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Theoretical Biology and Medical Modelling BioMed Central

Research Open Access The velocity of the arterial pulse wave: a viscous-fluid shock wave in an elastic tube Page R Painter

Address: Office of Environmental Health Hazard Assessment, California Environmental Protection Agency, P. O. Box 4010, Sacramento, California, 95812, USA Email: Page R Painter - [email protected]

Published: 29 July 2008 Received: 11 April 2008 Accepted: 29 July 2008 Theoretical Biology and Medical Modelling 2008, 5:15 doi:10.1186/1742-4682-5-15 This article is available from: http://www.tbiomed.com/content/5/1/15 © 2008 Painter; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Background: The arterial pulse is a viscous-fluid shock wave that is initiated by blood ejected from the heart. This wave travels away from the heart at a speed termed the pulse wave velocity (PWV). The PWV increases during the course of a number of diseases, and this increase is often attributed to arterial stiffness. As the pulse wave approaches a point in an artery, the pressure rises as does the pressure gradient. This pressure gradient increases the rate of blood flow ahead of the wave. The rate of blood flow ahead of the wave decreases with distance because the pressure gradient also decreases with distance ahead of the wave. Consequently, the amount of blood per unit length in a segment of an artery increases ahead of the wave, and this increase stretches the wall of the artery. As a result, the tension in the wall increases, and this results in an increase in the pressure of blood in the artery. Methods: An expression for the PWV is derived from an equation describing the flow-pressure coupling (FPC) for a pulse wave in an incompressible, viscous fluid in an elastic tube. The initial increase in force of the fluid in the tube is described by an increasing exponential function of time. The relationship between force gradient and fluid flow is approximated by an expression known to hold for a rigid tube. Results: For large arteries, the PWV derived by this method agrees with the Korteweg-Moens equation for the PWV in a non-viscous fluid. For small arteries, the PWV is approximately proportional to the Korteweg-Moens velocity divided by the radius of the artery. The PWV in small arteries is also predicted to increase when the specific rate of increase in pressure as a function of time decreases. This rate decreases with increasing myocardial ischemia, suggesting an explanation for the observation that an increase in the PWV is a predictor of future myocardial infarction. The derivation of the equation for the PWV that has been used for more than fifty years is analyzed and shown to yield predictions that do not appear to be correct. Conclusion: Contrary to the theory used for more than fifty years to predict the PWV, it speeds up as arteries become smaller and smaller. Furthermore, an increase in the PWV in some cases may be due to decreasing force of myocardial contraction rather than arterial stiffness.

