Radial and Longitudinal Motion of the Arterial Wall: Their Relation to Pulsatile Pressure and Flow in the Artery Dan Wang Old Dominion University

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2018 Radial and Longitudinal Motion of the Arterial Wall: Their Relation to Pulsatile Pressure and Flow in the Artery Dan Wang Old Dominion University

Linda Vahala Old Dominion University

Zhili Hao Old Dominion University

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Repository Citation Wang, Dan; Vahala, Linda; and Hao, Zhili, "Radial and Longitudinal Motion of the Arterial Wall: Their Relation to Pulsatile Pressure and Flow in the Artery" (2018). Mechanical & Aerospace Engineering Faculty Publications. 69. https://digitalcommons.odu.edu/mae_fac_pubs/69 Original Publication Citation Wang, D., Vahala, L., & Hao, Z. (2018). Radial and longitudinal motion of the arterial wall: Their er lation to pulsatile pressure and flow in the artery. Physical Review E, 98(3), 032402. doi:10.1103/PhysRevE.98.032402

This Article is brought to you for free and open access by the Mechanical & Aerospace Engineering at ODU Digital Commons. It has been accepted for inclusion in Mechanical & Aerospace Engineering Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. PHYSICAL REVIEW E 98, 032402 (2018)

Radial and longitudinal motion of the arterial wall: Their relation to pulsatile pressure and ﬂow in the artery

Dan Wang,1 Linda Vahala,2 and Zhili Hao1 1Department of Mechanical and Aerospace Engineering, Old Dominion University, Norfolk, Virginia 23529, USA 2Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529, USA

(Received 21 March 2018; published 4 September 2018)

The aim of this paper is to analyze the radial and longitudinal motion of the arterial wall in the context of pulsatile pressure and flow, and to understand their physiological implications for the cardiovascular system. A reexamination of the well-established one-dimensional governing equations for axial blood flow in the artery and the constitutive equation for the radial dilation of the arterial wall shows that two waves—a pulsatile pressure wave in the artery and a radial displacement wave in the arterial wall—propagate simultaneously along the arterial tree with the same propagation velocity, explaining why this velocity combines the physical properties and geometries of both the blood and the arterial wall. With consideration of their coupling, the governing equations for the radial motion and longitudinal motion of the arterial wall are derived separately. The driving force for the radial motion of the arterial wall arises from its longitudinal motion. The longitudinal motion of the arterial wall is the standard longitudinal elastic wave with two driving forces: one associated with the pulsatile flow rate and the other associated with the radial motion. These derived governing equations shed insights on some recent experimental findings in the literature, including the correlation of measured arterial wall stiffness (solely based on the radial motion) to its longitudinal motion, the decreasing trend of arterial wall distensibility from the aorta to the periphery, the underlying mechanism of the longitudinal motion pattern of the common carotid arterial wall, and the motion pattern variation in longitudinal motion with aging and diseases.

DOI: 10.1103/PhysRevE.98.032402

I. INTRODUCTION been found to be of equal amplitude to the radial motion of the arterial wall [14]. The longitudinal motion of the arterial The motion of the arterial wall accompanies the pulsatile wall was investigated for the purpose of complementing the pressure and flow in the artery, and plays a critical role in arterial wall stiffness based on the radial motion and devel- blood circulation throughout the arterial tree [1–4]. Therefore, oping new clinic indexes for vascular diseases [6,7,15,16]. the motion of the arterial wall has been investigated for its clinical implications of cardiovascular diseases (CVDs) Toward this end, several studies were conducted on finding and vascular diseases [5–8]. As the main focus of extensive the underlying mechanism of the longitudinal motion pattern research, the radial motion of the arterial wall has formed the of the arterial wall, but the conclusions from different studies basis for estimation of the arterial wall stiffness, mostly in are contradictory [13,15,17]. For instance, Ahlgren et al. [17] terms of pulse wave velocity (PWV), with the longitudinal found that the wall shear stress (i.e., local blood flow) is not motion being neglected [5,9]. Nowadays, PWV is a well- the main driving force of the longitudinal motion, while Au established independent risk predictor of adverse cardiovascu- et al. [13] found that the local blood flow is closely related lar events. However, current established methods for measur- to the first anterograde motion of the longitudinal motion. ing the arterial wall stiffness cannot discern small preclinical These studies [13,15,17] investigated the longitudinal motion atherosclerotic or arteriosclerotic changes in a comprehensive of the arterial wall purely from the physiological perspective manner [6]. Yet, detecting these preclinical aberrant changes (by altering one or two factors through drug administration in the arterial tree could be beneficial in developing novel and utilizing statistical analysis and/or machine learning to strategies to prevent the progress of vascular diseases [6]. establish their correlation to the longitudinal motion). Yet, the Compared with its radial motion, the longitudinal motion of determinants of the bidirectional pattern of the longitudinal the arterial wall has been suggested to be an earlier and more motion of the arterial wall remain unclear [11,15], although sensitive measure of vascular diseases [10,11]. Experimental new clinical indexes have been created from the measured studies have recently found that the measured arterial wall longitudinal motion of the arterial wall via machine-learning stiffness solely based on the radial motion is correlated to the techniques [16]. longitudinal motion amplitude of the arterial wall [8,12,13]. Since the pulsatile pressure and flow in the artery are For the purpose of devising new clinic indexes for detecting adjacent to the arterial wall and accompany the motion of the preclinical aberrant changes and finding its influence on the arterial wall, intuitively, a direct relation should exist between radial motion of the arterial wall, the longitudinal motion of these two parameters in the artery and the radial and longi- the arterial wall needs to be studied. tudinal motion of the arterial wall. Then, the question arises: Thanks to the advancement in imaging techniques over the How are pulsatile pressure and flow related to the radial and past decade, the longitudinal motion of the arterial wall has longitudinal motion of the arterial wall? From the mechanical

