In pregnant patients with a divided uterine cavity, the decidual tissue on the nonpregnant side may be discharged prior to the delivery of the fetus. The pregnancy can continue if the uterine contractions and vaginal bleeding are controlled and the fetus is not in distress. In pregnant patients with a divided uterine cavity, the decidual tissue on the nonpregnant side may be discharged prior to the delivery of the fetus. The pregnancy can continue if the uterine contractions and vaginal bleeding are controlled and the fetus is not in distress. She had a history of a resected vaginal septum at age 12. The ESHRE/ESGE  classification of the uterine malformation was U2C1V2 Figure .5 SimiTocolysis was discontinued at 23 weeks of gestation, and the patient was discharged at 24 weeks of gestation. She had regular outpatient follow‐up every 2 weeks. At 30 weeks and 5 days of gestation, the patient presented with vaginal bleeding and was observed to have a shortened cervical length of 16 mm. She was admitted to the hospital and administered betamethasone.At 33 weeks and 6 days of gestation, the rupture of membrane occurred spontaneously, and spontaneous labor started subsequently. A male infant weighing 1,684 g  was delivered. The infant's Apgar scores were 9 and 9 at 1 and 5 minutes, respectively. The postpartum period was uneventful, and the patient was discharged on postpartum day 4. Although the neonate was admitted to the neonatal intensive high care unit (NICU), he developed without any complications and was discharged home at the age of 66 days.3When pregnancy occurs in a uterus with a divided cavity, decidual tissue can be discharged with vaginal bleeding from the nonpregnant uterine cavity prior to the fetus delivery. When uterine contractions and bleeding are controlled properly and the fetal well‐being is confirmed, the pregnancy can continue even in cases of severe blood loss.In a normal uterus, the endometrium develops into decidua and becomes thicker due to the presence of progesterone during the first trimester of pregnancy. In the second and third trimesters, the decidua is compressed against the uterine wall due to the enlargement of the fetal and villous tissues. As gestation progresses, the decidua thins and fuses with the amniotic and villous membranes to form a thin fetal membrane.Vaginal bleeding and uterine contractions in the second trimester may be due to various conditions, including placental previa, placental abruption, chronic abruption‐oligohydramnios sequences, and subchorionic hematoma.I and my co‐authors have no conflict of interest.MU: Contributed to the treatment and first draft and finalization of the manuscript. SO: Contributed to the first draft and finalization of the manuscript. AY and SN: Contributed to patient care. YS and MO: Contributed to pathological examination and manuscript regarding the pathological image. EM and SA: Supervised the case report.Prior to submission, appropriate consent for publication of images and data has been obtained.Fig S1Click here for additional data file.


Stationary motorized cycling assisted by functional electrical stimulation (FES) is a popular therapy for people with movement impairments. Maximizing volitional contributions from the rider of the cycle can lead to long-term benefits like increased muscular strength and cardiovascular endurance. This paper develops a combined motor and FES control system that tasks the rider with maintaining their cadence near a target point using their own volition, while assistance or resistance is applied gradually as their cadence approaches the lower or upper boundary, respectively, of a user-defined safe range. Safety-ensuring barrier functions are used to guarantee that the rider’s cadence is constrained to the safe range, while minimal assistance is provided within the range to maximize effort by the rider. FES stimulation is applied before electric motor assistance to further increase power output from the rider. To account for uncertain dynamics, barrier function methods are combined with robust control tools from Lyapunov theory to develop controllers that guarantee safety in the worst-case. Because of the intermittent nature of FES stimulation, the closed-loop system is modeled as a hybrid system to certify that the set of states for which the cadence is in the safe range is asymptotically stable. The performance of the developed control method is demonstrated experimentally on five participants. The barrier function controller constrained the riders’ cadence in a range of 50 ± 5 RPM with an average cadence standard deviation of 1.4 RPM for a protocol where cadence with minimal variance was prioritized and used minimal assistance from the motor  in a separate protocol where power output from the rider was prioritized. Stationary cycling assisted by functional electrical stimulation (FES) can lead to long-term benefits for people with movement impairments due to neurological conditions such as stroke, spinal cord injury, traumatic brain injury, cerebral palsy, multiple sclerosis, and others The kinematics of the rider’s legs during stationary cycling are such that applying FES to their muscles produces non-negligible torque only in certain regions of the crank cycle. To maximize torque production, stimulation patterns often feature discontinuous jumps triggered as a function of the crank angle by discrete logic variables. The interaction of the resulting continuous-time and discrete dynamics results in a hybrid control system. Barrier functions, or control barrier functions (CBF), can be used to design controllers for hybrid systems that ensure safety by rendering sets of states either forward invariant or asymptotically stable In this work, we extend the development of Experimental trials were performed on five able-bodied participants to demonstrate the effectiveness and versatility of the developed control system. The barrier function controller was shown to effectively constrain the cadence to a range of 50 ± 5 RPM for all but a negligible amount of time, and to outperform the controller in The development in this work is based on the hybrid systems framework described in When the state is in the flow set Remark 1. Previous results cf.  have ana1 denotes the cycle’s measurable crank angle, and z2 is the calculable angular velocity . The system has control inputse > 0 is the known electric motor control constant relating input current to output torque. The torque generated from FES inputs to the rider’s muscles Analogous to The following properties of the cycle-rider system in Property 1. The inertial term is upper- and lower-bounded as V > 0 is a known constant.Property 2. The centripetal-Coriolis parameter is upper-bounded as G > 0 is a known constant.Property 3. The torque generated by gravity is upper-bounded as P1, cP2 > 0 are known constants.Property 4. The torque generated by the rider’s viscoelastic tissues is upper-bounded as b > 0 is a known constant.Property 5. The torque due to viscous damping is upper-bounded as d > 0 is a known constant.Property 6. The torques generated by system disturbances are bounded so that vol > 0 is a known constant.Property 7. Due to physical limitations of the rider, the volitional muscle torque is bounded so that Property 8. For each Property 9. The set-valued mapping 2d where the control inputs are zero, thereby forcing the rider to pedal on their own volition . When the cadence is slower than z2d, FES assistance is provided before assistance from the electric motor. When the cadence is faster than the setpoint, only the electric motor is used because creating resistive torque with FES by stimulating antagonistic muscles is undesirable. The rider or clinician can modify the amount of control effort provided and the size of the controlled regions using parameters adjusted for the specific individual.2 from a constant setpoint z2d > 0,To formalize the control objective, we define the tracking error e as the deviation of the cadence state zL < 0 < eH are user-defined constants. The FES control inputs attempt to constrain the rider’s cadence to the secondary set L < eFES < 0.The primary control objective is to guarantee a safe and effective therapy by constraining the rider’s cadence to the safe set The goal is to synthesize, in a systematic way, controllers that render a given set of states uniformly globally asymptotically stable (UGAS) while using the minimum required effort inside the set of interest.Lemma 1. Let the functions ∗ such that Then the set-valued mapping Note that there is no division by zero in 2d > 0. Our development is based on the design procedure described in In this section, the electric motor control input is designed to ensure that the safe set Equivalent to the original definition, we have b1 > 0 is a control gain. In the stability analysis of e acts as a Lyapunov function for the closed-loop dynamics outside the set While barrier functions are typically associated with forward invariance, they are naturally extensible to enforcing the stronger property of asymptotic stability. Asymptotic stability is beneficial for real-world applications since it guarantees robustness to perturbations from the safe set. For continuous-time systems, the stronger asymptotic stability condition is embodied in the definition of a zeroing CBF (ZCBF) ae on the barrier function candidate can be restrictive, and is not necessary to obtain theoretical guarantees. Because the value of the function M in Be is unknown and cannot be implemented in a control law, we use γe instead of the typical ZCBF-based selection when defining e in Remark 2. The definition of a ZCBF requires that u, we employ Lyapunov-based robust control techniques to develop a worst-case upper bound of the inner product in The regulation map in 1 − c3, asThe product u depends on the motor control input ue and the muscle control inputs uM. Because the value of the subsequently designed muscle control input will jump at discrete instances, it is desirable to decouple the motor input from the muscle input to ensure the continuity of the motor controller. A continuous motor controller will be more predictable and comfortable for the rider. Using Property 8, there exists a constant cM > 0 such thatIn SelectingThe following result summarizes the preceding development and explains the utility of 1, k2, and k3 satisfy the gain conditions in 2d > 0. Then, for any Proposition 1. Assume kThe constraint used to define Since the parameters in 0 = 0, the controller is a minimum norm controller and the motor is off whenever possible while still ensuring safety. When the rider requires additional assistance, selecting a positive u0 leads to the motor being biased to assist, while a negative u0 leads to additional resistance and a more challenging training program. However, there is no theoretical obstacle to using a more complex nominal controller .Remark 3. During the experiments in m and the timer variable τ will be used to define a hybrid system in To describe the FES control input, we define a concatenated state vector As discussed in 2d. However, it is always the case that H. By combining nominal assistance from FES with nominal resistance from the electric motor, a more intense training program can be designed where the rider must work against resistive torque from the electric motor to stay near the setpoint. We provide experimental results for a higher intensity configuration of the control system in When the nominal controller in m according to the rule in D > 0, at which point a jump occurs and each signal in σM is updated. Governing jumps by a dwell-time prevents multiple jumps from occurring in the same time instant, and models a computational implementation of the switching signals, where the values of the logic variables would be updated periodically at a fixed sampling frequency. In practice, the dwell-time τD will be the sampling frequency.We model the closed-loop system as a hybrid system u is defined in The hybrid system D,Jumps