Radiation dermatitis (RD) is one of the most common side effects of radiation therapy. However, to date, there is a lack of both specific treatments for RD and validated experimental animal models with the use of various sources of ionizing radiation (IR) applied in clinical practice. The aim of this study was to develop and validate a model of acute RD induced using proton radiation in mice. Acute RD (Grade 2–4) was obtained with doses of 30, 40, and 50 Gy, either with or without depilation. The developed model of RD was characterized by typical histological changes in the skin after irradiation. Moreover, the depilation contributed to a skin histology alteration of the irradiated mice. The assessment of animal vital signs indicated that there was no effect of proton irradiation on the well-being or general condition of the animals. This model can be used to develop effective therapeutic agents and study the pathogenesis of radiation-induced skin toxicity, including that caused by proton irradiation. Cancer treatment is currently mostly based on surgery, chemotherapy, and radiation therapy ,2,3. ForDuring the course of conventional radiotherapy, 95% of patients develop some grade of RD, which is a toxic skin reaction to ionizing radiation (IR) ,12. A siToday, there are no specific treatments for RD. Therefore, a crucial task for modern biomedicine is the development of agents to prevent and treat radiation burns ,16. To aThe aim of this study was to develop an RD model on the dorsal side of the mouse’s body after a single local exposure to proton radiation using focused delivery at the Bragg peak with an energy of 88 MeV. Location of the energy release peak of the proton beam is directly in the skin of the animal in order to reduce radiation damage to the underlying tissues;Model reproducibility of more than 90%;Absence of a long latent period (up to 2 weeks);Pronounced staging of RD formation;100% survival of animals within 30 days of exposure.A series of experiments were performed to select the optimal conditions for a mouse model of proton-radiation-induced RD. When developing the experimental model of RD, the following parameters of the beam and experimental conditions were taken into consideration: absorbed dose, position of the energy release peak of the monoenergetic proton beam, positioning and fixation of animals, method of depilation or its absence, duration of latent period and stages of RD, model reproducibility , assessment of vital signs ), and animal survival. Experimental groups of animals were divided into subgroups: nondepilated ; depilated via a physical method (trimmer) , and depilated via a chemical method (cream) . As a result of the experiments, a model of proton-induced RD in mice was developed that meets the following requirements:We think that the developed model of radiation-induced skin damage will help to expand the knowledge about the pathogenesis of proton-radiation-induced RD and can be used to evaluate the effectiveness of preventive and therapeutic agents.When modeling skin wounds in animals, the hair cover is usually removed. To date, there are no detailed unified criteria for choosing the method of animal depilation, particularly in RD modeling. In this regard, we investigated the effects of physical (trimmer) (CD) and chemical (depilatory cream) (CChD) hair removal methods on the skin of animals that had not been exposed to proton radiation . The skiAs can be seen in Thus, we think that the physical method of depilation should be preferred in RD modeling, taking into account the absence of initial skin reactions in animals, potential contribution of depilation to the severity of the wound, and the rate of skin recovery in irradiated mice. The effect of depilation method on the severity of the RD grade and healing time of radiation-induced skin damage in experimental animals are considered in At this stage of the work, the optimal proton beam energy was selected to create a mouse model of RD. The minimum energy of the proton beam released from the accelerator in the Prometheus proton therapy complex is 30 MeV. At this energy, the depth of particle penetration into soft tissues is less than 10 mm, while the width of the energy release peak is less than 0.5 mm, which causes significant difficulties both in dose estimation and irradiation. At this energy, a 0.2 mm error in animal positioning or beam approach can result in a dose delivery error of 50–80%, which is unacceptable for this type of experiment. To avoid these inaccuracies, a Plexiglas phantom filled with water was placed in the beam path. The phantom was 60 mm thick, which, accounting for the Plexiglas, corresponded to a water equivalent thickness (WET) of 61.8 mm. The position of the peak was checked using a parallel plate chamber, taking into account the thickness of the outer wall of the chamber placed behind the water phantom, where the animals were subsequently fixed. The peak position was checked in the energy range of 87 MeV to 92 MeV in 0.1 MeV increments. Calculations performed in the planning program  that is a part of the Prometheus complex showed that the energy required to pass this phantom and stop the maximum energy release on the skin of the animal was about 88 MeV .Consequently, at energy of 88 MeV, the peak width at 90% of the dose level was 1.2 mm, which allowed us to plan and perform irradiation with a skin dose accuracy of at least 10%. The value of linear energy transfer (LET) for proton radiation was approximately 5–10 keV/μm. Since the transverse size of the beam was about 8 mm when it was released from the accelerator and an irradiation field of 10 × 10 mm was required when working with animals, plans for uniform irradiation via proton beam scanning over a given area were created in Prometheus complex software . Taking into account the selected optimal values of energy and irradiation field, a dosimetric study involving a biological object (mouse) was carried out. The experimental animal was euthanized via cervical spine dislocation. After that, a skin flap was cut off from the animal’s back, and an EBT3 dosimetry film was placed over and under it . This prAt energy values of 88 MeV, the maximum energy release was located directly in the epidermis and dermis and did not affect underlying tissues and organs. The dosimetric assessment using the EBT film placed under the skin of the animal showed no significant difference in the dose distribution at two variations in the animal position: 45° and 90°. However, from the technical point of view, the optimal position is the 90° angle, which was used in the RD modeling in this work.At this stage of the work, the severity of the RD grade and the time of spontaneous wound healing of radiation-induced skin damage, depending on the absorbed dose and presence/absence of the hair coat as well as the method of depilation, were studied. The groups of animals were irradiated at doses of 30 Gy , 40 Gy , or 50 Gy (ChD50). It was shown that on the seventh day after a single exposure to local proton radiation at doses of 30 and 40 Gy, all animals that were not depilated  had no skin reaction (Grade 0) to proton radiation. In the D30 experimental group, 63% of mice showed no skin reactions, representing Grade 0, and 37% were Grade 1. In D40, 44% of mice showed no skin reactions to proton irradiation (Grade 0). The other animals in this group had RD Grades 1 to 3 with Grade 1 being predominant, which was noted in 31% of the animals. In the ChD50experimental group, 55% of mice showed no skin reactions to proton irradiation, and 45% of the animals had skin toxicity, represented by Grade 1. By the 14th day after exposure to proton radiation, it was found that 12.5% and 27% of the mice in ND30 and ND40, respectively, had no skin reactions. The other animals in these groups showed RD Grades 1 to 3 with a prevalence of Grade 1. In experimental groups D30 and D40, more than 50% of the mice were Grade 2, and Grade 3 was observed in 9% and 15% of the animals. The most severe skin reactions were seen in ChD50: 67% of the mice showed Grade 4 skin reactions accompanied by ulceration and necrosis, and 33% of the mice showed radiation-induced skin toxicity corresponding to Grade 3. It was found that by the 21st day after proton exposure, the animals in all experimental groups had radiation-induced skin toxicity corresponding to Grades 2–4 according to the RTOG classification. Only in experimental group ND30 did a larger proportion of animals (60%) have RD Grade 2, while in the other groups, the animals showed more severe skin wounds corresponding to Grade 3 and Grade 4. Only in D30 did 100% of the animals show radiation-induced skin toxicity corresponding to Grade 3. In experimental groups ND40 and D40, regardless of whether the hair was removed, the majority of animals  were Grade 3, while the other animals in these groups were Grade 4 of RD. In the experimental group ChD50, by the 21st day, the number of animals categorized as Grade 4 RD increased to 89%.Starting from day 21, the assessment of the animals’ skin condition was performed once a week. At this stage of RD development, visual assessment of the severity degree is difficult due to the ongoing reparative process. The duration of observation was 70 days because this time is enough to monitor the course of acute RD. At this stage of the work, we carried out visual assessment of the development of the reparative stage of RD in mice depending on the absorbed dose and method of depilation.As shown in As can be seen in The duration of observation of the reparative phase after different methods of depilation or its absence was 70 days. During this period, complete healing of radiation-induced skin damage and hair regrowth were achieved only in ND30 mice, and in other groups, reparative processes continued . It should be noted that in the reparative phase as well as in the destructive one, there was a pattern: with an increase in the absorbed dose the grade of RD and the time of spontaneous wound healing of the wound increased as well.Skin histology was assessed on the 14th day after proton exposure in the nondepilated irradiated groups , in the depilated using a physical method (trimmer) irradiated groups , and in control nonirradiated groups . Overall, the proton beam irradiation affected skin histology in a dose-dependent manner, with physical depilation producing additional substantial impairment. The epidermal and dermal thicknesses were significantly larger in depilated and irradiated mice (D30 and D40 groups). Moreover, the skin of the depilated and irradiated mice was characterized by a loss of follicular units, as well as by the presence of epithelial hyperplasia and hyperkeratosis. The epidermal thickness was significantly thicker in depilated and irradiated mice . Similarly, in the depilated and irradiated groups, the dermal thickness was significantly thicker compared to that in both control groups. The number of follicular units between the CND group and irradiated groups showed approximately two- to three-fold differences. All mice irradiated with either 30 Gy or 40 Gy showed no sign of changes in the number of follicular units compared to the depilated control mice (CD group). The scores of epithelial hyperplasia and parakeratotic hyperkeratosis showed significant differences between of the control (CND and CD) and irradiated groups. We assessed the vital signs of the experimental animals relative to intact animals according to the following criteria: 30-day survival, body weight dynamics, morphological composition of peripheral blood, and weight coefficients of internal organs .50/30) of mice, as well as changes in body weight and time of survival within 70 days after irradiation. In our work, by the 30th day, the survival in all groups of animals was 100%; moreover, all animals survived up to the end of the experiment (70 days). We conducted a study to investigate the effects of single local proton irradiation at doses ranging from 30 to 50 Gy on the 30-day survival (LDThe study of body weight change showed no significant differences between the groups of nondepilated (ND30 and ND40) and chemically depilated (ChD50) animals compared to the control group (CND and CChD) A,C. In tHematological data  of the peripheral blood of the white male SHK mice before irradiation and 7, 21, and 70 days after a single exposure to local proton radiation at doses from 30 to 50 Gy compared to the control  were studied . DespiteIn the study, we studied the changes in relative masses of spleen and thymus in % of experimental animal weights 7, 21, and 70 days after a single exposure to local proton radiation at doses from 30 to 50 Gy  compared to those of intact animals . The data are graphically presented in There were no significant changes in the relative mass of the spleen in either experimental group or intact mice throughout the experiment (70 days). No significant changes were observed in the thymus mass in any experimental group at 7 days after irradiation, while a decrease in organ mass was observed in all groups except ND30 and CD at 21 days compared to the control. In the ChD groups, this decrease remained significant up to 70 days. Thus, the reduction in the relative mass of the thymus was presumably associated with an increase in destructive processes and a decrease in the lymphoid cell population in response to proton irradiation, which was sustained until the end of the experiment in animals subjected to chemical depilation.In this study, a protocol was developed and an experiment was conducted to model proton-induced radiation dermatitis on the dorsal side of an animal for the first time. The model of local proton-induced dermatitis on the back of a mouse was obtained through a single exposure of anesthetized animals fixed perpendicular (90°) to a proton beam with an energy of 88 MeV. The absorbed dose ranged from 30 to 50 Gy, and the linear energy transfer (LET) value was approximately 5–10 keV/μm. The recommended irradiation field size is 10 × 10 mm. The developed radiation dermatitis model has several advantages for its in vivo application. Firstly, the single exposure enables the avoidance of subjecting the animal to multiple anesthesia procedures, thus minimizing the stress induced by transportation and repeated immobilization, thereby adhering to the requirements set forth in Directive 2010/63/EU of the European Parliament and Council of the European Union on the protection of animals used for scientific purposes. Secondly, the local administration directly onto the animal’s skin, facilitated by the presence of the Bragg peak, prevents damage to the spinal column and adjacent neural and muscular tissue. Thirdly, localizing the target area of radiation exposure at the center of the animal’s back allows the minimization of further animal interference in the process of applying the therapeutic agent and the wound healing stage.When studying in vivo model wounds, it is important to minimize the contribution of depilation to skin damage. As previously noted, there are no standardized methods for animal hair depilation when modeling skin damage, specifically RD. Furthermore, there is a lack of data on the necessity of removing or preserving the hair coat of animals before modeling RD.There are limited data on the suitability of using depilatory products on animals, but some studies have shown that both physical and chemical methods of depilation do not lead to an increase in bacterial load and prolonged wound healing in rat and mouse models of skin damage ,27. In aPhysical methods of hair removal also have their drawbacks, as any hair removal method is associated with potential operator error, leading to injuries, general discomfort, stress, and poor well-being of the animals. However, a study  demonstrIn this study, we conducted an investigation into the effects of physical and chemical hair depilation methods on the skin of intact and experimental animals exposed to proton radiation. Based on the obtained results, we suggest that the physical method of depilation is preferable compared with the chemical method, as it minimizes skin damage prior to irradiation. We concluded that since the choice of depilation method in RD modeling should aim to minimize the contribution of depilation to the formation of radiation-induced skin damage, it is advisable to avoid hair removal altogether. This is supported by the observation that nondepilated animals exhibited alopecia on day 14, which may serve as an additional criterion for the development of RD.The impact of local proton irradiation on animal skin has been examined in a limited number of studies, which makes this issue particularly relevant to modern radiation biomedicine. Additionally, no studies have been found on modeling the RD caused by proton irradiation on the back. However, in recent decades, the effects of other types of ionizing radiation on the skin have been fairly well described. For example, it was demonstrated that when BALB/c mice were irradiated with a dose of 36 Gy for three consecutive days (12 Gy/day) using a source of X-ray radiation, the peak of skin damage was observed 7–10 days after irradiation . In anotAs a result of the series of experiments, it was demonstrated that a single dose of proton beam irradiation from 30 Gy to 50 Gy led to radiation-induced skin toxicity in 100% of experimental animals. The duration of the latent period varied from 7 to 11 days depending on the absorbed dose. The duration of the destructive stage in all groups was 21 ± 3 days. In the course of development of radiation-induced skin damage, pronounced staging of RD formation  was observed in the mice. The severity of the RD grade differed among the groups, and the homogeneity of RD development in each particular group ranged from 60% to 100%. The highest grade of RD in the most animals  was noted in group ChD50, in which the animals were chemically depilated and irradiated at a dose of 50 Gy. In group ND30, radiation-induced skin damage was, on the contrary, the least and corresponded to Grade 2 in 60% of the animals. It should be noted that as the absorbed dose increased, the severity of RD increased from Grade 2 to 4. In addition, the presence of a hair coat in the animals contributed to a reduction in RD grade severity, so the development of alopecia may serve as another criterion for RD evaluation. In turn, depilation increased the severity of RD. Thus, our obtained results are consistent with the literature data ,20,32,33Skin changes in response to IR have been well studied via histological examination. Several studies have shown that exposure to IR leads to increases in the thicknesses of the epidermis and dermis; decreases in the number of blood vessels, sebaceous glands, and hair follicle density; as well as epidermal hyperplasia and parakeratosis ,35,36. I56Fe on the body weight of C57BL/6 mice at 3, 40, and 112 days after exposure. However, in other studies [56Fe at doses of 1–4 Gy. Additionally, when rats were irradiated with protons at doses of 7.2, 8.5, 9.2, and 11.2 Gy and X-ray at doses of 5.6, 6.3, 7.3, and 8.6 Gy to the abdominal area, there was no observed change in the body weight of the animals [To obtain reliable experimental results in animal studies, it is important to consider the animal welfare indicators that can influence the experiment outcomes . One of  studies ,43, a de animals . In our 50/30) is the most widely used and well-characterized criterion for assessing acute and chronic radiation damage [The study of the effects of radiation on animal survival after exposure for 30 days . The analysis of the relative mass of the thymus revealed a significant decrease in organ mass at 21 days in all groups, except for ND30 and CD. Moreover, in the groups subjected to chemical depilation, this decrease remained significant until 70 days. We propose that the decrease in relative thymus mass is associated with enhanced destructive processes and a reduction in cellular lymphoid population in response to proton irradiation, which was maintained in animals subjected to chemical depilation until the end of the experiment. From our study, it is evident that the thymus, unlike the spleen, responds to the effects of localized proton radiation; thus, studying the changes in relative thymus mass can serve as a valuable diagnostic marker.Mice were the object of the study, since it has been shown that skin toxicity in humans and mice is similar in nature and appearance . White oThe experiments were conducted in compliance with the ethical standards for work with laboratory animals according to the protocol approved by the ITEB RAS Commission on Bioethics and Biological Safety . All research involving animals in the ITEB RAS was carried out according to Directive 2010/63/EU of the European Parliament and Council of the European Union on the protection of animals used for scientific purposes, according to which the presence of painful procedures, anesthesia methods, the maximum expected severity of procedures, and the fate of the animals after the experiment were determined.Hair depilation was performed on the dorsal side of the animals one day before irradiation using one of two methods: chemical depilation using a depilatory cream  or physical depilation using a pet hair trimmer . The surface area was 15 × 15 mm. In the groups of animals subjected to chemical depilation, the hair on the dorsal side was evenly coated with depilatory cream. The duration of exposure to the cream was about 180 s. At the end of the exposure time, the hair was removed with a spatula together with the cream. After that, the skin was washed with warm distilled water (37 °C). In the group of animals that had their hair removed via shearing, the hair was shaved with a pet trimmer the day before irradiation without additional manipulation. The day after hair removal, the skin condition was captured using a camera .The mice were anesthetized with a combination of Zoletil 100  and Xyla , which does not affect animal vital signs and is relatively safe for minor manipulations with animal biomodels ,67,68. IAnimals were irradiated at the “Physical-technical center” branch of the P.N. Lebedev Physical Institute of the Russian Academy of Science (PTC LPI RAS)  in the Prometheus proton therapy complex, which is a Russian-made high-technology medical complex for the treatment of tumors using the remote radiotherapy technique. This complex is based on a compact proton synchrotron with a beam energy of 30 to 330 MeV .® 10000XL  flatbed photofilm scanner was used to digitize the EBT3 radiochromic film.Experimental verification of the actual peak position and absorbed dose estimation were performed using Gafchromic EBT3 film  and PTW dosimetry equipment complex : a Unidose webline electrometer with PinPoint 3D Chamber TM31022 and BraggPeak Chamber TM34073 ionization chambers. Measurements were carried out with an allowance for the distance to the sensitive volume and the recombination of charged particles outside the chamber, since the measurements were performed in air without a buildup cap. An EpsonClinical evaluation of each animal was performed daily for 70 days after irradiation in order to record clinical manifestations of RD. Two independent, double-blinded researchers graded radiation-induced dermatitis and hyperpigmentation after irradiation according toxicity criteria to the Radiation Therapy Oncology Group (RTOG) :Grade 0: No change;Grade 1: Faint erythema, dry desquamation, epilation, decreased sweating;Grade 2: Tender or bright erythema, moderate edema, patchy moist desquamation;Grade 3: Moist desquamation in areas other than in skin folds, pitting edema;Grade 4: Ulceration, hemorrhage, necrosis;Grade 5: Death.Animals were photographed from the dorsal side once a week using a camera. The following settings were used: aperture, f 1.8, sensitivity, ISO 320; shutter speed, 1/262 s. The focusing was performed automatically. The shooting scale was 1:2. A tripod with an LED lamp was used for even lighting in a dark room without natural light. Photographs were cataloged, reviewed, and processed using ImageJ software v1.53t .Skin samples were taken on 14th day after exposure to proton radiation from both control (unexposed) and experimental animals. For light microscopy, skin samples were fixed in a 4% paraformaldehyde solution, dehydrated with gradient alcohols, and embedded in paraffin