Cardiovascular Blood Flow Simulation

By Alison L. Marsden

Cardiovascular blood flow simulations offer non-invasive means to quantify hemodynamics in the heart and major blood vessels for patients with cardiovascular disease. This  data is not readily available from standard clinical measurements, yet it can offer key insights into disease progression and subsequent physiologic response, and thus aid in clinical decision-making. In much the same way that simulations have impacted aeronautical design, they now bring—for the first time—quantitative, predictive tools to surgical and treatment planning for a range of cardiovascular diseases in children and adults.

Simulations are performed in a multi-step process involving a rich and diverse array of mathematical techniques. The process generally starts with analysis of medical imaging data and proceeds through model construction, meshing, solution of the governing equations, boundary condition assignments, data assimilation and error propagation, and statistical correlation with clinical outcomes. A wide variation of mathematical and computational disciplines must be integrated throughout this modeling process. Simulations are already showing an impact in the clinical setting, and have the potential to improve outcomes for both children and adults affected by a multitude of diseases.

Patient-specific cardiovascular simulations typically begin with acquisition and processing of medical imaging data. Computed tomography (CT) or magnetic resonance imaging (MRI) is generally segmented using 2D or 3D level set, thresholding, or region-growing algorithms, which often necessitate manual intervention. Recent efforts to improve automated segmentation algorithms have focused on machine-learning algorithms, including deep learning and shape analysis. These improvements are crucial to enabling high-throughput processing of models for large patient cohorts in clinical studies. The segmented data allows for the construction of a 3D anatomic model that represents a region of interest of a particular patient’s vascular anatomy. Model construction requires care to ensure that boolean and smoothing operations maintain fidelity to the original image data.

Patient-specific simulations usually require time-dependent solutions of the partial differential equations (PDEs) governing blood flow, the Navier-Stokes equations, and complex geometries, which commonly require parallel, high-performance computing. Recent advances in simulation methodologies have increased efficiency and enabled heightened physiologic realism. Yet the addition of fluid structure interaction (FSI) to model vessel wall deformation makes simulations more challenging and computationally expensive. Finite volume or finite element methods with appropriate boundary and initial conditions allow the governing equations to be solved numerically. The cardiovascular community widely uses finite element methods with unstructured meshes due to complex geometries and FSI. Much of the recent work has employed stabilized (streamline-upwind/Petrov–Galerkin, or SUPG) methods with linear finite elements, but higher-order spectral elements, unstructured finite volume methods, and immersed boundary techniques offer attractive alternatives.

Since only a portion of a patient’s anatomy can be included in the 3D model—both due to computational expense and limits of image resolution—boundary conditions must be applied at inlets and outlets of the model to accurately represent the vascular network outside of the 3D domain. The choice of boundary conditions is of paramount importance in cardiovascular simulations, as conditions upstream and downstream of the 3D model greatly influence the local flow dynamics. The current state-of-the-art cardiovascular simulations implicitly couple the Navier-Stokes equations in the 3D domain with reduced-order models of circulatory physiology, forming a closed loop system. Electrical circuits, in which flow is analogous to electric current and pressure drop is analogous to voltage, offer a convenient analogy. Organ compartments, including the heart, systemic, and pulmonary circulations, are constructed from circuit elements, resulting in a system of ordinary differential equations (ODEs). Solution of this coupled PDE/ODE system poses numerical challenges. These challenges include flow reversal at the outlets, the need for specialized pre-conditioners for the linear system of equations, and ill conditioning arising from the dominance of a few eigenvalues coming from the coupled boundaries [4].

There is rising interest in clinical data assimilation and uncertainty quantification in cardiovascular simulations. Because physicians are unlikely to trust simulations for clinical use unless they are accompanied by confidence intervals on model predictions, models must account for myriad sources of uncertainty. These include noise in image data, measurement errors associated with cardiac catheterization, phase contrast MRI, and natural physiologic variation. One must first perform parameter estimation in the presence of uncertainty to assimilate clinical data into models. Propagation to model outputs and uncertainty quantification for model outputs happens next. Bayesian approaches are particularly attractive for this purpose, including multi-level Monte Carlo methods and compressed sensing [5]. Application of optimization to design surgical geometries and devices, including derivative free-pattern search methods incorporating surrogates, has proven to be particularly effective and efficient. Computational performance is of utmost importance in this context, as each function evaluation may require hours of run time on a parallel cluster, and algorithms can quickly become intractable without special care to maximize efficiency.

