# A novel, FFT-based one-dimensional blood flow solution method for arterial network

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## Abstract

In the present work, we propose an FFT-based method for solving blood flow equations in an arterial network with variable properties and geometrical changes. An essential advantage of this approach is in correctly accounting for the vessel skin friction through the use of Womersley solution. To incorporate nonlinear effects, a novel approximation method is proposed to enable calculation of nonlinear corrections. Unlike similar methods available in the literature, the set of algebraic equations required for every harmonic is constructed automatically. The result is a generalized, robust and fast method to accurately capture the increasing pulse wave velocity downstream as well as steepening of the pulse front. The proposed method is shown to be appropriate for incorporating correct convection and diffusion coefficients. We show that the proposed method is fast and accurate and it can be an effective tool for 1D modelling of blood flow in human arterial networks.

## Keywords

Fast Fourier transform (FFT) Perturbation method 1D arterial haemodynamics Pulse wave propagation## 1 Introduction

Mathematical and numerical modelling of blood flow in a human arterial network allows researchers to understand various flow related phenomena and disorders. It can help us to understand the genesis and progression of various diseases in the cardiovascular system, and it also provides us with a platform for developing methods for detecting such diseases. The arterial haemodynamics is influenced essentially by pulse wave propagation phenomena. At present time, one of the most effective and fruitful ways to understand wave phenomena in an arterial network is through one-dimensional (1D) flow modelling. Recent works on 1D blood circulation modelling have demonstrated its accuracy in predicting various flow quantities (Mynard and Nithiarasu 2008; Swillens et al. 2008; Alastruey et al. 2011; Gamilov et al. 2014; Sazonov et al. 2017).

The core of the 1D modelling is based upon numerical solution of nonlinear partial differential equations (PDE) derived, for example, in Formaggia et al. (2001), Sherwin et al. (2003). There are many numerical schemes/time integration methods available, including Finite Difference (FD) (Olufsen et al. 2000; Smith et al. 2002; Reymond et al. 2009; Saito et al. 2011), Finite Element (FE) (Formaggia et al. 2003; Sherwin et al. 2003; Mynard and Nithiarasu 2008; Alastruey et al. 2011), Finite Volume (FV) (Cavallini et al. 2008; Delestre and Lagrée 2013) and Discontinuous Galerkin (DG) methods (Matthys et al. 2007; Marchandise et al. 2009; Alastruey et al. 2011). Some of them are compared in the benchmark paper by Boileau et al. (2015). A rather fast scheme is proposed in Carson and Van Loon (2017) which is based on an Enhanced Trapezoidal rule method (Kroon et al. 2012). Many of these works appear to use slightly different mathematical models and constitutive relations as shown later in the present work.

Majority of the existing models, however, assume the convection velocity to be average over the cross section of an artery. Although this approximation is generally accepted, such approximations appear to give incorrect results, even for a steady-state flow. This error is pronounced in highly time-dependent flows, such as the pulsating flow experienced by an arterial network. In addition, space–time methods do not account correctly for the vessel skin friction of an arbitrary pulsating wave in an artery. The viscous effects should be based on the use of the Womersley solution (Womersley 1955) that is too complicated for the 1D space–time approach to handle. Other unsolved issues of 1D space–time approach include difficulties in implementing multi-elements Windkessel and/or non-reflective outlet boundary conditions, incorporate a robust viscoelastic vessel wall model, curvature of arteries and mass loss in smaller arteries and vessel walls.

An attempt to account for the vessel skin friction more accurately is made in the work by Bessems et al. (2007) where the authors consider differential velocity values near the wall region (a viscous layer) and the central core. Thus, the authors have introduced an additional degree of freedom. This approach can improve the accuracy of the computed skin friction to a certain extend but is less accurate than an approach based on the use of the Womersley solution. Another approach to account accurately for the skin friction in a space–time scheme is proposed in Reymond et al. (2009). The authors utilize a waveform obtained at a previous hear-beat cycle, perform the fast Fourier transform (FFT) for every element, build the Womersley solution by computing the Bessel functions of complex argument for every harmonic component, calculate contribution of every harmonic to the skin friction and perform the inverse Fourier transform. For every subsequent cycle, the accuracy of computed skin friction approaches to that of the Womersley solution. This method requires additional computations for every element.

An alternative to space–time method is proposed in Flores et al. (2016), which is based on linearization of the 1D equations and expanding the solution using the Fourier series. In this approach, the problem of wave propagation in an arterial network is solved analytically in the frequency domain, separately for every harmonic component. The pressure and flow rate waveforms are then calculated at any point of the network by numerically computing the inverse Fourier transform to the analytical solution obtained. In this approach, the skin friction can be accurately incorporated via the Womersley solution. Also, the viscoelastic properties of the vessel wall can be included without handling the complexities of the governing equations (Alastruey et al. 2011). As indicated in Flores et al. (2016), this approach allows one to rapidly investigate the role of individual physical properties of a cardiovascular system subjected to a pulsatile waveform. It is important to note that the Fourier transform and calculations carried out in frequency domain are the natural ways for dealing with the wave phenomena. Such important concepts as phase and group velocities can be employed in these methods to explain the distinctive features of wave the pulse propagation and reflection. The frequency domain approach has been successfully used in Sazonov et al. (2017) for developing a non-invasive, aortic aneurysm detection method using a waveform analysis.

