Photoplethysmography (PPG) is a non-invasive photometric technique that measures the volume changes in arterial blood. Recent studies have reported limitations in developing and optimising PPG-based sensing technologies due to unavailability of the fundamental information such as PPG-pathlength and penetration depth in a certain region of interest (ROI) in the human body. In this paper, a robust computational model of a dual wavelength PPG system was developed using Monte Carlo technique. A three-dimensional heterogeneous volume of a specific ROI  was exposed at the red (660 nm) and infrared (940 nm) wavelengths in the reflectance and transmittance modalities of PPG. The optical interactions with the individual pulsatile and non-pulsatile tissue-components were demonstrated and the optical parameters  were investigated. Results optimised the source-detector separation for a reflectance finger-PPG sensor. The analysis with the recorded absorbance, reflectance and transmittance confirmed the maximum and minimum impact of the dermis and bone tissue-layers, respectively, in the formation of a PPG signal. The results presented in the paper provide the necessary information to develop PPG-based transcutaneous sensors and to understand the origin of the ac and dc components of the PPG signal. Photoplethysmography (PPG) is a non-invasive technique that uses light for measuring the volumetric changes in blood associated with the cardiac cycle in vascular tissue beds . In PPG,SpO2) . The art . The arIn recent years, there has been a plethora of interest in extending the application of PPG beyond pulse oximetry, for example, usage of the PPG in the assessment of vascular mechanics, blood pressure, blood viscosity, pulse transit time estimation, pulse rate variability, assessment of tissue perfusion etc. ,7,8,9,10Previous mathematical models of PPG and pulse oximetry based on Beer-Lambert law, simple diffusion theory or random walk theory ,15,16 weIn this paper, a light-tissue interaction-based model of a dual wavelength PPG sensor is presented where a heterogeneous tissue-volume resembling human finger has been exposed at the commonly used optical wavelengths: red (660 nm) and infrared (940 nm). Monte Carlo method has been chosen for the analysis which is a stochastic approach and is extensively used for simulating light propagation through biological tissues . The metThe anatomical feature of the tissue model is presented in The ‘pulse’ was simulated in the tissue by increasing the arterial blood volume in the dermis during systole by twice as much as in diastole . The sysThe optical parameter responsible for attenuating light propagation through tissue is its absorption coefficient on below ,39:(1)μConsidering the absorbance of light through arterial and venous blood with different concentrations of the absorbers oxyhaemoglobin μThe absorption coefficients of water, oxyhaemoglobin and deoxyhaemoglobin (at a hematocrit of 45%) were adapted from literature ,42,43,44Since the dermis is responsible for the pulsatile volumetric changes in the PPG signal, and also each dermal sublayer is composed of different chromophore distribution, the contribution of the individual components in the absorption was considered, as already expressed by Equations (2)–(5). However, due to unavailability of precise vascular distribution for muscle and fat tissue, and also for the fact that those layers are much thicker than skin, the absorption coefficient of these layers were considered as bulk. The finger bone  is supposed to be different anatomically from cranial bone, especially in the blood content, which eventually would create differences in their optical properties. However, the main concern in the present study is the relative absorbance of light in red and infrared wavelength and not the absolute absorbance, which allowed the compromise of the bone optical properties.A flowchart for the basic steps of the Monte Carlo (MC) simulation for propagation of light through tissue in a finger-PPG configuration is presented in As shown, a photon packet with an initial direction and position co-ordinate was launched onto the tissue surface. The initial statistical weight of the photon packet was  scatter , i.e., was dedicated for the simulation. A MATLAB  platform was chosen for coding and a multi-thread programming environment was used for facilitating the MC simulation.I refers to the mean weight of the detected photon packets, which is termed as ‘reflectance’ or ‘transmittance’ at respective geometries. Expressing the ‘pulse’  by As stated before, the MC model was executed in two modalities. In the reflectance mode, the optical source and detector were placed 5 mm apart whereas in the transmittance mode, the those were placed at two opposite surfaces of the tissue site . Incidence of a Gaussian beam of 1 mm radius was simulated to the tissue surface that propagated through the tissue to be detected at a circular detector of radius 1 mm. The MC model was executed to quantify the distribution of scattering events, and to calculate the mean optical path and penetration depth at red and infrared wavelengths. The model was also employed to record the ‘intensity’ of the detected photon packets. Here, ‘intensity’ R) as:The ratio of the red and infrared relative normalised intensities was quantified by the ‘ratio of ratios’  and (9), the normalised absorbances io of RM :(10)RM=The distribution of the scattering events in a dual wavelength reflectance PPG setting (similar to reflectance pulse oximetry) simulated by MC is shown in 8. The average time taken for each simulation was 2 hours.Comparatively, the distribution of scattering events through the finger tissue volume at −2, which was about 105 times higher than the IR diastolic reflectance, i.e., 5.27 × 10−8. Apparently, both the reflectance and transmittance for red light increased very slowly with arterial oxygen saturation, whereas for IR it gradually decreased. The diastolic intensities were slightly higher than the systolic intensities in all cases. The calculated normalised pulsatile reflectance , the reflectance geometry was chosen. The source-detector separation was varied from 1 mm to 10 mm, with an 1 mm gap between two consecutive detections, and MOP and MD were recorded at each d for d were 1010. The simulations were performed at both the red and infrared wavelengths at systolic and diastolic states. MOP increased almost linearly with d. The maximum optical path taken by any photon packet was 60 mm. Penetration depth increased sharply with an initial increase of d, and for higher separations it increased slowly.For a comprehensive demonstration of the variation of the mean optical path (MOP) and the mean penetration depth (MD) of photons through finger tissue as functions of source-detector separation  is illustrated in The maximum absorbed photons of both red and infrared wavelengths at systole and diastole were localised in the epidermis. The minimal absorption was found in the stratum corneum, then in fat, bone, muscle and dermis. Interestingly, unlike any other layer, in dermal layer the difference between the infrared and red absorbances was very high in both diastole and systole  in clinical setting for continuous monitoring of arterial blood oxygen saturation. In the computational experiment, the subject was considered at rest, and free from any external influences such as motion, electromagnetic interference etc. The effect of the physiological parameters of the pulsatile components such as blood volume and oxygen saturation, and those of the non-pulsatile components such as tissue thickness or melanin concentration have been separately studied in this paper for detailed understanding of the realistic model.PPG is usually used for assessing the peripheral oxygenation and perfusion, rather than deep tissue monitoring, thus shorter source-detector separations (<1 cm) were simulated in the work. An important observation from the reflectance PPG simulation was the wavelength-dependence of the optical path and penetration depth for higher source-detector separations. In the dual wavelength PPG-applications such as pulse oximetry, it is usually assumed that the operating wavelengths interrogate the same sampling volume . However follows :(14)SaOThe dissimilarities in the distribution of scattering at red and infrared wavelengths were caused by the higher scattering co-efficient at the lower optical wavelength. The full scattering profile in the transmittance geometry showed the maximum scattering events taking place in the bone layer, followed by the dermal layer. On the other hand, the relative absorbance was maximum in the dermal layer and minimal in the bone layer. Results confirmed that despite exhibiting a high scattering, the bone layer had a minimum impact on the detected PPG intensity. There has been no study so far that has simulated a tissue volume including the bone optical properties in reflectance and transmittance optical modalities, thus no information was available on the contribution of the bone layer in the light-tissue interactions in a system. The quantification of the relative absorbance and the modulation ratio of different tissue layers, therefore, would be beneficial for understanding PPG and other similar optometric sensor technologies.The ‘pulse’ simulated in the model, i.e., the difference between systolic and diastolic intensities, was caused by the absorption by the pulsatile tissue compartment, contributing to the ac component of the PPG signal. The diastolic intensity, on the other hand, represented the absorption by the non-pulsatile tissue compartment producing the dc component of the PPG signal. The ratio of ratios plotted against arterial oxygen saturation resembled the calibration curve which is the technical characterisation of a pulse oximeter. The strong positive correlation between the calibration curves of the two modalities indicated that the sensor geometry does not significantly influence the efficiency of a pulse oximeter for The non-pulsatile parameters such as tissue thickness and skin pigmentation often raise questions regarding the PPG measurement. The influence of these parameters were assessed using the present model. For a person with a very light-couloured skin, the detected light intensity was seen to be much higher (3–5 times) compared to a person with very dark coloured skin because of the obvious reason that the epidermal melanin is a very strong absorber of light. The decay of intensity exponentially fell with the increasing melanin concentration. The optical pathlength in tissue also was found to decrease with the increasing absorbance by melanin even though no change in scattering took place, indicating the dependence of the optical path on the absorption property of the tissue. However, no change in penetration depth was found with the increasing melanin which establishes the fact that penetration depth does not depend on the absorption coefficient of the tissue. Due to the changes in the optical pathlength, the photons will be likely to interrogate slightly different sample volumes with increased melanin concentration, resulting in an increase in noise in the measurement. In the experiment with the increasing thickness of the fat layer in the finger, no significant change in the detected light intensity was visible. Therefore, the relativistic measurement should not be affected for different finger thicknesses in a transmittance PPG setting such as pulse oximetry.all non-pulsatile tissue compartments had to be considered which was never addressed in earlier studies. To the authors’ knowledge, the computational model presented in this paper is the first attempt to investigate the optical interaction with all pulsatile and non-pulsatile tissue compartments of the finger, providing an insight into the formation of both dc and ac PPG signal. The results from both the reflectance and transmittance geometrical settings exhibited a localisation of photons in the dermal tissue because of the vascular network present in this layer. Dermal vasculature, which consists of capillaries, arterioles and venules, originates from the deep blood net dermis adjacent to the subcutaneous fat and extends up to the papillary dermis. It is well-established that the pulsatility is observed only in arterioles [Previous research works in this field primarily focussed on the optical interactions with the cutaneous tissue structure only which is responsible for the pulsatile signal (ac component). Recent advancements in PPG research have explored the dc part of the PPG signal for potential applications ,7,56. Toterioles , which wterioles .n and b are the system-dependent constant terms. On the other hand, for any other tissue region that contains a large artery, for example, forehead or forehand, the stress-strain relationship between the pressure and the radius (r) of the artery can be written as:A basic model of finger PPG has been presented in this paper. To solve the problems related to PPG, for example, the motion artifacts, pulse transit time (PTT), the effect of large artery etc., future modifications would be implemented in the model. To produce the pulsatile signal, the mechanical properties of the blood vessels need to be incorporated in the model. Future plan includes the variation in the blood volume in accordance with the blood pressure. In a tissue volume such as finger, i.e., in absence of a large artery, the simulated volume nditions ,59,60: and different tissue physiology  in the presence of multiple absorbers . The results identified the optimal source-detector separation for a reflectance mode finger-PPG sensor. Depth-specific analysis of the tissue-volume revealed the maximum and minimum impacts of the dermis and bone, respectively, in the formation of a PPG signal. Results explained the origin of the pulsatile and non-pulsatile PPG components which is an invaluable information for designing a wearable finger PPG probe for continuous physiological measurements. All results presented in the paper contribute to the basic knowledge required for the development and advancement of the novel PPG-based transcutaneous sensors.