occur when the timer state τ grows to τ+ = z), the timer τ resets to zero at jumps (τ+ = 0), and σ is equal toThe jump map D when the state z1 is precisely on the boundary of m may or may not jump. Performing such a regularization leads to some robustness properties due to the fact that The set-valued case in Remark 4. The gain conditions in 2d > 0. Then the safe set Theorem 1. Consider the closed-loop cycle-rider system e in e is a valid barrier function candidate D ensures that for any solution ϕ to Proof. Since Be is a Lyapunov function for the restricted hybrid system e. It can be shown that there existsr. Using Property 1, there are class 1 and α2 such that The conditions in To conclude that To eliminate the possibility that maximal solutions escape in finite time by flowing, we first let ϕ be a solution to wing cf. . Thus, e2d where no assistance is provided, with discontinuous control effort being applied on the boundary of the region. In contrast to the barrier function controller, the electric motor controller for the 3M controller is coupled with FES stimulation via the angular position state z1, so that the motor is inactive whenever FES is active, and vice versa.The developed barrier function controller was tested on five participants and compared against uncontrolled volitional pedaling and the 3-Mode (3M) controller developed in The barrier function controller can be configured for various purposes based on the needs of the rider. Generally, there is a trade-off where smaller user-defined cadence ranges lead to greater applied control effort. Protocol A was designed to investigate whether the controller can reduce the variance in the rider’s cadence by constraining their cadence within a small range. Such a trial provides a point of comparison with the 3M controller and uncontrolled volitional pedaling and generates data where the controller is more active. Protocol B was designed to show how assistance from the motor can be reduced by selecting a wider safe range, thereby encouraging more volitional contributions from the rider. In fact, Protocol B featured a nominal amount of resistance from the motor, making the program more challenging and requiring additional power output from the rider.Due to COVID-19 related difficulties in scheduling participants with neurological conditions, the trials for this demonstration were done with able-bodied subjects. Each participant gave written informed consent approved by the University of Florida Institutional Review Board (IRB201600881). Participants 1-3 were male, participants 4 and 5 were female, and all ranged in age from 21–29 years old.The experimental testbed consisted of a stationary recumbent tricycle (TerraTrike Rover) with a 250 W, 24 V motor (Unite Motor Co.) coupled to the drive chain as described in 2d = 50 RPM. The safe set boundary for the barrier function controller was encoded by eL = − 5 RPM and eH = 5 RPM, while eFES = − 3 RPM. The inactive region for the 3M controller was 48–52 RPM, which is comparable to the range 50–55 RPM that was used for the experiments in The primary testing procedure (Protocol A) consisted of 180-s tests for each of the three configurations  under consideration. The volitional pedaling trial was always first, followed by a random selection of either the barrier function or 3M controller. The riders were asked to track a setpoint of zThe 180 s tests started with a 20 s ramp-up phase where the rider sat passively while the motor brought their cadence to the setpoint. To ensure that the presented data represented steady-state operation, the ramp-up and an additional 20 s after it was excluded from each dataset in post-processing. For each configuration, there was a separate warm-up run before the recorded session so the rider could become accustomed to the controller.Measurements of the position of the rider’s legs with respect to the cycle were used to determine the regions of effective torque transfer scle see  for moreL = − 12 RPM, eH = 10 RPM, and eFES = − 6 RPM. Because the rider (Participant #2) was able-bodied, we chose to make the program more challenging by adding nominal controllers with L and eH were displayed on-screen for this trial. In Protocol C, Participant #1 was asked to provide no volitional effort for both the 3M and barrier function controllers (Additional trials were conducted to highlight unique aspects of the barrier function controller, which were performed with only one participant. Protocol B was designed to prioritize power output from the rider over cadence tracking. For this alternative trial, the width of the safe range was larger, with etrollers . BecauseThe results for Protocol B, where there were nonzero nominal controllers and a wider safe range, are displayed in H, eL, and eFES, yet are inactive when the cadence is near the setpoint. Such a ramp-up is demonstrative of how barrier function methods can ensure a gradual transition from a nominal controller, which should be active on the interior of the safe set, to an invariance-ensuring controller on the boundary of the safe set. For Protocol A, the width of the nominal control range was fairly small, while the width was increased for Protocol B.The results for Protocol A demonstrate that the barrier function controller can assist a rider in tracking the cadence setpoint while constraining their cadence within a user-defined range. In some situations, increasing the training intensity for the rider will be preferred over improved cadence tracking. The relevant statistics are the assistive and resistive torques produced by the electric motor. Higher assistive torque indicates that less power is being produced by the rider. The barrier function controller is versatile and can be tuned to provide more or less interference from the control inputs. A continuous motor controller is more comfortable for the rider. Continuity is the primary difference between the barrier function and 3M controller. The 3M controller is discontinuous whenever the cadence crosses the boundary of the inactive zone, or the angular position crosses the boundary of the FES stimulation regions otocol C . The 3M There were some small deviations from the safe set because the asymptotic stability of This paper developed new FES and motor controllers that encourage the rider of a stationary cycle to provide volitional effort while constraining their cadence within a user-defined range. Using theoretical advances for barrier functions, the controllers are minimally invasive while transitioning gradually to a safety-ensuring controller on the boundary of the safe set. The control inputs are selected from regulation maps with sufficient regularity to ensure that optimal selections are locally Lipschitz functions of the cadence error. Robust control tools were used to develop the regulation maps, which are subsets of an original, uncertain map. The uniform global asymptotic stability of the user-defined safe set was certified with a hybrid system analysis. In the future, the performance of the controller can be improved by extending the development to a more complete dynamic model which accounts for muscle activation effects such as electromechanical delay in the rider’s muscles.Experimental results showed that the control system improved the rider’s cadence tracking and effectively constrained their cadence within the safe set. The versatility of the controller was demonstrated with trials featuring two different objectives: improved cadence tracking or more power output from the rider. A significant next step is to perform more rigorous experiments on a set of participants with neurological conditions. These riders have a reduced ability to provide volitional contributions, so the results are expected to be significantly different from those of able-bodied riders, and would be of interest in more clinically focused literature.


Intentionally intensifying the light scattering of medium molecules can allow the detection of suspended nanoparticles under conditions not suitable for conventional optical microscopies or laser particle counters. Here, we demonstrate how the collective light scattering of medium molecules and nanoparticles is imaged in response to the power, frequency, and oscillating direction of the incident light wave electric field, and how this response can be used to distinguish between nanoparticles and microparticles, such as viruses or bacteria. Under conditions that the medium light scattering is intensified, suspended nanoparticles appear as magnified shiny moving dots superimposed on the quasi-steady background of medium light scattering. Utilizing the visual enlargement resulted from the enhanced light scattering and possible light interference, we can detect directly suspended nanoparticles that are much smaller than visible light wavelengths even in unopened water bottles or other large containers. This suggests new approaches for detecting nanoparticles with many potential applications. To detect particles smaller than the wavelength of the light source, the technique is limited to a small optical focus4, as shown in Fig. Light scattering has been widely used for detecting suspended particles in gaseous and liquid media6. When a medium is far away from its critical state, density fluctuations in the medium are so small that even if a focused solar light beam is used, the light scattering by the medium still cannot be noticed. Historically, light scattering is observed when fine particles are added into the pure medium or in a colloidal solution, as described by the Tyndall effect7. With the advent of laser technology, however, it has become easier to observe light scattering by a pure medium. In the 1970s. An Ar-ion laser was used to study light scattering by pure water to characterize its physical properties such as the light scattering depolarization and anisotropy8. In the last few years, lasers have become more powerful, more compact, and more cost-effective. The new developments in laser technology and image sensing technology make it possible to further utilize the light scattering of pure medium for particle detection and analysis.The observation of light scattering by a medium without any suspended particles can be traced back to the mid-nineteenth century. However, the phenomenon can only be observed under certain conditions of the medium, such as critical opalescence due to strong fluctuation in the density or refractive index of the medium near the critical staten medium molecules of polarizability n can be calculated using the Avogadro number and the molar mass in the corresponding volume of the medium. At the detector position, the electric fields emitted from the nanoparticle and from each medium molecule due to the far-field radiation can be expressed as 9 is 1.47 × 10–12 µm3 and the polarizability of air10 is 2.12 × 10–11 µm3 which are several orders of magnitude smaller than the 4.28 × 10–8 µm3 polarizability of a 5 nm diameter dielectric particle calculated with an equation for an isotropic spheric particle11. Without the presence of the nanoparticle, the total intensity of light scattered by the n light scattering medium molecules can be expressed as 12 can be used here for better accuracy. When both the nanoparticle and n medium molecules are present, the total electric field reaching the detector is When light shines on nanoparticles suspended in a medium, the charges in the nanoparticles and the medium molecules move in sync with the incoming electric field of incident light wave, resulting in a collection of induced oscillating dipoles with the same frequency and orientation as the incident electric field. Each induced oscillating dipole emits electromagnetic radiation in all directions. The intensity of the emitted radiation depends on the direction of the light scattering as described as Rayleigh scattering. From the standpoint of an individual nanoparticle or medium molecule, the wave vectors of the incident light and the scattered light to a detector are depicted