Accurate trajectory tracking is a critical property of unmanned aerial vehicles (UAVs) due to system nonlinearities, under-actuated properties and constraints. Specifically, the use of unmanned rotorcrafts with accuracy trajectory tracking controllers in dynamic environments has the potential to improve the fields of environment monitoring, safety, search and rescue, border surveillance, geology and mining, agriculture industry, and traffic control. Monitoring operations in dynamic environments produce significant complications with respect to accuracy and obstacles in the surrounding environment and, in many cases, it is difficult to perform even with state-of-the-art controllers. This work presents a nonlinear model predictive control (NMPC) with collision avoidance for hexacopters’ trajectory tracking in dynamic environments, as well as shows a comparative study between the accuracies of the Euler–Lagrange formulation and the dynamic mode decomposition (DMD) models in order to find the precise representation of the system dynamics. The proposed controller includes limits on the maneuverability velocities, system dynamics, obstacles and the tracking error in the optimization control problem (OCP). In order to show the good performance of this control proposal, computational simulations and real experiments were carried out using a six rotary-wind unmanned aerial vehicle (hexacopter—DJI MATRICE 600). The experimental results prove the good performance of the predictive scheme and its ability to regenerate the optimal control policy. Simulation results expand the proposed controller in simulating highly dynamic environments that showing the scalability of the controller. The applications are not restricted to the industrial field, as nowadays robots are equipped with sophisticated and intelligent algorithms that are useful for performing complex task in unstructured and natural environments with a high degree of autonomy. The integration between robots and humans through service robotics has the objective of facilitating activities of daily life or improving repetitive tasks. As such, we can find mobile manipulators that are used to load transportation in structured environments ,2, robotrocesses ,6. (ii) rkplaces ,8. (iii)pulation ,10. (iv)s (UAVs) .The use of unmanned aerial vehicles have grown in a wide range of fields and has been extensively studied by the scientific community due to the versatility and the possibility to perform the fully autonomous tasks that humans are unable to. The extremely agility of the UAVs with four or more rotors has attracted attention due to their ability to vertically take off and land in confined or difficult to access spaces, which eases construction and enables having a higher payload capacity, which are useful in critical missions such as transportation, rescue, search, surveillance and disaster assistance. In these applications, the execution time is an extremely important feature due to the task only being executed for high-speed trajectories. To take advantage of UAVs, it is necessary to guarantee the safe execution of the application; however, dynamic environments are spaces with highly probabilities of collisions, where obstacle information is unknown at the beginning of the task, and thus the UAV must be capable of avoiding the obstacles using online-scheme information. Obstacle avoidance is an important feature that guarantees the correct functionality of UAV applications. In this context, research efforts are still needed to safely introduce UAVs as a technological tool with great potential to solve various current problems such as a collision avoidance control structure that can perform high-speed trajectories in dynamic environments.(i) Outdoor environments where many studies have employed UAVs with onboard sensors, e.g., Mellinger et al. presented collaborative grasping to logistic applications [(ii) Indoor environments are the counterpart applications, developed in cluttered indoor experiments, examples of which include the study by Song and Hsu who presented navigation based on factor graph optimization (FGO) [Depending on the environment, UAV applications can be divided in two groups: ications ; Bircherications ; search ications ; and an ications . (ii) Inon (FGO) ; Paredeson (FGO) ; and Sanon (FGO) . Due to on (FGO) ; and Gupon (FGO) . This coon (FGO) ,22. Moreon (FGO) ; Phong-Non (FGO) ; and Alion (FGO) .(i) System actuation: An inherently under-actuated system due to there only being four control inputs over the six degrees of freedom (DOF). (ii) Dynamics behavior: high nonlinearities produced by translational and rotational dynamics due to the considerable size of the system. (iii) Dynamic model mismatch: Unmodeled dynamics, uncertainties and disturbances produced for high-speed trajectories and the possibility of external forces such as wind in outdoor environments. All of these problems incited the special attention of researchers from different fields. As a consequence, a wide variety of control strategies have been proposed for the problem of trajectory tracking for hexacopters. Initially, some linear control methods were presented, and these works considered small angles’ assumptions that are necessary for linear control techniques, and the most common structures were presented by Alaimo et al., who developed a proportional–integral and derivative controller (PID) under linear assumptions in UAVs with an hexacopter configuration [The use of more rotors can produce high costs in production, a considerably increased size of the UAV and complex dynamics. Nowadays, it still a challenge to design effective controllers that ensure good performance in multi-rotors at high-speed trajectories. Precise trajectory tracking is an important feature for multi-rotors operating in real-world environments due to the considerable size of the structure and possible collision with obstacles. The main reasons that make this configuration more complex than the others are presented below: guration  and Saliguration .Linear models are only valid around the hover position, providing slow movements where the controllers cannot guarantee the convergence to the desired trajectories. In order to improve the performance and to satisfy the requirements of high-speed trajectories, a large number of studies have used the principles of nonlinear control, e.g., Chen et al., who proposed a nonlinear trajectory controller for UAVs based on backstepping and nonlinear observer  and WangSE(3) ; MellingSE(3) . The proSE(3) ,33. In rSE(3) ,35,36,37(i) rapidly exploring random tree (RRT), a formulation which generates a tree rooted at the initial configuration of the system and using random samples to create edges between feasible points. The work by Achtelik et al., who presented an RRT algorithm with a modification rapidly exploring random belief tree (RRBT), which was used to generate the collision free path using quadrotor UAVs [(ii) The belief roadmap, which is a probabilistic version of the roadmap, which uses the Kalman filter for the planning motion, and has the ability to generate trajectories by avoiding obstacles; this structure was presented by [(iii) The minimum snap trajectory generation this algorithm includes navigation through corridors whilst considering constraints on velocities and accelerations, the solution to which is based on the generation of optimal trajectories that minimizes a functional cost confirmed by square norm of the snap, an application of which was presented in [Unlike multi-rotor control techniques, obstacle avoidance has not been highly developed by the research community. Obstacle avoidance is a relatively new field and most works have used the concept of artificial potential fields, a method which is based on the generation of repulsive and attractive forces that allow the movements of the system around obstacles. The literature presents the extensive applications with UAVs, but these works revealed the problem of a local minimal that cannot guarantee the execution of the task ,39. Nowator UAVs ; (ii) Thented by . (iii) Tented in . All theGround mobile robots: where Sani et al. presented nonlinear model predictive control (NMPC) to solve the competitive games between ground robots , anotherRobotic arms and manipulation: In the last year, this field of research has been the focus of many studies due to the complexity of the systems and the scalability of the NMPC structure. The work of Osman et al. presented a task-space controller based on (NMPC) to control a mobile manipulator 10 (DOF) .Aerial mobile robots: this field has also been the subject of a large number of studies due to the high use of aerial vehicles in real-world applications, some applications of which were studies Neunert et al., who developed an unconstrained nonlinear predictive model for generation and tracking trajectories for the AscTec Firefly hexacopter ; Aoki etTaking into account the problems of no-dynamics constraints and only the prior knowledge of environments, the results thus produce unfeasible control actions and problems in dynamic environments with highly probabilities of collisions during the task execution. To fully exploit hexacopters’ capabilities and take advantage of the available computation power, optimization-based control techniques are becoming suitable for real-time problems. Model predictive control (MPC) has been used in many fields of robotics due to the accuracy of its tracking trajectories, robust performance and the ability to include constraints in the controller scheme. In function of the field, the general applications can be divided into the following groups:This kind of controller has had a huge impact on the field of robotics thanks to the predictive behavior and ability to introduce system constraints in the controller scheme. The main idea of this controller is to generate the control actions trying to minimize a cost function over a prediction horizon. The cost function is solved using constrained optimization techniques; however, the resolution of constrained optimization problems is computationally expensive. Due to the recent developments in hardware and nonlinear optimization solvers, nonlinear predictive control is computationally applicable in many fields and has received the attention of a hexacopter control field ,51,52,53System dynamics: a large number of research platforms cannot be controlled through torque commands; as such, this work developed a reduced dynamic model of the hexacopter using general control velocities, which is possible with the incorporation of low-level PID schemes in mathematical representation. The PIDs guarantee the required velocities generated by the high-level controllers. The mathematical representation was developed through an Euler–Lagrange formulation and the data-driven technique known as dynamic mode decomposition (DMDc). Both approaches were identified using experimental data from the (DJI MATRICE 600 PRO); furthermore, this work presents accuracy comparative results between both techniques and selects the best formulation to build the controller scheme.NMPC formulation: this includes the differential kinematics and the reduced dynamic representation; furthermore, due to the ability of the NMPC controller, the system constraints can be included in the optimization problem. The constraints included in this work are: control action limits, control rate of change, and the system dynamics and static and dynamic obstacles. The nonlinear optimization problem was solved using CasADI framework and the problem was transformed into a nonlinear programming problem (NLP) form using the direct multiple-shooting method. The CasADI optimization toolkit guarantees a fast convergence with the possibility of extending the solution to onboard hardware implementation.Results: