A Physics Based Approach to the Pulse Wave Velocity Prediction in Compliant Arterial Segments

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A physics based approach to the pulse wave velocity prediction in compliant arterial segments

Alexander S. Liberson n, Jeffrey S. Lillie, Steven W. Day, David A. Borkholder

Rochester Institute of Technology, James E. Gleason Building, 76 Lomb Memorial Drive, Rochester, NY 14623-5604, United States article info abstract

Article history: Pulse wave velocity (PWV) quantification commonly serves as a highly robust prognostic parameter Accepted 12 September 2016 being used in a preventative cardiovascular therapy. Being dependent on arterial elastance, it can serve as a marker of cardiovascular risk. Since it is influenced by a blood pressure (BP), the pertaining theory can Keywords: lay the foundation in developing a technique for noninvasive blood pressure measurement. Previous Blood pressure studies have reported application of PWV, measured noninvasively, for both the estimation of arterial Pulse wave velocity compliance and blood pressure, based on simplified physical or statistical models. A new theoretical Nonlinear 1D model model for pulse wave propagation in a compliant arterial segment is presented within the framework of fl Convective uid phenomena pseudo-elastic deformation of biological tissue undergoing finite deformation. An essential ingredient is Hyperelasticity the dependence of results on nonlinear aspects of the model: convective fluid phenomena, hyperelastic constitutive relation, large deformation and a longitudinal pre-stress load. An exact analytical solution for PWV is presented as a function of pressure, flow and pseudo-elastic orthotropic parameters. Results from our model are compared with published in-vivo PWV measurements under diverse physiological conditions. Contributions of each of the nonlinearities are analyzed. It was found that the totally nonlinear model achieves the best match with the experimental data. To retrieve individual vascular information of a patient, the inverse problem of hemodynamics is presented, calculating local orthotropic hyperelastic properties of the arterial wall. The proposed technique can be used for non-invasive assessment of arterial elastance, and blood pressure using direct measurement of PWV, with account of hyperelastic orthotropic properties. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction blood vessel elasticity and finite deformation are vitally important to match static testing results for an internal pressure load and an Historically PWV has conceptually been based on the Moens– axial tension of a cylindrical artery under physiological conditions Korteweg equation, which predicts PWV based on an acoustical (Cox, 1975; Demiray et al., 1988; Fung et al., 1979; Zhou and Fung, approach applied to a fluid structure interaction (FSI) of blood flow 1997). with a linear elastic cylindrical artery (O'Rourke, 2005). More The link between PWV, or pulse transit time (PTT), and blood accurate is a nonlinear traveling wave model developed for a pressure (BP) has been previously investigated based on statistical compliant thin-walled linear elastic tube filled with an incom- regression models, or empirical representation of an incremental pressible fluid (Liberson et al., 2014; Lillie et al., 2014). This model isotropic elastic modulus as a function of a transmural pressure accounts for the convective ideal fluid flow interacting with a (Chen et al., 2000, 2009; Muehlsteff and Schuett, 2006; Zhang linearly elastic aortic vessel. However assumptions regarding lin- et al., 2009). We propose a physics based mathematical model that predicts the general relationship between PWV and BP with a ear deformation of an arterial vessel are not precisely applicable to rigorous account of nonlinearities in the fluid dynamics model, living blood vessels as is proved in numerous related publications blood vessel elasticity, finite deformation of a thin anisotropic wall (Chuong and Fung, 1983; Cox, 1975; Demiray et al., 1988; Fung and a longitudinal pre-stress. The combined effect of BP and blood et al., 1979; Holzaphel, 2000; Holzaphel et al., 2000; Humphrey, flow on PWV is derived and correlates with clinical evidence 1995; Humphrey, 1999; O'Rourke, 2005; Takamizawa and Hayashi, presented in Nurnberger et al. (2003) and Salvi et al. (2013).In 1987; Vaishnav et al., 1972; Zhou and Fung, 1997). Nonlinearities in order to retrieve individual vascular information of a patient, we present the PWV based solution for the inverse problem of n Corresponding author. Fax: 585-475-7710. hemodynamics, calibrating individual orthotropic hyperelastic E-mail address: [email protected] (A.S. Liberson). arterial properties based on typical diagnostic measurements and http://dx.doi.org/10.1016/j.jbiomech.2016.09.013 0021-9290/& 2016 Elsevier Ltd. All rights reserved.