Although the method proposed in Flores et al. (2016) is an excellent progress in terms of speed, it has many restrictions. To obtain an analytical solution using this method, every vessel (or its segment between two junctions) is approximated by a cylindrical pipe of constant cross section. Thus, the solution obtained is not general for realistic tapering vessels, found in arterial networks. Another restriction is that the linear algebraic equations have been derived manually for solving the linear problem and thus the method is not easy to employ on an arbitrary arterial network. Finally, the nonlinear effects are not considered by Flores et al. (2016). Therefore, despite its advantages, it is not competitive against the existing 1D modelling methods that use computational algorithms, especially in terms of robustness and accuracy.

In the present work, we propose the generalization of the FFT method by developing an effective and accurate procedure for solving the equations for a tapering vessel or vessel with abnormalities such as aneurysm and stenosis. Also, we account for nonlinear terms in the 1D equations through a nonlinear correction to the linear solution. In this way, we can capture the effect of increasing pulse wave velocity (PWV) downstream as well as steepening of the pulse front. In addition, in the implementation of the proposed method, the system of algebraic equations for every harmonic component is built automatically for any arbitrary arterial network. Finally, the nonlinear equations we employ contain the correct convection and diffusion coefficients.

This paper is organized into the following sections. In Sect. 2, the governing 1D arterial network equations are presented and the nonlinear and viscous terms in these equations are analyzed. Here, different constitutive relations are examined and compared. In Sect. 3, the proposed perturbation method is described, which reduces the nonlinear equations to a set of linear partial differential equations (PDE) which, in turn, are reduced to the ordinary differentials equations (ODE) by the Fourier transform method. An effective method for integration of the derived ODEs for an arbitrary tapering vessel is presented in Sect. 4 and generalized to rapid geometrical variations in vessels in the same section. In Sect. 5, the boundary condition required at inlet, outlet and vessel junctions are described. Here also the method for obtaining a linearized solution for an arbitrary arterial network is explained. In Sect. 6, a method for computing the second-order nonlinear corrections is presented for a single tapering vessel, for the boundary conditions and for full solution in an arterial network. Section 7 compares the present numerical results to those obtained by established 1D numerical methods and experimental results. In the final section, we outline the advantages and future prospects of the proposed approach. Some auxiliary material is presented in Appendix, including generalization of the Womersley solution for flow in a flexible pipe (“Appendix 1”).

## 2 Governing equations for 1D arterial network

*p*(

*x*,

*t*) is the pressure;

*q*(

*x*,

*t*) is the flow rate;

*A*(

*x*,

*t*) is the lumen cross-sectional area; subscripts

*t*and

*x*denote partial derivatives with respect to time

*t*and axial coordinate

*x*, respectively; \(\rho\) and \(\nu\) are the blood density and kinematic viscosity, respectively; \(\alpha\) and \(\gamma\) are dimensionless coefficients defined and discussed in Secs. 2.1 and 2.2. Various forms of the constitutive relations \(A = A(p)\) are considered in Sect. 2.3.

Equations (1)–(3) form a basis for the 1D blood flow modelling. Note that these equations do not account for losses in the vessel wall due to its viscoelastic properties and also for blood flow rate losses in small lateral arterial vessels. In the following subsections we make some remarks on nonlinear and viscous terms in the governing PDEs.

### 2.1 The \(\alpha\) parameter

*u*is the flow velocity profile; the integral is taken over the lumen cross section

*S*. The \(\alpha\) coefficient is called the Coriolis parameter in Formaggia et al. (2001) and is also known as Boussinesq coefficient (Simakov and Kholodov 2009). Its value is assumed to be unity in most of existing models (Mynard and Nithiarasu 2008). This is actually valid for a flow with a uniform velocity profile \(u\equiv {\bar{u}}\), which has an infinite velocity gradient at the wall and, hence, the infinite value of the wall shear stress (WSS). Note that as shown in Sherwin et al. (2003), only for \(\alpha =1\), (1)–(2) can be rewritten in terms of the mean velocity \({\bar{u}} = q/A\) as

### 2.2 Friction coefficient \(\gamma\)

*a*can be calculated via the equation

*r*denotes the partial derivative with respect to

*r*, which is the polar coordinate in the lumen cross section

*S*(\(r\in [0,a]\)). A uniform velocity profile gives \(\gamma =\infty\) as \(u_{r}(a)=\infty\).

For the Poiseuille flow \(u_{r}(a)=4{\bar{u}}/a\), the \(\gamma\) parameter takes a value of 4, which is used in most of the works (e.g. Mynard and Nithiarasu 2008). However, a pulsating flow is characterized by the high near-wall velocity gradient \(u_{r}(a)\) much higher than in the steady-state flow. Therefore an effective value of the \(\gamma\) coefficient, higher than four is used as well. For example, a value of 11 for \(\gamma\) is used in Alastruey et al. (2012a, b). This value is calculated in Smith et al. (2002) by fitting experimental data presented in Hunter (1972).

