A Novel Analytical Approach to Pulsatile Blood Flow in the Arterial Network

Abstract

Haemodynamic simulations using one-dimensional (1-D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. We propose a novel linear 1-D dynamical theory of blood flow in networks of flexible vessels that is based on a generalized Darcy’s model and for which a full analytical solution exists in frequency domain. We assess the accuracy of this formulation in a series of benchmark test cases for which computational 1-D and 3-D solutions are available. Accordingly, we calculate blood flow and pressure waves, and velocity profiles in the human common carotid artery, upper thoracic aorta, aortic bifurcation, and a 20-artery model of the aorta and its larger branches. Our analytical solution is in good agreement with the available solutions and reproduces the main features of pulse waveforms in networks of large arteries under normal physiological conditions. Our model reduces computational time and provides a new approach for studying arterial pulse wave mechanics; e.g.,  the analyticity of our model allows for a direct identification of the role played by physical properties of the cardiovascular system on the pressure waves.

Appendices

Appendix 1: Derivation of Generalized Darcy’s Elastic Model (GDEM)

Equations (1), (2), (3) and (4) are obtained from the linearized momentum balance equations:

$$\rho \frac{\partial \mathbf { v}}{\partial t } = -\mathbf { \nabla } p - \mathbf { \nabla } \cdot \sigma ,$$

(48)

where $\mathbf {\nabla } \cdot \sigma$ represents the divergence of the viscous stress tensor. We consider a Maxwell fluid – the simplest fluid that presents viscoelastic behaviour – for which the fluid velocity and stress tensor are related by

$$t_r \frac{\partial \mathbf { \sigma }}{\partial t}=- \eta \nabla \mathbf v - \mathbf { \sigma },$$

(49)

where $t_r$ is the Maxwell relaxation time and is given by the ratio of the viscosity, $\eta$, and the elastic modulus, G; i.e., $t_r=\frac{\eta }{G}$. In the limit of zero relaxation time, Eq. (49) reduces to the constitutive equation for Newtonian fluids. We assume that the radial velocity is much smaller than the axial velocity, so that the momentum balance equations (48), together with the constitutive equations (49) are:

$$t_r \rho \frac{\partial ^2u}{\partial t^2} + \rho \frac{\partial u}{\partial t} = -\frac{\partial p}{\partial x} - t_r\frac{\partial ^2 p}{\partial t \partial x} + \eta \left( \frac{\partial ^2u}{\partial r^2} +\frac{1}{r}\frac{\partial u}{\partial r}\right) ,$$

(50)

$$\frac{\partial p}{\partial r} =0.$$

(51)

Equation (51) implies that pressure is only a function of $x$ and t, and adjusts instantaneously to any point of a luminal cross-sectional area.

We then transform Eq. (50) to the frequency domain. For simplicity of notation we define $k=k(\omega )$, such that $k^2=\frac{\rho }{\eta }\left( t_r \omega ^2 + i\omega \right)$ and $B(x, \omega )=\left( \frac{1-i\omega t_r}{\eta } \right) \frac{d{\hat{p}} }{dx}$. We obtain the following equation for the axial velocity ${\hat{u}}(x,r,\omega )$ in frequency domain,

$$r^2\frac{\partial ^2 {\hat{u}}}{\partial r^2}+r \frac{\partial {\hat{u}}}{\partial r} + k^2r^2{\hat{u}}=B r^2.$$

(52)

This is a Bessel equation of order zero, whose general solution is

$${\hat{u}}(x,r,\omega )=aJ_0(kr)+bN_0(kr)+ {\hat{u}}^p(x,\omega ),$$

(53)

where $J_0$ is the Bessel function of order zero of the first class and $N_0$ is the Bessel function of order zero of the second class, also known as Neumann function of order zero. The particular solution ${\hat{u}}^p(x,\omega )$ is given by

$${\hat{u}}^p=\frac{B}{k^2}=\frac{1}{i\omega \rho } \frac{d{\hat{p}}}{dx},$$

(54)

and the general solution for ${\hat{u}}(x,r,\omega )$ is

$${\hat{u}}(x,r,\omega )=aJ_0(kr)+bN_0(kr)+\frac{1}{i\omega \rho } \frac{d{\hat{p}}(x,\omega )}{dx}.$$

(55)