Nomenclature H, h Thickness of the wall at a zero stress and loaded conditions respectively (m) PWV Pulse wave velocity (m/s) F Axial pretension force (N) σ σ FSI Fluid structure interaction θ, z Circumferential and axial Cauchy stress components (Pa) BP Blood pressure Eθ, Ez Circumferential and axial Green-Lagrange strain A Cross sectional area (m2) components "# u Axial flow velocity (m/s) a11 a12 p Transmural pressure (Pa) A symmetric tensor of material constants (Pa) a12 a22 ρ Density of incompressible fluid (kg/m3) R,r Internal wall radii in a zero stress and loaded condi- Subscripts tions respectively (m) η Ratio of the wall deflection to R (r,θ,z) Radial, circumferential and axial components of a – CMK Moens Korteweg speed of propagation (m/s) corresponding 3D vector

an optimization technique. This study is a continuation of the conservative form previous work (Liberson et al., 2014; Lillie et al., 2014) in the ∂U ∂U þHUðÞ ¼ 0 ð3Þ context of in-vivo validation and application of the proposed ∂t ∂z methodology to continuous, noninvasive blood pressure mea- where surements. The proposed technique can be used for non-invasive 2 3 þ η assessment of arterial elastance, directly related to the direct η u 1 ¼ ¼ 4 2 5 ð Þ measurement of PWV, with account of hyperelastic orthotropic U ; H pη 4 u ρ u properties of a biological vessel. We find the eigenvalues of HðUÞ to be real and distinct. PWV is associated with the forward running wave velocity, i.e. the largest eigenvalue (Hirsch, 2006), hence it is identified as 2. Theory sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þη fl PWV ¼ uþ pη ð5Þ 2.1. Dynamics of incompressible ow in a compliant vessel 2ρ

One dimensional models simulating blood flow in arteries The partial derivative Pη indicates sensitivity of pressure with effectively describe pulsatile flow in terms of averages across the respect to the wall normal deflection η, and has a clear inter- section flow parameters. Although they are not able to provide the pretation as tangent (incremental) moduli in finite strain inelas- details of flow separation, recirculation, or shear stress analysis, ticity. System (1) and (2) is typically closed by defining an explicit they should accurately represent the overall and average pulsatile algebraic relationship between pressure and normal deflection. flow characteristics, particularly PWV. Derivations of one dimen- For instance, in case of a small deformation and linear elastic – sional models can be found in a number of papers, see for instance response, where E is the Young's modulus, v the Poisson coef- fi η Cascaval (2012), Liberson et al. (2014), and Lillie et al. (2014), and cient, and pressure relates to the circumferential strain via are not repeated here. EH E p ¼ η; E ¼ ð6Þ Conservation of mass and momentum results in the following R 1ν2 system of one dimensional equations Eqs. (4) and (5) can be transformed to the simplified form ∂A ∂ derived differently in Cascaval (2012), Liberson et al. (2014), and þ ðÞ¼uA 0 ð1Þ ∂t ∂z Lillie et al. (2014), sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ∂u ∂ u2 p ¼ þ þη; ¼ Eh ð Þ þ þ ¼ 0 ð2Þ PWV u cMK 1 cMK ρ 7 ∂t ∂z 2 ρ 2 R Under the assumption u{c ; η{1 (linearized approach) Eq. The model described by (1) and (2) is applied exclusively to the MK (7) converts into the Moens–Korteweg equation for the forward large arteries, where viscosity is important only in the regions of a and backward traveling waves. In the general case, Eq. (5) should boundary layer, and does not practically have an impact on PWV be supplemented by appropriate constituent equations for a (Caro et al., 2012; Parkhusrt et al., 2012). As it follows from hyperelastic anisotropic arterial wall, accounting for finite monographs of Caro et al. (2012) and Li et al. (1981) the intro- deformation. duction of visco-elastic, rather than elastic vessel wall properties results in a slight increase in wave speed, but affects noticeably the 2.2. Hyperelasticity of the vessel wall attenuation of a waveform. For an impermeable thin walled membrane, neglecting inertia It is assumed that arterial wall is hyperelastic, incompressible, – forces, the vessel pressure strain relationship is maintained by anisotropic, and undergoing finite deformation. After a few origi- equilibrium condition as a function p ¼ p η , based on relevant nal loading cycles (preconditioning) the arterial behavior follows constitutive relations. Noting that A ¼ πR2ð1þηÞ2, and assuming some repeatable, hysteresis free pattern with a typical exponential that transmural pressure is a smooth function of a wall normal stiffening effect regarded as pseudo elastic (Fung et al., 1979; deflection (derivative pη ¼ ∂p=∂η exists at any point), the total Holzaphel et al., 2000). Numerous formulations of constitutive system of equations can be presented in the following non- models for arteries have been proposed based on polynomials

