A Distributed Lumped Parameter Model of Blood Flow

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

Estimating pulmonary arterial remodeling via an animal-specific computational model of pulmonary artery stenosis

Pulmonary artery stenosis (PAS) often presents in children with congenital heart disease, altering blood flow and pressure during critical periods of growth and development. Variability in stenosis onset, duration, and severity result in variable growth and remodeling of the pulmonary vasculature. Computational fluid dynamics (CFD) models enable investigation into the hemodynamic impact and altered mechanics associated with PAS. In this study, a one-dimensional (1D) fluid dynamics model was used to simulate hemodynamics throughout the pulmonary arteries of individual animals. The geometry of the large pulmonary arteries was prescribed by animal-specific imaging, whereas the distal vasculature was simulated by a three-element Windkessel model at each terminal vessel outlet. Remodeling of the pulmonary vasculature, which cannot be measured in vivo, was estimated via model-fitted parameters. The large artery stiffness was significantly higher on the left side of the vasculature in the left pulmonary artery (LPA) stenosis group, but neither side differed from the sham group. The sham group exhibited a balanced distribution of total distal vascular resistance, whereas the left side was generally larger in the LPA stenosis group, with no significant differences between groups. In contrast, the peripheral compliance on the right side of the LPA stenosis group was significantly greater than the corresponding side of the sham group. Further analysis indicated the underperfused distal vasculature likely moderately decreased in radius with little change in stiffness given the increase in thickness observed with histology. Ultimately, our model enables greater understanding of pulmonary arterial adaptation due to LPA stenosis and has potential for use as a tool to noninvasively estimate remodeling of the pulmonary vasculature.

Non-invasive Estimation of Pressure Drop Across Aortic Coarctations: Validation of 0D and 3D Computational Models with In Vivo Measurements

Blood pressure gradient ( Δ P ) across an aortic coarctation (CoA) is an important measurement to diagnose CoA severity and gauge treatment efficacy. Invasive cardiac catheterization is currently the gold-standard method for measuring blood pressure. The objective of this study was to evaluate the accuracy of Δ P estimates derived non-invasively using patient-specific 0D and 3D deformable wall simulations. Medical imaging and routine clinical measurements were used to create patient-specific models of patients with CoA (N = 17). 0D simulations were performed first and used to tune boundary conditions and initialize 3D simulations. Δ P across the CoA estimated using both 0D and 3D simulations were compared to invasive catheter-based pressure measurements for validation. The 0D simulations were extremely efficient ( ∼ 15 s computation time) compared to 3D simulations ( ∼ 30 h computation time on a cluster). However, the 0D Δ P estimates, unsurprisingly, had larger mean errors when compared to catheterization than 3D estimates (12.1 ± 9.9 mmHg vs 5.3 ± 5.4 mmHg). In particular, the 0D model performance degraded in cases where the CoA was adjacent to a bifurcation. The 0D model classified patients with severe CoA requiring intervention (defined as Δ P ≥ 20 mmHg) with 76% accuracy and 3D simulations improved this to 88%. Overall, a combined approach, using 0D models to efficiently tune and launch 3D models, offers the best combination of speed and accuracy for non-invasive classification of CoA severity.

Accelerated simulation methodologies for computational vascular flow modelling

Vascular flow modelling can improve our understanding of vascular pathologies and aid in developing safe and effective medical devices. Vascular flow models typically involve solving the nonlinear Navier-Stokes equations in complex anatomies and using physiological boundary conditions, often presenting a multi-physics and multi-scale computational problem to be solved. This leads to highly complex and expensive models that require excessive computational time. This review explores accelerated simulation methodologies, specifically focusing on computational vascular flow modelling. We review reduced order modelling (ROM) techniques like zero-/one-dimensional and modal decomposition-based ROMs and machine learning (ML) methods including ML-augmented ROMs, ML-based ROMs and physics-informed ML models. We discuss the applicability of each method to vascular flow acceleration and the effectiveness of the method in addressing domain-specific challenges. When available, we provide statistics on accuracy and speed-up factors for various applications related to vascular flow simulation acceleration. Our findings indicate that each type of model has strengths and limitations depending on the context. To accelerate real-world vascular flow problems, we propose future research on developing multi-scale acceleration methods capable of handling the significant geometric variability inherent to such problems.

