Hemodynamics of heart diseases through data-driven reduced order models
by Caterina Balzotti and Pierfrancesco Siena
Heart disease is a leading cause of death in the world and encompasses a wide range of ailments such as coronary artery disease (CAD), arrhythmias, congenital heart defects, and heart muscle or valve disease. CAD and arrhythmias deserve particular attention as they affect a large number of people.
Coronary artery disease consists of the narrowing or blockage of one or more coronary arteries due to the accumulation of plaque in their walls. As a result, the damaged arteries are unable to supply the heart with oxygen-rich blood, often leading to heart attack. The typical procedure to treat CAD is the coronary artery bypass graft (CABG), an invasive surgical procedure that circumvents the blockage by diverting blood flow.
A heart arrhythmia is an irregular heartbeat that may be too fast (tachycardia) or too slow (bradycardia). Atrial fibrillation (AF) is the most common type of arrhythmia, characterized by a fast and irregular heartbeat. During an AF episode, the left atrium contracts in an irregular and ineffective way, and this is believed to lead to thrombus formation.
For both cases, studying the hemodynamics can provide a complete knowledge of the blood flow for different configurations. A virtual platform for a fast and reliable evaluation can lead medical doctors to a conscious choice for the appropriate treatment.
Modeling the blood flow
From a mathematical point of view, blood flow is governed by the Navier-Stokes (NS) equations. Depending on the disease considered, one may need to use the steady-state NS or the dynamic NS.
The development of fast and accurate numerical models to simulate hemodynamics in CABG has attracted the interest of the medical community, in order to provide concrete support in surgical planning. A typical issue that arises in the analysis of blood flow in CABG is the treatment of the outlet conditions. Indeed, imposing a Dirichlet condition generally results in inaccurate flow predictions. To handle this, one can consider an optimal control problem on the outlet boundary, to optimally match the simulated flow with real measurements. As a first step in the patient-specific simulation of hemodynamics in CABG, we consider an optimal control problem based on the steady-state NS equations [1, 2].
On the other hand, arrhythmias are characterized by irregular heartbeat, so they require the use of time-dependent NS equations. For patients with AF, the risk of thrombosis appears to be related to blood stasis in the left atrium appendage (LAA), a protruding cavity located in the left atrium. To assess the blood stagnation in the LAA, some clinical studies suggest considering atrial geometric parameters such as the LAA volume, ostium area or depth. Recent studies [3, 4] have shown the connection between the moments of the blood age and the LAA blood stasis, suggesting to couple the time-dependent NS equations with the dynamics of these moments.
Both for the CABG and AF hemodynamics, we consider parametric NS equations where the parameter represents different boundary conditions. Our goal is to simulate blood flow as the boundary conditions vary.
Data-driven reduced order models to simulate hemodynamics in CABG and left atrium
Numerical simulations of blood flow dynamics may provide practical support to doctors as long as they are fast enough to run in real (or nearly real) time. The literature on numerical methods to discretize the NS equations is well established and includes, for instance, finite volumes and finite element methods. The solutions recovered through these techniques are called high-fidelity solutions to the full order model (FOM). However, as the number of cells of the mesh grows, the computational time required to solve the FOM becomes easily demanding.
The main challenge we propose here is the reduction of computational times without affecting the accuracy of the results. Our approach is based on data-driven reduced order models (ROM) [5], which are non-intrusive methods able to reconstruct unknown dynamics from a dataset of high-fidelity solutions. A ROM solution is defined as a linear combination between reduced bases and modal coefficients. The most common technique used to recover the reduced basis space is the proper orthogonal decomposition [6], which extracts the essential information from the space generated by the high-fidelity solutions. Several methodologies can be used to evaluate the modal coefficients, such as interpolations, regressions or neural networks.
For the optimal control problem on the CABG the modal coefficients are computed through feedforward neural networks (NNs). Let us consider a set of parameters used for training the NN and the corresponding FOM solutions. By calibrating the NN, the latter learns from data and is then able to build a map from the set of parameters to the set of modal coefficients. In this way, we can compute the modal coefficients corresponding to new parameters and recover the reduced solution. The advantage of using NNs mainly involves the computational cost. Indeed, we get a FOM-to-ROM speed up of the order of 1e+06, which is 4 times higher than the one obtained using the POD-Galerkin approach proposed in [1]. In the figure below, we show a qualitative comparison between ROM and FOM solutions, corresponding to an accuracy order of 1e−04.
Simulating the hemodynamics in the left atrium is more challenging since the time-dependent NS equations are required and the computational cost becomes easily unmanageable. In this case, to compute the modal coefficients, we choose an interpolation technique based on radial basis functions. The computational gain is of the order of 1e+02 while the accuracy is of the order of 1e–01. In the figure below we compare the evolution in time between FOM and ROM solutions of the blood age, a variable used to determine the stasis. We focus only on the appendage, since it is the most relevant part of the left atrium. The obtained results are very similar from a qualitative point of view.
Conclusions
A data-driven and non-intrusive reduced order model is adopted for the study of the haemodynamics for two different heart diseases. As expected, the speed-up achieved during the online phase is very large and the level of accuracy of the ROM is comparable with intrusive approaches. It represents a good starting point to develop a platform for a real-time evaluation of different settings directly in the hospital.
References
[1] Z. Zainib, F. Ballarin, S. Fremes, P. Triverio, L. Jiménez-Juan, G. Rozza. Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation. Int. J. Numer. Methods Biomed. Eng. 37 (2021).
[2] C. Balzotti, P. Siena, M. Girfoglio, A. Quaini, G. Rozza. A data-driven Reduced Order Method for parametric optimal blood flow control: application to coronary bypass graft. To appear in Commun. Optim. Theory (2022).
[3] J. Dueñas-Pamplona, J. G. García, F. Castro, J. Muñoz-Paniagua, J. Goicolea, J. Sierra-Pallares. Morphing the left atrium geometry: a deeper insight into blood stasis within the left atrial appendage. Appl. Math. Model. 108 (2022).
[4] J. Dueñas-Pamplona, J. G. García, J. Sierra-Pallares, C. Ferrera, R. Agujetas, J.R. López-Mínguez. A comprehensive comparison of various patient-specific CFD models of the left atrium for atrial fibrillation patients. Comput. Biol. Med. 133 (2021).
[5] J. S. Hesthaven, G. Rozza, and B. Stamm. Certified reduced basis methods for parametrized partial differential equations. Springer (2016).
[6] A. Chatterjee. An introduction to the proper orthogonal decomposition. Curr. Sci. (2000).