Pesticides aerial application may results in the drift to neighboring areas if some application technology is not well executed. This phenomenon should be minimized to reduce environmental risks and agricultural production costs. This work aimed to investigate the interaction of environmental conditions, surrounding distance, and application conditions influencing spray drift in aerial applications. Sampling data from aerial sprays were collected during three agricultural years (from 2012 to 2014) in fields cultivated with sorghum, millet, soybean, corn, and cotton. The following variables were evaluated: application swath width, application rate, distance from the applied field, wind speed, relative humidity, and temperature. The estimated Pearson’s correlations and path analysis identified that application rate and distance from the applied field and application were the variables that most influenced drift. Equations relating spray drift in function of distance from the applied field and application rate were adjusted in function of the variable, and a response surface model was constructed to estimate drift. The major pesticide class sprayed with aircraft in the Brazilian Cerrado was insecticide, followed by fungicide. This scenario shows the potential hazard risk of spray drift over the environment. The concentration of the drift deposits decreased as surrounding distance and application rate were increased. A mathematical equation of drift prediction was established, where the variables that contributed most to drift deposits were surrounding distance and wind speed. Thus, it is very important to monitor and respect the wind speed limits during the aerial spraying, mainly when there is any risk potential associated with pesticide exposure over the downwind direction. Pesticide application corresponds to a most considerable portion of the production cost with annual crops. The concern about the use of these products in agriculture has significantly increased in recent years, especially regarding human health and environmental contamination . This isApplication technology aims to efficiently and economically apply the correct amount of the active ingredient to the desired target, without affecting the environment . Several-1, proper target coverage is more difficult . Th. Th33]. The correlation network generated from Pearson’s correlation matrix revealed a strong negative relation between temperature and relative humidity when studied apart . HoweverDespite its relevance, the Pearson’s correlation coefficient (r) may lead to misinterpretations about the relationship between two variables, and may not be a real measure of cause-and-effect. A high or low estimate of r between two variables may be the result of the effect that a third variable or a group of variables have on that pair, leading to the non-exact relative importance of the direct and indirect effects of these factors. Therefore, path analysis was carried out, which investigates the cause-and-effect relationship. This analysis provides detailed knowledge of the influences of each variable on the main dependent variable, discarding the effects of the others .The matrix of correlation estimates presented severe multicollinearity, being necessary to add a constant k = 0.10 to the diagonal of this matrix to correct this effect . AfterwaThe results of path analysis considering spray drift as the main dependent variable and the others as explanatory variables are expressed in According to the results, air temperature did not affect aerial spray drift. Results from other trials showed that droplet size and wind speed are the factors that most influenced drift distance . There i2 = 0.72).Surrounding distance and application rate were the most adequate variables to indirectly predict spray drift in this study conditions, as shown by path analysis. Thus, the adjustment of regression models considered these variables as explanatory . Both vaThe drift process in a spray application depends on several factors acting together. Thus, a response surface model was constructed considering simultaneously the distance from the applied field and the application rate to predict aerial spray drift, as shown in Based on the results of the path analysis, wind speed is the primary climatic variable with a positive cause-and-effect relation with drift, although of low magnitude. The low estimate of the direct effect of wind speed on drift can be explained by the small variation of this variable . Fig 8 sThe major pesticide class sprayed with aircraft in the Brazilian Cerrado was insecticide, followed by fungicide. This scenario shows the potential hazard risk of spray drift over the environment.The concentration of the drift deposits decreased as surrounding distance and application rate were increased.A mathematical equation of drift prediction was established, where the variables that contributed most to drift deposits were surrounding distance and wind speed. Thus, it is very important to monitor and respect the wind speed limits during the aerial spraying, mainly when there is any risk potential associated with pesticide exposure over the downwind direction.S1 Table(TXT)Click here for additional data file.


Methods of testing and describing the recrystallization process in ice cream systems were characterized. The scope of this study included a description of the recrystallization process and a description and comparison of the following methods: microscopy and image analysis, focused beam reflectance measurement (FBRM), oscillation thermo-rheometry (OTR), nuclear magnetic resonance (NMR), splat-cooling assay, and X-ray microtomography (micro-CT). All the methods presented were suitable for characterization of the recrystallization process, although they provide different types of information, and they should be individually matched to the characteristics of the tested product. The most important factor that determines frozen food quality is the course of crystallization . CrystalIce cream is a multiphase physicochemical system originating from the dispersion of individual components in different phases ,12. The The recrystallization process occurs at a constant temperature during long storage, especially above the glass transition temperature ,17,18. HHydrocolloid stabilizers are used in food production to modify water-binding capacity, freezing rates, ice crystal formation, and rheological properties ,11,12,24Different forms of carrageenan are commonly used as stabilizers. The kappa carrageenan form is mostly used to stabilize dairy products, but it may also be applied to control crystal growth in sorbet production ,25. The The aim of this review was to outline the basic characteristics of the measurement method, sample preparation, and equipment required to show and describe ice crystals during and after the recrystallization process using the following methods: FBRM, OTR, NMR, splat cooling, microscopy analysis, and X-ray microtomography.Control of the ice crystallization process is mostly conducted in an empirical way, mainly due to a lack of experimental data. Research on ice crystal size distribution (CSD) is not simple, especially because of the possibility of melting under unstable conditions and also when it is about a type of ice cream product that contains three phases . The focThe real-time particle size analysis technique of FBRM works by focusing a laser beam directly down the probe tip through a sapphire window. The optical part is rotated about an axis to the probe (2 m/s), so that the beam traces out the circular path (reflected light is detected in the probe). The probe tip is inserted, at an angle, directly into the process streams, to ensure that particles can flow easily across the probe window where the measurement takes place . The lasGenerally, the FBRM instrument can acquire thousands of chord lengths per second. On a counter board, these lengths are classified into a series of size ranges that are expressed as a distribution, referred to as a chord length distribution (CLD). This technique considers the shape and dimensions of the particles. However, the CLD value does not gives information about morphology of the particles and it is less useful for the characterization of a crystal lattice in frozen products ,7,27.Amamou et al. (2010)  presenteArellano et al. (2012)  demonstrThe recrystallization process in ice cream using the FBRM technique was investigated by Ndoye and Alvarez (2015) . They coThe main advantage of the FBRM technique seems to be its suitability for on-line measurements of high solid-concentration suspensions and for following rapid crystallization kinetics. However, it only provides information about the total number of ice crystals and the changes of diameters of ice crystals, and therefore the shape and the changes of ice crystal location are measured.It was shown that the dynamics of rheological measurements can be used for characterization of textural properties and structure of foamed dairy emulsion. The dynamic storage modulus G’ (elastic response) and loss modulus G” (viscous behavior) can provide information about properties of viscoelastic materials.Stanley et al. (1996)  proved tWildmoser et al. (2004)  used rheEisner et al. (2005)  examinedSensory quality closely correlates with viscoelastic properties of products. Tsevdou et al. (2015)  correlatContent, purity, and molecular structure of a sample can be determined using an analytical chemistry technique called nuclear magnetic resonance (NMR). When the NMR technique is used it is possible to quantitatively analyze mixtures containing known compounds. For unknown compounds, NMR can either be used to match against spectral libraries or to infer the basic structure directly. The NMR technique can be used to determine molecular conformation in solution and to study physical properties at the molecular level such as conformational exchange, phase changes, solubility, and diffusion. In order to achieve the desired results a variety of NMR techniques are available. In addition, NMR is versatile and has the potential to be nondestructive, which makes it a potential tool in quality control of various products, including milk-based desserts .The principle of the method relates to the fact that many nuclei possess spin and all nuclei are electrically charged. If an external magnetic field is applied, energy transfer is possible from the base energy to a higher energy level . The energy transfer takes place at a wavelength that corresponds to the radiofrequencies, and when the spin returns to its base level, energy is emitted at the same frequency. The signal that matches this transfer is measured in many ways and processed in order to obtain an NMR spectrum for the charged nucleus ,37.The NMR technique has been shown to be an appropriate method to calculate the amount of unfrozen water in a food sample. Lucas et al. (2004)  examinedHagiwara et al. (2006)  investigIt has been discovered that NMR could be used to determine fat globule size in ice cream and to determine the effect of the formulation on hard ice creams’ structure ,41,42. LThe NMR technique does not involve any thermal processes to assess the amount of ice and thus can be performed at a stable temperature. It is also important that the relaxation parameters of water were used to provide information on the water/non-water molecule interactions. This technique may be a valuable tool for understanding how various stabilizers affect the three-dimensional vein network and recrystallization processes.In order to determine the ice recrystallization inhibition (IRI) activity of ice cream stabilizers, the capillary method or splat-cooling assay can be performed . AlthougThe IRI activity of different substances such as AFP and some polysaccharides can be demonstrated using “splat” assay. This is not a new method and has been used for nearly twenty years. In these experiments, a small volume of sample was usually expelled from a height of 1.5 to 3 m onto a metal