according to the standard histological protocols. The sections were stained with hematoxylin and eosin, and the images were captured with an Axiovert 200 microscope, 40× magnification . The skin sections were examined for morphology via quantification of the number of follicular units, and assessment of epidermal and dermal thickening, epithelial hyperplasia, follicular epithelial hyperplasia and parakeratotic hyperkeratosis was performed using ImageJ v1.53t . A minimum of 10 fields for each skin slide were analyzed. The thickness of epidermis and dermis was measured using scale bars. The epithelial hyperplasia and parakeratotic hyperkeratosis were scored as “0” , “1” (moderate), “2” (marked), or “3” (strong).Animals were weighed immediately before irradiation and then on the 7th and 21st days and on the day immediately before euthanasia (day 70).Blood samples were taken from the animals by a tail tip incision above the clear vein before irradiation and then on the 7th and 21st days after irradiation and on the day immediately before euthanasia (day 70). Blood samples were taken for analysis on a DH36 Vet hematology analyzer  to measure the following parameters: white blood cell, lymphocyte, granulocyte, and platelet count. In addition, animal mortality within 30 days after a single exposure to local proton radiation was assessed, and pathomorphological examination of internal organs  was performed, including isolation, morphological assessment, and weighing.t-test. Differences were considered statistically significant at p ≤ 0.05. The obtained data were processed using GraphPad Prism 8.0.1 , Microsoft Excel 2016 , and ImageJ v1.53t  software.The experimental data are presented as the mean ± standard deviation (M ± SD). The statistical significance of differences between the values in experimental groups was determined using the Mann–Whitney U test and Student’s An in vivo model of RD induced using a single local proton beam irradiation with an energy of 88 MeV was developed. Optimal conditions, including proton beam parameters, animal fixation, positioning, and depilation method, were selected during model development. The developed protocol of irradiation in the dose range of 30 to 50 Gy allowed the establishment of an RD model in mice with a short latent period (7–11 days), pronounced stages of formation, characteristic histological skin changes, and no impact on animal well-being . We think that the optimal model for evaluation of pathogenesis and testing of new preventative and therapeutic agents for proton-induced RD is Grade 2 to early Grade 3, obtained at a dose of 30 Gy in nondepilated animals. At this stage, the external manifestations of RD are more pronounced, while the toxicity only affects the epidermis and dermal elements without disrupting the integrity of the basal membrane.


This paper analyzes a two-commodity perishable queueing inventory system with random common lifetimes commodity-wise, positive service time, and individual ordering policy. We have considered two types of customers: priority customers and non-priority customers. There is a buffer of finite capacity for the priority customers and a pool with infinite capacity made available for the non-priority customers. The arrival of these customers is occurred by two independent Poisson processes. The lifetimes of each of the commodities and service times follow independent exponential distributions. The matrix geometric approach is used to analyze the model's steady-state behavior and we computed various system performance measures. This article examines the effects of the common lifetime parameters on the measures of performance, such as average cycle time, the average number of reorders due to the realization of the common lifetime of commodities, etc. Also, a sensitivity analysis of the system parameters on the performance measures is carried out. Further, a cost function is constructed based on the performance measures, and its convexity is examined numerically. Inventory control is one of the most predominant business processes leaned on by businesses, as they reconcile their undertakings coordinated with purchase, sales, and strategic marketing activities. One of the factors that affect the prevailing inventory management is the group of items stocked. The management of inventory systems with multi-commodity items poses arduous challenges compared to their single-item counterpart inventory systems, ensued by the collection of inventoried items and correlated activities.Diligent management of perishable products is imperative and calls for companies to be run judiciously. These obsolete products impose time and cost premiums to change or even worse, to be disposed of. Such major problems are rampant in the food or healthcare industry where products perish easily on the manufacturing line, in storage, or during delivery. This precipitates intricacies in the distribution of inventory as per customer demands. Besides, there are variations in demand, lifetime, or customer behavior. Smart inventory management of perishable products obviates wastage, with an augmented likelihood of distribution of goods fulfilling customer demands.In perishable inventory systems, items with finite lifetimes can be divided into two categories. In the first category, each of the items has an independent lifetime, and in the second category, items that survive a random common lifetime in inventory. In the second category, when the random common lifetime realizes all items of that commodity perish together. There are numerous applications of inventory models in this category. Food items such as fresh fruits, vegetables, meats, pharmaceuticals produced in batches, photographic materials, and even electronic things like memory chips are examples of perishable inventory items that require batch replacement when their shelf life runs out. Another example of a common lifetime inventory situation is cloud data storage. Nowadays cloud storage has become an essential part of any business operation. On cloud storage, the data can be accessed only for the period it is purchased, for example, on over-the-top (OTT) platforms, where the content they provide has a limited span in terms of relevancy and viewer desire. The main objective of this research is to analyze the effect of the random common lifetime parameters on the measures of performance, such as average cycle time and average number of reorders due to the realization of the common lifetime of commodities of a two-commodity perishable inventory system.The rest of the paper is organized as follows: Section 2Stochastic inventory system with single and multi-commodity literature is very extensive. In this section, we provide some of the important research work carried out in two commodity stochastic inventory systems and perishable inventory systems. A two commodity inventory system in which each demand is for one unit of commodity-1, one unit of commodity-2, or one unit of each of the commodities with prefixed probabilities was studied by Some of the notable research work carried out in two commodity perishable inventory literature are as follows. A two-commodity perishable inventory system at a service facility, where each item in inventory followed an exponentially distributed lifetime, was examined in 3i at the beginning of a cycle is i is exponentially distributed with rate p or exit the service area with probability Consider a two-commodity perishable queueing inventory system at a service facility. Demands coming to the system occur according to two independent Poisson processes with rates Note:•In our model, inventory reaches its maximum level of 3.1For the mathematical formulation of the model, we define the following: At time t,t, whereLet The infinitesimal generator matrix of the Markov chain of the system is of the formI denote an identity matrix of appropriate order. e is a column vector of one's with appropriate order. The notation 0 represents a zero matrix of appropriate order. With these notations, the sub matrices of Q are as follows:3.2To get the stability condition, we examine the Markov chain Let e is a column vector of 1's. Substituting the values of Φ and A in equation Then Φ satisfiesFrom the set of equations where Substituting values of Thus we get all values of 3.3Lemma 3.3.1The queueing inventory system described in Section-3 is stable if and only ifThe termsandwere defined in the previous section andeis a column vector of 1's of corresponding order.ProofThe LIQBD description of the model shows that the queueing inventory system is stable  if and o3.4x denote the steady state probability vector of the generator Q. Then x satisfies the conditionx as x are, Assuming that the stability condition in section R is the minimal non-negative solution  Expected number of type-I (non priority) customers in the system2) Expected number of type-II (priority) customers in the system3) Expected inventory level of commodity-14) Expected inventory level of commodity-25) Expected inventory level of commodity-1 immediately on realization of CLT6) Expected inventory level of commodity-2 immediately on realization of CLT7) Probability that the server is busy processing a demand of commodity-1 alone8) Probability that the server is busy processing a demand of commodity-2 alone9(a) Probability that the server is idle(b) Probability that the server is busy10) Probability that commodity-1 alone reaches its reorder level before realizing CLT11) Probability that commodity-2 alone reaches its reorder level before realizing CLT12(a) Expected reorder rate of commodity-1 before realizing CLT12(b) Total Expected reorder rate of commodity-112(c) Average number of reorders due to realization of CLT of commodity-1 in a cycle13(a) Expected reorder rate of commodity-2 before realizing CLT13(b) Expected reorder rate of commodity-213(c) Average number of reorders due to realization of CLT of commodity-2 in a cycle14(a) Expected loss rate of priority customers in the system14(b) Expected loss rate of non priority customers in the system4.1t. The rank of a customer is defined as r if that customer is the t and his rank decreases as and when the customers ahead of him leave the system. The customers join after the marked customer cannot change the rank, the transitions in In this section, we calculate the average sojourn time of a type-1 customer in the pool. We consider the Markov process Now, the sojourn time of a marked customer, who joins the queue as the Therefore the average waiting time of a general (type-I) customer in the system is4.2t. The rank N(t) of the customer is r (finite) if he is the t. His rank decreases to 1 as the customers ahead of him leave the system after completing service. The state space of the process is Here, we calculate the average sojourn time of a priority customer. In this case when at least one customer in the buffer, the server continuously serve customers from the buffer until the buffer becomes empty. To calculate the average sojourn time of a type-2 customer, we examine the Markov process b.Now, the waiting time of type-2 customer, who enters the queue as the 4.3A cycle time for commodity-1 is defined as the time until replenishment of inventory starting from the maximum inventory level. That is, duration between two consecutive The symbol The infinitesimal generator of (a) The distribution of the cycle time of commodity-1 is of phase type with representation (b)The mean cycle length of commodity-1 is (c) Probability that the inventory level reaches the reorder level (d)Probability that commodity-1 realizes common lifetime before inventory level reaches 4.4We compute the cycle time for commodity-2 in a similar way as that of the cycle time for commodity-1. A cycle time of commodity-2 is defined as the time until replenishment of inventory starting from the maximum inventory level. That is, duration between two consecutive (a) The distribution of the cycle time of commodity-2 is of phase type with representation (b) The mean cycle time of commodity-2 is (c) Probability that the inventory level reaches the reorder level (d) Probability that commodity-2 realizes common lifetime before inventory level reaches 4.5To evaluate the mean time