We propose a method that quantitatively evaluates the typicality of a hematoxylin-and-eosin (H&E)-stained tissue slide from a set of immunohistochemical (IHC) stains and applies the We define the typicality of the H&E-stained tissue slides by the ratio of the probability density of the IHC staining patterns on low-dimensional embedded space. Employing a multiple-instance-learning-based convolutional neural network for the construction of the subtype classifier without the annotations indicating cancerous regions in whole slide images, we select the training data by referring to the evaluated typicality to improve the generalization ability. We demonstrate the effectiveness of the instance selection based on the proposed typicality in a three-class subtype classification of 262 malignant lymphoma cases.In the experiment, we confirmed that the subtypes of typical instances could be predicted more accurately than those of atypical instances. Furthermore, it was confirmed that instance selection for the training data based on the proposed typicality improved the generalization ability of the classifier, wherein the classification accuracy was improved from 0.664 to 0.683 compared with the baseline method when the training data was constructed focusing on typical instances.The experimental results showed that the typicality of the H&E-stained tissue slides computed from IHC staining patterns is useful as a criterion for instance selection to enhance the generalization ability, and this typicality could be employed for instance selection under some practical limitations. Malignant lymphomas have more than 70 subtypes, and pathologists are required to identify the subtype from a set of microscopic images of a specimen that is invasively extracted from a patient to determine the treatment of the patient . In a pacase corresponds to instance in the context of instance selection. If the criterion indicating how typical the H&E-stained tissue slide of each instance is known, the instance selection can be applied to the training data based on such typicality measures. In this paper, we propose a method that selects appropriate training data based on the typicality of the H&E-stained tissue slides of malignant lymphomas.For the construction of the subtype classifier, we use a set of WSIs of H&E-stained tissue specimens with slide-level subtype labels. The generalization ability of the classification model depends on the training data, and the constructed classifier often overfits the training data such that the generalization ability is degraded when the size of the training data is small. To improve the generalization ability while using a limited number of training data, it is necessary to construct appropriate training data. In the context of instance selection, it is known that the removal of atypical instances from the training data of a classification model can enhance the generalization ability , 14. EveIn general, a database for digital pathology manages information of each instance as a pair of a WSI and the corresponding medical record wherein patient metadata, the set of names of IHC stains used for the diagnosis, the identified subtype, and other findings are described by pathologists. Although these medical records include no information about the typicality of the morphological features in observing H&E-stained tissue slides, the set of IHC stains used for the definitive diagnosis is determined based on the candidates of the subtypes that are inferred by the first observation of the H&E-stained tissue slide. We note that the same set of IHC stains is not always used to diagnose instances wherein the definitive diagnosis is of the same subtype. A specific set of IHC stains is commonly used in almost all instances wherein the definitive diagnosis is of the same subtype; however, some additional IHC stains are used in many of the instances. Herein, we hypothesize that the morphological features observed in the H&E-stained tissue slides are typical for the corresponding subtype when only the set of IHC stains that is common among the instances of the same subtype is used for the definitive diagnosis. When the morphological features are typical for the subtype and hence only the subtype is inferred as the candidate at the first observation of the H&E-stained slide image, the pathologist then selects only the optimal set of IHC stains that are required to confirm that the candidate subtype is correct. In contrast, if the morphological features of the H&E-stained tissue slide are atypical for the subtype, and hence, multiple subtypes are inferred as candidates at the first observation, additional IHC stains are required to identify the subtype for the definitive diagnosis. Based on this hypothesis, we quantitatively evaluate how typical the H&E-stained tissue slide is by the set of IHC stains used in the diagnosis and employ the evaluated typicality as the criterion for the instance selection of the training data.In this paper, we propose a method that quantitatively evaluates the typicality of the H&E-stained tissue slides and selects the training data for the construction of subtype classifiers by referring to the evaluated typicality in order to improve the generalization ability. We demonstrate that instance selection based on the proposed typicality improves the generalization ability of the subtype classification model by a three-class classification experiment with 262 cases of malignant lymphoma. In the experiment, WSIs of the H&E-stained tissue sections were used, wherein a partial region of the tissue specimen was cancerous, and the WSIs had no pathologist annotation indicating cancerous regions. We employed a multiple-instance-learning-based (MIL-based) convolutional neural network (CNN) that could automatically focus on image patches of cancerous portions from bags that comprised sets of image patches extracted from the entire tissue slides, and the proposed method provided the criteria for selecting the appropriate training data for the MIL-based subtype classification.We proposed a method for evaluating the typicality of H&E-stained tissue slides for the criteria in the instance selection, which was derived from the sets of IHC stains used in the pathological diagnosis of malignant lymphomas.We analyzed the sets of IHC stains in the dataset consisting of clinical lymphoma cases and revealed the relationship between the subtypes and the sets of IHC stains, wherein we could observe similar sets of IHC stains between different subtypes and various sets of IHC stains within a subtype.We demonstrated an MIL-based CNN for classifying three types of malignant lymphoma tissues, of diffuse large B cell lymphoma (DLBCL), angioimmunoblastic T-cell lymphoma (AITL), and classical Hodgkin’s lymphoma (CHL) to confirm the effectiveness of the typicality-based instance selection.The contributions of this paper are as follows:Malignant lymphoma in digital pathologyMalignant lymphomas comprise a group of blood cancers that develop from lymphocytes—a type of white blood cell —and theiMultiple instance learning In cases without a pathologist’s annotation of the cancerous regions in the WSIs, MIL techniques have demonstrated excellent performance in pathological image analyses [analyses –22. MIL-Several MIL methods for aggregating features of the image patches in a bag into the bag’s features have been proposed , 23. We N be the number of instances of malignant lymphoma, and we define K-dimensional one-hot vector for the multi-class label of K subtypes, and L-dimensional multi-label binary vector that represents the IHC staining patterns, i.e., the combination of IHC stains used for the definitive diagnosis. In a medical record of malignant lymphomas, a list of IHC stains used in the definitive diagnosis is provided for each instance, and their expression patterns are described in the form of positive or negative. Each element of n-th patient indicates that CD20 and CD30 were used in the diagnosis, the multi-label vector Let n-th WSI are denoted by b-th bag   based onm, n) component is L-dimensional vectors that represent the IHC staining patterns in the M-dimensional space based on the distances in order to make it easier to estimate probability density distribution of IHC stains. Because MDS embeds the input feature vectors into low-dimensional space while preserving the relationship of the distances among instances, instances having similar IHC stains become close to each other even in the embedded space.The Hamming distance k-th subtype is computed from k-th component of the vector k is represented as:G is a Gaussian kernel function, and w is a bandwidth parameter. When n-th instance, its typicality k is calculated as the ratio of probability density for subtype k to probability densities of all subtypes:We employ kernel density estimation  to estimInstance selectionThe typicality that we defined can be considered as the difficulty measure of inferring subtype candidates only from the H&E-stained tissue slide in the diagnosis. Typical instances would have the classifiable subtype-specific features in the H&E-stained WSIs because the pathologist could accurately infer the candidate subtype. This quantitative measure can be employed as a typicality criterion for instance selection. As mentioned in Section 1, if we know how typical each instance is, we can apply instance selection to a pathological image dataset to improve the generalization ability of the subtype classification, in which instances having the atypical appearance of H&E-stained WSIs are removed from the training data , 14. In k. Figure Q-dimensional feature vector b are aggregated as a weighted sum, and the feature vector is obtained as In our classification experiment, we employ an attention-based MIL-CNN as the classification model , 24. Thementclass2pt{minimWe first apply MDS to IHC staining patterns for embedding them to a low-dimensional space and typicalities of all the instances are then calculated. In the classification experiments, we select instances according to the calculated typicality, and the classification performances with several instance selections are compared.Database Our database of malignant lymphomas comprises Calculation of typicality In this work, we considered 35 types of IHC stains, which were used for at least 10 instances because the IHC stains that were hardly used in the diagnostic routine could result in noise in the typicality calculation. That is, the size of the IHC staining pattern w for the kernel density estimation such that the probability density of each subtype is distributed evenly between the low and high values. We manually adjusted w in this study, but it should be automatically adjusted for a much larger dataset.entclass1pt{minimaSubtype