Multiscale modelling in biofluidynamics: Application to reconstructive paediatric cardiac surgery

Modeling single ventricle physiology: review of engineering tools to study first stage palliation of hypoplastic left heart syndrome

First stage palliation of hypoplastic left heart syndrome, i.e., the Norwood operation, results in a complex physiological arrangement, involving different shunting options (modified Blalock-Taussig, RV-PA conduit, central shunt from the ascending aorta) and enlargement of the hypoplastic ascending aorta. Engineering techniques, both computational and experimental, can aid in the understanding of the Norwood physiology and their correct implementation can potentially lead to refinement of the decision-making process, by means of patient-specific simulations. This paper presents some of the available tools that can corroborate clinical evidence by providing detailed insight into the fluid dynamics of the Norwood circulation as well as alternative surgical scenarios (i.e., virtual surgery). Patient-specific anatomies can be manufactured by means of rapid prototyping and such models can be inserted in experimental set-ups (mock circulatory loops) that can provide a valuable source of validation data as well as hydrodynamic information. Such models can be tuned to respond to differing the patient physiologies. Experimental set-ups can also be compatible with visualization techniques, like particle image velocimetry and cardiovascular magnetic resonance, further adding to the knowledge of the local fluid dynamics. Multi-scale computational models include detailed three-dimensional (3D) anatomical information coupled to a lumped parameter network representing the remainder of the circulation. These models output both overall hemodynamic parameters while also enabling to investigate the local fluid dynamics of the aortic arch or the shunt. As an alternative, pure lumped parameter models can also be employed to model Stage 1 palliation, taking advantage of a much lower computational cost, albeit missing the 3D anatomical component. Finally, analytical techniques, such as wave intensity analysis, can be employed to study the Norwood physiology, providing a mechanistic perspective on the ventriculo-arterial coupling for this specific surgical scenario.

Using of porous portion to simulate pulmonary resistance in the computational fluid dynamic models of Fontan connection

In this study, we performed computational fluid dynamic (CFD) simulations in a patient-specific three-dimensional extracardiac conduit Fontan connection. The pulmonary resistance was incorporated in the CFD model by connecting porous portions in the left and right pulmonary arteries. The pressure in the common atrium was set as boundary conditions at the outlets of the pulmonary arteries. The flow rate in the innominate veins and the inferior vena cava (IVC) was set as inflow boundary conditions. Furthermore, the inflow rate of IVC was increased to 2 and 3 times of that measured to perform another two simulations and the resistance provided by the porous portions was compared among these three conditions. We found out that the pulmonary resistance set as porous portion in the CFD models remains relatively steady despite the change of the inflow rate. We concluded that, in the CFD simulations for the Fontan connections, porous portion could be used to represent pulmonary resistance steadily. The pulmonary resistance and pressure in the common atrium could be acquired directly by clinical examination. The employment of porous portion together with pressure in the common atrium in the CFD model could facilitate and accurate the set of outlet boundary conditions especially for those actual pulmonary flow splits was unpredictable such as virtual operative designs related CFD simulations.

Numerical blood flow simulation in surgical corrections: what do we need for an accurate analysis?

BACKGROUND

Computational fluid dynamics has been increasingly used in congenital heart surgery to simulate pathophysiological blood flow, investigate surgical options, or design medical devices. Several commercial and research computational or numerical codes have been developed. They present different approaches to numerically solve the blood flow equations, raising the question whether these numerical codes are equally reliable to achieve accurate simulation results. Accordingly, we sought to examine the influence of numerical code selection in several complex congenital cardiac operations.

MATERIAL AND METHODS

The main steps of blood flow simulations are detailed (geometrical mesh, boundary conditions, and solver numerical methods) for congenital cardiac operations of increasing complexity. The first case tests different numerical solutions against an analytical, or exact, solution. In the second case, the three-dimensional domain is a patient-specific superior cavopulmonary anastomosis. As an analytical solution does not exist in such a complex geometry, different numerical solutions are compared. Finally, a realistic case of a systemic-to-pulmonary shunt is presented with both geometrically and physiologically challenging conditions. For all, solutions from a commercially available code and an open-source research code are compared.

