动脉中脉搏波传播分析英文版.doc

ANALYSE OF PULSE WAVE PROPATION IN ARTERIEAbstract: Based upon the blood vessel of being regarded as the elasticity tube, and that the tissue restricts the blood vessel wall, the rule of pulse wave propagation in blood vessel was studied. The viscosity of blood, the elastic modulus of blood vessel, the radius of tube that influenced the pulse wave propagation were analyzed. Comparing the result that considered the viscosity of blood with another result that did not consider the viscosity of blood, we finally discover that the viscosity of blood that influences the pulse wave propagation can not be neglected; and with the accretion of the elastic modulus the speed of propagation augments and the press of blood stream also heightens; when diameter of blood vessel reduces, the press of blood stream also heightens and the speed of pulse wave also augments. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of human disease.IntroductionThe blood vessels geometric form and mechanical characters are able to change because of the location pathological changes, and this change inevitably induces the aberrance that the characters of pulse wave propagation. From the later stage of the fifties of the 20th century more and more people have studied it. If we well and truly identify the pathological changes from the pulse wave signal that is checked without wound, this is undoubtedly an amazing contribution to medical science. However if we achieved this purpose, we must have understood the rule of pulse wave propagation in arteries system.When blood is flowing in the blood vessel, the pressure of blood, the speed of blood, the flux and the diameter of the blood vessel do the periodic change with time, and the wave form is complicated. Pulse wave is formed by heart beat along arteries blood vessel and the periphery blood flow propagation medium-the elasticity of arteries, the size of the blood vessel, the density and viscosity of blood etc, especially, it has the tight relationship with the arteries elasticity, caliber and thickness. The experiment finds the bigger the elasticity of the arteries blood vessel is(namely the bigger the accilimation is), then the less the spread of the pulse wave is; the faster and faster along the main arteries, aorta usually and smaller arteries. For a long time, people have done a large amount of work in study on the speed of pulse wave 1,2. The earliest quantitative description of pulse wave speed was educed by the British doctor and physical scientist Thomas Young 1 in 1808. In the article “Discussed the Blood Flow” in 1808, he firstly expounded the relation between the pulse wave speed and the blood vessel elasticity. Euler in 1755, Korteweg in 1878, Lambert in 1958, respectively, brought forward the non-linear distribution parameter model, analysed the non-linear characteristic of pulse wave propagation in the arteries, explained the kinds of sounds like pop in the arteries, analysed the forming and spread of the shock wave in the arteries. Modern analysis of the pulse wave propagation began from the fifties of the 20th century, and the most representative researchers from the fifties to the seventies are Womersley, McDanald, Bergel, Fung and Gang Xiao-tian3-5, who have set up the linear model of the pulse wave in the arteries, and developed the Windkessel model, discussed the law of arteries pulse flow in this basis. A lot of scholars in our country have made a large amount of study in the pulse wave propagation. In 1998, Wang Ying-xiao, Wu Wang-yi6 firstly studied the propagation of the pressure and flow pulse wave in the arteries of human body, on the basis of the heart and dynamic coupling of blood vessel; in 2000, Yao Da-kang 7 discussed the impact that the viscosity of blood to pulse wave propagation and analysed the influence that blood viscosity to the speed and decaying of the pulse wave and wave; Xie Guan-mo8 set up the theory model of the non-linear blood flow pulse wave propagation in the arteries in 2001. In order to analyse the qualitative law, we do certain simplifying in the article. Firstly the viscosity of blood is neglected, making its linearization and educing its universal law. On this basis, studying the impact of the propagation of pulse wave by the blood viscosity. We suppose the blood as Newton fluid, then the blood circulation obeys Navier-Stokes equation. The difficulty that discussion of non-linear particularity of the blood flows in arteries is the Navier-Stocks equation that describes the blood movement has the non-linear migration acceleration, and it brings very great difficulty to solve the equation. Firstly, we analyse the influence of flow character by the non-linear migration acceleration, secondly, we analyse the influence of flow character by the viscosity item in the equation.1 Influence of Flow Character by Migration Acceleration1.1 Equation of pressure wave propagationThe viscosity item is neglected in the equation of motion, namely, we suppose the blood as the inviscid ideal fluid, we also think it is one-dimensional to flow, at the same time, we assume the arteries blood vessel is the thin wall Hooke elastomer, and the blood vessel is supposed as the symmetrical elastic pipe in the nature with its radius R. So the law of blood propagation in arteries contents to Eq.(3) (1) (2)In these equations, is the speed of flowing of the blood, is the pressure of the blood, is the density of the blood, respectively are coordinate and time variables, is the average area of section, which changes with different coordinate and time. Define the relationship between the stress and the strain as the flowing equation:Where is the elastic modulus of blood vessel. So the area of section is The equation (2) is changed into (3)1.2 Movement equation of blood vessel wall Usually, blood vessel is regarded under the function of internal pressure (suppose that outside pressure is zero), and the radial stress is far smaller than the axial stress and circumfereftial stress, so it is ignored. The blood vessel is restrained by the connective tissue, so the axial stress, stain are all very small. Under the function of blood pressure, the radial expand of blood vessel is remarkable. We ananlyse the relationship with the circumferential stress and the internal pressure p of the blood vessel wall, and consider the movement equation of blood vessel wall in the section of arteries in balance under the function of the internal pressure. Womersley thinks that the connective tissue around the arteries is the additional layer with symmetrical density, and introduces the effective thickness of the blood vessel wall:Where, and, respectively, denote the density, radius, thickness of layer around blood vessel; ,and , respectively, denote the density, radius, thickness of blood vessel wall. In fact, it means that there is a mass attaches to blood vessel wall when we introduce the effective thickness of the blood vessel wall, but this part does not participate in the radial elastic transmutation of the blood vessel wall. We only consider the section area of blood vessel changes with the blood pressure. The axial motion restrained by the surrounding tissue is very small, and can be ignored. The movement equation of the blood vessel wall is (4)Applying the relationship with the stress and the strain, Eq.