classification by MIL-CNN We perform multi-class image classification, wherein each of the 262 slides is classified as AITL, DLBCL, or CHL. All the glass slides were digitized using a WSI scanner Aperio ScanScopeXT  at 20x original magnification (0.50 ResNet50  as the fintermediate), and atypical groups. Each group is randomly divided into five subgroups, as shown in Fig. Figure In Experiment II, we investigate whether we could improve the generalization ability of the subtype identification by changing the ratio of the number of data with different typicality in the training data. In contrast to Experiment I, we intend to confirm that the instance selection for the training slides based on their typicality improves the generalization ability of the subtype classification. In this experiment, first, all the dataset is randomly divided into five subgroups for fivefold cross-validation, as shown in Fig. In all the classification experiments, the training dataset is split into 75% training and 25% validation instances. During the testing, the trained models at epochs at which the model demonstrates the smallest losses for the validation data are employed.We applied MDS to all 262 instances, and Fig. Table Experiment IWith the instance selection mentioned in the experimental setting, we performed subtype identification of AITL, DLBCL, and CHL using only H&E-stained tissue images. Table typical slides would have typical subtype-specific features in the images, while atypical slides have insufficient image features for subtype identification. It was confirmed that the typicality measure computed from the IHC staining patterns had a certain relationship with the difficulty of the subtype classification from H&E-stained WSIs.Table Experiment II Next, the second instance selection presented in Fig. Table Visualization of attention weight We visualized the attention weights computed by our best classifier in Experiment II. The validity of attention-based MIL for DLBCL has already been confirmed in the literature [terature . Figure In this paper, we proposed a method for improving the generalization ability of a classification model that classifies the subtypes from H&E-stained WSIs of malignant lymphomas. We defined the typicality of an H&E-stained WSI from a set of IHC stains described in a medical record that indicates how typical its appearance is. The proposed typicality can be employed as a criterion in the instance selection to improve the generalization ability of the subtype classification. We performed subtype classification of malignant lymphomas wherein datasets were selected according to the computed typicality in order to investigate the relationship between the typicality calculated from the IHC staining pattern and the subtype classification difficulty and improve the generalization ability. The results confirmed that the typicality of subtypes was related to the classification performance, and we showed that instance selection based on typicality would be useful for better construction of the subtype classifier. By considering the typicality of the case, we constructed an appropriate dataset for realizing a higher classification performance with the small number of training data as compared to the baseline. For the application of the proposed typicality, instance selection on a large dataset is considered for reducing costs for the scanning of WSIs and the training of models. Another application is the instance selection for class-imbalance dataset such that large-class instances are sampled to adjust to the small class, wherein random sampling can degrade the model performance by sampling atypical instances. The proposed typicality, which is calculated from a set of IHC stains, can be used as a measure in various instance selections.


When producing packaging from corrugated board, material weakening often occurs both during the die-cutting process and during printing. While the analog lamination and/or printing processes that degrade material can be easily replaced with a digital approach, the die-cutting process remains overwhelmingly analog. Recently, new innovative technologies have emerged that have begun to replace or at least supplement old techniques. This paper presents the results of laboratory tests on corrugated board and packaging made using both analog and digital technologies. Cardboard samples with digital and analog creases are subject to various mechanical tests, which allows for an assessment of the impact of creases on the mechanical properties of the cardboard itself, as well as on the behavior of the packaging. It is proven that digital technology is not only more repeatable, but also weakens the structure of corrugated board to a much lesser extent than analog. An updated numerical model of boxes in compression tests is also discussed. The effect of the crushing of the material in the vicinity of the crease lines in the packaging arising during the analog and digital finishing processes is taken into account. The obtained enhanced computer simulation results closely reflect the experimental observations, which prove that the correct numerical analysis of corrugated cardboard packaging should be performed with the model taking into account the crushing. The environmental challenges faced by the contemporary world, such as an ecologically healthy lifestyle, thoughtful consumption of goods, and waste disposal in order to cause less damage to the environment, can be included in the three Rs, namely reduce, reuse, and recycle . A globaHowever, it should not be forgotten that in light of all these features and requirements for cardboard packaging, it must first meet a certain load-bearing capacity, which is strictly related to the mechanical properties of paperboard/corrugated board. The producers of cardboard packaging constantly maintain a deep interest in effective, economical, and straightforward solutions, which, in turn, leads to the intensive development of scientific research closely related to this branch of industry.Physical examinations are fundamental in assessing the strength of corrugated board boxes; therefore, a number of typical tests have been developed to standardize the process of characterizing the mechanical properties of corrugated board. The box compression test (BCT) ,3,4,5,6 To evaluate the strength of a corrugated board box, analytical formulae developed through the years and found in the literature can be utilized, starting from the simplest ,18,19, uThe basic raw materials used in the production of paper  are cellThe last stage of cardboard production is obtaining the appropriate sheet format. First, a cut along the sheet is performed and then lengthwise in a lateral direction. During this stage, it is also possible to crease (crush cardboard for bending). There are two possible variants of performing creases or die-cutting in the cardboard, namely analog ,46 and dCurrently, the packaging design process does not take directly into account the effects of crushing the material; the most common concerns are with global correction (safety) factors. If, however, the decrease in the load capacity of the packaging due to the crushing of the creasing lines was carefully measured, the effect could be taken into account much more accurately in calculations. This would make the calculations dependent, for example, on the type of machine or type of finishing process. It is also possible to test some particularly important productions or series on weaker (and therefore cheaper) materials on digital devices, which crush the cardboard only to a small extent or not at all.In the case of digital processes, die-cutting is performed using a laser while creasing is achieved either by means of specific laser cuts or a 3D-printed polymer on a replaceable carrier wound on a drum inside the finishing device. The former crease type enables control of the laser penetration into the corrugated board. However, due to material weakening (cutting the liners), it cannot always be applied to automatically folding boxes because the folding and gluing machines require a bending force within a certain predetermined range. The use of the latter type of creasing does not weaken the creasing lines but requires a preparation phase (printing the creasing rules), which, however, is still much faster and cheaper than preparing dies for conventional cutting tables. Both techniques are optimal for small and medium batches of boxes. The printing or coating of the paper has no influence on the digital process, although it is appropriate for cardboard up to the maximum B or C flute depending on the technique used.This paper presents the results of laboratory tests on corrugated board and packaging produced using both analog and digital technologies. Cardboard specimens with digital and analog creases are subject to various mechanical tests , which aFor obtaining measurement data from the outer surfaces of the sample during testing, video extensometry can be applied. This procedure relies on measuring the relative distances between pairs of points tracked across images acquired at specific load values ,56. It iIn this paper, we compare the results of laboratory tests such as the ECT, SST, TST, and BNT to examine the impact of creases on the mechanical properties of cardboard and corrugated board boxes. The results allow for the preparation of an improved numerical model of the packaging in which all the observed and measured creasing effects in the area of the creasing line are taken into account, including those effects in the numerical analysis that allow us to obtain promising results.In this study, an innovative device for corrugated board testing was used, namely the BSE System . The sysThe ECT is one of the most widely known mechanical testing methods used on corrugated board samples. It reflects the compressive behavior of the corrugated board in the cross-machine direction. There are multiple international standards for this testing method, although the general principle is similar, i.e., the sample of corrugated board of a particular size is placed along its cross-machine direction between rigid steel plates. An additional two steel blocks support the sample to hold it vertically, and the sample is statically compressed in a mechanical press.The particular testing setup used for the ECT depends on the part of the world where the test is conducted. For instance, one may use ISO 3037, ISO 13821, FEFCO 8, or TAPPI T839. In this paper, the ISO 3037 standard was applied for ECTs using the BSE System from Femat ; see FigThe SST is an innovative mechanical testing method applied to corrugated board samples. It reflects the static transversal shear stiffness of the corrugated board tested and is able to capture the crushing