RESULTS

In the first case, as the mesh or solver numerical method is refined, the simulation results for both codes converged to the analytical solution. In the second example, velocity differences between the two codes are greater when the resolution of the mesh were lower and less refined. The third case with realistic anatomy reveals that the pulsatile complex flow is very similar for both codes.

CONCLUSIONS

The precise setup of the numerical cases has more influence on the results than the choice of numerical codes. The need for detailed construction of the numerical model that requires high computational cost depends on the precision needed to answer the biomedical question at hand and should be assessed for each problem on a combination of clinically relevant patient-specific geometry and physiological conditions.

Lagrangian analysis of hemodynamics data from FSI simulation

We present the computation of lagrangian-based flow characterization measures for time-dependent, deformable-wall, finite-element blood flow simulations. Applicability of the algorithm is demonstrated in a fluid-structure interaction simulation of blood flow through a total cavopulmonary connection (Fontan procedure), and results are compared with a rigid-vessel simulation. Specifically, we report on several important lagrangian-based measures including flow distributions, finite-time Lyapunov exponent fields, particle residence time, and exposure time calculations. Overall, strong similarity in lagrangian measures of the flow between deformable and rigid-vessel models was observed.

Hemodynamics analysis of patient-specific carotid bifurcation: a CFD model of downstream peripheral vascular impedance

The study of cardiovascular models was presented in this paper based on medical image reconstruction and computational fluid dynamics. Our aim is to provide a reality platform for the purpose of flow analysis and virtual intervention outcome predication for vascular diseases. By connecting two porous mediums with transient permeability at the downstream of the carotid bifurcation branches, a downstream peripheral impedance model was developed, and the effect of the downstream vascular bed impedance can be taken into consideration. After verifying its accuracy with a healthy carotid bifurcation, this model was implemented in a diseased carotid bifurcation analysis. On the basis of time-averaged wall shear stress, oscillatory shear index, and the relative residence time, fractions of abnormal luminal surface were highlighted, and the atherosclerosis was assessed from a hemodynamic point of view. The effect of the atherosclerosis on the transient flow division between the two branches because of the existence of plaque was also analysed. This work demonstrated that the proposed downstream peripheral vascular impedance model can be used for computational modelling when the outlets boundary conditions are not available, and successfully presented the potential of using medical imaging and numerical simulation to provide existing clinical prerequisites for diagnosis and therapeutic treatment.

Computational modelling to optimize the hybrid configuration for hypoplastic left heart syndrome

OBJECTIVES

Hybrid palliation for hypoplastic left heart syndrome (HLHS) is associated with mortality and late ventricular dysfunction. Increased ventricular workload and coronary perfusion limitation may be the important factors. Using mathematical modelling, this study investigated the effects of differing hybrid configurations on the demands on this single ventricle circulation.

METHODS

A multicompartmental Windkessel model of hybrid HLH-aortic atresia circulation was adopted, with a time-varying elastance representing ventricular functionality. The effects of diameter increases in bilateral pulmonary artery bandings (PABs) (+0.5, 2.5-4 mm) and ductal stent (+1, 4-10 mm) on cardiovascular haemodynamics, systemic oxygenation and ventricular energetics were assessed.

RESULTS

Simulations showed that an increase in PAB diameter of 2.5-4 mm resulted in an increased Q (0.61-2.66), and diastolic stent backflow (-0.2 to -0.78 l/min) with reduced systemic perfusion (0.82-0.77 l/min) and diastolic pressures (48.3-41.2 mmHg). Arterial and venous saturations increased, SaO2 (%) was 62-88 and SvO(2) 41-65. To maintain mean systemic pressures, substantial increases in cardiac output (1.3-2.8 l/min) and ventricular stroke work (576-1360 mmHg ml) were required. A decrease in the ductal stent diameter over the range 10-7 mm had a negligible haemodynamic effect: reduced systemic systolic pressure (77-72 mmHg) and increase in ventricular stroke work (781-790 mmHg ml). When the ductal diameter was restricted to <7 mm, it resulted in a significant reduced systemic flow and increased stroke work. Optimal hybrid configuration was defined at PAB 3 mm and ductal stent ≥7 mm.