the experiments were conducted in real-world outdoor environments, where the system has a model mismatch and external disturbance product by air flows and delay in system communications. The experiments were developed through the hexacopter platform (DJI MATRICE 600 PRO) where multiple reference trajectories were selected in order to verify the performance of the controller. The metrics used in the experiments are the following: tracking accuracy, computational time and disturbance rejection. Additionally, the simulation results show the scalability in simulated high dynamic environments, considering the identified dynamics and obstacles; these results will be used as a start point in future research due to the fact that the hexacopter aerial platform does not have any sensor to measure obstacles during the experiments.Run NMPC structures are realizable in modern computers and guarantee the execution of complex tasks but still require more computational time than the schemes mentioned beforehand. It is necessary to show performance under tracking accuracy and the ability to extend this technique to dynamic environments in order to add obstacles in the optimization problem. With all the aforementioned controllers and the significant results in many areas, this work presents the nonlinear model predictive control to fast trajectory tracking in dynamic environments applied in hexacopter platforms. The controller best generates the control policy that moves the hexacopter to the reference trajectory while avoiding obstacles through the environment. To construct the approach, this work developed the important contributions presented below:This work is structured as follows. This section presents the instantaneous kinematics and the dynamic model. The dynamic model was formulated in maneuverability velocities space using two different formulations: the Euler–Lagrange and dynamic mode decomposition. After the formulation process, this work presents the validation results in order to verify the performance of each formulation.The robot has the ability to move through the inertial frame fined as:η˙xη˙yη˙zfined as:η˙xη˙yη˙zConsidering that This section presents the dynamic model, which has been developed considering low-level PIDs controllers that only generate longitudinal movements in m and the moment of inertia I around the vertical axis g represents the gravitational acceleration. The Lagrange formulation is obtained by the subtraction of the kinetic One of the approaches to find the mathematical model is the Euler–Lagrange formulation. This formulation is a convenient analytical method for obtaining the dynamic model and studying the physical phenomena of the hexacopter. The Euler–Lagrange formulation uses the total energy of the system confirmed by kinetic and potential energy. The total kinetic energy is defined by the following equation:Now, applying the Euler–Lagrange formulation:quations:mηx¨−dymcTo simplify the notation, this work compacts Equation , resultiDue to the fact that the hexacopter platform can be controlled through reference maneuverability velocity commands, this work converts the generalized force and torque inputs of the dynamic model  into refμl˙μm˙μn˙μl˙μm˙μn˙A similar concept was applied to the vertical force attached to the body frame The same approach can be extended to the attitude dynamic model presented in , which iAfter all the considerations presented beforehand, this work introduces the general structure of low-level PD controllers combining Equations , 19) an an19) anEquation  can be wFinally, with all the transformations considered beforehand, Equations  and 6)  6) are sThis new formulation represents the evolution of the hexacopter’s general velocities with the reference maneuverability velocities as the input of the system. This representation is useful because commercial aerial platforms can be directly controlled with maneuverability velocities, ignoring the low-level control. Equation  can be cl is an instant of measure and Model identification cost: this function was formulated using the subtraction between the reference maneuverability This section presents the identification and validation of the dynamic parameters, as this work uses real experimental data to estimate these values in order to use the model in the proposed controller. This work transforms the mathematical model  in the lWith Equation , it is pConsidering the performance index function presented in , the optThis work used the experimental information of the hexacopter platform DJI MATRICE 600 PRO to identify the dynamic parameters. The aerial system has a low-level flight controller called A3, which is confirmed by three global positioning systems (GPS) and three inertial measurement unit systems (IMUs). Furthermore, this system provides highly precise measurements from the system states. The identified parameters are presented in The validation process of the dynamic parameters is a very important factor; hence, the model needs to estimate the behavior of the system with different signals in order to represent the real dynamics of the system. The validation signals are shown in With all the previously presented results and the validation of the dynamic parameters, this work formulates the identified model using Euler-Lagrange as a vector-function presented as:This section presents the second approach of the dynamic model formulation which, due to the hexacopter platform, only has internal low-level PIDs; and the system tries to stay close to the hover-position with small angular variations in roll and pitch angles s and the formation of augmented matrices, resulting in a new formulation of the system as follows:s number of snapshots.One of the emerging techniques to identify systems by experimental information is that of dynamic mode decomposition (DMD) ,56,57, wFor estimation purposes, the system can be formulated as:One of the best methods to find the solution of the Frobenius norm is the singular value decomposition (SVD); therefore, applying this method to the augmented data matrix resulting as To approximate the values of the matrices Equation  can be wEquation , consideThe values of the matrix This section presents the identification and validation process of the dynamic model using the DMD formulation. The values of matrices Algorithm 1 Identification process Using DMDInput: DJI MATRICE 600 PRO measurements Output: Matrices     Identdo                                  end Ident    returnThe approximation values of the unknown matrix are presented in The signals used for the identification process are presented in The estimated model displays good behavior due to the fact it replicates the signals obtained from experimental data. The noise in the angular maneuverability velocity In the same way presented in the validation process using the Lagrange formulation, this section presents the validation process under DMD formulation, the results of which are shown in Finally, considering the approximation values of the unknown matrices, the dynamic model can be written in a vector-function form as:This section presents the comparative results between the Euler–Lagrange  and dynaWith the previously presented results, this work uses the DMD estimated model to construct the proposed controller. Combining Equations  and 36)36), the This section presents the formulation of the proposed controller, where the main objective is the trajectory tracking over the desired reference trajectory using an hexacopter platform and due to the dynamic effects and the important size of the system, this work includes obstacle avoidance to improve the behavior of the system in dynamic environments. This work presents the nonlinear model predictive controller (NMPC); furthermore, the constraints included in the optimization problem were: generalized system dynamics, bounded maneuverability velocities, rate of change of control velocities and obstacles in the dynamic environment. The main objectives of the proposed controller are: to solve the control problem associated with the trajectory tracking subject to the system and the environment and system constraints. The control problem is solved by the scheme shown in t and the prediction horizon T. Due to the prediction behavior of this scheme, the NMPC formulation requires index performances over the prediction horizon T that guarantees the solution of the control problem and the system constraints. The generation of the performance functions are presented below.In order to solve the NMPC proposed scheme, this work formulates the generalized dynamics  as a prel considering the following property: This subsection presents the generation of the cost function over the evolution instants Trajectory Stage Cost: this part of the function is structured by the control error between the desired trajectory Control stage cost: this part of the cost function defines the variations between the maneuverability control velocities T was formulated with following property: Terminal trajectory cost: this term considers the control error between the reference trajectory at the last step of the prediction horizon T, where The cost function at the last step of the prediction horizon This work uses Equations  and 40)40) to geThe performance index function  guaranteDue to the hexacopter platform having maximum maneuverability velocities, this work proposes control input constraints on the NMPC structure, in addition to further constraints on the successive difference of the maneuverability velocities which were included to prevent aggressive behaviors. The input constraints are defined in  in orderz and Z the number of obstacles detected by the system, the representation of the obstacles is shown in This section presents the obstacle constraints in order to track the desired trajectory and avoid obstacles in the dynamic environments. The obstacles can be defined as The function of the distance between obstacles and the hexacopter has the following property: aints in , and wheCasADI as a nonlinear optimization framework [Considering the performance index presented before  and the P:P(ηref(ation in . The sysramework .