of the board. In the second, updated generation of this testing method, the sample is square. The specimen is supported on two opposite corners and force is applied vertically at the other opposite corners.There are no international standards, which regulate the testing setup for the SST. In this paper, the tests were performed by employing the BSE System from Femat ; see FigThe TST is an innovative mechanical testing method applied to corrugated board samples and reflects torsional stiffness. During the test, the longitudinal sample is clamped at the bottom and top and twisted statically by rotating the upper clamp around the direction of the sample. The test is performed along two directions of the cardboard, i.e., two samples must be cut from one material, in machine and cross-machine directions.There are no international standards regulating the setup for the TST. In this paper, the tests were performed using the BSE System from Femat ; see FigThe BNT is one of the most widely known mechanical testing methods used on corrugated board samples and reflects bending behavior. There are multiple international standards for this testing method. In this paper, four-point bending is considered, although the general principle in all examination standards is similar, i.e., the sample of corrugated board of a particular size is placed longitudinally with two perpendicular rigid supports. During the test, two upper perpendicular edges of the traverse beam statically press the sample and cause four-point bending. The test is performed along two directions of the cardboard, i.e., two samples must be cut from one material, in machine and cross-machine directions.The particular testing setup used for the BNT depends on the part of the world where the test is conducted. For instance, one may use ISO 5628 or TAPPI/ANSI T 836. In this paper, the ISO 5628 standard was used for four-point bending with the BSE System from Femat ; see FigThe BCT is the most widely known mechanical testing method and reflects the compressive strength of a box. There are multiple international standards for this testing method, although the general principle is similar, i.e., the box sample is placed between two rigid steel plates and compressed by pressing the upper plate on the box.The international standards one may use for the BCT setup include ISO 12048, ASTM D642, or ASTM D4169-16. In this paper, the ISO 12,048 standard was adopted for the BCT by employing a mechanical press from Femat ; see FigIn this work, we used samples provided by a packaging manufacturer equipped with both packaging finishing technologies, namely analog and digital machines. The first is a traditional method based on flatbed die-cutting of packages and boxes. The technique involves cutting the boxes with a die to reproduce the designed shape of the packaging. The die has a knife with appropriately profiled blade shapes (different for cutting and creasing) and pressed with great force perpendicularly into the substrate (straight through). Lines of specific shapes can also be grooved, i.e., pressed into the surface. The die allows the cut packages to assume the desired external shapes, as well as to make holes and perforations. Rubber is often used around creasing knives to minimize the possibility of tearing the material. Unfortunately, the rubbers press on the material and can thus crush the cross-section of the corrugated board.2 lasers [The newest methods of punching and creasing come down to a process based entirely on CO2 lasers  or hybri2 lasers . In digiSamples prepared on analog and hybrid machines were made of the same materials. Therefore, it was possible to make an objective comparison of the two technologies and their influence not only on the material parameters of the corrugated board itself, but also on the strength of the packaging cut from those materials.In the study, three single-walled corrugated boards of B flute were considered, henceforth labeled B360, B370, and B380.The first corrugated board, B360, consists of the following papers:●testliner 130 g/●fluting 100 g/●testliner 130 g/The second corrugated board, B370, consists of the following papers:●kraftliner brown 135 g/●fluting 100 g/●kraftliner brown 135 g/The third corrugated board, B380, consists of the following papers:●kraftliner white coated 145 g/●fluting 100 g/●duplex white 135 g/Whether the delivered samples were made of the same material was first checked, and the actual grammage and thickness of all samples were measured  . The modEach vertical wall of the box model was divided into two partitions: (a) the edge area where the material models account for the crushing of the corrugated board and (b) the remaining areas where the material was not crumpled see b. In ordIn order to correctly model the progressive plasticization of the cardboard packaging, a nonlinear iterative analysis based on the Newton–Raphson procedure was used. In order to induce buckling from the plane of each vertical wall of the box, small imperfections were introduced into the model, which were determined in the linear perturbation buckling analysis.i is the torsion stiffness i is the bending stiffness 1, TST2, BNT1, and BNT2 parameters obtained from the BSE System are presented. For samples perpendicular to the crease, only the SST, TST1, and TST2 were undertaken; for the ECT, BNT1, and BNT2, the samples perpendicular to the crease would not produce beneficial results.The results from cardboard tests see  for eachThe results for crease-free region samples and samples with the crease along the sample length were compared to verify the influence of the crease lines, in particular production technologies. The data are presented in The results for crease-free region samples and samples perpendicular to the crease were compared to verify the influence of the crease lines, in particular production technologies. The data are presented in The results from the BCTs see  for eachThe results for boxes with analog crease lines and digital crease lines were compared to verify the influence of the crease lines on the compressive strength of the boxes due to the particular production technology. The data are presented in The box tested had a so-called offset between the horizontal crease lines. This causes two peaks on the compression plots in typical force vs. displacement graphs; see Based on all mechanical measurements of corrugated board carried out in the BSE System, four material models were created: (i) corrugated board from analog-cut packaging, (ii) cardboard crushed by an analog cut and creasing process, (iii) corrugated board from digitally cut packaging, and (iv) cardboard crushed by a digital (hybrid) creasing process. All parameters (elastic and plastic) are summarized in Numerical validation consisted of checking whether the A model, which takes into account the effect of crushing the corrugated board within 25 mm of all  creasing lines, appears to be more precise than the traditional B model, in which these effects are ignored. The confrontation of the numerical results with the experimental results see  providesIn the numerical model, all four vertical walls were directly loaded (the displacement control approach was used); thus, the offset effect  was not 1, TST2, BNT1, and BNT2, the samples lost from a few percentage points to up to 35% of their values. For the counterpart digital cases, the losses were usually meaningfully smaller, from a few percentage points to 31%.In 1 are much lower (9–13.5%) while the values for TST2 are a few percentage points lower (1–3%) for digital crease lines. Moreover, the values for BNT1 are also a few percentage points lower (3–5%), and the values for BNT2 are a few percentage points lower (2–4%) for digital crease lines. To sum up, in most cases, the samples with digital crease lines have lower decreases in the values of the mechanical parameters, if one compares the mechanical properties of no-crease samples and samples with creases along the sample length.Comparing the analog and digital crease line results in detail, the values for the ECT are similar while the values for the SST are a few percentage points lower (~2%) for digital crease lines. Further, the values for TST1, and TST2 in the analog cases, the samples lost from a few percentage points up to 30% of their values. For the counterpart digital cases, the losses were usually meaningfully smaller, from a few percentage points to about 11%.In 1 are 1.5–3.3% lower while the values for TST2 are 11–18.8% lower for digital crease lines. To sum up, in all considered cases, the samples with digital crease lines have lower decreases of mechanical parameter values, if one compares the mechanical properties of no-crease samples and samples with creases perpendicular to the sample length.Comparing the analog and digital crease line results in detail, the values for the SST are 3–9% lower for digital crease lines. The values for TSTWhile the samples made using the digital method presented lower decreases in their mechanical properties than those made by the analog method, it may be thought that the values are still small (1–5%). However, the values have an important impact on box compression strength. The main cost in producing packaging is the corrugated board itself. Therefore, employing the digital method can provide benefits in using lower quality and less expensive material, while still giving the box the strength required; it is less susceptible to crushing than more expensive and better quality materials. The cost of making the cardboard via the digital method is probably higher but remains more competitive due to the cheaper materials used in the final packaging.In For B380, the analog cardboard had a grammage of about 30 g more in comparison with the digital cardboard, but the box made of material obtained from digital technology still had higher strength (2.8%). It was also observed that the drop of the strength curve after the first peak in digital technology products is much smaller than in analog technology. The percentage difference between the analog and digital first peak is 2.8%, increasing to about 10% for the second peak see .To the authors’ knowledge, similar studies analyzing the effects of analog and digital creasing lines on the performance of corrugated board and boxes have not been published in the scientific literature. Therefore, the results presented here are unique and cannot be compared with the previous research. The results evidently demonstrate that digital technology is not only more repeatable, but also causes far less damage to the structure of corrugated board than analog technology.Similarly, the authors have not been able to find adequate research in the literature in which numerical models take into account the