CONCLUSIONS

In this model, increasing the PAB diameter, or a stent diameter <7 mm, substantially increased single ventricle workload and reduced systemic perfusion and diastolic pressure. This may compromise myocardial oxygen demand-supply, particularly in the setting of retrograde-dependent coronary perfusion.

Computational fluid dynamics models and congenital heart diseases

Mathematical modeling is a powerful tool to investigate hemodynamics of the circulatory system. With improving imaging techniques and detailed clinical investigations, it is now possible to construct patient-specific models of reconstructive surgeries for the treatment of congenital heart diseases. These models can help clinicians to better understand the hemodynamic behavior of different surgical options for a treated patient. This review outlines recent advances in mathematical modeling in congenital heart diseases, the discoveries and limitations these models present, and future directions that are on the horizon.

Treatment planning for a TCPC test case: a numerical investigation under rigid and moving wall assumptions

The hemodynamics in patients with total cavopulmonary connections (TCPC) is generally very complex and characterized by patient-to-patient variability. To better understand its effect on patients' outcome, CFD models are widely used, also to test and optimize surgical options before their implementation. These models often assume rigid geometries, despite the motion experienced by thoracic vessels that could influence the hemodynamics predictions. By improving their accuracy and expanding the range of simulated interventions, the benefit of treatment planning for patients is expected to increase. We simulate three types of intervention on a patient-specific 3D model, and compare their predicted outcome with baseline condition: a decrease in pulmonary vascular resistance obtainable with medications; a surgical revision of the connection design; the introduction of a fenestration in the TCPC wall. The simulations are performed both with rigid wall assumption and including patient-specific TCPC wall motion, reconstructed from a 4DMRI dataset. The results show the effect of each option on clinically important metrics and highlight the impact of patient-specific wall motion. The largest differences between rigid and moving wall models are observed in measures of energetic efficiency of TCPC as well as in hepatic flow distribution and transit time of seeded particles through the connection.

COMPUTATIONAL FLUID DYNAMICS OF TWO PATIENT-SPECIFIC SYSTEMIC TO PULMONARY SHUNTS

Tetralogy of Fallot is the most common cyanotic congenital heart defect. For severe cases, inserting a systemic to pulmonary shunt, which distributes part of systemic artery blood into the pulmonary artery, is the preferable palliative surgery. Based on the computed tomography images and three-dimensional geometry technologies, two patient-specific anatomical options of systemic to pulmonary shunts including the aorta to pulmonary shunt (APS) and innominate artery to pulmonary shunt (IPS) have been simulated for computational fluid dynamics. The objective of this study was to predict the hemodynamics within the shunts and confirm, through patient-specific simulations, the shunt with the optimal performance. Results indicated that both options created high velocity gradients and pressure gradients at the proximal end of the shunts. Obvious flow recirculation appeared at the inner region near the proximal end of the shunts. Part of the reverse flow from the descending aorta, left subclavian artery, left carotid artery and innominate artery was driven into the shunts during the diastolic period. The IPS provided better balanced and more adequate blood flow distributions between the systemic and pulmonary circulations. The APS provided slightly excessive pulmonary blood flow which can ultimately result in cardiac failure and pulmonary hypertension.