in Fig. mentclass2pt{minimmentclass2pt{minimmentclass2pt{minim as see ""3\documeEquation  shows th12. The scattering angle The light scattering intensity is a time average that covers many electric field oscillating cycles. If considering only the light scattering itself, a nanosecond period is sufficiently long because it covers approximately 500,000 cycles for light of 530 nm wavelength. However, when calculating the light scattering intensity sensed by an image sensor, the period for the average could be much longer. The frame rate of image sensors currently ranges from 10 to 1000 frames per second roughly. If an incident light source is not coherent, 12 particles per cubic centimeter, interference between lights scattered from particles can be ignored. Under the condition, the total light scattering intensity can be calculated, in a simplified approach, by summing the intensities from all light scattering units in the scattering volume of interest. For mono-dispersed particles, the total light intensity in a light scattering volume can be expressed ask different particle sizes, the total light intensity in a light scattering volume can be expressed asTotal light scattering intensity in a scattering volume is often used to calculate the brightness of a light scattering image. When the distance between the particles suspended in a medium is larger than the wavelength of the incident laser, that is, when the particles are not highly concentrated, i.e., less than 10For poly-dispersed particles, the total light scattering intensity perceived by an image sensor is a function of the size distribution of the suspended particles. The intensity can be used further to determine the Rayleigh ratio.th power of the wavelength, Several experiments are performed to evaluate the brightness of the light scattering image under different laser irradiation. Figure entclass1pt{minimaFigure Experiments are also performed to compare the light scattering by pure water and by particle-free air. In the test, each medium is irradiated with both blue and green laser beams at the same time. The power of these two lasers is adjusted to about 20 mW, and the laser beam diameter is adjusted to about 1 mm. A longer exposure time of the image sensor is used to make the image of the light scattered from the air more visible. As shown in Fig. Another important feature of Rayleigh scattering is that the scattering intensity varies with the angle entclass1pt{minimaFigure mentclass2pt{minimmentclass2pt{minim2 nanosphere particles under conditions that the light scattering by water is almost invisible and clearly visible. It clearly shows that when the water light scattering is more visible, the particles look brighter and larger. In fact, the light scattered by water presents as a fine shiny background, and light scattered by the particles presents as moving bright dots on the shiny background. By setting the threshold intensity in the image sensor, the water scattering image can be filtered out without changing the nanoparticle light scattering image, as shown in Fig. Figure 2 nanospheres, and 5 nm gold nanospheres at different angles of 2 and 5 nm Au particles are not visible. The 300 nm PSL particles behave differently from the smaller nanoparticles.If the particles are much smaller than the wavelength of the light source used for the particle detection, the intensity of scattered light should follow Rayleigh scattering as described in Eq. . The briA more detailed measurement of particle light scattering with the change of angle 13. The method can roughly distinguish certain viruses and bacteria in a simple, non-intrusive way.Light scattering by 300 nm PSL particles suspended in water is further tested with lasers of different wavelengths. When the angle 4 particles per milliliter. The particle detection for the bottled water was repeated several times over several months. No significant change is observed between the detections. Therefore, the particles may be not from deterioration of the water cleanliness or the degradation of the bottle material. The particles are more likely to come from the manufacturing process. Several other bottles of drinking water from different manufactures and different storage conditions were tested. It shows a big difference in terms of particle concentration and particle size. Some of the particles are likely coming from bottle materials, such as micro-plastics, and some of the particles are likely living organisms.Encouraged by experimental observations mentioned above, a method is proposed to detect nanoparticles under the condition that the light scattering of the suspension medium is sufficiently intensified. The method is mainly for detecting nanoparticles suspended in large containers where optical microscope settings are not suitable. One of the applications tested is to check the cleanliness of bottled drinking water even without opening the cap sealed by the manufacturer. Figure One of the bottles of drinking water is checked again one week after drinking half of the water in the bottle. The initial condition of the bottled drinking water is like the one shown in Fig. Detection of particles packed originally or collected in different transparent containers in different liquids or air is also performed with interesting observations. Since the method does not require the transfer of the media from one container to another container and does not require the particles located in a small optical focal space, it can be mounted or equipped with more flexibility. It will lead to many unique applications in the future.The intensified light scattering of the suspension medium can be intentionally utilized for the detection of nanoparticles. When light scattering by the medium is intensified, the light scattering by the particles suspended in the medium becomes more intensified, and constructive interference may occur between the light scattered by the particles and the light scattered by the medium. When using a pulse detection approach, the benefits of the light scattering medium may be limited, because strong light scattering by the medium may result in a lower signal-to-noise ratio or a loss in the dynamic range of the detection. However, when using the image detection approach, this is no longer a problem, because the image detection approach is more like an interferometer, where the reference and signal are on the same image. On the other hand, the intensified light scattering of the medium can also be the result of using lasers of a higher power. As indicated in Eq. , a highe14, available for computer vision or machine learning, can be utilized to determine particle count or particle number concentration from light scattering images of particles. In addition, it is possible to further determine the particle size distribution. Several adjustable measurement functions can be utilized to determine the relationship between particle concentration and particle size distribution as shown in Eq. ..6). UndeOverall, an image detection method is proposed to detect suspended nanoparticles that are much smaller than the wavelengths of visible light. The method utilizes the intensified light scattering of the suspension medium to make the nanoparticles easier to detect by an image sensor. Since this method does not rely on optical lenses to focus or observe light in a small space, it is more flexible for the detection of particles suspended in different conditions. As demonstrated experimentally, the method can be used to directly detect suspended nanoparticles contained in large containers, such as sealed glass or plastic bottles, and distinguish between small nanoparticles and large microparticles. It provides a new direction for the further development of new particle detection technologies as well as many new applications.Experiments performed in the work use compact 12 mm diameter laser modules of several selected wavelengths. The laser power ranges from 5 to 50 mW. The laser modules are obtained from online stores such as Thurlab, Amazon, and eBay with specified wavelengths and powers. The laser beam passes through a linear polarizer and iris of adjustable diameter from 0 to 5 mm. The output power of the laser to the water was adjusted using two polarizers align or against each other. The laser power incident to the sample is checked using a portable laser power meter . The laser and polarizer are mounted on a rotatable station so that it can change the angle between the beam polarization plane and imaging plane while keeping the incident power the same. No condenser lens is used. In the system, the sensor array used is a 2 MP Sony IMX 291 image sensor with a pixel size of 2.9 The water used for testing medium light scattering and particle sample preparation comes from a semiconductor fabrication cleanroom purified for wafer processes. The clean air used is the air passing through a H13 HEPA filter. The standard particles used in the experiment are 5 nm gold nanospheres , 40 nm silicon oxide nanospheres , and 300 nm PSL spheres . The particles were diluted in water through a transfer using droplets from the original containers or using a cleaned syringe without any syringe filter to avoid any undesired contamination. When the incident laser is set at the angle Light scattering from particles and medium. Without the presence of the large nanoparticle, the total intensity of scattered light by n small particles isSince the s to Eq.  as7\docun small light scatters are present, the total intensity to reach the detector When both the large particle and


Eliminating hepatitis C requires addressing issues other than medicines or therapies. Public health policies focused on the hepatitis C virus (HCV) must be emphasized and worked to know the impacts on its epidemiologic dynamics. This research aims to provide a tool to evaluate and simulate alternatives by redefining policies meeting specific needs in each country towards the HCV elimination target by 2030.The development of a gamified model with 24 public health policies focused on HCV was conducted to evaluate the impact of measures in the disease epidemiologic dynamics. The Let’s End HepC (LEHC) project encompassed key populations  in Austria and other countries, presenting prospects for every year from 2019 to 2030. The LEHC epidemiological model comprised an integrated solution for HCV, with adaptive conjoint analysis (ACA) and Markov chains constituting its main processes.Despite Austria’s efforts towards achieving the HCV elimination goal by 2030, the LEHC model forecast quantitative analysis predicts that it is still not enough to meet the target; however, prospects are very optimistic if public health policies are adapted to the country’s needs, being possible to achieve the goal as early as 2026.In Austria, the LEHC tool allowed to forecast the HCV elimination year anticipation to 2026, instead of being achieved after 2030. This target will only be valid if adequate management of the 24 public health policies focused on this pathology is further implemented. Hepatitis C is a silent disease that afflicts people all over the globe. This virus affects people according to unknown timetables, which imposes an important role on health services in terms of human resources, technology and capital. Its evolution is well-documented in the natural history of the disease .Globally, the hepatitis C virus (HCV) is the seventh leading cause of hepatic mortality , often rOne of the HCV characteristics is its concentration of higher prevalence rates among given population segments: people who inject drugs (PWID) –5, peoplTo address the public health issue of HCV, the WHO has defined an elimination goal for the disease by 2030, aiming at reducing disease incidence by 90% and mortality by 65% . This taProjections for the HCV prevalence have been considered to be between 0.5% and 3.5% in Europe , 16 and The LEHC tool aims to evaluate and simulate diverse adaptations of public health policies according to the modeled