(Vaishnav et al., 1972), exponential functions (Chuong and Fung, into a family of relating parametric problems. We introduce a load 1983; Fung et al., 1979; Holzaphel, 2000; Zhou and Fung, 1997), log parameter 0rτr1, and substitute p-τp; f -τ f considering functions (Takamizawa and Hayashi, 1987), or their mixtures continuous successive loading of the vessel by internal pressure τp (Humphrey, 1995), applied to the pseudo elastic strain-energy and axial load τf from zero to nominal values. Assume the solution density formulation. In a comparison paper (Humphrey, 1999)itis of (15) depends continuously and is differentiable with respect to concluded that the exponential descriptor of the passive behavior the parameter τ. Starting from the known answer for a certain of arteries, due to Zhou–Fung, is “the best available”. According to value of the parameter (τ ¼ 0 at a stress free case), the solution of -Fung et al. (Chuong and Fung, 1983; Fung et al., 1979; Zhou and the equation for other values of the parameter may be obtained by Fung 1997) the strain energy density function W for the pseudo integrating the rate of change of the solution with respect to the elastic constitutive relation may be presented in the form parameter. To find the rate of strain components with respect to τ, 1 take the natural log of Eq. (15) W ¼ cðeQ 1Þð8Þ 2 pR lnλz þQ þln sθ ¼ ln τþln where c is a material coefficient, and Q is the quadratic function of cH – fi fl f the Green Lagrange strain components. For the nite in ation and lnλz þQ þln sz ¼ ln τþln ð16Þ extension of a thin walled cylindrical artery the following strain c energy function is used and differentiate both sides (the dot above variable means τ 2 2 partial derivative by ), which yields Q ¼ a Eθ þ2a EθE þa E ð9Þ 11 12 z 22 z _ MEðÞE ¼ b ð17Þ where c; a11; a12; a22 are material constants. The Cauchy stress components in circumferential and axial directions are: where 2 3 2 ∂W 2 Q sθ σθ ¼ λθ ¼ cλθe sθ; sθ ¼ a Eθ þa E 2s2 þa þ2sθs þa ∂Eθ 11 12 z 6 θ 11 þ z 12 7 ð10Þ 6 1 2Ez 7 2∂W 2 Q MEðÞ¼4 5; σz ¼ λ ∂ ¼ cλ e sz; sz ¼ a12Eθ þa22Ez sz 2 z Ez z 2sθs þa þ2s þa z 12 1þ2E z 22 ! ! z With the geometry of the reference state determined, we define R, sθ Eθ τ Z, H – as an internal radius, axial coordinate and a wall thickness in E ¼ ; b ¼ ð18Þ E sz a stress free configuration, r, z, h – internal radius, axial coordinate z τ fi and a wall thickness in a physiologically loaded con guration. The As it follows from Eq. (18), τ¼0 is a singular point. To find the corresponding principal stretch ratios are. strain rates at infinitesimally small τ the following asymptotic