2470-0045/2018/98(3)/032402(8) 032402-1 ©2018 American Physical Society DAN WANG, LINDA VAHALA, AND ZHILI HAO PHYSICAL REVIEW E 98, 032402 (2018) perspective, this work derives the governing equations of than 1. Then, the boundary layer is thin and the blood flow the radial and longitudinal motion of the arterial wall, and velocity profile is close to plug flow. Therefore, the pressure analyzes how the radial (ur ) and longitudinal (ux ) motion and velocity are assumed to be uniform over the entire cross of the arterial wall is related to the pulsatile pressure (p) section. and flow rate (Q) in the artery. Comparison of the analyzed As shown in Fig. 1, the 1D governing equations of axial results with some recent experimental findings in the literature pulsatile flow in a distensible artery are then derived from con- [6,13,17] sheds insights on the physiological implications servation of mass (equation of continuity) and conservation of of the radial and longitudinal motion of the arterial wall to momentum (Navier-Stokes equation) along the axial direction the cardiovascular system. As such, they may serve as the (x axis) of the artery [1]: basis for developing model-based clinical indexes from the measured radial and longitudinal motion of the arterial wall ∂A ∂(AU ) for inferring the cardiovascular system condition, instead of + = 0, (1a) ∂t ∂x machine-learning-based clinical indexes. ∂U ∂U 1 ∂p f The rest of the paper is organized as follows. In + U =− + Sec. II, without considering the longitudinal motion, the well- ∂t ∂x ρb ∂x ρbA established one-dimensional (1D) governing equations for (TA) (CA) (PG) (VF), (1b) axial blood flow in the artery and the constitutive equation for the radial dilation of the arterial wall are reexamined for the role of the radial motion of the arterial wall in pulsatile where U(x, t) and p(x, t) denote the blood flow velocity pressure and flow in the artery. In Sec. III,bytreatingthe and blood pressure averaged over the entire cross section, A x t arterial wall as a thin-walled tube and considering radial- respectively; ( , ) is the cross-section area of the lumen; and f denotes the friction force in the blood flow. The blood longitudinal coupling in its motion, the governing equations flow rate, Q = AU, is the product of the average blood for the radial motion and longitudinal motion of the arterial flow velocity and the cross-section area. The terms in the wall are derived separately. The physiological implications of Navier-Stokes equation are the temporal acceleration (TA), the derived governing equations are discussed and utilized to convective acceleration (CA), pressure gradient (PG), and the explain some recent experimental findings in the literature in viscous force (VF). Given the small disturbances and the plug Sec. IV. The last section concludes the gained physiological flow, the CA is negligible [1]. When the Womersley number implications of the radial and longitudinal motion of the is much larger than 1, the VF is also small, relative to the TA arterial wall. and the PG [1,19]. The above two equations contain three time-varying pa- II. ROLE OF THE RADIAL MOTION OF THE ARTERIAL rameters: blood pressure (p), blood flow velocity (U ), and WALL IN PULSATILE PRESSURE AND FLOW cross-section area (A) of the artery, which varies with radial IN THE ARTERY dilation of the arterial wall. To complement the above two equations, the constitutive equation of the arterial wall (or A. Governing equations for axial blood flow in the artery arterial wall compliance) [9] is included in Eq. (1a): and the radial motion of the arterial wall The blood circulation in the arterial tree involves complex ∂A(t ) 2πr0∂ur (t ) three-dimensional (3D) fluid-structure interaction between the CA = = , (1c) blood flow in the artery and the motion of the arterial wall ∂p(t ) ∂p(t ) [18,19]. To better understand the physiological implications of the hemodynamic parameters and arterial wall motion where ur (t ) is the radial displacement of the arterial wall and parameters to cardiovascular diseases, 1D modeling of the r0 is the artery radius at diastolic pressure Pd .Bothr0 and blood flow in an artery has been well established and validated CA are related to the transmural pressure in the arterial wall: by experimental studies to some extent [1]. The fundamental PT = Pext–Pd , with Pext being the pressure outside the arterial assumptions underlying the 1D modeling are summarized wall. below [1,19]: In Eq. (1c), a purely elastic, linear relation between the (1) The artery has a straight, uniform, circular shape and pulsatile pressure and the radial displacement is assumed entails axis-symmetric blood flow. for the arterial wall. Furthermore, the radial displacement (2) The blood in the artery is an incompressible homoge- amplitude, ur0, is assumed to be much smaller than the arterial neous Newtonian fluid. radius r0 (i.e., ur0/r0 1). By taking the first-order derivative (3) The density ρb and viscosity μb of the blood are of Eq. (1c) with respect to t and x, respectively, the following constant. relations between the radial displacement and the pulsatile (4) Small disturbances (i.e., arterial pulses) with the angu- pressure are obtained: lar frequency ω and the wavelength λ take place in an artery with the radius being much smaller than the wavelength (i.e., ∂p 1 ∂A 2πr0 ∂ur r0 λ) and thus no blood flow along the radial direction = = , (2a) occurs. ∂t CA ∂t CA ∂t (5) As to large arteries (e.g., common√ carotid artery and ∂p 1 ∂A 2πr0 ∂ur = = = . (2b) aorta), the Womersley number α r0 ρbω/μb is much larger ∂x CA ∂x CA ∂x

