Reduced Order Modeling for Real-Time Stent Deformation Simulations of Transcatheter Aortic Valve Prostheses.

Computational modeling can be a critical tool to predict deployment behavior for transcatheter aortic valve replacement (TAVR) in patients with aortic stenosis. However, due to the mechanical complexity of the aortic valve and the multiphysics nature of the problem, described by partial differential equations (PDEs), traditional finite element (FE) modeling of TAVR deployment is computationally expensive. In this preliminary study, a PDEs-based reduced order modeling (ROM) framework is introduced for rapidly simulating structural deformation of the Medtronic Evolut R valve stent frame. Using fifteen probing points from an Evolut model with parametrized loads enforced, 105 FE simulations were performed in the so-called offline phase, creating a snapshot library. The library was used in the online phase of the ROM for a new set of applied loads via the proper orthogonal decomposition-Galerkin (POD-Galerkin) approach. Simulations of small radial deformations of the Evolut stent frame were performed and compared to full order model (FOM) solutions. Linear elastic and hyperelastic constitutive models in steady and unsteady regimes were implemented within the ROM. Since the original POD-Galerkin method is formulated for linear problems, specific methods for the nonlinear terms in the hyperelastic case were employed, namely, the Discrete Empirical Interpolation Method. The ROM solutions were in strong agreement with the FOM in all numerical experiments, with a speed-up of at least 92% in CPU Time. This framework serves as a first step toward real-time predictive models for TAVR deployment simulations.

Dynamic Coronary Blood Flow Velocity and Wall Shear Stress Estimation Using Ultrasound in an Ex Vivo Porcine Heart.

PURPOSE: Wall shear stress (WSS) is a critically important physical factor contributing to atherosclerosis. Mapping the spatial distribution of local, oscillatory WSS can identify important mechanisms underlying the progression of coronary artery disease. METHODS: In this study, blood flow velocity and time-varying WSS were estimated in the left anterior descending (LAD) coronary artery of an ex vivo beating porcine heart using ultrasound with an 18 MHz linear array transducer aligned with the LAD in a forward-viewing orientation. A pulsatile heart loop with physiologically-accurate flow was created using a pulsatile pump. The coronary artery wall motion was compensated using a local block matching technique. Next, 2D and 3D velocity magnitude and WSS maps in the LAD coronary artery were estimated at different time points in the cardiac cycle using an ultrafast Doppler approach. The blood flow velocity estimated using the presented approach was compared with a commercially-available, calibrated single element blood flow velocity measurement system. RESULTS: The resulting root mean square error (RMSE) of 2D velocity magnitude acquired from a high frequency, linear array transducer was less than 8% of the maximum velocity estimated by the commercial system. CONCLUSION: When implemented in a forward-viewing intravascular ultrasound device, the presented approach will enable dynamic estimation of WSS, an indicator of plaque vulnerability in coronary arteries.

Stent underexpansion is associated with high wall shear stress: a biomechanical analysis of the shear stent study.

Coronary stent underexpansion is associated with restenosis and stent thrombosis. In clinical studies of atherosclerosis, high wall shear stress (WSS) has been associated with activation of prothrombotic pathways, upregulation of matrix metalloproteinases, and future myocardial infarction. We hypothesized that stent underexpansion is predictive of high WSS. WSS distribution was investigated in patients enrolled in the prospective randomized controlled study of angulated coronary arteries randomized to undergo percutaneous coronary intervention with R-ZES or X-EES. WSS was calculated from 3D reconstructions of arteries from intravascular ultrasound (IVUS) and angiography using computational fluid dynamics. A logistic regression model investigated the relationship between WSS and underexpansion and the relationship between underexpansion and stent platform. Mean age was 63±11, 78% were male, 35% had diabetes, mean pre-stent angulation was 36.7°±14.7°. Underexpansion was assessed in 83 patients (6,181 IVUS frames). Frames with stent underexpansion were significantly more likely to exhibit high WSS (> 2.5 Pa) compared to those without underexpansion with an OR of 2.197 (95% CI = [1.233-3.913], p = 0.008). There was no significant association between underexpansion and low WSS (< 1.0 Pa) and no significant differences in underexpansion between R-ZES and X-EES. In the Shear Stent randomized controlled study, underexpanded IVUS frames were more than twice as likely to be associated with high WSS than frames without underexpansion.

Semi-Automatic Reconstruction of Patient-Specific Stented Coronaries based on Data Assimilation and Computer Aided Design.