(4) is changed intoThe equation (2),(3),and (5) are the basic equation to study the strain. Making the following change of variables, suppose, , , , We can get (6) (7) (8)1.3 Solving equationAdopting small perturbation laws to solve above-mentioned equations. Taking, , Into Eqs.(6)-(8) and only choosing relation with , we can get the linear equation group: (9) (10) (11)Expunging, we get the following four-order differential equations: (12)Similarly, on the basis of the Womersiey, we suppose the form of strain wave as follows: (13)Where, is the speed of wave, . Taking Eq.(13) into (12), we can get the chromatic dispersion connection of pulse wave like that: (14)From Eq.(14), the conclusion can be got that the chromatic dispersion exists in the course of the strain wave propagation along blood vessel and high frequency harmony wave velocity is obviously higher than the low frequency harmony wave velocity. So the propagation equation of the strain wave is (15)Where,.here,being the harmony wave number of the round frequency ,represents the special periodicity of wave, and corresponds with the phase difference between the two points that is apart unit distance in the course of harmony wave propagation.2 Viscosity Item Influence to Pulse Wave PropagationWhen we analysis the viscosity item influence to pulse wave propagation, the inertia item of blood can be neglected, the basic equation of the blood circulation is simplified: (16)Where, is the movement viscosity of the blood. According to the wave propagation equation (15) of not considering the blood viscosity, suppose .Under this prompting, the flow speed of blood is . Predigesting this equation, get (17)This is an imaginary quantum Bessel equation, and its solution isSo the speed of the blood stream is (18)Doing the same change of variable, take it into Eq.(9), (19)Where, . Get (20)Under the circumstances, the full of the viscosity liquid elasticity tube is restrained by strong limit along the axis, we can getWhere, , is Poisson ratio, Womersley number is. From above equation, when the blood viscosity is considered, is not only related with the elastic modulus of blood vessel and viscosity of the blood etc, but also related with frequency of the Womersley number tends to infinite, tends to zero, and .From the above analyse, no matter the blood viscosity is considered or not, the propagation of the pulse wave is related with the coefficient of propagation. The coefficient can be measured through medical means. Therefore, its necessity to labor the propagation of the wave by the coefficient.2.1 Analysing result Taking the former parameter into ,get (21a) (21b)The rapidity of the pulse wave is related with many kinds of physiology factors of the human body. We combine the equation to do the analysis for several kinds of the most direct and most important factors.From Fig.1, with the increasing of coefficient , the amplitude of the wave is reduced while considering the blood vessel viscosity; when not considering the blood viscosity, the amplitude does not change in the course of propagation. High frequency swing of wave reducing proves the blood viscosity makes undulation lose its energy, and the higher the frequency is, the bigger the ullage is.(a) No considering the viscosity of blood(b)Considering the viscosity of bloodFig.1 The influence of the wave propagation by difference k2.2 Elasticity of arteriesThe arteries elasticity is a physics foundation of forming the pulse wave. From fig.2(a),elastic modulus does not influence to the amplitude of the wave when the viscosity is not considered, and the amplitude is totally coincident. When considering the viscosity, from the expression (21) of the pulse rapidity and Fig.2(b),we can find when the elastic modulus of the arteries wall increases, the speed of the pulse wave also increases .Displaying on clinic, the blood vessel stiffens with the growth of age or other reasons, the elastic modulus accretion, the amplitude of wave accretion, the pressure accretion, then the propagation rapidity of the pulse wave obviously becomes large. For example, datum10 indicates the 20-year-old young people whose average rapidity of pulse wave is 6.1m/s,but the 80-year-old gaffer whose average rapidity of pulse wave is 8.5m/s,and the increase value of the pressure endured by the blood vessel has a maximum.From Eq.(21) and Fig.3,the amplitude of wave does not change when the viscosity is not considered; the pressure swing of wave increases, propagation rapidity of the pulse wave largens when the viscosity is consider and the radius of the blood vessel lessens. Displaying on clinic, the adult arteries become narrower with the growth of the age or the scleratheroma etc, and the pressure of arteries heightens. From the analyzing wave rapidity formula, the viscosity of blood makes the propagation of pulse wave have put-off on phase.(a) No considering the viscosity of blood(b)Considering the viscosity of bloodFig.2 The influence of the wave propagation with different elastic modulus(a) No considering the viscosity of blood(b)Considering the viscosity of bloodFig.3 The influence of the wave propagation with different radius of blood vesselConclusion In conclusion, the propagation rapidity of the pulse wave depends on the physical characteristic and geometrical characteristic of arteries, especially, the elasticity characteristic of arteries. Besides above-mentioned factors, the pulse wave rapidity is influenced by blood density, viscosity Poisson ratio and the restraint by surrounding tissue and other kinds of factors, and there factors are closely related with human bodys other physiological pathology factors, which make the law of pulse wave propagation become more and more complicated. Results finally show, whether the blood viscosity is considered or not, the amplitude of corresponding frequency reduces to some extent while the blood viscosity is considered. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of huma