(i) Real-world experiment: the results presented in this section were developed using the nonlinear model predictive controller and environment obstacles are not considered in this experiments due to the hexacopter platform not having the required sensor for real-world measurements. (ii) Simulation experiment: simulation experiments show the effectiveness of the proposed controller in highly dynamic simulation environments with simulations of the dynamic behavior of the hexacopter in order to make it more realistic and prove the scalability of the controller.This section presents the simulations and experimental results to validate the proposed controller for the hexacopter platform DJI MATRICE 600 PRO. The experiments were performed considering: Several experiments on trajectory tracing were performed in order to demonstrate the performance of the proposed controller. In order to demonstrate the effectiveness of the optimization structure, the controller was implemented at onboard computer of the hexacopter through the software Matlab, additionally the hexacopter platform is shown in The experiment was performed in the city of Ambato, in the province of Tungurahua, Ecuador, and started at 09:36 on 12 January 2022. The wind velocity at the time of the experiment was approximately 10.1 km/h, as shown in The desired reference trajectory for the hexacopter and the initial conditions are defined in To implement the proposed controller, the constant values of positive definite matrices are defined in The results of the experiment are illustrated in The previously presented results show the performance of the controller that guarantees the execution of the reference trajectory over different axes.The control signals represented in The time required to solve the optimal control problem is shown in This section presents the simulation result of the proposed controller. In order to improve the results, this experiment was performed using the identified dynamics in Equation  with theThe desired reference trajectories and initial values over the frame The desired values of the reference trajectory are the same as the proposed in real-world experiments; therefore, the results present similar behavior. The controller constant values are the same as those presented in The trajectory described by the hexacopter platform and the reference trajectory are presented in The values of the control errors presented in The control signals are presented in The time required to solve the proposed controller and the Euclidean distance are presented in The experiments presented in the last section show that the nonlinear model predictive scheme solves the control problem associated with the trajectory tracking subject to the system and environment constraints. Specifically, the real-world experiments showed the robustness of the proposed controller due to the excessive wind velocity during the experiment which acted as an external disturbance. Despite the excessive wind velocity, the controller generates the adequate smooth control actions guaranteeing that the steady state error that converges towards values close to zero with the dependency of the external disturbances. The use of CasADI such as an optimization framework guarantees the fact that the computational time always stays below the sample time, which is one of main problems in NMPC schemes due to the highly computational time that these solutions required, as similar computing times were presented in .On the other hand, the simulation results show the ability of the proposed scheme to generates the adequate control signals guaranteeing the corrected tracking trajectories while the system avoids obstacles during the simulation. The presence of Gaussian noise provides uncertainty in the measurements demonstrating the power of the controller as it maintains the distance to obstacles in consideration of the safety radius. One of the main differences presented in this experiment was the control error, which tends to increase when the hexacopter avoids obstacles satisfying the system constraints. Furthermore, the computational time presents peaks produced by the hard constraints included in the optimization problem, and despite the hard constraint, the controller finds a sub-optimal solution to guarantee a low computational time. Other works have used the concept of soft constraints, low-level programming language and embeddable optimization methods such as (SQP) in order to increase the convergence of the optimization algorithm and reduce the computational time ,62,63,64This work carried out the identification of a dynamic system using the dynamic mode decomposition with control technique to develop the optimal control problem for trajectory tracking with obstacle avoidance, a decision which was made due to the precision of the comparative results between the Euler–Lagrange and the dynamic mode decomposition formulations. The optimal control problem was translated into a nonlinear programming problem (NLP) using the multiple shooting technique and considering the control actions and systems states as decision variables, which is a technique that improves the computational time and the convergence of the algorithm. The nonlinear model predictive control (NMPC) generates the maneuverability velocities policy that allows the hexacopter platform to track the reference trajectory while avoiding obstacles presented in the environment. The NMPC formulation was developed in consideration of the reference desired trajectory and constraints in the system such as the bounded control actions, the generalized dynamics and static and dynamic obstacles. Furthermore, the NMPC problem was solved through the nonlinear programming framework CasADI, which solves high-dimensional optimization problems and has advantages that can be expanded to a low-level programming language to improve the computational time.The simulation results show how the hexacopter avoids both static and dynamic obstacles whose locations vary over time while executing the trajectory tracking task, and always maintaining a safe distance defined as a radius of repulsion. Since there is no sensor for obstacle measurements in the hexacopter platform, the real experimental tests were not carried out with obstacle avoidance; however it was verified that although the experimental tests were carried out during the period of time in which the average wind speed was approximately 10 km/h, the behavior of the control law on the hexacopter system fulfills the desired task, so that the control errors in stable state converge closer to zero. In addition, it can be observed that the steady state error asymptotically converges towards values closer to zero under conditions wherein the control actions are saturate when they reach the established restrictions. On the other hand, the computational time of the proposed control algorithm is maintained under 100 ms in real experimentation due to the optimization technique used to solve the control problem. During the simulation tests, it is clearly verified that the computational time remains under 100 ms; however, when obstacle avoidance occurs, this time increases but remains under the specified sample time with a sup-optimal solution. Other works use the concept of soft constraints to guarantee the rapid convergence of the optimization algorithm, as this work will use the same concept to improve the computational time and the possible expansion to visual-servoing systems under an MPC structure.


A complex drug treatment might pose a barrier to safe and reliable drug administration for patients. Therefore, a novel tool automatically analyzes structured medication data for factors possibly contributing to complexity and subsequently personalizes the results by evaluating the relevance of each identified factor for the patient by means of key questions. Hence, tailor-made optimization measures can be proposed.I_with) and a version without (GI_without) integrated key questions for the personalization of the analysis, while the control group (GC) did not use any tools (routine care). Four to eight weeks after application of the tool, the benefits of the optimization measures to reduce or mitigate complexity of drug treatment were evaluated from the patient perspective.In this controlled, prospective, exploratory trial the tool was evaluated with nine general practitioners (GP) in three study groups: In the two intervention groups the tool was applied in a version with . Thereby, the number of optimization measures recommended by the GP had no significant influence (P = 0.167).A total of 126 patients regularly using more than five drugs could be included for analysis. GP suggested 117 optimization measures in GThe study suggests that an automated analysis considering patient perspectives results in more helpful optimization measures than an automated analysis alone – a result which should be further assessed in confirmatory studies.DRKS00025257 (17/05/2021).The trial was registered retrospectively at the German Clinical Trials register under DRKS-ID The online version contains supplementary material available at 10.1186/s12875-022-01757-0. A complex drug treatment results from various factors of a drug treatment or the medication process, that potentially make the administration of a drug treatment difficult for patients. Many of these so-called complexity factors are associated with non-adherence or administration errors , suggestAll currently available approaches address health professionals, but do not consider individual patient perspective. However, it is well known that patient centeredness is essential to effectively tailor health care to patient needs, thereby fostering the acceptance and sustainability of potential changes in drug treatment  while prTherefore, an electronic tool was developed that is applied by health professionals but attempts to comprehensively consider the patient perspective . It autoThe objective of this pilot study was to evaluate the benefit of the developed tool in primary care. It was hypothesized that optimization measures recommended to the patient by the health professional are more often considered helpful by the patient when based on an automated and personalized complexity analysis compared to an exclusively automated complexity analysis.I_with) used the full version of the tool [I_without) used a version of the tool with limited functionality that did not propose key questions and, therefore, lacked personalization of the automated results. Group 3 (GC) was a control group representing routine care where physicians did neither receive a training in complexity reduction nor use any version of the tool. In each group, complexity was assessed at two points in time  was performed by the GP either as part of a regular patient consultation, or at a specifically arranged appointment (depending on the preference of GP and patients), the second analysis was performed by a study assistant in person or by telephone and was planned four weeks after t0. The study was conducted from May to October 2018 after receiving unrestricted ethical clearance , the University of Witten/Herdecke (57/2018), and the respective responsible authorities of the practice locations). The study was carried out in context of the Declaration of Helsinki and the data protection laws of the German state and the respective federal states as well as the European General Data Protection Regulation (GDPR), which came into force during the course of the study. The trial was registered retrospectively at the German Clinical Trials register under DRKS-ID DRKS00025257 (17/05/2021).The tool was exploratively evaluated in a controlled, prospective, pilot study with general practitioners (GP) allocated to three study groups. Group 1 . Among the GP of each federal state one was randomly assigned to one of three groups by the study team using the algorithm at www.randomization.com (first generator). This way each study group was supposed to contain one GP of each federal state, but one GP withdrew study participation and was replaced . Before inclusion of patients, all GP were trained in the study protocol and the GP allocated to GI_with and GI_without received a face-to-face training in the use of the respective tool version.For inclusion GP had to meet the following criteria: routine use of the nationally standardized medication schedule  and access to the internet. Nine GP were recruited in three different German federal states .I_with and GI_without, GP analyzed the current drug treatment with the respective tool version and consequently took measures to mitigate or reduce the complexity. The selection of the optimization measures, the communication of the measures  and eventually also the implementation was left to the responsibility and discretion of the GP. Thereby, GP could refer to the optimization measures suggested by the electronic tool or also implement their own measures. The drug regimen, all complexity factors identified, the time required for the analysis, and, if applicable, answers to key questions as well as the optimization measures chosen were documented in the tool by the GP (respective buttons were integrated in the tool). Additionally, GP could also document whether they carried out an optimization measure that was not suggested by the tool. Subsequently, GP in GI_with and GI_without were asked by the tool, whether they believed that the analysis had simplified