Residual stresses reduce pulse wave velocity in arteries

Mechanics Of Soft Materials (2020) 2:7 https://doi.org/10.1007/s42558-020-00022-7 ORIGINAL PAPER Residual stresses reduce pulse wave velocity in arteries P. Mythravaruni1,2 · K. Y. Volokh1 Received: 3 January 2020 / Accepted: 29 April 2020 © Springer Nature Switzerland AG 2020 Abstract In this note, we develop simple analytical formulas to estimate the effect of residual stresses on the pulse wave velocity in blood vessels. We combine these formulas with three constitutive models of the arterial wall. Particularly, we consider the Fung model and two models accounting for the dispersion of collagen fibers via 8 and 16 structure tensors accordingly. The residual stresses come into play with a description of the initial kinematics—the opening angle. Our numerical examples reveal that residual stresses reduce the pulse wave velocity. The latter effect becomes especially pronounced at high values of blood pressure. Keywords Residual stresses · Pulse wave velocity · Constitutive models 1 Introduction Blood pressure is one of the significant health indicators. Many factors such as physical exercise, emotional, and physiological states affect blood pressure. Excessive fall and rise in blood pressure leads to accompanying disorders, hypotension and hypertension, respectively [1–3]. Conventionally, blood pressure is measured by releasing the external pressure that is imparted to the arm by an inflating cuff, which brings the blood flow to halt temporarily [4]. This cannot be used for continuous monitoring of blood pressure. Continuous monitoring is possible only through invasive procedures that rely on the measurements of intra-arterial pressure [5]. These techniques put patients at a disadvantage of exposure to risk of complications and, hence, call for intensive care [6]. Blood pressure measurement alone cannot provide any conclusive prediction of the health status of heart. Pulse wave velocity (PWV) is described by the velocity with which the blood pressure pulse propagates in arteries. Pulse wave velocity measurement is suitable for continuous and noninvasive monitoring of blood pressure [7–10]. The MoensKorteweg equation [11] was used √ together with Hughes equation [12] to correlate the blood pressure to the measured pulse wave velocity (PWV): PWV = Eh0 /(2ρR0 ) and E = E0 exp(ζp), where p is the blood pressure, E0 is the elastic modulus at zero blood pressure, ζ is a material coefficient of artery, ρ is the blood mass density, and h0 and R0 are the initial wall thickness and radius of artery. The Moens-Korteweg equation was developed based on the assumptions that the artery wall was thin and its thickness and radius were constant. In reality for human arteries, the thickness to radius ratio varies with blood pressure [13] and the thickness to radius ratio is much higher than the range which can be√used for thin shell [14]. Recently, it was proposed [15], instead of the Moens-Korteweg equation, to use formula PWV = (s/ρ)dp/ds, where s is the inner cross-sectional area of the artery, together with Fung’s hyperelastic constitutive model [16] for the arterial wall. This new formula is valid for arteries with thick walls and with thickness to radius ratio varying with blood pressure.  P. Mythravaruni 1 Faculty of Civil and Environmental Engineering, Technion – I.I.T., Haifa, Israel 2 Present address: IIT Madras, Chennai, Tamil Nadu, India 7 Page 2 of 6 Mechanics Of Soft Materials (2020) 2:7 In the present work, we make the next step and develop simple analytical formulas to estimate the effect of residual stresses—prestresses—on the pulse wave velocity in blood vessels. We combine these formulas with three constitutive models of the arterial wall. Particularly, we consider the Fung model and two models accounting for the dispersion of collagen fibers via 8 and 16 structure tensors accordingly. The residual stresses come into play with a description of the initial kinematics—the opening angle. We note in passing that residual stresses can affect mechanical behavior of materials in various and unexpected ways. We refer to [20] and [21] for the effect of residual stresses on surface wave propagation and bifurcation in soft materials and structures accordingly. 2 Pulse wave velocity We refer to [17] for the general theoretical background and notation. With account of axial symmetry, nontrivial equilibrium equations for the arterial wall in cylindrical coordinates reduce to ∂σrr ∂ψ ∂ψ − λr , (1) r = λϕ ∂r ∂λϕ ∂λr where σrr is the principal Cauchy stress in the radial direction and ψ(λr , λϕ , λz ) is the strain energy density depending on the principal stretches in directions of cylindrical coordinates r, ϕ, z. With account of incompressibility, λr λϕ λz = 1, we can introduce a reduced strain energy function w(λ, λz ) = ψ(λr , λϕ , λz ) = ψ(λ−1 λ−1 z , λ, λz ). (2) Then, we calculate ∂ψ ∂λr ∂ψ ∂λϕ ∂ψ ∂λz ∂w = + + ∂λ ∂λr ∂λ ∂λϕ ∂λ ∂λz ∂λ ∂ψ 1 ∂ψ = − 2 , ∂λϕ λ λz ∂λr or ∂w ∂ψ ∂ψ λϕ − λr =λ . ∂λϕ ∂λr ∂λ (3) (4) Now, Eq. 1 can be rewritten as follows ∂w ∂λ ∂σrr =λ . (5) r ∂r ∂λ ∂λ Further transformation of the equilibrium equation requires calculation of ∂λ/∂r. For this purpose, we use the incompressibility condition in the form γ λz (r 2 − a 2 ) = R 2 − A2 , (6) where γ = 2π/(2π − ω) and ω is the artery opening angle after the cut; r, a and R, A are the radial coordinate and internal radius of the artery in the current and referential configurations accordingly. Consequently, we have λ = γ rR −1 = γ r(γ λz (r 2 − a 2 ) + A2 )−1/2 , (7) and, by the direct calculation, ∂λ = λ(1 − λz γ −1 λ2 ). r ∂r Substitution of Eq. 8 in Eq. 5 yields (8) ∂σrr 1 ∂w = . ∂λ 1 − λz γ −1 λ2 ∂λ (9) Integrating Eq. 9 we get for the blood pressure on the arterial wall  λa w dλ p(λa ) = , −1 λ2 − 1 λb (λa ) λz γ (10) (2020) 2:7 Mechanics Of Soft Materials Page 3 of 6 7 Table 1 Material constants for Fung’s model [16] c c1 c2 c3 c4 c5 c6 26.95 kPa 0.0089 0.9925 0.418 0.0193 0.0749 0.0295 where w ≡ ∂w/∂λ; λa = γ a/A; λb = γ b/B; and  λb = λ2a A2 B −2 + (1 − A2 B −2 )γ λ−1 z (11) by virtue of the incompressibility condition. According to Fung [13], the pulse wave velocity is calculated as follows  s ∂p PWV = , ρ ∂s (12) where s = π a 2 is the artery internal cross-section area and ρ is the blood mass density. In terms of the boundary stretch λa , we can rewrite Eq. 12 as follows  λa ∂p , PWV = 2ρ ∂λa (13) where the pressure derivative is calculated from Eq. 10 analytically λa A2 ∂p w  (λa ) w  (λb ) = − . −1 2 2 −1 ∂λa λz γ λa − 1 λz γ λb − 1 λb B 2 (14) Thus, we finally have (compare also with [18])     λa λa A2 w (λa ) w (λb )  − PWV = . 2ρ λz γ −1 λ2a − 1 λz γ −1 λ2b − 1 λb B 2 (15) 3 Constitutive models of arterial wall We consider the following three models of the arterial wall. The Fung constitutive model [16] is defined by the following strain energy density function w(λ, λz ) = 2 2 c c1 −2 −2 c2 2 exp λ λz − 1 + λ −1 2 4 4 2 c (...truncated)