λθ ¼ r=R; λz ¼ dz=dZ; λr ¼ h=H ð11Þ relations deduced from the Taylor expansions in the proximity of a stress free state (Oð…Þ – stands for the asymptotic approximation) Assuming isochoric deformation incorporate the incompressi- _ 2 _ 2 bility condition as Eθ ¼ Eθ τþO τ ; Ez ¼ Ez τþðτ Þ _ _ 2 _ _ 2 λzλθλr ¼ 1 ð12Þ sθ ¼ a11Eθ þa12Ez τþO τ ; sz ¼ a12Eθ þa22Ez τþOðτ Þ – 2 Q 2 The Green Lagrangian strain components relate to the princi- λz ¼ 1þOðÞτ ; Q ¼ O τ ; e ¼ 1þOðτ Þð19Þ pal stretch ratios of Eq. (12) by should be substituted into Eq. (15). As a result we arrive at the 1 2 τ¼ M E_ ðÞ¼b ; E ¼ λ 1 ; ði ¼ θ; z; rÞð13Þ linear equation for the strain rates at 0, 0 0 0 where i 2 i "#0 1 pR a11 a12 cH For the membrane thin walled cylindrical artery undergoing M ¼ ; b ¼ @ A ð20Þ 0 a a 0 f finite inflation and axial deformation, the load–stress relations 12 22 c follow from the static conditions Once E is known at τ¼0, Eq. (18) can be integrated to give pr pRλθ pR σ ¼ ¼ ¼ λ2λ θ θ z EðÞ¼τþΔτ EðÞþτ M 1ðÞτ bðτÞΔτ ð21Þ h Hλr H F The above process continues to provide solutions for τ¼1. Once σz ¼ λz ¼ f λz ð14Þ 2πRH strain components are known, the Cauchy stress components can A substitution back into Eq. (10) yields the desired relations: be calculated from Eq. (14), and stretch ratios and load–deflection relations – from Eq. (11). 1 Q pR cλ e sθ ¼ Alternative approaches of solving hyperelastic anisotropic finite z H strain problems can be found in Holzaphel et al. (1996) and cλ eQ s ¼ f ð15Þ z z Humphrey (2003) using finite element approximations, and in The solution of Eqs. (9), (13) and (15) results in a load–strain Wineman and Waldron (1995) and Zidi and Cheref (2002), based λ ¼ r ¼ r R on finite difference methods. relations, which with account of the identity θ R R þ1 ¼ ηþ1converts into the p ¼ p η function, required by (5) to predict a wave front speed of propagation, i.e. PWV. 3. Methods 2.3. Numerical computation of a tangent moduli in a finite strain hyperelasticity To validate the algorithm, the theoretical prediction of the relationship between circumferential Lagrangian stress compo- ¼ λ ∂W λ To solve the nonlinear deformation problem ((15), (13) and (9)) nent Tθ θ∂Eθ and circumferential stretch ratio θ is compared the theory of continuation method is applied (Grigolyuk and with experimental results in Zhou and Fung (1997). Zhou Fung's Shalashilin, 2012). It enables the transformation of algebraic, work was selected because in a comparison paper (Humphrey, functional or differential equations into initial value problems by 1999) it is concluded that the exponential descriptor of the passive introducing a parameter, and embedding the particular problem behavior of arteries, is “the best available”. Each loading curve

using a single elastic constant, which is a-priori independent on 0.006 0.007 0.008 100 0.005 pressure.