epidemiological characteristics of the disease and the needs of a country to achieve the HCV elimination goal by 2030. By modeling the impact of the dynamics between both referred elements, it is possible to generate a landscape on HCV epidemiology for every year until 2030. Therefore, the model is also aligned with the sustainable development goals timetable. The study of the 24 health policies impact and their different levels of implementation resulted in different estimates on HCV epidemiology for different population groups: PWID, blood products, vertical transmissionThis project is being promoted by the Public Health Unit of the Health Sciences Institute of the Catholic University of Portugal. Along with Austria, other countries were included in the project’s initial modeling phase  while others are also being currently worked.The LEHC model has the potential to forecast and gamifyThe LEHC epidemiologic model integrated a total of 1100 active Markov chainsIt is assumed that the statistical data used for the LEHC model are correct, considering that they are either collected from peer-reviewed journals or have their source in official data of countries or regions. In the case where it was not possible to retrieve data from these sources, proxies had to be produced, with the assumption that the leading experts in each country of the LEHC project are able to provide an accurate overview of the epidemiological reality.However, it is known that even data that were subject to peer review have often existed solely in micro-populations while being related to the methodology used in data collection. It must be noted that even official data from countries has some frailties, such as the difference of the total number of inhabitants by over one million individuals, the source or country on the origin of the data, the opinion of the most acknowledgeable experts, which is always a personal view based on vast experience.www.letsendhepc.com/)Around 100 key opinion leaders (KOL) from each NAB have also participated in surveys based on an adaptive conjoint analysis (ACA) approach, taking a global average of 2h for completion. The resulting data were used to develop the modeling component related to the 24 public health policies   were asshttps://suchthilfe.wien) were used along with the aforementioned reports.The PWID demography was based on two reports , 29, witData regarding the annual probability of death were based on the working group drug-related fatalities (2017) report (“Arbeitsgruppe drogenbezogene Todesfälle (Sterbejahr 2017)”, ) for theValues for HCV prevalence and HCV incidence were based on scientific reports , 29. A rhttps://suchthilfe.wien). For the period between 1997 and 2018, diagnose rates for both genders, and all age groups were assumed to be 1% for F0–F4 stages and 80% for decompensated cirrhosis and HCC stages.Diagnosis rates were considered to be 0% before 1997. For the following years, diagnosis rates were produced according to data from the “Suchthilfe Wien” . Until 1996 it was assumed that therapies were not yet available.The annual probability of an infected individual to be treated for the first time against HCV if already retained in care was based on the “Suchthilfe Wien”  was not feasible. Data from the Ministry of Justice  were used for the period between 1997 and 2018. The RIC rates for both genders and all age groups were assumed to be 20% for F0–F4 stages, 70% for decompensated cirrhosis stage and 60% for the hepatocellular carcinoma stage.Annual probabilities of individuals being treated for the first time and/or retreated for HCV, along with the respective sustained virologic response rates were produced under the advice of the National Advisory Board.This population group encompasses the total number of individuals that have ever received a blood transfusion.https://www.roteskreuz.at/). A total of 350,000 blood products were used in 2017. It was estimated that each patient who receives blood products uses an average of three blood preserves, according to clinical estimates. Values were assumed to be increasing in the period between 1950 and 2010. Due to the implementation of new programs and therapies, it was assumed that, for the remaining years, a decreasing rate should be applied to the identified trend.Demography values were based on data from Red Cross Austria  appoint a total population of 8,679,051 individuals, with 29,348 being infected with HCV and an occurrence of 607 new cases is to be registered. Forecasts show that in 2019 around 16,626 individuals will be diagnosed, with 12,402 being linked to care, from which 5836 will be under treatment and 1064 cured cases are expected to happen. As for the different liver disease stages, model results for this year show that there will be 4934 cases of compensated cirrhosis, 337 decompensated cirrhosis and 206 cases of hepatocellular carcinoma. In addition, 11 liver transplantation cases are expected, as well as 202 liver-related deaths.The LEHC model forecasts that HCV elimination will not be achieved by 2030 see Fig. . In thatThe LEHC model estimates that HCV prevalence values will steeply decrease until 2025 and slightly decrease until 2030. Assuming the elimination line as 10% of the modeled HCV prevalence value in 2015, the WHO target will not be achieved by 2030  –23, 32. Empirical evidence resulting from the LEHC project corroborates the view that the path for eliminating hepatitis C until 2030 requires a higher focus on finding and maintaining patients within the grasp of the healthcare system. Given the high prevalence and incidence rates among population groups, such as PWID and prisoners, it is important that measures focus on the micro-elimination concept.It is also important to consider the risk of overlapping populations, especially between PWID and prisoner populations, and to a much lesser extent between PWID and blood products. It was not possible to find published epidemiological data that could separate these populations in the remaining LEHC countries. As a result, in the modeling process, a proxy was created whereby it is assumed that about 60% of the prison population would have been HCV-infected in the past by i.v. drug use, subtracting this number from the PWID population. A similar situation was applied to the relationship between blood products and PWID, seeking to correct double interpretation situations.Considering the modeling results, there are several situations in which the number of patients with compensated cirrhosis is higher than the total number of HCV infected individuals. This situation is verified as people who are successfully treated for HCV will still have pre-existing conditions such as the cirrhotic state. Considering the increasing number of people undergoing treatment and thus cured, it is natural that even with new HCV infection cases, in a given moment there will be less HCV infected individuals than individuals with compensated cirrhosis.In addition, diagnosis and linkage to care measures are paramount to meet the HCV elimination target, being aligned with the sustainable development goals. In Austria, it has been confirmed that 15 out of the 24 public health policies considered for the study are currently fully implemented.Despite the fact that other countries are also included in the LEHC project, it was considered that introducing external readings of comparison between country policies and epidemiological indicators could be an inducement of bad practices for health authorities. In the LEHC project’s developed work in 12 countries, the dynamics of public health policies were verified to be very independent, even in common issues.The LEHC project accomplished the modeling of hepatitis C in terms of public health policy impact on the disease epidemiology for the remnant population and other population groups in Austria. The dynamics of this tool allowed the gamification of public health policies, which provided an insight into how different extents of specific measures have an impact on the HCV epidemiology.This model has the ability to forecast the extent to which a specific public health policy might exert influence on a particular population group, in a pool of 24 public health policies focused on HCV considered in the LEHC project. This is essential in identifying patterns that have the potential to not only allow countries to address given policies to achieve the HCV elimination goal by segmenting the population but also to know the why, who, what, where and when. The tool allows rehearsing public health policy concepts before implementing them in the field, thus anticipating the outcomes.In the light of the cure cascade system, modeling results appoint awareness and prevention as the dimensions in which public health policies might have the broadest impact in lessening HCV prevalence values in Austria. Therefore, given these areas of influence, there is a clear indication that achieving the HCV elimination target will require additional efforts besides medication development. Diagnosing HCV in individuals who are unaware of their condition and linking them to healthcare services will determine the success of achieving the HCV elimination goal.According to the modeling results, the elimination target will not be met by 2030 in Austria. Nonetheless, in the case that all the 24 public health policies are fully implemented, it will not only be possible to meet the elimination target, but it might happen as soon as 2026. As a result, it is seen that policies are fundamental in providing insights into combating HCV in order to achieve its elimination.The quality of the modeling always depends on data quality used as input, on the quality of the analysis and on the knowledge about the public health policies’ impact in the field. In LEHC (or any other model), it is paramount to deal with the issue of quality as well with data availability.Since there are no confidence intervals in modeling, the fitting test was performed by overlapping the results obtained by the LEHC modeling tool, on the known epidemiological curve or admitted into a given geographic region, in this case, a country. In the LEHC tool, only countries where this curve was at least 95% similar in the previous years were selected for publishing data; however, it must be noted that even national data are often limited in their reading due to being either very regional , from samples sometimes incident on populations that do not represent the country (studies with prisoners or PWID), or epidemiological studies that do not cover the entire population. This dependence on adjusted fitting based on known epidemiological evidence is a limitation of the tool, but endogenous to the modeling process. No country which worked in the LEHC project showed strong robustness of epidemiological data.Considering that the LEHC model was based on the best available data from Austria, it is also important to state that obtaining national data was challenging, which imposed the need of producing proxies from Austrian regions, to extrapolate and interpolate national data, and to create statistical series having proxies as a basis composed by other secondary data series. Therefore, the Austrian LEHC model is the best possible to achieve, considering the data available. A call for action on the research and publication of epidemiological data regarding hepatitis C in Austria is of the utmost importance in filling in data gaps. This project is fully available to analyze and integrate all data that the scientific community provides for this project.In perspective, the dynamic vision for this project should contemplate an annual review for integrating new official data and other epidemiological data that might have been produced in the meantime. Similarly, the authors comprise the modeling results of LEHC for 31 December 2018, in which it is based on statistical data available up to the referred data and this being designated as “LEHC: An integrated solution for HCV—application in Austria”.


