Background: Surgical interventions can cause severe fluid imbalances in patients undergoing cardiac surgery, affecting length of hospital stay and survival. Therefore, appropriate management of daily fluid goals is a key element of postoperative intensive care in these patients. Because fluid balance is influenced by a complex interplay of patient-, surgery- and intensive care unit (ICU)-specific factors, fluid prediction is difficult and often inaccurate.Methods: A novel system theory based digital model for cumulative fluid balance (CFB) prediction is presented using recorded patient fluid data as the sole parameter source by applying the concept of a transfer function. Using a retrospective dataset of n = 618 cardiac intensive care patients, patient-individual models were created and evaluated. RMSE analyses and error calculations were performed for reasonable combinations of model estimation periods and clinically relevant prediction horizons for CFB.Results: Our models have shown that a clinically relevant time horizon for CFB prediction with the combination of 48 h estimation time and 8–16 h prediction time achieves high accuracy. With an 8-h prediction time, nearly 50% of CFB predictions are within ±0.5 L, and 77% are still within the clinically acceptable range of ±1.0 L.Conclusion: Our study has provided a promising proof of principle and may form the basis for further efforts in the development of computational models for fluid prediction that do not require large datasets for training and validation, as is the case with machine learning or AI-based models. The adaptive transfer function approach allows estimation of CFB course on a dynamically changing patient fluid balance system by simulating the response to the current fluid management regime, providing a useful digital tool for clinicians in daily intensive care. In this regard, monitoring of vital signals and periodic assessment of fluid status are crucial. In daily clinical practice, the amount of body fluids gained or lost is usually estimated by calculating the external daily fluid balance . This rePossible CFB courses during a patient’s stay in the ICU have been described previously for critically ill, hyperhydrated patients  and can Based on the calculations for the daily fluid balance, individual goals are usually set for the patient to achieve optimal fluid management during the continued stay in the ICU. However, these goals may change due to alterations in the patient’s health status and the resulting adjustments to ongoing therapy. Therefore, setting adequate and achievable fluid balance goals requires profound and experienced knowledge. In addition to an intake-output-calculation, both currently applied therapies and planned interventions must be considered to estimate a specific fluid balance goal. Therefore, assessing and predicting the fluid balance can be challenging.Recent approaches to predict various physiological parameters using artificial intelligence , deep orIn this work, we present a first-of-its-kind digital model for predicting the patient-specific CFB trend over clinically relevant therapeutic timespans based on single patient CFI and CFB time series data. The algorithm should not only serve to improve the adjustment or optimization of therapies after cardiac surgery, but also to reduce the workload in intensive care units, as estimating the current CFB course is very time-consuming. The control theory model is intended to provide a basis for a more generally applicable algorithm.n = 2061 adult patients collected between 2011 and 2017 was used for sample selection. All patients underwent elective cardiac surgery for coronary artery bypass grafting and/or surgical valve replacement with subsequent admission to the cardio-thoracic intensive care unit at the Medical University of Graz, Austria. The study was approved by the Ethics Committee of Medical University of Graz in accordance with the Declaration of Helsinki (vote EK 30-076 ex17/18).A retrospective data set with a total of n = 926 patients were excluded because length of stay was less than 4 days and hemodynamic stabilization after surgery takes approximately 3 days , which is often used to describe the behavior of a physical or biological system. The TF allows the analysis of the dynamic properties of a system that models the systems output for each possible input function.n = 2. The general mathematical model of the system within the Z-domain is given by Eq. In the discrete-time domain, the TF can be determined using a discrete-time input time series U[k] and the corresponding output time series Y[k]. The Z-transforms of both signals (U[z] and Y[z]) are related to give the TF of the patient, P[z]. Following the control system theory, the transfer function should be chosen to correspond to the lowest-order differential equation to describe the behavior of a system. Basically, from a physiological point of view, oscillation and instability of the response to a fluid input is possible, resulting in the model to be at least a second order system. However, higher order systems led to a worse prediction in general. Possible reasons could be overfitting or the low variability of the input data, which makes an identification of a system depending on multiple parameters more difficult. Since it is not possible to increase the variability of the input data artificially, the transfer function with the empirically determined best result in prediction was selected. The model can basically be compared to a physical two tank water level system. The exponent n see Eq.  that detThe basic idea of CFB prediction using a system theory based modeling approach is shown in 0.For model implementation and prediction MATLAB (version 9.9 - R2020b Update 1 on Windows 10) and the system identification library (version 9.13) were used. The “iddata” function defines the transposed CFI and CFB as a system object. The number of poles of the TF model P[z] was set to 2, the number of zeros to 1, the sampling time was set to 60 s. By entering the “iddata” object and selected parameters into the “tfest” function, a TF object is created by non-linear least-squares search based updates to minimize a weighted prediction error norm. As the CFB is directly calculated from the CFI and CFL there is no delay from input to output assumed and therefore feedthrough must be activated by setting the “Feedthrough” parameter of “tfest” true. This object, along with the modeled and extrapolated input function, was used to simulate and predict the CFB trajectories. The resulting estimated TF with the previously described defaults can be seen in following Eq. Fluid data was imported from an SQL data base and included intakes  and losses  with their corresponding time points. The cumulative time series data were interpolated individually dependent on the specific type of administration or type of fluid loss using the “interp1” function for the entire length of stay at a sampling rate of one data point per minute. Interpolation was performed by inserting the last available value that followed the cumulative shape of the fluid data.Gravity infusions that did not have a documented ending timestamp were assumed by clinicians with a flow rate of about 16 mL per minute until the total administered volume was reached. Short infusions and drinking were included as bolus. Insensible losses such as estimating the volume lost by sweating were not considered in the data. Urine data was linearly interpolated to the next available documented entry with the simplified assumption that the bladder has filled evenly in a linear fashion during this time.The resulting individual CFI and CFL time series were subtracted to calculate the corresponding CFB to identify the TF. The described procedure for preprocessing the input and output data can be seen in detail in the flow chart in estimation time) and the last 24 h of each window used for prediction (prediction time). The sliding window was incrementally shifted by 2 h. Since fluid balance is constantly changing due to dynamically changing patient characteristics, the TF has to be periodically recalculated. A sliding window approach was chosen for TF estimation for (i) computational reasons and (ii) clinical considerations, because patients may become unstable again, which affects subsequent estimates. The length of the sliding window was set at 48, 72, 96, 120 and 144 h, with the first 24, 48, 72, 96 and 120 h used for TF estimation  within RStudio . The simulated (predicted) CFB course was compared with the real CFB within the maximal prediction period of 24 h. The root mean squared error (RMSE) of the last 30 min between both curves was calculated to determine the change of prediction error over time for the selected prediction times of 8, 12, 16, 20 and 24 h. This results in different prediction error combinations for model estimation (24–120 h) and model prediction (8–24 h) for a patient. Depending on the length of stay of each patient, not all combinations were available for all patients  is provided to the model. As can be seen in The model presented only estimates the behavior of the entire patient system by analyzing fluid intakes and losses and does not simulate a mechanistic fluid compartment system. Distributions of the fluids in the compartments and changes over time are therefore not included. Increased fluid retention due to, for example, a sodium excess or the accumulation of fluids in edemas leads to changes that are usually not relevant in the prediction window and can be therefore partially neglected. However, it must be emphasized at this point that the goal of this work was to find an algorithm that could be used with the existing data in a realistic scenario in daily clinical practice and that does not require additional subjective assessments or other biomarkers.In summary, the presented digital model is not based on machine or deep learning and therefore does not require large amounts of data for training and validation of the model. It allows for early recognition of a change in the CFB course, which leads to faster adjustment of treatment. The system theory model, based on single patient data, thus shows high potential for adapting and optimizing fluid management in critical care situations for a clinically feasible prediction window and could be further developed into a decision support system for fluid balance and imbalance profiling. With the here presented approach it is therefore possible to support fluid management with already existing data - with the mentioned limitations. Its advantage is its ability to reduce workload by, for example, taking over complex calculations or informing responsible clinicians of trends in changing health parameters.Additional clinical studies are needed to further improve the presented approach in terms of methodological accuracy and clinical validity by considering additional relevant clinical factors. The next steps toward a decision support system to assist clinicians with fluid management are the introduction of a feedback mechanism. The use of a dynamic, non-linear function instead of the presented TF and its embedding in a semi-closed loop could allow both the estimation of CFB trend in response to different fluid intake regimes and support the identification of the most appropriate CFI trajectory to achieve targeted CFB goals when treating individual patients.