The FDH model is utilized to determine the material constants 0.003 noninvasively using PWV vs. transmural pressure. Experimental 80 data of Histand and Anliker (Histand and Anliker, 1973; Pedley, ﬁ 2008), who introduced short trains of arti cial, high-frequency 60 0.009 waves to measure their propagation speed in the canine aorta of 0.002 canines at different static pressures was selected. This experiment was selected because it recorded PWV over a wide range of Pressure [mmHg] 40

pressures in the aorta. The noninvasive extracted material con- 0.004

0.007 0.008 stants for mongrel dogs, are then compare to an invasive method 20 0.005 0.006 obtained from a biaxial static load of a canine aorta for 5 mongrel dogs performed by Zhou and Fung (1997). 51015 20 Axial Force [N] 4. Results and discussion Fig. 1. The model predictions of stiffness match experimental in vivo data which is essential for accurate PWV determination. (a) Dependence of circumferential stress 4.1. Algorithm validation on circumferential stretch ratio during inflation and extension. The comparison of Zhou–Fung test to our model shows close correlation R2¼0.99. (b) Contour lines representing an overall residual of solved equations for different combinations of The mechanical properties (c¼124.5 kPa, a11 ¼0.264, a12 ¼ transmural pressure and extension force. 0.379, a22 ¼0.0461) are identified in Zhou and Fung (1997) from experimental data on mongrel dogs canine aorta, subjected to inflation and longitudinal stretch within the physiological range. The theoretical curves are calculated based on the algorithm described above, and presented in Fig. 1 by continuous lines. The 10 stress–strain relationship recorded by protocol (2) in Zhou and Fung (1997) is presented by markers for different values of axial stress. The circumferential stress becomes typically nonlinear around λθ ¼ 1:3, increasing its slope monotonically as shown in 8 Fig. 1a. The theoretical prediction is in a good correspondence with experimental data within the entire strain range, with the corre- 2 lation coefficient R exceeding 0.99 forq allffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cases.

PWV [m/s] 6 2 þ 2 Contour lines of an overall residual R1 R2, where R1 and R2 are the individual relative residuals of each of the Eq. (15),are presented in Fig. 1b. The accuracy of numerical computation within physiological range of transmural pressure and axial 4 extension force is better than 1%, which justifies application of a 50 100 150 200 250 numerical scheme as in Eq. (21). An accurate quantification of arterial stiffness is a key factor affecting precision of a PWV Pressure [mmHg] prediction. Fig. 2. The nonlinear model produced the best fit of the PWV vs. transmural pressure function. Dash lines indicate theoretical prediction. Square markers 4.2. Material properties Identification by fitting PWV curve illustrate the total set of experimental points (Wineman and Waldron, 1995; Zidi and Cheref, 2002). Solid square markers correspond to the subset of experimental points used for calibration. Using the properties extracted from the nonlinear fi In order to nd relationship PWV vs. BP it is essential to model the lower (solid) line shows the effect on PWV using the partially nonlinear identify the associated material properties of the vessel. Typically model, combining hyperelasticity with small deformation.

