Modelling of the acoustic properties of the larger human blood vessel
An acoustic model of a larger human blood vessel is developed in order to study the properties of an acoustic field produced by a stenotic narrowing in the vessel. This model takes into account the basic features of noise generation and propagation in human thorax from the source (turbulent pressure...
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Інститут гідромеханіки НАН України
1998
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| Cite this: | Modelling of the acoustic properties of the larger human blood vessel / A. O. Borisyuk // Акустичний вісник. — 1998. — Т. 1, N 3. — С. 3-13 — англ. |
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| author_facet | Borisyuk, A.O. |
| citation_txt | Modelling of the acoustic properties of the larger human blood vessel / A. O. Borisyuk // Акустичний вісник. — 1998. — Т. 1, N 3. — С. 3-13 — англ. |
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| description | An acoustic model of a larger human blood vessel is developed in order to study the properties of an acoustic field produced by a stenotic narrowing in the vessel. This model takes into account the basic features of noise generation and propagation in human thorax from the source (turbulent pressure fluctuations in blood flow) to the receiver resting on the skin. The low Mach number turbulent wall pressure models by Corcos, Chase, Ffowcs Williams and Smol'yakov-Tkachenko are used for describing the random noise sources in the vessel. The obtained relationships permit analyzing of dependence of the acoustic field in the thorax on the blood flow and vessel parameters, and give the possibility of finding the characteristic signs of presence of a stenotic narrowing in the vessel.
З метою дослідження властивостей акустичного поля, яке генерується стенотичним звуженням у кровоносних судинах людини, побудовано акустичну модель великої судини. Дана модель враховує основні особливості генерації і поширення шумів у грудній клітці людини від джерела (турбулентних пульсацій тиску у кровотоці) до сенсора, розташованого на шкірі. Для опису випадкових джерел шумів у судині використані моделі Коркоса, Чейза, Ффокс-Вільямса і Смольякова-Ткаченка для турбулентного пристінного тиску при малих числах Маха. Отримані співвідношення дозволяють проаналізувати залежність акустичного поля у грудній клітці від параметрів коровотоку й судини і дають можливість установити характерні ознаки наявності стенотичного звуження в судині.
С целью исследования свойств акустического поля, генерируемого стенотическим сужением в кровеносных сосудах человека, построена акустическая модель крупного сосуда. Данная модель учитывает основные особенности генерации и распространения шумов в грудной клетке человека от источника (турбулентных пульсаций давления в кровотоке) к датчику, расположенному на коже. Для описания случайных источников шумов в сосуде использованы модели Коркоса, Чейза, Ффокс-Вильямса и Смольякова-Ткаченко для турбулентного пристеночного давления при малых числах Маха. Полученные соотношения позволяют проанализировать зависимость акустического поля в грудной клетке от параметров коровотока и сосуда и дают возможность установить характерные признаки наличия стенотического сужения в сосуде.
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UDC 533.3+611.539MODELLING OF THE ACOUSTIC PROPERTIESOF THE LARGER HUMAN BLOOD VESSELA. O. BOR ISYUKInstitute of Technical Acoustics, Department of Electrical Engineering,Dresden University of Technology, Dresden, Germany,(permanent a�liation: Institute of Hydromechanics of NAS of Ukraine, Kyiv)Received 27.08.98An acoustic model of a larger human blood vessel is developed in order to study the properties of an acoustic �eld producedby a stenotic narrowing in vessel. This model accounts of the basic features of the generation and propagation of noisein the human chest from the source (turbulent pressure
uctuations in blood
ow) to a receiver resting on the skin. Thelow Mach number turbulent wall pressure models of Corcos, Chase, Ffowcs Williams and Smol'yakov {Tkachenko areused to describe random noise sources in the vessel. The relationships obtained permit one to analyse the dependenceof the acoustic �eld in the thorax on the parameters of the blood
ow and the vessel, and give the possibility of �ndingcharacteristic signs of the presence of a stenotic narrowing in vessel.INTRODUCTIONThe diagnosis of stenotic obstructions of vesselsis of a great concern to the medical clinician. Themost common method for obtaining information ona stenosis is through the use of arteriograms. Thisis, however, an invasive technique which is often ex-pensive, uncomfortable for the patient, involves riskof infection and bleeding, and it is usually only usedwhen the disease is rather advanced and accompa-nied by other symptoms. Of particular value there-fore are alternative diagnostic procedures which arenon-invasive. Since a stenosis creates a number ofabnormal
ow conditions in the vessel, several non-invasive techniques based on these abnormalities havebeen suggested in the last few decades [1 { 8].One such technique is called the method ofphonoangiography. It was initially proposed by Leesand Dewey [8] and has subsequently been studied andapplied extensively by many researchers [9 {20]. Thismethod uses the pressure �eld induced in vessels andperceived at the skin surface as sound. The mean sta-tistical characteristics of the sound �eld can then befound and analysed in order to obtain the informationabout stenosis (such as the presence, location, shape,characteristic dimensions, etc.). However, quantita-tive diagnosis of a stenosis is only possible if the fun-damental mechanisms of vascular sound generationand transmission are known. There is a number ofstudies [8 {12, 19 {22], which suggest that the mostprobable sources of blood motion sound are turbulentpressure