Clinical interest in patient-specific cardiovascular modeling is rapidly accelerating, with recent application to numerous diseases in children and adults. These diseases include congenital heart disease, coronary artery disease, Kawasaki disease, cerebral and abdominal aneurysms, and medical device design. In 2014 the FDA granted the first approval of cardiovascular simulations for routine clinical use with the introduction of fractional flow reserve by computed tomography (FFRct), created by HeartFlow, Inc. FFRct is an assessment of the functional severity of coronary artery stenosis, and clinical trial results have demonstrated excellent agreement with invasive measurements as well as a reduction in the number of invasive procedures performed on patients [1]. Current studies are examining surgical methods for coronary artery bypass graft surgery and causes of vein graft failure.

Simulations have also impacted pediatric cardiology with the introduction of novel surgical methods for single ventricle heart patients, a severe condition in which a child is born with only one functioning heart ventricle. Surgical methods for the Fontan procedure (the third stage surgery performed in single ventricle palliation) were changed based on simulation results to adopt an offset design in order to avoid flow collision, and, more recently, a “Y-graft” design that improves flow distribution. The “Y-graft” was designed and optimized via simulations before translation to clinical use at Lucile Packard Children’s Hospital at Stanford University [3]. A recently-introduced concept for stage-one single ventricle surgery, called the “Assisted Bidirectional Glenn,” now aims to improve outcomes in neonates by incorporating an ejector pump to increase pulmonary blood flow [2]. In this context, multiscale simulations provide a means to develop novel surgical methods, for which it may be ethically infeasible to test directly in patients.

Numerous challenges remain as tools for cardiovascular simulations mature. Among these is the ongoing challenge of validation against in vivo data, which is often hindered by ethical considerations of invasive data collection, particularly in children. Future simulations should include far more detailed models of biological and physiologic responses, such as the chemistry of blood clot formation, vascular growth and remodeling in response to changing mechanical loads, auto regulatory feedback mechanisms, and vascular mechanotransduction. These mechanisms will require specialized numerical methods capable of handling increasingly-complex multiscale problems.

Finally, because cardiovascular software development is a complex undertaking, readers might benefit from the exploration of software tools available through the open source SimVascular project, which provides a complete open source pipeline from image data and patient-specific model construction to meshing, boundary condition assignment, and simulation results, and is freely available for academic purposes.

References
[1] Douglas, P. M., Pontone, G., Hlatky, M. A., Patel, M. R., Norgaard, B. L., Byrne, R. A.,…Rogers, C. (2015). Clinical outcomes of fractional flow reserve by computed tomographic angiography-guided diagnostic strategies vs. usual care in patients with suspected coronary artery disease: the prospective longitudinal trial of ffrct: outcome and resource impacts study. European Heart Journal. Advance online publication.

[2] Esmaily-Moghadam, M., Hsia, T. Y., & Marsden, A. L. (2015). The assisted bidirectional Glenn: a novel surgical approach for first-stage single-ventricle heart palliation. The Journal of Thoracic and Cardiovascular Surgery, 149(3), 699–705.

[3] Marsden, A. L., Bernstein, A. J., Reddy, V. M., Shadden, S., Spilker, R. L., Chan, F. P.,…Feinstein, J. A. (2009). Evaluation of a novel Y-shaped extracardiac Fontan baffle using computational fluid dynamics. Journal of Thoracic and Cardiovascular Surgery, 137, 394–403.

[4] Marsden, A. L., & Esmaily-Moghadam, M. (2015). Multiscale modeling of cardiovascular flows for clinical decision support. Applied Mechanics Reviews, 67(3), 1–11.

[5] Schiavazzi, D. E., Arbia, G., Marsden, A. L., Baker, C., Hsia, T. Y., & Vignon-Clementel, I. E. (submitted 2015). Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation. International Journal of Numerical Methods in Biomedical Engineering.

Alison Marsden is an associate professor in the Department of Pediatrics and Department of Bioengineering at the Institute for Computational and Mathematical Engineering at Stanford University.