Velocity profiles in the human ductus venosus: a numerical fluid structure interaction study

The veins distributing oxygenated blood from the placenta to the fetal body have been given much attention in clinical Doppler velocimetry studies, in particular the ductus venosus. The ductus venosus is embedded in the left liver lobe and connects the intra-abdominal portion of the umbilical vein (IUV) directly to the inferior vena cava, such that oxygenated blood can bypass the liver and flow directly to the fetal heart. In the current work, we have developed a mathematical model to assist the clinical assessment of volumetric flow rate at the inlet of the ductus venosus. With a robust estimate of the velocity profile shape coefficient (VC), the volumetric flow rate may be estimated as the product of the time-averaged cross-sectional area, the time-averaged cross-sectional maximum velocity and the VC. The time average quantities may be obtained from Doppler ultrasound measurements, whereas the VC may be estimated from numerical simulations. The mathematical model employs a 3D fluid structure interaction model of the bifurcation formed by the IUV, the ductus venosus and the left portal vein. Furthermore, the amniotic portion of the umbilical vein, the right liver lobe and the inferior vena cava were incorporated as lumped model boundary conditions for the fluid structure interaction model. A hyperelastic material is used to model the structural response of the vessel walls, based on recently available experimental data for the human IUV and ductus venous. A parametric study was constructed to investigate the VC at the ductus venosus inlet, based on a reference case for a human fetus at 36 weeks of gestation. The VC was found to be [Formula: see text] (Mean [Formula: see text] SD of parametric case study), which confirms previous studies in the literature on the VC at the ductus venosus inlet. Additionally, CFD simulations with rigid walls were performed on a subsection of the parametric case study, and only minor changes in the predicted VCs were observed compared to the FSI cases. In conclusion, the presented mathematical model is a promising tool for the assessment of ductus venosus Doppler velocimetry.

Modeling hemodynamics in vascular networks using a geometrical multiscale approach: numerical aspects

On the one hand the heterogeneity of the circulatory system requires the use of different models in its different compartments, featuring different assumptions on the spatial degrees of freedom. On the other hand, the mutual interactions between its compartments imply that these models should preferably not be considered separately. These requirements have led to the concept of geometrical multiscale modeling, where the main idea is to couple 3D models with reduced 1D and/or 0D models. As such detailed information on the flow field in a specific region of interest can be obtained while accounting for the global circulation. However, the combination of models with different mathematical features gives rise to many difficulties such as the assignment of boundary conditions at the interface between two models and the development of robust coupling algorithms, as the subproblems are usually solved in a partitioned way. This review aims to give an overview of the most important aspects concerning 3D-1D-0D coupled models. In addition, some applications are presented in order to illustrate the potentialities of these coupled models.

Computational fluid dynamics in congenital heart disease

Computational fluid dynamics has been applied to the design, refinement, and assessment of surgical procedures and medical devices. This tool calculates flow patterns and pressure changes within a virtual model of the cardiovascular system. In the field of paediatric cardiac surgery, computational fluid dynamics is being used to elucidate the optimal approach to staged reconstruction of specific defects and study the haemodynamics of the resulting anatomical configurations after reconstructive or palliative surgery. In this paper, we review the techniques and principal findings of computational fluid dynamics studies as applied to a few representative forms of congenital heart disease.

A dimensionally-heterogeneous closed-loop model for the cardiovascular system and its applications

In the present work a computational model of the entire cardiovascular system is developed using heterogeneous mathematical representations. This model integrates different levels of detail for the blood circulation. The arterial tree is described by a one dimensional model in order to simulate the wave propagation phenomena that take place at the larger arterial vessels. The inflow and outflow locations of this 1D model are coupled with lumped parameter descriptions of the remainder part of the circulatory system, closing the loop. The four cardiac valves are considered using a valve model which allows for stenoses and regurgitation phenomena. In addition, full 3D geometrical models of arterial districts are embedded in this closed-loop circuit to model the local blood flow in specific vessels. This kind of detailed closed-loop network for the cardiovascular system allows hemodynamics analyses of patient-specific arterial district, delivering naturally the appropriate boundary conditions for different cardiovascular scenarios. An example of application involving the effect of aortic insufficiency on the local hemodynamics of a cerebral aneurism is provided as a motivation to reproduce, through numerical simulation, the hemodynamic environment in patients suffering from infective endocarditis and mycotic aneurisms. The need for incorporating homeostatic control mechanisms is also discussed in view of the large sensitivity observed in the results, noting that this kind of integrative modeling allows such incorporation.

