A Distributed Lumped Parameter Model of Blood Flow with Fluid-Structure Interaction

Abstract

A distributed lumped parameter (DLP) model of blood flow was recently developed that can be simulated in minutes while still incorporating complex sources of energy dissipation in blood vessels. The aim of this work was to extend the previous DLP modeling framework to include fluid-structure interactions (DLP-FSI). This was done by using a simple compliance term to calculate pressure that does not increase the simulation complexity of the original DLP models. Verification and validation studies found DLP-FSI simulations had good agreement compared to analytical solutions of the wave equations, experimental measurements of pulsatile flow in elastic tubes, and in vivo MRI measurements of thoracic aortic flow. This new development of DLP-FSI allows for significantly improved computational efficiency of FSI simulations compared to FSI approaches that solve the full 3D conservation of mass and momentum equations while also including the complex sources of energy dissipation occurring in cardiovascular flows that other simplified models neglect.

Introduction

The forces imparted on cardiovascular structures play an important part in cardiovascular pathology (Resnick et al. 2003) and due to their predictive capabilities, image based computational fluid dynamics (CFD) has the potential to enhance diagnosis from medical imaging, inform clinical treatment planning, and perform hypothesis based research (Taylor and Steinman 2010; Marsden and Esmaily-Moghadam 2015; Marsden and Feinstein 2015; Liang et al. 2018). However, the speed of CFD simulations is a limiting bottleneck, particularly with the increasing desire to perform numerous simulations in large patient populations, estimating boundary conditions (Xiao et al. 2014), uncertainty quantification (Boccadifuoco et al. 2016, 2018; Bozzi et al. 2017), parametric design studies (Gundert et al. 2012), and machine learning. Mirramezani and Shadden recently made an important contribution to the field of image based cardiovascular CFD by developing a distributed lumped parameter (DLP) model of blood flow that can be simulated in minutes while still incorporating complex 3D sources of energy dissipation in blood vessels (Mirramezani and Shadden 2020). While a DLP model of an arterial network is one of the oldest computational tools to study hemodynamics (Westerhof et al. 1969; O’Rourke and Avolio 1980; Van De Vosse and Stergiopulos 2011), by systematically including non-linear sources of energy dissipation this newly developed DLP model has comparable accuracy to 3D CFD simulations while only taking minutes to simulate (Mirramezani and Shadden 2020), A defining feature of cardiovascular fluid dynamics is that the cardiovascular structures interacting with blood flow are highly deformable but due to the increased complexity, most image based CFD models do not include fluid-structure interaction, including the DLP modeling framework of Mirramezani and Shadden.

The goal of this work is to extend the DLP model of blood flow to include fluid-structure interactions (FSI). This was performed with a compliance model that retains the reduced order nature of the original DLP formulation, and does not increase simulation complexity. After describing the new DLP-FSI model, we perform both verification studies and experimental validation studies.

Methods

First, the DLP-FSI mathematical model is described. Second, details of the specific cases used for verification and validation are presented.

Distributed Lumped Parameter Fluid Structure Interaction Model

Our description of the DLP-FSI model follows the derivation of lumped parameter models of vascular flow used by Milišić and Quarteroni (Milišić and Quarteroni 2004). The flow of blood in arteries is described by the 3D incompressible Navier-Stokes equations with an elastic wall law,

$$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\mu \frac{{\partial }^2{u}_i}{\partial x_j^2},$$ (1)

which can readily be integrated over the cross-sections of the vessel to arrive at the 1D Navier-Stokes equations (Barnard et al. 1966; Olufsen 1999)

$$\begin{array}{c}\frac{\partial A(x,t)}{\partial t}+\frac{\partial Q(x,t)}{\partial x}=0\\ \frac{\partial Q(x,t)}{\partial t}+\frac{\partial }{\partial x}\left(\alpha \frac{Q{(x,t)}^2}{A(x,t)}\right)+\frac{A(x,t)}{\rho }\frac{\partial P(x,t)}{\partial x}=-K\frac{Q(x,t)}{A(x,t)}\\ P(x,t)-P_0=\sqrt{\pi }\frac{Eh}{A_0}\left(\sqrt{A(x,t)}-\sqrt{A_0}\right),\end{array}$$ (2)

where $A$ is the cross-section area of the vessel, $x$ is length along the vessel centerline, $t$ is time, $Q$ is flow rate, $\alpha$ is a momentum flux factor, $P$ is pressure, $K$ is a viscous loss factor, $E$ is elastic modulus, $h$ is vessel wall stiffness, $P_0$ is a reference pressure, and $A_0$ is a reference area.

The 1D Navier-Stokes equations can be linearized around a reference state $(A=A_0,P=P_0,Q=0)$,

$$\begin{array}{c}\frac{\partial A(x,t)}{\partial t}+\frac{\partial Q(x,t)}{\partial x}=0\\ \frac{\partial Q(x,t)}{\partial t}+\frac{A_0}{\rho }\frac{\partial P(x,t)}{\partial x}=-\frac{K}{A_0}Q(x,t)\\ P(x,t)=\frac{Eh}{A_0{D}_0}\left(A(x,t)-A_0\right)+P_0,\end{array}$$ (3)

which simplifies solution and maintains the essential features of arterial flow and wave propagation (Sherwin et al. 2003). The linearized-1D Navier-Stokes equations can then integrated along the vessel length so that variables are now only functions of time,

$$\begin{array}{c}\frac{dV(t)}{dt}=Q_1(t)-Q_2(t)\\ \frac{d\widehat{Q}(t)}{dt}+R\widehat{Q}(t)=P_1(t)-P_2(t),\end{array}$$ (4)

$V$ is the vessel volume, $\widehat{Q}$ is the average flow rate in the vessel, $L$ is the impedance,

$$\begin{array}{c}V(t)={\int }_0^{l}A(x,t)dx\\ \widehat{Q}(t)=\frac{1}{l}{\int }_0^{l}Q(x,t)dx\\ L={\int }_0^l\frac{\rho }{A_0}dx,\end{array}$$ (5)

and $R$ is the resistance which will be described later.