Please cite this article as: Liberson, A.S., et al., A physics based approach to the pulse wave velocity prediction in compliant arterial segments. Journal of Biomechanics (2016), http://dx.doi.org/10.1016/j.jbiomech.2016.09.013i A.S. Liberson et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5 to identify material properties the fitting process is used mini- 13 mizing the least square difference between the measured and S3 – – 12 predicted stress strain (or load displacement curves) (Chuong S1 and Fung, 1983; Fung et al., 1979; Zhou and Fung, 1997). PWV 11 S2 based methodology evaluating arterial stiffness by minimizing the S4 difference between measured and model based predicted wave 10 S5 velocities, is presented in Shahmirzadi et al. (2012). Theoretical 9 PR model used in Shahmirzadi et al. (2012) is the Moens–Korteweg model, which is based on a linear elastic behavior of the wall 8 undergoing acoustic interaction with internal flow. As a conclusion PWV [m/s] 7 the paper affirms “acceptable performance except the region of a soft wall”. Due to the strong evidence of an essential hyperelastic 6 nonlinearities coupled with finite deformations (Fung, 1990; Gri- golyuk and Shalashilin, 2012; Hirsch, 2006; Histand and Anliker, 5 1973; Holzaphel et al., 1996; Humphrey, 2003; Lillie et al., 2014; 4 Pedley, 2008; Shahmirzadi et al., 2012; Sherwin et al., 2003; 50 100 150 200 250 Wineman and Waldron, 1995; Zidi and Cheref, 2002), no limita- Pressure [mmHg] tions are applied to the physical and geometric aspects of an arterial wall deformation in the present work. The current article Fig. 3. The material properties of a canine aorta of mongrel dogs extracted from investigates possibility for the arterial wall properties identifica- static measures (Zhou and Fung, 1997) predict PWV vs. BP distributions in close proximity to those extracted from direct PWV vs. BP experiments (Histand and tion using noninvasive PWV based approach. There is a noticeable – fi Anliker, 1973; Pedley, 2008). S1 S5 and present result (PR), associated material formal similarity between material parameters identi cation constants are shown in Table 2. problems based on measurements of a static pressure vs. stretch ratio, and a static pressure vs. PWV. In both cases the tangent (incremental) moduli is a key factor affecting modeling. In both 11 σ σ cases the components of Cauchy stress ( θ and z) contribute to Tz= 0.0 kPa the solution. 10 Tz=26.3 kPa Histand and Anliker results are presented in Histand and Tz=52.6 kPa Anliker (1973), O'Rourke (2005) and Pedley (2008), and repro- Tz=78.9 kPa duced in Fig. 2 by square markers as PWV plotted against a single 9 independent variable–transmural pressure. The theoretical model, recast in the form (λθ ¼ ηþ1Þ, 8 sffiffiffiffiffiffiffiffiffiffiffi λ mod θ PWV ¼ uþ ρpη ð22Þ 7

2 PWV [m/s] is calibrated based on a subset of four experimental points, shown 6 in Fig. 2 by solid squares, and verified across the total set of eight fl experimental points, shown by square markers. The mean ow 5 velocity u was estimated using the ratio of u=PWV ¼ 0:2 recom- fi mended by Pedley (2008). The Ptting process was based on ð exp modÞ2 4 minimization of the sum of squares i PWVi PWVi , where exp mod 50 100 150 200 250 PWVi and PWVi are the measured and modeled pulse wave velocities corresponding to the i-th experimental data point. The Pressure [mmHg] Nelder–Mead algorithm available from the Matlab Optimization Fig. 4. Simulation results show that within a physiological range (Horny et al., Toolbox was utilized as an optimization tool. The Fung's model 2013), longitudinal pre-stress load effects PWV by 3%. Tz denotes the axial phy- material properties [c, a11,a12,a22] serve as control variables. siological Lagrangian stress.

Table 1 Dash line indicates theoretical prediction based on the totally Contribution of nonlinearities into the quality of the best fit. nonlinear model, i.e. hyperelastic characterization accounting for the finite deformation. Using properties identified based on the Model FDH SDH SDE MK totally nonlinear characterization, the partially nonlinear model, Error function 0.0147 0.0242 1.08 5.16 combining hyperelasticity with a small deformation, was used to create PWV distribution, shown by the solid line in Fig. 2. Giving material properties of an arterial wall, the chart indicates proximity of two models in terms of PWV distribution. Table 2 fi Material constants, obtained from a biaxial static load of a canine aorta for The quality of a tting process accounting for different non- 5 mongrel dogs (cases S1–S5), performed by Zhou and Fung (1997), and from the linearities is presented in Table 1, quantified in terms of the error present results based on data collected with PWV vs. transmural pressure in function (sum of squares). Four different models have been com- mongrel dogs in Histand and Anliker (1973) and Pedley (2008). pared: the finite deformation hyperelastic (FDH), small deforma-

S1 S2 S3 S4 S5 Present result tion hyperelastic (SDH), small deformation linear elastic (SDE), and the Moens–Korteweg (MK) model. Material parameters have been c [mmHg] 901 945 372 1152 1297 1595 identiﬁed for each model independently, based on a best ﬁtpro- a 11 0.320 0.289 0.338 0.175 0.200 0.1205 cedure. SDE model is obtained from the Fung's model, accounting a 0.451 0.397 0.441 0.314 0.292 0.8080 22 for the linear terms only in a Taylor series expansion of Fung's a12 0.068 0.073 0.010 0.041 0.039 0.001 exponential function (8) of a potential energy of deformation. The

