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Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries
Irene Vignon-Clementel, C. A. Figueroa, K. E. Jansen, C. A. Taylor
TL;DR
Arterial simulations need outflow conditions that represent downstream vasculature without assuming periodic flow or pressure, because physiological and pathological phenomena can be transient. The paper couples three-dimensional fluid–wall dynamics to lumped-parameter outflow models, including a fully transient RCR condition, and finds that transient and periodic conditions can produce significantly different solutions when aperiodicity is large. The approach is demonstrated in idealized and patient-specific cardiovascular models.
Problem
Existing resistance or periodic impedance outflow conditions do not adequately represent arterial flow and pressure when heart rate, respiration, physiology, or flow patterns vary in time.
Method
The paper extends a coupled multidomain method by directly coupling three-dimensional blood-flow and vessel-wall equations to lumped-parameter outflow models, including a fully transient RCR condition.
Results
10% maximum pressure difference occurs between periodic and fully transient RCR conditions in the stenotic carotid example during the most aperiodic cycles.
Takeaways & Limitations
Boundary conditions that make no assumption of periodicity are postulated to be more appropriate when aperiodicity is large, while the methodology models heart-rate, respiratory, and complex-flow transients.
Takeaways & Limitations
Validation remains incomplete because in vitro and in vivo validation is still needed, and simultaneous patient-specific pressure and flow measurements are difficult to obtain.
Abstract
from arXiv · showhide
The simulation of blood flow and pressure in arteries requires outflow boundary conditions that incorporate models of downstream domains. We previously described a coupled multidomain method to couple analytical models of the downstream domains with 3D numerical models of the upstream vasculature. This prior work either included pure resistance boundary conditions or impedance boundary conditions based on assumed periodicity of the solution. However, flow and pressure in arteries are not necessarily periodic in time due to heart rate variability, respiration, complex transitional flow or acute physiological changes. We present herein an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena. We have applied this method to compute haemodynamic quantities in different physiologically relevant cardiovascular models, including patient-specific examples, to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease. The relevance of using boundary conditions that accommodate transient phenomena compared with boundary conditions that assume periodicity of the solution is discussed.
1 Introduction
The paper addresses the limitations of resistance and periodic impedance outflow conditions for arterial simulations with time-varying, non-periodic flow and pressure. It extends coupled multidomain modeling to transient lumped-parameter boundary conditions and applies the approach to physiologically relevant and patient-specific cases.
- Motivation: Prescribed outlet flow or pressure is impractical to obtain and synchronize across outlets, and may be unknown in treatment-planning applications.These issues are especially important when modeling wave propagation and when outlet flow distribution is part of the desired solution.
- Motivation: Resistance conditions can distort wave propagation by forcing flow and pressure in phase and producing aberrant pressures during flow reversal.Linearized one-dimensional downstream models offer a more suitable strategy but commonly assume periodic solutions.
- Motivation: Arterial flow and pressure can be non-periodic because of heart-rate variability, respiration, acute physiological changes, and transitional or turbulent flow.The paper therefore directly couples zero-dimensional lumped models to three-dimensional blood-flow and vessel-wall equations.
- Contributions: The method differs from prior work by using three-dimensional elastodynamics, patient-specific multi-branched geometries, and coupled 3D-0D simulations of dynamic cardiovascular changes.The targeted phenomena include heart-rate variability, respiratory effects, and non-periodic flow associated with congenital or acquired vascular disease.
- Contributions: The coupled multidomain method is extended with lumped-parameter outflow conditions that accommodate non-periodic phenomena, and transient and periodic boundary conditions are compared.Applications include carotid and Glenn models with pulsatility changes, stenosis-related transitional flow, and variable inflow.
2 Methods
The method couples three-dimensional fluid and deformable-wall dynamics to analytical downstream-domain models through interface fluxes. A fully transient RCR Windkessel condition relates outlet pressure to current and past flow, allowing transient downstream behavior without assuming periodicity.