A decision must now be made about how to integrate the elastic vessel wall law. In the classical lumped parameter formulation this integration is completed as

$$\widehat{P}=\frac{V-V_0}{C}+P_0,$$ (6)

where $C$ is the vessel compliance, $\widehat{P}$ is the average pressure in the vessel, and $V_0$ is the reference vessel volume

$$\begin{array}{c}C={\int }_0^l\frac{A_0{D}_0}{Eh}dx\\ \widehat{P}(t)=\frac{1}{l}{\int }_0^{l}P(x,t)dx.\end{array}$$ (7)

This is now a system of three equations and seven unknowns $\left(V,Q_1,Q_2,Q,P_1,P_2,P\right)$, of which only two are prescribed as boundary conditions. To mathematically close the classical lumped parameter model, it is assumed that the average pressure and flow rate are equal to the pressure and flow rate at the inlet and outlet respectively.

$$\widehat{P}\cong P_1,\widehat{Q}\cong Q_2.$$ (8)

The classical lumped parameter model can be simplified to

$$\begin{array}{c}C\frac{d{P}_1}{dt}=Q_1-Q_2\\ L\frac{d{Q}_2}{dt}-R{Q}_2=P_1-P_2.\end{array}$$ (9)

This is justified as a physiologically appropriate assumption because the differences in pressure and flow rate between the inlet and outlet of a vessel are typically small. However, in a compliant vessel there can be significant damping of flow between the inlet and outlet, and in certain diseases such as stenotic vessels there can also be large changes in pressure between the vessel inlet and outlet. Additionally, the classical lumped parameter model artificially dissipates forward traveling waves while amplifying backwards waves (Milišić and Quarteroni 2004).

Differing from the classical derivation, we begin by substituting the elastic wall law into the integration of cross-sectional area

$$V={\int }_0^l\frac{A_0{D}_0}{Eh}\left(P-P_0\right)+A_{0}dx.$$ (10)

We then assume that the variation of pressure over the vessel can be described by shape functions, $N_i(x)$, and the pressures at the vessel inlet and outlet

$$P=\left[\begin{array}{c}{N}_1(x)N_2(x)\end{array}\right]\left[\begin{array}{l}{P}_1\hfill \\ P_2\hfill \end{array}\right].$$ (11)

The shape functions always add to equal one, and also equal one at their corresponding node and equal zero at the opposite node. For most vessels we assume the shape functions are linear, but for vessels with sudden expansions of area it would be more appropriate to use piecewise functions (Figure 1).

Figure 1:

A. For a typically shaped vessel the shape functions are linear, while B. for a vessel with sudden expansions the shape functions are piecewise.

Equation 11 can be substituted into Equation 10

$$V={\int }_0^l\frac{A_0{D}_0}{Eh}\left(\left[N_1(x)N_2(x)\right]\left[\begin{array}{l}{P}_1\hfill \\ P_2\hfill \end{array}\right]-P_0\right)dx+V_0,$$ (12)

and as $N_1+N_2=1$ the constant reference pressure $\left(P_0\right)$ can also be multiplied by the shape functions

$$V={\int }_0^l\frac{A_0{D}_0}{Eh}\left[N_1(x)N_2(x)\right]\left(\left[\begin{array}{l}{P}_1\hfill \\ P_2\hfill \end{array}\right]-P_0\right)dx+V_0.$$ (13)

Now, variables without spatial dependence can be brought outside the integral

$$V={\int }_0^l\frac{A_0{D}_0}{Eh}\left[N_1(x)N_2(x)\right]dx\left(\left[\begin{array}{c}{P}_1\\ P_2\end{array}\right]-P_0\right)+V_0,$$ (14)

and we have arrived at the integrated elastic wall law

$$V=\left[\begin{array}{c}{C}_1{C}_2\end{array}\right]\left(\left[\begin{array}{l}{P}_1\hfill \\ P_2\hfill \end{array}\right]-P_0\right)+V_0,$$ (15)

where there are two weighted compliance terms associated with either the vessel inlet or outlet

$$C_i={\int }_0^l\frac{A_0{D}_0}{Eh}{N}_i(x)dx.$$ (16)

The vector multiplication in equation 15 can be carried out to calculate the changes in vessel volume associated with the corresponding inlet/outlet pressure as

$$V=V_1+V_2+V_0,$$ (17)

where

$$V_i=C_i\left(P_i-P_0\right).$$ (18)

The lumped parameter model of blood flow is now

$$\begin{array}{c}\frac{d{V}_1}{dt}+\frac{d{V}_2}{dt}=Q_1-Q_2\\ L\frac{d\widehat{Q}}{dt}+R\widehat{Q}=P_1-P_2\\ V_i=C_i\left(P_i-P_0\right),\end{array}$$ (19)

which is a mathematically under-determined system of four equations with seven unknowns (two of which are prescribed as boundary conditions). To mathematically close the system, we assume that vessel volume changes due to pressure at the inlet are related to conservation of mass between inflow and the average vessel flowrate.