From these results we can conclude that assuming coefficients \(\alpha\) and \(\gamma\) to be constant is very approximate. Assumption of an \(\alpha\) coefficient for an uniform velocity profile and the \(\gamma\) coefficient for the Poiseuille velocity profile in majority of existing studies should be changed to represent a more accurate pulsatile nature of the flow. A more rigorous approach can be based on convolution with functions \(\alpha (t)\) and \(\gamma (t)\) in the corresponding terms of the equations. These functions can be determined from the Womersley solution. However, this would complicate tremendously the space–time numerical schemes employed in the 1D modelling approach. Due to the reasons stated above, the nonlinear and viscous terms used in (2) by many of the space–time approaches are not very accurate. However, proposed method can easily incorporate different forms of convection and diffusion coefficients.

### 2.3 The constitutive relation

*A*(

*p*) is a linear function, the first term is nonlinear with respect to

*p*. Derivative \({{\mathrm {d}}A}/{{\mathrm {d}}p}\equiv A'(p)\) can be referred to as the cross-sectional vessel compliance. The following dependence

*p*(

*A*) is proposed in Formaggia et al. (2001) and Sherwin et al. (2003)

*h*being the wall thickness and \(E^{\prime }=E/(1-\sigma ^{2})\) being the plate/shell analogue of the Young’s modulus and \(\sigma\) is the Poisson ratio. For an incompressible material \(\sigma =0.5\) and therefore \(E' = \frac{4}{3} E\). In Eq. (9), the \(\beta\) parameter is assumed to be independent of

*A*, which is valid if the vessel wall stiffness \(hE'\) is assumed to be independent of strain.

*A*we see that function

*A*(

*p*) is obviously nonlinear, i.e.,

*p*(

*A*) are considered below, and they are also nonlinear. Nevertheless, all of them can be expanded around the equilibrium state: \(\varDelta p=p-p_{0}\):

*p*and subscript 0 means that the value is calculated at the equilibrium state. In order to further analyse pressure–area relationship, lets introduce the pulse wave velocity (PWV) of a small perturbation to the equilibrium state \(c_0\) as

*Constant wall stiffness model*Differentiating function (10) and using expansion (13) we obtain

*Olufsen’s model*Model proposed by Olufsen (1999) (used in, eg, in Vignon and Taylor 2004) can be written in the form:

*Power law*Both the above models are particular cases for the power law model proposed in Mynard et al. (2010), i.e.

*b*parameter is proposed:

*b*parameter obeys Eq. (20) or a similar law, then the nonlinearity parameters \(\delta _1,\delta _2,\ldots\) are different in different parts of the arterial system.

In the power law relationship, if \(b>1\) then \(\delta _1<1/2\) and the vessel wall becomes stiffer when the vessel expands during the pulse propagation. This dependence is indicated by works on arterial wall elastic properties (Holzapfel et al. 2000; Ogden and Saccomandi 2015). If \(b=2\) then \(\delta _1 = 0\) and the *A*(*p*) dependance becomes linear up to the third-order terms with respect to \({\hat{p}}\). There are other similar models in which the *p*(*A*) dependence is described in terms of elementary functions.

*Armentano et al’s model*One of the earliest models is proposed in Armentano et al. (1995). Here, the lumen diameter, \(D = \alpha + \beta \ln p\), where \(\alpha\) and \(\beta\) are constants. In terms of \(p_0,A_0\) and \(c_0\) the pressure-area relationship can be written as:

*Kholodov’s model*An exponential/log dependence is proposed in Kholodov (2001) and employed in Gamilov et al. (2014), Simakov and Kholodov (2009), Vassilevski et al. (2015). For \(A\ge A_0\) the dependence is exponential and is described by

It appears that not all studies use identical pressure-area relationships, and the nonlinear parameter \(\delta _1\) varies from positive to negative values depending on the model used. These variations between studies indicate that the accuracy of the nonlinear part of the dependence *p*(*A*) is not well established and needs further investigations. Computational results of different studies can agree with each other only if nonlinear effects are small. A small nonlinear term in the *A*(*p*) dependence can be accounted through a correction to the linear problem. Thus, instead of the detailed study on dependance *A*(*p*) or *p*(*A*), one should focus only on nonlinear coefficients \(\delta _n\), especially \(\delta _1\), in large arteries, where pressure and area variations during the heart cycle are relatively large.

### 2.4 Matching conditions at vessel junctions

*s*. Equation (25) is linear.

*p*, rather than the total pressure as the matching condition (Olufsen 1999; Reymond et al. 2009). A correction to the dynamic part of the pressure in the matching condition which depends on the angle between the parent and daughter vessels is discussed in Formaggia et al. (2003). The possibility to account for an additional pressure loss at junction is considered in Mynard and Valen-Sendstad (2015). This additional term, responsible for the pressure loss in the matching condition, is nonlinear with respect to velocity.

In summary, we can conclude that there are significant differences between studies in dealing with nonlinearity and often the nonlinearity is not satisfactorily represented. These variations are acceptable as long as the effect of nonlinearity is small. Since majority of the existing computational models assume limited nonlinearity in the arteries, we believe that a linearized perturbation method with nonlinear correction term is suitable for solving blood flow equations in a human arterial network.