uctuations in the
ow.At present several low Mach number models of thepressure
uctuations are available. They can be usedin constructing an acoustic model of a lager humanblood vessel. Such a model must correctly describethe rheological properties of blood and the nature of
the
ow in the vessel, the physical and geometricalcharacteristics of the vessel, the stenotic obstruction,the structure and acoustical properties of the humanbody tissue, etc. As a result, it will correctly de-scribe the generation and transmission of noise fromthe source to the receiver resting on the skin whichis necessary for solving the inverse problem (viz. lo-cating pathology by changes in the characteristics ofnoise �eld picked up from the chest surface of a givenpatient).Analysis of the scienti�c literature shows that thecreation of an acoustic model of a separate vessel isstill far from complete. To the author's knowledge,at present only a few works [17,22 {25] can be re-ported in which the simplest models are introduced.Some of them are the models of a larger airway ofthe human respiratory system [17,23 {25] (neverthe-less, these models can be easily adapted to the case ofa larger blood vessel for the corresponding parame-ter values). Although undoubtadly important, theseworks have, however, some disadvantages. Namely,in the model suggested by Wodicka et al. [23,24] thesource of sound is represented by a determined load-ing at the inner surface of an in�nite circular pipe,and the body is introduced as an in�nite homoge-neous medium of known density, sound speed anddamping coe�cient. Such an approach does not takeinto account neither the �niteness of the human bodyvolume nor the random statistical nature of loading.A few years later Vovk et al. [17] have outlined asimple qualitative model which looked much similarto that developed by Wodicka et al. [23,24].In the more recent paper by Vovk et al. [25] two�nite coaxial circular cylinders with random turbu-lent pressures at the surface of the inner cylinderwere considered. This is more realistic approach com-c
A. O. Borisyuk, 1998 3
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13
Fig. 1. Acoustic model of a larger human blood vesselpared with the previous ones. However, here the au-thors have not considered the presence of a stenot-ic constriction in the vessel. In addition, they haveused the Corcos model [26] for turbulent wall pres-sure
uctuations, the disadvantages of which are wellknown [27, 28].Wong et al. [22] have been likely the �rst who madethe attempt to describe a vascular stenosis. Theirmodel consisted of two �nite, joined in series, isolat-ed elastic cylinders excited by inner random turbu-lent forces. One cylinder simulated an arterial steno-sis, and the other was a poststenotic segment of anartery. In general, such approach looks interesting.However, to be used in practice, it requires some clar-i�cations and completions to be made. Namely, theboundary conditions for the cylinders and the tur-bulent wall pressure �eld, as well as the in
uence ofthe body tissue on a sound �eld have to be discribedmore adequately.This paper presents an acoustic model of a largerhuman blood vessel. This model takes account of allthe above mentioned basic features of the generationand transmission of noise in the human chest fromthe source to the receiver, and permits considerationof a stenotic narrowing in the vessel. The formulationof the model and the corresponding assumptions withrespect to the ranges of its applicability are given inSection 2. Section 3 brie
y describes the incompress-ible turbulent wall pressure models which are usedin the study. The analytical solution for the noise
�eld in the thorax is constructed and qualitativelyanalysed in Section 4. The numerical analysis of thenoise �eld and its dependence on the parameters ofthe vessel and the blood
ow is carried out in Section5. Finally, the conclusions of the investigation aresummarized in Section 6.1. ACOUSTIC MODELThe nature and character of the blood loading invessels, as well as the geometry and the physicalproperties of the blood vessels and surrounding bodytissue are extremely complicated. This is the rea-son for the absence of satisfactory information aboutthem. Under these circumstances precise modellingof the vascular sound generation and transmission isextremely di�cult, and at present the question canbe only about making approximate gradual steps toconstructing an acoustic model of a larger blod ves-sel. They must be based on the generally acceptedassumptions and available data [1, 2, 17, 22{ 25].Taking this into account we restrict ourselves toconsideration of the axisymmetric, quasy-steady, ful-ly developed turbulent
ow in the vessel, two-phaseacoustic medium as the body tissue and the cylin-drical shape for the human chest. These allow us toformulate the acoustic model of a larger blood ves-sel. The geometry of the model is shown in �g. 1.Here the human thorax is represented by a �nite cir-cular cylinder, of height H and radius R, �lled withan acoustic medium of density �0 and sound speedc0 and surrounded by air. The vessel is simulated bya �nite coaxial circular pipe, of the same height Hand radius a (a�R). The turbulent blood
ow inthe pipe is characterised by mean velocity U deter-minable as the ratio of the
ow rate, averaged over acardiac cycle, to the cross-sectional area of the ves-sel. Blood has density � and sound speed c. Turbu-lent pressure