In spatial frequency domain imaging (SDFI), tissue is illuminated with sinusoidal intensity patterns at different spatial frequencies. For low spatial frequencies, the reflectance is diffuse and a model derived by Cuccia et al. (doi 10.1117/1.3088140) is commonly used to extract optical properties. An improved model resulting in more accurate optical property extraction could lead to improved diagnostic algorithms.To develop a model that improves optical property extraction for the diffuse reflectance in SFDI compared to the model of Cuccia et al.We derive two analytical models for the diffuse reflectance, starting from the theoretical radial reflectance The model based on the PCBC resulted in the lowest errors, improving median relative errors compared to the model of Cuccia et al. by 45% for the reflectance, 10% for the reduced scattering coefficient and 64% for the absorption coefficient.For the diffuse reflectance in SFDI, the model based on the PCBC provides more accurate results than the currently used model by Cuccia et al. The amplitude modulation To obtain ,–To extract tissue optical properties from SFDI measurements of Two analytical models exist for SFDI, the model of Cuccia et al. for the diffuse regime2,As an alternative to directly solving the diffusion equation for a uniform plane illumination, a tissue “impulse response” can be obtained by computing the reflectance as a function of radial distance, ,In general, diffusion theory assumes infinite media. For semi-infinite media such as tissue, the refractive index mismatch between the sample and the medium above it results in a significant fraction of radiant energy being reflected back into the sample upon interaction with the boundary. Analytical expressions for The reflectance as a function of radial distance is then found as Performing the integrations for 3Since the model of Cuccia et al. was based on the PCBC, we first compare We also determined the relative errors in the values of 3.1We simulated a pencil beam illumination and collected photons versus radial distance from the source with a 0.001-mm bin size and 4We first compared the PCBC . While tAn example of the reflectance versus While .5  mm−1 . Using R5We developed a new model for the diffuse reflectance in SFDI that reduces the error in the estimated reflectance and the extracted optical properties compared to the currently used model of Cuccia et al.We compared the resulting equations for two models for –There has been a debate in literature on whether or not the absorption coefficient should be incorporated in the diffusion coefficient for the analytic solutions to the PCBC and EBC.Apart from analytical models, approaches exist to extract optical properties from SFDI measurements based on MC simulations—either employing look-up-tables or machine-learning algorithms. MC simulations can include the details of the optical setup that is used  and look-up-tables and machine-learning algorithms can improve the speed of optical property determination. For medical applications where speed is essential, such as endoscopy,In conclusion, we investigated the diffuse reflectance in SFDI by comparing the currently used model of Cuccia et al. to two new models based on integrating the theoretical response function


While cluster synchronisation appears crucial to healthy brain function, it may also be pathological if it leads to unbreakable local synchronisation which may happen at extreme topologies, with implications for epilepsy research, wider brain function and other domains such as social networks.Significant research has investigated synchronisation in brain networks, but the bulk of this work has explored the contribution of brain networks at the macroscale. Here we explore the effects of changing network topology on functional dynamics in spatially constrained random networks representing mesoscale neocortex. We use the Kuramoto model to simulate network dynamics and explore synchronisation and critical dynamics of the system as a function of topology in randomly generated networks with a distance-related wiring probability and no preferential attachment term. We show networks which predominantly make short-distance connections smooth out the critical coupling point and show much greater metastability, resulting in a wider range of coupling strengths demonstrating critical dynamics and metastability. We show the emergence of cluster synchronisation in these geometrically-constrained networks with functional organisation occurring along structural connections that minimise the participation coefficient of the cluster. We show that these cohorts of internally synchronised nodes also behave  Significant research has investigated how the brain network, leads to diseases such as epilepsy. To date, most of this work has explored connectivity on the macroscale; between whole brain regions. Here we explore how the topology of mesoscale networks, between cortical columns, could affect functional behaviour. We use a model of a simple oscillator; the Kuramoto model, with a variety of networks to investigate how the network topology affects how these oscillators synchronise with each other. It is hypothesised that healthy brain operates at a critical point where regions can both synchronise and desynchronise with each other easily which is dependent on the strength of connections between nodes. We show that the topology dramatically affects this behaviour and that certain patterns allow this critical behaviour to exist over a wider range of connection strengths, making the system more robust. We show the oscillators synchronise in clusters, which themselves behave as if they were individual oscillators, and show interactions with other clusters. While this behaviour may be necessary for healthy brain function, at extreme topologies this may become pathological if it leads to unbreakable cluster synchronisation with implications for epilepsy research and other domains such as social networks. Synchronisation is a widely studied phenomenon across a huge range of scientific domains and across an equally long period of time. One of the most widely used models in the study of synchronisation between coupled oscillators is the Kuramoto model ,2. The KAs the field of network science has expanded in recent years, so too has work on the application of Kuramoto oscillators to networks –6. The aOne of the properties which makes systems of coupled Kuramoto oscillators so widely studied is that they exhibit criticality . A systeWithin the field of neuroscience there has been an explosion in the use of network science to probe the function of the brain in health and disease . CommuniAs the ability of the brain to transition fluidly between states is posited to underlie normal functioning of the brain, so too has disruption in this ability, which can be quantified by metastability, been hypothesised to result in brain dysfunction in disease –25. ThisMost of this work has been done on structural and functional networks derived from tractography and functional MRI (fMRI) respectively, which looks at brain networks on a region-to-region scale, or macroscale. However, particularly within the field of epilepsy research, there has been work looking at how network effects at the mesoscale level underpin epileptogenicity –33. In aOur aim with this work was to therefore explore the effects of topology on the dynamical behaviour of a mesoscale model of neocortex. In doing so we will be investigating a spatially distributed network of nodes, where interaction between nodes is subject to a delay determined by the distance between nodes, and a simulated conduction velocity. Our interest was in how significantly the behaviour of the system was altered by changes in topology, and whether this gives any insights into how local topology could affect phenomena such as epileptogenicity or seizure spread.u and v is given by:Geometrically constrained, directed, unweighted, random networks were created by arranging 1600 nodes in a 40 by 40 hexagonal grid , wrappedE represents the Euclidian distance between two nodes. Edges are added to the network until reaching a total of 25600, giving an edge density of 1%. Edge density is held constant for every generated network. At low values of η, nodes make connections over a wide range of distances, while at high values nodes have a strong preference to make connections only with nearby neighbours.Where θ, and its evolution for a given oscillator θi is given by:Each node is represented by a Kuramoto oscillator. The model is represented by a single variable; the phase of the oscillator, λ represents the global coupling strength, ωi represents the intrinsic frequency of the given oscillator, A represents the randomly generated adjacency matrix, and Si represents stochastic noise. Each oscillator is randomly assigned an intrinsic frequency from a normal distribution with a mean frequency of 60Hz and standard deviation of 3Hz. Noise is included as a random phase deviation and per timestep is drawn randomly from a normal distribution with mean 0 and standard deviation of 0.04 radians.Where η ranging from 1 to 5. Each node in the network is represented by a Kuramoto oscillator. The distance between nodes is set at 0.5mm, and the conduction velocity as 4m/s. These numbers were chosen as they represent an approximate distance between cortical columns in human neocortex [Networks were randomly generated at values of eocortex , and a ceocortex ,40. To aθi is taken at the current timepoint, (t), while the phase of every other node, θj, is taken at time (t−dtij), where dtij is the conduction delay for the edge connecting nodes i and j. It is calculated using the conduction velocity above and the Euclidian distance between the nodes.Where for each node the phase η 16 different, random networks were created and for each network simulations were run over a range of 16 different coupling strengths. For each simulation the intrinsic frequencies of the nodes were randomised.Simulations were run for 10 seconds, with a time step of 0.1ms using an Euler solver. For each value of η and λ. For η, 1 represents a low value, 3 a medium value and 5 a high value. For λ, we use 4 as a low value, 28 as a medium value and 60 as a high value.As shown in r, for the system as a whole, as the mean of the complex phase vectors of all the nodes at each time point:To analyse the data, we calculate an order parameter, φ represents the mean phase of the system. The mean of the order parameter, r, over the whole simulation can be considered as the synchrony of the network, while the standard deviation of r represents the metastability of the network (η = 0) we see the characteristically described behaviour of a system of coupled Kuramoto oscillators, with a critical transition point as the coupling strength is increased, along with a peak in metastability  as coupling strength increases there is a resistance to global synchrony. There is no sharp transition point and the metastability of the network continues to rise to significantly higher levels over a wide range of coupling strengths. This additionally manifests itself when looking at the average synchrony