plate  that had been cooled in liquid nitrogen or dry ice. The sample drop froze upon hitting the metal plate, forming a thin (splat) wafer of ice. This wafer was later transferred to a microscope stage at a high sub-zero temperature where the sample recrystallized over time and annealing temperature . Different modifications of this method are possible, for example, the sample could be snap frozen between two cover slips. This technique was complemented by crossed polarizers with a dissecting microscope and ice-binding activities of some substances that were examined by Raymond and Knight (2003) . The splThe most popular and universal method of testing and describing the recrystallization process is direct light microscopy observation followed by image analysis of the ice crystals. When the work was started with another technique such as oscillatory thermos-rheometry or splat cooling assay, to complete it the work was usually supported by an appropriate microscopy technique, a cryo-scanning electron microscope or a microscope with a polarizer ,44. PhysDonhowe and Hartel  ,17 exami50) and the slope of cumulative distribution at X50. The recrystallization rate was calculated as the slope of linear regression of the curve plotted with values of X50 for each cycle. Using this method, they proved there was significant retardation of recrystallization with the addition of sodium alginate and xanthan. The same technique  was first used to explain the IRI activity of kappa carrageenan hydrolysates in model sucrose solutions [Regand and Goff (2003)  presenteolutions  Table 150 and thThe method described above makes it possible to analyze not only the changes in diameter of ice crystals but also their shape and location. On the basis of the images it is possible to analyze the mechanism of recrystallization. From the shape of the ice crystal, it is sometimes easy to read that accretion between adjacent crystals occurred . NumerouWhen the growth rate of ice crystals in samples with the IRI active substances is small, there is no difference in morphology or size of ice crystals, as was observed by Kamińska-Dwórznicka et al. (2016)  for kappThe technique of microscopy and image analysis has brought positive results in ice crystal microstructure investigations. However, it is a two-dimensional technique and it does not characterize multidimensional structure of ice cream mix before and after freezing. During the recrystallization process, the microstructure of ice cream  changed The micro-CT method is based on the same assumptions as classic tomography. However, by using a smaller radiation spot, it is possible to obtain a higher resolution of the reconstructed image ,58. ThisGuo et al. (2017)  presenteThe technique of microscopy and image analysis allows one to describe ice crystal microstructure. From the images of the ice crystals we can easily obtain information about the size and the changes of their shapes and location during storage at different temperature and time conditions. The images can be easily analyzed using specific computer software. The main disadvantages of this method are difficulties in the preparation of samples and its influence on the repeatability of results. The technique of X-ray microtomography seems to offer a new possibility in the analysis of the recrystallization process as a non-destructive method that shows ice cream samples in 3D, but has some difficulties with the final interpretation of images. The FBRM (focused beam reflectance) technique is fully automated and provides results more easily and faster than simple image microscopy and image analysis. It is a suitable method for in situ measurements, and it allows sample preparation to be avoided because the measurement is conducted by the probe immersed in the ice cream mixture. However, it can only provide information about changes in the diameters of crystals, without shape and location analysis. The OTR (oscillatory thermo-rheometry) technique is a method in which viscoelastic properties of ice cream closely correlate with the sensory quality, and hence it provides information about the shelf-life of ice cream without recrystallization changes. However, the changes in size or shapes and location of ice crystals are not measured. Nuclear magnetic resonance (NMR) is an effective method to evaluate the amount of unfrozen water in a food sample. Hence, it is a valuable tool for understanding the impact of a stabilizer on the recrystallization processes without providing any information about sizes of ice crystals and locations. Splat-cooling assay is the oldest method to describe the amount, sizes, and morphology of ice crystals. It is a method with a very specific technology for sample preparation and is not suitable for different types of frozen food.All of the discussed methods are suitable for describing the recrystallization processes, although they provide different types of information, and they should be matched individually to the characteristics of the tested product.


By using Langevin dynamics simulations, we study how semiflexible rings that are topologically linked respond to mechanical stretching. We use both constant-force and constant-velocity pulling protocols and map out how the mechanical tension affects observables related to metric quantities such as the longitudinal extension or span, and topology-related ones such as the length of the linked portion. We find that the average extension of linked rings, once divided by that of a single equivalent ring, is nonmonotonic in the applied force. We show that this remarkable feature becomes more prominent as the link complexity is increased, and originates from the different stretching compliance of the linked portion and the rest of the rings’ contour. By comparing the results of different pulling protocols, we also establish the best one for telling apart different types of links from their tensile response. Inter-chain entanglement can have major effects on the physical properties of polymer solutions or melts. An apt illustration is offered by biological systems, and particularly interphase chromosomes whose level of intermingling—Which could interfere with their capability to reorganize in later stages of the cell cycle—Is kept in check by active (enzymatic) and passive  mechanisms ,2,3.In systems where the free ends of the polymers can be chemically bridged, the aforementioned inter-chain entanglement can be trapped in the form of proper topological links . Again, For these systems, as well as in the context of molecular nanomanipulation, a relevant question is whether and how the response to mechanical stretching depends on their linked state.Besides the pratical implications, this problem is also conceptually appealing. In fact, it hinges on the interplay of topological constraints that act internally to the considered molecules and mechanical constraints that are imposed externally.This interplay has long been studied both theoretically and experimentally for single polymer chains ,12,13,14To our knowledge, the problem has remained practically unexplored for the case of linked rings or rings, except for a general overview presented in Ref. .Here, we tackle it with a systematic study based on theory and numerical simulations of the stretching response of two identical semi-flexible rings that are topologically interlocked. In particular, we shall address the following questions: How does the stretching response of two linked rings compare to that of a single ring with the same total contour length? How do the force–extension profiles vary with link type? How does the size of the linked portion depend on the pulling load and link type?To answer these questions, we carry out hundreds of stretching simulations using Langevin dynamics and compute conventional metric observables such as the projected distance of the anchoring points, as well as topology-oriented ones such as the length of the linked region.Inspired by the typical setups of micromanipulation techniques based on atomic force microscopy (AFM)  or opticThe model, stretching protocols, and observables are introduced in As model ring polymers, we consider semiflexible circular chains of otential  and a tri and j, In these expressions, As it is customary in coarse-grained canonical simulations, we took the thermal energy of the system, ith bond vector, with i-th bead. The bending rigidity parameter, K, is set to correspond to a nominal persistence length of The semiflexible character of the chain is instead accounted for by the following term:z Cartesian direction is controlled via a pulling potential The initial linked state of the rings is preserved at all times of the dynamical evolution by excluded volume effects. These steric interactions are enforced by a potential energy term, i-th bead, the stochastic equation of motion ism is the bead’s mass, The dynamical evolution of the system is described within a Langevin scheme. For the The equations of motion  are inteThe link types considered here are sketched in The repertoire of links considered here consists of two-component links only, and using Rolfsen’s notation, includes: the simplest (Hopf) link, For each initial configuration, the integration is carried out with the LAMMPS simulation package with standard values for the friction coefficient and beads mass  and by uion time .z coordinate. Next, on the partner ring, we pick as anchor the bead with the largest z coordinate. In the following, these two anchors are labelled as “A” and “B”, respectively . The actual rightmost and leftmost beads at any given time of the simulation are those actually used to define the longitudinal extension of the ring, Z, defined as the absolute difference of their z coordinates.Note that because of the competition between configurational entropy and tensile energy, during the dynamical evolution of the system, the anchoring points A and B do not necessarily remain those with the smallest and largest z components of the positions of anchoring points A  and B, respectively.In the constant-force protocol, the pulling potential takes on the formWe considered 11 values of the force in the range z direction by the time-dependent potentialIn the constant velocity protocol, the rooting point A is held fixed in space while the anchoring point B is progressively displaced along the z coordinate of point B at time t and at the beginning of the pulling action, In this expression,  t=0 see b.For this protocol, we set the spring constant equal to linked portion—Presents conceptual challenges akin to those encountered for defining the knotted portion of entangled rings or open chains [A key element of interest is how the applied tension affects the length of the region where inter-chain linking resides. Identifying this region—Which we term the n chains ,28,29,30More precisely, to pinpoint the linked portion, one needs to overcome two different hurdles. The first is being able to define the linked state for two open curves . The second is to use the very same definition within a physically- and computationally-viable algorithm that locates the smallest portion of the two curves where the entanglement resides.The former issue has been tackled by recent studies  that successfully dealt with open curves by using the Gauss linking number , by closThe second challenge (i.e. locating the linked portion) was addressed in our recent study of Ref. . The metTo better illustrate the viability of the top-down approach, we shall contrast it with a simpler search strategy motivated by the following considerations of the strong stretching limit.When the catenane is fully elongated, the linked portion is maximally tight (short) and located in the midpoint of the stretched catenane.z directions are suppressed everywhere except in the region accommodating the essential crossings. One can hence bracket the linked portion by considering two perpendicular planes that slide inward from the anchoring points up to the point where further sliding would