uctuations at the inner surface of thevessel (r=a) radiate sound waves that can be detect-ed at the body surface (r=R) by a special detector.The mean statistical characteristics of these soundscan then be determined and analysed in order to di-agnose the vessel condition.The corresponding axisymmetric mathematicalproblem is governed by the two-dimensional waveequation in the radial, r, and axial, z, coordinatesr2(r;z)p0 � 1c20 @2p0@t2 = 0 (1)and the following boundary conditionsp0jr=a = pt; p0jr=R = 0; p0jz=�H=2 = 0: (2)4
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13Here p0(r; z; t) and pt(z; t) are the acoustic and tur-bulent pressures, respectively, and the origin of thecylindrical coordinate system is taken in the centreof the cylinders. The random turbulent pressure �eldpt(z; t) is assumed to be temporally stationary andspatially homogeneous.The last two conditions in (2) are due to thatthe wave resistance of the body tissue is muchhigher than that of air (i. e., �0c0�9000 kg/m2s,(�c)air�420 kg/m2s).In generating noise by constricted vessel, many fac-tors play signi�cant role [29, 30]. These are the steno-sis geometry and the vessel geometry, velocity andwaveform of the
ow, viscosity and density of the
u-id, etc. The most important of these are the ratioof the minimum cross sectional area, A, to the unob-structed lumen area, A0, of the vessel (or the stenosisseverity, (1�A=A0)�100%) and the mean
ow veloc-ity. The above formulated model permits considera-tion of a simple stenotic narrowing which is charac-terized by the ratio A=A0, viz.intact vessel : a = a0;narrowed vessel : a =
a0; (3)(
2=A=A0, 0<
<1), and, via the mass conservationcondition in the narrowed and intact arteries, viz.uA = UA0; (4)takes into account the corresponding changes in themean
ow velocity in the constriction, viz.u = U (A0=A) = U=
2: (5)The arterial narrowing (3) causes not only the in-crease in the blood
ow energy (i. e. from E0�U2 inthe normal artery to E�u2 in the diseased artery)but also redistribution of the turbulence energy onthe
ow scales (i. e. from eddies of dimensions oforder D=2 convecting at speeds U and Uc in the in-tact pipe to eddies of dimensions of order d=2 con-vecting at speeds u=U=
2 and uc=Uc=
2 in the ob-structed pipe). These changes in the sound sourcestructure will cause both the increase in the radiatedacoustic power levels and the appearance of the newfrequency components in the sound �eld characteris-tics which are reported in periodicals. Consequently,these sound �eld variations can be used as the indi-cators of changes in the vessel state.2. TURBULENT WALL PRESSURE MODELSMuch work has been made in the past to de-scribe pressure
uctuations beneath an incompress-ible turbulent boundary layer [27, 31]. However, at
Fig. 2. Wavenumber-frequency spectrum �p(k; !) of tur-bulent wall pressure with !=constthe present time only several empirical and semiem-pirical models for the wavenumber-frequency spec-trum of turbulent wall pressure are available [26,32{ 34] which are more or less acceptable for prac-tice [27]. The �rst of these was proposed by Cor-cos [26]. Following his ideas, the cross-spectrumSp(�; !) of a statistically stationary and homogeneousone-dimensional wall pressure �eld, pt(z; t), at two ar-bitrary space-time points (z; t) and (z + �; t+ � ) canbe written asSp(�; !) = P (!)A(!�=Uc)e�i!�=Uc ; (6)where P (!) and A(!�=Uc) are the power (or frequen-cy) spectrum and normalized to unity cross-spectrum(i. e. A(0)=1) of pressure
uctuations, respective-ly. In practice [26, 27, 31], A(!�=Uc) is frequently ap-proximated by exponential decay function,Sp(�; !) = P (!)e��j!�=Uc je�i!�=Uc ; (7)in which � is a parameter chosen to yield the bestagreement with experiment. Substituting formula (7)into expression relating cross-spectrum, Sp(�; !), towavenumber-frequency spectrum, �p(k; !), i. e.�p(k; !) = (1=2�) 1Z�1 Sp(�; !)e�ik�d�; (8)yields the Corcos model for �p(k; !):�p(k; !) = P (!) ��[(kUc=! � 1)2 + �2] : (9)Function (9) describes quite well the structure ofthe wavenumber-frequency spectrum of the wall pres-sure only in the range of the convective wavenum-ber, k�kc=!=Uc, where �p(k; !) is sharply peaked,owing to the convective nature of the turbulence(see �g. 2). Consequently, when the total noise �eld5
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13generated by turbulence is dominated by the convec-tive wall pressure components, then the Corcos modelgives satisfactory predictions of noise. However, whenthe subconvective wall pressure components, k�kc,dominate, then the Corcos model highly overpredictsthe actually generated noise levels. This is becausethe spectral levels at wavenumbers below the convec-tive peak in spectrum (9) are 25� 35 dB higher thanthe available experimental data [27, 28, 31]. The oth-er disadvantage of expression (9) is that it violatesthe k2 dependence of the low-wavenumber spectrumas k approaches zero [27].More recent models by Chase [32], FfowcsWilliams [33] and Smol'yakov{Tkachenko [34] havebeen designed to describe the subconvective regionmore accurately. The various approaches have beenused herewith. In contrast to the purely empiricalCorcos model, which has been constructed proceedfrom examination of published experimental data,the semiempirical Chase model [32] was based on con-sideration of contributions of mean shear and pureturbulence to the spectrum of the wall pressure. Hismodel, adapted to the case of one-dimensional