PURPOSE: The interplay between geometry and hemodynamics is a significant factor in the development of cardiovascular diseases. This is particularly true for stented coronary arteries. To elucidate this factor, an accurate patient-specific analysis requires the reconstruction of the geometry following the stent deployment for a computational fluid dynamics (CFD) investigation. The image-based reconstruction is troublesome for the different possible positions of the stent struts in the lumen and the coronary wall. However, the accurate inclusion of the stent footprint in the hemodynamic analysis is critical for detecting abnormal stress conditions and flow disturbances, particularly for thick struts like in bioresorbable scaffolds. Here, we present a novel reconstruction methodology that relies on Data Assimilation and Computer Aided Design. METHODS: The combination of the geometrical model of the undeployed stent and image-based data assimilated by a variational approach allows the highly automated reconstruction of the skeleton of the stent. A novel approach based on computational mechanics defines the map between the intravascular frame of reference (called L-view) and the 3D geometry retrieved from angiographies. Finally, the volumetric expansion of the stent skeleton needs to be self-intersection free for the successive CFD studies; this is obtained by using implicit representations based on the definition of Nef-polyhedra. RESULTS: We assessed our approach on a vessel phantom, with less than 10% difference (properly measured) vs. a customized manual (and longer) procedure previously published, yet with a significant higher level of automation and a shorter turnaround time. Computational hemodynamics results were even closer. We tested the approach on two patient-specific cases as well. CONCLUSIONS: The method presented here has a high level of automation and excellent accuracy performances, so it can be used for larger studies involving patient-specific geometries.

Biomechanics of Transcatheter Aortic Valve Replacement Complications and Computational Predictive Modeling.

Transcatheter aortic valve replacement (TAVR) is a rapidly growing field enabling replacement of diseased aortic valves without the need for open heart surgery. However, due to the nature of the procedure and nonremoval of the diseased tissue, there are rates of complications ranging from tissue rupture and coronary obstruction to paravalvular leak, valve thrombosis, and permanent pacemaker implantation. In recent years, computational modeling has shown a great deal of promise in its capabilities to understand the biomechanical implications of TAVR as well as help preoperatively predict risks inherent to device-patient-specific anatomy biomechanical interaction. This includes intricate replication of stent and leaflet designs and tested and validated simulated deployments with structural and fluid mechanical simulations. This review outlines current biomechanical understanding of device-related complications from TAVR and related predictive strategies using computational modeling. An outlook on future modeling strategies highlighting reduced order modeling which could significantly reduce the high time and cost that are required for computational prediction of TAVR outcomes is presented in this review paper. A summary of current commercial/in-development software is presented in the final section.

A multi-domain shear-stress dependent diffusive model of cell transport-aided dialysis: analysis and simulation.

Kidney dialysis is the most widespread treatment method for end-stage renal disease, a debilitating health condition common in industrialized societies. While ubiquitous, kidney dialysis suffers from an inability to remove larger toxins, resulting in a gradual buildup of these toxins in dialysis patients, ultimately leading to further health complications. To improve dialysis, hollow fibers incorporating a cell-monolayer with cultured kidney cells have been proposed; however, the design of such a fiber is nontrivial. In particular, the effects of fluid wall-shear stress have an important influence on the ability of the cell layer to transport toxins. In the present work, we introduce a model for cell-transport aided dialysis, incorporating the effects of the shear stress. We analyze the model mathematically and establish its well-posedness. We then present a series of numerical results, which suggest that a hollow-fiber design with a wavy profile may increase the efficiency of the dialysis treatment. We investigate numerically the shape of the wavy channel to maximize the toxin clearance. These results demonstrate the potential for the use of computational models in the study and advancement of renal therapies.

Global Sensitivity Analysis for Patient-Specific Aortic Simulations: The Role of Geometry, Boundary Condition and Large Eddy Simulation Modeling Parameters.

