Three-dimensional Modeling of Ore-forming Elements and Mineralization Prognosis for the Yechangping Mo Deposit,Henan Province,China

Three-dimensional geochemical modeling of ore-forming elements is crucial for predicting deep mineralization.This approach provides key information for the quantitative prediction of deep mineral localization,three-dimensional fine interpolation,analysis of spatial distribution patterns,and extraction of quantitative mineral-seeking markers.The Yechangping molybdenum(Mo)deposit is a significant and extensive porphyry-skarn deposit in the East Qinling-Dabie Mo polymetallic metallogenic belt at the southern margin of the North China Block.Abundant borehole data on oreforming elements underpin deep geochemical predictions.The methodology includes the following steps:(1)Threedimensional geological modeling of the deposit was established.(2)Correlation,cluster,and factor analyses post delineation of mineralization stages and determination of mineral generation sequence to identify(Cu,Pb,Zn,Ag)and(Mo,W,mfe)assemblages.(3)A three-dimensional geochemical block model was constructed for Mo,W,mfe,Cu,Zn,Pb,and Ag using the ordinary kriging method,and the variational function was developed.(4)Spatial distribution and enrichment characteristics analysis of ore-forming elements are performed to extract geological information,employing the variogram and w(Cu+Pb+Zn+Ag)/w(Mo+W)as predictive indicators.(5)Identifying the western,northwestern,and southwestern areas of the mine with limited mineralization potential,contrasted by the northeastern and southeastern areas favorable for mineral exploration.

Numerical three-dimensional modeling of earthen dam piping failure

A physically-based numerical three-dimensional earthen dam piping failure model is developed for homogeneous and zoned soil dams.This model is an erosion model,coupled with force/moment equilibrium analyses.Orifice flow and two-dimensional(2D)shallow water equations(SWE)are solved to simulate dam break flows at different breaching stages.Erosion rates of different soils with different construction compaction efforts are calculated using corresponding erosion formulae.The dam's real shape,soil properties,and surrounding area are programmed.Large outer 2D-SWE grids are used to control upstream and downstream hydraulic conditions and control the boundary conditions of orifice flow,and inner 2D-SWE flow is used to scour soil and perform force/moment equilibrium analyses.This model is validated using the European Commission IMPACT(Investigation of Extreme Flood Processes and Uncertainty)Test#5 in Norway,Teton Dam failure in Idaho,USA,and Quail Creek Dike failure in Utah,USA.All calculated peak outflows are within 10%errors of observed values.Simulation results show that,for a V-shaped dam like Teton Dam,a piping breach location at the abutment tends to result in a smaller peak breach outflow than the piping breach location at the dam's center;and if Teton Dam had broken from its center for internal erosion,a peak outflow of 117851 m'/s,which is 81%larger than the peak outflow of 65120 m3/s released from its right abutment,would have been released from Teton Dam.A lower piping inlet elevation tends to cause a faster/earlier piping breach than a higher piping inlet elevation.

Oxygen tension modulates cell function in an in vitro three-dimensional glioblastoma tumor model

Hypoxia is a typical feature of the tumor microenvironment,one of the most critical factors affecting cell behavior and tumor progression.However,the lack of tumor models able to precisely emulate natural brain tumor tissue has impeded the study of the effects of hypoxia on the progression and growth of tumor cells.This study reports a three-dimensional(3D)brain tumor model obtained by encapsulating U87MG(U87)cells in a hydrogel containing type I collagen.It also documents the effect of various oxygen concentrations(1%,7%,and 21%)in the culture environment on U87 cell morphology,proliferation,viability,cell cycle,apoptosis rate,and migration.Finally,it compares two-dimensional(2D)and 3D cultures.For comparison purposes,cells cultured in flat culture dishes were used as the control(2D model).Cells cultured in the 3D model proliferated more slowly but had a higher apoptosis rate and proportion of cells in the resting phase(G0 phase)/gap I phase(G1 phase)than those cultured in the 2D model.Besides,the two models yielded significantly different cell morphologies.Finally,hypoxia(e.g.,1%O2)affected cell morphology,slowed cell growth,reduced cell viability,and increased the apoptosis rate in the 3D model.These results indicate that the constructed 3D model is effective for investigating the effects of biological and chemical factors on cell morphology and function,and can be more representative of the tumor microenvironment than 2D culture systems.The developed 3D glioblastoma tumor model is equally applicable to other studies in pharmacology and pathology.

