SIMULATION OF 3D TRANSIENT FLOW PASSING THROUGH AN INTESTINAL ANASTOMOSIS BY LATTICE-BOLTZMANN METHOD

1Dr.Sc, Professor of Department of Computer Engineering, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine 2Dr.Sc, Professor, Head of Department of Computer Engineering, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine 3Dr.Sc, Professor of Department of Computer Engineering, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine


NOMENCLATURE
x y z , , -continuous Cartesian coordinates of threedimensional space; t -time of simulation; v -elocity of the elemental volume of liquid at the mesoscopic level; v ω -vector of velocity of the elemental fluid volume in the node ω; ( ) , , , , f x y z v t -density distribution function; f ω -density distribution function at node ω ; u -vector of macroscopic velocity; w ω -weight ratio in the node ω;

INTRODUCTION
Life science nowadays is a rapidly developing area of research. The urgent need for obtaining new results in this field of knowledge arose long ago but was constrained by the considerable complexity of research objects. Biological processes, which, as a rule, are the main objects of the study of life sciences, are characterized by variability, which is caused by the dependence on many parameters. However, the complexity of measuring these parameters is another important challenge that impedes progress in this area. In most cases, this complexity is due to the fact that there are difficulties with direct access to the object of measurement. Another negative factor is that the measuring instruments significantly affect the investigated process. Therefore, in this area, simulation has been and remains one of the most effective methods of research. This work describes the mathematical modeling of fluid motion with fine-grained inclusions through anastomoses in the lower section of the human digestive tract. The importance of this study is based on the fact that this section of the digestive tract is characterized by high peristaltic activity and, accordingly, high sensitivity to surgical intervention. A traditional approach, in this case, is to solve a boundary value problem based on a system of equations containing the Navier-Stokes equation and the flow continuity equation. The process of solving this nonstationary nonlinear boundary-value problem is cumbersome and requires a large amount of computational resources since rapid change in parameters requires a high degree of discretization and imposes additional conditions on the convergence of computational methods. Therefore, in recent years, alternative approaches have been developing rapidly using neural networks, cellular automata and other means of description for complex processes. Among the abovementioned methods, the method based on solving the lattice Boltzmann equation in each node of the discrete area is important and is called the lattice Boltzmann method.

PROBLEM STATEMENT
This work describes the development and research of a mathematical model of fluid motion with fine-grained inclusions through the anastomosis area of the lower part of the digestive tract. The objective functions of such a model are to determine the velocity field, the pressure distribution and the characteristics of sticking of finegrained inclusions in the anastomosis area. To simplify the simulation process and achieve the possibility of mathematical modeling in real time, the lattice Boltzmann method is applied. The Boltzmann equation for 3D simulation describes the probability of the fact that the elemental volume of a liquid at a time t will be in a cube with a center at the point with the coordinates ( ) , , x y z and the lengths of the edges , dx dy, dz , and the velocity of the motion of this elemental volume will be in the range from v to dv [1].
In general, the equation is represented by the expression: The function of the right-hand side of the equation (1)  , , , , f x y z v t Ω describes the nature of the collisions of elemental volumes during the movement of the fluid. In general, the operator of collisions is given by the following expression: where τ -a constant that determines the time scale required to establish a local equilibrium, δ -a parameter whose value depends on the presence and the compacted location of the fine-grained inclusions in the node at the moment of collision, ( ) ( ) , , , , eq f x y z v t -equilibrium function of the Maxwell-Boltzmann density distribution. Equations (1) and (2) describe the fluid behavior at each point of the investigated area at mesoscopic level. Therefore, there is a problem of transition from mesoscopic to macroscopic level, on which the fluid parameters are represented by the velocity field and the pressure distribution.
For such a transition, we will apply a Chapman-Enskog expansion [2] adapted to the conditions of this model, which forms the connection between the kinematic viscosity and the Boltzmann equation parameters: The Chapman-Enskog expansion allows replacing the solution of a boundary-value problem based on the Navier-Stokes equation by solving a set of Boltzmann equations, each of which describes the dynamics of the movement of elementary fluid volumes in the nodes of a discrete lattice covering the study area.