of nodes with their immediate neighbours  highlights the effect of delays in the slightly off 0 phase difference between modules seen at the critical coupling strength in low networks . MoreoveWhat we see in this work is an example of cluster synchronisation, whereby discrete groups of nodes within a network synchronise internally with each other, and distinctly from the rest of the network. This is an area of research which has received significant attention recently due to its intuitive relationship with neural function . The braRecent work has focussed on describing a theoretical basis for the emergence and stability of such clusters –49, suggThere are however notable implications from this work. Firstly, we have looked at networks which are spatially arranged, with delays in transmission between nodes, in line with real-world networks. The networks we have looked at have been entirely random, with no structure or modularity imposed on them and are neither scale-free nor small world. What stood out in early exploration of the network morphospace , where wη, and thus increasing the clustering coefficient and path length and decreasing efficiency, had the effect of both significantly increasing metastability and widening the range of coupling strengths over which it was raised. This would suggest these networks are much more robust to changes in coupling strength while maintaining the ability to function at a critical point, with an ability to locally synchronise and form coalitions.Given the importance placed on metastability in efficient cognitive function it is notable that the effect of increasing η≥4. The clusters are spatially arranged; that is nodes which are close together are more likely to belong to the same module. A proportion of the nodes within a cluster remain stably within the cluster, while nodes which border clusters are more likely to switch between clusters repeatedly during a simulation. This results in the high degree of metastability seen in these simulations, as a significant proportion of the nodes in the network as a whole are recurrently switching synchronisation between different clusters.As we move to explore the functional behaviour with changing topology, we can see the emergence of functional clusters; sets of nodes which synchronise closely with each other but not with the rest of the network. This behaviour only emerges around the critical coupling strength of the system, and only in networks with In Figs By looking independently at the clusters within the networks over time we can also show the dynamic and metastable behaviour of these clusters. By calculating the mean phase of each cluster, we can analyse how the clusters interact with each other and show that the cluster as a whole behaves much like an individual node and they show weak coupling to neighbouring clusters. Each cluster has its own mean phase and intrinsic frequency, which proceeds relative to other clusters and shows gradual phase divergence from other clusters.Two weakly coupled Kuramoto oscillators show a characteristic pattern of phase divergence. When their phases are between 0 and π radians apart there is a slow phase divergence, as the sine of the phase differences acts to slow the phase progression of the faster node and speed up the progression of the slower node–it acts to make the nodes converge in phase. As the phase difference exceeds π radians, the sine of the difference now acts to speed up the faster node, and vice versa, acting to make the phases diverge and leading to rapid phase change until the phase difference returns to 0 radians. We can see in In addition, at the point that the behaviour transitions to repulsion and we see the rapid phase divergence between clusters there is a simultaneous rapid internal desynchronisation within the clusters. Given the stability of some nodes within clusters this can be explained as the bulk of the metastable nodes desynchronising from the cluster. As the phase difference between the clusters returns to the range 0 - π radians, this is associated with internal resynchronisation and the metastable nodes joining the same or a different cluster to previous.η networks show rapid increase in cluster interaction strength, and hence global synchronisation.Again, fitting with the idea that these more geometrically constrained networks show more robust critical behaviour we can calculate the interaction strength between clusters and show the weak coupling emerging at the critical coupling strength but then only very gradually increasing over a wide range of coupling strengths. In contrast the low Considering these results from the perspective of a model of neocortex it highlights the importance of network topology in allowing healthy neural functioning. Geometrically constrained connections are both biologically more cost-efficient, but also allow local synchronisation, and critical dynamics over a wider range of coupling strength resulting in a more functionally robust network. We know most real-world networks, and particularly brain networks show small-world and scale-free properties ,53. We wη networks may also become pathological if they entrain stable, local synchronisation that is robust against external input. In contrast to a generalised network problem, this would be a focal network disorder. What would make this more difficult to recognise in existing work is that the networks would likely still demonstrate high metastability due to interactions between clusters and the behaviour of large cohorts of nodes at the boundaries between clusters. Exploring the ability of external input to break cluster synchronisation would be important to help elucidate at what point increasing η results in pathological functional behaviour, where a network will generate an area of local synchrony which is persistent and unbreakable.A further avenue of research would be to explore the concept of external control of synchronisation. If such a sheet of neocortex operates at a critical point such that nodes can easily synchronise and desynchronise to transfer information, then it suggests that such a network should be susceptible to synchronising with an external input applied to a set of nodes. If we consider rapid global synchronisation of the whole network to be pathological–we need metastability in healthy brain function–then high While this work is theoretical it also raises hypotheses subject to investigation in real world data. In the brain, particularly in a disease such as epilepsy, it presents questions as to whether subtle changes in connectivity at the mesoscale underlie epileptogenicity. In acquired epilepsy following TBI or stroke, is abnormal rewiring leading to topological changes which result in abnormal synchronisation, either by increasing the distribution of edge length and therefore predisposing to strong synchrony over a wide area or is rewiring extremely local resulting in a very high clustering coefficient and unbreakable local synchronisation. Could similar processes underlie the epileptogenicity of cortical dysplasia or dysembryoplastic neuroepithelial tumours?η networks. It may simply be a phenomenon that within highly clustered networks local synchronisation will happen spontaneously if there is sufficient strength of interaction between participants. It may also be that sufficiently extreme local topology may render such areas prone to local synchronisation that cannot be broken by external influence. Ultimately, further research on cluster synchronisation and robustness of local synchronisation may prove a fruitful avenue of research in brain and social networks.Outside the brain there are implications in social networks. Much is made of the existence of echo chambers in social networks, which much discourse aimed at the use of algorithms and confirmation bias as contributing to their formation ,55. Yet S1 FigA reproduction of (PNG)Click here for additional data file.S2 Figa) As seen in b) The low participation coefficient is correlated with a decrease in the mean distance of member nodes of a module to its centre of mass, showing the module becoming confined in space.(PNG)Click here for additional data file.S3 Figa-c) Show high η networks, d-f) show medium η networks. In b) which shows a high η network at the critical coupling strength we see the modules have clustered in space (particularly relative to a and e), and there is only a relatively small number of black inter-module connections reflecting a low participation coefficient. Lower η networks show a much higher proportion of black inter-module connections, as do the high η networks at extreme connection strengths.In single simulations a timepoint is taken and the network is shown with each node coloured according to the module it belongs to at that time point. Structural connections between nodes belonging to the same module are coloured the same as the nodes. Connections between nodes in different modules are shown in black. The connection strength varies from low to high, from left to right. (PNG)Click here for additional data file.S4 Figa-c) Here we show an η = 5 network at three different coupling strengths, 4, 28 and 60. In green we show the phase divergence between the two largest modules, in blue we show the total number of nodes which change module assignment within that time window. Of note, in b we see the pattern of weakly interacting modules with slow then fast phase divergence. There are peaks of nodes assignment changes which occur shortly after a fast divergence phase at a time when the modules are internally resynchronising. d-e) show the mean and standard deviation of the number of assignment changes over a whole simulation as a function of connection strength for a range of η. f) Shows the mean distance of modes from the centre of mass of their assigned module as a function of connection strength and η. In high η networks, through the critical coupling point, the mean distance decreases showing that modules are clustering spatially.(PNG)Click here for additional data file.S5 FigBy rows, high η networks at the top to low η networks on the bottom row. By columns, low coupling strengths, then critical and high coupling strengths from left to right. Distribution of absolute phase differences between detected modules over latter half of simulations to exclude initial transients. At low coupling strengths, there is a uniform distribution of phase differences reflecting the lack of interaction between modules. At the critical coupling strength, in the high η network, you can see that modules spend most time with a phase difference around π/2 and 3π/2, which reflect the peaks of the sine function–and thus the period of greatest interaction strength. As η reduces there is a push towards synchrony and the modules spend most of the time in a very narrow phase range with peaks centred just off 0 and 2π radians, reflecting the effect of delays. At the maximum coupling strength these peaks of phase difference move to 0 in the low η networks, while the high η networks mimic the low η networks at lower coupling strengths.(PNG)Click here for additional data file.S1 Code(ZIP)Click here for additional data file.