Three-Dimensional Analytical Modeling of Axial-Flux Permanent Magnet Drivers

In this paper, the axial-flux permanent magnet driver is modeledand analyzed in a simple and novel way under three-dimensional cylindricalcoordinates. The inherent three-dimensional characteristics of the deviceare comprehensively considered, and the governing equations are solved bysimplifying the boundary conditions. The axial magnetization of the sectorshapedpermanent magnets is accurately described in an algebraic form bythe parameters, which makes the physical meaning more explicit than thepurely mathematical expression in general series forms. The parameters of theBessel function are determined simply and the magnetic field distribution ofpermanent magnets and the air-gap is solved. Furthermore, the field solutionsare completely analytical, which provides convenience and satisfactoryaccuracy for modeling a series of electromagnetic performance parameters,such as the axial electromagnetic force density, axial electromagnetic force,and electromagnetic torque. The correctness and accuracy of the analyticalmodels are fully verified by three-dimensional finite element simulations and a15 kW prototype and the results of calculations, simulations, and experimentsunder three methods are highly consistent. The influence of several designparameters on magnetic field distribution and performance is studied and discussed.The results indicate that the modeling method proposed in this papercan calculate the magnetic field distribution and performance accurately andrapidly, which affords an important reference for the design and optimizationof axial-flux permanent magnet drivers.

Three-dimensional magnetotelluric modeling in the spherical and Cartesian coordinate systems:A comparative study

With the increase in the coverage area of magnetotelluric data,three-dimensional magnetotelluric modeling in spherical coordinates and its differences with respect to traditional Cartesian modeling have gradually attracted attention.To fully understand the influence of the Earth’s curvature and map projection deformations on Cartesian modeling,qualitative and quantitative analyses based on realistic three-dimensional models need to be examined.Combined with five representative map projections,a type of model conversion method that transforms the original spherical electrical conductivity model to Cartesian coordinates is described in this study.The apparent resistivity differences between the spherical western United States electrical conductivity model and the corresponding five Cartesian models are then compared.The results show that the cylindrical equal distance map projection has the smallest error.A meridian convergence correction resulting from the deformation of the map projection is introduced to rotate the Cartesian impedance tensor from grid north to geographic north,which reduces differences from the spherical results.On the basis of the magnetotelluric field data,the applicability of the Cartesian coordinate system to western and contiguous United States models is quantitatively evaluated.Precise interpretations of the contiguous United States model were found to require spherical coordinates.

Implementation of a particle-in-cell method for the energy solver in 3D spherical geodynamic modeling

The thermal evolution of the Earth’s interior and its dynamic effects are the focus of Earth sciences.However,the commonly adopted grid-based temperature solver is usually prone to numerical oscillations,especially in the presence of sharp thermal gradients,such as when modeling subducting slabs and rising plumes.This phenomenon prohibits the correct representation of thermal evolution and may cause incorrect implications of geodynamic processes.After examining several approaches for removing these numerical oscillations,we show that the Lagrangian method provides an ideal way to solve this problem.In this study,we propose a particle-in-cell method as a strategy for improving the solution to the energy equation and demonstrate its effectiveness in both one-dimensional and three-dimensional thermal problems,as well as in a global spherical simulation with data assimilation.We have implemented this method in the open-source finite-element code CitcomS,which features a spherical coordinate system,distributed memory parallel computing,and data assimilation algorithms.

Background removal from global auroral images:Data-driven dayglow modeling

Global images of auroras obtained by cameras on spacecraft are a key tool for studying the near-Earth environment.However,the cameras are sensitive not only to auroral emissions produced by precipitating particles,but also to dayglow emissions produced by photoelectrons induced by sunlight.Nightglow emissions and scattered sunlight can contribute to the background signal.To fully utilize such images in space science,background contamination must be removed to isolate the auroral signal.Here we outline a data-driven approach to modeling the background intensity in multiple images by formulating linear inverse problems based on B-splines and spherical harmonics.The approach is robust,flexible,and iteratively deselects outliers,such as auroral emissions.The final model is smooth across the terminator and accounts for slow temporal variations and large-scale asymmetries in the dayglow.We demonstrate the model by using the three far ultraviolet cameras on the Imager for Magnetopause-to-Aurora Global Exploration(IMAGE)mission.The method can be applied to historical missions and is relevant for upcoming missions,such as the Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)mission.