$$\frac{d{V}_1}{dt}=Q_1-\widehat{Q}.$$ (20)

After some algebraic rearrangement, our lumped parameter system of blood flow in a single vessel is

$$\begin{array}{c}\frac{d{V}_1}{dt}=Q_1-\widehat{Q}\\ \frac{d{V}_2}{dt}=\widehat{Q}-Q_2\\ L\frac{d\widehat{Q}}{dt}+R\widehat{Q}=P_1-P_2\\ V_i=C_i\left(P_i-P_0\right).\end{array}$$ (21)

We believe improvements of this lumped parameter model over the classical lumped parameter model are three-fold: 1. The pressure drop over a vessel is now a function the average flow in that vessel, not just the flow at the outlet, 2. The changes in vessel volume are now a function of pressure at the vessel inlet and outlet, not just the vessel inlet, and 3. This lumped parameter model does not artificially dissipate (amplify) forward (backward) traveling waves (Appendix). Point 2 may be particularly important when there will be large pressure gradients such as in a stenotic vessel. It should also be noted that the classical lumped parameter model can be derived by this method if the first shape function equals one and the second shape function equals zero.

Lastly, the resistance term ($R$) in equation 21 is calculated using the recently developed method of Mirramezani and Shadden (Mirramezani and Shadden 2020).

$$R=\frac{8\mu }{\pi }{\int }_0^L\max \{\gamma ,\zeta \}\frac{1}{r^4}dx+\sum _{i=1}^n\frac{\rho K_t}{2A_{0,i}^2}{\left(\frac{A_{0,i}}{A_{s,i}}-1\right)}^2\left|Q\right|$$ (24)

Where the first term is the linear resistance and the second term is the non-linear Bernoulli type resistance. The $\gamma$ and $\zeta$ modify the Poiseuille resistance depending on whether viscous losses due to curvature or unsteadiness are greater. $\gamma$ is defined as:

$$\gamma =0.1033\sqrt{K}{\left(\sqrt{1+\frac{1.729}{K}}-\frac{1.315}{sqrt(K)}\right)}^{-\frac{1}{3}}$$ (25)

Where $K$ is the Dean number calculated from the Reynolds number $(Re)$, radius $(r)$, and vessel curvature $(a)$.

$$K=Re\sqrt{\frac{Re}{a}}$$ (26)

$\zeta$ is calculated from Womersly flow (Zamir 2000)

$$\zeta =\left|1+\frac{2}{\text{A}}\frac{J_1(\text{Λ})}{J_0(\text{Λ})}\right|$$ (27)

Where $\text{Λ}$ is calculated from Womersly’s number,

$$\text{Λ}=r\sqrt{\frac{\rho }{\mu }}\frac{(i-1)}{\sqrt{2}}$$ (28)

and $J$ is a Bessel function of the first kind.

In the non-linear resistance term to account for sudden expansions, $A_s$ is the local area minimum, $A_0$ is the average of the maximum areas before and after the stenosis, and $K_t$ is an empirically determined correction factor, 1.52. In this work we neglect the non-linear branching model included in the original DLP framework as it was found to have negligible contribution (Mirramezani and Shadden 2020).

We now shift focus to extending this model of flow for a single vessel to a network of vessels. We begin with a junction of three vessels which is the most common case in an arterial network. The conservation of mass equations at the junction node for the parent vessel (vessel 1) and daughter vessels (vessels 2 and 3) can be summed

$$\frac{d{V}_1}{dt}+\frac{d{V}_2}{dt}+\frac{d{V}_3}{dt}={\widehat{Q}}_1-Q_1+Q_2-{\widehat{Q}}_2+Q_3-{\widehat{Q}}_3.$$ (29)

Mass must be conserved between the compliance terms (Figure 2B) therefore

$$Q_1-Q_2-Q_3=0,$$ (30)

and the conservation of mass equation at the junction can be simplified to now only include the average flowrates in the parent and daughter vessels

$$\frac{dV}{dt}={\widehat{Q}}_1-{\widehat{Q}}_2-{\widehat{Q}}_3,$$ (31)

where

$$V=V_1+V_2+V_3.$$ (32)

The change in volume associated with changes in pressure at the junction node can be written

$$V=C\left(P-P_0\right)$$ (33)

where

$$C=C_1+C_2+C_3.$$ (34)

Note that this analysis is equivalent to realizing that compliance terms add in parallel.

Figure 2:

A. a junction of multiple blood vessels can be reduced to B. it’s circuit equivalent by distributing lumped parameters for each vessel. The compliance terms from each vessel associated with the junction node can then be added in parallel to calculate a single compliance term for the junction node.

The general form for a junction of $n$ vessels is achieved by enforcing mass conversation at the vessel junction

$$\text{Σ}{\widehat{Q}}_{\mathit{\text{in}}}-\text{Σ}{\widehat{Q}}_{\mathit{\text{out}}}=\frac{\partial V}{\partial t},$$ (35)

where $V$ is the volume associated with changes in pressure at that junction. The compliance associated with a junction node is

$$\text{C}=\sum _{i=1}^n{\int }_0^L\frac{A_0{D}_0}{Eh}{N}_i(x)dx$$ (36)

where $n$ is the number of vessels connected to that junctions.

Typical boundary conditions in cardiovascular simulations such as flowrate, pressure, or impedance can be applied to this model of a vascular network. The implementation of flowrate and impedance boundary conditions is straightforward from Equation 21 and a pressure boundary condition is equivalent to prescribing the volume of an inlet or outlet node compliance term. In the case of a constant pressure boundary condition the volume of the compliance term would also be held constant and conservation of mass requires that the average flowrate in associated vessel is equal to the flowrate at the outlet.