The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation. Conclusions for the impacts of the base stiffness, position, and amount of combined mass on the change of the vibration cycle graphically in non-dimensional presentations were demonstrated1. A brand-new factor iteration method to analyze the Duffing equation as having powerful and high-order nonlinearity was suggested2. It was demonstrated that the nonlinear frequency gives rise to accurate outcomes, in contrast to the linearized approach, parametrized perturbation procedure, and variational repetition procedure suggested by Prof. He. In order to arrive at analytical approximations and numerical explanations for the cubic-quintic Duffing-van der Pol oscillator, a number of methodologies, considering the prospective purposes in manufacturing, integrated circuit technology, physical processes, and natural science was employed3. The parametric Duffing oscillator stability evaluation in light of its numerous uses in science and engineering was examined4. Both resonance and non-resonance situations were examined. The perturbed solutions and the stability analysis were graphically verified by numerical approximations. Using the combination method, a uniform explanation of the quintic Duffing equation was reported5. This method involved developing the restoring force in Chebyshev's nonlinear differential equation and approximating it with a cubic Duffing oscillator where the constants for the linear and cubic components change on the primary amount.The computational asymptotic methodologies of nonlinear applications seemed to be of great significance to engineers and scientists as nonlinear science advances so quickly. However, we can easily use simulation studies to find solutions of linear systems. It is still extremely challenging to solve nonlinear problems theoretically. One of the perfect examples of a Hamilton system was a Duffing oscillator. Simple generalizations of these oscillators, including cubic-quintic Duffing oscillators, haven't been thoroughly researched7. Ganji et al.8 developed a cubic-quintic Duffing oscillator approach to approximate the performance of an inflexible rod swaying on a circular surface with no sliding. Khah and Ganji9 used the energy balancing approach to analyze the previous problem8. The new methodology showed high effectiveness and convenience and lacked the necessity for linearization or tiny perturbation. Additionally, a uniformly rigid rocking rod was analyzed10.In real life, there were occurrences where objects were not steadily attached to their framework but instead let rock or drop on the supporting material. These include unsafe supplies in transportation, petroleum cracking towers, air distillation columns, liquid gas tanks, and nuclear fuel cells in reactors. The most prominent example of how structures behave when they rock was certainly when they shake during earthquakes. Despite their familiarity and apparent simplicity, the rocking and overturning of stiff bodies in response to foundation stimulation pose difficult problems. The main motivation for understanding the issue has been the possible use of the rocking problem to prevent machinery, furniture, and structures from toppling over and endangering people when subjected to shaking during an earthquake. Satisfactory results of the isolation system rocking were obtained in Refs.11. For instance, even for very low time delays, a classical Duffing structure with delayed speed response displays an endless variety of regular movements; yet if the duration delay vanishes, the dynamics of this system are extremely straightforward. Delay feedback control, in contrast, has proven to be one of the most reliable and adaptable methods for controlling chaos in nonlinear dynamic systems12. Over the past few decades, controlled mechanical systems with time delays have received a lot of attention. Additionally, researchers in various disciplines, including biology, population dynamics, industrial machinery dynamics, and neural networks, have paid close attention to the dynamics of delay prototypes13. Through the use of a Duffing equation with delayed speed response, Wang and Hu14 presented research work on the viability of perturbation approaches as the multiple scales technique, and the Lindstedt–Poincaré methodology, to mention a few. Tunç15 provided the necessary conditions. Through building a Lyapunov function, a new finding was generated that incorporated and enhanced certain related results already found in the pertinent literature. An excited Van der Pol-Duffing oscillator nonlinearity was suppressed using time-delayed position and velocity16. The time delay served as an additional safeguard compared to the nonlinearly vibrating system under consideration. Technologies with a temporal delay have recently been the aim of various investigations; therefore, the current study is particularly examined.In industrial control mechanisms, time delay is common. It may become unstable or function poorly if there is a time delay, making it more difficult to accurately examine the system. In other words, time delays are a constant feature of controlled system feedback and have a significant impact on their dynamics17 utilized the Poincaré–Lindstedt technique to arrive at a roughly restricted solution for the hybrid structure. It was discovered that the approximate solution together with the RK4 were comparable. In reality, the presence of a little parameter in the scheme under study is required for all perturbation methods. Consequently, without this parameter, the problem is somewhat restricted. The Mathematician Chinese Prof. He18 discovered a novel perturbation technique that was not dependent on such a little parameter. This method allows for the placement of a small artificial embedding parameter where 19 and 20 used an extended frequency concept combined with the HPM and Laplace transform to analyze a parameterized Duffing equation to arrive at a valid constricted formula of solution. There have been recent efforts that were connected to the current manuscript23.As is commonly held, the majority of practical and technical implementations demand nonlinear equations. Perhaps, functional, differential, integral, or integro-differential equations make up these equations. These equations are quite challenging to obtain an exact solution. Subsequently, it becomes fundamental to employ numerical solutions in different directions. Regardless of the analytical approach, numerical solutions are more informative during specific time intervals. Therefore, perturbation approaches have developed in a variety of ways, from the standard strained factor approach to methods including those that depend on so many time scales. He et al.Given the implications of the above-mentioned features, and because of their extreme sensitivity to dynamic loading, differences in geometry, and dissipation difficulties, evaluating the rocking and overturning responses of rigid blocks to earthquakes is a challenge. This paper introduces the literature on traditional and cutting-edge rocking motion theories. This work has the goal of examining the motion behavior of rigid rocking rods with no slippage. The governing equation of motion (EOM) is regulated as an ordinary differential equation (ODE) contract with classical mechanics. The current study uses a connection between the Laplace transform and homotopy perturbation methods to identify perturbed solutions. The association of the time delay and its relationship with the improved HPM is what gives the interesting paper its uniqueness. It is possible to perform stability analysis using a nonlinear frequency. The residual of the text is constructed as follows: Finding a rough solution for nonlinear oscillation with an extended frequency is the focus of “10, the EOM of the uniform rigid rocking rod has been derived and taken the following form:Returning to our previous paperEquation  should jmentclass2pt{minim23. It was compared to the previous results to show how well the applied time delay suppressed nonlinear oscillations of the structure. Nevertheless, when the ideal time delay was considered, the generated theoretical and computational inspections indicated that the nonlinear position and nonlinear velocity were the best at suppressing the vibration. Furthermore, Saeed et al.24 suggested a straightforward technique for designing the ideal loop delay amounts in a such way that enhances the system profile. Additionally, Saeed et al.25 looked at the effectiveness of time-delayed linear and nonlinear feedback controllers for position, speed, and acceleration. According to investigators, the time-delayed cubic acceleration control was the best at suppressing bifurcations and reducing vibration. Therefore, in light of the aforementioned achievement, we lay prominence on the square time delay in the position and velocity of the existing model.As is well known, time-delayed control was suggested to manage the nonlinear vibrations. The loop delay can significantly contribute to both stabilizing/destabilizing the structure, as was earlier shownFrom now on, the EOM becomes:It is preferable to picture the initial conditions as follows:26, the HPM can provide a wide range of approximation solutions. One of these methods leads to a conventional solution that includes secular terms, and the cancellation of these secular terms provides a trivial solution that is not required. An alternative one produces a consistently satisfactory solution by using the expanded frequency conception; unfortunately, the obtained