160 Before concluding we analyze the inﬂuence of longitudinal pre- u = 0.00 m/s stress load on PWV. According to Fung (1990) longitudinal pre- 140 u = 0.40 m/s stress may well play a dominant role on wall stress distribution u = 0.80 m/s and effective arterial stiffness. To illustrate the effect of a long- 120 u = 1.20 m/s itudinal force on PWV, as a result of a numerical simulation, the u = 1.60 m/s u = 2.00 m/s variation of PWV due to the variability of a longitudinal pre-stress 100 force is presented in Fig. 4. According to simulation within the realistic physiological range of a longitudinal stress (Horny et al., 80 2013), the relative deviation in PWV does not exceed 3%.

60 4.3. Stability of the model convergence Distensibility (1/MPa) 40 Theoretically a least square procedure can lead to a non-unique set of material parameters in case of a non-convex objective 20 function, particularly dealing with a large numbers of control variables. In general the problem depends on a convexity of an 0 80 100 120 140 160 180 200 objective function, and possibly on a particular least square numerical algorithm and starting point of optimization. In our case Transmural Pressure (mmHg) stable convergence has been obtained using the Nelder–Mead Fig. 5. Dependence of distensibilty on transmural pressure and a flow velocity. algorithm available from the Matlab Optimization Toolbox. We have used different initial points, corresponding to the five cases latter results in a quadratic expression for the potential energy, presented in the Table 2, for numerical minimization. In all cases typical for classical linear orthotropic theory of elasticity. In this all varying material constants, except a22, converged to the same case the number of varying parameters was reduced to three: values, which are presented in the last column of the Table 2. The c*a11, c*a12, and c*a22, since coefficient c appears only as the quality of the fit is characterized by the least square value 0.011 in amplification factor for orthotropic constants a11,a12 and a22. each case. The coefficient a22 varied from 0.299 to 0.505, which did The Moens–Korteweg model is a classical model using a single not affect the distribution of the PWV vs. transmural blood pres- elastic constant, which serves as a control variable.The finite sure, i.e. the quality of the fit. deformation hyperelasticity (FDH) model and small deformation hyperelasticity (SDH) model have the highest quality of fitting 4.4. Prediction of compliance and distensibility of a hyperelastic process, creating practically the same model based regression in segment Fig. 2 within the physiological range of BP. The small deformation with linear elasticity (SDL) model and Moens–Korteweg (MK) Arterial stiffness, or its reciprocals, arterial compliance and model cannot be used to predict dependence of PWV on BP. Recall distensibility, may provide indication of vascular changes that that the SDL model accounts only for the convective flow type predispose the development of major vascular disease. In an iso- nonlinearity (Lillie et al., 2014), whereas the MK model is a totally lated arterial segment filled with a moving fluid, compliance is linear model, predicting PWV is independent of pressure. defined as a change of a volume V for a given change of a pressure, It is interesting to compare mechanical properties predicted and distensibilty as a compliance divided by initial volume using the PWV based methodology applied to the FDH model, to (O'Rourke, 2005). As functions of pressure the local (tangent) those identified in vivo by traditional static load. In (Zhou and compliance C and distensibility D are defined as (Meindrs and Fung, 1997) the thoracic aortas of five mongrel dogs were subjects Hoeks, 2004) of a mechanical test where each specimen was stretched biaxially. dV C dV As a result of the test measurements, material constants from the C ¼ ; D ¼ ¼ ð23Þ dP V VdP Zhou–Fung hyperelastic model were obtained in Zhou and Fung (1997) for canine aortas of mongrel dogs using the least square Eq. (23) determine arterial wall properties as local functions of minimization procedure. The corresponding five data sets being transmural pressure, contrary to the assessment using the dia- – replicated from Zhou and Fung (1997) fill the columns S1–S5 in stolic systolic pulse pressure. the Table 2. The last column represents the material properties of a We present Eq. (23) in the following equivalent form canine aorta of a mongrel dog, identified based on an experi- dV=dη 2V C 2 C ¼ ¼ ; D ¼ ¼ ð24Þ mentally measured PWV vs. BP, obtained by Histand and Anliker dP=dη Pη V Pη (1973) (Pedley, 2008), and the current article methodology. We For the linear elastic response and small deformation, as it have estimated the proximity of material properties predicted by follows from Eqs. (6) and (7), the described methodology and traditional static test in terms of PWV vs. BP distributions. Fig. 3 illustrates distributions of PWV for ¼ EH ¼ ρ 2 ð Þ Pη 2 cMK 25 the five Zhou–Fang's static test cases, presented by S1–S5 columns R of the Table 2, and its counterpart relating to the properties which in turn transforms Eq. (24) to the classical Bramwell–Hill extracted by the present method (the last column of Table 2). relations (Caro et al., 2012) Except the case S3 which is characterized by anomalously low V 1 material coefficient c, the relative variation in PWV prediction C ¼ ; D ¼ ð26Þ ρc2 ρc2 does not exceed 4% of all other cases within the range of trans- MK MK mural pressure presented in the chart. The focus of this work was Generalization of a Bramwell–Hill equations, based on an to create and validate a nonlinear model for predicting PWV in empirical exponential relationship between transmural pressure compliant arterial segments. It is recognized that the arterial and a cross section area, was presented in Pedley (2008). Below properties will change due to aging and disease state (O'Rourke, the classical results are generalized and discussed for the case of a 2005). The changes that may be required for the model especially hyperelastic arterial wall with account of finite deformation and as the arterial wall thickens is the focus of a future work. flow velocity.