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Introduction Womersley [9] gave the solution for the Lambossy model. Following the opening of the aortic valve in early systole, In addition, Womersley incorporated the elastic modulus blood pressure in the aorta rises rapidly as does the veloc- into the description of the wall of the model artery. ity of blood flow. This increase in blood pressure and Assuming that the change in the tube's radius is small, momentum travels the length of the aorta and is also that the tube is tethered to surrounding structures and that passed on to blood in arteries that branch from the aorta the mass of the tube's wall is negligible, the expression for (e.g. carotid, brachial and mesenteric arteries). The phe- the PWV for the Lambossy-Womersley model is nomenon of rapidly increasing pressure and velocity spreading from the aortic root to distal arteries is termed 2 2 32//32 cc/()/(),≈− Jiaa Ji (2) the pulse wave. 0 2 0 1/2 3/2 where α = R(ωρ/μ) , and Jm(i α) is the Bessel function The pulse wave is an example of a traveling wave in a of order m and variable i3/2α: fluid. Other examples are tsunamis and sound waves including sonic booms. In each of these, momentum n=∞ n 32/ mn+ 2 moving in the direction of the wave increases pressure ∑ ()(−+12imnna /) /[( )!!]. n=0 ahead of the wave's peak, and this increased pressure increases momentum ahead of the wave's peak. The speed 2 3/2 3/2 When α Ŭ 8, -J2(i α)/J0(i α) ≈ 1, and c/c0 is approxi- at which the peak of a shock wave moves depends on 2 3/2 3/2 2 mately 1. When α << 8, -J2(i α)/J0(i α) ≈ iα /8, and c/ physical properties of the fluid, dimensions of the space c0 is approximately proportional to R. For many years, the that bounds the fluid and physical properties of the result of Womersley has been accepted as a good approx- bounding material. For a tsunami moving through a imation for the PWV in relatively small arteries where - region of ocean of depth d, the assumptions that sea water 3/2 3/2 J2(i α)/J0(i α) differs significantly from 1 [10-12]. is incompressible and that viscous forces are negligible lead to the predicted speed (gd)1/2, where g is the acceler- Some features of arterial blood flow are not well described ation due to gravity [1]. For a shock wave moving through by the Lambossy-Womersley model. For example, the rel- an incompressible, non-viscous fluid of density ρ in a ative rate of increase in the pressure gradient that results cylindrical elastic tube of wall thickness h and elastic mod- from the power of myocardial contraction is not described ulus E, the speed, denoted c0, predicted by Korteweg [2] accurately by the simple harmonic pressure gradient and Moens [3] is assumed in the model. Furthermore, Womersley did not incorporate damping of the pressure wave in his model 1/2 c0= [(Eh)/(2ρR)] ,(1)before solving for wave velocity. Womersley also introduced a number of approximations before arriving at the The product Eh is the ratio of tension in the tube's wall to expression for the PWV. Therefore, we investigate the PWV the fractional amount of circumferential stretching, and R in a model that (1) contains a parameter that describes the is the radius of the tube measured from the central axis to relative rate of rise of the pressure gradient, (2) includes the inner face of the wall. an expression for damping of the pressure wave and (3) requires fewer assumptions and approximations in the The speed of the arterial pulse wave is commonly termed derivation of an expression for the PWV. the pulse-wave velocity (PWV). It has been shown to increase during the course of certain diseases, and this A model for the leading part of the pulse wave increase has generally been attributed to "arterial stiff- We start with the solution for the rigid-tube model of ness" [4-7]. This interpretation is consistent with Equa- Lambossy. A rigorous derivation of the solution for blood tion (1), the Korteweg-Moens Equation, which is based in volume flow rate, flow velocity and shear force has been part on the assumption that the fluid is not viscous. published by Painter et al. [13]. The velocity of flow at distance r from the central axis of the artery is Lambossy [8] introduced a model for arterial blood flow in which viscosity results in shear force on the inner wall =−itw 32/// 12 32 uA[/()][rw ie1 Ji0 ( { wr /} m rJi )/(0 a )]. of an artery and the pressure gradient is a simple har- The volume flow rate is monic function of time, A eiωt. In this model, A is a constant, and ω is the frequency of oscillation. Other QARei=−[/()][()/()].pwraa2 itw JiJi32//32 (3) constants in the model are he viscosity of blood, denoted 2 0 3/2 3/2 3/2 μ, and the density, denoted ρ. The arterial wall is a We define I(i α) = [-J2(i α)/J0(i α)]/(iωρ) and write straight, rigid cylindrical tube of radius R.

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I(iβR) is approximately equal to 1/(aρ), and K is approxi- 232itw / mately equal to 0. QAReIi= pa(). (4)