In order to determine the values of a and b we impose the following boundary conditions: the axial velocity, u, has to be finite at $r=0$, and zero at the average radius, $R_0$. This gives Eq. (1) that allows for the computation of velocity profiles. $K_L$ is a local dynamic permeability in frequency domain given by Eq. (2). Inverse Fourier transformation of Eq. (1) allows one to obtain the velocity profiles $u(x,r,t)$ in time domain. Averaging Eq. (1) over the cross sectional area gives a generalized Darcy’s law in frequency domain, namely,

$${\hat{U}} (x,\omega ) = -\frac{ K(\omega )}{\eta } \frac{\partial {\hat{p}}(x,\omega )}{\partial x} ,$$

(56)

where ${\hat{U}} (x,\omega )$ is the axial velocity averaged over the average cross-sectional area $A_0$. The dynamic permeability, $K(\omega )$, is simply the average of the local dynamic permeability over the average cross-sectional area and is given by Eq. (4). The $x$-dependence of ${\hat{U}} (x,\omega )$ comes from the pressure gradient. Equation (3) follows from Eq. (56) by assuming that the flow is approximately $Q(x,t) \approx A_0 U(x,t)$. The approximation of the area by its average $A_0$ is necessary in order to keep a linear relation between flow and pressure gradient in frequency domain.

The fluid velocity ${\mathbf{v}} = u(x,r,t) \, \hat{\imath}+ v(x,r,t) \, \hat{r}$ satisfies the continuity equation

$${\frac{\partial \rho }{ \partial t} + \mathbf{\nabla } \cdot \left( \rho \mathbf v \right) =0.}$$

(57)

For incompressible fluids in cylindrical coordinates, this one is given by

$$\frac{\partial u}{ \partial x} + \frac{1}{ r} \frac{\partial (r v)}{ \partial r} =0.$$

(58)

Averaging this equation over the mean cross-sectional area gives

$$\frac{\partial U}{ \partial x} + \frac{2\pi }{ A_0} R_0 v_{r=R_0} =0,$$

(59)

where U(x, t) is the axial velocity averaged over the mean cross-sectional area. We consider that the fluid and wall velocities are equal at the average radius, i.e., $v_{r=R_0} =\left. \frac{\partial R}{ \partial t}\right| _{R_0}$, which leads to

$$\left. \frac{\partial U}{ \partial x} + \frac{2\pi }{ A_0} R_0\frac{\partial R}{ \partial t}\right| _{R_0} =0$$

(60)

and, in terms of the flow $Q(x,t)=A_0 U(x,t)$, it becomes

$$\left. \frac{\partial Q}{ \partial x} + 2\pi R_0\frac{\partial R}{ \partial t}\right| _{R_0} =0.$$

(61)

A relationship between P and R is required to write the local radius of the vessel in terms of the local blood pressure. Here we consider a relationship between the pressure and the elastic deformation of the tube, $\Delta R$,22,31

$$p - p_\mathrm{ext}=\frac{Eh}{1-\nu ^2}\frac{\Delta R}{R^2_0} ,$$

(62)

where $p-p_\mathrm{ext}$ is the transmural pressure, E is the Young modulus, h is the vessel thickness, and $\nu$ the Poisson ratio, that we take as $\nu =1/2$ (i.e., we assume the arterial wall to be an incompressible material). Around the radius at diastole, $R_\mathrm{d}$, this can be approximated as

$$p - p_\mathrm{d}=\frac{4}{3}Eh\frac{R-R_\mathrm{d}}{R^2_\mathrm{d}}.$$

(63)

where $p_\mathrm{d}$ is the pressure at diastole. This type of ‘tube law’ has been extensively used in the literature.2, 4, 5,8,11,20,26,30,33,35,42–44,54,55

We take the time derivative of Eq. (63) in order to find an expression for the time derivative of the radius in terms of the time derivative of the pressure to be used in Eq. (61). We evaluate it at the average radius, ${R_0}$, and obtain

$$\left. \frac{\partial R}{ \partial t}\right| _{R_0} =\left. \frac{3R^2_d}{4Eh}\frac{\partial p}{ \partial t} \right| _{R_0}= \frac{3R^2_d}{4Eh} \frac{\partial p}{ \partial t}.$$

(64)

Equations (61) and (64) give Eq. (5), in frequency domain, with $C=\frac{3 \pi R_0 R^2_d}{2Eh}$.