cut in more than two points. The region between the two planes cuts the catenane in two arcs that constitute—Within this second scheme—The linked portion  have a sigmoidal trend, with an inflection point  and (f) for datapoints picked in two different force ranges: We have used Langevin dynamics simulations to study the mechanical stretching of a pair of topologically interlocked semiflexible rings. To our knowledge, a systematic analysis of such systems has not been carried out before, and therefore we addressed the problem by considering different types of links and pulling protocol (constant force and constant velocity) and by measuring various metric and topological observables. Besides the longitudinal span of the catenanes, these include the contour length of the linked portion; that is, the region of the linked rings where the topological entanglement is concentrated.We find that the force–Extension curves of all link types, once normalised with respect to one of a single ring of equivalent contour length, are non-monotonic. Specifically, they feature a minimum when the reduced force, This counter-intuitive non-monotonicity is reminiscent of the one found in knotted rings , and canThe observed phenomenology indicates that it is the competition between the entropy of the contour length stored in the linked region and the tensile energy of the elongated linked pair that causes the intriguing non-monotonic behaviour of the force–extension curves and its dependence on the link topology.The comparison of data from static constant-force simulations and the constant-velocity ones is finally used to characterise the out-of-equilibrium response of the linked pairs. Specifically, we observe that at intermediate forces, These results can have valuable implications in the context of possible future force-spectroscopy experiments aimed at distinguishing different link types from the tensile response. We envisage that, for optimal resolution, such approaches should adopt a constant-force pulling protocol, or a constant velocity one where sufficiently high tensile forces are reached.The present study could be extended in several directions. We believe it would be particularly interesting to analyse the tensile response of non-binary links, particularly multi-component catenanes and Brunnian links , which o


The industrial exploitation of high value nanoparticles is in need of robust measurement methods to increase the control over product manufacturing and to implement quality assurance. InNanoPart, a European metrology project responded to these needs by developing methods for the measurement of particle size, concentration, agglomeration, surface chemistry and shell thickness. This paper illustrates the advancements this project produced for the traceable measurement of nanoparticle number concentration in liquids through small angle X-ray scattering (SAXS) and single particle inductively coupled plasma mass spectrometry (spICPMS). It also details the validation of a range of laboratory methods, including particle tracking analysis (PTA), dynamic light scattering (DLS), differential centrifugal sedimentation (DCS), ultraviolet visible spectroscopy (UV-vis) and electrospray-differential mobility analysis with a condensation particle counter (ES-DMA-CPC). We used a set of spherical gold nanoparticles with nominal diameters between 10 nm and 100 nm and discuss the results from the various techniques along with the associated uncertainty budgets. Nanoparticles are increasingly used in innovative products manufactured by advanced industries and provide enhanced, unique properties of great commercial and societal value. The demand for high performance materials places increasingly stringent tolerances on the properties of nanoparticles. Advances in industrial production and research have made it possible to manufacture new kinds of nanoparticles, which open new possibilities for technological progress in, for example, drug delivery, medical imaging and electronics ,2. This Although some studies on concentration measurements for single methods exist ,14,15, nGold nanoparticles with nominal diameters of All samples were prepared as aqueous dispersions of spherical particles. The gold particles with a nominal polydispersity X obtained from one measurement, Z, Y added to sample X to prepare blend X added to internal standard Y to prepare blend Z added to internal standard Y to make calibration blend Y added to the primary standard solution Z to make calibration blend B, Here Small-angle X-ray scattering (SAXS) and single particle inductively coupled plasma mass spectrometry (spICPMS) are defined as reference methods. Here the dynamic mass flow method of spICPMS was used. Both methods fulfill the requirement as methods of exceptional scientific status which are sufficiently accurate to stand alone in the determination of number concentration for the certification of a reference material. The two methods also have a firm theoretical foundation so that systematic error is negligible relative to the intended use . The labSAXS measurements were performed at the four-crystal monochromator beamline (FCM) of PTB  with thew was determined in a traceable way. The capillaries were scanned with a step size of w. The incoming flux T, which is a total transmission of the sample and the capillary, t. The scattering from the particles is proportional to the square of the contrast The lower section was filled with Fluorinert FC-3283, acquired from Iolitec , which has a density of ±0.2)μm2 . The meaFor the calculation of the contrast, a gold density of was used ,21,22. ASpICPMS measured number concentrations of inorganic particles using an Agilent 8900 ICPMS/MS instrument . The instrument used the MassHunter4.3 software (Agilent) and is capable of microsecond detection, which makes the analysis in single particle mode possible. For analyses in fast transient analysis (TRA), the newly developed single particle application module of the ICPMS MassHunter software (G5714A) was used. For these analyses, a dwell time of C) in the sample was derived from Equation  equipped with a violet diode laser as a light source . Then, using an Analysis of variance (ANOVA) table, the day-to-day reproducibility and the repeatability within one day were calculated. The uncertainty associated with the detection threshold parameter set by the user was estimated experimentally by varying this parameter over a reasonable range and looking at the effect this had on the measured particle number. Regression analysis gave the gradient directly. The overall budget also included the uncertainty associated with weighing for the preparation of sample dilutions. The results provided by PTA represent the particle number concentration in the original samples, derived by multiplying the values obtained with the NTA3.2 analysis software and the dilution factor.The instrument used in the measurements is a Zetasizer Nano from Malvern Instruments using a He–Ne laser  or given as input to the program. The instrument scattering intensity was calibrated with measurements of pure toluene at several timepoints during the period of the measurements and was not observed to change significantly  compared to the intensity size distribution (I)— which is closest to what the DLS measures.The estimation of concentration (volume or number) is very dependent on the refractive index ere used . Even ifSince the software does not provide number concentrations and as the broad distributions result in large uncertainties when estimating number concentrations from the intensity, we conclude that the DLS is not the method of first choice for estimating number concentrations. The resulting size, polydispersity index and volume concentrations are given in tables using the “normal resolution” algorithm as well as the “high resolution” algorithm see  25,26].,26.25,26N of the particles was derived from the absorption of the particles according to the formula:L is the optical path and n and extinction coefficients of Spectra for UV-vis were acquired in quartz cuvettes using a LAMBDA 850 spectrophotometer . Samples were analyzed over the wavelength range The DCS technique measures the time of sedimentation of particles through a liquid medium, exposed to a centrifugal field, where the sedimentation time is a function of the particle size and density. The Stokes’ diameter of the particles is determined based on a spherical approximation, assuming knowledge of the mean density of particles. Because the average density of the Stokes’ volume of the particles was unknown, the DCS results were combined with the particle hydrodynamic diameters measured by DLS and analyzed with the cumulant method (see appendix). Details for the DLS methods used for DCS measurements are given in the w/w) sucrose gradient in water  equipped with an LED laser emitting light with a wavelength between The particle number concentration was measured by integrating the measured weight-based size distribution and dividing the measured mass by the average mass of a single particle. The results acquired with the DCS method are shown in the The ES-DMA-CPC measurements were carried out using an experimental setup consisting of a pressure-based flow controller , a liquid flow meter , a three-way valve, a mass flow meter , an ES source , a tube furnace , a DMA  and a CPC .A lab-built sample injection system based on a pressure flow controller was used to deliver the suspension to the electrospray generator. The nanoparticle (NP) suspension was injected into the ES through a  reduced , while lBy measuring the suspension flow rate through the capillary The number concentrations of NPs in the suspensions The articles . HoweverSeveral of the investigated methods provided the mean diameter of the particles which has a strong influence on the subsequent number concentration determination. The measured diameters are given under assumption of the spherical shape of the particles and are shown in In PTA and DLS measure the hydrodynamic radius of the particles, which includes the particle itself, the ligand layer and the associated hydration layer. Therefore they can not be directly compared to other three methods. Nevertheless, they provide useful information about performance of the methods. The PTA method agreed within uncertainties with DLS for the Au30 sample. For the Au100 sample the PTA deviated by a factor of ≈1.27 from the DLS. It has to be mentioned that despite different measurands, DLS agreed within uncertainties with all other methods for the Au10 and Au30 samples, which is a consequence of large uncertainties of the method. However PTA, which provides much lower uncertainties than DLS, agreed within uncertainties with all other methods for the Au100 sample. For DLS, the use of intensity size distribution delivers values for diameters up to 28% larger compared to the results obtained by use of the number size distribution. A comparison of the number-based size distribution measured by SAXS showed that the DLS size distribution measurement broadening is due to instrumental broadening. Because of this, the number-based size distributions measured by SAXS were used to estimate the number concentration by DLS in the next step.The number concentrations with corresponding standard uncertainty The comparison highlighted the strengths and limitations of the two reference methods. SAXS is highly accurate, but its accuracy is highly reliant on the exact knowledge of the difference of the electron density of the nanoparticles and the solvent. The second limitation was the time needed for a measurement—suspensions of gold particles with a diameter q-range in the setup. For the Au30 sample, SAXS provided accurate values with a relative standard uncertainty of The spICPMS method could not access the number concentration of the Au10 sample, as part of the particle population was difficult to resolve from the background signal. The Au10 sample could only be measured with SAXS as a reference method, but