A multi-dimensional CFD framework for fast patient-specific fractional flow reserve prediction

Fractional flow reserve (FFR) is considered as the gold standard for diagnosing coronary myocardial ischemia. Existing 3D computational fluid dynamics (CFD) methods attempt to predict FFR noninvasively using coronary computed tomography angiography (CTA). However, the accuracy and efficiency of the 3D CFD methods in coronary arteries are considerably limited. In this work, we introduce a multi-dimensional CFD framework that improves the accuracy of FFR prediction by estimating 0D patient-specific boundary conditions, and increases the efficiency by generating 3D initial conditions. The multi-dimensional CFD models contain the 3D vascular model for coronary simulation, the 1D vascular model for iterative optimization, and the 0D vascular model for boundary conditions expression. To improve the accuracy, we utilize clinical parameters to derive 0D patient-specific boundary conditions with an optimization algorithm. To improve the efficiency, we evaluate the convergence state using the 1D vascular model and obtain the convergence parameters to generate appropriate 3D initial conditions. The 0D patient-specific boundary conditions and the 3D initial conditions are used to predict FFR (FFRC). We conducted a retrospective study involving 40 patients (61 diseased vessels) with invasive FFR and their corresponding CTA images. The results demonstrate that the FFRC and the invasive FFR have a strong linear correlation (r = 0.80, p < 0.001) and high consistency (mean difference: 0.014 ±0.071). After applying the cut-off value of FFR (0.8), the accuracy, sensitivity, specificity, positive predictive value, and negative predictive value of FFRC were 88.5%, 93.3%, 83.9%, 84.8%, and 92.9%, respectively. Compared with the conventional zero initial conditions method, our method improves prediction efficiency by 71.3% per case. Therefore, our multi-dimensional CFD framework is capable of improving the accuracy and efficiency of FFR prediction significantly.

Non-invasive estimation of pressure drop across aortic coarctations: validation of 0D and 3D computational models with in vivo measurements

Purpose

Blood pressure gradient (Δ P ) across an aortic coarctation (CoA) is an important measurement to diagnose CoA severity and gauge treatment efficacy. Invasive cardiac catheterization is currently the gold-standard method for measuring blood pressure. The objective of this study was to evaluate the accuracy of Δ P estimates derived non-invasively using patient-specific 0 D and 3 D deformable wall simulations.

Methods

Medical imaging and routine clinical measurements were used to create patient-specific models of patients with CoA (N = 17 ). 0 D simulations were performed first and used to tune boundary conditions and initialize 3 D simulations. Δ P across the CoA estimated using both 0 D and 3 D simulations were compared to invasive catheter-based pressure measurements for validation.

Results

The 0 D simulations were extremely efficient (~15 secs computation time) compared to 3 D simulations (~30 hrs computation time on a cluster). However, the 0 D Δ P estimates, unsurprisingly, had larger mean errors when compared to catheterization than 3 D estimates (12.1 ± 9.9 mmHg vs 5.3 ± 5.4 mmHg). In particular, the 0 D model performance degraded in cases where the CoA was adjacent to a bifurcation. The 0 D model classified patients with severe CoA requiring intervention (defined as Δ P ≥ 20 mmHg) with 76% accuracy and 3 D simulations improved this to 88%.

Conclusion

Overall, a combined approach, using 0 D models to efficiently tune and launch 3 D models, offers the best combination of speed and accuracy for non-invasive classification of CoA severity.

Beyond CFD: Emerging methodologies for predictive simulation in cardiovascular health and disease

Physics-based computational models of the cardiovascular system are increasingly used to simulate hemodynamics, tissue mechanics, and physiology in evolving healthy and diseased states. While predictive models using computational fluid dynamics (CFD) originated primarily for use in surgical planning, their application now extends well beyond this purpose. In this review, we describe an increasingly wide range of modeling applications aimed at uncovering fundamental mechanisms of disease progression and development, performing model-guided design, and generating testable hypotheses to drive targeted experiments. Increasingly, models are incorporating multiple physical processes spanning a wide range of time and length scales in the heart and vasculature. With these expanded capabilities, clinical adoption of patient-specific modeling in congenital and acquired cardiovascular disease is also increasing, impacting clinical care and treatment decisions in complex congenital heart disease, coronary artery disease, vascular surgery, pulmonary artery disease, and medical device design. In support of these efforts, we discuss recent advances in modeling methodology, which are most impactful when driven by clinical needs. We describe pivotal recent developments in image processing, fluid-structure interaction, modeling under uncertainty, and reduced order modeling to enable simulations in clinically relevant timeframes. In all these areas, we argue that traditional CFD alone is insufficient to tackle increasingly complex clinical and biological problems across scales and systems. Rather, CFD should be coupled with appropriate multiscale biological, physical, and physiological models needed to produce comprehensive, impactful models of mechanobiological systems and complex clinical scenarios. With this perspective, we finally outline open problems and future challenges in the field.