- Fluid–structure formulation: The computational framework combines three-dimensional Navier–Stokes blood flow with elastodynamics of the arterial wall over a decomposed upstream domain.The primary fluid variables are velocity and pressure, while the primary wall variable is displacement.
- Fluid–structure formulation: Fluid and wall velocities are matched kinematically at the vessel boundary, while wall body forces are derived from the opposing blood traction under a thin-wall assumption.Fixed wall-domain edges complete the stated structural setup.
- Multidomain coupling: The multidomain decomposition separates the computational domain from downstream domains and rewrites the coupled weak formulation using upstream variables only.Continuity of stress and mass fluxes makes downstream effects appear in interface boundary fluxes through model-dependent momentum and continuity operators.
- Multidomain coupling: The downstream model acts through coupling operators on the interface, while the numerical domain solves for velocity and pressure with prescribed inlet velocity profiles.The downstream representation may use lumped models, one-dimensional blood-flow equations, or vessel-wall deformation models.
- Time-varying outlet boundary condition: The RCR Windkessel model represents a proximal resistance in series with a parallel capacitance and distal resistance, with optional time-varying terminal pressure.It is an ordinary differential-equation boundary model that can accommodate transient phenomena.
- Time-varying outlet boundary condition: The fully transient RCR condition relates pressure at time t to the flow history from the simulation start through time t.The time constant τ controls how rapidly the system responds to changes in the input function, and the resulting operators enter the interface coupling terms.
- Discretization: The continuous equations are discretized spatially with a stabilized finite-element method and temporally with a semi-implicit generalized α-method adapted for fluid–solid interaction.This supplies the numerical time-and-space treatment for the coupled formulation.
3 Numerical simulations and results
The simulations use fully transient RCR outflow boundary conditions to model periodic and non-periodic cardiovascular flow in deformable and patient-specific geometries. Results demonstrate cycle-to-cycle pressure and flow variability from heart-rate changes, stenosis, and complex anatomy.
- Simulation setup: The model prescribes inlet velocity profiles while representing downstream domains with fully transient RCR boundary conditions.This setup is used to compute velocity and pressure for each case.
- Verification and initialization: A periodic carotid simulation verified the RCR relationship and mass conservation using outlet flow and mean-pressure Fourier analyses.The extracted quantities were evaluated over the final cardiac cycle.
- Verification and initialization: The RCR condition incorporates flow-history memory through a time constant τ = 1.1 s, while convergence remains sensitive to the initial outlet pressure and boundary-condition type.Starting from 97 mmHg rather than 68 mmHg advanced the simulation to approximately cycle 4 of the reference convergence sequence.
- Heart-rate variability: With aperiodic carotid inflow, outlet pressure was also aperiodic, pressure waves lagged flow waves, and waveform features matched reported physiological values.The reported features included a first systolic peak near 0.13 s and an augmentation index around -20%.
- Carotid bifurcation with stenosis: In a patient-specific carotid bifurcation with 68% stenosis, transitional non-axisymmetric flow and cycle-varying outlet flow and pressure were obtained.Mean outlet pressures were 70 mmHg in the internal carotid and 90 mmHg in the external carotid.
- Carotid bifurcation with stenosis: Repeating a single periodic inflow produced negligible cycle-to-cycle outlet variation, indicating that most observed outflow non-periodicity arose from inlet-flow non-periodicity.The comparison used seven repeated cycles in the stenosed bifurcation model.
- Patient-specific Glenn model: The patient-specific Glenn model reproduced intricate pulmonary flow features, physiologic wall shear stress, and realistic pressure levels and gradients.The calculated mean pressure was 10.7 mmHg.
4 Discussion
The fully transient RCR boundary condition better accommodates aperiodic arterial flow than periodic assumptions, while remaining useful for short-term predictions and patient-specific simulations. Its main limitations are incomplete simultaneous clinical measurements and the need for further validation.
