^{1,2}

^{2,3}

^{2,3}

^{3}

^{1}

^{2}

^{3}

A computational method of reduced complexity is developed for simulating vascular hemodynamics by combination of one-dimensional (1D) wave propagation models for the blood vessels with zero-dimensional (0D) lumped models for the microcirculation. Despite the reduced dimension, current algorithms used to solve the model equations and simulate pressure and flow are rather complex, thereby limiting acceptance in the medical field. This complexity mainly arises from the methods used to combine the 1D and the 0D model equations. In this paper a numerical method is presented that no longer requires additional coupling methods and enables random combinations of 1D and 0D models using pressure as only state variable. The method is applied to a vascular tree consisting of 60 major arteries in the body and the head. Simulated results are realistic. The numerical method is stable and shows good convergence.

Blood flow involves pressure and flow waves that propagate through the vascular system. As a compromise between computational demand and physical detail, one-dimensional (1D) network models have been used to study pressure and flow waveforms under normal and pathological conditions [

These 1D network models consist of elements that locally describe the relation between pressure and flow. Relations between pressure, area, and flow in the blood vessels are given by the 1D wave propagation equations, that is, 1D partial differential equations of mass and momentum which are derived by integrating the Navier-Stokes equation over the cross-sectional area of the blood vessel [

To solve the system of equations derived from the 0D and 1D models and simulate propagation of pressure and flow waves through the vascular system, various, rather complex, algorithms exist. Regarding the 1D wave propagation equations, all numerical methods start from the same relation between pressure, area, and flow or cross-sectional mean velocity. First differences between the numerical methods arise with the state variables chosen to remain. With area and pressure related via a constitutive relation of the vessel wall, the result is either a pressure-flow [

The contribution of this study is to develop a simplified numerical method in which pressure is the only state variable. In this approach, the 1D wave propagation and 0D lumped model equations are cast into the same form. As such, 1D and 0D models are combined without the need to specify additional coupling equations. This allows for flexible model building from 0D and 1D elements for simulation of pressure and flow in a vascular network. For illustration, the numerical method proposed is applied to simulate pressure and flow waveforms in a vascular tree composed of 60 major arteries in the body and the head.

In large arteries, blood pressure ^{3}·s^{−1}), wall shear stress ^{2}) are related by 1D equations of mass and momentum. When neglecting leakage through the vessel wall as well as gravitational forces, the balance of mass and momentum is given by (derivations can be found in, e.g., Hughes and Lubliner [^{2}·Pa^{−1}) denotes the vessel area-compliance, and ^{−3}) denotes the blood density.

Wall shear stress

The mass and momentum equations are completed with expressions for area (

The contribution of the peripheral vasculature at each arterial terminus is lumped in a three-element windkessel model [

(a) Network model of 60 major arteries used to test the numerical method proposed. Adopted from Mulder et al. [

Proposed discretization for 1D wave propagation and 0D Windkessel elements. Notice the reversal of the flow in the second node with respect to the conventional discretization (indicated by superscript c).

To determine pressure and flow in the vascular network, the 1D pressure-flow relations for blood vessels in (

For each of the subelements of the windkessels, two nodal point pressures and two nodal point flows are defined (Figure

Before, Huberts et al. [

First, the wave propagation equations are linearized using estimates of area compliance, cross-sectional area, wall shear stress, and convective acceleration as obtained from a previous time step (indicated by symbol

By defining both flows as being directed inwards, continuity of pressure and flow at the 0D-1D interfaces and the 1D-1D interfaces (e.g., bifurcations) is automatically satisfied in the process of assembling the element equations into the large system of equations (Appendix

Note that any nonlinearity with respect to flow makes stiffness matrix

After pressures and flows are computed, the simulation proceeds to the next time step. The process is repeated until cardiac cycle time

Schematic overview of the numerical method used to compute pressures and flows. During a “time loop,” a cardiac cycle with cycle time