ow,is �p(k; !) = �2v3�[cMk2K�5M + cTk2K�5T ];K2i = (! � Uck)2=(hiv�)2 + k2 + (bi�)�2;i = M;T; (10)With the dimensionless coe�cients hM �hT �3,cT =0:0474, cM=0:0745, bT =0:378 and bM =0:756,recommended by Chase, expression (10) can predicta convective pressure level which agrees well with theexperimentally measured, and it displays the k2 de-pendence in the low-wavenumber domain.Ffowcs Williams [33] started from Lighthill's acous-tic analogy and assumed that the velocity sourceterms were of the general Corcos form. His �nal ex-pression for the wavenumber-frequency spectrum hasthe k2 dependence in the low-wavenumber range andaccounts of the e�ects of compressibility. However,it contains a number of unknown constants and func-tions to be determined experimentally. To date, theseremain unknown, but Hwang & Geib [35] have pro-posed a simpli�ed version, which neglects the e�ectsof compressibility and assumes a speci�c form for theremaining functions. The one-dimensional version oftheir expression, slightly adjusted to agree with theCorcos parameters, is�p(k; !)=P (!)[kUc=!]2 ��[(kUc=!�1)2 + �2] ; (11)Smol'yakov and Tkachenko [34] �tted exponen-
tial curves to their measured cross-spectral densities.However, in contrast to Corcos, who directly mul-tiplied his pure longitudinal and pure lateral cross-spectra, they took the compbined cross-spectrum tobe of more sophisticated form, and Fourier trans-formed their expression. The one-dimensional versionof their wavenumber-frequency spectrum is�p(k; !) = 0:025P (!)A(!)h(!)(Uc=!)2��[F (k; !)��F (k; !)];A(!) = 0:124[1� 0:2=!� + (0:2=!�)2]1=2;!� = !��=U;F (k; !) = [A2 + (1 � Uck=!)2]�3=2;�F (k; !) = 0:995�A2 + 1 + (1:005=m1)��[(m1 � Uck=!)2 �m21]��3=2;m1 = (A2 + 1)=(1:025 +A2);h(!)=[1�0:153A(A2 + 1)=((1:025+ A2)��(0:02 + A2))1=2]�1; (12)
Although Smol'yakov and Tkachenko gave argumentsand reported experimental results supporting theirmodel, expression (12) violates the k2 behaviour ofthe spectrum �p(k; !) at low wavenumbers.3. ANALYSIS OF THE SOUND FIELD IN THETHORAXThe solution to the formulated boundary prob-lem (1), (2) is obtained by taking the Fourier trans-form, de�ned here with the conventiong(k; !) = 1(2�)2 H=2Z�H=2 1Z�1 g(z; t)e�i(kz�!t)dzdt; (13)and expanding the acoustic pressure p0(r; z; !) intoan in�nite series of trigonometric functions and cylin-drical Bessel functions of zero order, viz.p0(r; z; !) = 1Xn=1�cos(�(1)n z)���A(1)n J0(�(1)n r) + B(1)n Y0(�(1)n r)�++sin(�(2)n z)���A(2)n J0(�(2)n r) +B(2)n Y0(�(2)n r)��; (14)6
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13where the axial wavenumbers �(1)n , �(2)n and radialwavenumbers �(1)n , �(2)n are given by the followingexpressions�(1)n =(2n�1)�=H; �(1)n =qk20�(�(1)n )2;�(2)n =2n�=H; �(2)n =qk20�(�(2)n )2: (15)In relationship (14), the acoustic modes (1)n (z)=cos(�(1)n z) and (2)n (z)=sin(�(2)n z) of thechest volume describe the symmetric and anti-symmetric parts of the random sound �eld p0(r; z; !),respectively (with respect to the plane z=0), and theBessel functions J0(: : :) and Y0(: : :) describe the ra-dial distribution of the �eld.In the form taken, solution (14) satis�es the two-dimensional Helmholtz equationr2(r;z)p0(r; z; !) + k20p0(r; z; !) = 0; (16)with acoustic wavenumber k0=!=c0 in the body tis-sue, and the boundary conditions (2) on the upperside, z=H=2, and lower side, z=�H=2, of the outercylinder. The unknown amplitudesA(j)n and B(j)n canbe obtained from the conditions (2) at the surfacesof the inner and outer pipes by the use of the or-thogonality properties of the acoustic modes (j)n (z)(j=1; 2), viz.H=2Z�H=2 (1)n (z) (2)m (z)dz = 8<: 0; for m = n0; for m 6= n;H=2Z�H=2 (j)n (z) (j)m (z)dz = 8<: H=2; for m = n0; for m 6= n:After �nding the coe�cients A(j)n and B(j)n , the�nal expression for the random acoustic pressurep0(r; z; !) takes the formp0(r; z; !) = 1Xn=1�cos(�(1)n z)���G(�(1)n ; r; R)=G(�(1)n ; r = a;R)�p(1)tn (!)++ sin(�(2)n z)���G(�(2)n ; r; R)=G(�(2)n ; r = a;R)�p(2)tn (!)�: (17)Here the term that de�nes the degree of excitation ofthe acoustic mode (j)n (z) by the random turbulent
pressure pt is given by the expressionp(j)tn (!) = 2H H=2Z�H=2 pt(�; !) (j)n (�)d�; (j = 1; 2);and G(�(j)n ; r; R) = Y0(�(j)n R)J0(�(j)n r)��J0(�(j)n R)Y0(�(j)n r)is a combination of the Bessel functions.Since in the framework of the model under con-sideration the acoustic pressure vanishes at the chestsurface, the basic parameter of the noise �eld record-ed from patients is the radial acceleration [17, 25],viz. wr(r; z; !) = �(1=�0)@p0(r; z; !)=@r == (1=�0) 1Xn=1[cos(�(1)n z)�(1)n ��(F (�(1)n ; r; R)=G(�(1)n ; r=a;R))p(1)tn (!)++ sin(�(2)n z)�(2)n ��(F (�(2)n ; r; R)=G(�(2)n ; r=a;R))p(2)tn (!)]; (18)where the function F (�(j)n ; r; R) is, like G(�(j)n ; r; R),written in terms of cylindrical Bessel functions, viz.F (�(j)n ; r; R) = Y0(�(j)n R)J1(�(j)n r)��J0(�(j)n R)Y1(�(j)n r); j = 1; 2:The spectral density Pw(r; z; !) of the random �eldwr(r; z; !) at the measurement point (r; z) can be ob-tained from the relationship of statistical orthogonal-ity [31], i. e.Pw(r; z; !)�(!�!0) = hw�r (r; z; !) wr(r; z; !0)i; (19)in which the brackets denote an ensemble average,�(: : :) is the Dirac delta function, and the asteriskdenotes a complex conjugate. When the radial accel-eration (18) is substituted into expression (19), andthe required operations are performed, the spectral7
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13density becomesPw(r; z; !) = 1Xn=1(Pw(r; z; !))n == (2=�0H)2 1Xn=1[cos2(�(1)n z)j�(1)n j2��(jF (�(1)n ; r; R)j2=jG(�(1)n ; r=a;R)j2)�(1)pn (!)++ sin2(�(2)n z)j�(2)n j2��(jF (�(2)n ; r; R)j2=jG(�(2)n ; r=a;R)j2)�(2)pn (!)]; (20)where �(j)pn (!) (j=1; 2) are the modal excitationterms, de�ned in terms of the wavenumber-frequencyspectrum of the turbulent wall pressure, �p(k; !),and the shape functions of the thorax volumejS(1)n (k)j2 = 4(�(1)n )2 cos2(kH=2)=[k2� (�(1)n )2]2;jS(2)n (k)j2 = 4(�(2)n )2 sin2(kH=2)=[k2� (�(2)n )2]2