effect of the crushing of the corrugated board with respect to analog and/or digital creasing. From the research and calculations performed here, an interesting conclusion can be drawn. In the case of model B, which does not take into account the crushing effect for box samples with digital creasing lines, the difference between prediction and the experiment was only 1–2%. However, when the crushing effects were not taken into account in the boxes with analog creases, the difference reached 13.7%. This observation makes it reasonable to take into account regions where the material is evidently different (crushed) in order to correctly model the behavior of the packaging. As creased material measurements are not subject to a common standard, it remains to take into account the drop in load capacity with a correction factor. An alternative is to produce the packaging while using digital tools to gain 10–13% of the load capacity of each package, at the expense of implementing a new technology in the production plant.This study verified the influence of two types of creasing line technologies, analog and hybrid digital, while applying digitally driven mechanical crease rules on the mechanical parameters of corrugated board and packaging. The research was carried out on corrugated board samples with edge crush resistance, shear stiffness, torsional stiffness, and bending stiffness measured for three types of three-layer board with different grammages. The same materials were used to test the boxes to determine and compare their compressive strength. All the measured mechanical properties of the corrugated board were used to build the correct constitutive models for both the undamaged and the crushed material for numerical analysis, which was performed to verify the packaging behavior during physical load-bearing tests.First, the laboratory data were analyzed to understand the difference in the deterioration of the mechanical properties of corrugated board and box when analog or digital creasing lines are used. It turned out that the use of digital crease lines can be beneficial, as they have a reduced impact on the shear and torsional stiffness of the cardboard. Although the decrease in bending stiffness is less, it is still noticeable. The drops in the edge crush tests are negligible. Likewise, boxes with digital crease lines are stronger compared to analog crease lines if box compression is analyzed. Producing crease lines by digital technology does not crush the corrugated board, due to which its material properties are higher and also give higher load-bearing capacity to boxes with such crease lines.Numerical simulations were then carried out to understand the impact of crushing in the vicinity of the creased edges of the packages on its load-bearing capacity. The most important conclusion that emerged after the validation while using two numerical models is the fact that only boxes made of corrugated board with hybrid creasing can be analyzed using a model that does not take into account the effects of cardboard crushing—in this case, the estimation error was within 2.02%. However, in order to correctly estimate packaging with analog creases, it is necessary to take into account the effects of the crushing of the corrugated board as a result of the analog finishing process and to model two material regions (intact and deteriorated due to creasing). In the case of boxes with analog creases, the numerical model, which did not take into account the effect of crushing, generated results as much as almost 12% higher than the model, which took these effects into account, which means that 12% of the load capacity is lost by clinging to traditional packaging finishing techniques. To sum up, modern methods of packaging finishing can not only help producers of corrugated boxes to digitalize their production, but can also reduce the weight of their products and thus save more trees that are for the common good.


The valuation adjustment mechanism (‘VAM’) agreement has recently been widely adopted in venture capital investment in emerging markets. The VAM agreement endows venture capital institutions a contractual right to reevaluate invested startup contingent on preset performance targets, which is crucial for the effectiveness of the VAM agreement by deeply affecting the strategy and fate of the startup. Motivated by exploring a rational performance target setting, this paper: 1)Firstly, extracts a general structure of the VAM agreement by cases analysis;2) Secondly, adopts a real options methodology to derive the option value held by venture capital institutions and how the pre-determined performance target affect the payoff of venture capital institutions;3)Thirdly, derives the expected time to achieve the given performance target and the behavior choice of entrepreneurs of startups; 4)Finally, by maximizing the contractual value of venture capital institutions with the participation constraints of the entrepreneur, derives the optimal performance target setting. The result finds that the option value of the VAM agreement is positively related to the performance target. It may partially explain why venture capital institutions tend to dominate overly high targets from a real options perspective. We also confirm the incentive effect of the VAM agreement that entrepreneurs tend to exert their best effort. Furthermore, the derivation of the optimal performance target shows that it is an increased function of the agreement period and a subtractive function of project risk. This paper contributes to the literature on contingent payment mechanisms and provides practical implications for the VAM agreement design. Since the low success rate of startups is primarily attributed to the shortage of funding , ventureAccording to the basic contractual logic giving venture capital institutions both downside protection and a favorable position if the investee proves its potential , convertThe VAM agreement offers a valuable mechanism to bridge the ex-ante valuation gap and align the startups’ interests with venture capital institutions. However, it has been documented excessive performance targets may lead to irrational operation of startups in pursuit of short-term benefits regardless of sustainable development –32. A moPrior studies have consented that performance target is crucial to the effectiveness of VAM agreement and found some attributes of investor, like bargaining power, agent’s self-interest, or conservatism, may affect the determination of performance target. However, two facts cannot be ignored for the rationality of the performance target. First, the effort of the invested startup is of the same significance in determining the performance target. Venture capital institutions may dominate the performance target, and the startup may have no choice but to accept an overly high standard. But if startups expect to fail to meet performance targets even with the best effort, the alternative option may be to reduce efforts to maximize their benefits. An excessively high target will undermine the incentive effect of the VAM agreement. Second, achieving performance targets is often subject to external factors such as macroeconomic and industry policy changes. When faced with changes in external conditions that may be positive or negative, there is a lack of intuitive understanding of how to set performance targets. To fill this research gap, this paper exploits the real options methodology to examine the economic value venture capital institutions acquire through the VAM agreement and startups’ behavioral choices to achieve their performance target. Assuming that venture capital institutions maximize their economic value within the constraint that startups expect to achieve their performance target, this paper derives an optimal performance target.Our study makes three primary contributions. First, our work contributes to the literature on the VAM agreement and, more generally, the contingent payment mechanism. By exploiting the real options methodology, we formally derive the economic value of the VAM agreement, which is positively related to the pre-determined performance target. The result may provide a theoretical explanation for previously documented overly high-performance targets imposed by investors. By introducing startups’ behavioral choices to achieve performance targets, we confirm the VAM agreement’s incentive effect and provide new insights into how to set optimal performance goals. Second, our findings are of practical implications for VAM contract design. By maximizing the payoff of venture capital institutions while the startups expect to achieve their performance target, the derivation of the optimal performance target suggests that it should be positively correlated with the duration of the VAM agreement and negatively correlated with internal and external uncertainties. Third, we extract the general structure and execution sequence of the VAM agreement from the analysis of two classic cases, which may provide a reference for further theoretical modeling.The rest of the paper is organized as follows. Section 2 analyzes the characteristics of the VAM agreement and extracts the general structure and execution sequence. Section 3 derives the economic value acquired by venture capital institutions through the VAM agreement and how that value relates to the pre-determined performance target. Section 4 analyzes the behavioral choices of startups and derives the optimal performance target. Finally, section 5 discusses limitations and future research directions, and section 6 concludes the paper.China has witnessed many investment events using the VAM agreement as the largest emerging market, including both win-win and failure cases. Thus, the VAM agreement is not only a simple investment tool but also profoundly impacts startups’ development strategy and destiny.Since the specific form of the VAM agreement is flexible, this section takes the VAM agreement signed by Morgan Stanley and Mengniu in 2002 and the VAM agreement signed by Morgan Stanley and Yongle in 2005 as cases to extract the general characteristics and structure of the VAM agreement. Mengniu and Yongle both adopted the VAM agreement for venture capital financing, while Mengniu ended the deal with a win-win situation for both sides, but Yongle failed.To introduce venture capital institutions to meet the financing needs of enterprises and create a flexible equity basis for an overseas listing, Mengniu first established the Cayman Islands company and Mauritius company. At the same time, Mengniu found Jinniu and Yinniu in the Virgin Islands. Among them, Jinniu shares are held by Mengniu executives, and Yinniu shares are held by senior executives related to the Mengniu business. Before