Automated generation of 0D and 1D reduced-order models of patient-specific blood flow

Three-dimensional (3D) cardiovascular fluid dynamics simulations typically require hours to days of computing time on a high-performance computing cluster. One-dimensional (1D) and lumped-parameter zero-dimensional (0D) models show great promise for accurately predicting blood bulk flow and pressure waveforms with only a fraction of the cost. They can also accelerate uncertainty quantification, optimization, and design parameterization studies. Despite several prior studies generating 1D and 0D models and comparing them to 3D solutions, these were typically limited to either 1D or 0D and a singular category of vascular anatomies. This work proposes a fully automated and openly available framework to generate and simulate 1D and 0D models from 3D patient-specific geometries, automatically detecting vessel junctions and stenosis segments. Our only input is the 3D geometry; we do not use any prior knowledge from 3D simulations. All computational tools presented in this work are implemented in the open-source software platform SimVascular. We demonstrate the reduced-order approximation quality against rigid-wall 3D solutions in a comprehensive comparison with N = 72 publicly available models from various anatomies, vessel types, and disease conditions. Relative average approximation errors of flows and pressures typically ranged from 1% to 10% for both 1D and 0D models, measured at the outlets of terminal vessel branches. In general, 0D model errors were only slightly higher than 1D model errors despite requiring only a third of the 1D runtime. Automatically generated ROMs can significantly speed up model development and shift the computational load from high-performance machines to personal computers.

Insights from computational modeling on the potential hemodynamic effects of sinus rhythm versus atrial fibrillation

Atrial fibrillation (AF) is the most common clinical tachyarrhythmia, posing a significant burden to patients, physicians, and healthcare systems worldwide. With the advent of more effective rhythm control strategies, such as AF catheter ablation, an early rhythm control strategy is progressively demonstrating its superiority not only in symptoms control but also in prognostic terms, over a standard strategy (rate control, with rhythm control reserved only to patients with refractory symptoms). This review summarizes the different impacts exerted by AF on heart mechanics and systemic circulation, as well as on cerebral and coronary vascular beds, providing computational modeling-based hemodynamic insights in favor of pursuing sinus rhythm maintenance in AF patients.

Framework for patient-specific simulation of hemodynamics in heart failure with counterpulsation support

Despite being responsible for half of heart failure-related hospitalizations, heart failure with preserved ejection fraction (HFpEF) has limited evidence-based treatment options. Currently, a substantial clinical issue is that the disease etiology is very heterogenous with no patient-specific treatment options. Modeling can provide a framework for evaluating alternative treatment strategies. Counterpulsation strategies have the capacity to improve left ventricular diastolic filling by reducing systolic blood pressure and augmenting the diastolic pressure that drives coronary perfusion. Here, we propose a framework for testing the effectiveness of a soft robotic extra-aortic counterpulsation strategy using a patient-specific closed-loop hemodynamic lumped parameter model of a patient with HFpEF. The soft robotic device prototype was characterized experimentally in a physiologically pressurized (50–150 mmHg) soft silicone vessel and modeled as a combination of a pressure source and a capacitance. The patient-specific model was created using open-source software and validated against hemodynamics obtained by imaging of a patient (male, 87 years, HR = 60 bpm) with HFpEF. The impact of actuation timing on the flows and pressures as well as systolic function was analyzed. Good agreement between the patient-specific model and patient data was achieved with relative errors below 5% in all categories except for the diastolic aortic root pressure and the end systolic volume. The most effective reduction in systolic pressure compared to baseline (147 vs. 141 mmHg) was achieved when actuating 350 ms before systole. In this case, flow splits were preserved, and cardiac output was increased (5.17 vs. 5.34 L/min), resulting in increased blood flow to the coronaries (0.15 vs. 0.16 L/min). Both arterial elastance (0.77 vs. 0.74 mmHg/mL) and stroke work (11.8 vs. 10.6 kJ) were decreased compared to baseline, however left atrial pressure increased (11.2 vs. 11.5 mmHg). A higher actuation pressure is associated with higher systolic pressure reduction and slightly higher coronary flow. The soft robotic device prototype achieves reduced systolic pressure, reduced stroke work, slightly increased coronary perfusion, but increased left atrial pressures in HFpEF patients. In future work, the framework could include additional physiological mechanisms, a larger patient cohort with HFpEF, and testing against clinically used devices.