Fig. 3. Spectra of the thorax surface radial accelerationat point (r=R, z=0) produced by turbulence in a nor-mal (solid lines, Rea0=3500) and narrowed (dashed lines,
=0:7; S=(1�
2)�100%=51%; Re
a0 =5000) vesselsat the
ow velocity U=0:7 m/s, as calculated for theCorcos (curves 1 and 2), Ffowcs Williams (curves 3and 4), Smol'yakov-Tkachenko (curves 5 and 6), andChase (curves 7 and 8) models
as �(j)pn (!) = 1Z�1 jS(j)n (k)j2�p(k; !)dk: (21)Thus, the spectral density, Pw(r; z; !), of a radi-al acceleration at the measurement point (r; z) is asum of individual mode contributions, (Pw(r; z; !))n.The modal spectral densities are determined bythe two factors. Firstly, this is the degree of ex-citation of the acoustic mode, (j)n (z), which isrepresented by the modal excitation term �(j)pn (!).This term depends on the amplitudes of the wallpressure components and their spatial correlationswith the mode (j)n (z). Secondly, this is the termj (j)n (z)j2 � j�(j)n j2 � jF (�(j)n ; r; R)j2=jG(�(j)n ; r=a;R)j2which is in fact the transfer function of the thorax.It contains the geometrical characteristics of the ves-sel and thorax, and the acoustical properties of thebody volume (such as cut-o� frequencies, acousticresonances), and describes propagation of the soundwaves from the source to the receiver.All information with respect to the
ow in the ves-sel is contained in the wavenumber-frequency spec-trum, �p(k; !), of the turbulent wall pressure, andhence, via formulas (20) and (21), in the spectrumPw(r; z; !). Any changes in the
ow structure willresult in the changes in the turbulent source struc-ture (i. e. increase of the turbulence energy and itsredistribution on the
ow scales), which are then re-
ected in the function Pw(r; z; !). Therefore, thismean statistical characteristic of the random signalrecorded from the chest surface can be used to diag-nose the vessel condition by changes in the structureof the noise �eld produced by the
ow. This propertyof the radial acceleration power spectrum is actuallyused by physician in diagnosing patients.4. NUMERICAL ANALYSIS AND DISCUS-SIONIn calculating the spectral density (20), the fol-lowing values of the geometrical and physical pa-rameters of the model have been used: a0=1 cm,R=0:2 m, H=0:4 m, �=1050 kg/m3, c=1500 m/s,�=4�10�6 m2/s, �0=300 kg/m3, c0=30 m/s,U =0:4� 1 m/s, f =!=2�=1 Hz� 2 kHz. Thesemagnitudes agree well with those cited in periodi-cals [23{ 25, 29, 30, 36], and are typical for patients.The predictions for the noise spectra10 lg[Pw(r; z; f)=(Pw(f))0] (where (Pw(f))0 == (2=�0H)2P (f); Pw(f)=4�Pw(!) [31]), ob-tained for the turbulent pressure
uctuations mod-els (9) { (12), are shown in �g. 3. Here the sol-8
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13
a bFig. 4. Behaviour of the shape function j S(j)n (k) j2 and wavenumber-frequency spectrum �p(k; !) with !=const:a { behaviour of j S(j)n (k) j2 and �p(k;!) with !=const under the integral sign in (21);b { approximate behaviour of di�erent models for �p(k;!) with !=const,1 { the Corcos model; 2 { Ffowcs Williams model; 3 { Smol'yakov {Tkachenko model; 4 { Chase modelid lines correspond to a normal vessel (Rea0 == U (2a0)=�=3500) and dashed lines correspondto a narrowed vessel (
=0:7, Re
a0 =u(2
a0)=� == Rea0=
=5000).One can see that the acoustic model predicts thatpipe constriction may be identi�ed by comparisonof the noise �elds produced by intact and obstruct-ed vessels. Since the
ow energy in a narrowedpipe (E�u2) is higher than that in a normal pipe(E0�U2), the noise �eld generated by the more pow-erful turbulent sources in a partially occluded vesselis of higher intensity compared to that from the lesspowerful sources in a normal vessel. The di�erencebetween the spectral levels of these noise �elds is sig-ni�cant and, as well as the noise levels themselves,depends on the turbulence model used.The physical basis of this dependence is illustratedin �g. 4. This �gure shows the wavenumber depen-dence of the shape function jS(j)n (k)j2 and spectrum�p(k; !) in the modal excitation term (21). In generalcase, the main contribution to the spectrum �(j)pn (!)comes from the main lobe of the oscillating functionjS(j)n (k)j2 at subconvective wavenumber k=�(j)n �kcand from the convective peak of the smooth function�p(k; !) at k=kc (see �g. 4, a), viz.�(j)pn (!) � (�(j)pn (!))subconv + (�(j)pn (!))conv== � Zsubconv + Zconv�jS(j)n (k)j2�p(k; !)dk: (22)As noted in the section 3, the turbulent wall pressuremodels by Corcos, Ffowcs Williams, Smol'yakov{Tkachenko, and Chase approximate the convective
domain of the spectrum �p(k; !) equally well, butthey di�er from each other in the subconvective range(see �g. 4,b). Consequently, when the total noise �eldfrom turbulence is dominated by the convective wallpressure components, viz.(�(j)pn (!))subconv � (�(j)pn (!))conv; (23)then all the turbulence models give the same predic-tions of noise, viz.�(j)pn (!) � (�(j)pn (!))conv == ((�(j)pn )conv)Corcos = ((�(j)pn )conv)F:W: == ((�(j)pn )conv)Sm:�Tk: = ((�(j)pn )conv)Chase:However, when the contribution from the subconvec-tive wall pressure components is either of the sameorder of magnitude as that from the convective com-ponents, viz.(�(j)pn (!))subconv=(�(j)pn (!))conv = O(1); (24)or dominates, viz.(�(j)pn (!))subconv � (�(j)pn (!))conv; (25)then, due to the di�erent values of the �rst term onthe right side in (22), the turbulence models (9) { (12)give di�erent predictions of noise.Inequality (23) takes place in case of high Machnumber