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(a) (b)

I!. T0=E 00hE 00

FIG. 1. Schematics of (a) pulsatile pressure p(t ) and axial blood ﬂow Q in the artery and (b) circumferential tension force Tθ in the arterial wall due to its radial dilation ur .

B. Pulsatile pressure wave in the artery equations: Substituting Eq. (2a) into Eq. (1a) yields the following ∂u ∂(AU ) relation: 2πr r + = 0, (9a) 0 ∂t ∂x ∂p ∂(AU ) C + = 0. (3) ∂U ∂U 1 2πr0 ∂ur f A ∂t ∂x + U =− + . (9b) ∂t ∂x ρ CA ∂x ρA Now, we take the derivative of Eq. (3) with respect to t and take the derivative of Eq. (1b) with respect to x, add up the By neglecting the two small terms (CA and VF) in Eq. (9b) two equations, and neglect the two small terms (CA and VF): and combining the above two equations, the governing equation for the radial motion of the arterial wall is obtained: ∂2p A ∂2p = . (4) 2 2 2 2 ∂ u A ∂ u ∂t ρbCA ∂x r = r . (10) ∂t2 ρ C ∂x2 Equation (4) indicates that there is a pulsatile pressure b A wave in the artery propagating along the arterial tree with the Equation (10) indicates that there is a radial displacement propagation velocity (vP ), commonly referred to as the pulse wave in the arterial wall (similar to a transverse wave in wave velocity: a string) propagating along the arterial tree. Comparison of A Eqs. (4) and (10) shows that the radial displacement wave in v2 = . (5) P ρ C the arterial wall and the pulsatile pressure wave in the artery b A propagate simultaneously (Windkessel effect) along the arte- By treating the arterial wall as a stretched thin-walled rial tree with the same propagation velocity. Mathematically, membrane in a circular shape with constant thickness h and the coincidence of the two waves arises from the definition constant density ρw, the radial displacement ur gives rise to a of the arterial wall compliance in Eq. (1c) and explains why circumferential tension force Tθ , as shown in Fig. 1(b).This the propagation velocity combines the physical properties and tension force can be expressed using an effective incremental geometries of the blood and the arterial wall. The experimen- circumferential modulus Eθθ of the arterial wall: tally observed similarity between arterial radius waveform and pulsatile pressure waveform [20] is coined in the arterial wall = ur = Tθ Eθθh r0 p. (6) compliance. r0 Note that this circumferential tension force needs to coun- terbalance the radial force arising from the pulsatile pressure, D. Driving force for the pulsatile flow in the artery p. Then, the arterial wall compliance is related to the BasedonEqs.(1b) and (9b), the driving forces for the physical properties and geometries of the arterial wall by pulsatile flow velocity (U ) or pulsatile flow rate (Q)inthe ∂A 2r A artery include the pressure gradient ∂p/∂x and the radial C = = 0 0 . (7) A ∂p E h displacement gradient, ∂ur /∂x. According to Eqs. (2b) and θθ (7), the pressure gradient is related to the radial displacement Substituting Eq. (7) into Eq. (5) leads to the well-known gradient by Moens-Korteweg formula [5]: ∂p E h ∂u = θθ r . (11) Eθθh ∂x r2 ∂x vP = . (8) 0 2r0ρb The pressure gradient is commonly related to the pulsatile flow rate Q by the vascular resistance R [21]: C. Radial displacement wave in the arterial wall Now, we derive the governing equation for the radial ∂p E h ∂u ∂u 8Lμ Q = R = θθ r r with R = b , (12) motion of the arterial wall by replacing the pulsatile pressure 2 4 ∂x r0 R ∂x ∂x πr0 with the radial displacement in Eqs. (1a) and 1(b). Substituting Eq. (2b) into Eqs. (1a) and (1b) gives rise to the following two where L is the arterial length.