Numerical simulations for computational hemodynamics in clinical settings require a combination of many ingredients, mathematical models, solvers and patient-specific data. The sensitivity of the solutions to these factors may be critical, particularly when we have a partial or noisy knowledge of data. Uncertainty quantification is crucial to assess the reliability of the results. We present here an extensive sensitivity analysis in aortic flow simulations, to quantify the dependence of clinically relevant quantities to the patient-specific geometry and the inflow boundary conditions. Geometry and inflow conditions are generally believed to have a major impact on numerical simulations. We resort to a global sensitivity analysis, (i.e., not restricted to a linearization around a working point), based on polynomial chaos expansion (PCE) and the associated Sobol' indices. We regard the geometry and the inflow conditions as the realization of a parametric stochastic process. To construct a physically consistent stochastic process for the geometry, we use a set of longitudinal-in-time images of a patient with an abdominal aortic aneurysm (AAA) to parametrize geometrical variations. Aortic flow is highly disturbed during systole. This leads to high computational costs, even amplified in a sensitivity analysis -when many simulations are needed. To mitigate this, we consider here a large Eddy simulation (LES) model. Our model depends in particular on a user-defined parameter called filter radius. We borrowed the tools of the global sensitivity analysis to assess the sensitivity of the solution to this parameter too. The targeted quantities of interest (QoI) include: the total kinetic energy (TKE), the time-average wall shear stress (TAWSS), and the oscillatory shear index (OSI). The results show that these indexes are mostly sensitive to the geometry. Also, we find that the sensitivity may be different during different instants of the heartbeat and in different regions of the domain of interest. This analysis helps to assess the reliability of in silico tools for clinical applications.

Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion.

We present an early version of a Susceptible-Exposed-Infected-Recovered-Deceased (SEIRD) mathematical model based on partial differential equations coupled with a heterogeneous diffusion model. The model describes the spatio-temporal spread of the COVID-19 pandemic, and aims to capture dynamics also based on human habits and geographical features. To test the model, we compare the outputs generated by a finite-element solver with measured data over the Italian region of Lombardy, which has been heavily impacted by this crisis between February and April 2020. Our results show a strong qualitative agreement between the simulated forecast of the spatio-temporal COVID-19 spread in Lombardy and epidemiological data collected at the municipality level. Additional simulations exploring alternative scenarios for the relaxation of lockdown restrictions suggest that reopening strategies should account for local population densities and the specific dynamics of the contagion. Thus, we argue that data-driven simulations of our model could ultimately inform health authorities to design effective pandemic-arresting measures and anticipate the geographical allocation of crucial medical resources.

Fluid-Structure Interaction Simulation of an Intra-Atrial Fontan Connection.

Total cavopulmonary connection (TCPC) hemodynamics has been hypothesized to be associated with long-term complications in single ventricle heart defect patients. Rigid wall assumption has been commonly used when evaluating TCPC hemodynamics using computational fluid dynamics (CFD) simulation. Previous study has evaluated impact of wall compliance on extra-cardiac TCPC hemodynamics using fluid-structure interaction (FSI) simulation. However, the impact of ignoring wall compliance on the presumably more compliant intra-atrial TCPC hemodynamics is not fully understood. To narrow this knowledge gap, this study aims to investigate impact of wall compliance on an intra-atrial TCPC hemodynamics. A patient-specific model of an intra-atrial TCPC is simulated with an FSI model. Patient-specific 3D TCPC anatomies were reconstructed from transverse cardiovascular magnetic resonance images. Patient-specific vessel flow rate from phase-contrast magnetic resonance imaging (MRI) at the Fontan pathway and the superior vena cava under resting condition were prescribed at the inlets. From the FSI simulation, the degree of wall deformation was compared with in vivo wall deformation from phase-contrast MRI data as validation of the FSI model. Then, TCPC flow structure, power loss and hepatic flow distribution (HFD) were compared between rigid wall and FSI simulation. There were differences in instantaneous pressure drop, power loss and HFD between rigid wall and FSI simulations, but no difference in the time-averaged quantities. The findings of this study support the use of a rigid wall assumption on evaluation of time-averaged intra-atrial TCPC hemodynamic metric under resting breath-held condition.

Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study.

The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models, in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings.

Non-Newtonian Effects on Patient-Specific Modeling of Fontan Hemodynamics.

The Fontan procedure is a common palliative surgery for congenital single ventricle patients. In silico and in vitro patient-specific modeling approaches are widely utilized to investigate potential improvements of Fontan hemodynamics that are related to long-term complications. However, there is a lack of consensus regarding the use of non-Newtonian rheology, warranting a systematic investigation. This study conducted in silico patient-specific modeling for twelve Fontan patients, using a Newtonian and a non-Newtonian model for each patient. Differences were quantified by examining clinically relevant metrics: indexed power loss (iPL), indexed viscous dissipation rate (iVDR), hepatic flow distribution (HFD), and regions of low wall shear stress (AWSS). Four sets of "non-Newtonian importance factors" were calculated to explore their effectiveness in identifying the non-Newtonian effect. No statistical differences were observed in iPL, iVDR, and HFD between the two models at the population-level, but large inter-patient variations exist. Significant differences were detected regarding AWSS, and its correlations with non-Newtonian importance factors were discussed. Additionally, simulations using the non-Newtonian model were computationally faster than those using the Newtonian model. These findings distinguish good importance factors for identifying non-Newtonian rheology and encourage the use of a non-Newtonian model to assess Fontan hemodynamics.