LITERATURE REVIEW
There are two main approaches for simulating the flow of fluid, which use the macroscopic and mesoscopic level of process description. A traditional approach to obtaining fluid flow parameters is involves applying numerical methods for solving a boundary value problem on the basis of the Navier-Stokes equation [3]. Methods of finite differences [4], methods of finite elements [5] and methods of finite volumes are widely used [6]. These methods have common disadvantages related to stability and convergence, since through discretization of the corresponding equations these methods are reduced to the solution of the system of linear algebraic equations by iterative methods. Despite the possibility of achieving high accuracy of the solution, it is often difficult to achieve the convergence of the iterative process, provided the variability of the input data and in areas of complex geometry. Unfortunately, multiple physical processes, including those processes occurring in biological objects, have precisely such characteristics [7]. The disadvantages of numerical methods also include the complexity of parallelizing the process of solving the boundary value problem. To overcome these and some other disadvantages, alternative approaches were explored. The lattice Boltzmann method, which describes the movement of elemental volumes of liquid at the mesoscopic level, is one of the most popular approaches. The first publications on the lattice Boltzmann method have a nearly 30-year history [8]. During this time, the method received a significant theoretical basis and became one of the popular methods of simulating fluid motion in biological objects [9]. The reason for the high popularity of lattice Boltzmann method is the fact that the calculations of fluid flow parameters are performed locally for each discrete point, which allows the use of parallel algorithms [10]. The method can also be effectively applied to simulate three dimensional areas with complex geometry [11] and to take into account the complex structure of the liquid [12]. These properties of the lattice Boltzmann method are the basis for choosing tools to study of anastomosis of the digestive tract [13,14].

MATERIALS AND METHODS
The mathematical model of ileum anastomosis of the human digestive tract is considered in this paper. The shape of a threedimensional area has the form of a rectangular parallelepiped with an internal cavity that corresponds to the investigated fragment of the ileum. The geometry of this anastomosis is based on the experience of practicing surgeons. The general view of the study area is shown in fig. 1.
Through the inner cavity flows a fluid that has a complex structure. Parameters of this fluid vary during the computational experiment in the following ranges: liquid  x y x ω of which are elements of the set: Each of the nodes belongs to one or three types: "wet" nodes or nodes in the fluid area, "dry" nodes or nodes of the environment and boundary nodes constituting the limiting surface.
We describe the evolution of elementary fluid volumes in the nodes of the area by a system of equations: where f ω -density distribution function in the direction of the mesoscopic velocity vector v ω , ( ) eq f ω -an equilibrium density distribution function corresponding to the vector v ω . We discretize the system of equations (3)  , , Marking the time step as t Δ , and spatial steps for each coordinate as we can produce the discrete version of the Boltzmann equation: Then equation (4) will look like: The simulation of the process of solving equation (5) is carried out in two stages.
1. Collision at the node ω : 2. Distribution of the obtained value of the distribution function ( ) , , , f x y z t t ω ω ω ω + Δ for the moment of time t t + Δ to the neighboring nodes of the grid: Neighborhood relationship identifies those nodes for which the interaction with the current node ω is given. For this model, the nature of such connections is unified and is called D3Q19 (fig. 2). Equation (6) includes an equilibrium density distribution function for a node ω .
We apply an expression for this function, which was first proposed in the work [15].
Let's modify equation (5), Having solved the equation according to the scheme (6), (7), we can determine the macroscopic parameters of the pressure distribution and velocity fields from the equations:

EXPERIMENTS
The main results of the simulations were obtained on the example of the classical spatial form of anastomosis, which is shown in fig. 1. During the simulation one of the important tasks was to determine the distribution of pressure and velocity fields in the study area, depending on the fluid parameters. Such research allows to determine areas with extreme values of measured parameters and, based on this information, formulate recommendations for choosing the geometric form of anastomosis, which would allow to critically reduce the risk of the so-called "blind bags" or unwanted additional cavities.
The second group of experiments aimed to determine the trajectories and the behavior of fine-grained inclusions in the fluid under study. The main parameters of the variation during these experiments were the size and number of particles, which can simultaneously be in the area we are investigating. To simplify the calculations, it is assumed that all the fine-grained inclusions have the same shape, size and density. The tasks of this group of studies were to determine the areas of accumulation of particles, which indicate the possibility of occurrence of stagnant phenomena in the application of a particular form of anastomosis.
The study used self-created software written in the algorithmic programming language Python in the environment PyCharm. Third-party software in the form of additional modules pyLBM-0.2.1, numpy-1.12.1, Cython-0.25.2, mpi4py-2.0.0, matplotlib-2.0.2 and others was also used. To prepare geometric forms of anastomosis, the package for creating 3D computer graphics Blender-2.78 was used.