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III. COUPLED RADIAL AND LONGITUDINAL MOTION When the time-variant terms in Eq. (18) are zero and the OF THE ARTERIAL WALL longitudinal motion term is neglected, the above equation reduces to Eq. (6) and consequently we have the arterial wall The arterial wall is treated as a stretched thin-walled mem- compliance: brane in a circular shape with constant thickness h and the constant density ρw. The wall thickness is much smaller than E p A 2r A θθ u = → C = = 0 0 . (19) the wall radius, h/r0 1. Figure 2 shows a small segment of 2 r A ρ r ρwh p Eθθh angle dθ and length dx of the arterial wall, which is at the w 0 transmural pressure PT , and is axially prestretched along its As such, the measured arterial wall stiffness (vP )inEq.(8) longitudinal direction. The circumferential and longitudinal ignores the viscous nature of the radial motion of the arterial tension forces in the arterial wall at diastolic pressure are T0θ wall and the inﬂuence of the longitudinal motion on the radial and T0x , respectively. As such, the arterial wall is analogous to motion. After removing the inertial term and the longitudinal a stretched spring moving along both directions, whereas the motion term, Eq. (18) becomes the commonly used Kelvin- arterial wall in Eqs. (1c) and (6) is treated as a stretched thin- Voigt material model for the arterial wall [4]: walled membrane with a purely elastic, linear relation of the pulsatile pressure to the radial displacement, and undergoes Eθθ + ηθθ dur = p 2 ur 2 . (20) only radial motion. The wall segment undergoes radial motion ρwr0 ρwr0 dt ρwh ur (t ) and longitudinal motion ux (t ) in a pulse cycle. Based on this equation, the elasticity and viscosity of the arterial wall are currently extracted from simultaneously A. Radial motion of the arterial wall measured pulsatile pressure waveform and artery radius wave- The arterial wall segment in Fig. 2 is subject to the follow- form at the same arterial location. Comparison of Eq. (20) ing forces along the radial direction: with Eq. (18) indicates that while the driving force for the (1) A radial force results from the pulsatile pressure: quasistatic radial motion is the pulsatile pressure, the dynamic radial motion (acceleration) of the arterial wall arises from F − = pr dθdx. (13) p r 0 the radial force component of the longitudinal motion of the (2) The arterial wall experiences a circumferential strain: arterial wall. εθθ = ur /r0 . Then, the restoring force from the arterial wall By replacing the pulsatile pressure with the radial displace- along the radial direction becomes ment using Eq. (1c), Eq. (18) becomes dε 2 =− + θθ d ur Eθθ 2πr0 ηθθ dur Fθ−r Eθθhεθθdθdx ηθθh dθdx , (14) + − u + dt 2 2 r 2 dt ρ r CAρwh ρ r dt w 0 w 0 where the second term is the viscous term with ηθθ being the Exx ∂ ∂ux ∂ur effective damping modulus of the arterial wall. = . (21) ρ ∂x ∂x ∂x (3) Because of its radial motion, the arterial wall deforms w along its longitudinal direction. The radial component from Accordingly, the arterial wall at any axial position behaves the longitudinal tension force of the wall segment is [22,23] as a second-order dynamic system and undergoes radial vibra- 2 tions, with the driving force from the longitudinal motion. ∂ ∂ux 1 ∂ur ∂ur By replacing the radial displacement in Eq. (18) with the Tx−r = Exxhr0dθ + dx, ∂x ∂x 2 ∂x ∂x pulsatile pressure using Eq. (1c), the governing equation for the pulsatile pressure acting on the arterial wall is obtained: (15) d2p E 2πr η dp where Exx is an effective incremental longitudinal modulus of + θθ − 0 p + θθ 2 2 2 the arterial wall. Since the second term is small relative to the dt ρwr0 CAρwh ρwr0 dt ﬁrst term in the square brackets, the overall radial component E r2 ∂ ∂u ∂u from the longitudinal tension force becomes = xx 0 x r . (22) Eθθρwh ∂x ∂x ∂x ∂ ∂ux ∂ur F − = E hr dθ dx. (16) x r xx 0 ∂x ∂x ∂x The pulsatile pressure variation on the arterial wall is caused by the radial force component of the longitudinal mo- According to Newton’s second law, the governing equation tion of the arterial wall. Comparison of Eq. (22) with Eq. (21) of the radial motion of the arterial wall segment is suggests that the pulsatile pressure acting on the arterial wall 2 and the radial motion of the arterial wall accompany each d ur ρ hr dθdx = F − + F − + F − . (17) other and exhibit the same dynamic nature. Taken together, w 0 dt2 θ r x r p r locally at any axial position, the radial motion of the arterial Substituting the expressions for the radial forces into wall and pulsatile pressure variation in the artery accompany Eq. (17)givesriseto each other and are both caused by the longitudinal motion of d2u E η du E ∂ ∂u ∂u p the arterial wall; systemically along the arterial tree, the radial r + θθ u + θθ r = xx x r + . displacement wave and pulsatile pressure wave propagation dt2 2 r 2 dt ρ ∂x ∂x ∂x ρ h ρwr0 ρwr0 w w are driven by the pressure gradient and the radial displacement (18) gradient.

