Chvala Aleksey Alekseyevich

Introduction

Researching of the principles which is underlying in a structure and functional principles of the system of blood circulation is extremely actual scientific problem. It is caused not only complexity of fundamental aspects, but also the practical interest connected with wide prevalence of vascular diseases. High frequency of occurrence and special weight vascular diseases demand improvement of knowledge in this area and development of new, effective ways of their diagnostics and treatment.

The review of existing researches

Elastic properties of arteries of various type are insufficiently studied It is connected by that during biomechanical tests the artery behaves in the beginning as an elastic or is viscous-elastic body, and since the certain moment reacts to deformation as plastic or is viscously-plastic. Methods of research of elastic and plastic components now does not exist. The decision of a problem of definition gemodinamic resistance of an arterial channel is connected with overcoming big difficulties. For this purpose it is necessary to know geometry and topology, and also elastic characteristics of each part of an arterial tree of the person. In the national and foreign literature there are only fragmentary data about some quantitative parameters of separate sites of an arterial channel of the big circle of blood circulation that is connected with special difficulties of registration of length, diameter and topological characteristics of arteries of fine diameter, especially intraorgan. Optimum way of the decision of the given question is mathematical modelling division of arteries up to a level micro channels. Thus it is necessary to consider the general and specific morpho-functional laws underlying the organization of arterial trees.

The task of work

The basic task is creation a specialized computer system for modelling gemodinamic structure of an arterial channel of the human, based on morphometrical data about extraorgan parts of an arterial channel and on the basis of genetic algorithm. The solved problem is exclusively important for modern medicine.

The theoretical analysis

A Genetic Algorithm is a computational model that emulates biological evolutionary theories to solve optimisation problems. A GA comprises a set of individual elements (the population) and a set of biologically inspired operators defined over the population itself. According to evolutionary theories, only the most suited elements in a population are likely to survive and generate offspring, thus transmitting their biological heredity to new generations. In computing terms, a genetic algorithm maps a problem on to a set of (binary) strings3, each string representing a potential solution. The GA then manipulates the most promising strings searching for improved solutions. A GA operates typically through a simple cycle of four stages: creation of a "population" of strings, evaluation of each string, selection of "best" strings, and genetic manipulation, to create the new population of strings. In each cycle a new generation of possible solutions for a given problem is produced. At the first stage, an initial population of potential solutions is created as a starting point for the search process. Each element of the population is encoded into a string (the chromosome), to be manipulated by the genetic operators. In the next stage, the performance (or fitness) of each individual of the population is evaluated, with respect to the constraints imposed by the problem. Based on each individual's fitness a selection mechanism chooses "mates" for the genetic manipulation process. The selection policy is ultimately responsible for assuring survival of the best fitted individuals. The combined evaluation/selection process is called reproduction. The manipulation process employs genetic operators to produce a new population of individuals (offspring) by manipulating the "genetic information", referred to as genes, possessed by members (parents) of the current population. It comprises two operations, namely crossover and mutation. Crossover is responsible for recombining the genetic material of a population. The selection process associated to recombination, assure that special genetic structures, called "building blocks", are retained for future generations. The building blocks then represent the most fitted genetic structures in a population. Nevertheless, the recombination process alone can not avoid the loss of promising building blocks in the presence of other genetic structures, which could lead to local minima. Also, it is not capable to explore search space sections not represented in the population's genetic structures. The mutation operator comes then into action. It introduces new genetic structures in the population by randomly modifying some of its building blocks. It helps the search algorithm to escape from local minima's traps. Since the modification introduced by the mutation operator is not related to any previous genetic structure of the population, it allows the creation of different structures representing other sections of the search space. The crossover operator takes two chromosomes and swaps part of their genetic information to produce new chromosomes. This operation is analogous to sexual reproduction in nature. After the crossover point has been randomly chosen, the portions of the parent strings P1 and P2 are swapped to produce the new offspring strings. Mutation is implemented by occasionally altering a random bit in a string. A number of different genetic operators have been introduced since this basic model was proposed by Holland. They are, in general, versions of the recombination and genetic alteration processes adapted to the requirements of particular problems. Examples of other genetic operators are: inversion, dominance, genetic edge recombination, etc. The offspring produced by the genetic manipulation process originate the next population to be evaluated. Genetic Algorithms can either replace a whole population (generational approach) or theirs less-fitted members only (steady-state approach). The creation-evaluationselection- manipulation cycle is repeated until a satisfactory solution to the problem is found. The description of the genetic algorithm computational model given in this section presented an overall idea of the steps needed to design a genetic algorithm. However, real implementations, as exemplified in the next section, have to consider a number of problem dependent parameters such as the population size, crossover and mutation rates, convergence criteria, etc. GAs are very sensitive to most of these parameters.