Progressive fragmentation of granular assemblies within rockslides: Insights from discrete-continuous numerical modeling

Rock fragmentation plays a critical role in rock avalanches,yet conventional approaches such as classical granular flow models or the bonded particle model have limitations in accurately characterizing the progressive disintegration and kinematics of multi-deformable rock blocks during rockslides.The present study proposes a discrete-continuous numerical model,based on a cohesive zone model,to explicitly incorporate the progressive fragmentation and intricate interparticle interactions inherent in rockslides.Breakable rock granular assemblies are released along an inclined plane and flow onto a horizontal plane.The numerical scenarios are established to incorporate variations in slope angle,initial height,friction coefficient,and particle number.The evolutions of fragmentation,kinematic,runout and depositional characteristics are quantitatively analyzed and compared with experimental and field data.A positive linear relationship between the equivalent friction coefficient and the apparent friction coefficient is identified.In general,the granular mass predominantly exhibits characteristics of a dense granular flow,with the Savage number exhibiting a decreasing trend as the volume of mass increases.The process of particle breakage gradually occurs in a bottom-up manner,leading to a significant increase in the angular velocities of the rock blocks with increasing depth.The simulation results reproduce the field observations of inverse grading and source stratigraphy preservation in the deposit.We propose a disintegration index that incorporates factors such as drop height,rock mass volume,and rock strength.Our findings demonstrate a consistent linear relationship between this index and the fragmentation degree in all tested scenarios.

The relationship between compartment models and their stochastic counterparts:A comparative study with examples of the COVID-19 epidemic modeling

Deterministic compartment models(CMs)and stochastic models,including stochastic CMs and agent-based models,are widely utilized in epidemic modeling.However,the relationship between CMs and their corresponding stochastic models is not well understood.The present study aimed to address this gap by conducting a comparative study using the susceptible,exposed,infectious,and recovered(SEIR)model and its extended CMs from the coronavirus disease 2019 modeling literature.We demonstrated the equivalence of the numerical solution of CMs using the Euler scheme and their stochastic counterparts through theoretical analysis and simulations.Based on this equivalence,we proposed an efficient model calibration method that could replicate the exact solution of CMs in the corresponding stochastic models through parameter adjustment.The advancement in calibration techniques enhanced the accuracy of stochastic modeling in capturing the dynamics of epidemics.However,it should be noted that discrete-time stochastic models cannot perfectly reproduce the exact solution of continuous-time CMs.Additionally,we proposed a new stochastic compartment and agent mixed model as an alternative to agent-based models for large-scale population simulations with a limited number of agents.This model offered a balance between computational efficiency and accuracy.The results of this research contributed to the comparison and unification of deterministic CMs and stochastic models in epidemic modeling.Furthermore,the results had implications for the development of hybrid models that integrated the strengths of both frameworks.Overall,the present study has provided valuable epidemic modeling techniques and their practical applications for understanding and controlling the spread of infectious diseases.

Total ionizing dose effect modeling method for CMOS digital-integrated circuit

Simulating the total ionizing dose(TID)of an electrical system using transistor-level models can be difficult and expensive,particularly for digital-integrated circuits(ICs).In this study,a method for modeling TID effects in complementary metaloxide semiconductor(CMOS)digital ICs based on the input/output buffer information specification(IBIS)was proposed.The digital IC was first divided into three parts based on its internal structure:the input buffer,output buffer,and functional area.Each of these three parts was separately modeled.Using the IBIS model,the transistor V-I characteristic curves of the buffers were processed,and the physical parameters were extracted and modeled using VHDL-AMS.In the functional area,logic functions were modeled in VHDL according to the data sheet.A golden digital IC model was developed by combining the input buffer,output buffer,and functional area models.Furthermore,the golden ratio was reconstructed based on TID experimental data,enabling the assessment of TID effects on the threshold voltage,carrier mobility,and time series of the digital IC.TID experiments were conducted using a CMOS non-inverting multiplexer,NC7SZ157,and the results were compared with the simulation results,which showed that the relative errors were less than 2%at each dose point.This confirms the practicality and accuracy of the proposed modeling method.The TID effect model for digital ICs developed using this modeling technique includes both the logical function of the IC and changes in electrical properties and functional degradation impacted by TID,which has potential applications in the design of radiation-hardening tolerance in digital ICs.