solution does not pass through the numerical solution. It is therefore required to modify Homotopy once again. Therefore, one is able to re-examine the fundamental Homotopy equation utilizing a recent development in place of the traditional extension to look into the consequences of the delay parameter which is superior at preventing bifurcations and decreasing vibration. Subsequently, we think that 16. The processes to get the needed solution are as follows:As clearly explained in our previous studies28:The HPM is built generally on the fundamental Homotopy formula20:Applying the technique of the previously comprehended examination, the established natural frequency can be increased as shown belowOne insert Eqs.  and 7)  7) into It is observed that the time delay of the zero-order solution is given as:The Homotopy Eq.  first-orUsually, the secular terms are dropped to create a phrase that is consistently acceptable. The coefficients of the circular functions Now, the first-order uniform formula is characterized asAccordingly, the approximate uniform formula of the fundamental equation of movement described in Eq.  may be dConsequently, the uniform estimated formula of Eq.  may be cActuality, the uniform estimated formula in Eq.  needs thEquation  represenThe approximated uniform solution, which is stated in Eq. , can be The above solution 17) has been drawn in Fig.  has beenentclass1pt{minimaIt is suitable to evaluate this procedure with the numerical solution (NS), which can be obtained using the computational approach that is identified to RK4 to evaluate the practicality of the formerly expanded frequency implications. The requirements for implementation are listed below. Therefore, the analytical solution (AS) as given by Eq.  is drawnThe computations demonstrated that the synthetic frequency has an amount , while the others are graphed in the fourth quartile. Consequently, there is symmetry in the curves drawn around the Equation  of the fs of Eq. , in whicThe purpose here is to depict the stability analysis as well as the phase portrait of the EOM in the absence of the time delay as given in Eq. . For thiThe equilibrium points happen at the points It follows that the fixed points are given at In considering the Taylor series, the functions 3, the eigenvalues As previously shown3, the final classification can be used for this methodology, as seen in Table The equilibrium point is traditionally considered to be stable if all eigenvalues, which are computed at the equilibrium points, have negative real portions. As previously shown29. One might also think of the explanatory variables as a function of This method is used to calculate the stability behavior of Eq.  for variFor this objective, one starts by presenting three different independently variable quantities corresponding to:Consequently, one gets:The solution of Eq.  may be cIn view of the Homotopy perturbation, Eq.  might beFor more accessibility, to achieve a precise formula, we restrict our analysis up to Substituting from Eqs. –26) int int26) iWith this methodology, it is appropriate to create the solution of Eq.  in the fHere, Substituting Eq.  into Eq.The removal of the undesired parts, in Eq. , at the It worth observing that the consistently reasonable formula of Eq.  can be rAgain, the exclusion of the unsought pieces, in Eq. , at the Equation  may be sd by Eq. , to findTo go back to the initial varying duce Eq.  with therate Eq.  by \docuAs usual, the solution of Eq.  may be fCombining Eqs.  and 37)37), one Solving Eqs.  and 36)36), one Curves of Fig. The variation of the function 10, the EOM was developed on the basis of the Euler–Lagrange theorem. This work applies a time-delayed square position as well as speed to examine the stability analysis of a rocking rigid rod in line with the perception of time delay. The subject of this inquiry is extremely significant since time-delayed technology has recently been the subject of several investigations. A coupling of the enlarged nonlinear frequency notion and the HPM is employed in view of the time delay. This investigation is significant since the time-delayed technology issue of this investigation is extremely relevant. An exponential term is also introduced to obtain a rough solution of the EOM in light of the time delayed factor. The temporally histories of the obtained solution have been plotted at various values of the used parameters. It is noted that the behavior of graphed curves has a periodic form, in which the phase plane plots of the obtained outcomes assert their stabilities. To validate the accuracy of the theoretical outcome, a comparison with the RK4 is provided. The present study allows a careful analysis of the conclusions reached using the analytical strategy of practical approximation. The time delay acts as an extra protection against the system potential for nonlinear oscillation. In contrast to earlier research, the methodology arrived at the current answer is notable for being effective, promising, and simple. The multiple scale method is used to test if it can be applied to additional nonlinear oscillators even with the organized nonlinear prototype approach. It has investigated how various regulatory constraints affected foundation vibration achievements.The present investigation looks at the subject of an inflexible rod revolving on a solid rod as a spherical surface with no sliding. In our previous study33 is suitable for use as a prototype for the latest research. With this methodology, the nonlinear EOM as shown in Eq. (Another non-perturbation procedure built on the well-known He's frequencyn in Eq.  is trans


Present understanding of protein structure dynamics trails behind that of static structures. A torsion‐angle‐based approach, called the representation of protein entities, derives an interpretable conformational space that correlates with data collection temperature, resolution, and reaction coordinate. For more complex systems, atomic coordinates fail to separate functional conformational states, which are still preserved by torsion angle‐derived space. This indicates that torsion angles are often a more sensitive and biologically relevant descriptor for protein conformational dynamics than atomic coordinates. Proteins naturally sample the conformational space by variation in torsion angles, which have a significantly lower energy barrier than changing bond lengths and angles. This powerful space shows correlations with the resolution of the data set, data collection temperature, space group and bound ligand, which goes beyond what is achievable using atomic coordinates.Beyond experimental data collection, we have computational tools to analyze structures. Individual crystal structures can be analyzed using tools such as https://rope.hginn.co.uk).This multi‐dataset analysis is named representation of protein entities (RoPE). The product of this analysis is a representation of protein conformational space (hereby referred to as RoPE space). What were previously considered subtle shifts in the protein's conformational state are starkly separated in RoPE space and provide a means for classification and comparison. RoPE is a potentially powerful tool that can also be used to drive experimental decision‐making to help experimentalists obtain data suitable to address complex biological questions dominated by protein dynamics. The algorithm is contained within a software package and available as an open‐source application  reference protein has been chosen. This decision is entirely avoided by using torsion angles, which report only on the internal relationships in the protein and does not require a reference. The following cases are reported in order of increasing complexity, where torsion angles remain useful as atomic coordinates become increasingly incapable of separating conformational states.2.2A time‐resolved experiment on the dynamics of photoexcitation of carboxymyoglobin to initiate loss of carbon monoxide consists of timepoints between −0.1 and 150 ps from the point of photoexcitation , followed by Create.The RoPE concept revolves around protein entities, which are defined by a primary sequence. In the case of alcohol dehydrogenase, this is the sequence defined in the UniProt database. From alcohol‐dehydrogenase and click Search PDB > Run. Click Download to fetch the PDB files from the PDB‐redo server into the local file system, which happens in the background.In order to automatically fetch proteins from the PDB, enter the entity by clicking on Back button to return to the Protein Entities menu. Click the toolkit symbol in the top right and click automodel. This will automatically turn all PDB files within the system into models. It will also automatically associate existing entities with parts of a model if they have over 80% sequence identity.Files loaded into the system do not automatically translate into the RoPE definition of models. Models are linked to PDB files and can carry additional metadata. In order to make models, click the Menu > Save. The web application will issue an invitation to save a rope.json file to the local filesystem, whereas the native application will save rope.json to the project folder. Note that for generating figures, navigating to Menu > Options brings up the option of changing the background to pure white.This is a good time to click back until returned to the main menu, and then click alcohol‐dehydrogenase and click on RoPE space. At the moment all models are unrefined and you will be asked if you wish to refine them. Click Yes. This is the right point to