## 3 Reduction of governing nonlinear PDEs to linear ODEs

### 3.1 Perturbation method

*m*. It is homogeneous for \(m=1\). The right-hand side for \(m>1\) depends on \(\varDelta p^{(m')},q^{(m')}\) calculated at previous iterations: \(m'<m\):

### 3.2 The Fourier transform method

*i*is the imaginary unity,

*n*th harmonic component and \(P_n(x)\) and \(Q_n(x)\), respectively, represent the

*n*th harmonic component (complex amplitudes of a harmonic wave at current

*x*). They can be calculated using the integral over the cardiac period

*T*, which is the direct Fourier transform of a periodic function, i.e.,

*n*th harmonic component of the state vector \({\varvec{u}}^{(m)}\). The linear operator contains only full derivatives, i.e.,

*N*is the number of harmonic components required. We do not need to solve these equations for negative

*n*as a real \({\varvec{u}}^{(m)}\) results in \({\varvec{U}}^{(m)}_{-n} = \left( {\varvec{U}}^{(m)}_n\right) ^*\), where \(*\) denotes a complex conjugate.

## 4 Integration of the ODEs

Since we are working with the basic state values of *A*, *q* and *c* here, we omit subscript 0 in this section and in Sect. 6 as it can be confused with the zeroth harmonic component. In these sections we follow \(c=c_{0}\) and \(A=A_{0}\) and \(q=q_{0}\) and regard them as independent of the wave amplitude. In the subsections below we consider the linear approximation, i.e. calculation of \(P_n^{(1)},Q_n^{(1)}\) and the nonlinear corrections are considered in Sect. 6.

### 4.1 Zeroth harmonic component

### 4.2 Nonzeroth harmonic components

*n*in variables \(P_n^{(1)}\), \(Q_n^{(1)}\) and \(\omega _n\) to make the equations easy to follow. For the linear approximation, we have the following ODEs:

### 4.3 Vessel of constant cross section and wall stiffness

The ODEs (40)–(41) do not admit an analytical solution in a closed form for a generic tapering vessel. Therefore, we first consider the simplest case of uniform vessel with constant parameters along the vessel axis.

*Solution as a sum of travelling waves*If

*A*,

*c*and \(\gamma\) are constant along the vessel, then we can write a general analytical solution in the linear approximation using the forward, \([P_f,Q_f]^{\mathrm{T}}\), and backward, \([P_b,Q_b]^{\mathrm{T}}\), harmonic travelling waves as particular solutions, i.e.,

*Effect of blood viscosity on wave propagation* In the presence of viscosity, the wavenumber is complex \(k = \omega \phi /c\). The real part of \(\phi\) affects the wave speed (slowing it down) and increases the total phase incursion \({\mathrm {Re}}\,(kL)\) in the segment of length *L*. Its imaginary part, \({\mathrm {Im}} \phi \le 0\), indicates the wave decay during propagation. The wave decay per unit of length equals to \(\exp \{-{\mathrm {Im}} k\}\).

*a*and the wave frequency \(f=\omega /2\pi\) is shown in Fig. 2(left) for \(\gamma\) calculated via Eq. (130) (see “Appendix 1”). One can see that its contribution to the wavenumber is essential for narrow vessels and at low frequencies. The wavenumber varies essentially along a tapering vessel at low frequencies. Therefore contribution of viscous losses looks to be more crucial for lower frequencies.

Due to viscous losses, the wave amplitude decays by a factor \(\exp \{-{\mathrm {Im}} kL\}\) after propagation through the vessel. Therefore, since the total phase incursion is small for low frequencies, the relatively large imaginary part of the wavenumber does not cause the essential decay of the wave in a vessel. This is indicated in Fig. 2(right) where plots of \(-{\mathrm {Im}} kL\) are depicted for \(L = 10\,\hbox {cm}\). One can see that the decay at the total length *L* is higher at a higher frequency.

*Transmission matrix*An alternative form of a general solution can be used if the vessel inlet conditions,

*P*(0),

*Q*(0), are given as

*P*(0) and

*Q*(0). Note that in Flores et al. (2016), two numbers:

*P*(0) and

*P*(

*L*) are selected to determine the total field in the vessel. Below we show that the choice of

*P*(0) and

*Q*(0) has some advantages for numerical implementation.

### 4.4 Tapering vessel

ODEs (40) and (41) do not admit a closed from analytical solution for arbitrary functions *A*(*x*) and *c*(*x*). After analysing different possible approaches, we have selected the following two approaches.

*Numerical integration of initial value problems*ODEs (40) and (41) can be integrated numerically. To compute entries of transmission matrix \({\varvec{T}}\), we should integrate ODEs (40) and (41) twice with the initial conditions, i.e.,

*Piecewise conic approximation*Eliminating

*Q*from ODEs (40) and (41), we obtain the second-order ODE with respect to

*P*. After obtaining a solution, we can calculate

*Q*as

*c*varies along the vessel approximately proportional to \(a^{-1/2}\), i.e. slower than the lumen radius. The variable \(\phi\) varies fast only in a narrow and strongly tapering vessel. If we approximate

*c*, \(\phi\) and \(k=\omega \phi /c\) by their averaged values, we have

A tapering vessel can be approximated by a sequence of truncated conic elements with values \(c,\phi\) and *k*, approximated by their average value over every element. Let \(\{x_0=0,x_1,\ldots ,x_{{N_e}-1},x_{{N_e}}=L\}\) be edge points of the cones. Here, \({N_e}\) is the number of conic elements.