Analysis of Inlet Velocity Profiles in Numerical Assessment of Fontan Hemodynamics.

Computational fluid dynamic (CFD) simulations are widely utilized to assess Fontan hemodynamics that are related to long-term complications. No previous studies have systemically investigated the effects of using different inlet velocity profiles in Fontan simulations. This study implements real, patient-specific velocity profiles for numerical assessment of Fontan hemodynamics using CFD simulations. Four additional, artificial velocity profiles were used for comparison: (1) flat, (2) parabolic, (3) Womersley, and (4) parabolic with inlet extensions [to develop flow before entering the total cavopulmonary connection (TCPC)]. The differences arising from the five velocity profiles, as well as discrepancies between the real and each of the artificial velocity profiles, were quantified by examining clinically important metrics in TCPC hemodynamics: power loss (PL), viscous dissipation rate (VDR), hepatic flow distribution, and regions of low wall shear stress. Statistically significant differences were observed in PL and VDR between simulations using real and flat velocity profiles, but differences between those using real velocity profiles and the other three artificial profiles did not reach statistical significance. These conclusions suggest that the artificial velocity profiles (2)-(4) are acceptable surrogates for real velocity profiles in Fontan simulations, but parabolic profiles are recommended because of their low computational demands and prevalent applicability.

The effect of respiration-driven flow waveforms on hemodynamic metrics used in Fontan surgical planning.

OBJECTIVE: Poor total cavopulmonary connection (TCPC) hemodynamics have been hypothesized to be associated with long-term complications in Fontan patients. Image-based Fontan surgical planning has shown great potential as a clinical tool because it can pre-operatively evaluate patient-specific hemodynamics. Current surgical planning paradigms commonly utilize cardiac-gated phase contrast magnetic resonance (MR) imaging to acquire vessel flows. These acquisitions are often taken under breath-held (BH) conditions and ignore the effect of respiration on blood flow waveforms. This study investigates the effect of respiration-driven flow waveforms on patient-specific hemodynamics using real-time MR acquisitions. METHODS: Patient-specific TCPCs were reconstructed from cardiovascular MR images. Real-time phase contrast MR images were acquired under both free-breathing (FB) and breath-held conditions for 9 patients. Numerical simulations were employed to assess flow structures and hemodynamics used in Fontan surgical planning including hepatic flow distribution (HFD) and indexed power loss (iPL), which were then compared between FB and BH conditions. RESULTS: Differences in TCPC flow structures between FB and BH conditions were observed throughout the respiratory cycle. However, the average differences (BH - FB values for each patient, which are then averaged) in iPL and HFD between these conditions were 0.002 ±â€¯0.011 (p = 0.40) and 1 ±â€¯3% (p = 0.28), respectively, indicating no significant difference in clinically important hemodynamic metrics. CONCLUSIONS: Respiration affects blood flow waveforms and flow structures, but might not significantly influence the values of iPL or HFD. Therefore, breath-held MR acquisition can be adequate for Fontan surgical planning when focusing on iPL and HFD.

High Coronary Shear Stress in Patients With Coronary Artery Disease Predicts Myocardial Infarction.

BACKGROUND: Coronary lesions with low fractional flow reserve (FFR) that are treated medically are associated with higher revascularization rates. High wall shear stress (WSS) has been linked with increased plaque vulnerability. OBJECTIVES: This study investigated the prognostic value of WSS measured in the proximal segments of lesions (WSSprox) to predict myocardial infarction (MI) in patients with stable coronary artery disease (CAD) and hemodynamically significant lesions. The authors hypothesized that in patients with low FFR and stable CAD, higher WSSprox would predict MI. METHODS: Among 441 patients in the FAME II (Fractional Flow Reserve Versus Angiography for Multivessel Evaluation II) trial with FFR ≤0.80 who were randomized to medical therapy alone, 34 (8%) had subsequent MI within 3 years. Patients with vessel-related MI and adequate angiograms for 3-dimensional reconstruction (n = 29) were propensity matched to a control group with no MI (n = 29) by using demographic and clinical variables. Coronary lesions were divided into proximal, middle, and distal, along with 5-mm upstream and downstream segments. WSS was calculated for each segment. RESULTS: Median age was 62 years, and 46 (79%) were male. In the marginal Cox model, whereas lower FFR showed a trend (hazard ratio: 0.084; p = 0.064), higher WSSprox (hazard ratio: 1.234; p = 0.002, C-index = 0.65) predicted MI. Adding WSSprox to FFR resulted in a significant increase in global chi-square for predicting MI (p = 0.045), a net reclassification improvement of 0.69 (p = 0.005), and an integrated discrimination index of 0.11 (p = 0.010). CONCLUSIONS: In patients with stable CAD and hemodynamically significant lesions, higher WSS in the proximal segments of atherosclerotic lesions is predictive of MI and has incremental prognostic value over FFR.