A schematic representation of the DLP-FSI framework is shown in Figure 3. Centerlines for the lumped parameter model were created with Mimics (Materialize, Leuven). The system of ordinary differential equations was discretized to a system of non-linear algebraic equations using a backwards Euler time-stepping scheme and solved in MATLAB using the Levenberg-Marquardt algorithm (Moré 1978). Pseudo-code describing the implementation is provided in the appendix.

Figure 3:

A. 3D segmentation of the thoracic aorta anatomy and B. a distributed lumped parameter representation of the thoracic aorta anatomy where every vascular segment has an associated inertia and non-linear resistance, and every vascular junction and boundary node has an associated compliance. AAo – Ascending aorta, Dao – Descending aorta, BA – Brachiocephalic artery, LCCA – left common carotid artery, LSA – left subclavian artery

Verification Case #1

The first verification case considered was a straight, circular tube with a length of 30cm. The tube diameter was 4 cm and material properties (wall thickness and elastic modulus) were selected to achieve a pulse wave velocity (PWV) of 300 cm/s. The inflow boundary condition was a square wave with a 0.1 second duration and the outlet boundary condition was zero pressure.

$$\begin{gathered} Q_{in}=Q_0\text{Π}\left(\frac{t-\frac{1}{2}}{\tau }\right) \\ {P}_{out}=0 \end{gathered}$$ (37)

The zero pressure outlet results in an open ended reflection site. When the wave reflection returns to the inlet there is no flow which is equivalent to a closed ended reflection site. As resistance in the tube was negligible, simulated results from DLP-FSI were compared to both analytical solutions of the wave equations without viscous drag (Zamir 2000) and also to the analytical solution for an inductor-capacitor (LC) circuit. The wave equation is a close representation of the physics of flow in an elastic tube, only neglecting non-linearity and viscous forces. The outlet flowrate and inlet pressure are

$$\begin{gathered} Q_{out,Wave}=\sum _{n=0}^{\infty }{Q}_0\text{Π}\left(\frac{t-\frac{1}{2}-\left(1+2n\frac{l}{c}\right)}{\tau }\right) \\ {P}_{in,Wave}=\sum _{n=0}^{\infty }{\text{Z}}_0{\text{Q}}_0\text{Π}\left(\frac{t-\frac{1}{2}-\left(2n\frac{l}{c}\right)}{\tau }\right) \end{gathered}$$ (38)

Where ${\text{Q}}_0$ is 100mL/sec, $\text{Π}$ is a square wave, $l$ is the tube length, $c$ is the pulse wave velocity, $\tau$ is 0.1 sec, and $Z_0$ is the tube’s characteristic impedance.

The LC circuit represents the analytical solution of this system after making the lumped parameter assumptions,

$$\begin{gathered} Q_{out,LC}=Q_0\left[1-\cos \left(\frac{c}{l}t\right)-H(t-\tau )\cos \left(\frac{c}{l}(t-\tau )\right)\right] \\ {P}_{in\text{,}LC}={\text{Z}}_0{\text{Q}}_0\left[\sin \left(\frac{c}{l}t\right)-H(t-\tau )\sin \left(\frac{c}{l}(t-\tau )\right)\right] \end{gathered}$$ (39)

where parameters are the same as in equation 30 and $H$ is a Heaviside step function.

Verification Case #2

The second verification case was of latex tube #1 described in the next section. The inflow waveform was generated from a series of complex exponentials with a period of 1 second. Higher frequency components had lower magnitudes. The inflow waveform is explicitly given by

$$Q_{in}=Q_0\left[1+\sum _{f=1}^5\frac{1}{\text{f}}{\text{e}}^{\text{j}2\text{πf}(\text{t}-\text{τ})}\right]$$ (40)

where $Q_{in}$ is the inlet flow waveform, $Q_0$ is 50 mL/sec, and $\tau$ is a time delay of 0.2 sec.

The outlet boundary condition was a three element Windkessel model with the proximal resistance $\left(R_p\right)$ matching the tube impedance, and the distal resistance $\left(R_d\right)$ and compliance $\text{(}C\text{)}$ set to give normal aortic pressures (120/80 mmHg). The complex impedance $\left(Z_{out}\right)$ of the outlet is

$$Z_{out}(f)=\frac{R_d+R_p+j2\pi f{R}_p{R}_{d}C}{1+j2\pi f{R}_{d}C}$$ (41)

The analytical solution was obtained from the wave equation with the reflection coefficient calculated from the tube impedance and the Windkessel impedance. This analytical solution neglects viscosity, non-linearity, and only included one reflected component. The reflection coefficient $(R)$ is calculated from the characteristic impedance of the tube and the distal impedance.

$$R(f)=\frac{{\text{Z}}_{\text{out}}-Z_0}{Z_{out}+Z_0}$$ (42)

The refection coefficient is then used to calculate a coefficient $(A)$ that ensures the inlet boundary condition is enforced

$$A(f)=\frac{{\text{Q}}_{\text{in}}(f)}{e^{j2\pi ft}-R(f)e^{j2\pi f\left(t-\frac{2l}{c}\right)}}$$ (43)

Lastly, the coefficients $R$ and $A$ are used to calculate the outlet flow rate, the inlet pressure, and the outlet pressure.