Appendix 2: Full Aorta Model

This appendix describes the analytical solution for the full-aorta model, which consists of 20 vessels representing the aorta and its first generation of larger branches. At the inlet of the ascending aorta, the flow rate measured in vivo was prescribed as the inflow boundary condition, $Q_{\mathrm{in}}(t)$. Terminal vessels were coupled to three-element lumped parameter models simulating blood flow and pressure in downstream vessels. Following the notation of Fig. 5, node 1 is of Type I, nodes 2, 3, 5, 6, 7, 8, 9, and 10 are of Type II, node 4 is of Type III, and nodes 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20 are of Type IV. According to Eqs. (29), (30), (31) and (32), the system of equations for the pressures at the nodes, in matrix form and in terms of the functions $\kappa _{1}^{i}$, $\kappa _{2}^{i}$ and $\kappa _{3}^{i}$ defined in Eq. (28), is given by: ${{\hat{\mathbf {p}}}}={\mathbf {K}}^{-1} {{\hat{\mathbf {Q}}}}$. Here ${{\hat{\mathbf {p}}}}$ is the 20-element vector for the pressures at the nodes, ${{\hat{\mathbf {Q}}}}$ is the vector of the inflow boundary conditions whose sole non-zero element is the first one and is given by $-\frac{{\hat{Q}}_{in}^{1}}{\cos (k_{c}^{1}l^{1})}$, and ${\mathbf {K}}^{-1}$ is the inverse matrix of ${\mathbf {K}}$, which is a response function of the system. ${\mathbf {K}}$ is given by:

$${\mathbf {K}}= \left( \begin{array}{cccccccccccccccccccc} a &{} \kappa _{2}^{2} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{11} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \kappa _{2}^{2} &{} b &{} \kappa _{2}^{3} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{12} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \kappa _{2}^{3} &{} c &{} \kappa _{2}^{4} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{13} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \kappa _{2}^{4} &{} d &{} \kappa _{2}^{5} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \kappa _{2}^{5} &{} e &{} \kappa _{2}^{6} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{14} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{6} &{} f &{} \kappa _{2}^{7} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{15} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{7} &{} g &{} \kappa _{2}^{8} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{16} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{8} &{} h &{} \kappa _{2}^{9} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{17} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{9} &{} j &{} \kappa _{2}^{10} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{18} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{10} &{} k &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{19} &{} \kappa _{2}^{20} \\ \kappa _{2}^{11} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} l &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \kappa _{2}^{12} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} m &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \kappa _{2}^{13} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} n &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{14} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{15} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{16} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{17} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} s &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{18} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} t &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{19} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} u &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{2}^{20} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} v \\ \end{array} \right) ,$$

where

$$\begin{array}{lllllll} a = \kappa _{3}^{1}-\kappa _{1}^{2}-\kappa _{1}^{11} , &{} &{} f = -\kappa _{1}^{6}-\kappa _{1}^{7} - \kappa _{1}^{15} , &{} &{} l= -\kappa _{1}^{11}-\frac{1}{z_{11}}, &{} &{} r=-\kappa _{1}^{16}-\frac{1}{z_{16}} , \\ b = -\kappa _{1}^{2}-\kappa _{1}^{3} - \kappa _{1}^{12} , &{} &{} g = -\kappa _{1}^{7}-\kappa _{1}^{8} - \kappa _{1}^{16} , &{} &{} m=-\kappa _{1}^{12}-\frac{1}{z_{12}} , &{} &{} s=-\kappa _{1}^{17}-\frac{1}{z_{17}} , \\ c = -\kappa _{1}^{3}-\kappa _{1}^{4} - \kappa _{1}^{13} , &{} &{} h = -\kappa _{1}^{8}-\kappa _{1}^{9} - \kappa _{1}^{17} , &{} &{} n=-\kappa _{1}^{13}-\frac{1}{z_{13}} , &{} &{} t=-\kappa _{1}^{18}-\frac{1}{z_{18}} , \\ d = -\kappa _{1}^{4}-\kappa _{1}^{5} , &{} &{} j = -\kappa _{1}^{9}-\kappa _{1}^{10} - \kappa _{1}^{18} , &{} &{} p=-\kappa _{1}^{14}-\frac{1}{z_{14}} , &{} &{} u=-\kappa _{1}^{19}-\frac{1}{z_{19}} , \\ e = -\kappa _{1}^{5}-\kappa _{1}^{6} - \kappa _{1}^{14} , &{} &{} k = -\kappa _{1}^{10}-\kappa _{1}^{19} - \kappa _{1}^{20} , &{} &{} q=-\kappa _{1}^{15}-\frac{1}{z_{15}} , &{} &{} v= -\kappa _{1}^{20}-\frac{1}{z_{20}} . \end{array}$$