*i*th element can be used to calculate the corresponding mean values:

*i*indicates that the value is calculated at point \(x_i\). Using (52) we can write the transmission matrix \({\varvec{T}}_i\) of every element as

*Accuracy of the piecewise conic approximation*To study the accuracy of the piecewise conic approximation, we compute the reflection \(R(\omega )\) and transmission \(S(\omega )\) coefficients of a tapering vessel, located between two cylindrical pipes with matched areas and wave velocities. The Right Carotid segment has been selected from the arterial network described in Mynard and Nithiarasu (2008) having simultaneously the largest length and highest lumen radius gradient: the length is \(L=9.4\hbox { cm}\), the inlet and outlet diameters, respectively, are \(D_1 = 6.75\hbox { mm}\) and \(D_2 = 3.5\hbox { mm}\). The vessel is approximated by a cone. The wave velocity dependence on lumen radius is taken to be approximated by Eq. (102) proposed in Blanco et al. (2015).

Three basic element lengths are selected \(h_b=2,1\) and \(0.5\hbox { cm}\) for the conic approximation. These lengths are adjusted to the vessel length by the rule: \({N_e} = \lceil L/h_b\rceil\), \(h = L/{N_e}\), where \(\lceil x\rceil\) means the closest integer equal or greater than *x*. Therefore the element lengths are adjusted to values \(h\approx 1.88,0.94\) and \(0.49\hbox { cm}\).

## 5 Boundary conditions

Consider a network containing \(N_s\) vessels and vessel segments with one inlet segment \(s=1\) and a number of terminating segments. In every segment the axial coordinate *x* is referenced from its inlet. The flow parameters \(P_{ns}(x) = P_s(x;\omega _n)\) and \(Q_{ns}(x) = Q_s(x;\omega _n)\) in the *s*th segment can be calculated by the methods described above if the inlet flow parameters \(P_{ns}(0)\) and \(Q_{ns}(0)\) are known.

Introduce the notations \({\hat{P}}_{ns}=P_{ns}(0)\) and \({\hat{Q}}_{ns}=Q_{ns}(0)\), where \(s=1,\ldots ,N_s\). Thus \({\hat{P}}_{ns},{\hat{Q}}_{ns}\) are inlet amplitudes for every segment of the network. To compute the flow in the entire network, we have first to calculate \(2N_s\) unknowns \([{\hat{P}}_{n1},{\hat{Q}}_{n1},\ldots , {\hat{P}}_{nN_s},{\hat{Q}}_{nN_s}]\), for every harmonic component. Hence, we have to derive \(2N_s\) equations with respect to these unknowns from the inlet, junction and outlet boundary conditions.

In Flores et al. (2016) the following \(2N_s\) unknowns are taken: \([P_{n1}(0),P_{n1}(L_1),\ldots ,P_{nN_s}(0),P_{nN_s}(L_{N_s})]\). This is suitable if the vessels are approximated by uniform pipes without nonlinear correction. In stead of dealing with \(P_{ns}(0)\) and \(Q_{ns}(0)\), we can actively use the transmission matrices \({\varvec{T}}\) to calculate solutions for tapering vessels. Also, this approach simplifies computation of nonlinear corrections.

*Inlet boundary condition*Let the periodic pressure waveform is set at the inlet of the network \(p^{\mathrm {in}}(t)=p^{\mathrm {in}}(t+T)\) . We regard \(p^{\mathrm {in}}(t)\) as a full pressure taken from direct measurements (i.e. sum of forward and the backward waves). Then the boundary conditions is

*Matching conditions at junctions*Consider the

*s*th segment of length \(L_s\) connected to \({\mathcal {D}}\) daughter segments at its outlet. Remind that \({\hat{P}}_{ns}\) and \({\hat{Q}}_{ns}\) are inlet flow parameters for the

*s*th segment for the

*n*th harmonic component. Now, the outlet flow parameters can be written in the linear approximation through the transmission matrix \({\varvec{T}}_{ns}\) for the

*s*th segment as

*s*as \({\mathfrak {D}}=\{d_1,\ldots ,d_{{\mathcal {D}}} \}\), the continuity of the total pressure and the mass conservation at junction can, respectively, be written as

*Matching conditions at junctions with merging arteries*In some junctions two parent arteries can merge to form a single daughter artery. For example, the left and right vertebral arteries merge to form the basilar artery which is a common daughter segment for the vertebral arteries. There are few similar backward bifurcations in the cerebral arterial system. Similar situation can occur in an arterial system with a bypass. In a general case, for the node with \({\mathcal {P}}\) parent segments and \({\mathcal {D}}\) daughter segments, we have the following set of equations

*Boundary condition at the outlets of terminating segments*In general, the Windkessel model is employed at the outlet of a terminating vessel (Alastruey et al. 2012a; Boileau et al. 2015; Carson and Van Loon 2017), which is analogous to an electric circuit containing resistors and capacitors. In the present approach proposed, for every harmonic component with frequency \(\omega _n\), the impedance load for a terminating segment

*s*can be defined by single complex number \(Z_{sn}^{\mathrm {out}}\) rather than by an ODE as in the traditional numerical schemes, i.e.,

*C*, the impedance equals \(Z_{ns}^{\mathrm {out}} = R_1 + R_2/(1+{i}\omega _n R_2C)\).