RESULTS
The main results are obtained through the creation and study of a mathematical model of fluid motion with finegrained inclusions in a three-dimensional complex domain. To determine the fluid parameters, a technology based on the application of the Boltzmann lattice method was used. The created model allows to get parameters of speed of a liquid with accuracy − ε < 5 10 m/s. However, the great variability of biological processes makes absolute values of field velocities less informative when evaluating biological phenomena. The simulation results presented in a graphical form allow us to qualitatively evaluate the field of fluid velocities and draw conclusions about the correction of the form of anastomosis. For example, the velocity field shown in fig. 3, displays areas that require shape correction.
The trajectory of the motion of fine-grained inclusions at each time of the modeling time was determined as the result of the action on the particle of three forces: the interaction force of the particle with the liquid, the force that is the result of collision of this particle with the adjacent particles and the force of gravity. Studies have shown a significant dependence of the location of anastomosis in space on the trajectory of fine-grained inclusions. We can observe on fig. 4  m. The distribution of particles at speeds is determined by the tint in m/s. From the picture, it is obvious that the position of anastomosis affects the nature of the movement of finegrained inclusions, which can lead to stagnant phenomena in the anastomosis area. The nature of such influence depends essentially on the relationship between the density of the liquid and the density of the fine-grained inclusions, as well as the dynamic viscosity of the liquid, which can vary in a certain range. Figure 5 shows the dependence of the percentage of "sticking" of fine-grained inclusions, depending on the dynamic viscosity of the liquid. This study used fine-grained inclusions with size d

DISCUSSION
Simulations of fluid movement in the digestive tract of a person have a certain history, both in our country [16] and abroad [17]. Relevant mathematical models have allowed a qualitative assessment of the phenomena studied. However, the high variability of biological processes and the complexity of geometric shapes did not allow practical application of simulation results. The current level of development of computer technology and new methods of parallel computing have significantly increased the accuracy and reduced computation time due to the parallel processing of large data sets. The approach to the simulation of physical processes proposed in this paper has a number of advantages, among which is the important fact that the complexity of the mathematical model does not depend on the shape of the area. The reason for such an effect is the locality of the calculations, the nature of which is unified for each node of the discretized area of the study. In addition, the local nature of data exchange with neighboring nodes in the area lies at the heart of creating algorithms that have the property of natural parallelism. The mentioned factors allowed to build software with the ability to interact online in real time. The advantage of this approach is accounting for the fine-grained inclusions in the liquid. This significantly increased the adequacy of the mathematical model and made it possible to estimate the effect of gravity on the processes associated with undesired accumulation of these inclusions Figure 5 -The percentage of "sticking" of fine-grained inclusions in certain areas of anastomosis. The practical value of this work is that by applying these research results it was possible to reduce 10-15% negative consequences of reconstructive operations on the digestive tract. The further direction of research includes increasing area of modeling and accounting for peristaltic oscillations, as the main source of fluid flow through the area of anastomosis. A separate aspect of the study is the consideration of possible peristaltic oscillations of the actual anastomosis area, which has not yet been used in practice and has not been considered in mathematical modeling. In the research plan, there is also the construction and research of models of reconstructive operations on different parts of the digestive tract.

CONCLUSIONS
The work involves the development and research of a mathematical model that describes the parameters of the functioning of anastomosis in the lower parts of the human digestive tract. The importance of these studies is due to a significant increase in such operations and the presence of negative consequences of surgical intervention. The proposed approach to the description of the motion of the fluid in the anastomosis area is based on the use of lattice Boltzmann method. Due to the use of this method and the application of modern software development technologies, the adequacy of the mathematical model in comparison with the traditional mathematical models based on the solution of the boundary value problem based on the Navier-Stokes equation and the flow continuity equation is significantly increased. Experiments have shown that this method allows us to calculate the volume velocity in anastomosis zone with an accuracy of 5 10 − ε < m/s for the geometric parameters of the area and physical parameters of the fluid given in the work for 2-3 minutes.
A feature of the proposed mathematical model is the fact that it takes into account the complex nature of the liquid, which includes fine-grained impurities. The character of the behavior of these impurities in the field of anastomosis, depending on the dynamic viscosity of the fluid, is investigated. The second advantage of the proposed approach is the possibility to reduce the simulation time by applying parallel algorithms for calculating local mesoscopic fluid parameters for each node in the investigated area. Thus, for the first time it became possible to obtain the results of model experiments using three-dimensional mathematical modeling in real time. Актуальність. Останнім часом істотно зросла кількість реконструктивних операцій на травному тракті людини. Результати таких операцій мають прогнозовані негативні наслідки, що пов'язані з порушеннями гідродинамічних процесів у зоні анастомозу. Ці негативні наслідки можливо частково усунути шляхом вибору форми анастомозу на основі математичного моделювання. Відомі математичні моделі є громіздкими і не дозволяють отримувати результати в реальному масштабі часу. Запропонований в роботі підхід з використанням решітчастого методу Больцмана дозволяє вирішити цю проблему.