Modularized and Parametric Modeling Technology for Finite Element Simulations of Underground Engineering under Complicated Geological Conditions

The surrounding geological conditions and supporting structures of underground engineering are often updated during construction,and these updates require repeated numerical modeling.To improve the numerical modeling efficiency of underground engineering,a modularized and parametric modeling cloud server is developed by using Python codes.The basic framework of the cloud server is as follows:input the modeling parameters into the web platform,implement Rhino software and FLAC3D software to model and run simulations in the cloud server,and return the simulation results to the web platform.The modeling program can automatically generate instructions that can run the modeling process in Rhino based on the input modeling parameters.The main modules of the modeling program include modeling the 3D geological structures,the underground engineering structures,and the supporting structures as well as meshing the geometric models.In particular,various cross-sections of underground caverns are crafted as parametricmodules in themodeling program.Themodularized and parametric modeling program is used for a finite element simulation of the underground powerhouse of the Shuangjiangkou Hydropower Station.This complicatedmodel is rapidly generated for the simulation,and the simulation results are reasonable.Thus,this modularized and parametric modeling program is applicable for three-dimensional finite element simulations and analyses.

Study of the geomagnetic field's regional gradients in Chinese continent using three-dimensional surface Spline model

We combined domestic ground-based and satellite magnetic measurements to create a regional three-dimensional surface Spline(3DSS)gradient model of the main geomagnetic field over the Chinese continent.To improve the precision of the model,we considered the data gap between the ground and satellite data.We compared and analyzed the results of the Taylor polynomial,surface Spline,and CHAOS-6(the CHAMP,?rsted and SAC-C model of Earth’s magnetic field)gradient models.Results showed that the gradients in the south-north and east-west directions of the four models were consistent.The 3DSS model was able to express not only gradients at different altitudes,but also average gradients inside the research area.The two Spline models were able to capture more information on gradient anomalies than were the fitted models.Strong local anomalies were observed in northern Xinjiang,Beijing,and the junction area between Jiangsu and Zhejiang,and the total intensity F decreased whereas the altitude increased.The gradient decreased by 21.69%in the south-north direction and increased by 11.78%in the east-west direction.In addition,the altitude gradient turned from negative to positive while the altitude increased.The Spline model and the two fitted models differed mainly in the field sources they expressed and the modeling theory.

Modeling Analysis of Factors Influencing Wind-Borne Seed Dispersal: A Case Study on Dandelion

A weed is a plant that thrives in areas of human disturbance, such as gardens, fields, pastures, waysides, and waste places where it is not intentionally cultivated. Dispersal affects community dynamics and vegetation response to global change. The process of seed disposal is influenced by wind, which plays a crucial role in determining the distance and probability of seed dispersal. Existing models of seed dispersal consider wind direction but fail to incorporate wind intensity. In this paper, a novel seed disposal model was proposed in this paper, incorporating wind intensity based on relevant references. According to various climatic conditions, including temperate, arid, and tropical regions, three specific regions were selected to establish a wind dispersal model that accurately reflects the density function distribution of dispersal distance. Additionally, dandelions growth is influenced by a multitude of factors, encompassing temperature, humidity, climate, and various environmental variables that necessitate meticulous consideration. Based on Factor Analysis model, which completely considers temperature, precipitation, solar radiation, wind, and land carrying capacity, a conclusion is presented, indicating that the growth of seeds is primarily influenced by plant attributes and climate conditions, with the former exerting a relatively stronger impact. Subsequently, the remaining two plants were chosen based on seed weight, yielding consistent conclusion.

Can preoperative planning using IRIS™three-dimensional anatomical virtual models predict operative findings during robot-assisted partial nephrectomy?