drug treatment for the respective patient or not. The GP of GC did not receive any support for analyzing and reducing complexity but were aware of the study´s focus and, thus, could have concentrated on the complexity of drug treatment as part of the routine patient consultation. The BMP of patients of GC was analyzed retrospectively with the electronic tool by the study team.In GI_with and GI_without were asked to rate if the optimization measures recommended by the GP at t0 have had an influence on the handling of their drug treatment . Thereby, the wording should indicate on whether the measures were indeed implemented by the patient. Conversely, patients of GC were asked whether the GP made any recommendations, gave any explanations or adapted the medication schedule in order to address the complexity of their drug treatment. If changes in the BMP were noticed at t1 that were not documented in the tool, the patients were asked whether or not they were made by the GP based on the initial complexity analysis (GI_with and GI_without) respectively the study visit (GC).The current drug treatment was reanalyzed by a study assistant (pharmacist). For this purpose, the study assistant obtained copies of the patients’ BMPs from the GP practices, which, if necessary, had been updated by the GP after the first analysis. The study assistant analyzed the BMPs of all patients with the full version of the electronic tool  in all study groups. Furthermore, the patients in G1 that it did not improve (no effect) or even worsened the handling of their drug treatment. Furthermore, optimization measures that patients could not remember at t1 were considered not helpful. The number of complexity factors and medications and the type and number of recommended and helpful measures were analyzed in all three groups. The time needed for the analysis with the electronic tool was evaluated in groups GI_with and GI_without.Primary outcome was the number of optimization measures per patient reported as helpful by patients. Counting absolute numbers was considered more suitable than for instance a percentage, because the use of percentages would have overestimated the ratings of patients with only small numbers of optimization measures proposed. Optimization measures were considered not helpful if patients indicated at tI_with and GI_without using Poisson regression; incidence rate ratios (IRR) were derived as exponentiated parameter estimates from the regression model. All other outcomes and socio-demographic data were analyzed using nonparametric tests . P-values less than 0.05 were considered as formally significant, yet all p-values should be regarded as exploratory. All analyses were carried out using Microsoft Excel 2016, IBM SPSS Statistics 25 and the R software environment in version 4.0.2 .All results as well as the socio-demographic data were reported descriptively . The primary outcome was compared between G0 were available, and for 126 patients  complete data on both visits were collected, allowing these patients to be included for statistical analysis . In GC the date of the written informed consent was used for this calculation, assuming that written informed consent was obtained at the date of t0.The actual interval between the first and the second analysis was 7.1 ± 2.1 weeks, 6.4 ± 1.8 weeks, and 6.1 ± 1.2 weeks in G0 to t1 in any group. Furthermore, the number of complexity factors did not differ significantly between the groups at t0, but it did at t1  for the average time required between the groups GI_with (6.4 ± 3.6 min) and GI_without (6.3 ± 4.3 min).There was no significant difference ) and GI_without ) . Patients in GI_with were three times more likely to rate an optimization measure as helpful than patients of GI_without . The number of optimization measures recommended by the GP . In both groups, more than half of the recommended measures were not remembered by the patients .There was a significant yet very small difference in the counts of optimization measures that were considered helpful by patients Fig. , betweenI_with, the most helpful optimization measures were the recommendations for actions , two times the switch of a product  and a measure that was not proposed by the electronic tool were perceived helpful by patients. Indeed, such optimization measures that were not primarily proposed by the electronic tool were recommended most frequently in GI_without, e. g. to ensure on the prescription that the drug of one manufacturer cannot be exchanged for a drug of another manufacturer. In GC, only two patients reported that they had been recommended one optimization measure each by their GP. The two recommended optimization measures were both considered helpful  and both measures would also have been recommended by the tool if it had been applied to the respective BMP.In GI_with expected 39 (84.8%) of their analysis to be helpful for the respective patient. In GI_without, significantly fewer analysis, i.e. less than one third  of the analysis were considered helpful by the performing GP .GP of GI_with) considered their analyses to be helpful more than 2.5 times as often as GP only using the automated analysis (GI_without). However, although patients and GP in GI_with subjectively valued the results of the intervention, it did not have any influence on the number of medications. The number of complexity factors identified indeed differed in all groups between t0 und t1, but there was no difference between the three groups, thus the intervention does not seem to have had any influence on this.When individual patient preferences were used to personalize an automated analysis, the patients considered significantly more optimization measures helpful than when this was not done. In addition, the GP whose analysis took the patients’ views into account  complexity of drug treatment. In a prospective, observational study a visual analogue scale was used to assess patient’s perception of the complexity of their drug treatment and the results strengthen the idea of considering the patient-perceived complexity of drug treatment, because no concordance was found in the complexity of the antiretroviral therapy perceived by patients and the one calculated using the MRCI [Given the small number of patients included and optimization measures that were actually remembered by patients at tthe MRCI .I_without had a significantly longer relationship with their GP than patients of the other groups. This might have led to a more trusting relationship with the GP and, thus, it is more likely that difficulties have already been discussed in routine care, which could have weakened the difference between GI_with and GI_without. Moreover, also the number of drugs was not equally distributed over the different groups with patients taking on average one drug more in GI_with. While the sheer number of drugs does not necessarily lead to a complex drug treatment [As an exploratory pilot trial, this study has several limitations. First, patients in Greatment , 15 and This pilot study suggests that the involvement of patients in the analysis and reduction of the complexity of a drug treatment leads to a needs-based assistance by GP, because patients considered more recommendations by their GP helpful if the analysis also took into account patients’ views. However, the observed effects are small and could have been influenced by different factors, e.g., differences between study groups or a selection bias due to the non-consecutive inclusion of patients.Additional file 1. Histograms that show the distribution of the number of complexity factors and the number of drugs among all three study groups.Additional file 2. Overview of the optimization measures that were considered as helpful for data analysis. The complexity factors identified as relevant for patients, thus leading to the proposal of optimization measures, and a description of the optimization measures suggested by the tool are given.


An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton–Jacobi–Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations. Continuous differentiability of the value function with respect to the initial datum is an important problem in optimal feedback control theory. Indeed, if the value function is  control  and 25]\documentThe underlying analysis demands stability and sensitivity analysis of infinite dimensional optimal control problems subject to nonlinear equations. For this purpose we utilize the theory of generalized equations as established by  and 23]23]. It iThe current investigations are to some degree a continuation of work the first author’s work on optimal feedback control for infinite dimensional systems. In –6 TaylorLet us also compare our work to the developments in the field of parametric sensitivity analysis of semilinear parabolic equations under control constraints. There are many papers focusing on stability and sensitivity analysis of finite time horizon problems along with pointwise control constraints, see e.g. , 26, 27,2(Ω). In  again thIn the finite dimensional case, there is, of course a tremendous amount of work on the treatment of the value function if it is not mentclasspt{minima mention  in this In order to achieve the goal we desire, we lay out the following setup. In Sect. entclass1pt{minimaHere we present a stability result for an abstract, infinite dimensional optimization problem which will be the building block for the results below. This result is geared towards exploiting the specific nature of optimization problem with differential equations as constraints. First existence of a dual variable will result from a regular point condition. Subsequently the Lipschitz stability result is obtained in two steps. In the first one, we rely on the relationship between the linearized optimality conditions and an associated linear-quadratic optimal optimization problem, with an extra convex constraint. This approach is useful since it provides the existence of solutions to the linearized system on the basis of variational techniques. However it dictates a certain norms for the involved quantities. These norms are too weak for our goal of obtaining Lipschitz continuity of the adjoint variables in such a manner that differentiability of the cost with respect to the initial conditions can be argued. Therefore, in a second step we exploit the specific structure of the optimality systems, using the fact that it is related to a parabolic optimal control problem, to obtain the Lipschitz continuity in the stronger norms. This two step approach is also present in some of the earlier work on stability and sensitivity analysis which was quoted in the introduction. But due to that fact these papers considered finite horizon problems it came as a byproduct which improved the regularity of the adjoints. In our work it is essential to reach our goal. This is why we decided to formalize this two step approach which was not done in earlier work. with a parameter dependent equality constraint, and a general constraint described by C is a closed convex subset of a real Hilbert space X. Further, W is a real Hilbert space and P is a normed linear space. In the application that we have in mind, the parameter q will appear as the initial condition in the dynamical system. The following Assumption int denotes the interior in the W topology. This regularity condition implies the existence of a Lagrange multiplier C at where see e.g.  such thaLet (positive definiteness)Condition  is a bitq at X to The stability result of olutions , see e.gmentclass2pt{minimf and e to the choice of the spaces in (Now we adapt the conditions on paces in .