3/2 3/2 3/2 2 We define PQ(i α) as J2(i α)/[(i α) /8] and note that Now consider the fluid motion when the force function, F 3/2 2 as R goes to zero the imaginary part of PQ(i α) vanishes = πR P, is 3/2 and |PQ(i α)| goes to 1. Substitution now gives F = feate-bz, QARePiJi= [/()]()/().pm4328 itw // a32 a(5) Q 0 where a, b and f are positive real-valued constants. At time Therefore, I(i3/2α) is approximately R2/(8μ) when α2 << 8 t, this expression defines the force function of distance z and is approximately 1/(iωρ) when α2 Ŭ 8. along the tube. A point defined by a particular value of z can be traced backward in time to a point at z = 0 on the The shear force (per unit area) at the inner wall of the tube force function at time t-z/c. In the absence of damping as is a result of shear forces, the value of the force function would be identical for these two points. Therefore, feate-bz = fea(t-z/c), so that b = z/c in the absence of damping. iωt 3/2 3/2 3/2 -μ [∂u/∂r|r = R = -[ A μ/(ρiω)]e i (α/R)J-1(i α)/J0(i α), 3/2 3/2 3/2 1/2 where i (α/R)J-1(i α) = dJ0(i [ωρ/μ] r)/dr|r = R. We It is assumed that the ratio of flow rate to force gradient is constant in our model of a segment of an artery that does note that -J (i3/2α) = J (i3/2α), which is written as (i3/2α/ -1 1 not contain a branch. An equivalent assumption is that 3/2 2)PS(i α). Substitution now gives the expression for the the value of R2 does not vary significantly with decreasing shear force (per unit area), [ A eiωtR/2]P (i3/2α)/J (i3/2α). pressure along the segment. As a consequence, it follows S 0 that flow rat, Q, is proportional to ea(t-z/c) in the absence of This expression is further simplified to a(t-z/c) damping, i.e. the flow wave is described by Q0e , where Q is the flow rate at t = 0 and z = 0. +− 0 −∂maa[/ur ∂ = [ ARe /]2 itiwqPS i q J0 Pi (32// )/( Ji32 ). rR= S 0 In the absence of damping, there is no loss of momentum (6) per unit length in the velocity wave. When there is damp- The ratio of shear force per unit length, -2πRμ[∂u/∂r|r = R, ing, Equations (7) and (8) imply that the point on the to volume flow rate is denoted K. velocity wave at z = 0 at t = 0 loses momentum at specific rate -K as it travels distance z = t/c in time t. Therefore, We now replace the harmonic pressure gradient by the momentum, velocity and flow rate are reduced by the fac- exponential gradient Aeat. A solution for this exponential tor e-Kz/c as the flow wave moves a distance equal to z, and a(t-z/c) -Kz/c pressure-gradient model is generated by substituting A for volume flow rate is proportional to Q0e e . By anal- ogy to the rigid-tube model where flow rate is propor- A and a for iω in Equations (4), (5) and (6). Note that, tional to the force gradient -∂F/∂z for small R, it is with these substitutions, the Bessel functions become real- assumed that, in the model where small changes in R are valued series in which each term is a positive real number. allowed, force gradient is also proportional to volume The solutions for flow rate and shear force per unit area in flow rate (for small R). Therefore force is likewise propor- the exponential pressure wave model are, respectively, tional to ea(t-z/c) e-Kz/c, and we write

4 at at -z(a + K)/c Q = [πAR /(8μ)]e PQ(iβR)/J0(iβR)(7) F = fe e .(10) and The force gradient is

at -∂F/∂z = [(a + K)/c]feate-z(a + K)/c.(11) -μ[∂ u/∂r]|r = R = [AR/2]e PS(iβR)/J0(iβR), (8) where β =(aρ/μ)1/2. The function I(iβR) is equal to Q Substitution into Equation (7) gives divided by the force gradient πR2Aeat. When Q = [(a + K)/c]feate-z(az+K)/cI(iβR). (12) aρR2/μ << 8, (9) For an incompressible fluid in a cylindrical elastic tube, conservation of mass requires that PQ(iβR), PS(iβR) and J0(iβR) are approximately equal to 1, I(iβR) is approximately R2/(8μ), and the function K is approximately equal to 8μ/(ρR2). When aρR2/μ Ŭ 8, -∂Q/∂z = 2πR(∂R/∂t). (13)

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2 1/2 Substitution into Equation (12) gives c ≈ c0 [2+8μ/(aρR )] .(23)

[(a + K)/c]2feate-z(a + K)/c I(iβR)- [(a + K)/c]feate-z(az+K)/ This result is not in agreement with Womersley's predic- c (∂I(iβR)/∂R)(∂R/∂z) = 2πR(∂R/∂t). (14) tion that the PWV is approximately proportional to c0 multiplied by R when aρR2/μ << 8. For an elastic tube with wall thickness equal to h and elastic modulus equal to E, For the case where aρR2/μ Ŭ 8, I(iβR) is approximately equal to 1/(aρ), and K is approximately equal to 0. Substi- 2 P = Eh(R-R0)/R . (15) tution into Equation (14) leads to the Korteweg-Moens expression,

Consequently, F = πEh(R - R0), and ∂F/∂z = πEh(∂R/∂z), which is rewritten as c2 ≈ Eh/(2ρR).