Gb of complex formation between a highly charged polyelectrolyte and a protein. The model introduced here comprises both the effect of released counterions and the uptake or release of water molecules during complex formation. The resulting expression for ΔGb is hence capable of describing the dependence of ΔGb on temperature as well as on the concentration of salt in the system: An increase of the salt concentration in the solution increases the activity of the ions and counterion release becomes less effective for binding. On the other hand, an increased salt concentration leads to the decrease of the activity of water in bulk. Hence, release of water molecules during complex formation will be more advantageous and lead to an increase of the magnitude of ΔGb and the binding constant. It is furthermore demonstrated that the release or uptake of water molecules is the origin of the marked enthalpy–entropy cancellation observed during complex formation of polyelectrolytes with proteins. The comparison with experimental data on complex formation between a synthetic (sulfated dendritic polyglycerol) and natural polyelectrolytes  with proteins shows full agreement with theory.A theoretical model is presented for the free energy Δ Schematic illustration of protein and polyelectrolyte is presented. Binding is accompanied by release of counterions and water molecules. For DNA being the best‐studied natural polyelectrolyte, this fact was already established in the 1970s of the last century and the review by Record et al. marks the progress made in this early work. Since this time the interaction of DNA with various proteins has been studied with great care by a number of groups. Interaction of synthetic polyelectrolytes with proteins has been a field of active research since a long time and a review of this work was given recently. Counterion release was established as the main mechanism operative in binding: The highly charged polyelectrolytes carry condensed counterions that are firmly bound and which do not contribute to the osmotic pressure within the system. For a negatively charged polyelectrolyte as, e.g., DNA the patches of positive charge on the surface of the protein become the multivalent counterion upon complex formation. A concomitant number of condensed counterions is released in this process. The gain of free energy thus effected is purely entropic. Up to now, the binding of DNA to various proteins is by far the best‐studied case. However, the scheme is far more general and counterions release is important for the interaction of proteins to a wide variety of highly charged natural polyelectrolytes such as RNA, or highly charged glycoaminoglycans as, e.g., heparin.Charge–charge interaction is of central importance for the formation of a complex between a polyelectrolyte and a protein.1 Thus using the theory of linked equilibria and binding polynomials. Already in 1969 Tanford called attention on the fact that the activity of water is bound to the activity of salt by virtue of the Gibbs–Duhem relation. The release or binding of counterions can be accompanied by a release or binding of water molecules the number of which being defined as Δw. This effect leads to an additional term in the expression for the binding free energy scaling as (ms/mw)Δw where ms denotes the molality of salt ions whereas mw is the molality of water (55.6). Tanford argued that this term is small for salt concentrations not too high. However, for salt concentrations in excess of 0.5 m, the release water will become important and modify the free energy of binding as shown for DNA interacting with natural proteins. In a series of precise and comprehensive experiments, Bergqvist et al. demonstrated that the release of water can reverse the weakening of the binding by added salt and lead to a nonmonotonous dependence of the thermodynamic binding constant Kb on the concentration cs of the added salt. A refined discussion of the release of water was given by Record and co‐workers who suggested that Δw is intimately related to the preferential adsorption of the ions to the surface of the biomolecule  has been a subject of intense discussion since the first paper by Lumry. However, the EEC is a clearly established fact as shown for biological systems by Jen‐Jacobson and co‐workers. Moreover, the EEC has been shown to be an established fact for synthetic systems by Liu and Guo. Recently, Schönbeck and coworkers presented a thorough study of the EEC on the interaction of cyclodextrin with guest molecules. All experimental studies done so far clearly reveal that the EEC is a general phenomenon which can be observed for a wide variety of synthetic and biological systems and which is related to the change of water networks during complex formation . If ξ > 1, a fraction 1–1/ξ of the counterions will be condensed to the linear chain. From these considerations a local concentration of condensed counterions cci can be defined that is typical of the order of one mol per liter. These ions must be treated as a reaction partner and considered in the stoichiometry of the reaction. The counterions balancing the charge of the macroion are therefore divided into a condensed part and a part interacting with the macroion via a screened Debye–Hückel interaction. In addition, hydration of the three components may change during complex formation and water (W) must be considered in the stoichiometry of the reaction as well. Hence, for a reaction of a protein P with an anionic polyelectrolyte PE to a complex PEP we havenci cations of type +M condensed to the polyelectrolyte have been released. It should be noted that anions may be released or taken up during binding as well. Therefore, Δnci denotes the net number of released ions. The quantity Δnw is the net change of bound water molecules during binding.We consider the equilibrium of a polyelectrolyte PE with a protein P in aqueous solution in the presence of monovalent salt ions with concentration Kb in terms of the concentrations of the reactantsIn the following, we formulate the measured equilibrium constant Then it can be shown that3(3)da± denotes the mean activity of the added salt, p = 2 for monovalent salt with molality m. The first term on the right‐hand side is related to the net release/uptake of ions Δnci whereas the release/uptake of water in the course of complex formation is treated in terms of the parameter Δw independent of salt concentration. Water does not constitute an independent component, however, but its activity is bound to the activity of the salt ions by virtue of the Gibbs–Duhem relation. The second term on the right‐hand side contains the ratio of the activity coefficients γPEP, γP, and γPE of the complex, the protein, and the polyelectrolyte, respectively. These activity coefficients are related to the Debye–Hückel interactions of the reaction partners. As shown by Record et al., a general analysis of the activity coefficients of rodlike polyelectrolytes and their complexes with proteins demonstrates that the term lnγPγPEγPEP gives a small but non‐negligible contribution to Kb. For the spherical macroions the term lnγPγPEγPEP was considered recently and shown to be negligibly in first approximation. Given these approximations it is expedient to replace the activity a± by the concentration [M+] of the monovalent ions. For the dilute solutions of the polyelectrolyte and protein under consideration here, [M+] may safely expressed through the concentration of added salt cs. Since experiments analyzing Δw are often done at high salt concentrations, however, the replacement of replace the activity a± by cs may become a stringent assumption and possible errors thus incurred must be kept in mind. Within the present approximations the molality m can be replaced by cs.Here Given the various assumptions and approximations, Equation  may now Integration of Equation  leads to53(5)Kb by theory. For counterion release embodied in the first term on the right hand side of Equation   Kb by tC can only depend on temperature but not on salt concentration. The binding constant Kb can be measured precisely by isothermal titration calorimetry (ITC) and leads to the free energy of binding throughFrom Equation  we see tThereforeGres = − RT · C is independent of salt concentration by virtue of Equation  cont cont8) c53 It quations  and (5) Kb depends on two decisive variables: i) the temperature T and ii) the salt concentration cs in solution. A comprehensive discussion of the complex formation must therefore comprise a set of experiments in which both variables are changed over the widest range possible. A suitable model must describe the dependence on both T and cs and contain mixed derivatives with regard to both quantities.Equation  demonstrcp is found. For many systems in which DNA is interaction with various proteins, Δcp is found to be negative whereas a positive Δcp is detected for several well‐studied synthetic polyelectrolytes. In these systems, |Δcp| ≫ |ΔSb| and both ΔHb and ΔSb exhibit a strong variation with temperature. This feature is a direct consequence of the fact that the free energy of binding ΔGb exhibits either a maximum or a minimum in the experimental range of temperatures where ΔSb = 0. Hence, two characteristic temperatures must be defined: The entropy ΔSb is zero at temperature Ts whereas ΔHb = 0 at a temperature Th. Here, Ts is the temperature where ΔGb exhibits an extremum while Th is the temperature where the binding constant Kb has a maximum or a minimum. This extremum of ΔGb and the large value of the specific heat are the thermodynamic origin of the EEC. ThereforeIn practically all systems in which polyelectrolytes form complexes with proteins, a large specific heat ΔCombination of both expressions leads to the well‐known generalized van't Hoff expression59cs=ΔHb/RT2 we get through differentiation of Equation (We now turn to a discussion of dependence of ΔEquation  and diffEquation (13)1RT2Hb(cs) scales linearly with salt concentration cs while Equation (Hb(cs) is also depending linearly on temperature. Therefore ΔHb must be given byH0 is an enthalpic contribution that depends neither on cs nor on temperature T. Therefore, the first derivative of ΔHb with regard to salt concentration cs is directly related to dΔw/dT. This fact shows directly that Δw cannot be a simple stoichiometric coefficient but is related to the free energy of binding. Since cs and T are independent variables, we obtain a Maxwell relationEquation  shows thEquation  states tHb depends linearly on both cs and T. In consequence, dΔcp/dcs must be a constant that does neither depend on T nor on cs. This constant is therefore characteristic for given system polyelectrolyte/protein. Hence, conditions (9), (14), and (15) require that ΔHb(cs) must be given by the following expressionT0 is a characteristic temperature specifying the dependence of Δw on T. Combining Equation , , and Δquations  and (20)Δquationsquations  we finalH0 and ΔS0, respectively, both referring to a new characteristic temperature T0. The last term in Equation . The prw is the part of the free energy of binding related to the release or uptake of water. It is hence interesting to compare the present treatment with the solute partitioning model (SPM) of Record and coworkers. The SPM treats the interaction of the ions with the dissolved biomolecule in terms of an ion‐specific interaction related to the Hofmeister series and the nonspecific lowering of the water activity by the salt ions. The protein and the polyelectrolyte are strongly hydrated and a part ΔBH2O of this shell of water molecules is released (or taken up) upon complex formation. The ions are now partitioned between the bulk solution and the hydration water on the surface. Thus, the equilibrium between these ions can be described by a partition coefficient Kp,+=(m+loc/m+bulk) for the cations where m+loc denotes the molality of the cations in the hydrated shell whereas m+bulk is the respective quantity in bulk. The anions are distributed in the same way characterized by the partition coefficient Kp,−. With these definitions, Δw can be expressed as For systems in which proteins interact with DNA, ΔBH2O is expected to a large negative number . The in3Gb on temperature and salt concentration and leads to characterization of the complex formation by a single set of parameters. The various terms in Equation . In this case the free energy of binding is purely entropic and does not exhibit any dependence on T. It should be kept in mind, however, that the present model comprises only two effects leading to complex formation, namely counterion release