with the relative standard uncertainty of The number-based concentration of gold nanoparticles was measured with spICPMS following the dynamic mass flow method, which achieved the standard measurement uncertainty  method, precise knowledge of the particle size and size distribution as well as the density of the material is essential for use in laboratory conditions. The major contribution to DCS uncertainty is composed of the uncertainties on the size and density of the calibrants [The DCS method could provide number concentration for all samples, even though the size had to be measured with DLS. If the diameter of the particles measured with the reference method SAXS was used, the obtained results were in better agreement with the values of the reference methods. As for DLS, DCS relies on the exact knowledge of the refractive index librants . Method The UV-vis method could provide reliable number concentration values as only high-density particles were used in the experiments presented, however this methods needs additional size measurements e.g., by TEM, in order to calculate the extinction cross-section. The estimation of the uncertainty for the UV-vis method relies on a comparison to gravimetric methods and theory ,32,33. TThe electrospray transmission efficiency in ES-DMA-CPC experiments depends on the suspension flow rate through the capillary and the suspension matrix. Compared to the primary methods, the deviation of the ES-DMA-CPC values was up to a factor of eight. Additionally, ES-DMA-CPC could only provide number concentrations for particles with a diameter of d<100nm . HoweverThe particles smaller than The number concentrations measured by PTA agree within the stated uncertainty with reference methods for both measured samples. Under the current conditions, UV-vis delivered results comparable with reference methods for all the measured samples, but with higher uncertainties. Notably, for this type of sample this is the most commonly used and cost-effective method, provided an additional method like TEM delivers the diameters of the particles. DCS provided much higher uncertainties in number concentration measurements and somewhat variable results. It showed good agreement with reference values for smaller particles. For Au100, DCS delivered values up to a factor of posed in  must be In general, it could be shown that reference methods agree on number concentration values within their uncertainties, which are better than


Health interventions often depend on a complex system of human and capital infrastructure that is shared with other interventions, in the form of service delivery platforms, such as healthcare facilities, hospitals, or community services. Most forms of health system strengthening seek to improve the efficiency or effectiveness of such delivery platforms. This paper presents a typology of ways in which health system strengthening can improve the economic efficiency of health services. Three types of health system strengthening are identified and modelled: (1) investment in the efficiency of an existing shared platform that generates positive benefits across a range of existing interventions; (2) relaxing a capacity constraint of an existing shared platform that inhibits the optimization of existing interventions; (3) providing an entirely new shared platform that supports a number of existing or new interventions. Theoretical models are illustrated with examples, and illustrate the importance of considering the portfolio of interventions using a platform, and not just piecemeal individual analysis of those interventions. They show how it is possible to extend principles of conventional cost-effectiveness analysis to identify an optimal balance between investing in health system strengthening and expenditure on specific interventions. The models developed in this paper provide a conceptual framework for evaluating the cost-effectiveness of investments in strengthening healthcare systems and, more broadly, shed light on the role that platforms play in promoting the cost-effectiveness of different interventions. •Health interventions which rely on joint delivery platforms become interdependent.•Such interventions must be evaluated as a portfolio to avoid allocative inefficiency.•There are three types of health system strengthening. There iIt was estimated in 2014 that around $274 billion spending on health is needed per year to reach the ambitious Sustainable Development Goals 3 targets on Women's and Children's health by 2030, of which around 75% of costs are for health system strengthening, with health workforce and infrastructure  as the main cost drivers . StrengtThe purpose of this paper is to develop an analytical methodology that can model the impact of HSS on the cost-effectiveness of interventions and can inform the optimal balance between spending on HSS and spending directly on interventions. The intention throughout is to retain the principle of cost-effectiveness analysis (CEA), of seeking to maximize the health benefits created by interventions subject to a fixed budget constraint . However2The importance of interdependencies between health interventions is reflected in the health systems literature, which recognizes that the optimal design of a health system crucially depends on the balance between different components of the health system . In this3Box 1Population-based health interventions: This platform captures all non-personal or population-based health services, such as mass media and social marketing of educational messages, as typically delivered by public health agencies.Community services: The community platform encompasses efforts to bring health care services to patients, meeting people where they live. It includes a wide variety of delivery mechanisms. Specific sub-platforms include the following: health outreach and campaigns ; schools ; community health workers, who may be based primarily in the community but also connected to first-level care providers, with ties to the rest of the system.Health centres: The health centre level captures two types of facility. The first is a higher-capacity health facility staffed by a physician or clinical officer and often a midwife to provide basic medical care, minor surgery, family planning and pregnancy services, and safe childbirth for uncomplicated deliveries. The second is a lower-capacity facility  staffed primarily by a nurse or mid-level health care provider, providing services in less-resourced and often more remote settings.First-level hospitals: A first-level hospital is a facility with the capacity to perform surgery and provide inpatient care. This platform also includes outpatient specialist care and routine pathology services (such as newborn screening) that cannot be feasibly delivered at lower levels.Referral and specialized (second- and third-level) hospitals: This platform includes not only centralized, general referral hospitals with the capacity to provide secondary and tertiary care, but also specialty facilities . The latter may be distinct facilities or specialized units within referral hospitals.Alt-text: Box 1Source: Watkins, D., D. Jamison, and A. Mills. "Universal health coverage and essential packages of care", in: Disease Control Priorities. New York, NY: The World Bank (2017).Health service delivery platforms are defined by the Disease Controls Priorities project as logistically related service delivery channels that collectively make up the organisational components of the healthcare system, and mark the point of contact between service users and healthcare providers, amongst which are included hospitals, health centres and community services, see 4Box 2Economies of Scope (EoS) are present when So for example, if the costs of providing intervention 1 are If the benefits of providing intervention 1 are This measure indicates the savings or gains of joint compared to separate production, as a percentage of joint production. It shows that providing both interventions together results in cost savings or outcome increases of 30%.The concept of economies of scope was originally developed to explain the benefits of diversification of multi-product firms. A hospital is the prime example of a multiproduct firm with its substantial economies of scope across the diverse healthcare interventions it delivers. Hospitals provide an infrastructure of general services and inputs  which are used and shared across a wide range of interventions. The costs of the platform can consequently be shared across all these interventions, resulting in efficiency gains that can be used to reduce aggregate costs or increase aggregate outcomes.Alt-text: Box 2The idea of a platform is closely related to the notion of economies of scope, which suggest that the production of two interventions by the same entity is more cost effective than separate production, see Surprisingly, most studies find either no or very minor economics of scope on hospital or health centre level, with some even suggesting that there are dis-economies of scope, implying that joint implementation is less cost-effective than separate implementation. The difficulty of finding empirical evidence of economics of scope is probably explained by the problem that they materialize only if the shared platform has the capacity to deliver the two interventions; if that is not the case, interdependencies may result in incidental reductions in outcomes or increases in costs of other interventions delivered by the same platform although it is difficult to find empirical evidence of these impacts . They ca5B.Standard CEA assumes that interventions are independent, in the sense that the costs (and benefits) of each intervention remain unchanged whatever other interventions are implemented. This is because a common focus of CEA in high-income countries is on situations where a single new technology, most commonly a novel drug, is being introduced into a health system, but the delivery platforms remain essentially unchanged . The stai for adoption yields incremental benefits vi and incurs incremental costs ci. Benefits vi can be measured in natural units (e.g. infections averted) or with a generic measure of health such as quality- or disability-adjusted life-years. The decision variables xi indicate the proportion of intervention i implemented (coverage). The model is solved to find the values for xi that maximize aggregate health from all interventions vi, subject to the budget constraint B. This yields the usual CEA marginality condition that interventions are accepted if i. In practice, the costs ci may include expenditure on health system strengthening (HSS) specific to intervention i, or some estimate of the contribution to the costs of an existing or newly established delivery platform shared with other interventions.Each candidate intervention Yet if several interventions rely on a delivery platform, they are implicitly interdependent, and it is often impossible satisfactorily to attribute to specific interventions the HSS costs that improve the quality or capacity of that platform. From a systems perspective, therefore, the costs and benefits of a specific intervention are often conditional on the other interventions chosen to share the common delivery platform. The existence of platforms introduces additional considerations into the standard CEA framework. A decision to implement a specific intervention may have positive or negative effects on the costs and benefits of other interventions depending on whether it helps share the costs of an underutilized existing platform, puts pressure on the capacity of an existing platform, or introduces a new platform. We will illustrate these three scenarios with worked examples, supported by mathematical formulations. Note that the interventions we have chosen for the scenarios, and their costs, are not real-world case studies and are meant for illustrative purposes only.66.1Investments in an existing shared platform can improve the technical efficiency of the