the system stays with inventory level The infinitesimal generator of Where The elements of matrix Thus the mean sojourn time in the maximum inventory level before attainment of CLT of commodity-1 is given by 4.6We evaluate the mean time in 4.7To compute the mean number of times the inventory level reaches its maximum level Let k revisits, then 4.8t before attainment of CLT for commodity-2. Other terms are same as in section We compute the mean number of revisits to and the matrices 5In this section we provide numerical illustration of the system performance measures with various values of the underlying parameters.Influence ofon various performance measuresWe study the influence of Influence ofon various performance measuresHere we study the influence of Influence ofon various performance measuresHere we study the influence of Influence ofon various performance measuresHere we study the influence of Influence ofon various performance measuresHere we study the influence of Influence ofon various performance measuresImage 1 on various performance measures and are given in Here we study the influence of 5.1Based on the performance measures obtained, we build a cost function for checking the optimality of the control parameters i customer, Due to the intricacy of the cost function it is difficult to obtain the global optimum values of For the given fixed values of the parameters and fixed values of Influence of common lifetime parameters on the cost functionHere, we study the effect of common lifetime parameters 6In this paper, we have studied a two commodity queueing inventory system at a service facility with a common lifetime for each of the commodities, zero lead time, and positive service time, and two classes of customers. Also, there is a finite capacity buffer for priority customers and an infinite pool for non-priority customers. The matrix geometric method is used for finding the steady state probability vector. We have studied various system characteristics such as the expected number of customers, the expected loss rate of customers, the expected waiting time of customers, expected inventory levels, expected reorder rates for commodities, mean sojourn time at the maximum inventory level before attainment of the common lifetime, mean number of revisits to the maximum inventory level before attainment of the common lifetime, and analyzed the mean cycle time for both commodities. We have presented numerical illustrations to show the effect of various parameters on performance measures. A cost function is constructed and analyzed numerically to find the optimum values of the control parameters S. Dissa: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. P.V. Ushakumari: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.The authors declare no conflict of interest.


To improve the mechanization level of rice planting, a new type of direct seeding device for rice was designed. The device's structural properties will be crucial in determining its seeding performance. Structure optimization in the current seed metering device design process focuses on a single or few indexes, resulting in improved individual performance but imbalanced overall performance. Therefore, a structure optimization method of the direct seeding device based on a multi-index orthogonal experiment was proposed in this study.First, the DEM-MBD coupling method observed the factors and levels that affected the performance overall. Second, a test platform based on the electric drive control model was constructed, and a multi-index orthogonal test was devised. Finally, the structural parameters of the seed metering devices were optimized based on matrix analysis.From the results, the primary and secondary levels of significance of factors were just as follows: hole diameter > hole number > adjustment angle. The following are the optimal parameters found by optimization analysis: the diameter of the hole was 12 mm, the number of holes was 10, and the adjustment angle was 80°. Validation tests were carried out and analyzed based on the optimal structural parameter combination. The qualification rate of seeds per hole, empty hole rate, average seed number, coefficient of variation of seed number, average hole spacing, and the variance coefficient of hole spacing are 93.07%, 0%, 9.39,14.04%, 22.84 cm, and 9.14%, respectively.In comparison to traditional design and structural parameter optimization methods for rice precision seed metering device, this study not just to provides an optimization scheme for improving the overall performance of rice precision seed metering device, but also serves as a technical reference for the development and design of new rice precision seed metering device. IncreasThe seed-metering device is an essential component for planting . The perScholars have studied rice direct seed metering devices, but most optimized the structure based on a single or a few indexes. To conclude, the available study uses less than three indices to measure the performance of the seed metering device in the existing research . AdditioThe structural parameters of the seed metering devices will influence their working quality, which in turn will influence the emergence and yield of rice seeds . Therefo22.1The seed metering device designed in this study is shown in The sowing rate adjustment mechanism should be adjusted to the theoretical sowing rate position before the seed metering device may function. Rice seeds complete the seed-filling process by virtue of gravity, seed pressure, and the rotation of the seed metering wheel. The seeds with an irregular posture that fall into the hole are cleaned by the brush and subsequently enter the retaining area, which is point E in The seed metering device designed in this paper adopts the form of a gradual change in hole depth to realize sowing rate adjustment and adopts double tracks to realize the dynamic adjustment of the sowing rate. The adjustment method has the benefits of being easy to modify and having high accuracy. At the same time, the forced seeding mechanism effectively overcomes the issue of rice seed hole creation under card seed and multicast rates. Compared to existing rice direct seeding devices, this device has several advantages, including continuous adjustment of the sowing rate, a simple adjustment process, a more comprehensive adjustment range, and higher precision.2.22.2.11, empty hole rate y2, average seeds per hole y3, coefficient of variation of seed number y4, average hole spacing y5, and the coefficient of variation of hole spacing y6 were selected as performance evaluation indexes in this study. Moreover, the empty hole rate is the ratio of the number of holes when the seed per hole is zero to the number of experimental groups. The seed metering device must meet the agronomic requirements, which include a qualification rate of seeds per hole of ≥ 85.0%, an empty hole rate not exceeding 5.0%, a coefficient of variation of seed number of ≤ 40.0%, a coefficient of variation of hole spacing of ≤ 30.0%, and an adjustable average hole spacing of 10~25 cm. The average seeds per hole is generally 3~8, but for cold areas in the north, the number of seeds in holes needs to be 8~10. The performance evaluation index is calculated as shown in Equation (1).According to the results of related research , the quaN is the theoretical number of seed metering; n1 is the number of qualified holes for seed metering; n2 is the number of the empty hole; Ni is the seeds per hole in the i-th hole; sd is the standard deviation of the seeds per hole; si is the hole spacing in the i-th hole; Sd’ is the standard deviation of the hole spacing.Where For the hole-type rice direct seed metering device, the probability of two seeds lying flat and one seed lying sideways entering the hole is the highest when the seeds are filled. However, the increasing hole diameter resulted in a difference in seed-filling number in every layer . So, theThis study conducted single-factor experiments based on the co-simulation approach to examine the impact of the structure parameters of the seed metering device on the performance index and identify the particular value range of the structure parameters. The effect of hole diameter, number, and adjustment angle on seed separator performance was investigated, and the range of variables was selected for orthogonal testing.2.2.2The seed metering device’s single index test can only represent the main and secondary order of the structural characteristics, as well as the level’s importance . The mulOptimization of structural parameters based on matrix analysisFi at the j level is kij. When the evaluation index is large or small, Kij equals kij or 1/kij, respectively. The index evaluation matrix M is shown in Equation (2).The matrix analysis approach was utilized in this work to further evaluate and deal with the orthogonal test findings in order to thoroughly improve the structural parameters of the seed metering device under different indices. The method can solve the unreasonable configuration and selection of structural parameters of the seed metering devices under multiple indexes . Accordii equals Secondly, we defined the factor layer matrix of the structural parameters of the seed metering device. When TFi in the orthogonal test is Si. When Si equals Finally, we defined the horizontal layer matrix of structural parameters of the seed metering device. The range of factor To sum up, the weight matrix γ of defining the structural parameters of the seed metering device is the product of the above three matrices, so the weight matrix γ is shown in equation (5).The specific steps for multi-index structure parameter optimization based on matrix analysis is shown in 2.3Longken 58 rice seeds from Northeast China were selected as experimental materials. The mean values of the long axis, short axis, thickness, 1000-grain weight, moisture content, and sphericity of rice seeds were 6.98 mm, 3.29 mm, 2.35 mm, 24.84 g, 23.63%, and 54.14%, respectively. The rice seeds were washed and prepped before the experiment so that they could meet the direct planting criterion. According to the results of related research , the com33.1To ensure the accuracy of the test results and the simplicity of seeing the assessment index, the Discrete element method (DEM) and multi-body dynamics (MBD) software were used to build a simulation platform in this paper . The cou3.1.1The design processes are as follows, based on the DEM-MBD coupling simulation test: (1) the number of fixed holes is determined to be 12, the adjustment angle is determined to be 80, and the hole diameters are changed to 10, 12, 14 and 16 mm, respectively, and the simulation analysis were carried out; (2) Take the result of higher qualified rate in the first step as the typical value of analyzing the diameter of the hole. Based on typical values of fixed hole diameters and adjustment angles, the variables for the number of holes are set to 8, 10, 12, and 14, respectively. (3) The higher qualified rate and lower coefficient of variation in the first and second steps are taken as typical values for analyzing fixed adjustment angles. Based on the typical values of the fixed hole diameter and the number of holes, the experiments were carried out by changing the adjustment angles to 50, 60, 70, and 80°, respectively. According to the relevant research, results of the pre-test and rice varieties used in the test, when the diameter of the hole is 10, 12, 14, and 16, the number of seeds per hole is 5 ~ 7, 8 ~ 10, 15 ~ 17 and 22 ~ 24, respectively, which are regarded as qualified. The number of seeds per hole discharged by the seed metering device should be recorded, and every 250 holes should be divided into one group. Each group should be repeated three times, and the average value should be taken. The design of the test factor level is shown in 3.1.23 kg/m3 by Density module  may be found by sorting the formula (6).According to equation (7), when the rotational and working speeds are constant, the hole spacing is only related to the number of holes. In other words, when the number of holes is 8, 10, 12 and 14, the theoretical hole spacing is 27.78, 22.22, 18.52, and 15.87 