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(a) (b) (c) ' _!1 T .(x+dx) u,

.6T.(x)' ----✓,' , : u,(x+dx) I u,(x) , 720077 --+, ,~ --+, ,.__ 0 X u.(x) u.(x+dx)

FIG. 2. Coupled radial motion (ur ) and longitudinal motion (ux ) of a small segment of angle dθ and length dx of the arterial wall. (a) A segment of artery of length dx. Q: the blood ﬂow rate in the artery; PT : the transmural pressure in the arterial wall; T0x : the longitudinal tension force in the arterial wall at diastolic pressure; T0θ : the circumferential tension force in the arterial wall at diastolic pressure; Tθ :the circumferential force due to the radial displacement. (b) Radial direction. h: arterial wall thickness; p: pulsatile pressure; r = r0 + ur :the arterial wall radius. (c) Longitudinal direction. ux (x)andux (x + dx): longitudinal displacements at the positions of x and x + dx, respectively; ur (x)andur (x + dx): radial displacements at the positions of x and x + dx, respectively.

B. Longitudinal motion of the arterial wall motion counteracts the force associated with the blood flow The arterial wall segment in Fig. 2 is subject to the follow- rate to reduce the longitudinal motion amplitude. ing forces along the longitudinal direction: (1) The longitudinal motion of the arterial wall gives rise IV. DISCUSSION to the overall tension force: A. Correlation of the longitudinal motion of the arterial wall 2 ∂ ux to the radial motion of the arterial wall F − = E hr dθ dx. (23) x x xx 0 ∂x2 Taivainen et al. [8] recently found that the longitudinal (2) Owing to the radial motion of the arterial wall, the motion amplitude of the common carotid wall were directly pulsatile pressure results in a longitudinal force: correlated with its distensibility and inversely correlated with its PWV. Based on Eq. (21), the driving force for the ra- ∂ur 2πr0 ∂ur dial motion is the longitudinal motion. Then, high longitu- Fp−x =−p r0dθdx =− ur r0dθdx. ∂x CA ∂x dinal motion amplitude translates to high distensibility. As (24) explained below, high distensibility indicates low circumferential modulus and thus is accompanied with low PWV, Note that the pulsatile pressure is further expressed in according to Eq. (8). terms of the radial displacement. Elastic arteries are significantly prestretched along their (3) The blood flow rate causes the wall shear stress τw on axial (longitudinal) direction, setting up their tension state (T0r the arterial wall. The overall shear force caused by the blood and T0θ ) at diastolic pressure. With this axial prestretch, the flow rate becomes [20] radial motion of the arterial wall and the pulsatile pressure variation happen at any axial position, and enable the propaga- 4ηbQ − = = tion of the radial displacement wave and the pulsatile pressure FQ x τwr0dθdx 3 r0dθdx. (25) πr0 wave along the arterial tree. It was found [24] that high axial E By summarizing the above equations, the governing equa- prestretch (or high longitudinal modulus xx) translates to a tion for the longitudinal motion of the arterial wall becomes large radial motion and thus a high distensibility, which is possibly due to the axial prestretch aligning collagen fibers d2u E ∂2u πE ∂u 4η Q to the axial direction, then leading to a low circumferential x = xx x − θθ u r + b , (26a) modulus (E ). According to Eq. (21), the combination of 2 2 r 3 θθ dt ρw ∂x A0ρw ∂x ρwπr0 h high longitudinal modulus and low circumferential modulus d2u E ∂2u p ∂u 4η Q translates to high distensibility. Conversely, low axial pre- x = xx x − r + b . (26b) 2 2 3 stretch corresponds to low longitudinal modulus and high dt ρw ∂x hρw ∂x ρ πr h w 0 circumferential modulus. Furthermore, given the same mean Equation (26) indicates that there is a longitudinal elastic blood pressure, a highly axially prestretched arterial wall can wave in the arterial wall propagating along the arterial tree, exhibit higher distensibility than a less axially prestretched with the propagation velocity solely related to the physical arterial wall [24]. Since a change in axial prestretch is typ- properties of the arterial wall. There are two driving forces ically accompanied by changes in artery radius and mean for the longitudinal motion: one associated with the radial blood pressure, all these factors adjust the circumferential and motion (or pulsatile pressure) and the other associated with longitudinal modulus of the arterial wall, and consequently its the blood flow rate. The force associated with the blood flow radial and longitudinal motion. rate is in the opposite direction to the force associated with the As the arterial wall stiffness increases from the aorta to radial motion. Thus, a high blood flow rate may cause a large the periphery [25], axial prestretch and artery radius decrease longitudinal motion, but the force associated with the radial with more distal arterial locations [26]. Low axial prestretch