- Practical implications: The implicit boundary condition enforces a pressure–flow relationship rather than prescribing either variable directly, supporting short-term surgical predictions.The examples showed good agreement with targeted values and available experimental data.
- Limitations: Patient-specific determination of more complex time-varying zero-dimensional boundary-condition parameters remains difficult because simultaneous in vivo pressure and flow measurements are technically challenging.The Glenn example used measurements acquired under different conditions, contributing to discrepancies in computed and measured SVC pressure pulsatility.
- Transient versus periodic conditions: A 10% maximum pressure difference occurred between periodic and fully transient RCR conditions during the most aperiodic carotid cycles.Pressure was overpredicted in some cycles and underpredicted in others, while flow waveforms were almost identical.
- Transient versus periodic conditions: Velocity patterns and pressure maps differed markedly between the two boundary conditions downstream of the stenosis.These differences were observed at 3.6 s in the computational domain.
- Numerical behavior: Fully transient RCR simulations required somewhat less effort to solve the nonlinear equations than periodic RCR simulations during aperiodic phenomena.When the solution was periodic, residuals and iterations per time step differed by less than 0.1%.
- Scope and interpretation: A boundary condition that makes no assumption about flow or pressure periodicity is considered more sensitive when non-periodicity is important.Arterial unsteadiness can arise from heart-rate variability, respiration, acute physiological changes, transitional flow, turbulence, or complex geometry.
- Limitations: Further validation with in vitro and in vivo data is still needed, although comparisons with clinical pressure measurements have begun.A variable-pulsatile-inflow flow-phantom experiment is proposed as a future validation approach.
5 Conclusions
The methodology models transient cardiovascular phenomena, including heart-rate variability, respiratory and cardiac asynchrony, and non-periodic effects from complex geometries and flows. Boundary conditions without periodicity assumptions can produce different solutions and may be more appropriate when aperiodicity is large.
- In a stenotic carotid bifurcation, naturally varying inflow produced large flow and pressure variations in both branches, suggesting amplification of inlet variability.
- A patient-specific Glenn model reproduced flow and pressure variations associated with naturally varying respiratory and cardiac rhythms.
- Boundary conditions making no periodicity assumption produced different numerical solutions from periodic boundary conditions and may be more appropriate when aperiodicity is large.
- The simulations captured heart-rate variability, respiratory and cardiac asynchronous variations, and non-periodic phenomena caused by complex geometries and flows.
- The approach can incorporate sophisticated lumped parameter models for time-varying phenomena such as cardiac perfusion and autonomic effects.
8 Figure captions
The figures document the Windkessel model, its frequency-domain verification and convergence behavior, and applications to non-periodic carotid, stenotic bifurcation, and Glenn simulations. Comparisons emphasize differences between fully transient and periodic RCR boundary conditions, particularly in pressure fields.
- Figure 1 presents the Windkessel electric analog used to represent the downstream model.
- Figure 2 shows normalized frequency content of outlet flow and pressure for a deformable carotid model with a fully transient RCR boundary condition.
- Figure 3 compares theoretical and simulated impedance, computed as P(ω) divided by Q(ω), for the fully transient RCR boundary condition.
- Figures 4 and 7 show convergence toward periodicity or a common solution after initialization from different cycles.
- Figures 5 and 6 connect measured cycle-to-cycle velocity variability with resulting non-periodic outlet flow and pressure over seven cardiac cycles.
- Figures 9–11 examine complex stenotic-bifurcation flow and compare transient versus periodic behavior in outlet pressure and flow.
- Figures 12–16 depict a patient-specific Glenn model, measured cardiac and respiratory variation, realistic haemodynamic fields, and comparison with catheter pressure.
- Figures 17–19 compare periodic and fully transient RCR simulations, showing small velocity differences but significantly different pressure maps.