platform, i.e. the effectiveness with which a given set of inputs is used to produce an output. Such investments could take the form, for example, of new information systems, workforce training, or improved laboratory facilities. The efficiency improvement will in turn affect the cost-effectiveness of existing interventions that use the platform, by either improving service quality and thus health outcomes, or reducing unit costs, or both. Improvements in the quality of the platform may improve the cost-effectiveness of many interventions, a classic example of economies of scope. They may also change the relative ranking of interventions with respect to their cost-effectiveness.y into the standard CEA formulation that indicates the expenditure on HSS. The effect of spending $y on HSS is to scale the effectiveness of all interventions using a platform by a dilution factor of y may capture the additional costs of a ‘pay for performance’ (P4P) initiative for nursing staff, imposing a cost in the form of nursing reimbursement, but also yielding more effectively delivered interventions across the board. Through the parameter y, HSS type1 is implicitly addressing the ease with which technical efficiency of a platform can be improved. The simplest case is to assume that y uniformly improves the effectiveness of all interventions . As a result, considering the joint provision of the interventions, all three interventions would be cost-effective using the country's current threshold of $200 per QALY.However, the additional $2 m could also be spent on HSS see . For exaWhile both HPV and HSS investment are cost effective using the country threshold ($200 per QALY), the additional $2 m would be better spent on HSS than the HPV vaccinations.6.2Changes in the way an existing shared platform is used can improve the mix of services it supports, and therefore the allocative efficiency of the platform, without necessarily increasing expenditure. Allocative efficiency implies that the health system delivers the optimal range and coverage of healthcare interventions. The second type of HSS therefore arises from the joint reliance of interventions on a shared platform in which some resource constraint limits the capacity to produce the optimal quantity and range of outputs. The objective of HSS in these circumstances is to relax such capacity constraints.There is a well-established literature on physical health system constraints that may affect the optimal range of interventions provided . AmongstL and BO for labour and for other spending.i. There are now two shadow prices L favours interventions that make relatively smaller demands on labour. The effect of relaxing constraint BL via HSS is to allow more interventions to be delivered and shift towards more cost-effective interventions that could not previously be implemented because of the opportunity cost they imposed by pre-empting use of the scarce labour resource. The key investment trade-off is between relaxing the labour constraint and relaxing the other constraint. Only when the two shadow prices HSS might therefore seek to relax the resource constraint by, for example, training more staff to fulfil skilled tasks. Following i the associated decision variables must satisfy An alternative formulation would be to consider a different method of delivering treatments which requires less intensive use of the scarce resource. This requires a simple adaptation of the above mathematical programme to include a revised set of delivery methods with parameters Consider the example in The decision maker considers joint implementation of TB second-line and CVD treatment. At first glance, sharing the fixed costs of the platform might have the advantage that it secures a reduction of the fixed costs per case for both interventions to $714 each and a favourable cost-per-QALY ratio for CVD treatment, compared to sole implementation see . HoweverHowever, CVD treatment makes cost-effective use of the platform, because it can share the fixed costs of the platform with TB treatment. The 5000 patients requiring CVD treatment can now be covered at a favourable cost per QALY of $136. If considered in isolation, the less intensive TB treatment is not cost-effective with the addition of CVD treatment, but the cost-effectiveness of the two treatments must be considered jointly. The addition of CVD treatment leads to a net reduction in aggregate costs-per-QALY from $180 to $175, and so the new scenario where TB and CVD treatment are delivered in conjunction is preferable to the scenario where only TB treatment is delivered. In effect, CVD treatment allows better use to be made of the fixed constraint, notwithstanding the adverse effects for patients receiving TB treatment.If policy makers do not want to accept the negative impact for TB second-line treatment, they can invest in the capacity of the platform (HSS 2) to reduce or relax the constraint. We assume that $2 m are spent, for example by hiring more nurses or giving them training. This allows treating the original 8500 cases with TB second-line treatment and improves the cost-effectiveness of both TB and CVD treatment. The joint cost-effectiveness increases to $144 cost-per-QALY, compared to the situation without HSS. The effect of HSS 2 is to improve the allocative efficiency of the delivery platform by reducing the impact of a binding constraint.6.3Finally, interdependencies can occur when an intervention introduces the need for a delivery platform that has not hitherto been in place, and that could be shared by other interventions. Investing in a new shared platform could support a number of existing or new interventions. This type of HSS differs from the other two types because it may expand the scope of interventions that can be considered for implementation. It may also alter the costs or benefits of existing interventions, leading to a reordering of interventions based on cost-effectiveness. An example for this type of HSS might be community services in the form of a new mobile HIV testing and counselling campaign, which creates a potential platform for other interventions including testing for malaria, hypertension and diabetes .zi are the associated decision variables, and CP is the fixed cost of the platform. The health benefits obtained from the first scenario must be compared with those from the second in order to decide whether or not to invest in the platform. The additional constraints ensure that mutually exclusive interventions (reflecting interventions that could be delivered with or without the infrastructure) are selected only once. That is, each intervention i can only be delivered with or without the platform, or not at all. Interventions that cannot be delivered with S (or P) will be attributed zero benefits To formulate mathematically, we need to first consider a scenario without the proposed platform;A succinct way of expressing the comparison of the two models is as follows:y is set to 1 if the platform is implemented, zero otherwise. The new constraint ensures that the set of interventions in P can be implemented only if the platform is put in place, the constant M merely being a sufficiently large constant.The new decision variable If the two interventions are both implemented, they share the fixed costs of the putative distribution system. That is, the $11 m cost of the platform can be shared amongst the total of 21,000 patients receiving the two interventions. Assuming an equal allocation across patients, this results in a reduction in fixed costs from $1294 (VL testing) and $880 (Oxytocin) to $524 per case, and a reduction in cost-per-QALY from $213 to $103 (VL testing) and $276 to $205 (Oxytocin). Note that this favourable cost-effectiveness can be secured only if Oxytocin is implemented alongside VL testing, sharing the fixed costs. The ‘package’ of VL testing and Oxytocin has a combined total cost of $18.95 m, which is higher than the $12.7 m required for VL testing alone. However, the package yields an aggregate cost-per-QALY of $155, below the threshold of $200, and therefore represents a good use of these additional funds, provided they can be secured.7Much of the HSS literature is implicitly concerned with the need to address opportunities and constraints presented by service delivery platforms. Many constraints on decision makers arise from the existence of platforms that impose an element of fixed costs on the health system . PlatforThis paper has examined three classes of HSS associated with delivery platforms. They are (1) HSS that improves the technical efficiency of an existing platform and is explicitly designed to increase the cost-effectiveness of a range of existing interventions, through improved quality, reduced unit costs, or both; (2) HSS that invests in the capacity of a platform when interventions compete for a delivery platform that has limited capacity, for example due to shortages in human resources. Such shortages lead to inefficiencies because economies of scope cannot be exploited. The marginal benefits of HSS are reflected in the reduced implicit prices of the constraints. Such HSS offers the ability to shift towards more cost-effective interventions, increase access to services, and therefore improve allocative efficiency; (3) HSS that introduces a new platform to support a new intervention that may create opportunities for cost-effective delivery of other interventions that could share the platform. Economies of scope arising from joint implementation make new interventions cost-effective compared to a situation where they would be implemented on their own. The paper illustrates these three classes of HSS with simplified examples, and how they can be modelled mathematically. The principal insight is that when there is reliance on a joint platform, interventions become interdependent, and must therefore be evaluated as a portfolio . Myopic With this paper, we expand on the work by The models developed in this paper are consistent with standard CEA but seek to model platforms and the associated cost structures more realistically. The models for HSS types 1 and 2 are more applicable in local decision-making situations, where the details of capacity constraints, service demands and other relevant factors related to interdependencies are likely to be very important. HSS type 3 will often be relevant to less incremental and more strategic decision-making. Such reforms will often require considerable investment, occur less frequently, and be taken at a more central budgetary level. However, the distinction between the models presented is to some extent artificial, because many realistic decision problems contain elements of all three types of HSS.We assume an average level of efficiency in the use of existing resources, following usual practice in most CEA . EckermaWe envisage that the next steps from our research may be the development of decision rules or a ‘checklist’ of the type or types of HSS that may accompany major adoption decisions. This would not only provide structure to the decision problem and break it down into distinct steps, but would also illustrate what kind of data and empirical evidence is required to help with informed decision making on HSS. National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Modelling Methodology at Imperial College London in partnership with Public Health England, and by the MRC Centre for Outbreak Analysis and Modelling (funding reference MR/K010174/1B). The views expressed are those of the authors and not necessarily those of the International Decision Support Initiative, the NHS, the NIHR or the Department of Health. RM received funding from the NIHR.International Decision Support Initiative , Bill & Melinda Gates Foundation, Department for International Development (UK). KH received additional funding from the Alec Morton received personal fees from Office of Health Economics and AstraZeneca unrelated to this work. The remaining authors have no conflicts of interests to declare.