cm, respectively.In this paper, the sowing test platform was set up as shown in 1, and when the last seed leaves, the sensor is recorded as t2. Therefore, we can make the following definition: the time T1 of the first hole is (t2+t1)/2, the time T2 of the second hole is (t4+t3)/2, and the interval time between the first point and the second hole is the difference between T2 and T1. The response speed of the sensor selected in this test is 2 μs, and the detection accuracy of the sensor meets the requirements through the pre-test. Combined with the derivation of relevant theoretical models and the pre-test results , the coefficient of variation of seed number was 27.98% (the highest), and the coefficient of variation of hole spacing was 13.71% (the highest). Therefore, the hole diameters of 12, 14, and 16 mm were selected as the orthogonal test factor levels in the following experiments. According to section 3.1.1, the hole diameter with the higher qualification rate is the typical value for subsequent single-factor analysis. The diameter of the hole was fixed at 14 mm in the subsequent single-factor test.As shown in 4.1.2When the seed metering device’s rotational speed is constant, the performance of the seed metering device can be improved by changing the number of holes. The specific influence of the number of holes on the performance of the seed metering device is as follows. From the analysis results, we can determine the factor level value of the orthogonal test, and the influence of different hole numbers on the performance of the seed metering device is shown in 1 first increases and then decreases, the coefficient of variation of seed number y4 first decreases and then increases, the coefficient of variation of hole spacing y6 first decreases and then increases, the average seeds per hole y3 and the average hole spacing y5 gradually decrease, and the empty hole rate y2 is zero. When the number of holes was 10, the qualification rate of seeds per hole was 94.11%, and the coefficient of variation of seed number was 17.95%. When the number of holes was 12, the qualification rate of seeds per hole was 88.23%, slightly lower than that when the number of holes was 10 (94.11%). However, the coefficient of variation of seed number (16.21%) is better than that when the number of holes was 10. According to section 3.1.1, the number of holes with the higher qualification rate and lower coefficient of variation of seed number was regarded as the typical value of subsequent single-factor analysis. Therefore, in the subsequent single-factor test, the number of holes was determined to be 12. When the number of holes was 8, the qualification rate of seeds per hole was 82.35% (the lowest). For precision direct seeding technology of rice, the qualification rate of seeds per hole can directly reflect whether the seed metering device meets the requirements of precision direct seeding. Therefore, the number of holes 10, 12, and 14 was selected as the orthogonal test factor level in the subsequent test.The results show that with the increase of the number of holes, the qualification rate of seeds per hole y4.1.3Different adjustment angles indicate different pre-filling seed areas. Too small pre-filling seed areas can affect the filling seeds of the seed metering device, and then affect the sowing performance of the seed metering device. The specific effects are explained as follows. The influence of different adjustment angles on the performance of the seed metering device is shown in 1 shows an upward trend, the coefficient of variation of seed number y4 and the hole spacing y6 shows a downward trend, the average seeds per hole y3 and the average hole spacing y5 show an increasing trend, and the empty hole rate y2 is zero. When the adjustment angle was 50, the qualification rate of seeds per hole (76.47%) was lower than 80%. According to the results of related research, if the qualification rate of seeds per hole is less than 80%, it will not meet the requirements of precision sowing. Therefore, the adjustment angles of 60°, 70°, and 80° were selected as the factor levels of the orthogonal test in the subsequent test.Through the analysis of 4.2According to Section 2.2, it is necessary to carry out the orthogonal test based on the single-factor test results after completing it. The orthogonal test results of structural parameter optimization of the seed metering device are shown in 4.2.12B1C3. That is, the diameter of the hole was 14 mm, the number of the hole was 10, and the adjustment angle was 80°.The range analysis of the orthogonal test results of the seed-metering device is shown in 2B1C3, A1B1C3, and A2B1C3, respectively. The average seeds per hole and hole spacing should be selected under the premise of a small coefficient of variation. The optimal level combination of average seeds per hole is A1B1C3, and the optimal level combination of average hole spacing is A2B1C3.Similarly, the optimum level combinations of the empty hole rate, coefficient of variation of seed number, and hole spacing were A4.2.2The variance analysis of the orthogonal test results of the seed metering device is shown in 4.3The matrix analysis method was used to optimize the structure parameters of the seed metering device, which solves the unreasonable problem of multi-factor optimization under multi-index. According to the theoretical analysis in section 2.3.3, the weight matrix of each evaluation index of the seed metering device can be calculated as shown in equations (8)–(13).The index of the qualification rate of seeds per hole should be maximized to ensure the best overall performance of the seed metering device. In contrast, the empty hole rate, the coefficients of variation of the seed number, and the hole spacing should be as low as possible. Combined with the structural parameters and experimental conditions, the calculated theoretical seeds per hole and hole spacing all meet the agronomic requirements. Therefore, the average seeds per hole and the hole spacing should be the parameters when the corresponding coefficient of variation is the smallest. That is, the average seeds per hole and the hole spacing should be the smallest. In summary, if the total weight matrix of multi-index evaluation is the average of a single index matrix, then there is a total weight matrix, as shown in Equation (14).1 is the highest weight of factor A, indicating that the first level of hole diameter in the structural parameter of the seed metering device has the most effect on the multi-index test results. Similarly, the first level in factor B has the most significant influence on the results of the multi-index test, and the third level in factor C has the most significant influence on the results of the multi-index test. Therefore, the optimal factor level combination of structural parameters of the seed-metering device is A1B1C3. The hole diameter, number of holes, and adjustment angle was 12 mm, 10, and 80°, respectively. The primary and secondary order of structural parameter effect on the complete assessment index is A > B > C.We can observe from the total weight matrix that A4.4The verification experiment occurred under identical test conditions using the ideal combination of structural characteristics, the experiment was repeated three times, as well as the average value was taken. The experimental results are shown in 5The connection between the evaluation index and each component was analyzed to determine the variables that impact the performance index of the seed metering device. To begin, the single-factor test serves to establish the value range of each structural parameter. The orthogonal test would then be run to determine the best combination of structural parameters. Finally, the validity of the modified structural parameters was tested, and the seed metering mechanism was fully evaluated. Following is a discussion of the main test results of a comprehensive analysis:1) The diameter of the hole affects the performance of the seed metering device Figure 62) As shown in the range analysis Figure 9< 0.001). There are extremely significant differences in the number of holes and average hole spacing among types (P< 0.001). The adjustment angles were 60° and 80° degrees, and the average hole spacing had a distinct effect (P< 0.001).3) As shown by the analysis of variance (4) Although if the qualification rate of seeds per hole is lower than that of the fourth group, the coefficient of variation of seed number and hole spacing is better, according to the findings of the verification test. The performance of a seed metering device is often determined by multiple indexes, and a single index does not always indicate superior performance. Thus, this also demonstrates the necessity and significance of this research.2, and the adjustment range is superior to the rice precision seed metering device now on the market.5) The maximum number of seeds per hole under the optimal parameter combination was 11, signifying that the seed metering mechanism described in this research can plant 11 seeds with a sphericity of 54.14% per hole. According to the agronomic requirements of rice direct seeding, the seed metering device designed in this study can achieve an accurate adjustment of 37~92 kg/hm6) This experiment cannot accurately represent the adaptability of the seed metering device, thus many other types of spherical rice seeds will be chosen to explore adaptation further forward. Moreover, the coupling simulation method utilized in this experiment may expedite the development cycle for seed metering device research. Simultaneously, the multi-factor and multi-index optimization approach that this study presents will be helpful to future researchers studying other relevant parameter optimization.7) The existing research methods mainly have three evaluation indexes, while this study selects six evaluation indexes. The more evaluation indexes, the more complex the optimization process becomes. The results obtained using this research method are superior to the existing relevant studies. This reflects the reliability of this research and highlights the necessity and importance of this research. The research methods and ideas are not limited to the rice direct seeding device, but can also be applied to optimizing parameters of other devices. Using this method has a significant effect on improving device performance and can improve intelligent plant protection.6Aiming at the problem of an unreasonable configuration of structural parameters optimization method of rice precision direct seeding metering device, a structural optimization method of rice direct seeding apparatus based on the multi-index orthogonal experiment was proposed in this paper. Firstly, the influencing factors and levels of the overall performance of the rice direct seeding device were assessed using the coupling simulation analysis method. Second, a test platform was constructed using the electric drive control model, and orthogonal tests based on numerous indices were created. Finally, a multi-index comprehensive optimization theoretical model based on the matrix analysis method was established, through which the optimal structural parameters of the seed metering device were determined, and the accuracy and qualitativeness of the optimization method of the seed metering device structure based on the multi-index orthogonal test was verified. The following are the main conclusions of this study.1) On the basis of the multi-index orthogonal test, a thorough optimization method for the structural parameters of the seed metering device was proposed. Experiments have confirmed the suggested method. The method combines theory with co-simulation, designs the test platform using the electric drive exact control theory model, and performs the test while avoiding precise control of the speed of the seeding device in the real operation by depending on the mechanical drive.2) The single-factor simulation test demonstrates that the seed metering device construction parameters have an effect on the performance evaluation index, and the orthogonal test parameter range was identified. From the results of a multi-index orthogonal test, the hole diameter has the most