-[(a + K)/c]feate-z(a + K)/c = πEh(∂R/∂z). (16) The mass of the arterial wall and surrounding tissue was not included in the above analysis. The effect of this mass Similarly, on the PWV can be assessed by writing a differential equation for its acceleration caused by the difference in fluid afeat e-z(az+K)/c = πEh(∂R/∂t). (17) pressure in the tube and the force per unit area of the inner wall on the fluid. The solution leads to the approximation Combining Equations (16) and (17) with Equation (14) 2 2 gives a description of the flow-pressure coupling (FPC) P ≈ (a [R-R0]hsρs + Eh)(R-R0)/R , associated with the pulse wave. The FPC equation will be solved for the PWV. This will be simplified by considering where hs and ρs are the thickness and density, respectively, two cases. The first is when aρR2/μ << 8. I(iβR) is approx- of the wall and surrounding tissue accelerated by the imated as R2(8μ), and the function K is approximated as increasing arterial pressure. 8μ/(ρR2). Combining Equation (16) and (17) with Equa- tion (14) leads to There are many published estimates of the modulus E for mammalian elastic arteries. Estimates from studies of the Eh/(2ρR) + 2feate-z(az+K)/c/(πρR2) = c2aK/(a + K)2. change in arterial radius with changes in pressure are usu- (18) ally between 106 dynes/cm2 and 107 dynes/cm2 [14,15]. 2 2 Therefore, the term a R hsρs may be small compared with Equation (18) is rewritten as Eh unless the rate of rise in the arterial pulse is very steep.

Eh/(2ρR)[1 + 4feate-z(a + K)/c/(πREh)] = c2aK/(a + K)2, Comparison with Womersley's derivation of the PWV Womersley [9] replaced the rigid tube of Lambossy with and combining this equation with Equations (10) and an elastic tube that expands or contracts in response to (15) leads to increasing or decreasing pressure of blood. If the tube is not tethered to surrounding tissue, it also moves axially in 2 2 response to frictional force of blood on the inner wall. Eh/(2ρR)[1 + 4(R-R0)/R] = c aK/(a + K) . (19) Womersley denoted the radial displacement of the inner When wall of the tube by ξ and the longitudinal displacement by ς. It is assumed that ξ and ς are described by harmonic functions: 4(R - R0)/R << 1, (20) ξ = D exp[iω(t - z/c)], (24) the above equation is approximated as 1 ς = E exp [iω(t-z/c)]. (25) c2 ≈ [Eh/(2ρR)](a + K)2/(aK). (21) 1 which is rewritten as The pressure of the fluid p is also assumed to be a harmonic function

cc2 //2 ≈++ aK2 Ka /. (22) 0 p = p1 exp [iω(t-z/c)]. (26) Because a/K = aρR2/(8μ) is assumed to be much smaller than 1, If c is a positive real number, this equation imposes a direction of flow for the pulse wave.

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In Equations (35) and (36) of the article by Womersley Womersley interprets the imaginary part of 1/c multiplied [9], the boundary conditions for the fluid in contact with by ω as the coefficient in the exponential damping func- the tube's inner wall are described by: tion of the wave as a function of distance. The exponential damping coefficient of time is therefore equal to the real part of c times the imaginary part of 1/c multiplied by ω. =+A1 iEw 11 C , (27) When α2 << 8, the real part of c is approximately c α/4, r0c 0 and the imaginary part of 1/c is approximately 4/(c0α). Therefore, the technique of Womersley leads to the pre- =+1 iRw A1 diction that the coefficient for the damping with time is iDw 1110{()},CF a (28) 2 c r0c approximately equal to ω and that this coefficient is approximately independent of the radius in small arteries. where ρ0 is the density of the fluid, C1 is a constant and This does not make sense because Equations (5) and (6) 3/2 3/2 3/2 F10(α) is 2J1(i α)/[i αJ0(i α)]. In Equations (37) and show that for α2 << 8 the damping coefficient is approxi- (38) of the article, the boundary conditions for the wall of mately 8μ/(ρR2), the expression used in the exponential the tube in contact with the fluid are described by: pressure gradient model to describe damping of pulse waves in small arteries. − −=−2 A111B swiE + D w D1 {( ) }, (29) hrrR c R2 There is another solution for the PWV in the above equations from the paper of Womersley. Combining equations (27) and (30) for the case where σ = 0 and E1 = 0 leads to −−m C 1 A R2 BEwsw2 i D −=w 2E {()}{1 iRF322aa+−+1 w 2 1 1}, 1 hRr 2 10 2 r c 2 r 2 R c 0 c c 2 2 2 c = ω /[iα F10(α)], (30) which can not be correct because it predicts that c does not where ρ is the density of the tube, σ is Poisson's ratio and approach the Korteweg-Moens velocity for large values of B is E/(1-σ2). R. Womersley interprets Equations (27)–(30) as a system of The equations of Womersley do not contain an expression homogenous linear equations with variables A , C , D 1 1 1 for the damping caused by shear force between the inner and E . Setting the determinant of coefficients equal to 0 1 wall of the tube and the moving liquid. If the expression gives the equation for c. A solution for c is easily found for shear force is added to the Womersley model, Equa- when σ = 0 and when the tube is tethered to surrounding tion (31) becomes structures so that E1 = 0. Combining Equations (27) and (28) gives Ri()w+ K2 A iDw = 1 {()},1 − F a 1102 2 =−1 RC1 − icwr0 D110{()}.1 F a (31) 2 c and Equation (32) becomes