032402-5 DAN WANG, LINDA VAHALA, AND ZHILI HAO PHYSICAL REVIEW E 98, 032402 (2018) in a peripheral artery translates to a low longitudinal modulus altered, many interdependent factors, including the radius, and a high circumferential modulus. Based on Eq. (21), the distensibility, and circumferential and longitudinal modulus driving force for the radial motion is reduced, but the stiffness of the carotid arterial wall, were also altered by catecholamine in the radial motion, which is proportional to the ratio of administration. As shown in Eq. (26), the longitudinal motion the circumferential modulus to the squared artery radius, is of the arterial wall is a collective effort of all these interdepen- increased. Consequently, Eq. (21) explains the reason that dent factors. Later on, Au et al. [15] investigated the influence the arterial wall distensibility decreases from the aorta to the of some physiological factors on the longitudinal motion of periphery [25,26]. Based on Eq. (22), the pulsatile pressure the carotid arterial wall in young healthy human subjects amplitude increases from the aorta to the periphery, because through sympathetic activation and vascular smooth muscle of the accompanying increased ratio of longitudinal modulus relaxation. This study also could not establish significant to circumferential modulus. correlation of the blood flow rate to the first anterograde motion in early systole by vascular smooth muscle relaxation. Similarly, the vascular smooth muscle relaxation alters not B. Bidirectional pattern of the longitudinal motion of the only the blood flow rate, but also other interdependent fac- common carotid arterial wall in healthy young subjects tors. Moreover, since administration of different drugs alters Most of the studies on the longitudinal motion focused regulatory control over the arterial tree, the axial prestretch on the common carotid artery, due to its easy access. This in the arterial wall might also be altered. It has been found artery has been shown to undergo a multiphasic bidirectional that the axial prestretch is not negligible in mechanobiological motion pattern in healthy young subjects: an anterograde processes, including its adaptation response to elevated blood motion (in the direction of the blood flow) in early systole, a pressure [29]. The axial prestretch dictates the tension state of retrograde motion (in the direction of opposing blood flow) in the arterial wall at diastolic pressure and thus determines wall late systole, and then a second distinct anterograde-retrograde thickness, radius, and the circumferential and longitudinal motion with a lower amplitude in diastole than the one in modulus. As such, the studies on the underlying mechanism of systole [14,27]. The study conducted by Cinthio et al. [14] the longitudinal motion of the arterial wall via drug adminis- demonstrated that the first anterograde motion peak occurs tration should take these interdependent factors into account. after the blood flow rate peak but before the radial motion peak. Thus, the blood flow rate in Eq. (26) is the dominant driving force for the longitudinal motion of the arterial wall. C. Motion pattern variation of the longitudinal motion of the After the blood flow rate recedes significantly and the first an- common carotid arterial wall with aging and diseases terograde motion peak is reached in systole, the longitudinal The longitudinal motion pattern of the carotid arterial restoring tension in the arterial wall starts to drive the arterial wall was found to vary with aging [27]. Different from its wall into its first retrograde motion in late systole. According counterpart in young healthy subjects, the longitudinal motion to Au et al. [13], the first retrograde motion is associated pattern of the common carotid artery in middle-aged and older with left ventricle rotation and thus an active regulatory factor subjects were found to include the appearance of two addi- may also contribute to this retrograde motion in late systole. tional distinct phases of motion [27]. Both the axial prestretch This factor also explains why the first retrograde motion and the distensibility are found to decline with aging [24]. has a relatively large amplitude, as compared with the first The low longitudinal tension (T0x ) at diastolic pressure due anterograde motion. to decreased axial prestretch goes down to a larger extent than With the aid of a low late systolic blood flow rate peak, the circumferential tension (T0θ ) at diastolic pressure. The low the longitudinal restoring tension moves the arterial wall into longitudinal tension translates to a low longitudinal modulus the second anterograde-retrograde motion in diastole. Based (Exx). At the same time, a late systolic flow velocity peak ex- on Eq. (21), the radial motion of the arterial wall should ists in the carotid flow velocity waveform and increases with register the second anterograde-retrograde motion. The end aging. Heffernan et al. [30] found that both increased pressure of the first retrograde motion occurs almost simultaneously from wave reflections and increased suction from the left with the dicrotic notch in the radial motion [14,17], indicating ventricle contribute to the late systolic flow augmentation. As that the radial motion does capture the second anterograde- such, it is the combination of the increased late systolic flow retrograde motion, but with a much smaller amplitude. This velocity peak and the much decreased longitudinal modulus is possibly due to the active regulatory factor from the left that drives the arterial wall to bounce forward and backward ventricle existing only in the longitudinal motion, but not in in diastole along the longitudinal direction. the radial motion. The longitudinal motion pattern of the carotid arterial wall To investigate the underlying mechanism of the longitu- was also found to vary with diseases. Tat et al. [31] found that dinal motion of the carotid arterial wall, Ahlgren et al. [28] the presence of carotid plaque can have significant influence altered certain hemodynamic parameters in anesthetized pigs on the longitudinal motion of the arterial wall, with signif- by administration of catecholamines and found that the lon- icantly greater anterograde motion amplitude with increased gitudinal motion of the carotid wall undergoes profound stenosis. Stenosis reduces the artery radius and increases the change in response to catecholamine. By administration of blood flow rate. According to Eq. (26), the dominant driving epinephrine, norepinephrine, and β-blockade (metoprolol) in force for the longitudinal motion of the carotid arterial wall anesthetized pigs, Ahlgren et al. [17] found that the longitudi- is significantly increased by stenosis. On the other hand, the nal motion of the carotid arterial wall has no correlation with same group of authors [11] found that, as compared with their the blood flow rate. However, when the blood flow rate was healthy counterparts, the first retrograde motion amplitude is