effect on the complete assessment index, while the adjustment angle has the lowest impact.1B1C3. In other words, the hole’s diameter was 12 mm, the number of holes was 10, and the adjustment angle was 80°. The verification tests showed that the qualification rate of seeds per hole, empty hole rate, average seed number, coefficient of variation of seed number, average hole spacing, and variance coefficient of hole spacing are 93.07%, 0%, 9.39,14.04%, 22.84 cm, and 9.14%, respectively, under the optimal structural parameters. These results have confirmed the accuracy of the proposed method.3) Following the multi-index comprehensive optimization method, the optimal factor level combination of the seed metering device’s structural parameters was A4) The existing research on rice direct seeding metering devices shows that the seed qualification rate per hole is less than 90%, the empty hole rate is over 3%, and the coefficient of variation of hole spacing is over 10%. The optimization method used in these studies typically focuses on no more than three evaluation indicators . However5) Most of the seed metering devices currently available have 8 holes, while the device studied in this research has 10 holes. The results indicate that when seeding with the same hole spacing, the seed metering device with 10 holes has a lower rotational speed. The lower the rotational speed, the smaller the impact on the device, and the more stable its performance.6) At present, a comprehensive optimization method of structural parameters based on a multi-index orthogonal test has been provided, which has been validated in a rice precision direct seeding metering system, and its application must be confirmed. The test method needs further improvement. For a more exact analysis, it is also important to enhance the test conditions on the bench, a challenge for future study. This approach enhances research in the area of detailed optimization of the structural parameters of a precision rice metering device under multiple indexes. The concepts and findings of this study provide techniques and references for optimizing the parameters of a precision rice seed metering device.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.Conceptualization, HaL, LL, and BY. Methodology, HaL, CW, and GW. Validation, HaL, ZM, and BY. Formal analysis, HaL, LL, and BY. Investigation, HuL, ZM, XA, and BY. Resources, LL, HaL, and GW. Writing—original draft preparation, HaL, CW, and BY. Writing—review and editing, HaL, and BY. Supervision, HuL, XA, and ZM. Funding acquisition, XA, ZM, and BY. All authors contributed to the article and approved the submitted version.


To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity supresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications. Newtonian fluid is the most famous model in the field of fluid dynamics, but only viscosity is considered. The Kelvin–Voigt Further, the Zener model corresponds to the generalized Kelvin–Voigt and Maxwell model, including the elastic term and the relaxation effect of a viscoelastic medium. Warnez and Johnsen Therefore, the behavior of a single or few bubbles in viscoelastic liquid has been well investigated; however, considering multiple bubbles from a macroscopic view is necessary for practical medical applications. The Westervelt equation The authors were involved in the first derivation of the KZK equation for a liquid containing multiple bubbles In this study, the KZK equation describing the weakly nonlinear propagation of focused ultrasound in a viscoelastic liquid containing multiple bubbles is derived. In Section 22.1The weakly nonlinear propagation of focused ultrasound in a liquid containing multiple bubbles is considered. In this study, the viscoelasticity of the liquid phase is considered in two regions: the entire liquid and the bubble–liquid interface see .Focused ultrasound is radiated from a sound source in a viscoelastic medium containing multiple bubbles . The cenThe KZK equation is derived by assuming quasi-plane propagation, i.e., weakly diffracted (focused or spreading) waves. In this study, two types of KZK equations are derived: two-dimensional (2D) and three-dimensional (3D) spatial forms, as in our previous work (a-1)The elasticity of the liquid phase is considered; however, the viscoelastic shells (a-2)The viscosity and elasticity of the gas phase are neglected because they are significantly smaller than those of the liquid phase.(a-3)The basic equations based on the mixture model (a-4)The difference of velocity between the gas and liquid phases is sufficiently small, and then the drag force (a-5)The initial void fraction is sufficiently small.(a-6)The viscosity and elasticity of the gas–liquid mixture are modeled by only those of the liquid phase, because the viscosity and elasticity of the gas phase are sufficiently smaller than those of the liquid phase according to Assumptions (a-2) and (a-5).(a-7)The bubble–bubble interaction (a-8)Bubble oscillation is spherically symmetric.(a-9)Bubbles do not collapse, appear, or coalesce.(a-10)The phase change (a-11)The distributions of the pressure and temperature of the gas inside the bubbles (a-12)The temperature of the liquid phase is assumed constant, whereas the temperature of the gas phase is assumed to fluctuate.(a-13)At initial state, the liquid is at rest and spatially uniform, except for the spatial distribution of bubbles. The nonuniformity of the bubble size In the theoretical analysis, the following assumptions are introduced for simplicity.2.2The conservation laws of mass and momentum based on the mixture model Eqs. The Keller–Miksis equation n is the material constant (e.g., The energy equation describing the thermal conduction at the bubble–liquid interface [10]2.3V are the dimensionless constants of The following scale parameters are introduced to consider the low frequency and long wave as in our previous studies The assumption of weakly diffracted waves given by Eq. The liquid viscosity The nondimensionalization of the energy Eq. 2.4t and x are extended to the near and far fields using the dimensionless wave amplitude xz are defined in the far field only asThe independent variables The gas density is assumed to be significantly smaller than the liquid density in the initial state:To incorporate the weak nonuniformity of the spatial distribution of bubbles in the initial state, void fraction 3The scale parameters of Eqs. 3.1From the approximations of Eqs. Next, the variable transformation is introduced as3.2From the approximations of 3.3K is given byFrom the approximations of 3.43.53.6In 3.7In As shown in 3.8The resultant KZK Eq. 44.1f and liquid pressure R; then, The KZK equation of the spatial 2D form of Eq. R direction with the truncation error of order Next, the KZK Eq. m are the effective duration and the rise time of pulses, respecrively The boundary condition at 4.2In 4.3The dispersion effect is represented in the form of the third derivative with coefficient 4.4In In 5Weakly nonlinear propagation of focused ultrasound in viscoelastic liquid containing multiple bubbles was investigated using the volumetric averaged equations of liquid containing multiple bubbles based on the mixture model As a result of the theoretical analysis based on perturbation expansion with the multiple-scales method In Section In our future work, theoretical extensions of the KZK equation will be conducted, such as incorporating blood vessels Shunsuke Kagami: Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing. Tetsuya Kanagawa: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision, Project administration, Funding acquisition.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Dental epidemiological studies are essential for analysing and evaluating the population’s health state and dental treatments provided, as well as for planning future oral health programme activities and interventions based on their findings. In order to determine the health condition of children and adolescents in connection to the prevalence of caries, caries indices (decayed and filled teeth (dft) for primary teeth and decayed, missing and filled teeth (DMFT) in permanent teeth) and periodontal indices ), oral exams of children and adolescents aged 6, 12 and 15 from selected schools were conducted. To assess the achievement of the oral health objectives set for Spain in 2020, these data were compared with those acquired at the national level. At 6 years of age, 278 children were examined, the prevalence of caries was 55.6%, the dft index was 2.77 (±3.44), the DMFT was 0.19 (±0.16), the restorative index (RI) was 4.62%, and the significant caries (SiC) index was 8.40 (±2.07). At 12 years of age, 208 students were examined, the prevalence of caries found was 65.86%, the DMFT index was 1.85 (±2.22), the RI was 36.63%, the SiC index was 5.43 (±2.07), and children without the presence of periodontal pathology was 59.13%. At 15 years of age, 165 students were examined, the prevalence of caries was 70.06%, the DMFT was 3.08 (±3.39), the RI was 42.42%, the SiC index was 8.10 (±2.55), and adolescents without periodontal disease was 47.90%. Conclusions: Melilla-born children and adolescents had higher caries indicators and indices than the corresponding national averages for Spain. Teenagers under the age of 15 have a particularly high frequency of dental caries. The investigation of the children’s origin is where there is the most disparity. Children of Berber descent have much higher values than children of European heritage. Epidemiological studies are the main source of information on the frequency and distribution of conditions, health determinants and risk factors in populations. They provide us with the necessary data to carry out preventive and therapeutic programmes for the population studied .In all the regions of Spain, with the exception of Melilla, there are epidemiological surveys on oral health. Their importance lies in the fact that they are the basis for decision-making by public health services. The measures adopted are aimed at achieving the objectives set by the Spanish Council of Dentists and Stomatologists for Spain for 2020 .The geographical importance of Melilla should be highlighted. This Spanish autonomous city of twelve square kilometres is located in the North of Africa. It can be seen on the map in Because of Melilla’s proximity to Morocco, the Muslim population has grown in recent years and has reached almost 50% of the registered population . MoreoveIn terms of oral and dental health care, the city’s only hospital provides a maxillofacial surgery service, and in 2020 a dental care service for special patients was incorporated.Melilla has four health centres for primary care, but only one offers dental care. Two dentists who are members of the Melilla College of Dentists are in charge of this service, and two hygienists also work there . The children’s oral health care programme is based on a public model regulated by Spanish law (RD 111/2008 of 1 January 2008) . The theThe main objective of this pioneering survey is to find out the main oral health problems in children and in this way propose a line of preventive and therapeutic action, which will make it possible to know if there are differences in oral health and hygiene habits between children and adolescents from different communities.In this way, we will analyse the prevalence of caries, caries rates, restoration rate and community periodontal index (CPI) in cohorts established by WHO and establish a comparison with the targets set by the Council of Dentists for the year 2020 [A sampling plan was drawn up for carrying out an oral health survey in the Autonomous City of Melilla, taking as a reference the general recommendations established by the WHO regarding surveys in this area of health .An oral health form was also developed based on the model proposed by WHO , with thIn order to carry out the survey in schoolchildren, an appropriate authorisation was obtained from the Provincial Direction of Education and Science of the Autonomous City of Melilla, with verbal authorisation from the directors of the educational centres and with signed informed consent and authorisation from the parents or legal guardians.The reviews were carried out in schools in Melilla. For this purpose, we divided the city map geographically into 3 zones: north, centre and south. With a sample universe of about 22,000, a margin of error of 5% and a confidence level of 95%, the final sample size should be 378 children and adolescents. Nevertheless, finally, the sample was as follows, as shown in A total of 651 children were examined, divided into the following cohorts: 278 children belonged to the cohort of 6-year-olds (3rd year of infant school), 208 belonged to the 12-year-old cohort (6th year of primary school), and 165 belonged to the 15-year-old cohort (3rd year of secondary school).