the introduction of venture capital institutions, Jinniu and Yinniu each accounted for 50% of the shares of the Cayman Islands company. In the first round of capital injection, Jinniu and Yinniu purchased 1134 class A shares and 2968 class A shares of the company, accounting for 9.4% of the equity and 51% of the company’s voting rights. Morgan Stanley, CDH, and CDC have injected the US $25.97 million, holding 48980 class B shares, accounting for 90.6% of the equity and 49% of the voting rights. The funds injected by the venture capital institutions are used to purchase 66.7% of the registered capital of Mengniu shares.After the first capital injection in 2002, venture capital institutions and Mengniu management signed a VAM agreement. The duration of the deal is one year. Suppose the agreed performance is achieved one year later. In that case, Mengniu management can convert the class A shares held by Mengniu into class B shares in the proportion of 1:10, accounting for 51% of the equity of the Cayman company, to achieve the consistency of equity and voting rights. If Mengniu’s performance fails to meet the target, venture capital institutions will obtain 60.4% of Mengniu’s absolute control. From 2002 to 2003, as Mengniu’s management achieved the performance target, the management accounted for 51% of the equity of the Cayman company, and the proportion of equity holding Mengniu Dairy increased to 67.32%. The ratio of equity containing Mengniu Dairy held by venture capital institutions was 32.68%. After the end of the first round of financing and agreement, in 2003, Morgan Stanley and other venture capital institutions purchased the convertible note the Cayman Islands company issued for the second round of funding with the US $35.23 million. The convertible note’s conversion price was only HK $0.74 per share, and signed the VAM agreement for three years. Since 2003, if the compound annual growth rate of Mengniu’s profit is not less than 50%, the venture capital institutions will transfer 78.3 million shares to the management of Mengniu. If the performance target is not achieved, the management of Mengniu will transfer the same amount of shares to the venture capital institutions. From 2003 to 2005, since the development of Mengniu far exceeded the performance target agreed in the agreement, the venture capital institutions paid the management of Mengniu a convertible note with the principal of US $5.98 million in 2005, which terminated the VAM agreement in advance, and the two sides achieved a win-win situation. Like Mengniu, Yongle carried out overseas restructuring in 2004. First, it established Retail Management in the Virgin Islands, whose shareholders and the shareholding ratio was the same as those of Shanghai Yongle in China. Then, it found Yongle Electric Appliance Sales in the Cayman Islands, becoming the main body for introducing venture capital institutions and overseas listing. After that, in 2005, Morgan Stanley and CDH purchased 363 million shares and 59.13 million shares of Yongle through their subsidiaries, MS Retail, and CDH, obtaining 27.36% equity. As a result, the venture capital institution and Yongle management have signed a VAM agreement. If the profit of Yongle in 2007 is more than 750 million yuan, the venture capital institutions will transfer 46.97 million shares to Yongle management. If the profit is less than or equal to 675 million yuan, the Yongle management will transfer 46.97 million shares to the venture capital institutions. If the profit is not more than 600 million yuan, Yongle management transferred 93.94 million shares to venture capital institutions. After the introduction of venture capital institutions, China Yongle was listed in Hong Kong. A total of 518 million shares were issued in Hong Kong, with a financing amount of 1.2 billion yuan. Since the launch of the VAM agreement, Yongle has entered a stage of rapid mergers and acquisitions expansion. Because it is challenging to integrate mergers and acquisitions units in a short period effectively, the cost of mergers and acquisitions synergy is high, and the operating cost is much higher than that of competitors Gome and Suning. The decline in performance led to a sharp decrease in Yongle’s share price. As the agreement expired, Yongle, which had been listed for only nine months, finally chose to be merged by Gome, ending the VAM agreement. Based on the above facts, the characteristics of the VAM agreement can be summarized as follows:First, the VAM agreements reconcile valuation differences between investors and investees. The financing parties involved in the VAM agreements are often startups in rapid development, which increase but lack cash flow. With narrow access to financing, high-growth startups introduce venture capital, reduce debt ratios, and gain value-added services from venture capital institutions in management, marketing, etc. When venture capital institutions inject money into startups and obtain equity, they first evaluate the project value and development prospects. However, the future development potential of the enterprise is full of uncertainty. In the early stage, the enterprise mainly faces product technology risk, management risk, and market demand risk. After successfully entering the growth period, the product enters the marketing stage, and the market share and future competition risk the profit. This makes it impossible for both sides of investment and financing to predict enterprises’ future development prospects. Although venture capital institutions can acquire specific information about enterprises through business plan screening, entrepreneur financing demonstration, due diligence, and other means, there is still a huge information asymmetry between venture capital institutions as financial investors and financing enterprises in professional fields, which also leads to differences in project evaluation between the two sides. The original intention of the VAM agreement was to reconcile the valuation differences between the investing and financing parties through the redistribution of rights afterward so that the investment process of venture capital institutions could proceed smoothly.Second, there is the incentive effect of the VAM agreement. After the investment, venture capital institutions usually occupy a seat on the board of directors to participate in the decision-making of significant events and obtain business information. However, as financial investors, venture capital institutions do not intervene in daily business activities, which entrepreneurs and the original management usually control. Venture capital institutions mainly care about financial returns. In contrast, entrepreneurs care about financial returns and pay attention to private benefits of control in enterprise operation, including a sense of achievement, reputation, and on-the-job consumption. Because the interests of entrepreneurs and venture capital institutions are not precisely aligned, there may be moral hazard issues. For venture capital institutions, the return on investment ultimately depends on business performance. Therefore, it is necessary to restrict the moral hazard tendency of entrepreneurs and encourage their performance creation. As shown in the cases of Mengniu and Yongle, the VAM agreement stipulates that when the enterprise’s operating results exceed the preset performance objectives, equity incentives will be given to entrepreneurs. If the goals are not met, the management’s shares will need to be transferred to venture capital institutions to realize the transfer of control to a certain extent. Because entrepreneurs attach great importance to enterprise control through equity incentives and punishment measures, the interests of entrepreneurs can be bound with venture capital institutions.Third, a reasonable performance target is crucial for achieving a win-win situation. The VAM agreement is an investment tool of an optional nature. Both the investment and financing parties bet on future uncertainties; the financing party exercises its rights if it meets certain trigger conditions, and the investor exercises its rights if the conditions are not met. Appropriate performance target setting can motivate entrepreneurs, but the pressure brought by the too high-performance target may make enterprises choose short-sighted and irrational business strategies. In the case of Yongle, the compound annual growth rate of Yongle’s profit from 2003 to 2005 is 40%. Even if this growth rate is maintained, the profit in 2007 will only reach 566 million yuan, far lower than the performance target of the equity award of 750 million yuan in the VAM agreement. The harsh performance target makes Yongle choose large-scale mergers and acquisitions expansion. Still, it is difficult to effectively integrate the mergers and acquisitions resources in the short term, increasing the management cost and reducing profitability. For venture capital institutions, the financial return ultimately depends on improving the enterprise’s fundamental profit level. Although the VAM agreement gives the venture capital institutions specific equity compensation when the performance does not reach the target, if the enterprise’s operation has been in trouble, it may also cause losses due to the decline in earnings per share. Therefore, a reasonable performance goal is necessary for venture capital institutions to obtain considerable financial returns and for entrepreneurs to win the right of control.Through the analysis of Mengniu’s and Yongle’s cases, we propose a general structure of the VAM agreement as follows:π0, the proportion of equity obtained by venture capital institutions is S (0<S<1). The agreement duration is T, and the performance target is π*.Both parties agree that if πt<π* within the duration of the agreement, the entrepreneur will transfer δ1 shares to the venture capital institutions. Conversely, the venture capital institutions transfer δ2 shares to the entrepreneur.Enterprise profit is the trigger condition of the agreement’s implementation, and equity is the adjustment object of the deal. The initial profit of the startup when it obtains the investment is The structure takes profit as the indicator of performance evaluation. It reflects that most agreements take simple and intuitive financial indicators as trigger conditions. In contrast, non-financial performance indicators or a combination of the two are rarely seen, and the profit level can most directly reveal the enterprise’s business performance. Therefore, according to the VAM agreement of Yongle, the enterprise profit is the performance target for the valuation adjustment of both parties.The structure takes equity as the object of valuation adjustments. Venture capital institutions are financial investors who pursue the maximization of interests. The equity reflects the overall value of the enterprise, so investors usually take the equity of the financing party as the bargaining chip of the VAM agreement to realize a change of a certain number of shares and even transfer of controlling rights according to the trigger conditions.