schedule a tea break.Once modeling is complete, now the Vagabond algorithm can be run to optimize the torsion angles to best match the atom positions. In order to do this, click on Back button to reveal the RoPE space. This is also a good point to save your refinement progress. Each icon represents one monomer of alcohol dehydrogenase. This RoPE space can be rotated with a left mouse‐button drag, zoomed with a right‐click drag, and panned with a Ctrl + left mouse button drag. This RoPE space currently lacks annotation but it is clear that there is some substructure to the occupied set of torsion angles in alcohol dehydrogenase. Molecules that are close together have similar conformations. We need to know more about what each structure contains, so click Back again to return to the entity menu.Once all proteins have been refined, click the Add to metadata and then choose from the Protein Data Bank in order to automatically download supporting information such as resolution, collection temperature and space group. Note that this uses the model name as a search term, and can fetch information when this matches the PDB code. Once this has downloaded, click on Add to database to add this to the known metadata.Click on back button and return to the RoPE space. It is now possible to add annotations to this space with the additional metadata. The coenzyme‐bound, ligand‐free holoenzyme of horse liver alcohol dehydrogenase often crystallizes in P1, so this can be assigned a different icon. Click on the rules menu and click on (+) new rule. The options at the top specify what kind of rule to add. To select a discrete value, click on the Change icon. In order to change what metadata title this responds to, click on Choose… and then click Space group. Enter the Options menu to change the properties of the metadata selection and the icon type. The default is to change the icon type on the basis of having any kind of assignment. However, all our models have a space group of some form, and we only want to select on P1. In order to narrow down the value, click Choose… to see the options. On the right‐hand side is the number of models for each assignment. Choose P 1 and this will return you to the options menu. Choose an icon type  and press OK, then Create, and then OK to see the effect on the RoPE space.Click the 7uee_A and 7uee_B are at the top of the V). On the other hand, P1‐structures with minimal separation are found at the pit of the “V” such as 1qv6_A and 1qv6_B. In order to focus on the P1‐structures only, click on the icon legend Space group = P1, on the right‐hand side, and click choose group. This will re‐run the RoPE algorithm on the specified subset of structures.The RoPE space now shows that two of the variations of direction are covered by P1‐only space groups which form a “V” shape. Mouse‐over the icons reveals the model name and chain number of each monomer. Each P1 model comprises a dimer, and each direction contains one monomer from each model. Inspection reveals that each monomer of a dimer is in roughly similar positions up each branch of the “V.” Those at the top of the “V” have the greatest variation in conformation between monomers  pointing toward the other P1 branch and click on explore axis.Each direction in RoPE space corresponds to a variation in all torsion angles across the entire molecule. The exact nature of these torsion angle motions is determined by principle component analysis (PCA) during clustering. This chooses torsion angle motions which show evidence of independent motion between structures, which is naturally limited by the sampling of the conformational space provided by the structures. In order to establish what these torsion angle motions mean, it is possible to draw the motion of a torsion angle on a reference structure. As an example, right‐click 7uee_B, and load the torsion angle changes associated with one of the principle component analysis (PCA) axes. If there are torsion angles missing from the 7uee_B structure then a dialogue window will give an outline of the problem. It is good to manually check for disordered regions or deletions/insertions which may distort the motion in the middle of the protein backbone. Torsion angles missing from the N‐ or C‐terminus, or in sidechains, are less detrimental to the overall motion of the protein. Press OK to dismiss the warning.This will bring up a structure of your reference, The slider at the bottom allows you to slide the position in RoPE space along your chosen PCA axis, ranging from −1 to 0 to +1. The colors of the torsion angle are proportional to the total change in Ramachandran angles for each Cα atom. Brightly colored residues therefore denote hinge regions in the structure. Note that the torsion angle motion does not correspond to any single motion between two structures, but uses the information across all data sets to derive the principle independent motions. It is also displayed independently of any partially correlated motions which may accompany such a transition. The same RoPE axis will also produce different atomic motions depending on the reference structure. However, inspection of this motion shows a collapse of the active site of the protein. Reference to an external model‐viewing program suggests a collapse of the distance between residues S118 and P296 across both sides of the active site to produce a closed conformation.back until returned to the Entity‐alcohol‐dehydrogenase menu and click Sequence. This loads the sequence from the UniProt ID. Use the left and right arrows in the screen to flick through successive pages. Click on residue 118 (S) and click CA (the Cα of residue 118), and click Yes to confirm you wish to measure from this atom. Flick to residue 296 residue CA (P) and confirm measurement to this atom. Wait for the distances to be calculated and then click Add to database.Let us examine this distance as a potential “marker” more carefully: the distance between S118 and P296 is a function of all backbone torsion angles in between these two residues. RoPE contains the ability to measure distances between corresponding atoms in two structures. Click rules menu, and (+) add rule to vary color on the 118CA to 296CA distance. Create this and return it to the RoPE space. It is immediately visible that there is wide variation across all structures of this reporter distance. Structures which are not in P1 are more likely to have wider distances, with some exceptions. First, there is far more variation across the lengths of the V branches than between them. The tips of the P1‐V contain the lowest distances of about 12.2 Å. However, comparison of monomers from each model shows a systematic increase in this reporter distance between chains assigned “A” to chains assigned “B”. This therefore demonstrates that when the protein crystallizes as a closed‐form dimer, there are nevertheless subtle and consistent differences between the subunits in this space group.You can now return to the RoPE space. Go to the Additional walkthroughs are available on the website, including how to import locally saved PDB files, and introducing custom metadata.5axes menu when viewing the RoPE space.The number of PCA axes vastly exceeds the number displayable at any one time. The first three most important axes are displayed by default, but the researcher should always bear in mind additional separation between structures on nondisplayed axes. The displayed axes can be altered in the In this version of RoPE, disordered regions in proteins are a source of breaking the correlation in the backbone. Ordered regions of the backbone are separately superimposed during PCA axis exploration as a reasonable alternative, but must be taken into account when evaluating motions.How many protein structures are needed for the analysis? This depends on the scientific question one is trying to answer and therefore how much sensitivity is needed. The success of disentanglement of independently varying motions, particularly separation of the salient and ambient motions, relies on good sampling and coverage of the conformational space of the protein. Repeating experiments under identical conditions is key to understanding the natural variation of the sample. Make use of the group selection feature to eliminate modes of motion that are irrelevant to the biological question, which may otherwise pollute the PCA result.6The software allows the researcher to easily access the analysis described in this paper for their own projects and proteins, and provides a framework in which additional analytical tools will be made available for biological interpretation. Additional features not described in this paper are documented on the RoPE website.77.1The UniProt sequence was obtained for trypsin and lysozyme, and the sequences were downloaded and referred to as an entity. Sequence similarity was used to search for and download similar protein structures from the PDB. Alternatively, PDB codes associated with a single multi‐dataset research project were downloaded for BAZ2B , sin(t), cos(t)). Unassigned angles acquired a unit vector of length zero. The cluster4x algorithm is applied to the differences in this two‐dimensional vector format. A correlation matrix was generated by calculating correlation coefficients between matched torsion angles between each pair of data sets. SVD, as implemented in Numerical Recipes in C ; data curation ; formal analysis ; investigation ; methodology ; software ; visualization ; writing – original draft ; writing – review and editing .The author declares no conflicts of interest.