(a)Inclusion criteria: All students from schools in Melilla included in the cohorts were included in the study.(b)Exclusion criteria: Children were not included in the study if they had any type of physical or mental disability or oral malformation such as cleft lip.Data collection was carried out by a single person. Intraexaminer agreement was established with a Kappa index of 0.90 according to the criteria of Landis and Koch .It was carried out between October 2019 and March 2020 in the classrooms of the centres themselves.The materials used were ambient light and direct light by means of an LED flashlight, a mirror LED torch, disposable intraoral flat mirror no. 5, WHO periodontal probe, gauze for drying the field, disposable latex gloves and masks.Several sociodemographic variables were collected in the overall study, such as gender, geographical area where the school or institute was located and socioeconomic level of the families.In this article, we focus on the variable that may be of most interest: the origin of the children. For this purpose, we recorded whether the child was of Berber or European origin. This criterion was established on the basis of the child’s first name and surname.1:European origin2:Berber origin3:Mixed origin of both culturesThe codes used were as follows:-Dental pathologies: presence or absence of caries (according to WHO 4th edition criteria), absence of teeth and presence of dental pathology (according to WHO 4th edition  criteria-Need for treatment (according to WHO 4th edition criteria ).-Periodontal pathologies: presence of bleeding on probing and presence or absence of calculus (according to WHO 4th edition criteria ).p < 0.05. The STATA V15 package was used  for data analysis.The database was cleaned to check the completeness and consistency of the information. Continuous variables are presented as means, medians and standard deviations. Categorical variables are presented as proportions and percentages. In order to identify the association between the presence of caries index and the independent variables, non-parametric Wilcoxon/Kruskal–Wallis tests (rank sums) were performed for continuous variables, and in the case of more than two categories, non-parametric tests were used for each pair (Wilcoxon method of non-parametric comparisons). Analysis of the chi-squared test was used to identify associations. In the case that the expected value of any of the cells in the continence table was less than five, Fisher’s chi-squared test for tetrachoric tables was applied. A statistically significant difference was considered to exist for By origin, in the three ages studied, around 60% of the population was of Berber origin while approximately 30% was of European origin.The results obtained for the overall caries rates in the three cohorts studied are shown in In all cohorts, the component of the dft/DMFT index with the highest presence is the carious component. The absent component is almost nil at all three ages, and the obturated component is most prevalent at 15 years of age.At 15 years of age, we found the highest DMFT index of the three cohorts with a value of 3.08 (±3.39).The restorative index does not reach 50% in any of the cohorts, and for the 5–6-year-old group it is only 4.62% with an SD of ±16.94.As we have seen in The 12-year-old children screened were distributed as follows: 76 were of European origin (36.54%), 116 were of Berber origin (55.77%), and 16 were of mixed marriage origin (7.69%).The 15-year-old study group is distributed as follows: 76 adolescents were of European origin (40.12%), 116 were of Berber origin (55.69%), and 16 were of mixed marriage origin (4.19%). We can verify that, at all ages, the children who presented the highest caries experience are the group of children whose origin is Berber. These children had approximately twice the caries prevalence of children of European origin, mainly at the age of 6 years in the primary dentition, where children of Berber origin had a prevalence of 65.95%, compared with 36.84% for children of European origin.Children of Berber origin present at all ages a higher dft/DMFT index than that found in children of European origin, with the greatest differences found at 6 and 15 years of age, becoming in the last cohort four times higher (DMFT of schoolchildren of Berber origin = 4.55 (±3.70) and the average DMFT of schoolchildren of European origin = 1.22 (±1.60)).Within this index, the carious component is the one that represents the greatest weight, being, once again, greater in children of Berber origin. This difference is notable in the 6-year-old cohort in the primary dentition and in the 15-year-old cohort, where it is almost three times greater in children of Berber origin.Of the three components of the index, the absent component is the one that represents the least volume at all ages. No statistically significant differences were found in any of them.In the filled component, statistically significant differences were only found in the primary dentition at 6 years of age and in the permanent dentition at 15 years of age. In the latter group, children of Berber origin have a higher rate of fillings than adolescents of European origin.At 6 years of age in the primary dentition, children from families of European origin present an average number of restorations of 18.80 (±33.76), while children of Berber origin only have an average number of restorations of 1.55 (±7.33).At age 12, the children with the highest rate are children of European origin with an average of 56.18 (±48.63). This is followed by children from mixed marriages with an average value of 50.00 (±50.00), and, finally, the children with the lowest value are children of Berber origin with an average of 25.46 (±39.44).At 12 years of age, we found that 59.13% of the children had a healthy periodontium, 25.48% of the children had haemorrhage or bleeding, and 15.38% of the children had calculus.We found no significant differences in the CPI according to the origin of the children and adolescents between the three groups studied.To begin with, we would like to comment that this study has a series of limitations to be considered for the interpretation of the results obtained. These are inherent to the type of study carried out and are minimised by the standardisation of the criteria as set out by the WHO in its recommendations .In terms of caries prevalence, the results obtained show that in all cohorts, children of Berber origin have higher values. It should be noted that at 6 and 15 years of age, children of Berber origin have twice the caries prevalence of children of European origin. In none of the cohorts and none of the ethnic groups are the targets set by the General Council of Dentists for the year 2020 [The dft/DMFT index is higher in children of Berber origin in all cohorts. At the age of 6 years, there is the least difference in the average dft/DMFT index between children of different ethnicities. The difference becomes more noticeable as the age of the children increases, reaching four times higher in the Berber population at 15 years of age.Within the DMFT index, the component that stands out the most is the carious component, since in all cohorts it is the one with the greatest weight, being higher, again, in children and adolescents of Berber origin.In the 2020  nationalIn Melilla in 2016, Hadi  publisheIn 2016, Gonçalves [The 1996 Ceuta survey  collecteAt 6 years of age in Ceuta, the percentage of children with DMFT > 1 was 84% in Muslim children compared with 50.7% in non-Muslim children. At this age, ethnicity statistically influenced the cod index (4.40 for Muslim children and 2.07 for non-Muslim children).By age 12, the percentage of children with DMFT > 1 was also higher in Muslim children (93.2%) than in non-Muslim children (79.7%). The DMFT index followed the same trend as in the previous cohort with statistically significant differences: 5.00 in Muslim children versus 3.26 in non-Muslim children.By the age of 15, the percentages in Ceuta are closer between the two groups, but the percentage of children with DMFT > 1 is still higher in Muslims than in non-Muslims . At this age, there is no statistically significant difference between the two groups in the DMFT index (5.30 in Muslim children and 4.20 in non-Muslim children).As it can be seen, the results of our study are similar to those found in Ceuta: In all cohorts, caries rates and caries prevalence are higher in children of Berber origin .The Berber ethnic group extends throughout North Africa, from the Canary Islands to Egypt ,23. HencThe closest Berber country to Melilla is Morocco, and many of the children included in our study have a deep-rooted culture from this neighbouring country . It is tIn Morocco, at 12 years of age, the DMFT was 4.82 and the prevalence of caries was 81.8%, while in the study in Melilla we obtained results of DMFT of 2.40 and caries prevalence of 73.8%.For the 15-year-olds, the results are more similar: In Morocco, the DMFT index was 5.54 with a prevalence of 86.67%, and in our study the result for this cohort was 4.55 with a prevalence of 86.67%.We can also compare the results for the Berber population studied in Melilla with results obtained in the second closest country to our city, Algeria , in 2013As we have seen above, the RI serves as an indicator of the quality of oral health services offered to the general population . In our This may be due to the fact that, in Melilla, children of Berber origin, as a general rule and with some exceptions, are associated with lower socioeconomic levels and reside in the most vulnerable areas , making The study conducted in 2016 on the oral health of Melilla’s 6-year-old children  does notIn the study by Gonçalves  on childThis fact indicates that adolescents of Berber origin in Melilla have easier access to dental treatment than the children at the CETI.In Ceuta, in 1996 , no relaThe studies carried out in Moroccan cities, likewise, do not reveal any data on the restoration rate of the child population, but only a study of the prevalence and rates of caries.In the CPI study for the 12- and 15-year-old cohorts, we observed no statistically significant differences between the two groups of children by origin. In both cohorts, the Oral Health Objectives set by the Council of Dentists for the year 2020 [In the study of the oral health of children in Ceuta in 1996 , the CPIThese results, on the whole, are similar to ours, as none of them show statistically significant differences, and non-Muslim children (those of European origin in our study) show better results than Muslim children (of Berber ethnicity in our study).In Morocco, the study by Zaoui  obtainedIn our study for the same age group, the results are similar in children of Berber origin. The only component that differs is bleeding on probing, as in our study, unlike that of Zaoui , it is lThe 2012 national survey in Morocco  establisThis epidemiological study shows an improved oral health status in terms of caries and periodontal disease in the infant–juvenile population of Melilla, which should be taken into account when developing oral health programmes for the affected population.The prevalence and rates of caries are considerably higher in children of Berber origin, who have a higher prevalence and rates of caries than children of European origin. Only 12-year-old children of both ethnicities and 15-year-old children of European origin meet the objectives related to periodontal health set by the Spanish authorities.