$$\begin{gathered} Q_{out}=\sum _{f=0}^5\text{A}(\text{f})(1-\text{R}(\text{f})){\text{e}}^{\text{j}2\text{πf}\left(\text{t}-\frac{\text{l}}{\text{c}}\right)} \\ {P}_{in}=Z_0\sum _{f=0}^5\text{A}(\text{f})\left[e^{j2\pi ft}+R(f)e^{j2\pi f\left(t-2\frac{l}{c}\right)}\right] \\ {P}_{out}=Z_0\sum _{f=0}^5\text{A}(\text{f})(1+\text{R}(\text{f})){\text{e}}^{\text{j}2\text{πf}\left(\text{t}-\frac{\text{l}}{\text{c}}\right)} \end{gathered}$$ (44)

Elastic Tubes Experimental Validation

The first set of validation cases considered were experimental measurements of pulsatile flow in elastic tubes in a cardiovascular flow loop (Ruesink et al. 2018). Experimental measurements were made with ultrasonic flow meters, pressure taps, and tube diameters were measured with high-speed cameras. Further information on the experimental measurements can be found in Ruesink et al., 2018 (Ruesink et al. 2018). The dimensions and material properties four tubes (two latex, tygon, and silicone) are described in Table 1. The pump generated inflow had a mean flow rate of 4 L/min and a frequency of 1Hz. The cardiovascular flow loop components were tuned to achieve a mean pressure of 70mmHg and a pulse pressure of 80mmHg. DLP-FSI simulations of these experiments used a cosine inflow wave form with the same mean flow and frequency and a two-element Windkessel outlet with parameters selected to match the experimental pressures.

Table 1:

Properties of elastic tubes from validation experiments

	Inner Diameter (mm)	Thickness (mm)	Elastic Modulus (MPa)	Theoretical PWV (m/s)
Latex #1	15.9	0.3	1.27	1.54
Latex #2	12.7	0.3	1.67	1.99
Tygon	19.1	3.2	2.07	5.88
Silicone	19.1	3.2	3.35	7.59

Deformable Stenosis Validation

The second validation case was a flexible stenosis model (Figure 6) that was experimentally characterized by Kung et al (Kung et al. 2011). Briefly, a silicone tube was manufactured with a rigid stenosis and rigid connectors at the inlet and outlet. The unloaded tube diameter was 2cm, the stenosis diameter was 0.88cm, and the wall thickness was 0.8mm. We estimated our linearized elastic modulus from pressure-diameter data of the flexible stenosis model (Kung et al. 2011). The inflow boundary condition was the experimentally measured flow waveform and the outflow boundary condition was a four-element Windkessel model. Flowrates were measured with 2D phase contrast MRI and pressures were measured with MR compatible pressure transducers. Lastly, we obtained flow and pressure waveforms at the model inlet and outlet from Figure 6 of Kung et al., (Kung et al. 2011) using the MATLAB function GRABIT. In addition to experimental data, we also compare our DLP-FSI results to 3D FSI simulations that were performed by Kung et al., using the coupled momentum method (CMM) for 3D FSI (Figueroa et al. 2006).

Figure 6:

A. Schematic diagram of the flexible stenosis model with the model’s unloaded dimensions. B. Pressure waveforms at the inlet and outlet show good agreement between experiments and simulations. C. Similarly, flow waveforms at the inlet and outlet show also good agreement between experiments and simulations.

In Vivo Aorta Validation

The third validation case was in vivo flow in the thoracic aorta of a healthy volunteer. The 3D anatomy was segmented from contrast enhanced magnetic resonance angiography. Flow waveforms were measured from 4D Flow MRI in the ascending aorta, descending aorta, and three aortic arch branch arteries. The ascending aorta flow waveform was used as the simulation boundary condition. Subject specific pressure measurements were not available so instead outlet three-element Windkessel parameters were tuned to match normal systemic pressures (120/80 mmHg). For comparison with simulated results, the MRI flow waveforms of the descending aorta and aortic branches were scaled to ensure mass conservation with the ascending aorta flow. We used constant material properties throughout the aorta, which were set to give a pulse wave velocity of 300 cm/s in the ascending aorta. Aorta cross-sectional areas along the aorta centerlines were reconstructed using the relationship established pressure-area relationship and interpolating pressure between nodes using shape functions

$$A=\frac{A_0{D}_0}{Eh}\left[\begin{array}{c}{N}_1(x)N_2(x)\end{array}\right]\left(\left[\begin{array}{c}{P}_1\\ P_2\end{array}\right]-P_0\right)+A_0.$$ (45)

A rigid walled DLP simulation was also performed to compare FSI results with.

Results

Verification Case #1

The first verification case was flow in a straight tube with a square inflow pulse and a zero pressure outlet. Results are shown in Figure 4. Results show that the analytical solution from the wave equations maintains sharp or high frequency components of the inflow waveform. In contrast, while both the LC circuit analytical solution and the DLP-FSI simulations correctly predicted the timing and direction of wave reflections, they dampened high frequency components of the inflow waveform. Excellent agreement was seen between the LC circuit analytical solution and DLP-FSI simulations however. Excellent agreement was expected because without curvature the DLP-FSI model is very similar to the LC circuit. This excellent agreement also indicates that viscous forces were negligible and that neglecting viscous forces was acceptable in the analytical solutions.

Figure 4:

A. Straight tube geometry for verification cases and B. LC circuit equivalent for verification case #1. Case #1: Verification results show that the DLP-FSI software is correctly solving the equations for an LC circuit analogy of hemodynamics. Differences are noted between the wave equation solution and DLP-FSI results for a square flow waveform with zero pressure outlet. Case #2: For more realistic cardiovascular boundary conditions excellent agreement was observed between the wave equation solution and DLP-FSI results.