Enhanced 4D Flow MRI-Based CFD with Adaptive Mesh Refinement for Flow Dynamics Assessment in Coarctation of the Aorta

4D Flow MRI is a diagnostic tool that can visualize and quantify patient-specific hemodynamics and help interventionalists optimize treatment strategies for repairing coarctation of the aorta (COA). Despite recent developments in 4D Flow MRI, shortcomings include phase-offset errors, limited spatiotemporal resolution, aliasing, inaccuracies due to slow aneurysmal flows, and distortion of images due to metallic artifact from vascular stents. To address these limitations, we developed a framework utilizing Computational Fluid Dynamics (CFD) with Adaptive Mesh Refinement (AMR) that enhances 4D Flow MRI visualization/quantification. We applied this framework to five pediatric patients with COA, providing in-vivo and in-silico datasets, pre- and post-intervention. These two data sets were compared and showed that CFD flow rates were within 9.6% of 4D Flow MRI, which is within a clinically acceptable range. CFD simulated slow aneurysmal flow, which MRI failed to capture due to high relative velocity encoding (Venc). CFD successfully predicted in-stent blood flow, which was not visible in the in-vivo data due to susceptibility artifact. AMR improved spatial resolution by factors of 101 to 103 and temporal resolution four-fold. This computational framework has strong potential to optimize visualization/quantification of aneurysmal and in-stent flows, improve spatiotemporal resolution, and assess hemodynamic efficiency post-COA treatment.

Machine Learning for Cardiovascular Biomechanics Modeling: Challenges and Beyond

Recent progress in machine learning (ML), together with advanced computational power, have provided new research opportunities in cardiovascular modeling. While classifying patient outcomes and medical image segmentation with ML have already shown significant promising results, ML for the prediction of biomechanics such as blood flow or tissue dynamics is in its infancy. This perspective article discusses some of the challenges in using ML for replacing well-established physics-based models in cardiovascular biomechanics. Specifically, we discuss the large landscape of input features in 3D patient-specific modeling as well as the high-dimensional output space of field variables that vary in space and time. We argue that the end purpose of such ML models needs to be clearly defined and the tradeoff between the loss in accuracy and the gained speedup carefully interpreted in the context of translational modeling. We also discuss several exciting venues where ML could be strategically used to augment traditional physics-based modeling in cardiovascular biomechanics. In these applications, ML is not replacing physics-based modeling, but providing opportunities to solve ill-defined problems, improve measurement data quality, enable a solution to computationally expensive problems, and interpret complex spatiotemporal data by extracting hidden patterns. In summary, we suggest a strategic integration of ML in cardiovascular biomechanics modeling where the ML model is not the end goal but rather a tool to facilitate enhanced modeling.

On the Periodicity of Cardiovascular Fluid Dynamics Simulations

Three-dimensional cardiovascular fluid dynamics simulations typically require computation of several cardiac cycles before they reach a periodic solution, rendering them computationally expensive. Furthermore, there is currently no standardized method to determine whether a simulation has yet reached that periodic state. In this work, we propose the use of an asymptotic error measurement to quantify the difference between simulation results and their ideal periodic state using open-loop lumped-parameter modeling. We further show that initial conditions are crucial in reducing computational time and develop an automated framework to generate appropriate initial conditions from a one-dimensional model of blood flow. We demonstrate the performance of our initialization method using six patient-specific models from the Vascular Model Repository. In our examples, our initialization protocol achieves periodic convergence within one or two cardiac cycles, leading to a significant reduction in computational cost compared to standard methods. All computational tools used in this work are implemented in the open-source software platform SimVascular. Automatically generated initial conditions have the potential to significantly reduce computation time in cardiovascular fluid dynamics simulations.