Low Coronary Wall Shear Stress Is Associated With Severe Endothelial Dysfunction in Patients With Nonobstructive Coronary Artery Disease.

OBJECTIVES: This study investigated the relationship between low wall shear stress (WSS) and severe endothelial dysfunction (EDFx). BACKGROUND: Local hemodynamic forces such as WSS play an important role in atherogenesis through their effect on endothelial cells. The study hypothesized that low WSS independently predicts severe EDFx in patients with coronary artery disease (CAD). METHODS: Forty-four patients with CAD underwent coronary angiography, fractional flow reserve, and endothelial function testing. Segments with >10% vasoconstriction after acetylcholine (Ach) infusion were defined as having severe EDFx. WSS, calculated using 3-dimensional angiography, velocity measurements, and computational fluid dynamics, was defined as low (<1 Pa), intermediate (1 to 2.5 Pa), or high (>2.5 Pa). RESULTS: Median age was 52 years, 73% were women. Mean fractional flow reserve was 0.94 ± 0.06. In 4,510 coronary segments, median WSS was 3.67 Pa. A total of 24% had severe EDFx. A higher proportion of segments with low WSS had severe EDFx (71%) compared with intermediate WSS (22%) or high WSS (23%) (p < 0.001). Segments with low WSS demonstrated greater vasoconstriction in response to Ach than did intermediate or high WSS segments (-10.7% vs. -2.5% vs. +1.3%, respectively; p < 0.001). In a multivariable logistic regression analysis, female sex (odds ratio [OR]: 2.44; p = 0.04), diabetes (OR: 5.01; p = 0.007), and low WSS (OR: 9.14; p < 0.001) were independent predictors of severe EDFx. CONCLUSIONS: In patients with nonobstructive CAD, segments with low WSS demonstrated more vasoconstriction in response to Ach than did intermediate or high WSS segments. Low WSS was independently associated with severe EDFx.

Patient-specific CFD modelling in the thoracic aorta with PC-MRI-based boundary conditions: A least-square three-element Windkessel approach.

The increasing use of computational fluid dynamics for simulating blood flow in clinics demands the identification of appropriate patient-specific boundary conditions for the customization of the mathematical models. These conditions should ideally be retrieved from measurements. However, finite resolution of devices as well as other practical/ethical reasons prevent the construction of complete data sets necessary to make the mathematical problems well posed. Available data need to be completed by modelling assumptions, whose impact on the final solution has to be carefully addressed. Focusing on aortic vascular districts and related pathologies, we present here a method for efficiently and robustly prescribing phase contrast MRI-based patient-specific data as boundary conditions at the domain of interest. In particular, for the outlets, the basic idea is to obtain pressure conditions from an appropriate elaboration of available flow rates on the basis of a 3D/0D dimensionally heterogeneous modelling. The key point is that the parameters are obtained by a constrained optimization procedure. The rationale is that pressure conditions have a reduced impact on the numerical solution compared with velocity conditions, yielding a simulation framework less exposed to noise and inconsistency of the data, as well as to the arbitrariness of the underlying modelling assumptions. Numerical results confirm the reliability of the approach in comparison with other patient-specific approaches adopted in the literature.

Coupled Morphological-Hemodynamic Computational Analysis of Type B Aortic Dissection: A Longitudinal Study.