(i)the restriction of (ii)the restriction There exists a neighborhood q.With these assumption holding we define the restricted linearized Lagrangianation of  and defi case of . The folx, and There exists x in X. For this purpose we exploit the symmetry of x in the small space Let us further setumptions –H5) hol hol\docuumptions  can be eMoreover  restrictlated to  is defin}U^, see ,  is a local solution of (Assume that –H7) hol holH7) hFor the proof we shall employ the following lemma in which Let atisfies , and thaC is a closed and convex, x can be uniquely decomposed as x minimizes x follows from (Since w=b~. By  the funckerE. By  the sequows from .The regular point condition implies the existence of a multiplier uch that  holds. S(i)The proof of the first assertion of the Theorem sumption  implies -q2P. By  and H5)\documentDontchev . In the (ii)A, E defined in ((Existence). Let, at first, fined in , 2.5). . \documeponds to  with \doEx0}. By  and :z asT, but T is fixed. We obtain the following estimate,L such thatp(0). Therefore it is not indicated. Since mentclass2pt{minimε). From  and (4.4β^-βY.By  we have ¯;U),and  follows.¯;U),and  there exWith the above proposition, the verification of –H7) is  is H7) iLet assumptions (A) hold and let Next we obtain one of the main results of this paper, the Fréchet differentiability of the local value function associated to ((Sensitivity of Cost) Let assumptions (A) hold and let Let nU.Using , we haveq in ((Sensitivity w.r.t. other parameters) We have developed a technique to verify the continuous differentiability of the local value function see e.g. , 15, conY so that the assertions of Theorem Utilizing the results established so far we now verify that the  value function ution to , 3.2c)  \documenFor convenience we restate Theorem Let assumptions (A) hold and let t tends to 0. Since The proof is similar to that of [Y, and that Y, we can pass to the limit in ..\documen, and by  we have,ion into  we obtaijectory,  follows.In this section we discuss the applicability of the framework in two specific cases. It should be noted that even for linear state equations, the sensitivity result for the constraint infinite horizon optimal control problem may be new. where Y. Thus (A1) holds. For We consider the optimal stabilization problem for the Fisher equation in an open connected bounded domain lyzed in , for exay, its boundedness is automatic. For the sake of illustration we verify the boundedness of the bilinear form of the second derivative on V embeds continuously into It can readily be checked that the nonlinearity (A3) and  are satiV is compactly embedded in Y, we obtain by the Aubin Lions lemma that Turning to (A4) we show that V embeds continuously into We turn to the assumption The specificity of this example rests in the fact that the second derivative is independent of the point were it is taken. Other nontrivial cases of analogous structure are reaction diffusion systems with bilinear coupling, see  where thConsider the system  and let V in Y, we obtain (A4).In order to verify (A4), we show that Now we verify (A5). We recall Remark We can also consider the optimal stabilization problem with cubic nonlinearity, i.e.  In this model, one can easily verify assumption (A1) is satisfied by taking Y which is uniformly stable. Assumption (A2) is satisfied under the same argumentation as in Fisher’s equation. Differentiability assumption (A3), and continuity assumption (A4) are satisfied along similar computations as in Sects. t, x),subject to


Light propagating along a reversed path experiences the same transmission coefficient as in the forward direction, independent of the path complexity. This is called the optical reciprocity of light, which is valid for not too intense scattering media as well. Hence, by utilizing the reciprocity principle, the proposed novel technique can achieve axially and laterally tunable focus, non-invasively, through a scattering media without a priori knowledge or modeling of its scattering properties. Moreover, the uniqueness of the proposed technique lies in the fact that the illumination and detection are on the same side of the scattering media. The information gets distorted within the speckle. However, the information is not lost, it just needs to be retrieved from the scattered speckle pattern. Starting from the late 1960s, many researchers have paved the way for this desirous goal of imaging through a scattering media with includes the works of Goodman et al.3, Leith and Upatnieks4, and Kogelnik and Pennington5.Light passing through a scattering media such as biological tissues undergo scattering which creates difficulties for an imaging system to produce quality images. Especially while using a monochromatic laser source, random interference produces speckles. Hence it is difficult to achieve optical focusing and imaging for its internal targets, which restricts in-depth imaging for medical diagnosis and treatment7. However, this technique suffers from the fact that it requires a lot of image processing tools, and post processing. Also it is not very effective for thick scattering medium, especially when the thickness is larger than the mean free path as the number of ballistic photons will be very few and the signal will be corrupted by non-ballistic photons as well11.The Optical Coherence Tomography technique utilizes the ballistic photons (photons that are not scattered) for imaging making it an important tool in many diagnostic measurementsmemory effect of speckles. Memory effect states that there exists a correlation between a speckle pattern even when the laser illumination is tilted within a given angular range. Freund14 used the scattering media as a lens based on speckle intensity correlation. This idea was extended by Bertolotti et. al13 to image a fluorescent object behind the media. Phase retrieval algorithms like Fienup’s phase retrieval algorithm15 were utilized by Katz et. al16 to retrieve the object from the autocorrelation of the speckle pattern. Singh et al17 introduced a point source as well in the object plane. The advantage was that with simple autocorrelation of the camera intensity, the image was reconstructed hence avoiding the use of a phase retrieval algorithm. Despite the advantage of utilizing the scattered light without any wavefront correction, this approach suffers from the limited memory effect range and axial decorrelation of the speckle effect16. Hence this technique is inadequate for imaging through thick specimens such as biological tissues.Another interesting approach is to exploit the 19, acoustic-Opto tomography21and ultrasound-guided optical imaging21. The aforementioned techniques are non-invasive with the capability to shift the achieved focus. However, the main disadvantage lies in the fact that the resolution of the focal spot depends on the ultrasonic wavelength is orders of magnitude inferior to the optical diffraction limit.Apart from these optical techniques, acoustic waves can also be used in conjunction with the optical wave for deep tissue imaging because of the deeper penetration of acoustic waves. To this end, we have techniques like photoacoustic tomography22 utilized a Spatial Light Modulator (SLM) to modulate the incoming phase of the beam. A camera was placed behind the scattering medium, and this acted as feedback to guide the wavefront pattern on the SLM. Finally, through optimization, a focused spot was obtained on the camera. Later Popoff et al23 came up with the idea of measuring the transmission matrix of a scattering media and used it to introduce the inverse phase on the SLM to get a focused spot behind the scattering media. This method requires an initial calibration to find the complete transmission matrix. There were several works done by using SLM as a phase compensator to image through a scattering medium 33. Among them, optical phase conjugation is a well-known technique wherein the light distribution is replicated by reversing the propagation direction of the detected field while conjugating its wavefront. Digital optical phase conjugation (DOPC) has been well established for focusing and imaging through complex or disordered media37.Optical Wavefront Shaping Technique is potentially a strong candidate for focusing light through a scattering media by shaping the incident wavefront employing some feedback from the behind of the scattering media. Vellekoop et al38, laid the theoretical foundation to a novel technique of focusing light through a scattering medium non-invasively, i.e., the illumination and detection are on the same side of the media  such that L = M.dl. We assume that (i) the refractive index contrasts are low which makes sure that the backward scattering is negligible, (ii) diffuser thickness is smaller than the transport mean free path, (iii) the diffuser’s absorption is negligible. In our simulation, we show that (i) it is possible to focus light in a thick diffuser. For this to happen the cost function is the phase correlation between the Input wavefront and that of the conjugate of the output wavefront as described by Eq. z plane which is z distance after the medium for which the cost function is described in Eq. (Simulations were performed in MATLAB. We followed the modeling as proposed and implemented by Zhu d in Eq.  (see “Mepbr) has improved by N.A of this scattering lens is 0.005, with spot size not smaller than Simulations were done for a total of 100 variables, being the input modes, whose values change from entclass1pt{minimaentclass1pt{minimaentclass1pt{minimapbr remains the same.We proceed further in shifting the focus obtained laterally by multiplying the phase mask obtained through optimization with the required phase gradient phase mask. The results are shown in Fig. pbr is Axial shifting of focus is obtained at a plane  the Eq.  . Here we also show the capability of the technique to scan the spot both axially and laterally. Figure Lens L5 images the plane PL1 on the CAM1 . The CAM1 is placed such that it is pixel to pixel matched with the SLM projection on the plane PL1. We use an off-axis reference beam to capture the interference pattern of the output speckles on CAM1 and process it using the off-axis holographic technique to retrieve the output speckle phase. This acts as feedback to the SLM. CAM2 is placed at varying z distances from the diffuser to verify whether we achieved focus at the desired location or not. However, we don’t use any information from this camera to modulate the input SLM phase.pbr has increased 6 times compared to the unfocused condition. Figure In the first set of experiments, we tried to focus light at a distance of 100 mm from the diffuser. The objective function to optimize the wavefront to achieve desired focus non-invasively is obtained from Eq. pbr has improved almost 5 times the unfocused case. Here also we have effectively used 40 × 40 pixels on the SLM by binning 27 × 27 as one pixel. Figure Second set of experiments are performed to scan the focal spot in axially. Again, using Eq. For most biological applications like imaging a specimen through a scattering tissue, it is important that the produced focal spot can shift within a desired field of view. Hence, in the next set of experiments, we show that it is possible to shift the obtained focal spots laterally as well. This can be done either by multiplying the optimized phase mask obtained with a linear grating or by shifting the pattern on the SLM which in turn will shift the focal spot due to the memory effect of speckles. We use the results from the first set of experiments we performed, i.e. by shifting the optimized phase mask used to produce the focal spot in Fig. In the above set of experiments, we concentrated on proving the concept of axial and lateral shifting of focus through a scattering medium non- invasively. We believe that there is room for improvement and future work to be done to study various parameters that need to be tuned to further