Inversion of the matrix ${\mathbf {K}}$ was done symbolically using Mathematica, which took less than a second on a standard laptop. However, the text length required to explicitly write ${\mathbf {K}}^{-1}$ and the pressures at the nodes is excessively large to show it in an Appendix. Once the pressures at the nodes were obtained, analytical expressions for pressure, the flow and velocity profiles in each vessel were calculated using Eqs. (14), (15), and (16) for vessel 1, and Eqs. (8), (10), and (12) for the rest of the vessels. For instance, for the first aortic segment, the pressure at the first node, $\hat{p}^{[1]}$, is necessary in Eqs  (14), (15), and (16) where $\hat{p}_{o}=\hat{p}^{[1]}$. This one is given by:

$$\begin{aligned} \hat{p}^{[1]} &= - \frac{n \kappa _{2}^{3} \mathcal {F}1 \hat{Q}_\mathrm{in}^{1} }{ \kappa _{2}^{2} \kappa _{2}^{13} \cos (k_{c}^{1} l^{1}) } - \frac{B \hat{Q}_\mathrm{in}^{1}}{ \kappa _{2}^{2} \kappa _{2}^{11} \kappa _{2}^{12} \cos (k_{c}^{1} l^{1}) } \\ &\quad \times \left[ \frac{ p \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{12} \mathcal {F}2 }{d m \kappa _{2}^{3} \kappa _{2}^{14} } - \frac{ \mathcal {F}1 \mathrm{T9} }{d m \kappa _{2}^{3} \left( \kappa _{2}^{13}\right) ^2 } \right] , \end{aligned}$$

(65)

with

$$\mathcal {F}1 = \frac{\mathcal {F}2 \mathrm{T7}}{n \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{14} } + \frac{d q \kappa _{2}^{6} \kappa _{2}^{13} \mathcal {F}3 }{n \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{15} },$$

(66)

$$\mathcal {F}2 =- \left( \frac{ F \mathcal {F}3 }{ p \kappa _{2}^{6} \kappa _{2}^{15} } + \frac{ r \kappa _{2}^{7} \kappa _{2}^{14} \mathcal {F}4 }{ p \kappa _{2}^{6} \kappa _{2}^{16} } \right) ,$$

(67)

$$\mathcal {F}3 = - \left( \frac{ G \mathcal {F}4 }{ q \kappa _{2}^{7} \kappa _{2}^{16} } + \frac{ s \kappa _{2}^{8} \kappa _{2}^{15} \mathcal {F}5 }{q \kappa _{2}^{7} \kappa _{2}^{17} } \right) ,$$

(68)

$$\mathcal {F}4 = - \left( \frac{ H \mathcal {F}5 }{r \kappa _{2}^{8} \kappa _{2}^{17} } + \frac{ t \kappa _{2}^{9} \kappa _{2}^{16} \mathcal {F}6 }{r \kappa _{2}^{8} \kappa _{2}^{18} }\right) ,$$

(69)

$$\mathcal {F}5 = \frac{\mathrm{N3}}{\mathcal {M}}- \frac{J \mathcal {F}6}{s \kappa _{2}^{9} \kappa _{2}^{18} },$$

(70)

$$\mathcal {F}6 = \frac{ \mathrm{N1} - \mathrm{N2} }{ \mathcal {M} }.$$

(71)

These quantities contain another set of definitions given by:

$$\mathrm{N1} = d l m n p q r s u \kappa _{2}^{2} \kappa _{2}^{3} \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{6} \kappa _{2}^{7} \kappa _{2}^{8} \kappa _{2}^{9} \kappa _{2}^{11} \left( \kappa _{2}^{12} \kappa _{2}^{14} \kappa _{2}^{15} \kappa _{2}^{16} \kappa _{2}^{17} \kappa _{2}^{18} \kappa _{2}^{20} \right) ^{2} \left( \kappa _{2}^{13} \right) ^{3} K,$$

(72)

$$\mathrm{N2} = d l m n p q r s v \kappa _{2}^{2} \kappa _{2}^{3} \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{6} \kappa _{2}^{7} \kappa _{2}^{8} \kappa _{2}^{9} \kappa _{2}^{11} \left( \kappa _{2}^{12} \kappa _{2}^{14} \kappa _{2}^{15} \kappa _{2}^{16} \kappa _{2}^{17} \kappa _{2}^{19} \kappa _{2}^{20} \right) ^{2} \left( \kappa _{2}^{13} \kappa _{2}^{18} \right) ^{3},$$