and uptake/release of water. Other factors that may contribute as, e.g., conformational changes of the protein upon binding are not included. They will in consequence be reflected in the terms ΔH0 and ΔS0 that are treated as adjustable parameters in the present model. Moreover, the present model is fully compatible with an additional term in Equation  on temperature is very weak whereas the magnitude of the free energy diminishes strongly with salt concentration cs. In our previous analysis, both features have been understood in terms of the number of released counterions Δnci and a temperature Ts that depends on salt concentration. No closed expression for ΔGb could be offered, however. Equation  if these data have a minute error only. However, extrapolation of the free energies of binding to a salt concentration of 1 m led for both systems to a constant value for ΔGres. This feature was already found by Dragan et al. for a large number of systems in which DNA interacts with various proteins. We could demonstrate in a recent paper that ΔGres is the free energy resulting at the direct contact between the polyelectrolyte and the protein.Equation  was usedEquation  was consMathLab routine cftool in which the pertinent parameters are fitted at once to the surface defined by the entire set of data ΔGb. This procedure leads to the parameters gathered in Table in two plots: Figure Gb measured for different salt concentrations as the function of temperature. Evidently, the slight curvature with T is correctly described by theory. Some data are less well described but it is important to note that a single set of parameters .T0 can already been read off the position of the maximum or minimum of ΔGb whereas Δnci can be derived by plots of ΔGb versus log cs  is fitted by the above procedure.It may appear doubtful to derive five parameters from the data displayed in Figure Equation  turns oubG against log cs may deviate from linearity when the third term in Equation  is dominated by the first term in Equation  on temperature. A second point in Figure cs measured at 310 K shown in Figure cs is increasing. Evidently, the conventional fit by a straight line used so often cannot not be entirely correct. Finally, the quantity ΔGres as calculated from the parameters in Table –1. From the experimental data we previously found a value of −20.3 kJ mol–1 (see the discussion in ref. [Equation . The las in ref. ). Given Figurem as the function of temperature whereas Figure Gb obtained at a temperature of 298 K as the function of cs. It should be noted that the dependence of ΔGb on temperature was only measured at a single salt concentration. Also, the dependence on cs is only available for a singe temperature which may restrict the accuracy of the derived parameters. The points in Figure cftool. The respective parameters are again gathered in Table It is interesting to compare the present theory to data taken from systems in which DNA interacts with proteins. Here we use the data of Datta and LiCata on the interaction of DNA with the polymerase Klentaq.7, 8RTΔncilncs + ΔH0 − TΔS0  and the blue solid line giving the full free energy of binding. The changes are of the order of a few kJ mol–1 and lead to the characteristic parabolic shape of ΔGb as the function of temperature. It is hence evident that Equation  which aEquation  can descEquation  adds up.Equation  is fullyEquation  with a tEquation  leads toEquation .Gres = ΔH0 − T0ΔS0 describes the free energy of the complex at direct contact. The present model allows us to split up this term into an enthalpic contribution ΔH0 and an entropic one ΔS0. The data gathered in Table As discussed previously, the residual free energy ΔGb in the experimental range of temperatures, Equation Gb in th4Gb of the complex formation between a polyelectrolyte and a protein in aqueous solution. The model covers the leading factors for binding, namely counterion release and the release or uptake of water during complex formation. Counterion release is covered by the first term in Equation (Gb on temperature. The term Δw interpreted so far as the number of released water molecules is related to the free energy of hydration and vanishes at a characteristic temperature T0. Theory is capable of describing the free energy ΔGb on temperature as well as on salt concentration in a fully quantitative manner. Equation (Gb together with the enthalpy of binding ΔHb and the respective entropy of binding TΔSb obtained through application of Equation (T. This finding which is typical for many complexes of polyelectrolytes with proteins can be easily explained in terms of the present model. Therefore the present model allows us not only to model the interaction of polyelectrolytes with proteins but also to understand a long‐standing and controversial subject in the field.We presented a simple phenomenological model for the free energy ΔEquation  whereas Equation  determinEquation  also shoEquation . The freT and cs. Moreover, these investigations should be done using salts that exhibit defined Hofmeister effects. A comparison with the SPM could then quantify the role of given ions for complex stability and compare this information to the rich literature on protein denaturation and also on work done on synthetic polymers.The present work also demonstrates that studies of complex formation between a polyelectrolyte and a protein should comprise the investigation of the binding constant as the function of both The authors declare no conflict of interest.


The current clinical interface for Varian's intrafraction motion review (IMR) is limited, providing only qualitative data for review at the treatment console. This study provides a method of extracting and interpreting data from combined log files for quantitative evaluation. Combined log files acquired during patient treatment and a parsing code was developed to scan the combined log file looking for unique identifiers pertaining to the data of interest. We were able to extract clinically relevant parameters from the log files including date and time, gantry angle, expected marker position, found marker position, pixel size, and detection result. This study details how to compare IMR data to Calypso investigating dual‐surrogates for intrafraction monitoring during treatment for other researchers to build on these methods. Understanding data recorded during treatment within the combined log files can be helpful in quality improvement of patient care by retrospectively reviewing intrafraction motion. Orthogonal or cone‐beam computed tomography (CBCT) imaging is common for modern linear accelerator‐based (linac) treatment, though soft tissue evaluation can be difficult to evaluate using these methods. To help with these challenges, it is common to insert fiducial markers such as gold seeds into or adjacent to the target to be used as surrogates for localization. These fiducials can also be used to monitor the target during treatment, though more sophisticated tools are required.For departments using Varian TrueBeam® linacs with Advanced Imaging, Intrafraction Motion Review (IMR) is a tool for monitoring fiducial positions throughout treatment. Fixed points in space with reference to isocenter, known as markers, are placed in reference to the fiducial location during treatment planning. IMR uses the position of the markers to calculate where the fiducials would be expected when taking kilovoltage (kV) images at associated gantry angles during treatment and an algorithm compares these expected locations to the detected fiducial location in the image. The images are taken orthogonal to the treatment axis, and the frequency of the imaging is determined by the user based on gantry angle, time, or number of delivered monitor units.While IMR has been shown to be useful by investigators,Although it is currently not a user‐friendly method for extraction and analysis, the expected and detected marker location data is recorded in the combined log files. The goal of this work is to share our method for IMR data extraction so that others can improve patient care in their own clinic. We also present our method for comparing IMR data with Calypso data positions for further research.22.1All IMR data is stored in combined log files on the linac console computer, which is compiled daily, archived to last several months, and then periodically deleted. These files are intended for service engineers to access and use, so the data is not exported automatically for clinical operations via PeerSync™ requiring manual extraction from the console computer. Access to the drive where the files are located is limited, so if the user lacks the appropriate rights, it may be necessary to contact the vendor to transfer the files.Combined log files track nearly all of the linac's operations throughout the day, making them difficult to extract only the IMR‐relevant data using manual export method as shown in Figure It is recommended that care is taken to ensure the correct location data is being extracted from the log file. For IMR, there are three sets of positions for each fiducial's X and Y coordinates. Labels for maker positions include the expected position noted as “ExpectedMarkerPosition” and found position as “MarkerSingleResult” for data analysis. While “MarkerGeometryCheck” also records fiducial positional data, it is not directly related to found or expected fiducial position and is not needed to determine planned versus treatment absolute fiducial positional differences. If one or more markers are not detected, they are recorded as such (NaN) in the combined log file.Fiducial position data is stored in terms of pixel location, making it is necessary to obtain pixel size information to convert to the International System of Units. The log file subdivides all positional and pixel information by fiducial and X and Y coordinates. The information collected using this method yields X and Y marker positions relative to the imaging panel. If pixel size is used to convert to units of millimeters, it should be noted that this distance is for the projection at isocenter, not at the imaging panel distance.Detection result is the classification of the color‐coded results displayed on the console during treatment . The stored data does not record found versus expected fiducial position difference, but the absolute X and Y positions and the detection result . However, such numerical detection positional differences can be calculated externally.Combined log files do not store any patient information, so if it is desired to correlate the data to a particular patient, a system for patient identification will need to be developed. To accomplish this, we used a time‐based system such that the time of each beam or arc was recorded during treatment in reference to that particular patient and fraction. Time information is also stored on each line of the log file with millisecond precision. If this is extracted for each segment of image data, it is then straightforward to match time information to patient data.Log into the console computer as an admin and copy the combined log files to a drive accessible on your network.Within the log file, locate the first instance of the unique identifier “ImageAcquisition” and copy the date and time from that row.Continuing down the file, the next unique identifier of interest is InvokeIP|ImageSource|AcqId” which represents gantry angle information in this row.Next is "|_InvokeIP|ExpectedMarkerPosition|AcqId|" which contains expected marker positions.The next row with "|_InvokeIP|InputSearchRegion|AcqId|" contains the pixel size information.The "|_InvokeIP|MarkerSingleResult|AcqId|" row will yield the expected marker position and seed state in relation to tolerance diameter set before treatment.Finally, convert from pixels to millimeters and calculate true geometric differences.A step‐by‐step guide of the complete process if performed manually is as follows. Steps 2–7 were automated by the authors for this study to increase the efficiency and accuracy of the data processing. Please note that the unique identifiers may have changed since publication and will vary depending on what data is required for your analysis.2.2While most users will only use a single method for intrafraction monitoring during treatment, Cetnar et al. performed a dual‐surrogate study to evaluate the accuracy of the IMR system.The Calypso system reports the fiducial positions in 3D in the linac vault coordinate system. The IMR system uses a kV x‐ray source and detector to image the patient and detects the Calypso beacon fiducials as markers on the 2D kV imager coordinate