Objective To evaluate the predictive validity of IRIS™(Intuitive Surgical®,Sunnyvale,CA,USA)as a planning tool for robot-assisted partial nephrectomy(RAPN)by assessing the degree of overlap with intraoperative execution.Methods Thirty-one patients scheduled for RAPN by four experienced urologists were enrolled in a prospective study.Prior to surgery,urologists reviewed the IRIS™three-dimensional model on an iphone Operating System(iOS)app and completed a questionnaire outlining their surgical plan including surgical approach,and ischemia technique as well as confidence in executing this plan.Postoperatively,questionnaires assessing the procedural approach,clinical utility,efficiency,and effectiveness of IRIS™were completed.The degree of overlap between the preoperative and intraoperative questionnaires and between the planned approach and actual execution of the procedure was analyzed.Questionnaires were answered on a 5-point Likert scale and scores of 4 or greater were considered positive.Results Mean age was 65.1 years with a mean tumor size of 27.7 mm(interquartile range 17.5-44.0 mm).Hilar tumors consisted of 32.3%;48.4%of patients had R.E.N.A.L.nephrometry scores of 7-9.On preoperative questionnaires,the surgeons reported that in 67.7%cases they were confident that they can perform the procedure successfully,and on intraoperative questionnaires,the surgeons reported that in 96.8%cases IRIS™helped achieve good spatial sensation of the anatomy.There was a high degree of overlap between preoperative and intraoperative questionnaires for the surgical approach,interpreting anatomical details and clinical utility.When comparing plans for selective or off-clamp,the preoperative plan was executed in 90.0%of cases intraoperatively.Conclusion A high degree of overlap between the preoperative surgical approach and intraoperative RAPN execution was found using IRIS™.This is the first study to evaluate the predictive accuracy of IRIS™during RAPN by comparing preoperative plan and intraoperative execution.

Modeling of multiphase flow in low permeability porous media:Effect of wettability and pore structure properties

Multiphase flow in low permeability porous media is involved in numerous energy and environmental applications.However,a complete description of this process is challenging due to the limited modeling scale and the effects of complex pore structures and wettability.To address this issue,based on the digital rock of low permeability sandstone,a direct numerical simulation is performed considering the interphase drag and boundary slip to clarify the microscopic water-oil displacement process.In addition,a dual-porosity pore network model(PNM)is constructed to obtain the water-oil relative permeability of the sample.The displacement efficiency as a recovery process is assessed under different wetting and pore structure properties.Results show that microscopic displacement mechanisms explain the corresponding macroscopic relative permeability.The injected water breaks through the outlet earlier with a large mass flow,while thick oil films exist in rough hydrophobic surfaces and poorly connected pores.The variation of water-oil relative permeability is significant,and residual oil saturation is high in the oil-wet system.The flooding is extensive,and the residual oil is trapped in complex pore networks for hydrophilic pore surfaces;thus,water relative permeability is lower in the water-wet system.While the displacement efficiency is the worst in mixed-wetting systems for poor water connectivity.Microporosity negatively correlates with invading oil volume fraction due to strong capillary resistance,and a large microporosity corresponds to low residual oil saturation.This work provides insights into the water-oil flow from different modeling perspectives and helps to optimize the development plan for enhanced recovery.

Integrated Hydrological Modeling of the Godavari River Basin in Maharashtra Using the SWAT Model: Streamflow Simulation and Analysis

Hydrological modeling plays a crucial role in efficiently managing water resources and understanding the hydrologic behavior of watersheds. This study aims to simulate daily streamflow in the Godavari River Basin in Maharashtra using the Soil and Water Assessment Tool (SWAT). SWAT is a process-based hydrological model used to predict water balance components, sediment levels, and nutrient contamination. In this research, we used integrated remote sensing and GIS data, including Digital Elevation Models (DEM), land use and land cover (LULC) maps, soil maps, and observed precipitation and temperature data, as input for developing the SWAT model to assess surface runoff in this large river basin. The Godavari River Basin under study was divided into 25 sub-basins, comprising 151 hydrological response units categorized by unique land cover, soil, and slope characteristics using the SWAT model. The model was calibrated and validated against observed runoff data for two time periods: 2003-2006 and 2007-2010 respectively. Model performance was assessed using the Nash-Sutcliffe efficiency (NSE) and the coefficient of determination (R2). The results show the effectiveness of the SWAT2012 model, with R2 value of 0.84 during calibration and 0.86 during validation. NSE values also ranged from 0.84 during calibration to 0.85 during validation. These findings enhance our understanding of surface runoff dynamics in the Godavari River Basin under study and highlight the suit-ability of the SWAT model for this region.