Combining Equations (27) and (29) gives 2 2 (-ω hρ + Eh/R )D1 = iωA1/(iω).

2 2 (-ω hρ + Eh/R )D1 = -ρ0cC1, (32) When ω2ρ << E/R2, the above equations are combined and approximated by and dividing by Equation (31) gives 2 2 2 Eh ()iKw+ c = {-ω Rhρ/(2ρ0) + Eh/(2ρ0R)}{1-F10(α)}. 2 = − c {()}/()1 Fi10 aw 2r0R iw 3/2 3/2 Substituting -J2(i α)/J0(i α) for 1-F10(α) gives 2 When α << 8, {1-F10(α)}/(iω) is closely approximated by 1/K. Furthermore, replacing {1-F (α)}/(iω) by 1/K and 2 2 3/2 3/2 10 c = {-ω Rhρ/(2ρ0) + Eh/(2ρ0R)}{-J2(i α)/J0(i α)}. replacing iω by the constant a gives Equation (21). There- fore, it appears that the difference between Equation (21) 2 Because ω Rhρ/(2ρ0) is small compared to Eh/(2ρ0R), we and the corresponding expression derived by Womersely have for c2 in the tethered, thin-walled tube model is due largely to his omission of the expression for damping due 2 2 ≈− 32//32 cc/()/().0 Ji2 aa Ji0 to friction between the wall of the tube and blood moving downstream in the pressure wave.