Verification Case #2

The second verification case was flow in a straight tube with an inflow waveform from an exponential series and a three-element Windkessel outlet. This verification case is more representative of boundary conditions typically used in cardiovascular flow simulations. Results are shown in Figure 4. Excellent agreement was seen between DLP-FSI simulations and the analytical solution from the wave equations. Simulations correctly predicted the time lag between inlet and outlet flow and pressure waveforms. Simulations also correctly predicted that flow pulsatility decreases from the inlet to outlet while pressure pulsatility increases. Similar to verification case #1, this excellent agreement again indicates that neglecting viscous forces was acceptable in the analytical solution.

Elastic Tubes Experimental Validation

The first set of cases for experimental validation was pulsatile flow in four different elastic tubes. Results are shown in Figure 5. Flow waveforms show that with increasing tube stiffness, the inlet-to-outlet time lag decreases and the flow waveform distortion decreases (Figure 5A). Diameter changes show reasonable agreement with experimental measurements (Figure 5B). With increasing tube stiffness, the diameter change decreases. However, for stiffer tubes the maximum diameter occurred earlier than simulations predicted. This could be due to differences between the flow waveforms used experimentally and the flow waveforms used in simulation or this could highlight a limitation of the DLP-FSI model as the experimentally observed phase shift was not predicted with simulation.

Figure 5:

Experimental validation of DLP-FSI with elastic tubes. A. Flow waveforms show that with increasing tube stiffness, pulse wave velocity increases and there is less distortion of the waveform over the tube length. B. Changes in diameter showed good agreement between simulations and experiments. D-E show changes diameter in greater detail for each tube.

Deformable Stenosis Validation

The second validation case was based on prior experiments of a deformable stenosis model (Figure 6A). DLP-FSI simulations overall had good agreement with experimental measurements as well as 3D CMM-FSI simulations (Figure 6B and 6C). The time delay between inlet and outlet waveforms as well as the degree of damping between the inlet and outlet flowrates were well predicted. DLP-FSI simulations overpredict the outlet pulse pressure by approximately 10%.

In Vivo Aorta Validation

The final validation case was simulation of a healthy aorta (Figure 7A), with comparison to measurements from MRI as well as a rigid walled simulation. Results are shown in Figure 4. Descending aorta flow simulated with rigid walls was in phase with the ascending aorta flow (Figure 7B). In contrast, descending aorta flow measured with MRI and simulated with FSI show a time-lag compared to the ascending aorta flow due to the elastic aortic walls. Additionally, descending aorta flow measured with MRI and simulated with FSI both show dampened systolic flow and increased diastolic flow compared to the rigid wall simulation. Simulated aortic pressures from rigid wall and FSI simulations had similar mean pressures, but rigid wall simulations had significantly greater pulse pressures (Figure 7C). As expected, FSI simulations predicted greater relative area change (RAC) in the ascending aorta than in the descending aorta due to the larger vessel having greater compliance (Figure 7D and 7E).

Figure 7:

Simulation of A. a healthy thoracic aorta. B. FSI simulations show improved agreement with descending aorta flow measured from MRI compared to rigid wall simulations. C. FSI simulations had smaller pulse pressures than rigid wall simulations. D. FSI simulations predicted larger area changes over a heartbeat in the ascending aorta than in the descending aorta. E. Relative area change (RAC) varied from approximately 60% in the ascending aorta to 35% in the descending aorta. Distance from inlet refers to distance from the ascending aorta inflow plane along the aorta centerline.

Discussion

Distributed lumped parameter (DLP) models have long been a computational tool to study hemodynamics (Westerhof et al. 1969; O’Rourke and Avolio 1980; Van De Vosse and Stergiopulos 2011) and variations of reduced order models have included either arterial compliance or non-linear sources of energy dissipation, In this manuscript we have shown that a recently developed DLP model that includes the complex sources of energy dissipation occurring in 3D cardiovascular flows can be extended to include vessel distensibility and fluid structure interactions (FSI). The DLP-FSI modeling framework described can calculate key biophysical phenomena such as vessel deformability and pulse wave propagation that could not be calculated with the original DLP model that assumed rigid walls. DLP-FSI simulations showed good agreement with analytical solutions of flow in an elastic tube, experimental measurements of flow in elastic tubes, and in vivo measurements of flow in the aorta. This new development allows for significantly improved computational efficiency of FSI simulations compared to FSI approaches that solve the full 3D conservation of mass and momentum equations (Figueroa et al. 2006; Borazjani et al. 2008; Liu and Marsden 2018). The DLP-FSI formulation also includes the complex sources of energy dissipation occurring in 3D cardiovascular flows that other simplified models neglect (Xiao et al. 2014). Additional novel aspects of this work include demonstrating how reduced order (lumped) arterial compliance models can be derived from assuming a pressure profile over the vessel and that unlike the classical lumped parameter model (Milišić and Quarteroni 2004), this approach symmetrically treats forwards and backwards traveling waves (Appendix).

Inclusion of FSI in cardiovascular simulations is critical for the most accurate simulations possible. As confirmed by our FSI simulations, distensible vessel walls dampen flow waveforms and introduce the physics of wave propagation that is not present in rigid wall simulations. Previous studies have also shown the importance of including distensible vessel walls for realistic cardiovascular simulations (Boccadifuoco et al.). FSI simulations also enable the calculation of important clinical metrics of vessel distensiblity such as relative area change that are linked to mortality in cardiovascular disease (Gan et al. 2007). DLP-FSI modeling could also be useful in other physiological systems (cerebrospinal fluid, respiratory, urinary tract, gastrointestinal, etc.) where key biological processes are also mediated by flow in elastic tubes.