ows (i. e. k(b)0 =kc=Uc=c�1 or k(b)0 =kc>1),whereas relationships (24) and (25) are associ-ated with low and extremely low Mach number
ows, respectively. Since blood
ow in the ves-sel is characterized by extremely low Mach numbers9
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13
Fig. 5. Spectra of the thorax surface radial accelerationat point (r=R, z=0) produced by turbulence in a nor-mal (solid line; Rea0=3500) and constricted (dashed line;
=0:7; S=51%; Re
a0=5000) vessels at the
ow velocityU=0:7 m/s, as calculated for the Chase model
Fig. 6. Spectra of the thorax surface radial accelerationat point (r=R, z=0) produced by turbulence in a nor-mal (solid line; Rea0=3500) and constricted (dashed line;
=0:6; S=64%; Re
a0=5833) vessels at the
ow velocityU=0:7 m/s, as calculated for the Chase model
Fig. 7. Spectra of the thorax surface radial accelerationat point (r=R, z=0) produced by turbulence in a nor-mal (solid line; Rea0=4500) and constricted (dashed line;
=0:7; S=51%; Re
a0=6429) vessels at the
ow velocityU=0:9 m/s, as calculated for the Chase model
(M =U=c < 10�3 for the physiological range of
owvelocities U <1 m/s), the case (25) takes place in thisstudy. This explains the di�erences between the noisepredictions for the various turbulence models.The most important feature of the data shownin �g. 3 is that one cannot precisely superpose thecorresponding dashed and solid lines by parallel dis-placement. This is due to the redistribution of theturbulence energy on the di�erent
ow scales andproduction of the new frequency components in thepower spectrum. The redistribution of the turbulenceenergy on the
ow scales in a narrowed pipe and thecorresponding production of the new frequency com-ponents in the acceleration power spectrum (20) isassociated with the displacement of the convectivepeak at kc=!=Uc in a normal vessel to k(d)c =
2kc ina narrowed vessel.The other feature of the curves in �g. 3 is connect-ed with the existence of the great amount of spec-tral peaks. The maxima in the spectra are identi-�ed with the acoustic resonances of the chest volume.These resonances are contained in the denominatorsjG(�(j)n ; r=a;R)j2 in expression (20), and their num-ber is determined by the chosen values of the acousticmodel parameters.As noted in the introduction, the authors of theacoustic model [25] have used the Corcos cross-spectrum (7) to describe random pressures at thevessel surface. The above given comparative anal-ysis of the noise predictions for the various turbulentwall pressure models shows that the Corcos spectrumwill overestimate the sound levels in the physiologi-cal ranges of the
ow parameter values. This makesinvestigators to be careful in applying this model todescribe turbulent sources in the vessel.The other models in the section 3 simulate thestructure of the near-wall turbulence in the subcon-vective region better than the Corcos model. The es-timates of the acoustic �elds obtained for the Chase,Ffowcs Williams, and Smol'yakov{Tkachenko spec-tra, and their comparison with the available exper-imental data show that the Chase model gives thebest predictions for these �elds [27]. It is thereforefurther used in this work for the analysis of the sound�eld.Having given the arguments in favour of the Chasemodel, let us consider �g. 5 and 6 which presentthe noise spectra (dashed lines) produced by con-stricted pipes of di�erent diameters (�g. 5:
=0:7,Re
a0 =5000, and �g. 6:
=0:6, Re
a0 =5833) at thesame
ow velocity. These curves demonstrate the in-
uence of the constriction severityS = (1�A=A0)� 100% = (1�
2)� 100%10
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13on the noise �eld. Their comparison shows that therole of this parameter is re
ected in the di�erent de-gree of turbulence of a
ow. The higher S (viz. thesmaller
) the higher the turbulence intensity in thenarrowed pipe, and therefore the higher the gener-ated noise intensity. This agrees well with the avail-able experimental data [20, 29, 30] which indicate thatstronger stenosis produces higher noise levels, andtherefore becomes easier to be detected by a physi-cian.Although the in
uence of mean
ow velocity, U , ona noise �eld generated is qualitatively similar to thatof a constriction severity (i. e., an increase of noiselevels due to an increase of a velocity), medical inter-pretation of this e�ect is di�erent. Namely, it is wellknown in medical practice that a stenosis in a givenpatient cannot be detected under resting
ow con-ditions (the patient rests), but it becomes detectableunder elevated
ow conditions (manual labour). Thisis because the di�erence between the levels of noisefrom diseased and normal vessels increases as the
ow velocity increases. This is demonstrated in �g. 5and 7 in which the noise spectra produced by intactvessel and the same narrowed vessel at di�erent
owvelocities are shown. The di�erence is seen to berather sensitive to small changes in the mean
owvelocity, U .CONCLUSIONSThe character of blood
ow in a larger vessel andthe properties of blood vessel and surrounding bodytissue are complicated, and the available informationabout them is restricted and incomplete. The appear-ance of a vascular stenosis causes signi�cant changesin the
ow structure, such as the appearance of a re-gion of a fully developed turbulence, increase of the
ow energy, redistribution of the energy on di�er-ent