improve the results. Currently, we used a thin scattering medium as a specimen. We required nearly 4000 iterations to perform the optimization. It took about 25 min for optimization since our SLM has a very slow response time (120 ms), and additionally, the phase becomes stable only after a total of 300 ms. So, for each random phase mask on the SLM, we need to wait around 300 ms for capturing the frame. The camera capturing process and the hologram reconstruction process altogether also takes time which is about 50 ms. Hence it delays the entire process by 350 ms. It was observed that the laser loses its phase stability from the initial state in about 20 min. Hence, it is essential to perform the entire optimization within this time frame. However, this is not a limitation to the technique. Today’s market is equipped with SLM that has a very fast response time of 5 to 10 ms. Also, it is important to mention the use of a Digital Micromirror Device (DMD) based system which is an array of micrometer-sized mirrors that can be individually controlled at a high refresh rate of up to 20 kHz could also be used to shape the wavefront. In addition to this, parallel processing and GPU-based programming can further boost the speed of the entire process. Indeed, using our technique, it is possible to focus light through on a thick specimen as well, but it will require more iteration and computational load as we will require more pixels on the SLM to efficiently control the phase of the light. We would like to extend our workflow towards this direction by optimizing our current setup that would accommodate the high-speed processing required for a thick specimen. We also hope that the current study encourages the research community to explore more in this domain.40 or the presence of loss43 In the linear regime, this suggests a definite relation, or symmetry, between the forward and the backward transmission when interchanging the source and detector. This symmetry is also valid for a forward and backward scattering of light in scattering media. Recently Szu et al.8 proved this point experimentally in a disordered medium like a multimode fiber. He measured the transmission matrix both in the forward and backward direction and found that indeed the symmetry exists, and the backward transmission matrix is the transpose of the forward transmission matrix. This lays the foundation for our technique. We employ an adaptive optics scheme-based detection for wavefront compensation to focus light in the scattering medium. In our scheme, we optimize the input wavefront applied to a scattering medium by measuring the phase of the light that passed twice through the same medium (on the way forward and on the way back). To obtain the encoded inverse wavefront that will be focused after passing the above-mentioned scattering medium we use the following modeling: we assume that the scattering medium can be modeled by a sequence of operator multiplications while the first is an operator of a random phase and the second is an operator of short free-space propagation of distance of dz being the average scattering length in the inspected tissue. This couple of operators is repeated until the number of repeats equals M = L/dz where L is the distance of the target behind the media from the point of illumination , which states that light propagating along a reversed path experiences the same transmission coefficient as in the forward direction, independent of the path complexityS can be written as:z is the free space distance, dz is the averaged free optical path (average distance between scattering events), P is a random phase factor (of the scatterer) and μ is the spatial frequency.Thus, by modeling the 1D case due to cause of simplicity, the forward scattering matrix Hence,S is unitary since it is the product of the multiplication of two unitary matrices. The operator matrix S was applied twice as there is double passing through the scattering medium (on the way forward and on the way back). On the way back, the scattering is in inverse order, and therefore the transpose operation was applied on the matrix S. We want to find the input field distribution vector Ein  such that after one passage a focus is obtained, i.e., the operator S produces a delta function in the middle of the spatial axis (zero position coordinate):It is to be noted that the matrix Thus, one can extract that:N is the number of spatial sampling points along the output axis.in equals to the middle column of the matrix operator of inverse S:From Eq. A is a unitary matrix then there is a known relation between A transpose (St) and inverse S (S-1).S denotes its conjugate transpose. By combining Eqs. Since Hence, we obtain the desired T for free-space propagation of distance of We extend our technique to axially tune the focus. In the following section, we will model the theoretical foundation for axial tuning of the focal point after the diffuser. .We assume focus after 1st pass and thus:Then, our output field (as detected by the holographic wavefront sensor) is given byHence, we can write:Thus,Due to the unitary property of the matrix Thus, by dividing Eq. Let us calculateOur free-space propagation matrix T is given bySo, in Eq. , what weIn the next step, we take the d in Eq. 24\documeUsing the same computation, one obtains the following:Similarly, we can find the value of:Thus, after putting the value of Eq.  and 26)26) in EqIf we tune the lateral location this means that Z = 0, i.e., we have only the scattering medium and no free space propagation. Also, since the focus will now be not exactly in the center the assumption of Eq. 45. PSO finds an optimum solution for a problem by iteratively improving the solution from a set of candidate solutions being random guess solutions to a problem. These random solutions are called particles. Each particle is assigned a ‘position’ and a ‘velocity’ vector. The search space is known as ‘swarm’. Hence, the name PSO. The flowchart describing the steps involved in PSO is shown in Fig. We used Particle Swarm Optimization Scheme (PSO) to perform the iterations. It is a stochastic optimization technique that mimics the social behavior of birds flocking together or fish schooling. It was introduced by Kennedy, Eberhart, and ShiWe have presented a novel technique to focus light through a scattering medium non-invasively. Uniqueness comes from the fact that we neither need to know the scattering properties a priori nor need to use any feedback mechanism from behind the medium. In the current proof of concept experiment we demonstrated that indeed, it is possible to tune and shift the focus axially and laterally by numerical optimization of the incident wavefront. This technique has a strong potential to make its impact in various fields such as biomedical imaging and diagnostics. The concept of reciprocity-induced symmetry and its implications has not been extensively explored. Hence, we hope that our technique in its current form can be further enhanced to implement it for various practical imaging applications.


Pleuronectes platessa) from the last 30 years. Fitting this model to length‐at‐age data can provide new insights in the dynamics of the environmental factors limiting individual growth and provides a useful tool for ecological research and management.Growth and growth limitation are important indicators of density dependence and environmental limitation of populations. Estimating individual growth trajectories is therefore an important aspect of understanding and predicting the life history and dynamics of a population. Variation in individual growth trajectories arises due to variation in the environmental factors limiting individual growth. This environmental limitation can vary over time, between cohorts and between individuals within a cohort. For a complete and accurate understanding of individual growth in a population, it is important to include all these sources of variation. So far, statistical models only accounted for a subset of these factors or required an extensive growth history of individuals. Here, we present a novel model describing the growth curves of cohorts in a population. This model is derived from a stochastic form of the Von Bertalanffy growth equation describing individual growth. The model is specifically tailored for use on length‐at‐age data in which the growth trajectory of an individual is unknown and every individual is only measured once. The presented method can also be used if growth limitation differs strongly between age or length classes. We demonstrate the use of the model for length‐at‐age data of North Sea plaice ( Individual growth is often strongly affected by the environment. This could result in large variation in body size over time and between individuals and cohorts. We present a method to fit growth curves and estimate the environmental limitation of growth, while accounting for these sources of variation. Following Dynamic Energy Budget theory  scaled by the limitation of the environment through assimilation (ft). Although individuals might vary slightly in the values of the Von Bertalanffy growth rate scalar and the asymptotic size due to genetic differences, we will only focus on environmental effects on growth because these effects can affect the growth of individuals through their life. It is commonly assumed that the asymptotic length is the only parameter in the Von Bertalanffy growth equation which depends on the environment  differences between individuals can be accounted for by incorporating a random effect for every individual, but this is only feasible in datasets with multiple observations per individual. We focus on datasets with single observations for each individual, where this approach would only lead to extreme overfitting. Therefore, the only option to account for individual variation caused by a shared environment in datasets with single measurements is by considering the length‐at‐age at the population level as a distribution rather than a single value .The different sources of variation in the length‐at‐age are entangled due to the auto‐regressive nature of individual growth processes in which the current growth rate depends on the growth history of an individual. As a consequence, variation in growth arises between individuals and cohorts and could fluctuate over time. All these sources of variation should be considered to obtain an accurate estimate of the Von Bertalanffy growth parameters of a specific species, even if we are only interested in one of the sources or variation or average growth parameters. Such a method is currently unavailable. To fill this gap, we derive a model that describes the length distribution at a given age for every cohort, which can be used for datasets with a single length and age observation per individual. Because we derived this model from a stochastic version of the Von Bertalanffy growth equation for single individuals Equation , it simu22.1μt) and variance (σt2) are allowed to vary over time:We start with the equation describing the growth trajectory of a single individual Equation . IndividTb. We can solve this equation by separation of variables and integration:By substituting this distribution in Equation , we obtaℓTb,0,i represents the length at birth of an individual. In addition, Wt represents a Wiener process, which describes the outcome of a continuous process with independent Gaussian increments . The integrals in this expression cannot be solved explicitly because the dynamics of the mean and variance of the environmental limitation  are not defined. If the environmental limitation was constant over time and space , individuals would follow a Von Bertalanffy growth curve toward a constant asymptotic length.The parameter T follows a Gaussian distribution as