032402-6 RADIAL AND LONGITUDINAL MOTION OF THE … PHYSICAL REVIEW E 98, 032402 (2018)

Equation (27) indicates that the pressure gradient causes au, I ax the blood ﬂow rate and then the longitudinal motion, which further causes the pulsatile pressure and then the radial Q motion. BasedonEq.(9b), the radial displacement gradient causes i ux the blood ﬂow rate. The rate of pulsatile pressure variation ap! ox drives the pressure gradient [32]: !).p

∂p 1 ∂p ap1 at =− . (28) ∂x vP ∂t FIG. 3. Cause-effect relation of the radial and longitudinal motion (in blue) of the carotid arterial wall to the hemodynamic param- As such, a closed-loop cause-effect relation exists among eters (in red) in the common carotid artery, with the arrow showing the radial and longitudinal motion of the arterial wall and the the cause-effect direction. hemodynamic parameters in the common carotid artery. much reduced but the first anterograde motion in the common E. Limitations of the study carotid artery does not show a difference in subjects with There are two major limitations of this study. One is the spinal cord injury. Similar to the study conducted by Au assumption that the arterial wall at any axial position is homo- et al. [13], the reduced first retrograde motion amplitude was geneous along the circumferential and longitudinal directions. attributed to left ventricle rotation in systole, which could The other is the assumption that the arterial wall is a passive produce the retrograde motion [11]. material and is not controlled by regulatory factors in a pulse cycle, except that left ventricle rotation is added to explain D. Timing of hemodynamic parameters in the artery the large retrograde motion in late systole. Nevertheless, the and the radial and longitudinal motion of the common derived governing equations for the arterial wall motion are carotid arterial wall helpful in shedding insights on some recent experimental findings in the literature and might aid in developing model- Figure 3 shows the correlation of the hemodynamic pa- based clinical indexes for detection and early diagnosis of rameters in the common carotid artery: pressure gradient, CVDs and vascular pathologies. pulsatile pressure, and the blood flow rate to the radial and longitudinal motion of the arterial wall. According to Eq. (11), the blood flow rate and the pressure gradient accompany each V. CONCLUSION other and thus are treated as two interdependent parameters. Similarly, the radial displacement wave in the arterial wall In this study, we have reexamined the 1D governing equa- is accompanied by the pulsatile pressure wave in the artery tions of axial blood flow in the artery and the constitutive and they are treated as two interdependent parameters too. equation of the arterial wall, derived the governing equations Propagating in two different media (blood versus arterial for the radial and longitudinal motion of the arterial wall with wall), the radial displacement wave and the pulsatile pressure consideration of their coupling, and utilized the derived gov- wave are separated in the figure. erning equations to explain some recent experimental findings According to the literature [32], the pressure gradient peak in the literature. The comparison of the derived governing occurs before the flow rate peak, with the latter occurring equations with the experimental findings in the literature gives before the pulsatile pressure peak: ∂p/∂x → Q → p, indi- rise to the following physiological implications of the radial cating that the pressure gradient causes the blood flow rate and and longitudinal motion of the arterial wall to the cardiovas- then the pulsatile pressure. According to Cinthio et al. [14], cular system: the blood flow rate peak occurs before the first anterograde (1) A pulsatile pressure wave in the artery and a radial motion peak of the carotid arterial wall: Q → ux . Since the displacement wave in the arterial wall simultaneously propa- radial motion of the arterial wall and the pulsatile pressure gate along the arterial tree with the same propagation velocity, are both caused by the longitudinal motion, the longitudinal which is determined by the physical properties and geometries motion peak of the arterial wall should occur before the radial of the blood and the arterial wall. motion peak and the pulsatile pressure peak: ux → p and (2) The radial motion of the arterial wall and the pul- ux → ur . Two experimental studies [14,27] observed ux → satile pressure variation in the artery arise from the radial ur at the carotid arterial wall. The relation of p → ur has force component of the longitudinal motion of the arterial been validated by many experimental studies [4]. However, wall, establishing the correlation of the measured arterial the experimental validation of ux → p cannot be found in wall distensibility with the longitudinal motion amplitude and the literature. Nonetheless, the peaks of these five parameters explaining the decreasing trend of the arterial wall stiffness for the carotid arterial wall should follow the following time and the increasing trend of pulsatile pressure amplitude from sequence: the aorta to the periphery. (3) The longitudinal motion of the common carotid ar- ∂p/∂x → Q → ux → p → ur . (27) terial wall is primarily driven by the blood flow rate in the

032402-7 DAN WANG, LINDA VAHALA, AND ZHILI HAO PHYSICAL REVIEW E 98, 032402 (2018) common carotid artery and is also regulated by left ventricle and the interindependent factors associated with the arterial rotation. wall itself. (4) The motion pattern variation of the longitudinal mo- In the future, a scaling analysis of the terms in the derived tion of the common carotid arterial wall with aging and dis- governing equations and experimental studies are needed to eases arises from changes in both the blood ﬂow rate pattern validate the above conclusions.