Bioprosthetic heart valves (BHVs) are commonly used as heart valve replacements but they are prone to fatigue failure; estimating their remaining life directly from medical images is difficult. Analyzing the valve performance can provide better guidance for personalized valve design. However, such analyses are often computationally intensive. In this work, we introduce the concept of deep learning (DL) based finite element analysis (DLFEA) to learn the deformation biomechanics of bioprosthetic aortic valves directly from simulations. The proposed DL framework can eliminate the time-consuming biomechanics simulations, while predicting valve deformations with the same fidelity. We present statistical results that demonstrate the high performance of the DLFEA framework and the applicability of the framework to predict bioprosthetic aortic valve deformations. With further development, such a tool can provide fast decision support for designing surgical bioprosthetic aortic valves. Ultimately, this framework could be extended to other BHVs and improve patient care. Valvular heart disease is clinically characterized either by gradual narrowing of the valve due to calcification of the leaflets or regurgitation through the valve due to insufficient valve closure2. Valve repair and replacement are two possible interventions to address diseased valves and prevent congestive heart failure or death. Based on the estimates from the American Heart Association, more than 2.5% of the United States population is affected by valvular heart diseases3. Heart valve replacement is common for patients suffering from valvular heart valve disease; over 90,000 prosthetic heart valves are implanted in the United States every year4. One of the most popular classes of replacement valves are surgical bioprosthetic heart valves (BHVs), fabricated from chemically-treated biological tissues2. They provide better hemodynamic characteristics than mechanical prostheses (the other most popular class), but are prone to fatigue failure, limiting their durability to 10–15 years. However, estimating the remaining life of a BHV directly from medical images is difficult. On the other hand, valve performance measures can be used by physicians to make better valve replacement decisions, preventing premature replacements or surgeries6.Semilunar valves  are structures that permit blood to be pumped into the aorta and pulmonary artery from the ventricles during systole, and prevent backflow into the ventricles during diastole7. Heart valve analysis using computational models has been extensively studied in recent years11. A review of computational modeling methods that have been developed to provide diagnosis from medical images for aortic valves is presented by Zakerzadeh et al.9. Several quantities of interest can be obtained and studied from these computational models. Two key quantities of interest are the coaptation area, which reflects of the degree of valve closure, and the effective orifice area (or open area), which reflects the degree of valve opening. The closure of the heart valve can be assessed by performing structural analysis of the valve geometry with appropriate boundary conditions to simulate the valve closure. Similarly, the effective orifice area can be computed by performing valve opening simulation or dynamic simulation of the valve for one complete cycle of the heart beat10. In this paper, we restrict our focus to valve closure simulations; however, our framework can be extended to dynamic simulations for obtaining other quantities of interest such as effective orifice area.Computational analysis of the heart valve can be an important tool in understanding the etiology of valvular diseases and can help clinicians in obtaining additional information that aid in therapeutic or valve replacement decisions. For example, the failure of aortic valves can be related to stress concentration in the leaflets of the BHV12 have been the de facto standard for geometry representation in mechanical computer-aided design (CAD). NURBS surfaces can be used to parametrically design complex geometric objects, while allowing easy modifications. Current state-of-the-art valve analysis approaches reconstruct the aortic heart valve geometry from computed tomography (CT) images using NURBS10. In addition, current BHVs are designed only for certain discrete population-averaged sizes. These geometries are then analyzed using shell formulations of finite element analysis after meshing to assess the valve performance5. One of the most promising new analysis technologies that can be also used for valve simulations is isogeometric analysis (IGA)13. IGA unites engineering analysis and design by eliminating the tedious process of finite element mesh generation from the design geometry. IGA uses the B-spline basis functions for both representing the geometry and for the analysis. Hence, the NURBS valve geometry can be directly used for both valve design and analysis using IGA17.An accurate representation of the heart valve geometry is essential to assess its performance. Non-uniform rational B-splines (NURBS)18. In sciences, the applications range from astronomy19, high-energy physics20 and material science23 to medical diagnostics24 and plant sciences25. In the domain of engineering, deep learning is the key enabler of the recent autonomous driving revolution26 along with other significant progresses in robotics27, design and manufacturing29 and prognostics and health management of engineered systems30. Along with recent algorithmic advances, the success of deep learning as a powerful function approximator could be attributed to the availability of large volume of data and advances in high-performance computing such as the Graphics Processing Units (GPUs).The complete pipeline for the design and analysis of heart valves is illustrated in Fig. 5. The stresses and strains during the heart valve closure can be directly computed from the heart valve deformations. However, the proposed methodology is general; it can be extended to analyze other key performance characteristics of heart valves and more complex valve simulations that include fluid structure interaction31.In this work, we leverage the advances in deep learning to overcome the computational overhead in analysis of heart valves and to make valve design decisions. Note that in this work, we only specifically refer to surgical aortic valves with a stent that is sutured to the aortic root. Specifically, we propose a deep-learning-based convolutional autoencoder architecture (referred to as DLFEA) for predicting the analysis output directly from the input heart valve geometry. This approach accelerates the analysis by acting as a surrogate model that can be trained using previous analyses of multiple simulations. In particular, as a proof of concept, we predict the final deformed closed shape of the heart valve and the coaptation area, a key quantity of interest for surgeons. Coaptation area has been widely used as a key valve performance metric in the design and diagnosis of heart valves and predicting it has been the focus of several previous studiesNURBS-aware convolution to directly input the valve geometry information to the deep-learning model; and (iii) statistical and anecdotal results that establish the accuracy and robustness of the proposed method. Please refer to the Related Works section in the Methods for a detailed discussion of our contributions in the context of recent advances in machine-learning based surrogate modeling for simulations. The results suggest that the DLFEA framework can be directly used in the design and optimization of patient-specific BHVs.We have the following specific contributions: (i) a deep learning framework, DLFEA, to predict the deformation biomechanics of aortic valves trained using isogeometric analysis simulation data; (ii) a novel geometric analysis tool called The DLFEA framework predicts the final deformations of the aortic valve and the valve coaptation area using the original undeformed geometry of the valve, the aortic pressure, and the material properties of the valve as input. We performed 90,941 valve closure simulations by varying the undeformed geometry, pressure, and material properties. Of these simulations, 72,753 simulations were used for training the DLFEA; 9,094 simulations for validating the training and hyperparameter tuning; and the rest for testing. We compare the performance of DLFEA framework on a test dataset , which were not used for training.32). An histogram of the Euclidean and Hausdorff distances between the predicted and simulated deformed valve geometry is shown in Fig. 2 and 0.9328, respectively  and the features of the input. In Fig. Finally, we also visualize the high-dimensional data manifold using t-distributed stochastic neighbor embedding (t-SNE)The results presented in the above section show that the DLFEA framework is able to accurately capture the deformation biomechanics to within reasonable error bounds. The maximum value of the displacement among all the valve leaflets from the simulations is 0.7642 cm. The maximum Hausdorff distance  between the predicted and simulated valve displacements is within 15% of this maximum displacement. In addition, the median and mean value is less than 5% of the maximum displacement.34. In addition, the deformations predicted by DLFEA are accurate and smooth enough that the strains calculated from them are also smooth without any oscillations. More interestingly, we find that the DLFEA is able to learn certain complex interaction characteristics of the leaflets and their deformations. The predictions by DLFEA capture the symmetry of the three leaflets without any explicit training constraints. The DLFEA also captures the contact characteristics between the three leaflets accurately without any explicit information about contact mechanics .The predicted and simulated deformations are visually quite similar with some minor regional differences that are within ≈10% error, which is usually within the measurement accuracy of non-invasive imaging modalities such as Echocardiography35 . In the t-SNE embedding of our DLFEA network . In order to further understand the efficacy of DLFEA in interpolating and extrapolating the coaptation area predictions for different parameter values, we performed an ablation study  represents a non-linear activation function, Deep Learning, a subset of machine learning approaches, has emerged as a versatile function approximator that can establish a reliable map between (possibly heterogeneous) inputs and outputs of complex phenomena. A deep neural network is made up of several layers 37, generative adversarial networks38, graph convolutions39, etc. Once the network architecture is defined, the network weights  are updated using the back-propagation algorithm40 based on minimizing a loss metric. At the end of training, the best weights and biases that generate the minimal prediction loss are chosen for the network. This network can then be used to predict the outputs for the test inputs.Recently, deep learning has been successfully deployed in several areas with newer and more sophisticated architectures such as variational autoencodersPhysics-consistency in deep learning: The overall idea is to merge the ideas of deep learning and physics by using physics-based features while training the deep learning models. For example, researchers have modified the loss functions to ensure some physical constraints are satisfied45. There has also been work on interpreting the predictions of the deep learning model based