δ1 and δ2 shares are respectively transferred to the other party based on whether the performance target is reached. In Mengniu’s first round of financing, the VAM agreement only stipulates incentives with no punitive measures, which is equivalent to δ1 = 0.The structure reflects entrepreneurs’ incentive and restraint mechanism of equity incentive and punishment. In the VAM agreement of Mengniu’s second-round financing, it is agreed that if Mengniu achieves the performance target, it can get an equity reward. Otherwise, it must transfer equal shares to the investor as punishment. Without losing generality, it is assumed that transferred shares are not necessarily identical, but The structure allows valuation adjustments to be performed in advance. The VAM agreement stipulates that the performance of the enterprise shall be inspected during the agreement period. If the performance increases significantly during the agreement period and meets the trigger conditions, the equity delivery shall be carried out in advance; otherwise, the delivery will be carried out at the end of the agreement period. If the agreement period is extended, it can wait for the achievement of the performance goal of the startup for a longer time.The general structure and execution sequence of the VAM agreement is shown in πt follows the stochastic process described by geometric brownian motion,Uncertainty about development prospects is a significant feature of startups. Both sides of investment and financing are faced with the uncertainty of future profit levels when signing the VAM agreement. Considering the impact of a series of uncertain factors such as technology and market demand on the operating performance of a startup, it is assumed that the profit flow μ is the drift term, which represents the expected growth rate of enterprise profits; σ is the volatility item, which indicates the risk level of enterprise profit; dWt is the increment of the Wiener process. It is assumed that the expected growth rate of corporate profits is mainly determined by the entrepreneurial effort Where πt at time t, then the conditional expectation of profit at time s is R. Thus, the project value is the expected present value of the future profit flow, and the project value Vt at time t is:The uncertainty of profit also brings the uncertainty of the startup’s value. Given the profit According to Ito’s lemma,Vt follows the same geometric brownian motion.Therefore, the evolution of project value The assumption of startups’ profit and value process in this paper is the same as Hsu’s (2010) , and geoVAM agreements give venture capital institutions the contractual right to adjust valuations so that when the deal ends, the venture capital institutions have two kinds of payoff outcomes based on whether the pre-agreed performance targets are met. One scenario is when performance targets are achieved, where venture capital institutions need to transfer part of their stake to the financier, but the startup has a higher valuation. Another scenario is when performance targets are not met, and equity compensation can be obtained from the financier, but the startup’s valuation declines. Given the structure of the VAM agreement and preset performance target, at the beginning of the agreement, the option value held by the venture capital institutions is the present value of the contingent payoff at the end of the agreement period. Therefore, we first derive the option value held by the venture capital institutions under the VAM agreement and put forward the following proposition, which is the basis for the decision-making of the venture capital institutions.Proposition 1: The option value of the VAM agreement is,Where,The proof is as follows:T), the value of a startup (VT) is π* during the agreement period, the payoff of the venture capital institutions at the end of agreement period is (s+δ1)VT, if the profit of the startup reaches π* during the agreement period, the payoff of the venture capital institutions at the end of agreement period is (s−δ2)VT.At the end of the VAM agreement 2dwI1=e12The option value of the VAM agreement is:Venture capital institutions maximize the value of agreements they hold by negotiating or dominating the core terms of performance targets. Therefore, we further examine the impact of performance targets on the agreement’s value and propose the following proposition.Proposition 2: The higher the performance target, the greater the option value held by the venture capital institutions.The proof is as follows:π1*>π2*,Under different performance targets, we compare the payoff of venture capital institutions and suppose π* = π1*, the payoff of the venture capital institutions at the end of the agreement period can be expressed as:When π* = π2*, the payoff of the venture capital institutions at the end of the agreement period can be expressed as:When Thus,Φ≥Φ.According to the nature of expectations, Inequality is strictly true in consideration of the following equation,And Proposition 2 proves that under the contract structure of the VAM agreement, the higher the performance target set by the venture capital institutions, the greater the option value of the agreement. An intuitive reason is that venture capital institutions benefit from rising share values when the performance target is met and benefit from compensation when performance declines, which locks in the risk to a certain extent. Therefore, venture capital institutions tend to set high-performance targets if they only consider the option value of the VAM agreement and do not consider whether the performance can be achieved.Based on the analytic solution of the option value of the VAM agreement obtained from Proposition 1, the effect of performance target and entrepreneurial effort on the option value of the VAM agreement are numerically illustrated in π* is taken as the optimization goal of entrepreneurs. After the agreement is reached, the entrepreneur chooses the effort degree μ to minimize the expected time to achieve the performance goal π* in the agreement period. Under this assumption, the following propositions are proved:As financial investors, venture capital institutions only focus on the monetary return and are value maximizers. Different from venture capital institutions, entrepreneurs do not simply regard venture projects as ordinary investments. Besides paying attention to financial returns, they also emphasize corporate control. Hsu (2010)  argues tProposition 3: Once the VAM agreement concludes, the entrepreneur chooses the highest level of effort.The proof is as follows:By Ito’s lemma,t obeys brownian motion with drift rate σ.The result of Tπ* be the time when the profit level reaches the performance target π*, lnπ0, the probability density function of Tπ* is,Let Tπ* is,And the Laplace transformation of π* is,It can be inferred that the expected time for the profit level to reach the performance target π*, entrepreneurs choose effort μ to minimize the expected time to reach the preset target π*, and differentiates For a given For entrepreneurs, the expected time to achieve the performance target is a monotonic decreasing function of the level of effort. Therefore, after the VAM agreement concludes, entrepreneurs will choose the highest level of effort.Based on the analytic solution of the expected time for entrepreneurs to achieve the performance target obtained in proposition 3, the effect of the performance target and entrepreneurial effort on the expected time to complete the target is numerically illustrated in It can be seen that the higher the performance goal setting, the longer the expected time to achieve the goal, while the higher the degree of effort of entrepreneurs, the shorter the expected time to complete the performance goal. Proposition 3 shows the VAM agreement can effectively encourage entrepreneurs to make the best efforts to maximize the interests of venture capital institutions.π* cannot be increased indefinitely. Considering the participation constraints of entrepreneurs, that is after the entrepreneurs do their best π* can not be greater than the agreement period, and thus π* has an optimal upper limit, which is given by proposition 4.Considering only the option value of the VAM agreement, it seems that venture capital institutions should set high-performance targets. But it is evident that performance targets Proposition 4:Assuming that venture capital institutions maximize their economic value with the participation constraints of entrepreneurs, the optimal performance target is set as,The proof is as follows:π* is given, it is known that the entrepreneur will choose the highest level of effort From proposition 3, once the performance target To meet the participation constraints of entrepreneurs, that is, to achieve the performance goals in the agreement period, there is Faced with the uncertainty of the business environment, rational entrepreneurs can only accept the performance goals that can be expected to be achieved within the agreement period. Therefore, to meet the participation constraints of entrepreneurs, venture capital institutions will set high-performance targets as far as possible to obtain maximum payoff under the contractual structure of the VAM agreements.T and the risk level σ as follows,Differentiates the agreement period It can be found that the optimal performance target setting should be an increasing function of the duration of the agreement, which means that if the period of the agreement is short, the performance target requirements should be reduced. On the other hand, if the performance target setting is high, the duration of the contract should be appropriately extended to expect the enterprise to operate rationally. In addition, the optimal performance target setting should be a decreasing function of the enterprise risk level, which means that the performance target requirements should be reduced in the face of high business risk.This paper extracts the general structure of the VAM agreement through a typical case analysis. It gives the optimal pre-determined profit target, contingent on which the venture capital institutions exercise a contractual right to adjust the startup’s valuation. The findings provide essential insight and principles for setting performance goals in VAM agreements. For example, venture capital institutions firms should not over-exploit negotiation advantages to