[1] M. Willemet and J. Alastruey, Ann. Biomed. Eng. 43, 190 [16] S. Qorchi, G. Zahnd, D. Galbrun, A. Sérusclat, P. Moulin, (2015). D. Vray, and M. Orkisz, IRBM 38, 219 (2017). [2] G. Baltgaile, Perspect. Med. 1, 146 (2012). [17] A.˚ R. Ahlgren et al., Ultrasound Med. Biol. 41, 1342 (2015). [3] D. Bia, I. Aguirre, Y. Zócalo, L. Devera, E. C. Fischer, and R. [18] J. R. Womersley, Philos. Mag. 46, 199 (1955). Armentano, Rev. Españ. Cardiol. (Engl. Ed.) 58, 167 (2005). [19] G. Morgan and J. Kiely, J. Acoust. Soc. Am. 26, 323 (1954). [4] R. K. Warriner, K. W. Johnston, and R. S. Cobbold, Physiol. [20] D. Hughes, L. A. Geddes, J. D. Bourland, and C. F. Babbs, Meas. 29, 157 (2008). in Acoustical Imaging, edited by A. F. Metherell, Acoustical [5] T. Pereira, C. Correia, and J. Cardoso, J. Med. Biol. Eng. 35, Imaging Book Series, Vol. 8 (Springer, Boston, MA, 1980), pp. 555 (2015). 699–707. [6] H. Yli-Ollila, T. Laitinen, M. Weckström, and T. M. Laitinen, [21] T. G. Papaioannou and C. Stefanadis, Hell. J. Cardiol. 46,9 Clin. Physiol. Funct. Imaging 36, 376 (2016). (2005). [7] S. H. Taivainen, H. Yli-Ollila, M. Juonala, M. Kähönen, O. T. [22] N. Etchenique, S. R. Collin, and T. R. Moore, J. Acoust. Soc. Raitakari, T. M. Laitinen, and T. P. Laitinen, Atherosclerosis Am. 137, 1766 (2015). 272, 54 (2018). [23] B. Bank and L. Sujbert, J. Acoust. Soc. Am. 117, 2268 (2005). [8] S. H. Taivainen, H. Yli-Ollila, M. Juonala, M. Kähönen, O. T. [24] L. Horný, M. Netušil, and T. Vonavková,ˇ Biomech. Model. Raitakari, T. M. Laitinen, and T. P. Laitinen, Clin. Physiol. Mechanobiol. 13, 783 (2014). Funct. Imaging 37, 305 (2017). [25] C. Fortier and M. Agharazii, Pulse 3, 159 (2015). [9] C. Leguy, E. Bosboom, H. Gelderblom, A. Hoeks, and F. Van [26] A. V. Kamenskiy, I. I. Pipinos, Y. A. Dzenis, C. S. Lomneth, De Vosse, Med. Eng. Phys. 32, 957 (2010). S. A. J. Kazmi, N. Y. Phillips, and J. N. MacTaggart, Acta [10] S. Svedlund and L. M. Gan, Clin. Physiol. Funct. Imaging 31, Biomater. 10, 1301 (2014). 32 (2011). [27] J. Albinsson, Advancements of 2D speckle tracking of arterial [11] J. Tat, J. S. Au, P. J. Keir, and M. J. MacDonald, Clin. Physiol. wall movements, Ph.D. thesis, Lund University, 2017. Funct. Imaging 37, 106 (2017). [28] A.˚ R. Ahlgren, M. Cinthio, S. Steen, T. Nilsson, T. Sjöberg, [12] H. Yli-Ollila, M. P. Tarvainen, T. P. Laitinen, and T. M. Laitinen, H. W. Persson, and K. Lindström, Am. J. Physiol. Heart Circ. Ultrasound Med. Biol. 42, 2873 (2016). Physiol. 302, H1102 (2011). [13] J. S. Au, D. S. Ditor, M. J. MacDonald, and E. J. Stöhr, Physiol. [29] L. Horný, T. Adámek, and M. Kulvajtová, Biomech. Model. Rep. 4, e12872 (2016). Mechanobiol. 16, 375 (2017). [14] M. Cinthio, A. R. Ahlgren, J. Bergkvist, T. Jansson, H. W. [30] K. S. Heffernan, W. K. Lefferts, and J. A. Augustine, Int. J. Persson, and K. Lindstrom, Am. J. Physiol. Heart Circ. Physiol. Hypertens. 2013, 920605 (2013). 291, H394 (2006). [31] J. Tat, I. N. Psaromiligkos, and S. S. Daskalopoulou, Ultrasound [15] J. S. Au, P. A. Bochnak, S. E. Valentino, J. L. Cheng, E. J. Stöhr, Med. Biol. 42, 2114 (2016). and M. J. MacDonald, Exp. Physiol. 103, 141 (2018). [32] D. McDonald, J. Physiol. 127, 533 (1955).

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