation (SCS) requires the implantation of a prosthesis with the related problems. Ablative radiofrequency used for somatic lower back pain is a destrutive procedure that cannot be used at the gangliar or epidural level in PSPS type 2. Pulsed radiofrequency does not require implants and no major complications are reported : howeverThe first results of our study have to be interpreted with caution. Nevertheless the results compare favorably with the scarce data about the application of PRF with the traditional monopolar or bipolar by two needle approaches in PSPS type 2 patients. We postulate that the introduction of the OCF electrode configuration could made the difference. The voltage in PRF literature has always been expressed in volts .We estimate that in this study the distance between the cathode and the anode was no more than 3–4 cm. Since the applied voltage was 25 V, the electric fields in the largest part of the spheroid must have been approximately 200–400 V/m. These values are concordant with the fields that are generated by transcutaneous PRF, a method that has been proven to be effective in a RCT study . The resOn this technique, the principal item to consider is the density of current in the pathologic area to treat. The relation that determine the density of electrostatic charge in the pathologic area is:where: J = density of electrostatic charge; Delta V = Voltage; R = Resistence; A = distribution area of charge.If we consider the same resistance in the two cases (monopolar system with ground plate and OCF), the two variables are the voltage and the distribution area of the charge.In this case, the ratio between the surface area of the needle and ground plate is much bigger than the difference of voltage in the two systems. The result is the increase in the density of the allostatic charge in the pathologic small area with OCF, also with the same electric field value and resistance as the conventional monopolar system. The increase in the electrostatic charge could explain the better results despite the same electric field value, and this could improve the clinical results and at the same time reduce the risk of iatrogenic damage of the nervous tissue.The results of PRF in PSPS type 2 performed with an electrode placed through the sacral hiatus was reported to be successful against pain in 32% of patients ; the eleSecondly, with this new electrode configuration there is no need to enter the inflamed or scar area. RF current easily crosses barriers such as scar tissue or other tissue barriers, leaving the current density grossly intact as long as dilution of the current for geometrical reasons does not take place. Using the OCF bipolar system effectively reduces this “geometrical dilution”  of the cWe believe that PRF is an innovative method that has already demonstrated good results in other types of painful pathologies so: it is time to develop an effective technique also for PSPS.This new modality of PRF in patients with PSPS type 2 seems not to be inferior to procedures done with a needle, for patients with radiculopathy without previous spinal surgery . FurtherThe raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.The studies involving humans were approved by the Medical Management Direction of ASST Settelaghi . The studies were conducted in accordance with the local legislation and institutional requirements. The participants provided their written informed consent to participate in this study. Written informed consent was obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article.AD and SC contributed to the clinical data collection and literature review. All authors contributed to the article and approved the submitted version.


The results showed that the irradiated curve was well matched to uniformly shifted nonirradiated curve for the cumulative mortality of all causes of death but not for the cumulative mortality of all solid tumors and prevalence of ovarian tumors as is. After adjusting lifetime mortalities, it was also well matched for all solid and ovarian tumors. The shifted days by irradiation were 71–116 days for all causes of death, 56–135 days for all solid tumors, and 41–140 days for ovarian tumors in the 1.9 Gy-irradiated group. The response was switched between irradiation at 35 and 105 days consistently for all the above indexes, supporting the hypothesis that radiation sensitivity differs between juvenile and adults. The shifted days of all causes of death showed a tendency of linear response to dose. This concept of shifting the age of death can be applied not only for all cause of death but also for mortality of all solid tumors after adjusting the magnitude. These findings contribute to the discussion on the application of the ‘shifting age of death’ concept to radiation protection.Recently, the question of whether cancer risk is only accelerated but not increased by radiation exposure has been raised. To explore this matter, we analyzed whether the cumulative mortality of irradiated mice could be explained by x-axis (age) shifted cumulative mortality of nonirradiated mice. We reanalyzed publicly available data on observed cumulative mortality or prevalence in irradiated female B6C3F ERR is 1 mice 1 mGy/day1 mice in Japan [Concerning these discussions about shifting the age of death, it is also important to expand the knowledge on mouse response, for which abundant data are available compared with those for humans. Epidemiological studies tracking the lives of people exposed to radiation are limited, but there are some data for mice maintained for life after irradiation under various conditions, such as dose, dose rate, and age at exposure. These experiments on maintaining mice for life have mainly been conducted in the Argonne National Laboratory , the Natin Japan  is the mTo evaluate the applicability of the shifting the age of death concept, herein we reanalyzed published data on high dose-irradiated mice  concerni1 mice irradiated at several doses (0.8–0.98 Gy/min) at different ages