set excessively high-performance targets so that entrepreneurs have reasonable expectations for achieving performance goals with the best efforts. Moreover, under the assumption of a continuous stochastic process of project value, we can obtain an accurate mathematical expression of the optimal performance target. However, the specific performance target clauses of VAM agreements vary widely. In addition to the most commonly used financial indicators, such as profits, revenue, etc., some non-financial indicators, such as the acquisition of patents, the retention rate of core scientific and technological personnel, and the industrialization of new technologies, will also be used as ‘bet-on’ targets. In this case, the principles we advocate for setting optimal performance goals may still apply. Still, the uncertain portrayal of non-financial indicators will be more complex, such as a random jump process, making it difficult to obtain a concise mathematical expression of the optimal performance target. In future research, it is an important research direction to adopt a more comprehensive approach to study the performance target setting of non-financial indicators in VAM agreements to more fully reflect practical operations.The VAM agreement is a prevalent investment tool in emerging markets with the agreement upon fulfilling a particular performance target, contingent on which the investor may exercise a contractual right to adjust the valuation by altering the relative shareholding or other financial positions. Motivated by exploring reasonable performance targets to improve the effectiveness of the VAM agreement and achieve a win-win situation for both investment and financing parties, this paper extracts the general framework of the VAM agreement through typical case studies and then exploits the real options method first to derive the impact of the performance target on the option value held by venture capital institutions, and then analyze the behavior choices and participation constraints of entrepreneurs under the VAM agreement. Assuming that the venture capital institutions maximize the value of the agreement they hold under the condition of the participation of entrepreneurs, that is, entrepreneurs have reasonable expectations for the achievement of performance goals, the paper derives the mathematical formula of optimal performance target. From the perspective of venture capital institutions, we derive the option value of the VAM agreement and prove that under the structure of the VAM agreement, the payoff of the venture capital institutions is the increased function of the preset performance target. This may explain that venture capital institutions tend to set higher performance targets from the perspective of real options, especially when they have a strong negotiating advantage. From a startup’s perspective, we derive the expected time to achieve performance goals under uncertain conditions and prove that the expected time is a subtractive function of the entrepreneur’s effort and an increased function of the performance target. This result confirms the incentive effect of the VAM agreements on entrepreneurs, i.e., entrepreneurs will make the most effort to achieve their performance goals. More importantly, this result provides a constraint to the search for rational performance goal setting, i.e., the expected time to complete the performance goal with the best effort of the entrepreneur should be at least within the agreement period. Thus, we derive an optimal performance goal setting, which is the increasing function of the agreement period and the subtraction function of project risk. To achieve a win-win situation, venture capital institutions should fully consider the participation constraints of entrepreneurs and the internal and external risks of the projects other than set excessively high-performance targets, which may lead to irrational speculation and reduce the effectiveness of VAM agreements.S1 Fig(TIF)Click here for additional data file.S2 Fig(TIF)Click here for additional data file.S3 Fig(TIF)Click here for additional data file.S4 Fig(TIF)Click here for additional data file.S5 Fig(TIF)Click here for additional data file.S1 Text(DOCX)Click here for additional data file.


Laser-driven metasails can enable reaching velocities far beyond the chemically propelled spacecrafts, which accounts for precise engineering of the acceleration and the stability degree of the lightsail across the Doppler-broadened band. All-dielectric metasurfaces have shown great promise toward the realization of low-weight photonic platforms suitable for integrating multiple functionalities. The most paramount factor in the stability analysis of lightsail is the coupling between displacement and rotation, which mainly determines the durability of the nanocraft against displacement and rotation offsets. In this work, the marginal stability conditions of laser-propelled lightsails have been extended by replacing the reflective elements near the edges portions of the sail with broad-band transmissive elements and applying a multi-objective genetic algorithm (GA) optimization to the proposed configuration. The presented design not only remarkably suppresses the amplitude of the oscillatory motion but also can decrease the center of the mass requirement of the lightsail while maintaining an acceptable acceleration time. Next, a configuration where the payload is at the non-illuminating side of the dual-portion sail is proposed to protect the payload from the intense laser beam. In this case, a spherical phase profile is imprinted across the reflective elements while it is being propelled by a multi-modal beam. More precisely, Breakthrough Starshot is an initiative aiming to send a swarm of nanoscale lightsails to probe Proxima Centauri B, an exoplanet orbiting in the habitable zone of the closest star system, by a flyby mission using an Earth-based high-power laser arrays10. Originally, optical forces have been employed for various applications in the area of microscopic nanoparticle manipulation,trapping and acceleration and thin membrane pressurizing16. The promising utilization of optical forces for mechanical manipulation of nanoscale objects has sparked a great interest in harnessing the radiation pressure of light for accelerating large-scale and lightweight spacecrafts namely, solarsails and lightsails19. Lightsailing technology is an emerging area that encounters different branches of science and engineering for designing nanocrafts for extended mission applications. Although the realization of such a mission does not violate any physical laws, it demands considerable improvements in the area of sail, Starchip, and laser array designing. Moreover, one of the crucial requirements for successful photonic propulsion is an ultimate design of lightsail to accomplish a stable beam-riding while minimizing acceleration time along with proper thermal management via radiative cooling21. Successful beam-riding requires the generation of sufficient restoring force and torques against the possible displacement and tilt of the beam with respect to the center of the sail28. Previously, it has been well understood that flat macroscopic structures are unequivocally unstable and any slight misalignment or displacement will cause the sail to deviate away from the center of the beam immediately. In order to introduce a self-stabilizing mechanism in the spacecraft, spherical, conical, and rigid parabolic structures were studied, and it has been proven that under the specific configurations, conical and parabolic structures can maintain in the beam area with oscillatory motions during the acceleration period23. However, challenges related to the material selection and conserving their initial rigid structure have made their realization very demanding. Recently photonic metasurfaces have shown a great promise toward realization of relativistic lightweight nanocrafts thanks to their unparalleled domination over the magnitude and direction of the reflected and transmitted beam’s wavefront30. Moreover, photonic metasurfaces enable anomalous reflection and transmission of the incident light by creating spatially variant phase discontinuity over the aperture of the sail, giving rise to prices government over the in-plane forces which is of paramount importance for designing self-stabilizing metasails. Reaching relativistic velocities calls for a gram-scale total mass of the spacecraft while retaining maximal control over the mechanical performance of the sail, which can be achieved by all-dielectric photonic metasails due to their unique properties such as ultra-thin and low-density structure. The flat all-dielectric photonic metasail can imitate the optical performance of an ideal macroscopic structure while satisfying the mass and thermal management requirements of the relativistic spacecraft, which are keys to a successful launch of lightsails. Nevertheless, all-dielectric photonic platforms hold great promise in bringing advanced multifunctional performances for extended mission applications that are not viable with other platforms. Recently there has been experimental demonstration of the beam-riding and material characterization of the lightsails which can pave the way toward realization of relativistic lightsails34. In this work, we have proposed a novel structure for extending the marginal stability basins and reducing the center of mass distance requirements of the metasail by applying a multi-objective optimization to the dual-pattern photonic metasail for the different center of mass distances. The exerted optical forces and torques are derived in this scenario, followed by a motion trajectory study to highlight the remarkable effect of the implemented technique on increasing the stability degree of the sail while minimizing the reduction in thrust as a result of substituting the reflective elements with transmissive ones. Moreover, we have investigated the possibility of deploying the Starchip at the back (non-illuminating side) of the lightsail and the required propulsion beam shape to ensure stable beam-riding performance. It should be emphasized that this type of arrangement of the sail brings several advantages including protection of the Starchip against the intense laser beam and the heat caused by it and also providing a new degree of freedom in designing dynamic and time-varying beams that can pave the way for realization of such nanoscale spacecrafts which can be further studied. As shown in Fig. Using the momentum of light, a propellant has initiated a new generation of gram-scale nanocrafts that can reach relativistic speeds within a few minutes of propulsionThe rest of the manuscript is organized as follows. In “37. Furthermore, the first