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Discussion Acknowledgements One possible source of errors in the analysis of this paper The author thanks Ann Young for many thoughtful discussions and thanks is the limited consideration of changes in arterial radius. Paul Agutter for helpful suggestions and editorial comments. Such changes are fully considered only when the Law of Laplace, Equation (15), is incorporated in an expression. References 1. Stein S, Okal EA: Seismology: speed and size of the Sumatra For this reason, caution should be exercised when using earthquake. Nature 2005, 434:573-574. the above results to interpret data from arteries in which 2. Korteweg DJ: Ueber die Fortpflanzungsgeschwindigkeit des there is a considerable change in the radius and the cross- Schalles in elastischen Röhren. Annalen der Physik 1878, 24:525-542. sectional area during the cardiac cycle. In large elastic 3. Moens AI: Die Pulskurve Leiden: E. J. Brill; 1878. arteries, this change in radius may be 10% or more from 4. Safar H, Chahwakilian A, Boudali Y, Debray-Meignan S, Safar M, Blacher J: Arterial stiffness, isolated systolic hypertension, and the median value [14], and this may be a source of error cardiovascular risk in the elderly. Am J Geriatr Cardiol 2006, that is of concern in certain contexts. 15:178-182. 5. Mayer O, Filipovsky J, Dolejsova M, Cifkova R, Simon J, Bolek L: Mild hyperhomocysteinaemia is associated with increased aortic In small arteries where pressure oscillations are of low stiffness in general population. J Hum Hypertens 2006, 20:267-71. magnitude, the above concern diminishes. In addition, as 6. Kullo IJ, Bielak LF, Turner ST, Sheedy PF 2nd, Peyser PA: Aortic arteries become smaller and smaller, the flow becomes pulse wave velocity is associated with the presence and quantity of coronary artery calcium: a community-based closely described by the equations for the rigid tube. Con- study. Hypertension 2006, 47:174-179. sequently, the damping function approaches the damping 7. Matsuoka O, Otsuka K, Murakami S, Hotta N, Yamanaka G, Kubo Y, Yamanaka T, Shinagawa M, Nunoda S, Nishimura Y, Shibata K, Saitoh function calculated from Poiseuille's Equation. Therefore, H, Nishinaga M, Ishine M, Wada T, Okumiya K, Matsubayashi K, Yano concern for errors in the analysis is less for the expression S, Ichihara K, Cornelissen G, Halberg F, Ozawa T: Arterial stiffness giving the PWV in small arteries than it is for the expres- independently predicts cardiovascular events in an elderly community – Longitudinal Investigation for the Longevity sion giving the PWV in large arteries. and Aging in Hokkaido County (LILAC) study. Biomed Pharma- cother 2005, 59(Suppl 1):S40-44. 8. Lambossy P: Oscillations forcées d'un liquide incompressible A plausible explanation for an increase in the PWV during et visqueux dans un tube rigide et horizontal. Calcul de la the course of a disease is an increase in arterial stiffness force frotement. Helvetica Physica Acta 1952, 25:371-386. leading to an increase in the parameter E. This explanation 9. Womersley JR: Oscillatory motion of a viscous liquid in a thin- walled elastic tube. 1: The linear approximation for long is largely based on the Korteweg-Moens equation. An waves. Philos Mag 1955, 46:199-231. increase in the elastic modulus, E, or the relative thickness 10. Caro CG, Pedley TJ, Schroter RC, Seed WA: The Mechanics of the Cir- of the arterial wall, h/R, would increase the PWV. How- culation Oxford, UK: Oxford University Press; 1978. 11. West GB, Brown JH, Enquist BJ: A general model for the origin ever, changes in other properties associated with arterial of allometric sc1ling laws in biology. Science 1997, 276:122-126. walls and surrounding tissue may also increase the PWV 12. Warriner RK, Johnston KW, Cobbold RS: A viscoelastic model of arterial wall motion in pulsatile flow: implications for Dop- and may be interpreted as increasing stiffness. One source pler ultrasound clutter assessment. Physiol Meas 2008, of arterial stiffness that is not considered in the above 29:157-179. analysis is production of heat as the arterial wall is 13. Painter PR, Eden P, Bengttson HU: Pulsatile blood flow, shear force, energy dissipation and Murray's Law. Theor Biol Med stretched [14]. Model 2006, 3:31. 14. Canic S, Hartley CJ, Rosenstrauch D, Tambaca J, Guidoboni G, Mikelic A: Blood flow in compliant arteries: an effective viscoelastic The above results show that the PWV can increase in small reduced model, numerics, and experimental validation. Ann arteries if the parameter a decreases. The parameter a Biomed Eng 2006, 34:575-592. describes the rate of increase in pressure during the initial 15. Marque V, Grima M, Kieffer P, Capdeville-Atkinson C, Atkinson J, Lar- taud-Idjouadiene I: Determination of aortic elastic modulus by rise as the pulse wave approaches a point in an artery. This pulse wave velocity and wall tracking in a rat model of aortic rise is determined by the rise in the ejection rate through stiffness. Vasc Res 2001, 38:546-550. the aortic valve in early systole. A number of heart disorders can affect this rate of rise. Examples include aortic ste- nosis, myocardial ischemia and certain conduction Publish with BioMed Central and every disorders. It appears plausible that, in certain diseases of scientist can read your work free of charge the heart, attributing an increase in the PWV to arterial stiffness may not be the correct explanation. The link "BioMed Central will be the most significant development for disseminating the results of biomedical research in our lifetime." between an increase in the PWV and increased risk of Sir Paul Nurse, Cancer Research UK myocardial infarction [7] may be due, at least in part, to myocardial ischemia. Your research papers will be: available free of charge to the entire biomedical community Competing interests peer reviewed and published immediately upon acceptance The author declares that they have no competing interests. cited in PubMed and archived on PubMed Central yours — you keep the copyright

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