system. In order to be able to compare the positions reported by the two separate monitoring systems, we translated the positions of the fiducials from the 3D Calypso coordinate system to the 2D IMR coordinate system.Three fiducials were analyzed for each patient during treatment. IMR 2D marker positions in the kV imager coordinate system, timestamp, and planar acquisition angle were recorded during treatment. Calypso System Data Converter Version 1.9 was used for 3D data extraction which reports 3D marker coordinates  of each fiducial. The synchronization of the two datasets was performed using the timestamps. If exact timestamp match was not found, the data set pairing was matched to the one with the shortest temporal difference. The average sampling time for our Calypso data reported coordinates was approximately 30 ms.The gantry angle reported at the matched timestamp was used to project the 3D coordinates of the fiducials into the 2D planar kV imager coordinate system using a cone‐beam geometry shown in Figure u,v) from Calypso are given bydSDD and dSAD are the x‐ray source‐to‐detector distance and the x‐ray source to rotation axis distance, respectively  are the 3D coordinates of the fiducials, and θ is the IMR planar image acquisition angle.The 90‐degree angle offset between the x‐ray source and the gantry angle was also taken into account when comparing data between the two systems. The projected 2D coordinates on the x‐ray detector  for all pair combinations between the systems. The pair combinations with the minimum distance were interpreted as the matched IMR and Calypso marker and fiducial pairs.The developed 3D to 2D projection code comparing IMR and Calypso data was validated with the data acquired using phantoms with embedded with Calypso fiducials shown in Figure 3u,v) of the fiducials with the Calypso data projected on the kV detector reference frame. The IMR detected coordinates and plotted in Figure u‐coordinate between the Calypso and IMR coordinates with Δu as function of kV source angle θ, the variation of this difference for each fiducial, and finally the variation of all fiducials in a single histogram. The bottom row is plotted for the v‐coordinate. The mean and standard deviation of the differences between the Calypso and IMR calculated u‐ and v‐coordinates are 0.0 (0.07) cm and 0.04 (0.05) cm, respectively. Validation testing was repeated using the Calypso Daily QA phantom yielding similar results calculated u‐ and v‐coordinates are 0.0 (0.04) cm and 0.04 (0.02) cm, respectively.Validation of the 3D to 2D projection code for IMR and Calypso data compared calculated coordinates (u and Δv as a function of kV source angle (θ), variation of the differences of each fiducial, and finally the variation of all fiducials in a single histogram in a patient cohort. The IMR algorithm was sometimes able to detect all three fiducials and marked them undetected or outside of the expected limit thresholds. In this analysis, we excluded the data when all three markers were not detected. For the patient data, the means of Δu and Δv are calculated as 0.01 cm and 0.00 cm, with standard deviations of 0.11 cm and 0.07 cm, respectively.Similarly, Figure 4We faced several logistical challenges in the process of data extraction. The first involved the verification of timestamps between the two systems and establishing a method for synchronization for our analysis. The second is the awareness that updates to software can affect the output in the combined log files. It is recommended that a quality assurance program be developed for in‐house software after changes have been made.Our final recommendation is to perform end‐to‐end tests using phantoms before using clinically to better understand how the data in the log files can be interpreted. We used vendor‐provided QA devices and anthropomorphic phantoms for our initial validation testing of our workflow which was helpful for making final improvements to our in‐house analysis. As an example, initial comparison data was not expected to have a large sinusoidal pattern between the two data sets. Upon investigation, we found differences in the offline review data for the imager position which we were able to correct and modify to better represent the data.5Through this process, we were able to develop ways to extract quantitative data helpful for us in understanding the accuracy of the IMR system and use it for process improvement within our clinical workflow. The current data available for clinical decision‐making is limited when using IMR. However, data is recorded in the combined log files, which can be utilized to better understand motion during patient treatment. Understanding data recorded during treatment within the combined log files can be helpful for retrospectively reviewing motion during treatment to continue to improve patient care.There are no conflicts of interest to disclose.Gavin Graeper – Made substantial contributions to the concept and design of the work, including acquisition, analysis, and interpretation of the data. Helped draft and revise the document through to its final submitted form. Agrees to be accountable for all aspects of the work. Ashley Cetnar – Made substantial contributions to the concept and design of the work, including acquisition, analysis, and interpretation of the data. Helped draft and revise the document through to its final submitted form. Agrees to be accountable for all aspects of the work. Ahmet Ayan – Made substantial contributions to the concept and design of the work, including acquisition, analysis, and interpretation of the data. Helped draft and revise the document through to its final submitted form. Agrees to be accountable for all aspects of the work. Michael Weldon – Made substantial contributions to the concept and design of the work, including acquisition, analysis, and interpretation of the data. Helped draft and revise the document through to its final submitted form. Agrees to be accountable for all aspects of the work.


The purpose of this study was to investigate the matching error that occurs when the Mobius3D fingerprinting system is applied in conjunction with an Elekta linear accelerator (LINAC) and to offer an acceptable and alternative method for circumventing this problem.To avoid the multileaf collimator (MLC) conflicting error in the Mobius3D fingerprinting system, we developed an in‐house program to move the MLC in the Digital Imaging and Communications in Medicine (DICOM) radiotherapy (RT)‐Plan to pertinent positions, considering the relationship between log data and planned data. The re‐delivered log files were calculated in the Mobius3D system, and the results were compared with those of corrected data . The results were then evaluated by comparing several items, such as point dose errors, gamma index (GI) passing rates, and MLC root‐mean‐square (RMS) values.t‐test showed that there were no significant differences within the uncertainty. In the case of the RMS test, the maximum difference found was approximately 0.08 mm.For the point dose error, the maximum difference found was below 2.0%. In the case of GI analysis of all plans, the maximum difference in the passing rates was below 1.4%. The statistical results obtained using a paired Student's Our results showed that all the mismatched log files were sufficiently acceptable within the uncertainty. We conclude that the matching error obtained when applying Mobius3D to an Elekta LINAC may be addressed using a simple modification of the fingerprinting system, and we expect that our study findings will help vendors resolve this issue in the near future. The new QA platform can automatically calculate 3D dose distribution in the patient Computed tomography (CT) and report any plan issues detected from various items , and 3D gamma analysis).When a machine log file is received into the Mobius3D platform, the platform's fingerprinting system is performed to match the log file and its pertinent plan data. The fingerprinting system is well‐established for almost all cases; however, we suffered a specific problem during the last year of our application of the Mobius3D platform in practice. The problem is that the log file collected from the Elekta LINAC cannot find a corresponding plan even if the log data were correctly collected. This issue has twice been reported to Varian Medical Systems , but an effective solution has not been presented. Currently, the vendor recommends re‐delivery of the plan to temporarily avoid this issue.In this study, the cause of the mismatch between the Mobius3D platform (simply referred to as “Mobius3D” in the remainder of this paper) and the log file from the Elekta LINAC was analyzed using a patient‐specific QA case that was not recognized by Mobius3D. The results of our analysis using the re‐delivered data were compared with the modified data from the case that was mismatched. We presented a method for reducing the mismatching issue between the two systems.22.12.1.1Mobius3D is a platform for patient‐specific QA using a patient CT dataset, independent beam models, and a simplified collapsed‐cone dose calculation algorithm. Mobius3D is divided into two parts: MobiusCalc and MobiusFX.MobiusCalc is a dose recalculation module based on an independent beam model and planned data extracted by the primary treatment planning system (TPS). In this module, the planned data are assessed using several items, such as the target dose coverage, DVH limits, 3D gamma comparison, and deliverability confirmation, which are then compared with machine delivery parameters. In Mobius3D, the MobiusCalc procedure is termed as “Plan Check.”MobiusFX is another dose recalculation module accompanying MobiusCalc; the difference between the two modules lies only in whether the dose calculation procedure employs collected data from a LINAC machine (a.k.a. log data) or the planned data. In Mobius3D, the MobiusFX procedure is termed as “QA Check.” The collected log data from the LINAC machine are described in Section 2.1.2.2.1.2In an Elekta LINAC, the machine parameter data are extracted via iCom communication, and the data from the LINAC are collected and transferred to the Mobius3D system using Mobius Log, which is a program provided by the vendor. The schematic workflow for the collection of log data using Mobius Log is shown in Figure 2.22.2.1Energy–must be equal to that of planned dataInitial X Jaw extents–within 1.1 cm of planned data (ignored for Elekta)Initial Y Jaw extents–within 1.1 cm of planned dataInitial Gantry angle–within 4.1° of planned dataInitial collimator angle–within 4.1° of planned dataInitial MLC positions–within the field, while allowing a 1‐cm tolerance of planned dataThe installed Mobius3D platform may receive several DICOM datasets from the TPS and several log files from the Mobius log program. To match the corresponding data between a DICOM dataset and its log data, Mobius3D provides a fingerprinting system; in other words, the QA check is performed using “passed” log data from the fingerprinting system that lie within their range of uncertainty. The conditions imposed on the fingerprinting system are listed below2.2.2The MLC conflicting issue occurs frequently when using Mobius3D with an Elekta LINAC, owing to the differences in the initial MLC positions over the 1‐cm tolerance of the fingerprinting system, even if the initial MLC positions are virtually the same for the planned and logged data. The three different types of Mobius3D visualized analyses for checking the initial MLC positions are shown in Figure 2.3To analyze the cause of the MLC conflicting issue, five mismatched cases were collected. Figure Unfortunately, the fundamental reasons for the MLC conflicting issue could not be discovered without the Mobius3D source code. As two indirect methods, in this study, the comparison study was designed as a below: (1) re‐delivery and acquisition of new log data that passed the fingerprinting system , and (2) adjustment of the initial MLC positions included in the DICOM RT‐Plan to pertinent positions within the 1‐cm tolerance range . Note that log file modification is also an option but not considered in this study owing to their own binary formats. The DICOM RT‐Plan was modified using MATLAB (The MathWorks) for each plan. The schematic workflow for calculating four different dose distributions using TPS, M3D, MFX, and Modified MFX is shown in Figure From the calculated results, point dose and gamma index (GI) analyses were performed, and the results were compared with those of the TPS. With respect to the point dose, a specific volume with a size of an A1SL ionization chamber  in