To proceed, determine Pη from Eq. (5) and substitute in Eq. (24), Cox, R.H., 1975. Anisotropic properties of the canine carotid artery in vitro. J. Bio- arriving at the following relations mech. 8, 293–300. Demiray, H., Weizsäcker, H.W., Pascale, K., Erbay, H.A., 1988. A stress–strain relation 1þη for a rat abdominal aorta. J. Biomech. 21 (5), 369–374. D ¼ ; C ¼ VD ð27Þ ρð Þ2 Fung, Y.C., 1990. Biomechanics Motion, Flow, Stress and Growth, 569th ed Springer, PWV u Germany. Fung, Y.C., Fronec, K., Patucci, P., 1979. Pseudo-elasticity of arteries and the choice of Fig. 5 illustrates the dependency of distensibility on pressure mathematical expression. Am. J. Physiol. – Heart C 237 (5), H620-31, Nov. and flow velocity. Since PWV is monotonically increasing with Grigolyuk, E.I., Shalashilin, V.I., 2012. Problems of Nonlinear Deformation: The pressure, distensibility is a decreasing function. 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The method shows potential to be used for Humphrey, J.D., 1995. Mechanics of arterial wall: review and directions. Biomed. Eng. 121, 1–162. non-invasive continuous long term blood pressure monitoring. Li, J., Melbin, J., Riffle, R., Noordercraaf, A., 1981. Pulse wave propagation. Cir. Res. 49, 442–452. Liberson, A., Lillie, J.S., Borkholder, D.A., 2014. Numerical solution for the boussinesq type models with application to arterial flow. JFFHMT 1, 9–15. Conflict of interest Lillie, J.S., Liberson, A.S., Mix, D., Schwartz, K., Chandra, A., Phillips, D.B., Day, S.W., Borkholder, D.A., 2014. Pulse wave velocity prediction and compliance assess- – Alexander Liberson, Jeffrey S. Lillie, Steven Day, and David A. ment in elastic arterial segments. Cardiovasc. Eng. Technol. 6 (1), 49 58, Dec. Meindrs, J.M., Hoeks, A.P.G., 2004. Simultaneous assessment of diameter and Borkholder declare that they have no conflict of interest. pressure waveforms in the carotid artery. Ultrasound Med. 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