*P*/

*Q*ratio in the forward propagating wave is \(P^f/Q^f = \tilde{Z}/\left( 1-{i}\zeta \right)\). Therefore the non-reflecting outlet impedance should be

*s*. Expressing \(P_{ns}^{(1)}(L_s)\) and \(Q_{ns}^{(1)}(L_s)\) through the transmission matrix \({\varvec{T}}_{ns}\) we obtain the linear equation for \({\hat{P}}_{ns}^{(1)}\) and \({\hat{Q}}_{ns}^{(1)}\) as

*Zeroth harmonic component*The inlet/junction/outlet conditions should be considered separately for the zeroth harmonic component. The component \(P_{0s}\) for the inlet segment is set by Eq. (56). At junctions, we have the matching conditions (65). As for the outlet conditions at terminating vessel segments, the outlet resistance \(R_{s}^{\mathrm {out}}\) can be set at all such segments as:

*Solution to the linearized problem* Conditions (58) [or (60)], along with conditions (65) and (72) form a closed systems of linear algebraic equations with respect to \({\hat{P}}_{ns}^{(1)}\) and \({\hat{P}}_{ns}^{(1)}\) for all \(n\ne 0\). Equations (60), (65) and (73) [or (74)] form a closed system for \(n=0\).

*n*we have the same solutions but complex conjugate. Then for monitored site \(x\in [0,L_s]\) of a selected vessel segment

*s*, the waveform can be computed through the transmission matrix \({\varvec{T}}_s\) and the inverse Fourier transform, (34),

## 6 Nonlinear corrections

*Nonlinear correction to solutions* Let inlet parameters \({\hat{P}}_{n}^{(1)}(0)\) and \(Q_{n}^{(1)}(0)\) for the *s*th vessel segment are found for all *n* (subscript *s* is omitted in this section). The next stage is to substitute them into relation (33) in order to calculate the right-hand sides for equations at the second iteration. We should mention here that the most expensive part is to compute the \(\alpha\) parameter in relation (33).

Also observe that in Fig. 1 the variation of \(\alpha\) is relatively small around the value 4/3 in contrast to the larger variation of the \(\gamma\) parameter. Also, note that the accuracy in the computation of the correction to the main approximation can be lower compared to the main approximation. Therefore from practical viewpoint, the easiest way is to approximate \(\alpha\) by a constant value, say, 4/3. Accurate computation of the \(\alpha\) parameter and study of its influence on the waveform will be addressed in the future works.

*n*is omitted here for brevity. To calculate entries of the \({\varvec{T}}\) matrix, we use the conic approximation described in Sect. 4.4. To calculate the quadratures, we employ the Trapezoidal rule using cones’ edge points \(\{x_{0},\ldots ,x_{{N_e}}\}\) as points of discretization. Now, the flow parameters at the outlet of the

*j*th element are given as

*Nonlinear corrections to the junction matching conditions*With the account for nonlinear corrections, the outlet flow parameters take the following form

*s*th segment, and \(Q_{ns}^{(1)}(L_{s}) = T_{21}^{ns} {\hat{P}}_{ns}^{(1)} + T_{22}^{ns}{\hat{Q}}_{ns}^{(1)}\). Now, substituting (87), we obtain \({\mathcal {D}}+1\) linear inhomogeneous equations as

*Solving the nonlinear problem* The second iteration of the network problem equations is different from the first iteration as inhomogeneous equations (90) instead of (65) are employed in the second iteration. If Murray’s law, (74), is used, then the corrected outlet flow rate also should be accounted. The total system of \(2N_s\) algebraic equations with respect to \({\hat{P}}_{ns}\) and \({\hat{Q}}_{ns}\) is still linear, but most of the equations are inhomogeneous. After solving all \(N+1\) systems of equations, the corrected waveforms can be computed in all necessary sites of a flow network.

## 7 Comparison with other numerical 1D schemes for an arterial network

### 7.1 Effect of accurate viscous term

First, we consider the effect of the more accurate account of viscous decay of the propagating pulse. For this purpose we compute propagation of a single Gaussian-shaped pulse along a uniform pipe having length of \(10\hbox { m}\), lumen radius of \(1\hbox { cm}\) and \(c_0=6.17\hbox { m/}s\), considered first in Alastruey et al. (2012a) and then used for comparison of various numerical 1D schemes in Boileau et al. (2015). The flow rate at the inlet is set as \(q^{\mathrm {in}}(t) = 1\times \exp \{-(t-0.05)^2/0.01^2\} \hbox { cm}^3\)/s. Non-reflection boundary condition is set at the outlet. The standard blood properties are used, i.e., \(\rho = 1.04\hbox { g}/\hbox {cm}^3\) and \(\nu = 0.04 \hbox { cm}^2\)/s. The velocity profile at inlet is taken as \(u(r,x,t) = {\bar{u}}(x,t)(\zeta +2)/\zeta \,\left[ 1 - (r/a)^{\zeta } \right]\) Alastruey et al. (2012a), Boileau et al. (2015) with \(\zeta = 9\) that corresponds to \(\gamma = 11\).