(73)

$$\mathrm{N3} = d l m n p q r t u v \kappa _{2}^{2} \kappa _{2}^{3} \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{6} \kappa _{2}^{7} \kappa _{2}^{8} \kappa _{2}^{11} \left( \kappa _{2}^{10} \kappa _{2}^{12} \kappa _{2}^{14} \kappa _{2}^{15} \kappa _{2}^{16} \kappa _{2}^{18} \kappa _{2}^{20}\right) ^{2} \left( \kappa _{2}^{13} \kappa _{2}^{17} \right) ^{3},$$

(74)

$$\mathcal {M} = -v \kappa _{2}^{18} \kappa _{2}^{20} \left( \kappa _{2}^{19} \right) ^{2} \mathrm{T1} + u \kappa _{2}^{20} \left( K \mathrm{T1} -t v \kappa _{2}^{17} \kappa _{2}^{18} \kappa _{2}^{20} \left( \kappa _{2}^{10}\right) ^{2} \mathrm{T2} \right) ,$$

(75)

$$\mathrm{T1} = \kappa _{2}^{20} J \mathrm{T2} - s t \kappa _{2}^{16} \kappa _{2}^{17} \kappa _{2}^{18} \kappa _{2}^{20} \left( \kappa _{2}^{9} \right) ^2 \mathrm{T3},$$

(76)

$$\mathrm{T2} = \kappa _{2}^{18} H \mathrm{T3} - r s \kappa _{2}^{15} \kappa _{2}^{16} \kappa _{2}^{17} \kappa _{2}^{18} \left( \kappa _{2}^{8}\right) ^2 \mathrm{T4},$$

(77)

$$\mathrm{T3} = \kappa _{2}^{17} G \mathrm{T4} - q r \kappa _{2}^{14} \kappa _{2}^{15} \kappa _{2}^{16} \kappa _{2}^{17} \left( \kappa _{2}^{7}\right) ^2 \mathrm{T5},$$

(78)

$$\mathrm{T4} = \kappa _{2}^{16} F \mathrm{T5} - d p q \kappa _{2}^{13} \kappa _{2}^{14} \kappa _{2}^{15} \kappa _{2}^{16} \left( \kappa _{2}^{6}\right) ^2 \mathrm{T6},$$

(79)

$$\mathrm{T5} = \kappa _{2}^{15} \mathrm{T6} \mathrm{T7} + n p \kappa _{2}^{12} \kappa _{2}^{14} \kappa _{2}^{15} \left( \kappa _{2}^{4} \kappa _{2}^{5} \kappa _{2}^{13} \right) ^2 \mathrm{T8},$$

(80)

$$\mathrm{T6} = d m n \kappa _{2}^{11} \kappa _{2}^{14} \left( \kappa _{2}^{3} \kappa _{2}^{12} \kappa _{2}^{13} \right) ^2 A - \kappa _{2}^{14} \mathrm{T8} \mathrm{T9},$$

(81)

$$\mathrm{T7} = p D - d \kappa _{2}^{13} \left( \kappa _{2}^{14}\right) ^2,$$

(82)

$$\mathrm{T8} = \kappa _{2}^{12} \kappa _{2}^{13} A B - l m \kappa _{2}^{11} \kappa _{2}^{12} \kappa _{2}^{13} \left( \kappa _{2}^{2}\right) ^2,$$

(83)

$$\mathrm{T9} = d C \kappa _{2}^{13} - n \kappa _{2}^{12} \kappa _{2}^{13} \left( \kappa _{2}^{4}\right) ^2,$$

(84)

that in turn contain a third set of definitions given by:

$$A = a l - \left( \kappa _{2}^{11} \right) ^{2},$$

(85)

$$B = b m \kappa _{2}^{11} - \kappa _{2}^{11} \left( \kappa _{2}^{12} \right) ^{2},$$

(86)

$$C = c n \kappa _{2}^{12} - \kappa _{2}^{12} \left( \kappa _{2}^{13} \right) ^{2},$$

(87)

$$D = d e \kappa _{2}^{13} - \kappa _{2}^{13} \left( \kappa _{2}^{5} \right) ^{2},$$

(88)

$$F = f q \kappa _{2}^{14} - \kappa _{2}^{14} \left( \kappa _{2}^{15} \right) ^{2},$$

(89)

$$G = g r \kappa _{2}^{15} - \kappa _{2}^{15} \left( \kappa _{2}^{16} \right) ^{2},$$