step in designing such relativistic nanocrafts is to properly outline the stable beam-riding requirements by studying the propulsion stage. According to the atmospheric transparency window choosing the ground-based laser wavelength as 1.3 3. The proposed designs will consist of two different unitcells, reflective and transmissive, with high overall spatial reflectivity and transmissivity in the Doppler-broadened propulsion spectrum, respectively. Nevertheless, both unitcells are designed to provide a wide phase span throughout the Doppler-broadened propulsion band. Based on the application, all-dielectric metasurfaces can operate both in reflective and transmissive regimes by offering a broad phase accumulation43. Moreover, sub-wavelength dielectric resonators enable the excitation of electric dipole (ED) and magnetic dipole (MD) resonances which can operate in the reflection with high efficiencies38. However, operating in either electric dipole (ED) or magnetic dipole (MD) does not provide a wide phase span for broadband spectrum applications. Among the available methods for rendering broadband reflecting metasurfaces, adopting a design based on the combination of guided-mode resonance (GMR) and magnetic dipole resonance (MD) seems to be ideal, which is enabled by adding a subwavelength dielectric layer leading to expansion of the reflection bandwidth and phase span46. As outlined in the previous section of the manuscript, in addition to the broadband reflective metasurface, a broadband transmissive metasurface in the Doppler-broadened spectrum is also deemed. Generally, transmissive metasurfaces can be achieved by spectrally overlapping the electric dipole (ED) and the magnetic dipole (MD) resonances of the all-dielectric nanoresonators to satisfy the Huygens’ condition. However, these kinds of designs suffer from low efficiency due to interelement field coupling, which makes them unsuitable for complex beam steering applications47. Using high aspect ratio nanopillars is shown to provide a viable solution toward realizing broadband all-dielectric metasurfaces with high efficiencies50. It should be noted that all of the simulations are done using MATLAB and the schematics are generated using Autodesk 3ds Max52. The tall nanodisks with high heights All-dielectric metasurfaces are great candidates for accommodating the laser propulsion requirements of the relativistic nanocrafts owing to their complex wavefront control ability and low-weight structure3. In order to design a polarization-insensitive reflective metasurface, we have selected c-Si nanodisks over ultra-thin c-Si and silica layers. Our simulation has been conducted within frameworks of the rigorous coupled wave analysis (RCWA) by illuminating the metasurface by plane waves with perpendicular (s-) and parallel (p-) polarization. Also, the experimentally measured refractive indices of c-Si and silica in the near-infrared are used53. It is envisioned that the temperature escalation of the sail as a result of a high-power incident beam may alter the optical properties and thermal emissivity of the utilized materials, which should be taken into account however, experimental data for such characterization does not exist. The two different goals of having high reflectivity and maximum phase span call for adopting a multi-objective genetic algorithm (GA) optimization to attain a near-optimal trade-off between the desired performance metrics. Next, the optimized dimensions of the reflective metasurface are the following: 54. Figure 47. In order to gain a clear picture of the obtained resonances, we have plotted the nearfield distribution of the magnetic and electric nearfields in the vicinity of the MD, GMR, and the hybrid Mie–Fabry–Perot resonances in the 6. It should be noted that due to the rotational symmetry of the designed reflective and transmissive building blocks, similar results can be obtained for the p-polarized normal incident beam. Another important factor in photonic metasail design is the angular sensitivity of the unitcell’s response, as the sail will experience oscillatory motion resulting in an oblique incidence of the propulsion beam. Further information on the oblique incidence’s effect on the building blocks’ response and, consequently, the performance of the lightsail is brought in “6. It should be noted that the absence of vibrational modes of silicon in the mid-infrared range makes its emissivity insufficient for the thermal survival of the nanocraft in the propulsion stage. Hence, adding a thin silica layer with high emissivity in the same spectrum (unlike silicon) can help reduce the heterostructure’s steady-state temperature during the acceleration stage1. It has been shown that by the addition of the silica layer, the maximal steady-state temperature of the proposed relativistic platform can be kept under its melting point6.Among the limited number of allowed materials to be used in the lightsail, we have chosen the Si-silica platform due to its prosperous properties such as ultra-low mass density and thermal durability against extreme temperaturesentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaTo account for imperfections in the fabrication process and their adverse effect on the response of the proposed all-dielectric building blocks, we have performed a tolerance analysis in section In this section, we briefly delineate the analysis platform that is used to evaluate the optomechanical functionality of the proposed nanocraft throughout the Doppler-broadened propulsion band. Further details are provided in the section 55. It should be noted that the large-scale structure of the lightsail consists of trillions of nanoscale building blocks which prohibits employing full-wave simulations; hence we embrace a simplified method centered on local periodicity assumption8. In this method, the reflection amplitude and phase of the periodic structure of unitcells are obtained using the rigorous coupled wave analysis (RCWA), and then the scattering performance of the macroscopic configuration of the nanoscale elements is predicted by the Generalized Snell’s law, which dwindles the computational complexities considerably56. It should be noted that the violation of the local periodicity, mainly for corner elements, and structural imperfections in the large-scale pattern of unitcells will result in spurious and parasitic scattering responses. However, considering their small number with respect to the total number of unitcells, the method is believed to offer sufficiently accurate results for unveiling the essential physics of the problem59. Assuming the sail, payload, and connecting booms as a rigid body, the dynamics of the nanocraft can be described by Newton Euler’s equations. Figure 60:61. It should be remarked that the imparted forces and torques possess a nontrivial dependency on the position and rotation of the incident laser beam, which is taken into account by adding an offset and altering the beam’s incident angle. The calculated parameters in the local coordinate system are transformed into the global coordinate system via a set of coordinate transformations to numerically solve the nonlinear system of equations using the Runge-Kutta method. A deeper insight into the stability conditions of the spacecraft can be achieved by projecting the dynamics of the sail to the transverse plane , which renders a 4D oscillator matrix, and obtaining its eigenvalues as62:2. In the next sections, two different configurations for the sail’s propulsion scenario are presented based on the imprinted phase gradient and incident laser beam profile. It should be noted that a full-wave RCWA simulation of reflective and transmissive supercells consisting of twelve unitcells for verifying the broad-band functionality and adopted optomechanical method is conducted in the section The centerpiece of the analysis method is calculating the imparted optical forces to the sail’s surface under the illumination of an unpolarized high-power laser beam using Maxwell’s stress tensor method (MST)8. Nevertheless, minimizing the amplitude of the sail’s oscillatory motion is crucial for preventing the sail from losing its path en route to its destination. Generally, metasails with parabolic or conical phase profiles under the incidence of a single Gaussian beam satisfy the marginal stability condition as this kind of arrangement yields In this section, we study a 68. In order to achieve a near-optimal trade-off between the stability and thrust, an evolutionary multi-objective optimization based on genetic algorithm (GA) namely controlled elitist non-dominated sorting genetic algorithm (CE-NSGA) which is implemented in MATLAB’s Global Optimization Toolbox is adopted70. Moreover, by choosing a simple Gaussian beam with a spot size of 8. In addition, Fig. Furthermore, acceleration time is of crucial importance in realization of the relativistic spacecrafts as it has a direct impact on the required power and size of the ground laser arrays to keep the beam diffraction-limited using the adaptive optics methodIn order to render the actual nonlinear system stable, one should conduct the motion trajectory study over the propulsion timescale to analyze the effects of relative ratios of displacement and rotation. Therefore, it is crucial to assess the mechanical performance of the system under initial displacement and rotation to estimate the acceleration, velocity, and amplitude of residual motion. As the sail accelerates, the major effect of the dispersion of the metasail building blocks, which yields fluctuating amplitude of displacement and rotation, can be unfolded by driving the in-plane forces  optimization is applied to obtain a near-optimal trade-off between the stability degree and net thrust of the lightsail. The optimized structure is not only able to effectively damp the oscillatory motion but also can decrease the required minimum center of mass distance, which is of great importance toward the realization of gram-scale nanocrafts. Next, we have proposed a novel configuration of the spacecraft where the central and outer building blocks have reflective and transmissive phase gradients, respectively, in which the payload is placed at the back-side of the lightsail to protect it from the high-power propulsion laser beam, which can severely damage the Starchip. In this case, a simple uni-modal Gaussian beam is not able to enable beam-riding stability. Instead, a multi-modal Gaussian beam, with its maximums at the corners of the sail, accelerates the lightsail. In this case, the calculated imparted forces and torques followed by a motion trajectory study verify the broadband functionality and its effect on extending the stability degree of freedom of nanocrafts.Supplementary Information.