and allowed to live their whole lifespan were reanalyzed. We analyzed the shifting ages regarding the effect of age at exposure on cumulative mortality using data from the 1.9-Gy irradiation group [The cumulative mortalities of the open data set derived from female B6C3Fon group  and day-es 0.8–0.8 Gy/min i) was defined as the ratio of the number of mice that died with an ovarian tumor (N*i) to the total number of mice that died in interval i (Ni): ρi = N*i/Ni [Age-specific cumulative mortality data for ovarian tumors have not been reported, although the age-specific prevalence of ovarian cancer has . The age= N*i/Ni . Since o= N*i/Ni , and theFirst, the cumulative mortalities of mice with all solid tumors were fitted to a logistic model and the Gompertz model as a typical example of sigmoid curves, both of which are commonly used for assessing survival. Since a better fit was obtained using the logistic model in all data of cumulative mortality of all solid tumors, according to the Akaike Information Criterion using R software , this moasym, xmid and scal are coefficients of function of the logistic model and were obtained by SSlogis function in R software [Asym is a numeric parameter representing the asymptote, and ximd is a numeric parameter representing the x value at the inflection point of the curve; thus, the value of the SSlogis function will be asym/2 at xmid. Scal is a numeric scale parameter on the input axis [asym for cumulative mortality of all cause of death was greater than one, the asym was fixed at 1 and the other two parameters were calculated. In the case of cumulative mortality of all solid tumors and cumulative prevalence of ovarian tumors, the asym was fixed at maximum raw data and the other two parameters were calculated when asym was greater than the maximum raw data. The comparison of free asym and fixed asym is shown in where, software . Asym isput axis . When thestimatedy and irradiatedy was at the minimum for each age point (x) of 250–1250 days depending on the data points of the data set [We analyzed the shift required in the number of days in age to match the irradiated curve and shifting the nonirradiated curve. The number of shifted days was determined when the residual sum of squares (RSS) between data set .For all solid cancers and ovarian tumors, the shifting age of death was calculated after aligning the lifetime mortalities as follows:The suitability of the model was judged by the above-mentioned RSS.Analysis of the cumulative mortality for all causes of death showed that the time of death in the irradiated group uniformly shifted, and thus was the same as the slope of cumulative mortality in the nonirradiated group . Comparey = 56.1x − 5.2, R2 = 0.993). Although a less linear dose response was observed for day 7 of irradiation , it appeared as a linear response when focusing only on mice irradiated with a dose of ≤1 Gy   , Table 3Finally, to examine the shifting day of site-specific tumors, we analyzed data from mice with ovarian tumors, including lethal and nonlethal tumors, which were available for each age group . The cumConsidering the cumulative prevalence of ovarian tumors, the adjustment factors did not show a linear response , and shiFor all causes of death, the cumulative mortality curves of irradiated groups were well-matched by uniformly shifting the curve of the nonirradiated group compared to all solid tumors and ovarian tumors. Although the curves for the cumulative mortality of all solid tumors and the cumulative prevalence of ovarian tumors did not match by simply shifting them uniformly, they became consistent after adjusting lifetime mortalities or prevalence. This may be due to the increased risk, which cannot be expressed in the model of earlier radiation risk, or changes of death causes (competition of death causes). This indicates that both discussions about the viewpoint of increase/earlier radiation risk and also the effects of competition for causes of death should be further explored.As shown in The differences in shifted days in all causes of death were milder compared with those in all solid tumors and ovarian tumors . This coFor all causes of death, the irradiation at day 0 showed a linear dose response, but the irradiation at day 7 showed a linear dose response when limited to 1 Gy or less. For ovarian tumors, the number of shifted days reached saturation at high doses of 0.95 Gy and 2.38Gy for mice irradiated at day 0 and 7, respectively. Although the linear relationship may be applicable to shifted days in mice, it is necessary to further investigate whether the concept of ‘shifting age of death’ is applicable to radiation protection.In the 1.9 Gy irradiation condition, the response was switched between 35 and 105 days consistently for all-cause mortality, all solid cancer mortality and ovarian cancer prevalence. Mouse ages of 0, 105 and 365 days correspond to human ages of 0, 20 and 30 years, respectively . The resOne limitation of this study is that since only published data were used, the irradiation conditions that could be explored were limited. If further raw data become available in the future, it will be possible to expand the analysis range, such as the dose response of all solid tumors and dose rate effects. It is important to continue to collaborate among epidemiological, biological and computational studies on radiation risk evaluation of individual cancers. For example, computational studies can suggest a model to address the relationship between the competition for cause of death and adding site-specific risks to estimate whole risk in animal experiments. This can contribute to the analysis of irradiation conditions without modifiable factors, such as smoking and epidemiological studies, which have a huge advantage to estimate risk in humans.Data supporting the findings of this study are available within the article. The details of the data sets used for this study are provided in Table 1.shifting_age_suppl_info_rrad006Click here for additional data file.