(90)

$$H = h s \kappa _{2}^{16} - \kappa _{2}^{16} \left( \kappa _{2}^{17} \right) ^{2},$$

(91)

$$J = j t \kappa _{2}^{17} - \kappa _{2}^{17} \left( \kappa _{2}^{18} \right) ^{2},$$

(92)

$$K = k v \kappa _{2}^{18} - \kappa _{2}^{18} \left( \kappa _{2}^{20} \right) ^{2}.$$

(93)

Appendix 3: Error Calculations

For the test cases presented in “Common Carotid Artery” to “Full Aorta Model” sections, the numerical solutions of pressure (p), pressure difference between inlet and outlet ($\Delta p$), volumetric flow rate (Q), and change in radius from diastole ($\Delta r$) given by the analytical GDEM were compared with corresponding values provided by computational 1-D and 3-D formulations. We used the following relative error metrics for p and Q:

$$\begin{aligned} \mathcal {E}^\mathrm{RMS}_{p}&= \sqrt{\frac{1}{n}\sum _{i=1}^n\left( \frac{p_i^{\mathrm{GDEM}}-\mathscr {P}_i}{\mathscr {P}_i} \right) ^2},&\mathcal {E}^\mathrm{RMS}_{Q}&= \sqrt{\frac{1}{n}\sum _{i=1}^n \left( \frac{Q_i^{\mathrm{GDEM}}-\mathscr {Q}_i}{ \max _j(\mathscr {Q}_j)}\right) ^2}, \end{aligned}$$

(94)

$$\begin{aligned} \mathcal {E}^\mathrm{MAX}_{p}&= \mathop {\max }_{\begin{array}{c} i \end{array}}\left|\frac{p_i^{\mathrm{GDEM}}-\mathscr {P}_i}{\mathscr {P}_i}\right|,&\mathcal {E}^\mathrm{MAX}_{Q}&= \mathop {\max }_{\begin{array}{c} i \end{array}}\left|\frac{Q_i^{\mathrm{GDEM}}-\mathscr {Q}_i}{\max _j(\mathscr {Q}_j)}\right|, \end{aligned}$$

(95)

$$\begin{aligned} \mathcal {E}^\mathrm{SYS}_{p}&= \frac{\max (p^{\mathrm{GDEM}})-\max (\mathscr {P})}{\max (\mathscr {P})},&\mathcal {E}^\mathrm{SYS}_{Q}&= \frac{\max (Q^{\mathrm{GDEM}})-\max (\mathscr {Q})}{\max (\mathscr {Q})}, \end{aligned}$$

(96)

$$\begin{aligned} \mathcal {E}^\mathrm{DIAS}_{p}&= \frac{\min (p^{\mathrm{GDEM}})-\min (\mathscr {P})}{\min (\mathscr {P})},&\mathcal {E}^\mathrm{DIAS}_{Q}&= \frac{\min (Q^{\mathrm{GDEM}})-\min (\mathscr {Q})}{\max (\mathscr {Q})}, \end{aligned}$$

(97)

where $p_i^{\mathrm{GDEM}}$ and $Q_i^{\mathrm{GDEM}}$ are the results obtained using the analytical GDEM at a given spatial location and time point i ($i=1,\ldots ,n$). At the same spatial location and time point i, $\mathscr {P}_i$ and $\mathscr {Q}_i$ are either the pressure and flow given by the linear 1-D model or the cross-sectional averaged pressure and flow calculated from the 3-D model. The number of time points n was determined by the 3-D solution. $\mathcal {E}^\mathrm{RMS}_{p}$ and $\mathcal {E}^\mathrm{RMS}_{Q}$ are the root mean square relative errors for pressure and flow; $\mathcal {E}^\mathrm{MAX}_{p}$ and $\mathcal {E}^\mathrm{MAX}_{Q}$ are the maximum relative errors in pressure and flow; $\mathcal {E}^{SYS}_{p}$ and $\mathcal {E}^\mathrm{SYS}_{Q}$ are the errors in systolic pressure and flow; and $\mathcal {E}^\mathrm{DIAS}_{p}$ and $\mathcal {E}^\mathrm{DIAS}_{Q}$ are the errors in diastolic pressure and flow, respectively. Flow errors were normalized by the maximal flow over the cardiac cycle to avoid division by small values of the flow. For the quantities $\Delta p$ and $\Delta r$ we used the same metrics as for the flow rate. All error metrics were calculated over a single cardiac cycle, using the numerical 1-D and 3-D results in the periodic regime.