Progressive false lumen aneurysmal degeneration in type B aortic dissection (TBAD) is a complex process with a multi-factorial etiology. Patient-specific computational fluid dynamics (CFD) simulations provide spatial and temporal hemodynamic quantities that facilitate understanding this disease progression. A longitudinal study was performed for a TBAD patient, who was diagnosed with the uncomplicated TBAD in 2006 and treated with optimal medical therapy but received surgery in 2010 due to late complication. Geometries of the aorta in 2006 and 2010 were reconstructed. With registration algorithms, we accurately quantified the evolution of the false lumen, while with CFD simulations we computed several hemodynamic indexes, including the wall shear stress (WSS), and the relative residence time (RRT). The numerical fluid model included large eddy simulation (LES) modeling for efficiently capturing the flow disturbances induced by the entry tears. In the absence of complete patient-specific data, the boundary conditions were based on a specific calibration method. Correlations between hemodynamics and the evolution field in time obtained by registration of the false lumen are discussed. Further testing of this methodology on a large cohort of patients may enable the use of CFD to predict whether patients, with originally uncomplicated TBAD, develop late complications.

Numerical methods for polyline-to-point-cloud registration with applications to patient-specific stent reconstruction.

We present novel numerical methods for polyline-to-point-cloud registration and their application to patient-specific modeling of deployed coronary artery stents from image data. Patient-specific coronary stent reconstruction is an important challenge in computational hemodynamics and relevant to the design and improvement of the prostheses. It is an invaluable tool in large-scale clinical trials that computationally investigate the effect of new generations of stents on hemodynamics and eventually tissue remodeling. Given a point cloud of strut positions, which can be extracted from images, our stent reconstruction method aims at finding a geometrical transformation that aligns a model of the undeployed stent to the point cloud. Mathematically, we describe the undeployed stent as a polyline, which is a piecewise linear object defined by its vertices and edges. We formulate the nonlinear registration as an optimization problem whose objective function consists of a similarity measure, quantifying the distance between the polyline and the point cloud, and a regularization functional, penalizing undesired transformations. Using projections of points onto the polyline structure, we derive novel distance measures. Our formulation supports most commonly used transformation models including very flexible nonlinear deformations. We also propose 2 regularization approaches ensuring the smoothness of the estimated nonlinear transformation. We demonstrate the potential of our methods using an academic 2D example and a real-life 3D bioabsorbable stent reconstruction problem. Our results show that the registration problem can be solved to sufficient accuracy within seconds using only a few number of Gauss-Newton iterations.

Numerical sensitivity analysis of a variational data assimilation procedure for cardiac conductivities.

An accurate estimation of cardiac conductivities is critical in computational electro-cardiology, yet experimental results in the literature significantly disagree on the values and ratios between longitudinal and tangential coefficients. These are known to have a strong impact on the propagation of potential particularly during defibrillation shocks. Data assimilation is a procedure for merging experimental data and numerical simulations in a rigorous way. In particular, variational data assimilation relies on the least-square minimization of the misfit between simulations and experiments, constrained by the underlying mathematical model, which in this study is represented by the classical Bidomain system, or its common simplification given by the Monodomain problem. Operating on the conductivity tensors as control variables of the minimization, we obtain a parameter estimation procedure. As the theory of this approach currently provides only an existence proof and it is not informative for practical experiments, we present here an extensive numerical simulation campaign to assess practical critical issues such as the size and the location of the measurement sites needed for in silico test cases of potential experimental and realistic settings. This will be finalized with a real validation of the variational data assimilation procedure. Results indicate the presence of lower and upper bounds for the number of sites which guarantee an accurate and minimally redundant parameter estimation, the location of sites being generally non critical for properly designed experiments. An effective combination of parameter estimation based on the Monodomain and Bidomain models is tested for the sake of computational efficiency. Parameter estimation based on the Monodomain equation potentially leads to the accurate computation of the transmembrane potential in real settings.

Transversally enriched pipe element method (TEPEM): An effective numerical approach for blood flow modeling.

In this work, we present a novel approach tailored to approximate the Navier-Stokes equations to simulate fluid flow in three-dimensional tubular domains of arbitrary cross-sectional shape. The proposed methodology is aimed at filling the gap between (cheap) one-dimensional and (expensive) three-dimensional models, featuring descriptive capabilities comparable with the full and accurate 3D description of the problem at a low computational cost. In addition, this methodology can easily be tuned or even adapted to address local features demanding more accuracy. The numerical strategy employs finite (pipe-type) elements that take advantage of the pipe structure of the spatial domain under analysis. While low order approximation is used for the longitudinal description of the physical fields, transverse approximation is enriched using high order polynomials. Although our application of interest is computational hemodynamics and its relevance to pathological dynamics like atherosclerosis, the approach is quite general and can be applied in any internal fluid dynamics problem in pipe-like domains. Numerical examples covering academic cases as well as patient-specific coronary arterial geometries demonstrate the potentialities of the developed methodology and its performance when compared against traditional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.