This study presents an extended evaluation of a numerical approach to simulate artifacts of metallic implants in an MR environment.The numerical approach is validated by comparing the artifact shape of the simulations and measurements of two metallic orthopedic implants at three different field strengths . Furthermore, this study presents three additional use cases of the numerical simulation. The first one shows how numerical simulations can improve the artifact size evaluation according to ASTM F2119. The second use case quantifies the influence of different imaging parameters (TE and bandwidth) on the artifact size. Finally, the third use case shows the potential of performing human model artifact simulations.The numerical simulation approach shows a dice similarity coefficient of 0.74 between simulated and measured artifact sizes of metallic implants. The alternative artifact size calculation method presented in this study shows that the artifact size of the ASTM-based method is up to 50% smaller for complex shaped implants compared to the numerical-based approach.In conclusion, the numerical approach could be used in the future to extend MR safety testing according to a revision of the ASTM F2119 standard and for design optimization during the development process of implants. Over the past decades, the number of patients with passive and active implants who need to be examined with magnetic resonance (MR) imaging has increased . In addiDue to the importance of MR-related image artifacts the ASTM F2119 standard was developed to provide a consistent method for testing these artifacts from medical implants. This standard defines basic spin echo and gradient echo sequences with clearly defined sequence parameters to make the artifacts between different test objects (TO) comparable and to get a basic understanding of how large the expected artifacts of medical implants will be .Nevertheless, the results of the ASTM F2119 test method are not transferable to clinical routine MR imaging applications since investigated artifacts are only evaluated under clearly defined conditions in phantom measurements with limited clinical relevance. Additionally, the artifacts according to ASTM F2119 are defined as the maximum distance from the edge of the implant to the fringe of the artifact measured only in the one slice with the largest artifact. This also leads to a lack of information, because there is no information about the TO location inside the artifact, which also depends on the orientation of the implant. Hence, for multi-componential medical implants of complex shape like orthopedic plates and screws, it is impossible to describe these artifacts in a three-dimensional space by only one distance. Furthermore, artifact investigation according to ASTM F2119 includes different test objects and slice orientations, thus constituting a time-intensive measuring procedure that requires a large amount of MR scanning time.To reduce the aforementioned limitations of the current ASTM method, in this work, a numerical framework for simulating MR image artifacts was extended and used to create an improved artifact investigation technique. This technique is based on a numerical simulation framework which was previously validated with small TOs with a simple geometry . In this3). The phantom was filled with a virtual medium  for all three field strengths. A set of spin echo  and gradient echo sequences  were simulated for image acquisition with the same parameters as described in ASTM F2119 [The MR image artifacts of two medical orthopedic implants were simulated with a resolution of 2 pixel/mm at different magnetic field strengths of 1.5 T, 3 T, and 7 T by using a numerical framework which was developed and validated in a previous study for test objects with simple geometry . Within It is well known that T1 and T2 times show variation with different magnetic field strengths . The bacε = 0.40, σ = 6.6 mS/m) and the TOs were fixed with fishing lines in the middle of the phantom  and the measured (Amea) artifact area were compared using the dice similarity coefficient (DSC) [To quantify the agreement between the simulated and the measured artifacts, the areas of the simulated (nt (DSC) . The DSCx and y direction until the highest DSC agreement between the two images was achieved. This final overlay is illustrated as an example in Fig. To achieve a better agreement for the superposition of these two artifact masks and to compensate minor positioning errors, the simulation image was shifted in an iterative procedure within the slice by ± 20 pixel in Note that in this context higher DSC values represent a better agreement between simulated and measured artifact sizes. As a general acceptance criterion for DSC, a value of larger 0.70 is considered to be a good agreement between simulation and measurement , 14.n = 36) and the 2 sequences (sample size per group n = 54). Afterwards, mean, standard deviation, maximum, and minimum of DSC were calculated for each group. The DSC was calculated for each of the 108 measurement-simulation pairs and the data were grouped according to the 3 different field strengths (sample size per group dart) was measured, and the test object was placed in the center of the artifact. The artifact size was then calculated based on this assumption.Based on the artifact masks, the artifact size calculation according to ASTM F2119 was performed. The standard defines an artifact as the distance from the edge of the TO the fringe of the artifact. In cases where the edge of the implant cannot be localized in the image due to large signal voids, an alternative artifact calculation is suggested by the ASTM standard. Here, the total artifact extent  and the bandwidth (BW) of a spin echo and gradient echo sequence on the size of artifacts were investigated in more detail. Five different echo times  and five different bandwidths  were used for both TOs and the entire set of slice and test object orientations. As in the validation simulations, these configurations were simulated for the three magnetic field strengths .The changes in artifact size for the specific test configurations were normalized to the lowest value of the parameter (BW: 100 Hz/px and TE: 10 ms) for each test configuration. This eliminates the test configuration specific artifact size that is caused by the orientation of the slice and test object. This comparison allows an evaluation based on the echo time and bandwidth that is independent of the configuration.The numerical approach also allows to simulate artifacts inside a human body model to better display the general appearance of the artifact in relation to the surrounding tissues and anatomy.Exemplarily, the human model “Duke”  from IT’IS foundation  was impoThe results section is divided into four major parts. The first part describes the evaluation of different T1 and T2 values on the artifact size whereas the second section describes the validation of the numerical approach. The third and the fourth part present the potential use cases of the numerical approach in comparison to the ASTM F2119 standard.The T1 and T2 times for the background medium in the final simulations were fixed to T1 = 900 ms, T2 = 50 ms for all three field strengths . Variation of the T1 and T2 times by a maximum of ± 50% in simulations did affect the resulting artifact sizes only with very little impact (averaged artifact size change of 1.9%). While the impact on the artifact size was considered neglectable, the variations of T1 and T2 lead to variations of the amplitude of the background signal (brightness). This signal change of the background medium does not affect the calculation of the artifact size, since a reference image with the same background medium was always used in the artifact calculation. Thus, with neglectable impact on artifact size, for the further simulations the T1 and T2 times of medium were fixed to the reported values.Figure Across all these different configurations, the average DSC was 0.74 with a standard deviation of ± 0.09. This value is above the acceptance criterion of 0.7 , 14, indTo investigate the difference between the artifact sizes of the ASTM-based and the numerical method, the artifact sizes for the corresponding configurations were compared and the percentage difference between the two methods was determined. Table It can also be seen in both figures that the TO was not correctly centered in the artifact, especially regarding the frequency-encoding direction. Here, the off-center shift for Fig. The second part of the evaluation focused on additional numerical simulations to show the influence of different scanning parameters on the artifact size. Figure The third use case shows the results of the human model simulations. Figure As shown by the averaged DSC 0.74), the artifacts show a high overlap between the simulations and the measurements. Nevertheless, there was a difference between the simulated and the measured MR image artifacts, which may be explained by inaccuracies in the exact placement of the TO and the selection of the correct slice position. Due to the placement of the TO with fishing lines, it was difficult to place the TOs in the phantom with the exact same orientation and at the same position as in the simulation. To reduce this mismatch for the artifacts, the simulations and measurements were overlaid based on the artifact’s masks by an iterative procedure. This procedure, however, only works for the positioning problems within the slice (in plane). The inter-slice positioning error could not be corrected by this procedure. As shown in Fig. , the artAs shown in the results, the ASTM F2119 standard method yields artifact sizes which are up to 50% smaller compared to the presented approach where the TOs were placed at the exact position. This may be explained by the fact that the artifacts had a larger extension in the frequency-encoding direction  and, theAdditionally, the numerical approach allowed an easy adjustment and evaluation of additional imaging parameters  which helped to quantify the influence of different imaging parameters on the resulting artifact sizes. In this study, it was shown that an increase of the TE leads to an increase of the artifact size for the gradient echo images due to an increase of intravoxel phase dispersion before the echo is regenerated . This lix2. There is a quadratic dependency, because the final simulations are performed on the 2D slice and only the preprocessing  is performed using the 3D data set [Furthermore, a higher bandwidth leads to a decrease of the artifact size especially for the spin echo sequence. The general artifact size reduction is caused by the lower spatial distortion of the signal for the SE and GRE images. This effect is stronger for the SE compared to the GRE, because here the total artifact is more dominated by the spatial distortion, whereas the GRE artifact is more dominated by the intravoxel dephasing . Howeverdata set . For thiA slight variation (DSC = 0.86) between the human model and the phantom simulations is shown in Fig. Numerical simulations are of potential interest for implant manufacturers to assess MR imaging artifact sizes during the development of their products and to design and optimize their implants based on these findings. Additionally, the use of human model simulations Fig.  can be oConcluding, the proposed method could also be used to extend the ASTM-based method, for example by determining the worst-case configuration or by obtaining more detailed information about the artifact by varying sequence parameters.A numerical method for the simulation and prediction of MR imaging artifacts generated by realistic and complex orthopedic implants was evaluated and validated. The method provides an accurate and detailed determination of the size and shape of artifacts as a function of field strength, echo time and image bandwidth. The method can be used for MR safety testing according to the ASTM F2119 standard as well as for design optimization during the development process of implants.