well. Because the expected value of the Wiener process is equal to zero, we can derive an expression for the expected mean length at time T of a cohort born at time Tb.In this method, we consider datasets in which every individual is only measured once. In the ideal situation, these individuals are selected randomly from the population. In such datasets, it is not possible to follow the growth trajectory of a single individual and fit the derived growth curve on single individuals. Instead, we describe the distribution of the length‐at‐age for a cohort. Because we assumed that the environmental limitation of growth follows a Gaussian distribution, the length of individuals in a given cohort at time EℓT,a), to make clear that this expected value is a statistic of the length‐at‐age distribution of a cohort rather than the length of single individuals. By using Equations   EℓT,a), T. Under this assumption, the equations become independent of the length distribution at birth and can be applied without knowledge about the full growth history of a cohort.We assume that samples are taken with an approximately constant time interval and therefore discretize the equations characterizing the length distribution of a given cohort at time μt and σt2). These quantities only appear within the integral from time T up to time T + 1. We therefore only have to make assumptions about the mean and variance of the environmental limitation between consecutive timepoints or measurements. We assume that the mean and variance of the environmental limitation between times T and T + 1 are well approximated by the average value of these quantities in the given time interval (μ¯T and σ¯T2). Under these assumptions, the model will approximate the growth dynamics if the interval between measurements becomes small relative to the average lifetime of an individual. Because we substitute the mean and variance of the environmental limitation by the average of these quantities over a time interval, they become independent of time in the domain of integration and we can solve the integrals in Equations μt and σtσ¯T2) has the same unit as the time constant (T). This arises because we model the length of an individual as a Brownian process which is a process with random increments. The variance of a Brownian process increases due to the random nature of the process and therefore depends on the length of the time between consecutive measurements in our model. In other words, the total variance in environmental limitation experienced by an individual increases (decreases) if the actual variation of the environment increases (decreases) or the individual experiences the environment for a longer (shorter) period of time.Interesting to note from this formulation is that the variance in environmental limitation over a given period (VℓT∞). This represents the variation in length that individuals would have after spending an infinitely large time in an environment with a given amount of variation in growth limitation (σ¯T2). At this asymptotic variation in length, the loss of variation in length due to growth  is equal to the gain in variation in length due to variation in the environment (12rBσ¯T2ℓ∞21−e−2rB). In other words, when a cohort reaches the asymptotic variation in length, the variation in length of a cohort does not change any further over time. From Equation , the length distribution at the youngest age at every time point , the length distribution of all other cohorts at the first time point  and the distribution of the environmental limitation between all time points , . Therefore, the maximum asymptotic length cannot be estimated separately with this method and is incorporated as a species‐specific scalar of the environmental limitation. We provided an R‐package 7b) predi), Table . In the e Croll,  that ince Croll, .EℓT,a) decreases if it exceeds the asymptotic length at some time step (μ¯Tℓ∞). This is a mathematical artifact of the model that predicts that individuals will shrink in size if they are too large to be supported by the environment. Although some species might shrink in size under bad conditions, this is not realistic for all species 7b), the μ¯T,c, σ¯T,c2) for every age or length class at every time step. It is important to note that this extension of the model only uses the mean length of a cohort to identify the length class and therefore all individuals in a cohort are always placed in the same length class. In addition, the number of observations per class decreases with an increase in the number of age or length classes. The incorporation of age or length classes can therefore make the model fit less accurate if there are no true differences in the environmental limitation of the selected classes.The second extension available in the R‐package deals with differences in environmental limitation between age and length classes. Differences in environmental limitation between age or length classes can arise if age or length classes show spatial segregation or differ in diet. The model allows specification of age or length classes and estimates separate means and variances in environmental limitation (2.3Pleuronectes platessa) obtained from the Beam Trawl survey (BTS). This survey is designed to monitor plaice in the North Sea and is consistently conducted in the third quarter (July to September) from 1990 onwards. The length of individuals is measured with at least 5 mm accuracy and the age of sampled individuals is obtained through otolith readings. We downloaded the datasets with individual ages and lengths recorded during the third quarter of 1990 to 2021 from the online ICES DATRAS data portal on the 1st of November 2021 . Starting values for the variance in length at the lowest age class and the variance in asymptotic length were set to the variance in the youngest and oldest age class, respectively (VℓTamin=801mm2y and σ¯t2ℓ∞2=5789mm2y).Starting values for the expected length at the first age in the dataset and the expected asymptotic length and the growth scalar were estimated by fitting a Von Bertalanffy growth model without considering differences between years and cohorts  estimated by Van der Veer et al. .We used a estimation of the maximum asymptotic length . This suggests that the mean and variance of the environmental limitation are likely to fluctuate between years. More precisely, the model with yearly varying environmental limitation suggests a weak downward trend in this limitation, indicating that the environmental limitation became stronger over time . This overestimation of the variance in environmental limitation arises because the fixed limitation model accounts for the variation in the asymptotic length between years in addition to the variation in asymptotic length within a year. As expected, the models also differ slightly in the estimated parameters defining the length distribution at the youngest age. The estimates of the Von Bertalanffy growth scalar of the models  are relatively close to estimates based on individual energy expenses . While such a decrease can realistically occur, it is sometimes a biologically impossible result. Repeating the analysis on this dataset with the restriction that the expected length of a cohort cannot decrease, yields very similar results (not shown). Nonetheless, this additional restriction should be handled with care, because early tests on simulated data showed that this restriction makes the model more dependent on the data points of young age classes.The estimated model parameters lead to predictions of the length‐at‐age distribution for every cohort, which differ most strongly for the older age classes Figure . On visuTo demonstrate the robustness of the model, we used a jackknife approach in which we repeated the analysis with the samples from 1 year omitted Figure . It is ny) is smaller compared to the average estimated variance in environmental limitation in the model without age groups (on average 0.0268y). This might suggest that the environmental limitation of individuals in the youngest age group is more similar to the environmental limitation of individuals in the same age group compared to the environmental limitation of individuals in other age groups. This could explain why the model with three age groups fits the data better compared to the model without age groups. In contrast, the average estimated variance in environmental limitation for the two oldest age groups (on average 0.0399y for age group 4–7 years and 0.0801y for age group 7–10 years) is larger compared to the average estimated variance in environmental limitation in the model without age groups (on average 0.0268y). This might suggest that the environmental limitation of some individuals in the two oldest age groups is more similar to the environmental limitation of individuals in other age groups compared to the environmental limitation of individuals in the same age group. It is important to note that the reported variation corresponds to the fitted variation in environmental limitation and not the variation in the individual sizes. The variation in individual size is a balance between the variation in size at the previous time step and the variation in environmental limitation. The variation in size is therefore likely to increase or decrease with age, depending on the variation in size at birth. This does not hold for the variation in environmental limitation as it is independent of size and age.Our estimation method can also be used on populations which consist of separate ecological groups. We demonstrate this using plaice, the distribution of which has been shown to shift away from the coast with increasing length or age ; it should therefore first be scaled by an estimate of the species specific maximum asymptotic length before it can be compared between species. Comparison of this scaled proxy for individual food availability between species might then reveal links such as shared resources or competition.Because our model simultaneously accounts for variation between individuals and cohorts and allows variation from the environment to fluctuate over time, the model can be used for a wide range of applications. First of all, the estimated mean and variance of the asymptotic length estimated by our model can be used as a summary statistic for environmental limitation under the assumption that the sources of variation are independent. Growth of individuals is likely limited by numerous factors, which are often unknown. Our method offers a summary statistic for the cumulative distribution of all these factors. Our method is especially appropriate to estimate individual limitation in growth due to limitation through food availability. General theory about individual energy allocation links the asymptotic length of this Von Bertalanffy growth equation to the energy ingestion by individuals  in our example of North Sea plaice. This suggests that our arbitrary division of the population into three age groups does not accurately represents the ecology of North Sea plaice. It does demonstrate that our model can be used to verify whether a suspected division in ecological groups is likely by comparing the estimated variance in environmental limitation of a fit without ecological groups with a fit with ecological groups as is done in the example for north sea plaice.Lastly, our method might also be applicable to management as it is able to model variation in growth through time and between individuals based on only a limited number of parameters ; formal analysis (lead); methodology (lead); software (lead); writing – original draft (lead); writing – review and editing . Tobias van Kooten: Conceptualization ; funding acquisition (lead); supervision (lead); writing – review and editing .The authors declare that there is no conflict of interest.