Combining field data and modeling to better understand maize growth response to phosphorus(P) fertilizer application and soil P dynamics in calcareous soils

We used field experimental data to evaluate the ability of the agricultural production system model (APSIM) to simulate soil P availability,maize biomass and grain yield in response to P fertilizer applications on a fluvo-aquic soil in the North China Plain.Crop and soil data from a 2-year experiment with three P fertilizer application rates(0,75 and 300 kg P_(2)O_(5) ha^(–1)) were used to calibrate the model.Sensitivity analysis was carried out to investigate the influence of APSIM SoilP parameters on the simulated P availability in soil and maize growth.Crop and soil P parameters were then derived by matching or relating the simulation results to observed crop biomass,yield,P uptake and Olsen-P in soil.The re-parameterized model was further validated against 2 years of independent data at the same sites.The re-parameterized model enabled good simulation of the maize leaf area index (LAI),biomass,grain yield,P uptake,and grain P content in response to different levels of P additions against both the calibration and validation datasets.Our results showed that APSIM needs to be re-parameterized for simulation of maize LAI dynamics through modification of leaf size curve and a reduction in the rate of leaf senescence for modern staygreen maize cultivars in China.The P concentration limits (maximum and minimum P concentrations in organs)at different stages also need to be adjusted.Our results further showed a curvilinear relationship between the measured Olsen-P concentration and simulated labile P content,which could facilitate the initialization of APSIM P pools in the NCP with Olsen-P measurements in future studies.It remains difficult to parameterize the APSIM SoilP module due to the conceptual nature of the pools and simplified conceptualization of key P transformation processes.A fundamental understanding still needs to be developed for modelling and predicting the fate of applied P fertilizers in soils with contrasting physical and chemical characteristics.

Modeling and analysis of air combustion and steam regeneration in methanol to olefins processes

Light olefins is the incredibly important materials in chemical industry.Methanol to olefins(MTO),which provides a non-oil route for light olefins production,received considerable attention in the past decades.However,the catalyst deactivation is an inevitable feature in MTO processes,and regeneration,therefore,is one of the key steps in industrial MTO processes.Traditionally the MTO catalyst is regenerated by removing the deposited coke via air combustion,which unavoidably transforms coke into carbon dioxide and reduces the carbon utilization efficiency.Recent study shows that the coke species over MTO catalyst can be regenerated via steam,which can promote the light olefins yield as the deactivated coke species can be essentially transferred to industrially useful synthesis gas,is a promising pathway for further MTO processes development.In this work,we modelled and analyzed these two MTO regeneration methods in terms of carbon utilization efficiency and technology economics.As shown,the steam regeneration could achieve a carbon utilization efficiency of 84.31%,compared to 74.74%for air combustion regeneration.The MTO processes using steam regeneration can essentially achieve the near-zero carbon emission.In addition,light olefins production of the MTO processes using steam regeneration is 12.81%higher than that using air combustion regeneration.In this regard,steam regeneration could be considered as a potential yet promising regeneration method for further MTO processes,showing not only great environmental benefits but also competitive economic performance.

MATHEMATICAL MODELING AND BIFURCATION ANALYSIS FOR A BIOLOGICAL MECHANISM OF CANCER DRUG RESISTANCE

Drug resistance is one of the most intractable issues in targeted therapy for cancer diseases.It has also been demonstrated to be related to cancer heterogeneity,which promotes the emergence of treatment-refractory cancer cell populations.Focusing on how cancer cells develop resistance during the encounter with targeted drugs and the immune system,we propose a mathematical model for studying the dynamics of drug resistance in a conjoint heterogeneous tumor-immune setting.We analyze the local geometric properties of the equilibria of the model.Numerical simulations show that the selectively targeted removal of sensitive cancer cells may cause the initially heterogeneous population to become a more resistant population.Moreover,the decline of immune recruitment is a stronger determinant of cancer escape from immune surveillance or targeted therapy than the decay in immune predation strength.Sensitivity analysis of model parameters provides insight into the roles of the immune system combined with targeted therapy in determining treatment outcomes.

Mathematical Modeling of Cell Polarity Establishment of Budding Yeast

The budding yeast Saccharomyces cerevisiae is a powerful model system for studying the cell polarity establishment.The cell polarization process is regulated by signaling molecules,which are initially distributed in the cytoplasm and then recruited to a proper location on the cell membrane in response to spatial cues or spontaneously.Polarization of these signaling molecules involves complex regulation,so the mathematical models become a useful tool to investigate the mechanism behind the process.In this review,we discuss how mathematical modeling has shed light on different regulations in the cell polarization.We also propose future applications for the mathematical modeling of cell polarization and morphogenesis.