The numerical method is applied to simulate hemodynamics in an arterial tree composed of the 60 major arteries in the body and the head (Figure ^{−3} and

Aortic inflow is considered as the only external flow and is prescribed according to the waveform as depicted in Figure

To assess the convergence behavior of the proposed numerical method with respect to temporal and spatial discretization, a series of simulations is done with combinations of element sizes (

For each of the simulations, the hemodynamic convergence norm

For most simulations pressure and flow have converged after 12 cardiac cycles (Figure

Convergence behavior for the proposed method for different time step and element sizes. Results obtained with the same time step, but different element sizes are indistinguishable.

Figure

Computed pressure and flow waveforms in the aorta, the arm, and the brain. Results are shown as obtained using the simulation with element size

As shown in Figure

(a) Influence of element and time step size on

In this study, a simplified numerical method was developed for time-domain simulation of blood pressure and flow waveforms in the vascular system that couples nonlinear one-dimensional (1D) wave propagation models for the blood vessels to zero-dimensional (0D) lumped (windkessel) models for the periphery using pressure as degree of freedom.

To show performance of the method in a physiologic setting, the method was applied to simulate hemodynamics in a vascular network containing the 60 major arteries in the body and the brain. The specific choice of vessel behavior, velocity profile, windkessel parameters, and essential boundary conditions was beyond the scope of this study. The pressure-area relations of the bloodvessels were assumed non-linear and convective acceleration was included to assess behavior of the method in solving the model equations in its most non-linear form.

The pressure and flow waveforms that were obtained with the method (Figure

Typically 12 cardiac cycles are needed to reach convergence when starting with zero pressure and flow conditions. This convergence is fairly independent on the element and time step size used to discretize the model equations (Figure

As listed in the Introduction section, many different numerical methods already exist to couple the 1D wave propagation equations for the large arteries to the 0D lumped windkessel equations for the peripheral part of the vascular tree. Usually, the wave propagation equations for the vascular segments are written in discrete form using finite/spectral-element or finite-difference schemes. Such methods have the disadvantage that bifurcations require additional coupling equations to be defined in terms of the Riemann invariants or penalty functions. Furthermore, equations of the peripheral model are usually incorporated by solving a characteristic equation (such as that of the three-element windkessel in (

In the numerical method proposed, the windkessel and wave propagation equations are cast into the same form to strongly couple them without the need for additional coupling equations or availability of a characteristic equation. In fact, any combination of windkessel (or lumped) elements and wave propagation elements is possible, allowing for a broader application to vascular networks that combine arteries, microcirculation (periphery), and veins.

The numerical method allows for easy extension with a lumped model of the heart to study arterioventricular interaction such as done by others [

To cast equations for the wave propagation model into the same form as those for the windkessel model, it is required that each discrete element contains only two nodes in which both flows are directed inwards. As a consequence, higher-order elements such as those used in, for example, spectral element discretization are no longer possible. This limitation on order of approximation, however, was found to have little influence as element size appeared to be of minor importance for convergence as well as for the pressure and flow curves obtained.

As indicated by the tangent of Figure

For the vascular network as presented in this study, we incorporated windkessel models for the periphery. However, the algorithm proposed is not restricted to this particular model choice. Other lumped element models such as structured tree models [

In conclusion, a novel numerical method is developed for computation of pressure and flow waveforms in the vascular system. Using pressure as only degree of freedom, 0D lumped (windkessel) elements and 1D wave propagation elements can be randomly combined without the need for additional coupling equations. This property facilitates flexible model building from 0D and 1D elements for a wide range of applications in studying vascular hemodynamics.

Application of definitions for nodal point pressures and flows as shown in Figure

To describe the linearized versions of the 1D balances of mass and momentum as given by (

Next, a switch is made to the proposed discretization as illustrated in Figure

Notice that (

This study was funded by the European Commission 7th Framework Programme (ARCH ICT-224390).