A distributed lumped parameter model of blood flow with fluid-structure interaction

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.

Non-invasive MRI Derived Hemodynamic Simulation to Predict Successful vs. Unsuccessful Catheter Interventions for Branch Pulmonary Artery Stenosis: Proof-of-Concept and Experimental Validation in Swine

OBJECTIVE

This study assessed the ability of hemodynamic simulations to predict the success of catheter interventions in a swine model of branch pulmonary artery stenosis (bPAS).

BACKGROUND

bPAS commonly occurs in congenital heart disease and is often managed with catheter based interventions. However, despite technical success, bPAS interventions do not lead to improved distal pulmonary blood flow (PBF) distribution in approximately 1/3rd of patients. New tools are needed to better identify which patients with bPAS would most benefit from catheter interventions.

METHODS

For 13 catheter intervention cases in swine with surgically created left PAS (LPAS), PA pressures from right heart catheterization (RHC) and PBF distributions from MRI were measured before and after catheter interventions. Hemodynamic simulations with a reduced order computational fluid dynamics (CFD) model were performed using non-invasive PBF measurements derived from MRI, and then correlated with changes in invasive measures of hemodynamics and PBF distributions before and after catheter intervention to relieve bPAS.

RESULTS

Compared to experimentally measured changes in left PBF distribution, simulations had a small bias (3.4 ± 11.1%), moderate agreement (ICC = 0.69 [0.24-0.90], 0.71 [0.23-0.91]), and good diagnostic capability to predict successful interventions (> 20% PBF increase) (AUC 0.83 [0.59-1.0]). Simulations had poorer prediction of changes in stenotic pressure gradient (ICC = 0.28 [- 0.33 to 0.73], r = 0.57 [- 0.04 to 0.87]) and MPA systolic pressure (ICC = 0.00 [- 0.52 to 0.53], r = 0.29 [- 0.32 to 0.72]).

CONCLUSION

While there was only weak to moderate agreement between predicted and measured changes in PA pressures and pulmonary blood flow distributions, hemodynamic simulations did show good diagnostic value for predicting successful versus unsuccessful catheter based interventions to relieve bPAS. The results of this proof of concept study are promising and should encourage future development for using hemodynamic models in planning interventions for patients with bPAS.

Accelerated Estimation of Pulmonary Artery Stenosis Pressure Gradients with Distributed Lumped Parameter Modeling vs. 3D CFD with Instantaneous Adaptive Mesh Refinement: Experimental Validation in Swine

Branch pulmonary artery stenosis (PAS) commonly occurs in congenital heart disease and the pressure gradient over a stenotic PA lesion is an important marker for re-intervention. Image based computational fluid dynamics (CFD) has shown promise for non-invasively estimating pressure gradients but one limitation of CFD is long simulation times. The goal of this study was to compare accelerated predictions of PAS pressure gradients from 3D CFD with instantaneous adaptive mesh refinement (AMR) versus a recently developed 0D distributed lumped parameter CFD model. Predictions were then experimentally validated using a swine PAS model (n = 13). 3D CFD simulations with AMR improved efficiency by 5 times compared to fixed grid CFD simulations. 0D simulations further improved efficiency by 6 times compared to the 3D simulations with AMR. Both 0D and 3D simulations underestimated the pressure gradients measured by catheterization (- 1.87 ± 4.20 and - 1.78 ± 3.70 mmHg respectively). This was partially due to simulations neglecting the effects of a catheter in the stenosis. There was good agreement between 0D and 3D simulations (ICC 0.88 [0.66-0.96]) but only moderate agreement between simulations and experimental measurements (0D ICC 0.60 [0.11-0.86] and 3D ICC 0.66 [0.21-0.88]). Uncertainty assessment indicates that this was likely due to limited medical imaging resolution causing uncertainty in the segmented stenosis diameter in addition to uncertainty in the outlet resistances. This study showed that 0D lumped parameter models and 3D CFD with instantaneous AMR both improve the efficiency of hemodynamic modeling, but uncertainty from medical imaging resolution will limit the accuracy of pressure gradient estimations.