ow scales, etc. These result in the increase ofthe acoustic power generated, production of the newfrequency components in the sound �eld characteris-tics, etc. In such situation, precise modelling of bothsound generation by a larger blood vessel and trans-mission of sound through the body tissue for diag-nostic purposes is di�cult, and only the �rst approxi-mate steps in adequate describing these processes canbe made. An attempt to do this was undertaken inthis study. The results obtained here are summarizedbelow.1. An acoustic model of a larger human blood vesselhas been developed. This model takes into ac-count the random statistical nature of the noisesources, the main features of the human chest
structure, and permits estimation of some e�ectsconnected with the presence a stenotic narrow-ing in the vessel.2. On the basis of this model, a relationship (20)has been obtained. It relates the mean statisti-cal characteristic of the noise �eld to the vesseldiameter and mean
ow velocity. This relation-ship also re
ects the in
uence of the geometricaland physical parameters of the human thorax onthe propagation of sound waves from the vesselto body surface.3. The model gives the possibility of illustrating themain features of the mechanism of noise produc-tion by a vessel constriction. These are the in-crease of the noise intensity and generation of thenew frequency components in the power spec-trum. The components are determined by themean
ow velocity in and the diameter of a con-striction.ACKNOWLEDGEMENTSThe author gratefully acknowledge the �nancialsupport of the Alexander von Humboldt Founda-tion (Germany) and useful comments given by Prof.V. Grinchenko and Prof. P. K�oltzsch in discussingthis work.1. Giddens D. P., Mabon R. F., Cassanova R. A. Mea-surements of disordered
ow distal to subtotal vascu-lar stenosis in the thoracic aorta of canines // Circ.Res.{ 1976.{ 39.{ P. 112{119.2. Tobin R. J., Chang I. D. Wall pressure spectra scal-ing downstream of stenoses in steady tube
ow //J. Biomech.{ 1976.{ 9.{ P. 633{640.3. Clark C. Turbulent velocity measurements in a modelof aortic stenosis // J. Biomech.{ 1976.{ 9.{ P. 677{687.4. Clark C. Turbulent wall pressure measurements in amodel of aortic stenosis // J. Biomech.{ 1977.{ 10.{P. 461{472.5. Khalifa A. M., Giddens D. P. Analysis of disorder inpulsatile
ows with application to poststenotic bloodvelocity measurement in dogs // J. Biomech.{ 1978.{11.{ P. 129{141.6. Gosling R. G., King D. H. Continuous wave ultra-sound as an alternative and complement to X-rays invascular examinations // Reneman,R.S. (ed.), Car-diovascular Applications of Ultrasound.{ New York:Elsevier, 1974.{ P. 266{282.7. Fitzgerald D. E., Carr J. Doppler ultrasound diagno-sis and classi�cation as an alternative to arteriogra-phy // Angiology.{ (.{ 1.{ P. 9.75)26283{2888. Lees R. S., Dewey C. F., Jr. Phonoangiography: anew noninvasive diagnostic method for studying ar-terial disease // Proc. Nat. Acad. Sci.{ 1970.{ 67.{P. 935{942. 11
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 139. Fredberg J. J. Pseudo-sound generation at atheroscle-rotic constrictions in arteries // Bull. Math. Biol.{1974.{ 36.{ P. 143{155.10. Fredberg J. J. Origin and character of vascular mur-murs: model studies // J. Acoust. Soc. Amer.{ 1977.{61.{ P. 1077{1085.11. Duncan, G. W., Gruber J. O., Dewey C. F, Jr., My-ers G. S., Lees, R. S. Evaluation of carotid stenosisby phonoangiography // New Eng. J. Med.{ 1975.{293.{ P. 1124{1128.12. Pitts W. H., III and Dewey C. F., Jr. Spec-tral and temporal characteristics of post-stenoticturbulent wall pressure
uctuations // ASMEJ. Biomech. Eng.{ 1979.{ 101.{ P. 89{95.13. Gavriely N., Palti Y., Alroy G. Spectral character-istics of normal breath sounds // J. Appl. Physiol.{1981.{ 50.{ P. 307{314.14. Kraman S. S. Determination of the site of productionof respiratory sounds by subtraction phonopneumog-raphy // Am. Rev. Resp. Dis.{ 1980.{ 122.{ P. 303.15. Charbonneu G., Raccineux J. L., Subraud M.,Tuchais E. An accurate recording system and itsuse in breath sounds spectral analysis // J. Ap-pl. Physiol.{ 1983.{ 55.{ P. 1120{1127.16. Cohen A., Landsberg D. Analysis and automat-ic classi�cation of breath sounds // IEEE Trans.Biomed. Eng.{ 1984.{ BME-31.{ P. 585{590.17. Vovk I. V., Grinchenko V. T., Krasnyi L. G. andMakarenkov A. P. Breath sounds: recording and clas-si�cation problems // Acoust. Phys.{ 1994.{ 40.{P. 43{48.18. Makarenkov A. P., Rudnitskii A. G. Diagnosis of lungpathologies by two-channel processing of breathingsounds // Acoust. Phys.{ 1995.{ 41.{ P. 234{238.19. Borisyuk A. O. Noise generation by
ow inpipes in presence of wall obstructions // Rep.Nat. Acad. Sci. Ukr.{ 1996.{ 11.{ P. 66{70 (inUkrainian).20. Borisyuk A. A. Noise generated by steady
ow in hu-man blood vessels in presence of stenoses // Bionika.{1998.{ 27{28.{ P. 144{151 (in Russian).21. Kim B., Corcoran W. K. Experimental measure-ment of turbulence spectra distal to stenosis //J. Biomech.{ 1974.{ 7.{ P. 335{342.22. Wang J., Tie B., Welkowitz W., Semmlow J. L.,Kostis J. B. Modeling sound generation in stenosedcoronary arteries // IEEE Trans. Biomed. Eng.{ 37.{1990.{ P. 1087{1094.23. Wodicka G. R., Stevens K. N., Golub H. L., Craval-ho E.G., Shanon D. C. A model of acoustic trans-mission in the respiratory system // IEEE Trans.Biomed. Eng.{ 1989.{ 36.{ P. 925{933.24. Wodicka G. R., Stevens K. N., Golub H. L., Shan-non D. C. Spectral characteristics of sound transmis-sion in the human respiratory system // IEEE Trans.Biomed. Eng.{ 1990.{ 37.{ P. 1130{1134.25. Vovk I. V., Zalutskii K. E., Krasnyi L. G. Acous-tic model of the human respiratory system //Akust. Phys..{ 1994.{ 40.{ P. 676{680.26. Corcos G. M. The resolution of pressure in turbu-lence // J. Acoust. Soc. Amer.{ 1963.{ 35.{ P. 192{199.27. Borisyuk A. O., Grinchenko V. T. Vibration andnoise generation by elastic elements excited by a tur-bulent
ow // J. Sound Vib.{ 1997.{ 204.{ P. 213{237.