Including the effects of FSI in an efficient, reduced order computational model opens opportunities for growth and remodeling simulation (Ambrosi et al. 2011; Valentín et al. 2013; Bellini et al. 2014; Zhang et al. 2016). Hemodynamic forces imparted by blood flow on vessel walls are key drivers of vascular growth and remodeling. However, the high cost of full 3D FSI simulations makes them impractical for growth and remodeling scenarios where multiple simulations need to be performed as hemodynamics change during the simulated time course. Using DLP-FSI simulations could remove this bottle neck for calculating the hemodynamic forces and blood flow distributions. We do note that DLP models cannot predict local distributions of wall shear stress which limits their utility for certain growth and remodeling applications (Morbiducci et al. 2020) however many growth and remodeling models are based on an axisymmetric vessel and do not require local stress distributions (Valentín et al. 2009).

Another important use of reduced order models is to lower the need to perform full 3D simulations that are more expensive. Tasks such as physiologically realistic boundary condition estimation, uncertainty quantification, parametric design, and generating data for machine learning all require numerous evaluations of the computational model. For boundary condition estimation, our results from the thoracic aorta (Figure 4) show that boundary conditions fit to a reduced order model that assumes rigid walls will not be accurate for use in a full 3D FSI simulation. In addition to estimation outlet boundary conditions more accurately, DLP-FSI simulations could also be used to estimate patient specific vessel wall stiffness, either from the change in vessel area over a heartbeat (Zambrano et al. 2018) or pulse wave velocity (Pewowaruk and Roldán-Alzate 2019). Our framework does permit for varying mechanical properties along the length of the vessel.

The greatest errors for the DLP-FSI simulations were comparing to analytical solutions for a square inflow pulse. The lumped parameter model dampened the high frequency components of the inflow wave form. This could be corrected by developing 1D FSI models that include more accurate sources of energy loss from the DLP model as proposed by Mirramezani and Shadden (Mirramezani and Shadden 2020). However, given that cardiovascular flow do not generally include sharp features this would not substantially improve accuracy but would greatly increase the simulation complexity. Areas where we would expect the DLP-FSI model to perform poorly would be in cases where the wall mechanical properties are heterogenous such as plaques and calcifications and modeling of the venous circulation where venous collapse and valves would need to be taken into consideration (Chow and Mak 2006). Additional areas for improvements to the DLP-FSI model could be using non-linear and viscoelastic material models to derive the pressure-area relationship (Reymond et al. 2009; Alastruey 2011), updating the vessel areas as a function of pressure in the conservation of momentum, or including a non-linear inertia term as has previously been derived from boundary layer theory. The incremental value of the additions is likely to be context specific. For instance, Battista et al., found the effects of viscoelasticity in the ovine systemic arterial network to be minimal (Battista et al. 2016).

The greatest limitations of this work are related to the broader challenge of verification and validation studies in hemodynamic simulations. The popular definitions of V&V attributed to Roache are: verification is “solving the equations right” and validation is “solving the right equations”, but both of these standard practices can be problematic for hemodynamic simulations for a variety of reasons (Steinman and Migliavacca 2018): 1. there are no theoretical solutions for the 3D Navier-Stokes equations in realistic anatomies much less fluid-structure interactions or biological mechanisms, 2. benchtop validation experiments may not include relevant physics such as vessel wall compliance and certainly do not include physiological and biological feedback mechanisms, and 3. in vivo measurements of the cardiovascular system have non-negligible errors. In light of these challenges, we have attempted to select verification and validation cases that balanced these different types of limitations. Software comparison studies also warrant mention, particularly in light of the comprehensive software comparison studies that have been performed by others (Xiao et al. 2014; Mirramezani and Shadden 2020). While software comparison studies can provide valuable information regarding the limitations and assumptions made by different modeling software, software comparison studies cannot replace verification and validation which is why we chose not to focus this work on comparison with a 3D FSI software.

Conclusion

This manuscript extended a recently developed distributed lumped parameter model of blood flow to include vessel distensibility and fluid structure interactions. The DLP-FSI modeling framework showed good agreement with analytical solutions of flow in an elastic tube, experimental measurements of flow in elastic tubes, and in vivo measurements of flow in the aorta. This new development allows for significantly improved computational efficiency of FSI simulations compared to FSI approaches that solve the full 3D conservation of mass and momentum equations while also including the complex sources of energy dissipation occurring in 3D cardiovascular flows that other simplified models neglect. Future work should investigate the DLP-FSI model’s capacity to handle more complicated geometries and inhomogeneous material properties.

Appendix

Symmetry of Forward vs. Backward Traveling Waves

By considering the “convergence” of 0D models to 1D models it has previously been shown that the classical lumped parameter model dissipates forward traveling waves while amplifying backwards traveling waves (Milišić and Quarteroni 2004). The analysis begins by assuming that the 1D linear system (Equation 3) is replaced by a series of 0D models with length $\text{Δ}x$. For the classical lumped parameter model this series of 0D models can be written as

$$\begin{array}{c}{C}^{\prime }\frac{d{P}_i}{dt}=-\frac{1}{\text{Δ}x}\left(Q_{i+1}-Q_i\right)\\ L^{\prime }\frac{d{Q}_i}{dt}=-R^{\prime }{Q}_i-\frac{1}{\text{Δ}x}\left(P_i-P_{i-1}\right),\end{array}$$ (A1)

where C’, L’, and R’ are the vessel’s compliance, impedance, and resistance per unit length. This can be regarded as a first order finite difference approximation of 1D linear system.