When correctly accounted for viscous term, i.e. when the \(\gamma\) parameter is calculated accurately based on Womersley’s solution (see Flores et al. 2016 and Sect. A), change in shape and amplitude of the propagating pulse as shown in Fig. 4 (left) is observed. One can see here that the pulse is decaying noticeably faster, its shape is modified with formation of pulse tail, and the pulse peak propagates slightly slower than in the case of constant \(\gamma\). Such big difference in pulse decay is obtained as we have considered a very narrow pulse instead of a realistic waveform, observed in arteries. This narrow pulse is rich in high-frequency harmonics with \(\gamma\) value greater than 11. In a realistic waveform, first five harmonic components contain 95% of the pulse energy, whereas in the narrow Gaussian pulse used in Alastruey et al. (2012a), Boileau et al. (2015), the number of harmonic components containing 95% of the pulse energy is 32. For a realistic waveform computed with the constant gamma \(\gamma = 11\) and based on Womersley profiles give approximately the same decay of the pulse peak as seen in Fig. 4 (right). Nevertheless there is some difference in the pulse shape and propagation speed between the constant \(\gamma\) approximation and Womersley’s profile-based treatment of the viscous term. A large difference is observed in the narrow negative peak in Fig. 4 (right) as it is formed by higher frequencies.

For the sake of comparison, a waveform propagation computed with the \(\gamma =4\) (the value often taken in 1D arterial network modelling) is plotted using the dashed line in Fig. 4 (right). One can see that \(\gamma =4\) underestimates the pulse peak decay in the pipe with \(a_0=1\hbox { cm}\). This indicates that in a large artery like aorta \(\gamma = 11\) is a reasonably effective viscous parameter, averaged over accurate \(\gamma\) values of the main harmonics constituting a typical waveform. At the same time, in a narrower vessel, the effective constant \(\gamma\) should be smaller than 11 as the Womersley’s profile-based \(\gamma\) values for the main harmonics are smaller (see Fig. 13).

### 7.2 Nonlinear effects in a uniform pipe

*Comparison with the analytical solution*First, we estimate the accuracy of the numerical integration (84) against the analytical solution for a uniform pipe and inviscid flow. Equation (32) may be written in the following form:

### 7.3 Accuracy for a tapering vessel

*1*—the inlet,

*2*—midpoint and

*3*—outlet of the first pipe and

*4*—the outlet of the tapering segment. The waveforms computed by the numerical scheme described in Carson and Van Loon (2017) are shown in Fig. 6 using coloured lines and waveforms computed by the proposed method are shown using dashed black lines. The inlet flow amplitude is taken sufficiently small with \(q_{\max }=0.1\hbox { cm}^3\)/s to ensure that the process is linear and \(\gamma =4\) is employed in both computations. One can see that agreement between proposed method and numerical computations is excellent and this proves the accuracy of the proposed method, at least for the linear process.

*3*(inlet of the tapering segment), in the vicinity of the main peak are shown in Fig. 7. They are shown in a normalized form with \(\varDelta {\hat{p}} = p/\rho c_0^2\) and \({\hat{q}} = q/(A_0c_0)\), which are proportional to the amplitude parameter, \(\varepsilon = q_{\max }/(A_0c_0)\). Therefore, dividing by \(\varepsilon\) we obtain normalized waveforms having approximately the same amplitude distorted only by the nonlinear factors. One can see that the pressure peaks are shifted forward with the increase in amplitudes. The waveforms computed by the proposed method behaves similarly to that of the numerical computation, but their shift is smaller when the amplitude approaches \(q_{\max }=10\hbox { cm}^3\)/s.

*p*and

*q*at a realistic amplitude \(q_{\max }\) and calculate the nonlinear correction by the formula,

Note that the nonlinear distortion of the pulse is high in this network due to a long (\(L=1\hbox { m}\)) pipe connected to a tapering segment. The consideration based on Riemann equation, describing formation of a shock wave confirms this. The nonlinear distortion is accumulated during its propagation along a long uniform pipe as the higher harmonics generated by the nonlinear effect propagate with the same speed along a uniform pipe. Any change in geometry like tapering, branching, PWV variation, etc. will cause discrepancy between the phase velocity of the different harmonics that will prevent accumulation of nonlinear distortion. Therefore, in the real arterial system such distortion should be smaller.