28. Martin N. C., Leehey P. Low wavenumber wall pres-sure measurements using a rectangular membrane asa spatial �lter // J. Sound Vib.{ 1977.{ 52.{ P. 95{120.29. Young D. F. Fluid mechanics of arterial stenoses //J. Biomech. Eng.{ 1979.{ 101.{ P. 157{175.30. Mirolyubov S. G. Hydrodynamics of stenosis // Mod-ern Probl. Biomech.{ 1983.{ N 1.{ P. 73{136 (in Rus-sian).31. Blake W. K. (ed.) Mechanics of Flow-Induced Soundand Vibration. In 2 vol..{ New York: Academic Press,1986.{ 954 p.32. Chase D. M. Modeling the wavevector-frequencyspectrum of turbulent boundary layer wall pres-sure // J. Sound Vib.{ 1980.{ 70.{ P. 29{67.33. Ffowcs Williams J. E. Boundary-layer pressures andthe Corcos model: a development to incorporate low-wavenumber constraints // J. Fluid Mech.{ 1982.{125.{ P. 9{25.34. Smol'yakov A. V., Tkachenko V. M. Model of a�eld of pseudosonic turbulent wall pressures and ex-perimental data // Sov. Phys. Akust..{ 1991.{ 37.{P. 627{631.35. Hwang Y. F., Geib F. E. Estimation of wavevector-frequency spectrum of turbulent boundary layer wallpressure by multiple linear regression // Proc. Inter-national Symposium on Turbulence-Induced Vibra-tions and Noise of Structures.{ 1983, Nov. 13{18,Boston.{ P. 13{30.36. Fredberg J. J., Holford S. K. Discrete lung sounds:Crackles (rales) as stress-relaxation quadrupoles //J. Acoust. Soc. Amer.{ 1983.{ 73.{ P. 1036{1046.APPENDIX: NOMENCLATUREH { height of thorax and length of blood vessel;R { radius of thorax;a0 { radius of intact blood vessel;A { minimum cross-sectional area of blood vessel;A0 { unobstructed lumen area of blood vessel;
2 { ratio of the minimum cross-sectional area tothe unobstructed lumen area of vessel;S { severity of stenosis;� { mass density of normal blood;�0 { mass density of the body tissue;� { kinematic viscosity of normal blood;c { sound speed in normal blood;c0 { sound speed in the body tissue;U { mean
ow velocity in the intact vessel;Uc { convective velocity in the intact vessel;u { mean
ow velocity in the stenosed vessel;v� { friction velocity;Rea0 { Reynolds number of mean
ow in the intactvessel;Re
a0 { Reynolds number of a
ow in the stenosedvessel;Recr { critical Reynolds number;r; z { radial and axial coordinates, respectively;t { time;! { circular frequency;12
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1998. �®¬ 1, N 3. �. 3 { 13f = !=2� { frequency;k { wavenumber in the
ow direction;k0 = !=c0 { acoustic wavenumber of sound wavesin the body tissue;k(b)0 = !=c { acoustic wavenumber of sound wavesin blood;�(1)n , �(2)n { axial wavenumbers;�(1)n , �(2)n { radial wavenumbers;wr { radial acceleration in sound wave;pt { turbulent wall pressure;p0 { acoustic pressure;
(1)n (z), (2)n (z) { acoustic modes of the thorax vol-ume;!(1)n , !(2)n { cut-o� frequencies;jS(1)n (k)j2, jS(2)n (k)j2 { shape functions;�(1)pn (!), �(2)pn (!) { modal excitation terms;�p(k; !) { wavenumber-frequency spectrum of tur-bulent wall pressure;P (!) { power (frequency) spectrum of turbulentwall pressure;Pw(r; z; !) { spectral density of radial accelerationin the measurement point (r; z).
13
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| id | nasplib_isofts_kiev_ua-123456789-870 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1028-7507 |
| language | English |
| last_indexed | 2025-12-07T15:15:08Z |
| publishDate | 1998 |
| publisher | Інститут гідромеханіки НАН України |
| record_format | dspace |
| spelling | Borisyuk, A.O. 2008-07-03T12:44:35Z 2008-07-03T12:44:35Z 1998 Modelling of the acoustic properties of the larger human blood vessel / A. O. Borisyuk // Акустичний вісник. — 1998. — Т. 1, N 3. — С. 3-13 — англ. 1028-7507 https://nasplib.isofts.kiev.ua/handle/123456789/870 533.3+611.539 An acoustic model of a larger human blood vessel is developed in order to study the properties of an acoustic field produced by a stenotic narrowing in the vessel. This model takes into account the basic features of noise generation and propagation in human thorax from the source (turbulent pressure fluctuations in blood flow) to the receiver resting on the skin. The low Mach number turbulent wall pressure models by Corcos, Chase, Ffowcs Williams and Smol'yakov-Tkachenko are used for describing the random noise sources in the vessel. The obtained relationships permit analyzing of dependence of the acoustic field in the thorax on the blood flow and vessel parameters, and give the possibility of finding the characteristic signs of presence of a stenotic narrowing in the vessel. З метою дослідження властивостей акустичного поля, яке генерується стенотичним звуженням у кровоносних судинах людини, побудовано акустичну модель великої судини. Дана модель враховує основні особливості генерації і поширення шумів у грудній клітці людини від джерела (турбулентних пульсацій тиску у кровотоці) до сенсора, розташованого на шкірі. Для опису випадкових джерел шумів у судині використані моделі Коркоса, Чейза, Ффокс-Вільямса і Смольякова-Ткаченка для турбулентного пристінного тиску при малих числах Маха. Отримані співвідношення дозволяють проаналізувати залежність акустичного поля у грудній клітці від параметрів коровотоку й судини і дають можливість установити характерні ознаки наявності стенотичного звуження в судині. С целью исследования свойств акустического поля, генерируемого стенотическим сужением в кровеносных сосудах человека, построена акустическая модель крупного сосуда. Данная модель учитывает основные особенности генерации и распространения шумов в грудной клетке человека от источника (турбулентных пульсаций давления в кровотоке) к датчику, расположенному на коже. Для описания случайных источников шумов в сосуде использованы модели Коркоса, Чейза, Ффокс-Вильямса и Смольякова-Ткаченко для турбулентного пристеночного давления при малых числах Маха. Полученные соотношения позволяют проанализировать зависимость акустического поля в грудной клетке от параметров коровотока и сосуда и дают возможность установить характерные признаки наличия стенотического сужения в сосуде. en Інститут гідромеханіки НАН України Modelling of the acoustic properties of the larger human blood vessel Моделювання акустичних властивостей великої кровоносної судини людини Article published earlier |
| spellingShingle | Modelling of the acoustic properties of the larger human blood vessel Borisyuk, A.O. |
| title | Modelling of the acoustic properties of the larger human blood vessel |
| title_alt | Моделювання акустичних властивостей великої кровоносної судини людини |
| title_full | Modelling of the acoustic properties of the larger human blood vessel |
| title_fullStr | Modelling of the acoustic properties of the larger human blood vessel |
| title_full_unstemmed | Modelling of the acoustic properties of the larger human blood vessel |
| title_short | Modelling of the acoustic properties of the larger human blood vessel |
| title_sort | modelling of the acoustic properties of the larger human blood vessel |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/870 |
| work_keys_str_mv | AT borisyukao modellingoftheacousticpropertiesofthelargerhumanbloodvessel AT borisyukao modelûvannâakustičnihvlastivosteivelikoíkrovonosnoísudinilûdini |