on physical conditions23.Incorporating partial differential equations (PDEs) in deep learning models: The key idea is to use the underlying governing equations such as Berger’s equation, Navier-Stokes equation, Cahn-Hillard’s equation, etc. to compute the residual for the sample. Since modern software systems can define these partial differential equations numerically in terms of automatic differentiable functions, it is easy to minimize these residuals. There are several recent works on learning from partial differential equations51.Generative vs. distinctive predictions: While there are methods in Deep Learning for generating and even predicting desired outputs, the underlying physics is often more strict. For example, given a set of physical conditions (such as loads on a well-defined geometry) will result in a deterministic desired output (such as displacements). Modeling them as a generative model is not consistent with the physics. On the contrary, the inverse problem, of defining the displacement of a given geometry and predicting the set of physics conditions is often ill-posed and could be consistently modeled as a generative model. There are several works showing the capability of Deep Learning methods to act as a surrogate51. These surrogates are modeled as distinctive (non-generative) networks, since the physics is deterministic and the problem is well-posed. On the contrary, there are some recent works46, which deal with stochastic PDEs or with ill-posed problems such as inverse design which demand the use of a generative model.Deep learning applications to create surrogate models for finite element analysis is a very recent area of research that specifically focuses on bridging physics based models and data-driven models. Some of the recent advances in this research area are summarized below:54 or estimating the constitutive model parameters for aortic wall56. For BHVs, while there are optimization based methods for design of transcatheter aortic valves58, machine learning methods have mainly been used for 3D reconstruction of the valve geometry59. Further, there are physics-informed deep learning approaches for modeling cardiovascular flows, which incorporate residual minimization of the PDEs using deep learning61. However, to the best knowledge of the authors, deep learning has not been used for analyzing the deformation behavior of bioprosthetic valves.In the application area of biomechanics, most of the existing works use simple deep learning or physics-consistent deep learning methods. Specifically, these methods have been applied for modeling the aorta and estimating the stress fieldsPredicting raw values vs. descriptors for biomechanics applications: The current state-of-the-art machine learning works in biomechanics directly predict the stress field. However, during the physics solve, the stresses are not directly obtained. While it is challenging to obtain displacements from stresses, it is straight-forward to obtain the stresses and strains from displacements. Even though, small variations in the displacements can lead to large oscillations in the strain computations, in our case . This is important when we need to model contact physics among the objects. This is necessary in BHVs while predicting the contact between the leaflets. As seen in the Figures shown in Fig. Accurate representation of 3D geometries: In general, there is a disconnect between the geometry, the physics domain mesh, and the data representation for training physics consistent deep learning model. Converting one form of data to another is computationally expensive and not accurate. To avoid this, often researchers use a structured mesh which could be expensive in case of representing geometries with complex geometric features such as the heart valve. On the contrary, we make use of a NURBS-aware convolution operation and isogeometric analysis to alleviate this issue.In this work, we leverage the advances in deep learning to model the analysis of BHVs and to accelerate their design. The efforts made in physics-consistent deep learning and deep learning applications to biomechanics motivates this work. While there are several works in this fast growing area, there are still some gaps which are yet to be filled. This paper addresses some of those gaps:Learning the deformation biomechanics of heart valves involves learning multiple physical phenomena by the DLFEA framework. First, the DLFEA needs to learn from the input 3-dimensional Euclidean space geometry and predict the deformed shape also in 3-dimensional Euclidean space. Next, it needs to learn the effect of loads and boundary conditions on the deformation. It should learn about the interaction between the leaflets during closure  to predict the coaptation area. Finally, it should learn the material behavior and the dependence of the deformation on the thickness of the leaflets used in the simulation.Any machine learning framework requires the identification of three main components: (i) data representation, (ii) model architecture, and (iii) training algorithm. In the following subsections, we describe the different components of our DLFEA framework. Please refer to the 62 or its extensions such as Octrees63 and multi-resolution voxels64. Further, another class of algorithms have been developed to learn from point clouds65. There are other approaches where the topology of the data is modeled as a graph to perform a graph convolution operation39. However, the above mentioned methods have high sample complexity or time complexity for learning. Moreover, there is loss of information while transforming the CAD model to other representations. Hence, integrating the learning paradigm with the CAD representation is crucial. To this end, we propose a new NURBS-aware convolution operation.Learning directly from 3-dimensional Euclidean space is an interesting notion that has been explored extensively in machine learning literature. There are traditional approaches of object recognition using a 3D volume occupancy grid66. This method was used earlier to perform fast and parallel GPU evaluation of NURBS surfaces. Here, using this representation of the leaflet, we perform the convolution operation directly using the NURBS surface without any loss of information, and can learn using traditional convolutional neural networks. A similar mapping is required for deconvolving the final deformed geometry from the textural representation to the global coordinates.The crux of the NURBS-aware convolution operation is to perform the convolution operations on the NURBS control mesh of the input geometry to obtain the valve performance measures. The control points of NURBS surfaces provide the 3D surface representation of BHVs. Hence, for extracting the geometry information from different heart valves, it is sufficient to utilize the control points directly for the convolution operation. The control points can be represented as a rectangular matrix representing the tensor-product structure of these valve surfaces. Since, these control points are physically significant and are represented in the 3D Euclidean space, they can be represented as three different matrices with each of them containing the position with respect to each coordinate as shown in Fig. 67.We use convolutional autoencoder-type architecture to obtain the deformed geometry for a given heart valve. The autoencoder architecture is composed of three main components: (i) an encoder network, (ii) a code layer and (iii) a decoder network see Fig. . An encom × n × 3, where m and n are the number of control points in both directions of the parametrization). Now, using this data representation, we convolve further using traditional convolutional layers to create the encoder block of the autoencoder37.We split the heart valve into three leaflets and for each leaflet, we use the NURBS-aware convolution operation where the NURBS surface is represented using three rectangular matrices . We fuse the material properties vector and pressure boundary condition of the heart valve (pressure vector) with the fused embedding (fully connected layer obtained earlier) of the leaflet geometry.Since the coaptation area is an important functional parameter used to determine the BHV health, we chose to predict this quantity of interest directly using the network. We also chose to predict the final deformed shape of the leaflet geometry, which can be used to obtain any other measures such as leaflet strains and also provide visual feedback of the deformed shape. The deformed geometry is predicted by using a decoder block which decodes the information from the fully connected layers used for fusing the information from thickness, pressure, and leaflet geometry. The decoder predicts the final deformed control points of the heart valve, which can be multiplied directly with the weights and knot vectors of the original geometry to get the deformed surfaces.The final requirement for effective learning is to introduce a linear/non-linear activation for each output. In the case of the coaptation area, a rectified linear unit (ReLU) is the best fit, since the coaptation area is always non-negative. The ReLU function is represented as follows:On the contrary, deformations of the leaflets could be negative at some locations, which makes ReLU a bad fit to use as an activation function for this output. Hence, linear activation function is used for the deformations. In the next subsection we outline the details of training the proposed network architecture.In the previous two subsections, we explained the embedding of the different physical attributes in the data representation and the machine learning model to enable effective learning. This is required for effective learning since the sample complexity and time complexity are still a challenge. Although compute capability is abundant especially with the advent of GPUs, there are still limitations on the amount of data that can be generated in a viable time frame, specifically when the data is generated from computationally heavy simulations. Hence, embedding the physical attributes in different possible ways is required for effective machine learning methods. In addition, this helps us leverage some of our physical understanding of the process, thereby reducing the learning complexity for the machine learning network.l, the modified loss function islbc represents the boundary condition incorporated loss function and u represent the displacements. In practice, we use l to be mean squared error:upred and utrue represent the predicted and true displacements respectively.Another important physical characteristics that we need to embed are the essential boundary conditions imposed on the geometry for valve closure. Fixed boundary conditions affect the deformation of the BHV simulations. While the fixed node has zero deformation, achieving zero predicted deformation up to an arbitrary precision is numerically difficult. Further, if the back-propagation algorithm tries to achieve that, the nodes with non-zero deformation are affected, making the overall deformation difficult to learn. We deal with fixed boundary conditions by weighing the loss with the true deformations. If the original loss function used is 68. Using arbitrary scalar values, we shift the BHV models to generate modified control points of a new BHV, which augment the original training data. We make use of a parametric BHV design algorithm (see In order to improve the sample complexity, we also perform data augmentationithm see  to generSupplementary informationVideo demonstrating our results