Utilizing the “water hammer” equations (Milišić and Quarteroni 2004; Parker 2009)

$$\left[\begin{array}{l}W\hfill \\ Z\hfill \end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1& \sqrt{\frac{L^{\prime }}{C^{\prime }}}\\ 1& -\sqrt{\frac{L^{\prime }}{C^{\prime }}}\end{array}\right]\left[\begin{array}{l}P\hfill \\ Q\hfill \end{array}\right],$$ (A2)

the 1D linear system can be rewritten as

$$\begin{array}{c}\frac{\partial w}{\partial t}=-\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\partial w}{\partial x}-\frac{R^{\prime }}{2L^{\prime }}(w-z)\\ \frac{\partial z}{\partial t}=\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\partial w}{\partial x}+\frac{R^{\prime }}{2L^{\prime }}(w-z),\end{array}$$ (A3)

where w and z are called the characteristic variables and physically represent the pressure of the forward and backwards traveling waves.

The finite difference scheme (Equation A1) can similarly be rewritten for characteristic variables as

$$\begin{array}{c}\frac{\partial w_i}{\partial t}=-\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(w_{i+1}-w_{i-1}\right)}{2\text{Δ}x}-\frac{R^{\prime }}{2L^{\prime }}\left(w_i-z_i\right)+\frac{\text{Δ}x}{2\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(z_{i+1}-2z_i+z_{i-1}\right)}{\text{Δ}{x}^2}\\ \frac{\partial z_i}{\partial t}=\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(z_{i+1}-z_{i-1}\right)}{2\text{Δ}x}+\frac{R^{\prime }}{2L^{\prime }}\left(w_i-z_i\right)-\frac{\text{Δ}x}{2\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(w_{i+1}-2w_i+w_{i-1}\right)}{\text{Δ}{x}^2},\end{array}$$ (A4)

which corresponds to a centered finite difference discretization of the characteristic system (Equation A3) with the addition of two terms that scale with $\text{Δ}x$, a dissipative term in the first equation and an anti-dissipative term in the second equation. This asymmetry indicates that the classical lumped parameter circuit artificially dampens forward travelling waves while amplifying backwards travelling waves.

In our lumped parameter scheme the pressure drop between nodes is due to the average flow between those two nodes so a series of 0D models can be written as

$$\begin{array}{c}\frac{C^{\prime }}{2}\frac{d{P}_i}{dt}=-\frac{1}{\text{Δ}x}\left(Q_{i+\frac{1}{2}}-Q_i\right)\\ \frac{C^{\prime }}{2}\frac{d{P}_{i+1}}{dt}=-\frac{1}{\text{Δ}x}\left(Q_{i+1}-Q_{i+\frac{1}{2}}\right)\\ L^{\prime }\frac{d{Q}_{i+\frac{1}{2}}}{dt}=-R^{\prime }{Q}_{i+\frac{1}{2}}-\frac{1}{\text{Δ}x}\left(P_{i+1}-P_i\right),\end{array}$$ (A5)

which corresponds to a centered finite difference approximation of the 1D linear system where the average flowrate between nodes is defined at the segment midpoint. Recognizing that the lumped parameter model was derived by assuming linear interpolation of pressure between nodes $(P_{i+\frac{1}{2}}=\frac{1}{2}\left(P_i+P_{i+1}\right)$, the system of equations A5 can also be rewritten in terms of the characteristic variables (w and z)

$$\begin{array}{c}\frac{\partial w_{i+\frac{1}{2}}}{\partial t}=-\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(w_{i+1}-w_i\right)}{\text{Δ}x}-\frac{R^{\prime }}{2L^{\prime }}\left(w_{i+\frac{1}{2}}-z_{i+\frac{1}{2}}\right)\\ \frac{\partial z_{i+\frac{1}{2}}}{\partial t}=\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}\frac{\left(z_{i+1}-z_i\right)}{\text{Δ}x}+\frac{R^{\prime }}{2L^{\prime }}\left(w_{i+\frac{1}{2}}-z_{i+\frac{1}{2}}\right),\end{array}$$ (A6)

which is a centered finite difference approximation of the characteristic equation (Equation A3). Importantly, the system of equations A6 does not contain the dissipative and antidissipative terms that resulted from the classical lumped parameter model (Equation A4). This result indicates that our lumped parameter model does not asymmetrically treat forward and backward traveling waves. The asymmetry of the classical lumped parameter model is visually appreciated by looking at the numerical stencils of the finite difference approximations generated by the classical lumped parameter model (Equation A1) and our symmetric lumped parameter model (Equation A5).

Figure A1:

A. The numerical stencil for the finite difference approximation derived from the classical lumped parameter model and B. the numerical stencil for the finite difference approximation derived from the symmetric lumped parameter model.

Rigid-walled DLP Model

The rigid wall DLP model used in the in vivo validation case was implemented as by enforcing mass conservation at vascular junctions

$$\text{Σ}{\widehat{Q}}_{in}-\text{Σ}{\widehat{Q}}_{out}=0,$$ (A7)

and conserving momentum over each vascular junction

$$L\frac{d\widehat{Q}}{dt}+R\widehat{Q}=P_1-P_2$$ (A8)

where the impedance term $L$ is calculated with equation 5 and the non-linear resistance term $R$ is calculated with equation 24. A schematic representation of the rigid DLP model would be similar to Figure 3B, but without only the resistors and inductors.