While the initial article was composed hastily by the editor in a journalistic style and contained no information on the nature of the new rays, the second publication was partly composed by the son of the editor, a physicist, who informed on some of the basic physical properties of the Röntgen rays.It is generally omitted that the press article announcing the discovery of Röntgen rays on the 5 Die Presse on Sunday January 5,1896 , because the required equipment is not powerful enough’. This proves that experimentation started in Vienna directly following the receipt of Röntgen’s material. Ernst L. had either witnessed firsthand this progress or he had heard about it during the previous two days. Even if the Crookes’ tubes available in Exner’s laboratory or elsewhere in Vienna were too feeble to produce discriminative photographic shadows, they might have been strong enough to excite to some degree fluorescent material placed in the vicinity of the radiant source. After contact with other researchers on the Sunday and Monday, Ernst L. was able to come back with the latest news, which found again their place starting on the front page of his father’s newspaper. On January 11, Exner confessed to Röntgen in a letter, that he still failed reproducing the experiment although he guessed, it must be quite simple [


I (0) versus nonstationary I (1) dichotomy which has been used in previous studies. All series appear to be highly trended and to exhibit high degrees of integration and persistence, especially in the case of CPI. Since the two variables have different degrees of integration in each of the two countries, fractional cointegration tests cannot be carried out. We assume instead weak exogeneity of each of them in turn and test for causality by regressing the other variable against lagged values of the weakly exogenous one. We find that the only significant relationship implies the existence of a lagged effect of prices on output in the case of the US, which suggests a dominant role for demand shocks.This paper aims to provide new evidence on the relationship between prices and output in both the US and the UK (which is important to discriminate between different macroeconomic theories) by focusing on the long run. For this purpose, it applies fractional integration and long-range dependence techniques that are more general than the standard modelling approach based on the stationary  The relationship between prices and output is crucial to understanding the nature of economic fluctuations and to be able to discriminate between rival macroeconomic models. A lot of the literature in this area has focused on whether prices are procyclical or countercyclical, i.e. whether they move in the same or in the opposite direction to output. This depends on the nature of the underlying shocks: aggregate demand and/or monetary policy shocks should produce procyclical behaviour, whilst aggregate supply (technology) shocks should result in countercyclical behaviour. Friedman and Schwartz , 1982 anBy contrast, in the Real Business Cycle (RBC) framework introduced by Kydland and Prescott  businessOn the whole, the evidence concerning the US is mixed  versus nonstationary I (1) dichotomy which has been used in previous studies since they allow the differencing parameter to be a fractional value as well as an integer and, therefore, are more general and flexible. After analysing the stochastic behaviour of each individual series, we examine their long-run linkages. Since the two variables have different degrees of integration in each of the two countries, fractional cointegration tests cannot be carried out for this purpose. We assume instead weak exogeneity of each of them in turn to carry out causality tests in a regression framework. Please note that the key contribution of the paper is represented by the chosen methodology which, as already mentioned, uses a more general modelling approach than the papers discussed above and enables us to establish both whether the effects of shocks to the individual series are transitory or permanent and whether they are linked by a long-run equilibrium relationship; instead it is not our aim to analyse instead the determinants of either inflation or output.The present paper aims to provide new evidence on the relationship between prices and output in both the US and the UK by focusing on the long-run rather than on cyclical fluctuations. For this purpose, it applies fractional integration and long-range dependence techniques; these are more general than the standard modelling approach based on the stationary The rest of the paper is structured as follows: Sect. I (0). In such a case the original series is said to be integrated of order 1 or I (1). Following the seminal work of Nelson and Plosser (I (0) nor I (1) but instead integrated of order d, where d can be any fractional value in the interval between 0 and 1, or even to be above 1. Gil-Alana and Robinson (I (d) with 0 < d < 1. Since then, fractional integration has been widely used for the analysis of macro series  stationary if its first two moments are independent of time. However, most macroeconomic series appear to be nonstationary. A standard approach to remove nonstationarity is to take first differences on the assumption that the differenced series will be stationary  Plosser  many papRobinson  examinedRobinson  dataset xt stands for either the observed data or the errors in a regression model that may include deterministic terms such as a constant or a linear time trend or weakly exogenous variables, d is a parameter to be estimated from the data providing a measure of persistence, and ut is an I (0) process that is assumed to be in turn a white noise or exhibiting (weak) autocorrelation. Note that the specification given by (1) includes the classical cases of I (0) and I (1) if d = 0 and 1, respectively, and if ut is an AutoRegressive Moving Average (ARMA) process with orders p and q for the AR and MA components, xt becomes an ARMA  and ARIMA  process, respectively, and an ARFIMA  one for non-integer values of d. By allowing d to take any real value, we make the model sufficiently general to include a variety of cases such as those of nonstationary though mean-reverting series when d belongs to the interval . The estimation of the differencing parameter d will be based on the Whittle function in the frequency domain using a parametric approach developed in Robinson (d ≥ 0.5) and thus it does not require preliminary differentiation in the case of nonstationary data; it has a standard N  limit distribution and it is the most efficient method in the Pitman sense against local departures from the null.Therefore, the model estimated in the empirical section is of the following form:Robinson . This mek > 0, where yt1 and yt2 stand for log CPI and log real GDP, respectively. We carry out the analysis using a simple version of the tests of Robinson (I (d) and d may be any real value. In this context, the limit distribution is unaffected by the inclusion of these terms or the specification of the I (0) error term.Fractional cointegration is the extension of the concept of fractional integration to the multivariate case. A necessary condition to test for it in a bivariate context such as ours is that the two individual series display the same degree of integration. Since this condition is not satisfied in our case (see below) cointegration tests cannot be performed; instead we analyse the relationship between the two variables by treating each of them in turn as weakly exogenous, i.e. the other variable is regressed against lagged values of the weakly exogenous one; in other words, we consider the following model:Robinson  that allThe quarterly series used are the consumer price index (CPI) and real gross domestic product (GDP) index 2015 = 100) for both the UK and the US. The sample period goes from 1975Q1 to 2020Q2. The data sources are the OECD Statistics for the CPI series, Eurostat for UK real GDP and the US Bureau of Economic Analysis for US real GDP. Figure 5 = 100 fyt is the time series under examination and ut is assumed to be I (0) and to follow alternatively a white noise or an autocorrelation process, in the latter case specifically the non-parametric model of Bloomfield  hypothesis cannot be rejected despite the fact that the estimated values of d are still below 1. Table Concerning the original series, Table d < 1) is found.As for the logged series  and 1.73 (k = 1 with Bloomfield errors); however, the slope coefficient is not significantly different from zero in any single case. As for the US results  contexts have recently been developed  model (see Pesaran and Shin


Different surgical techniques have been proposed to reconstruct combined anterior cruciate (ACL) and lateral collateral ligaments (LCL). Although these surgical techniques are reliable and reproducible, the number of autologous grafts needed for the reconstruction could be a limiting factor, especially when patients present with multi-ligament knee injuries and the posterior cruciate ligament is also torn. In addition, some of these techniques are not easy to master and have a steep learning curve. We present a surgical procedure that has been used over the last 18 years to reconstruct combined ACL and LCL injuries and has become a reproducible, feasible and time-efficient procedure to approach combined ACL and LCL injuries using an ipsilateral hamstring autograft. Anterior cruciate ligament (ACL) tears associated with injury to the lateral collateral ligament (LCL) of the knee are disabling , 2. In tIn our practice, the diagnosis is by clinical assessment, and confirmed imaging by magnetic resonance imaging (MRI). Valgus stress radiographs are used to quantify the asymmetric lateral joint line widening.With the patient supine and under spinal anesthesia, the knee to be operated is kept at 90° flexion with a support on the lateral aspect of the proximal 1/3 of the thigh and another one under the foot. After exsanguination, the tourniquet is inflated to 300 mmHg. A routine diagnostic arthroscopy of the knee is performed to confirm the diagnosis. If necessary, meniscal injuries are addressed. The ACL stumps are resected. The ACL femoral tunnel is performed by an outside-in ACL guide positioned at a slightly posterior point (5 mm) to the origin of the LCL in the femur, under arthroscopic control. Initially, the bone tunnel has a 6 mm diameter and will be adjusted later according to the diameter of the harvested graft. The fibular head is approached through a 6–7 cm longitudinal incision. The common peroneal nerve is identified and protected before proceeding with the dissection to bone. The diameter of the tunnel is normally 5–6 mm depending on the diameter of the gracilis graft. Through a 5 cm longitudinal incision on the pes anserinus, the gracilis and semitendinosus tendons are harvested with an open stripper, keeping them attached to their insertion on the tibia. Using the incision used to harvest the hamstring tendons, a tibial tunnel 6 mm in diameter is drilled at a 55° angle using a tibial guide under direct arthroscopic control. Again, the tunnel diameter will be adjusted after measuring the diameter of the final graft. After all bone tunnels have been drilled, the length of ACL and LCL graft are measured. Then, the gracilis tendon is detached from its tibial insertion to build the ACL and LCL grafts. According to the length of the tendons, the ACL graft could have a double or triple semitendinosus tendon plus a strand of gracilis. In contrast, the LCL has a strand of gracilis long enough to loop the fibular head and return to the femoral tunnel. The tibial and femoral tunnel diameter is adjusted according to the diameter of the final graft start performing a 6 mm bone tunnel diameter, and then adjust the diameter according to the diameter of the graftCombined reconstruction of ACL and LCL is technically demanding. Some authors have proposed surgical techniques involving more than one femoral tunnel to perform a combined ACL and LCL reconstruction and use different tendon grafts –9, 11. TThis surgical technique broadens the use of autografts to reconstruct combined ACL and LCL injuries and is an exciting alternative for bicruciate ligament reconstruction associated with an LCL tear.