### 7.4 Comparison with experimental results

All the pipes in the experimental setup, except one with the aneurysm have a constant diameter. Therefore, the calculation of the transmission matrix \({\varvec{T}}\) is straightforward using (45). The element size for nonlinear corrections is taken equal to 2 cm. For a 14 cm segment with an aneurysm, the element size is taken equal to 0.5 cm, both for the linear problem and for nonlinear corrections. The viscosity factor \(\gamma\) is taken frequency dependent and is computed by the approximation, (130). The nonlinearity coefficients are: \(\alpha = 4/3\), \(\delta _1 = 1/2\). In the experiments described in Sazonov et al. (2017), single pulses are used because of the strong reflection from the outlet of the pipe network. As in the proposed method we have to deal with periodic signals, the period is taken large (4 s), and non-reflecting boundary condition is set at the outlet of the pipe network. The small discrepancy between computed and measured waveforms in Fig. 9 is caused by the uncertainty of mechanical parameters of some elements of the setup and uncertainty of the inlet boundary conditions.

### 7.5 Application of the method to arterial networks

*A*and the wave speed

*c*[or the \(\beta\) parameter in relation (9)] require interpolation along the length. Often

*A*(

*x*) is linearly interpolated along the vessel length (Mynard and Nithiarasu 2008) and the same approach is followed for \(\beta\) parameter if its inlet and outlet values are given. In some works, the authors approximate only

*A*(

*x*) and apply one of the known approximations for

*c*(

*a*) (recall \(a = \sqrt{A/\pi }\) is the lumen radius). Such approximations are mainly employed in 1D modelling, which can be written in the following forms:

*a*should be in centimetres, and the obtained PWV

*c*is in m/s. Approximation (101) is proposed in Olufsen (1999) and its parameters for systemic arteries are listed in Mynard and Smolich (2015). Approximation (102) is proposed in Blanco et al. (2015) and approximation (103) is proposed in Reymond et al. (2009). Here, 20

*a*equals to the lumen diameter

*D*expressed in mm. These approximations for

*c*(

*a*) are plotted in Fig. 10(left).

Different types of lumen radius variation, adapted by different authors, are depicted in Fig. 10(right). They are: the area is linearly interpolated as in Mynard and Nithiarasu (2008) (blue), the radius is linearly interpolated as shown in red and exponential interpolation of area is shown in green. One can see that the linear interpolation of the area produces a convex and non-realistic shape. Thus, other types of interpolation of *A*(*x*) are better approximations. Nevertheless, for a typical artery, the difference between inlet and outlet radii is much smaller and discrepancies between the interpolations are hardly important.

For assessing the accuracy of the proposed method, we compare our results against the numerical modelling results of the human arterial network using a well established method, described in Mynard and Nithiarasu (2008). Here we use the arterial network described in Mynard and Smolich (2015), which contains 107 blood vessel segments as shown in Fig. 11. The dependence of the PWV on lumen radius is calculated using Eq. (102). The flow rate \(q^{\mathrm {in}}(t)\) is imposed at the inlet of the first segment as in Blanco et al. (2015); Boileau et al. (2015) with the heartbeat period of \(T = 1\hbox { s}\). All the tapering segments are approximated by truncated cones. We have set constant values \(\gamma = 4\) and \(\alpha =1\) as in the computational model Mynard and Nithiarasu (2008). The constitutive relation is taken in the form of relation (9) and thus the nonlinearity parameter \(\delta _1=0.5\). The three-element Windkessel (lumped) model is employed as terminal boundary conditions.

## 8 Discussion and conclusions

Many of the currently used numerical models for arterial flow do not account correctly for the viscous friction, especially for a pulsating flow. It is obvious that several different values of viscosity coefficient \(\gamma\) are employed by currently used models. The nonlinear convection term used in the mathematical models in the past is not rigorous for a pulsating flow. In addition, a variety of constitutive laws are employed for vessel walls, some of which resulted in nonlinear terms in the governing equations. All these variations in mathematical model suggest that there is no consensus among researchers on how to deal with nonlinear effects when solving for flow in arterial networks. Since the nonlinearity is often small, all the treatments used in the literature appear to work without too much disagreement in certain flow regimes (low amplitude waves). However, we have proposed a generalization of the nonlinear terms through corrections that may be applicable for all flow regimes.

A novel method, based on the perturbation technique and fast Fourier transform (FFT) in solving the 1D blood flow equations, has been proposed in the present work. The proposed method makes the FFT competitive against traditional space–time numerical schemes in terms of both robustness and speed. In contrast to the FFT approach described in Flores et al. (2016), the proposed method can be applied to an arbitrary arterial network, containing tapering vessels, vessels with stenosis and aneurysms, and to a high amplitude waveform in which the nonlinear effects are relevant. As demonstrated by the results, the proposed method is faster than competing methods and it is accurate. It accounts for viscous effects more accurately than any existing space–time methods and more importantly the viscous coefficient, \(\gamma\), is automatically calculated for different flows and physical conditions. The proposed method simplifies boundary conditions required at the terminal vessels. Thus, we believe that this method can be an alternative and potentially a more effective tool for 1D modelling of blood flow in arterial networks.

Although the proposed method is a substantial improvement to the existing methods, it requires further development in the following area. Further attention is required to deal with viscoelastic effects, blood mass loss due to smaller branches, porous nature of arteries and application to clinical environment.

## Notes

### Acknowledgements

Authors acknowledge the help of and support of Dr Jason Carson, Innovation Research Fellow funded by Medical Research Council UK, for helping with some of the numerical computations.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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