Theoretical Analysis And Numerical Simulation Of Embryo Transfer

In current world,since the number of infertility patients has increased year by year,infertility has become one of the public health problems to be solved.The advent of assisted reproductive technology has brought hope to infertile patients and helped tens of thousands of families get their children.In vitro fertilization-embryo transfer technology,also known as IVF technology,is the most widely used assisted reproductive technology today.One drawback,however,is its high price but low success rate.In the past 20 years,domestic and foreign studies have found that the last step in this technology,the embryo transfer(ET)process,is a key factor in determining the success rate of IVF.Since clinical in vivo experiments have been greatly restricted by physiology and ethics,the mechanism of embryo motion in the uterine cavity during ET has not been well understood.Therefore,based on the fundamental theory of fluid mechanics,the transfer mechanism of transferred fluid and embryo in ET process is studied by mathematical method.In this paper,three mathematical models are established to analyze the fluid flow in the uterine cavity.The first model is a simplified model of one-component laminar flow at a small Reynolds number,and the flow state of high-viscosity fluid is estimated by obtaining an approximate analytical solution of this model.The second model is an approximate model of one-component laminar flow under the critical Reynolds number.By establishing the vorticity migration model and combining with the analytical solution of the first model,the flow state of low-viscosity fluid is estimated.The third model is general equations for two component.The equations are calculated numerically,and the important control parameters of the ET process are analyzed.The results of this paper are of practice instruction to the clinical implementation of ET and contribute to the improvement and optimization of ET technology.In the first part of this paper,the transferred fluid and the intrauterine fluid are treated as the fluids of the same physical property,which represents the use of a transfer fluid having similar physical properties to intrauterine fluid in clinical ET.It is reasonable to assume the liquid as an incompressible viscous fluid and the flow process is considered as follows:the initial stationary fluid is driven by an external force acting at a certain point and produces a planar flow,of which the vorticity is concentrated.The vector form of the vorticity transport equation iswhere ?,u,x and v represent vorticity,velocity vector,position vector and kinematic viscosity,respectively.Since the Reynolds number of the flow is small,the second term(convection term)in Eq.(1)can be ignored by using the Stokes approximation.So,Eq.(1)is linearized into a diffusion equation:According to the multipole expansion of the solution of the vorticity-flow function equation and the dimensional analysis of Eq.(1),the self-similarity form solution of Eq.(2)can be obtained as follows:?=1/tg(?)Re*cos(n(?+?0)),where ? is the self-similarity variable,g is the dimensionless function to be solved,and r=|x|.Ref is the amplitude of the dimensionless force,satisfying where f is the ratio of the magnitude of the force per unit volume to the density,while KL and Kt represent the indices of the length and time dimensions of f,respectively.The general solution of Eq.(2)can be obtained by solving the ordinary differential equation,which is obtained by substituting(3)into linearized Eq.(2).According to the far field condition(?)the coefficient in the general solution of Eq.(2)can be determined,thus,the particular solution of Eq.(2)is obtained.In Eq.(5),A is a circular area of radius R and contains all the vorticity.In this paper,the particular solution obtained above is used to estimate the flow characteristics of ET.The results show that a pair of vortices is formed on both sides of the force point,and the vortex pair diffuses gradually over time but does not migrate.In addition,the approximate analytical solution of the one-component flow model is the solution of the linear differential equation,which conforms to the superposition principle.Therefore,the smooth wall boundary condition representing the uterine fundus is obtained according to the particular solution and the mirror method.The results show that the fundus of uterus limits the original flow and directs the flow to both sides of the fundus;the initial circular vortex pair is compressed and deflect toward upstream of the force point.The solution of equation(1)is obtained under the restriction of small Reynolds number,and can only characterize the diffusion of vortices in the flow.Therefore,an approximate description of vorticity migration is presented in this paper.According to the symmetry of the flow during embryo transfer,the vortices should move in a straight line along the direction of force.Based on the solution of equation(1),the vorticity on this line is always 0.Therefore in this paper,we assume that the vorticity migration velocity approximates to the potential velocity outside the centralized region of vorticityAssuming that the initial position of the vortex is the origin,the migration distance can be then obtained asThe solution of equation(2)is ?I(x,y,t)in the case of instantaneous force.According to the superposition principle of solutions combining with the consideration of vortex migration and time delay,the solution in case of continuous force can be obtained in the integral formFormula(6)and the corresponding expression of flow function can characterize the migration and diffusion of vortices.These formulas established in this paper can be named as the approximate model of one-component laminar flow below the critical Reynolds number.The results show that the streamline of one-component flow with low viscosity is mainly distributed on the right side of the action point of force,and the streamlines concentrate around the x-axis.The vortices obviously migrate to the direction of force.In the second part of this paper,a two-component flow is considered and a general form of its governing equations is established.(?)Here,the subscripts '1' and 'm' indicate the transferred liquid and mixture,respectively,and the superscript 'T' indicates the matrix transposition.?,?,u,D and Y represent density,dynamic viscosity,velocity vector,mass diffusion coefficient and mass fraction of the fluid,respectively.The variables in the equations are ?m,?m,um,Y1,p and Vl,and all these six variables are functions of time t and position(x,y,z).Therefore,Eqs.(7)is a set of nonlinear equations,which should be solved numerically.According to the specific implementation process of ET,we found that the Peclet number(the ratio of the diffusion term to the convection term)of the third equation in the Eqs.(7)is far less than 1,thus,the diffusion term could be ignored.By replacing the mass fraction with the volume fraction,Eqs.(7)becomes the Homogeneous Multiphase model.Therefore,we can employ the computational fluid dynamics software Fluent to calculate the ET process numerically.By comparing the numerical results with the experimental results,it can be concluded that the numerical results of the transferred liquid distribution pattern are basically consistent with the experimental results.In order to understand the flow mechanism of the ET process and the movement characteristics of the embryo,we've analyzed the viscosity of the transferred liquid,the injection speed,the distance between the catheter tip and the fundus,and the effect of catheter withdrawal.The results show that high-viscosity transferred liquid not only prevents the embryo from moving toward the cervix but also promotes the embryo closer to the fundus.When the injection volume of transferred liquid is the same,the distribution pattern of transferred liquid is not sensitive to the injection speed,but the low injection speed can reduce the driving force during injection.In the process of catheter withdrawal,high viscosity transferred liquid can effectively reduce the distance of the embryo moving upstream and avoid the embryo being discharged from the uterine cavity.The optimal protocols for ET are obtained by analyzing numerical results:the viscosity of the transferred liquid should be close to or greater than the viscosity of the intrauterine fluid,the catheter tip should be placed in the middle of the uterine cavity,the injection speed and the withdrawal of the transfer catheter should be slow.These protocols will provide theoretical guidance for the clinical implementation of ET and help improve the success rate of assisted reproduction.