P-sets (P stands for Packet) is a set model with dynamic characteristics, which is obtained by introducing dynamic characteristics into Cantor set and improving Cantor set. According to the fact that the characteristics of class I big data are completely consistent with the basic characteristics of P-sets, this paper gives research on theory and application on class I big data from the view of mathematics. Here we introduce Class I big data which need some new definitions of data block, microdata and data link. Based on these concepts, decomposition theorem of data block and microdata relation theorem are given, and then attribute reasoning theorem and microdata intelligent discovery and the intelligent secure acquisition algorithm of microdata are also proposed. By using these theoretical results, the applications of secure acquisition of microdata are presented. In summary, P-sets mathematical model provides a new theory and method for studying class I big data. The attribute set Data element Big data Big data is a new and independent field formed by the mutual penetration and intersection of mathematics, statistics, probability theory, computer technology and humanities. As applications, big data and its attributes are interdependent, and the dynamic changes of big data keep pace with those of its attribute set, which is the dynamic nature of big data. Big data and its sub-data have dynamic characteristics. The attribute Characteristics 1–4 are hidden in big data, which cannot be lost or avoided.1 established a basic mathematical model called PEFS to evaluate and optimize the stability of the food system. A prediction model was estimated and evaluated using vector autoregressive model with time series data of long- and short-term interest rates2. Based on big data, Fan et al.3 proposed a grey predictive mathematical model which used to analysis on the effect of Escherichia coli infection on patients with lupus nephritis. However, there is little research on big data features 1–4 in these references, and they focus on combining big data with other disciplines and only give application research, but ignore the theoretical research of big data, pay no attention to the mathematical concept connotation and dynamic characteristics contained in big data, and no one combines big data with mathematics to give theoretical research of big data from the perspective of mathematics.Many scholars have devoted themselves to the application research of combining mathematical models with big data, and have achieved some achievements. Based on the mechanism analysis of big data technology in the food supply chain. Li et al.4 obtained P-sets by introducing dynamic characteristics into Cantor set ISupplementing some attributes in IIDeleting some attributes in IIIIf attributes are supplemented and deleted in Shi5. An algebraic model of P-sets is proposed6. By introducing assistant set into function P-sets to expand the function P-sets7. More dynamic characteristics and applications of P-sets are discussed10. P-information fusion and application based on P-sets are obtained14. The inverse P-sets is proposed21, which is the dual form of P-sets. Function P-sets is given23, which is the functional form of P-sets; and the function inverse P-sets is also proposed26, which is the functional form of inverse P-sets. However, some of these achievements generalize or expand P-sets, while others discuss their applications in dynamic information systems using P-sets as tools, such as intelligent data mining, risk tracking and recognition, and intelligent image recognition. no one has applied the P-sets mathematical model to big data theory and application research in the available literature.The mathematical characteristics of the P-sets such as quantitative characteristics, algebraic characteristics, geometrical characteristics, random characteristics, and theory applications are studied by scholars. By using P-sets, L. A. Zadeh fuzzy set is improvedP-sets and class I big data have the same dynamic characteristics and attribute "conjunction" characteristics. In view of I-III, it is easy to get that P-sets is a new mathematical model and method for studying big data. It provides theoretical support for studying the structure, characteristics and basic theory of class I big data.The purpose of this paper is to give research on theory and application of Class I big data by using the mathematical model with dynamic characteristics.Cantor set If attribute set mentclass2pt{minimCantor set X,If attribute set F¯≠∅; in , \documeThe set pair which is composed of internal packet set From  and 11)11), we g is referFrom  to 8), , 8), we Under the condition of 1. if =∅, from , we get:entclass1pt{minima2. if  Formula , \docume Formula .Based on 1 and 2, we can complete this Proposition.Under the condition of The proof is similar to Proposition From Formula –8) and  and 8) aUnder the condition we continuously add attributes to 1. 2. 3. The characteristics of 4. The characteristics of 5. The dynamic characteristics of Formula  are the 6. The dynamic characteristics of Formula  are the  Formula  \documenLet If deleting attribute This simple fact and logical feature can be accepted by ordinary people. The relationship among mentclass2pt{minimAgreement: we call that By using characteristics and concept of P-sets in "If We call Under the condition of supplementing attributes in Under the condition of deleted attributes in From  to 15),,15), we  in " in "3) i1. if attribute set 2. If mentclass2pt{minim(Decomposition Theorem outer The sufficient and necessary conditions for Its proof is similar to Theorem The necessary and sufficient conditions for the existence of From Formula  to 3) a a3) and The necessary and sufficient conditions for the existence of Its proof is similar to Theorem ((Propositions 3 and 4 are obtained from  to 21)::21):By utilizing the concepts and the model in "Suppose  is referSuppose  is referIf  is refern. Where  represenFrom (24) to (26), we get:, -microdata are given, it is a part of In this section, only the intelligent decomposition and secure acquisition algorithms of entclass1pt{minimaThe detailed process of the algorithm is as follows:(1) Big data (2) (3) Attribute reasoning is established:(4) (5) Given the standard microdata (6) (7) Ellipse curve (8) (9) (10) The algorithm ends.By using the preliminary concept in "“commodity profit estimation”;The example in this section comes from Make mentclass2pt{minimThe "—" in Table t}αtF in  generateBy using  in "Attr gives\doThe attribute of By using Table Ellipse curve model and its encryption decryption algorithm27:Call If If Taken Pm be the plaintext of A is the encryptor of plaintext Pm and the sender of ciphertext Cm; B is the decryptor of ciphertext Cm and the accepter of plaintext A, A; Let I Information encryption process: Then II Information decryption process: Using B respectively.A chooses 1. 2. After The encryption–decryption of Important description of application examples1. The application example of the paper is taken from the "commodity profit estimation" block in "commodity big data", which is the compression and simplification of the original case. Because of the special nature of the example, some information can not be made public. The profit value 2. In the competition of commodity market, if the profit value 3. The security of ellipse curve cryptography comes from the difficulty of solving discrete logarithms, point set Confirmation of application examplesThe applications and the methods of the examples have been applied and confirmed in 8 business districts in Zhengzhou, China. The profits mentclass2pt{minimBig data is a branch of theory and application research with mathematical concept connotation, which has infiltrated into many application researches and aroused people's interest. What is the structure and characteristics of big data? Can it be classified? What is the basis for classification? In the existing literature, these problems have not been discussed. In this paper, the theory and application of Class I big data are presented by using the p-sets mathematical model with dynamic characteristics and attribute “Conjunction” characteristics. Restore the mathematical characteristics of big data with mathematical methods and models, and understand the mathematical structure and logical characteristics of big data. These characteristics are hidden in big data and do not attract people's attention. Finally, the paper selects the "financial security" data block in the "financial big data" as an example, uses the big data block theory given in this paper, and combines the elliptic curve to give the security algorithm of financial information encryption and decryption.In the research of big data, most scholars combine mathematical models with big data to carry out application research, rarely pay attention to the theoretical research of big data, who do not recognize the dynamic and logical characteristics of big data, which lack strict reasoning and proof. In this paper, the author not only gives the theoretical research of big data by using mathematical models, but also pays attention to the combination of theoretical research with application research. Perhaps the research and results presented in this paper provide a research idea and mathematical method for researchers who are engaged in the fundamental theory and application of big data. The infiltration of mathematical concepts into the study of big data may make people acquire new theoretical understanding of big data, and it becomes an inevitable process from the simple study application to the theoretical study of big data.In the inverse P-sets, attribute Function P-sets and function inverse P-sets are mathematical models and methods to study class I big data in function form and class II big data in function form respectively. All of P-sets, inverse P-sets, function P-sets, function inverse P-sets are the preparation for studying the application of big data, understanding the characteristics of big data, and giving mathematical models as well as methods.