Background: The failure and the success rate of an implant depends on biomechanical factors, esthetics and painless sterile implant surgery conditions, out of which stresses applied to the bone and its surrounding, bone-implant interface, material characteristics of the implant used and the strength of the bone and its surrounding are the important factors. This study aimed to evaluate the stress distribution of divergent collar design (DCD) and convergent collar design (CCD) implants placing them in four different densities of the bone . The evaluation of the stress distribution of DCD and CCD was performed using the 3D finite element method (FEM), by placing them in four different bone densities. In addition to this, a comparison of the effect of DCD and CCD in terms of stress distribution in the bone was also done.Materials and methods: The software used to process the geometric characteristics of the missing first molar in the mandibular section were Ansys, version 19.2, CATIA, version 5, and Solidworks (Dassault Systèmes). Using these software, three models were designed and successfully restored using an all-ceramic crown implant. The first model was a geometric model of the first molar mandibular bone section, the second model was a cylindrical implant (4x10 mm) with a DCD and CCD, and the third model had titanium alloy (Ti-6Al-4V) properties incorporated into the implant.Results: The D1 bone model showed the lowest stress concentration compared to D2, D3, D4. The DCD showed the lower stress and strain concentrations as compared to the CCD in the contiguous crestal bone in all the densities of the bone in both vertical and lateral or oblique loadings. The DCD with the D1 bone showed the least stress concentration around the crestal bone region. The results of this study also showed that the maximum von Mises stress was observed in the crestal region or the neck of the implant for both the convergent and divergent collar implant designs in all the four densities of the bone.Conclusion: Before a patient trial of a new implant design or a new implant material, finite element analysis (FEA) gives us a clear picture of what will be the patient bone response when an implant will be placed and loaded. FEA also gives us an opportunity to test a new implant material without putting a patient at risk. In this study, four different types of bone were incorporated with two different implant collar designs. Each implant assembly was subjected to vertical as well as oblique forces. The response of each bone type for the titanium alloy implant was recorded. A color-coded response for the magnitude and the location of the maximum stress received by the bone was observed. Maximum stresses were seen in the crestal region. As this is a computer-based model, dynamic loading was not possible. This study provided us with the possible outcome in patients under a static load. Further studies can be conducted in vivo to record dynamic loading responses as well as long-time loading responses. Dentistry in the modern era has advanced to a great extent. Replacement of missing teeth with the osseointegrated supported prostheses called dental implants has been introduced. The aim of implant dentistry is to rehabilitate a patient’s missing teeth with dental implants in order to achieve normal mastication, esthetics, phonetics, health and comfort . But allIn the treatment planning for a dental implant, quality and quantity of the recipient’s bone is utmost important to be considered for a better prognosis. A thick, dense cortical bone is found in the mandibular anterior alveolar arch, followed by mandibular posterior and anterior maxillary alveolar arches. The posterior maxillary alveolar bone has the lowest density .The four standard bone densities  to place the implants were given by Lekholm and Zarb in 1985. As reported by Jaffin and Berman, the D4 bone showed 55% of failure rate [The stress can be originated at the implant crestal module, where the crestal bone makes contact with the implant. The initial bone quality and quantity are the reason for the mechanical immobilization in the healing process and also give a better stress distribution and transmission at the implant-body interface . The creFor designing the best dental implants, stress transferred to the surrounding bone has to be precisely analyzed . In implIn the bone around an implant, different mechanical stresses such as von Mises stress that comprises tensile stress, compressive stress and the maximum shear stress can be found ,12. von In the present study, the pattern of load distribution like vertical, horizontal or oblique loading in a section of the mandibular bone with a missing first molar was used, which was restored with a ceramic crown-based implant. Also, a geometrical model of the first mandibular molar area bone section, a cylindrical implant (4x10 mm) with a divergent and convergent collar design and an implant model with titanium alloy (Ti-6Al-4V) properties incorporated were developed in this work.A 3D bony section of a mandibular model was prepared and a dental implant was designed with software CATIA, version 5  and SolidWorks . The models were subjected to pass through finite element stress analysis with software Ansys, version 19.2  and SolidWorks; this determined the response of the bone density to a particular collar design.The first step to construct the geometry of the model involved modelling different types of bone densities , with similar morphological architect of the bone as in the mandibular first molar region area using CATIA V5 software. Figure Cortical and cancellous bone properties were incorporated into the bone model as shown in Figures The average height and width of the mandibular section model were 28 and 12mm. Figure A V-shape thread was incorporated into the implant model that encircled the implant body with a depth of 0.36mm, pitch 0.8mm and width 0.25mm [The two implant collar designs (DCD and CCD) were also designed with SolidWorks and had the same height (2mm) and same diameter (4mm) at the implant collar interface. The diameter of the implant in the collar abutment interface model for CCD was 3.5mm and for DCD was 4.5mm. An abutment conical and convergent in shape, and having a diameter of 3mm at the junction of the abutment and collar with a height of 5mm was designed along with a porcelain prosthetic crown for the missing mandibular first molar replacement. All units of the models were assembled and subjected to abutment mesh and molar mesh with the SolidWorks software as shown in Figures The convergent collar implant mesh is shown in Figure The divergent collar implant mesh is shown in the Figure After designing the geometrical models, they were fixed at the base and supported at the distal and inferior side of the mandibular bony section to mimic the action of the muscles and ligaments. Then, a vertical force of 400N was applied at the centric fossa and a lateral force of 400N was applied onto the lingual inclines of the buccal cusp of the molar teeth in the linguo-buccal direction at a 30° angle. Basically, two implant models were prepared for each density , with the divergent and convergent implant collar designs, which were tested for the von Mises stress by applying a vertical and lateral or oblique load of 400N on each model. The application of the load is shown by a schematic diagram in Figure The von Mises stress value measures the point of deformation in dental implants ductile in nature. The failure of the dental implant is seen when the von Mises stress is greater than its yield strength . TherefoThe analysis of models was done using the finite element method; the properties of materials used in this method are summarized in Table Inclusion/exclusion criteriaModels whose properties matched the material properties enlisted in Table 2). The minimum and maximum stress is marked by blue and red color bands, respectively. It was observed that all the models had the maximum stress at the crest of the bone and collar of the implant. The von Mises stress field for the D1 divergent collar with vertical loading is shown in Figure In this study, stress was analyzed using Ansys software. In this process, von Mises stress field traces could be observed in the form of color-coded bands as shown in Figures The D1 bone shows the minimum stress concentration compared to other bone densities. The divergent collar design shows the minimum stress concentration compared to the convergent collar design in both the vertical and oblique loading.2) in the crestal bone of the CCD and DCD in four densities during vertical and oblique loading (400 N). It can be seen that the divergent collar with D1 density shows the minimum stress while the convergent collar with D3 and D4 densities shows the maximum stress on the vertical load.Tables Overall, the D1 bone with the divergent collar shows the minimum stress concentration and the D4 bone with the convergent collar shows the maximum stress concentration in the crestal bone region during vertical and oblique loading. The stress concentration for vertical loading was lesser than that for oblique loading.The reason for osseointegration failure and bone loss in the pre-implant is the micromovement of a dental implant that is embedded inside the available bone and the amount of pressure on the bone that is surrounding the endosseous dental implant, as seen by van Steenberghe et al. .The most essential factor to design good quality, successful implants is the density, that is, the quality of the edentulous bone as stated by Holmes and Loftus . In addiWe used a model for finite elemental analysis in a mandibular section of the bone with a dental implant and ceramic crown to observe the effect of the stress on different densities of the available bone by designing a CCD and DCD. Each layer of the finite element model is assigned with some particular materials and characteristics, which can be changed by the stress developed in different bone qualities. FEM is based on some assumptions like loading/boundary conditions and material properties to simulate the architect and surface morphology of the dental implant.In this study, the use of convergent collar designs showed destructive stresses and pathologic micro-strains around the crestal compact bone under vertical and oblique loading. These collar designs also demonstrated pathologic micro-strains in the crestal compact bone in the low-density cancellous bone under vertical and oblique loading. This can be explained by “saucerization” being particularly prone to occur in the first year of function. The results also showed that there is an increase in the level of stress around the neck of the implant with a decrease in the bone density. This is supported by the study of Sevimay and Turhan .It should be noted that values observed in this FEA study might be worse in actual clinical situations. For a long-term bone maintenance, the divergent implant collar appears to be a favorable choice. By increasing the surface area and decreasing forces, stresses can be reduced as stress is a result of force divided by area. The divergent implant collar had a surface area greater than the convergent collar design and hence it transferred the lowest stress levels to the crestal compact bone under the same load, while the convergent collar had a smaller surface area and transferred the highest stress to the adjacent compact bone . The comFEA is a powerful tool to design an implant to have a better prognosis in prosthodontic treatment planning and rehabilitation . The limIn this research article, von Mises stresses were seen to be maximum in the bone crest or the neck of the implant for both the convergent and divergent collar implant designs in all the four densities of the bone. The D1 bone model showed the lowest stress concentration compared to D2, D3, and D4. The DCD showed the lower stress and strain concentrations as compared to the CCD in the contiguous crestal bone in all the densities of the bone in both vertical and lateral or oblique loading. The DCD with the D1 bone showed the least stress concentration around the crestal bone region. In the convergent collar implant, the implant crestal module is closer to the bone than in the implants with divergent collars. Therefore, the stress concentration in the convergent collar implant may have a greater effect on crestal bone resorption as compared to that in other collar designs.