Experimental researches

On the basis of the actual material which has been saved up by employees of faculty of human anatomy of Donetsk medical university, have been carried out researches of blood system in view of features of topology for each body of the person. All model of a blood channel has treelike structure (the cyclic structure of vessels is present only at fabrics of intestines). In treelike models each vessel can be parent only for two vessels, i.e. one vessel leaves only two vessels of unequal diameters. Dependences of diameter of a greater proceeding vessel on diameter of a parent vessel are necessary for construction of model, dependence of diameter of a smaller proceeding vessel on diameter of a parent vessel and dependence of length of all branch on diameter of a parent vessel which have been received by means of regression analysis (Fig.1).

Figure 1 - Dependence of length of a blood branch on internal diameter of a parent artery

The algorithm of reception of diameters of vessels of all model can be presented of next steps:
1. Calculates two vessels, starting from parent one
2. Each of them is accepted to a parent artery
3. For each of the received vessels action 1 repeats
This process proceeds until diameters of counted vessels do not become less than the set limit. The sufficient limit of accuracy necessary for the decision of practical problems{tasks}, the size of diameter of vessels up to 0.1 mm (a level of capillaries) is. For definition of lengths of counted vessels treelike model is consideredas a unit, instead of as sites independent from each other. The model is considered upside-down (from a level of capillaries to a parent artery). For each body in an organism of the person the laws of an arrangement of vessels operate, therefore at modelling a blood channel features of morphological structure of each body are considered. For vessels of a kidney of function for creation of model look like (1). For other bodies similar functions are received.

(1)

Dmax (D) - dependence of diameter of a greater proceeding vessel on diameter of a parent vessel, Dmin (D) - dependence of diameter of a smaller proceeding vessel on diameter of a parent vessel, Lv (D) - dependence of length of all branch on diameter of a parent vessel. For an estimation of resistance of separate blood vessels is used next equation:

(2)

where: R - hydrodynamical resistance, is blood viscosity, L - length of a vessel, D - internal diameter of a vessel, 128 and pi are constants. For calculation of volumetric speed of a current and pressure in them it is necessary to consider character of connection of vessels (consecutive or parallel). At consecutive connection of vessels their resistance develop:

(3)

and at parallel connection there are return sizes of resistance:

(4)

Resistance intraorgan vessels down to a level of division of capillaries was also use of data of model of division of the vessels, described above. Except for treelike structure in an organism of the person in fabrics of intestines there is a cyclic model of a blood channel. Construction of model for such type of vessels differs from treelike model, but any cyclic model comes to an end with the branched out network of treelike vessels. Usually internal diameter of cyclic vessels is no more 0.7 mm. At smaller diameter the cyclic structure passes in treelike.

Figure 2 - Program Vasculograph

Result of these researches became development of program Vasculograph (рис.2). It allows to build the planar image of a blood channel based on a data which received experimental by of corrosion preparations of bodies of the person or received on the certain mathematical dependences. The program can define geometrical parameters of a blood channel such as length, width and some others.

Conclusions

In arterial channel the laws similar to laws in an electric circuit and change of resistance of one site inevitably operate leads to change of a current not only in this site, but also in system as a whole. Therefore in conditions of plural defeat of vessels that is almost always observed at an atherosclerosis, the doctor cannot predict precisely enough, what changes of blood circulation will happen at this or that operative manual so, cannot choose an optimum variant of operation for each concrete patient. In the future in clinical conditions practical application of gemodenami model will enable objective planning of actions on reconstruction of pathologically changed arterial channel. Using a program complex will help to lower probability of intraoperational and early postoperative complications.

References

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Note: Presented abstract is not a final version of master's dissertation. The full version is planned to be completed by December 2006. For the final version of master's work please e-mail the author.