the planning target volume was defined for five DICOM datasets, and all the average values pertaining to each of the volumes were compared with each other. The dose errors were calculated according to Equation . Data were compared as follows: null hypothesis: H0 = 0 (no difference).In addition, the root‐mean‐square (RMS) errors for the MLC position between the plan and log data values were calculated to quantitatively evaluate the differences between the failed and passed log data. The failed log data originated from a comparison with the modified RT‐Plan in our study, and the passed log data originated from a comparison with the unmodified RT‐Plan. The RMS error was calculated at all control points and the tolerance was set to 0.5 mm. All the comparison results included a statistical analysis performed using a paired Student's 3The point dose results obtained from M3D, MFX, and Modified MFX for the five QA cases are displayed in Table p‐value (p = 0.840) was not rejected; that is, it was confirmed that there was no significant difference between the passed and failed log data. As a result of comparing the RMS values for the MFX and the modified MFX, which were evaluated to confirm the positioning of the MLC, it was confirmed that the RMS values lay within 0.5 mm of the MLC position, and the difference of the maximum RMS values was 0.08 mm, which implies that the values were numerically similar. The results of the statistical analysis indicated that there was no statistically significant difference between the RMS values (p = 0.972).The GI passing rates of the results calculated by M3D, MFX, and Modified MFX were compared with those calculated by the TPS and are displayed in Table 4We performed an analysis to determine the cause of the MLC conflicting error that occurs when Mobius3D fingerprinting is applied in conjunction with an Elekta LINAC, even if the planned and delivered beams are virtually identical. Five mismatched cases were collected and analyzed for identifying the differences in the MLC positions and their related dose distributions, calculated using various methods. Our results confirmed that there was no significant difference between the passed and failed log data; nevertheless, the failed log data was filtered by the fingerprinting system of Mobius3D. This means that the visualization in Mobius3D is only visible between the position of the first collected log data and the RT‐Plan, but the initial position of all log data  is verified by the fingerprinting system.This outcome can be attributed to the difference of an initial MLC position between a trigged log information and a planned data. To the best of our knowledge, the log information with respect to the MLC positions for Elekta machines was collected every approximately 0.3 s with a value of control point index. In this study, we found that the value of the initial point index might not change even when the MLC position moves according to the next index in all plans. At this point, the fingerprinting system can detect an MLC conflicting error if the logged initial MLC position  is collected by more than 1‐cm tolerance of the planned data. Notably, the MLC conflicting errors could be caused due to the fast movement of MLCs between the initial and subsequent positions. An example of MLC position changes according to the collected MLC positions in a log file is shown in Figure In addition, an RMS calculation was used to compare the set of the RT‐Plan of the TPS with the passed logs of the set of modified RT‐Plans and the failed logs. The values of the two results were found to be similar. The reason for this was that only the initial point changed. However, the RMS values modify the RT‐Plan of the TPS, implying that even larger differences may appear. Nevertheless, we found no significant difference since the modified method itself was adjusted to less than 1 cm, the MLC matching tolerance.Two additional cases were excluded from this study: the pertinent positions for the modified MFX calculation that could not be found; that is, the cases in which the difference between the initial and the next MLC position was higher than twice the fingerprinting tolerance . These cases were successfully passed from the assessment of the conventional patient‐specific QA.With respect to the frequency of occurrence in the study conducted in our institution, the MLC conflicting error was detected approximately 3% of the time. The QA procedure was re‐performed at such occurrences, following the vendor's recommendation.5We analyzed the causes of log file errors that may occur in Mobius3D software and Elekta machine log files and analyzed the dose errors that may be caused by incorrect log files by modifying the RT‐Plan file of patient‐specific QA. As a result, we confirmed that the dose difference was negligible, and we conclude that the problem can be resolved by using a simple modification of the fingerprinting system, such as a correct log data collection corresponding to a change of control point index. In the near future, we expect that vendors will be able to resolve the frequent MLC conflicting issues using the Elekta LINAC machine. Consequently, we anticipate that users’’ time and effort  expended in addressing this problem will be significantly reduced with future updates of Mobius3D.The authors have no conflict of interest to declare.Min Cheol Han and Tae‐Suk Suh conceived the presented idea. Yu‐Yun Noh and Han‐Back Shin carried out the experiment and analyzed the data. Min Cheol Han, Jihun Kim, and Yu‐Yun Noh designed the model, and Yu‐ Yun Noh and Min Cheol Han wrote the manuscript consultation with Jin Sung Kim and Tae‐Suk Suh. All authors provided critical feedback and helped shape the research, analysis, and manuscript.


We couple the mixed variational problem for the generalized Hodge–Helmholtz or Hodge–Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a “rough” surface. The low-frequency robustness of the potential formulation of Maxwell’s equations makes this model a promising starting point for Galerkin discretization. Let ry parts  and suppentclass1pt{minimanon-static electromagnetic transmission problems). The hypothesis that the first Betti number is zero is used in Sect. The hypothesis that the second Betti number is zero is only used to prove injectivity of the coupling problem for Hodge–Laplace operators. It can be dropped without any changes to the following development for couplings involving the Hodge–Helmholtz operator  with the first-kind boundary integral equation (BIE) arising from (3b) using these recently developed Calderón projectors for the Hodge–Helmholtz and Hodge–Laplace operators. The proof of the well-posedness of the coupled problem relies on T-coercivity  and is iated to a with thLet Development of trace-related theory for Lipschitz domains and detailed definitions for the surface differential operators found in , 8, 10. Based on the continuous and surjective extensionsThe duality pairings enter Green’s formulas  re re16b) rproblems , Sec. 6.problems  that theinfinity , Sec. 3.ectively , Sec. 7.In the case of the classical electric wave equation, the boundary integral operators involved in the Caldéron projectors enjoy a hidden symmetry: there exists a compact linear operator ,Γ), cf. , Lem. 6.cancellation, up to compact terms, of the operators lying on the off-diagonal of the block operator matrix associated to the coupled variational system introduced in Sect. We will extend this result to the integral operators defined for the scaled Hodge–Helmholtz/Laplace equation to better characterize the structure of . The symThere is a compact linear operator This proof utilizes a strategy found in [On the one hand, using the scalar Helmholtz equation  and recad, using  togetherd, using , we alsooperties , Sec. 5 For all In the following calculations, the boundary integrals are to be understood as duality pairings. Since On the other hand, the same observation implies that There exists a compact operator Recall that ult from  with LemAs consequence of Proposition In this section, we derive a variational formulation for the system (3a)–(4b) which couples a mixed variational formulation defined in the interior domain to a boundary integral equation of the first kind that arises in the exterior domain.As proposed in , we intr formula  in \docufirst exterior Calderón identity given on the first line of  eigenvalues of the scaled Hodge–Laplace operator with constant coefficient non-trivial solution In the following proposition, we call eader to , where teader to , 29, 30 Suppose that  formula  for the The proof follows the approach in [ows that  holds foWe recognize in the equivalent operator equationrojector  whose im problem , Sec. 3  problem , Sec. 3.entclass1pt{minimaNow, the null space of the interior Caldéron projector mentclass2pt{minimSuppose that We show in , where tned from .Using the classical Hodge decomposition, a general inf-sup condition for Hodge–Laplace problems posed on closed Hilbert complexes was derived in . HoweverA decomposition with similar properties can be designed for the space As before, the extra regularity of the range, in this case provided by [The subspaces A very useful property of this pair of decompositions is stated is shown in [Another benefit of this pair of regular decompositions will become explicit in the poof of Lemma It follows from [We use the direct decompositions introduced in Sect. The coupled variational problem (30) translates into the operator equationThe symmetry revealed in Sect. Our goal is to design an isomorphism V is T-coercive:V, then the operator b associated with uniformly elliptic operator of even order A is injective is given in [The authors of  refer toing that  holds wigiven in . T-coercgiven in .In deriving the following results, it will be convenient to denote In this section, we take up the challenge of finding a suitable isomorphism ation in .We start with devising an isomorphism Under the assumption that the first Betti number of Notice that since it also follows from the Poincaré-Friedrichs inequality that For any By showing that Let Now, suppose that We now turn to the design of an isomorphism for the Neumann trace space For any We proceed as in proposition Suppose that In the following, we will write The main result of this work, stated in Theorem ribed in . It reliblock diagonal ofis T-coercive.The first claims that the For any frequency The proof of this proposition will rely on several steps: Lemmas The second proposition states that the off-diagonal blocks are compact operators. The proof of that fact relies on definitions and results that belong to the next technical section. It will materialize as the last piece of the puzzle that completes the proof of the T-coercivity of For any frequency The main result immediately follows from the two previous propositions.For any The proof will amount to the validation that the choices of parameters in the previous propositions The following corollary is immediate upon applying Theorem The system operator Injectivity, guaranteed when Equipped with the isomorphism fined in  and assoFor any frequency As The complex inner productsLet us denote the two components of the isomorphism For any frequency The jump condition  yield \dmentclass2pt{minimIn the next lemma, we prove coercivity of the lower diagonal block of the coupling operator For any frequency The jump conditions  yield \dEquipped with the previous three lemmas, we are now ready to prove Proposition For any parameters Finally, The off-diagonal blocks remain to be considered. We will show that, up to compact perturbations, a suitable choice of parameters in the isomorphisms The isomorphisms  side of .From the jump properties , we havee, as in , we eval. When summing the real parts of 4445Many lying on  to 12d)4445Many To go further, we need to settle for a choice of parameters in the volume and boundary isomorphisms. Choose  vanish, because we have Under this choice of parameters, the terms in lying on , 10b) a a vanishlying on  once mormixed formulation of the variational form of the Hodge–Helmholtz and Hodge–Laplace equation with first-kind boundary integral equations. Well-posedness of the coupled problem was obtained using a T-coercivity argument demonstrating that the operator associated to the coupled variational problem was Fredholm of index 0. When In Sect. The symmetrically coupled system (30) offers a variational formulation of the transmission problem (3) in well-known energy spaces suited for discretization by finite and boundary elements. It is therefore a promising starting point for Galerkin discretization.