A multiscale bidirectional coupling framework

The lung is geometrically articulated across multiple scales from the trachea to the alveoli. A major computational challenge is to tightly link ODEs that describe lower scales to 3D finite element or finite volume models of airway mechanics using iterative communication between scales. In this study, we developed a novel multiscale computational framework for bidirectionally coupling 3D CFD models and systems of lower order ODEs. To validate the coupling framework, a four and eight generation Weibel lung model was constructed. For the coupled CFD-ODE simulations, the lung models were truncated at different generations and a RL circuit represented the truncated portion. The flow characteristics from the coupled models were compared to untruncated full 3D CFD models at peak inhalation and peak exhalation. Results showed that at no time or simulation was the difference in mass flux and/or pressure at a given location between uncoupled and coupled models was greater than 2.43%. The flow characteristics at prime locations for the coupled models showed good agreement to uncoupled models. Remarkably, due to reuse of the Krylov subspace, the cost of the ODE coupling is not much greater than uncoupled full 3D-CFD computations with simple prescribed pressure values at the outlets.

Optimization of Shunt Placement for the Norwood Surgery Using Multi-Domain Modeling

An idealized systemic-to-pulmonary shunt anatomy is parameterized and coupled to a closed loop, lumped parameter network (LPN) in a multidomain model of the Norwood surgical anatomy. The LPN approach is essential for obtaining information on global changes in cardiac output and oxygen delivery resulting from changes in local geometry and physiology. The LPN is fully coupled to a custom 3D finite element solver using a semi-implicit approach to model the heart and downstream circulation. This closed loop multidomain model is then integrated with a fully automated derivative-free optimization algorithm to obtain optimal shunt geometries with variable parameters of shunt diameter, anastomosis location, and angles. Three objective functions: (1) systemic; (2) coronary; and (3) combined systemic and coronary oxygen deliveries are maximized. Results show that a smaller shunt diameter with a distal shunt-brachiocephalic anastomosis is optimal for systemic oxygen delivery, whereas a more proximal anastomosis is optimal for coronary oxygen delivery and a shunt between these two anatomies is optimal for both systemic and coronary oxygen deliveries. Results are used to quantify the origin of blood flow going through the shunt and its relationship with shunt geometry. Results show that coronary artery flow is directly related to shunt position.

The multi-scale modelling of coronary blood flow

Coronary flow is governed by a number of determinants including network anatomy, systemic afterload and the mechanical interaction with the myocardium throughout the cardiac cycle. The range of spatial scales and multi-physics nature of coronary perfusion highlights a need for a multiscale framework that captures the relevant details at each level of the network. The goal of this review is to provide a compact and accessible introduction to the methodology and current state of the art application of the modelling frameworks that have been used to study the coronary circulation. We begin with a brief description of the seminal experimental observations that have motivated the development of mechanistic frameworks for understanding how myocardial mechanics influences coronary flow. These concepts are then linked to an overview of the lumped parameter models employed to test these hypotheses. We then outline the full and reduced-order (3D and 1D) continuum mechanics models based on the Navier–Stokes equations and highlight, with examples, their application regimes. At the smaller spatial scales the case for the importance of addressing the microcirculation is presented, with an emphasis on the poroelastic approach that is well-suited to bridge an existing gap in the development of an integrated whole heart model. Finally, the recent accomplishments of the wave intensity analysis and related approaches are presented and the clinical outlook for coronary flow modelling discussed.

Rat airway morphometry measured from in situ MRI-based geometric models

Rodents have been widely used to study the environmental or therapeutic impact of inhaled particles. Knowledge of airway morphometry is essential in assessing geometric influence on aerosol deposition and in developing accurate lung models of aerosol transport. Previous morphometric studies of the rat lung performed ex situ provided high-resolution measurements (50-125 μm). However, it is unclear how the overall geometry of these casts might have differed from the natural in situ appearance. In this study, four male Wistar rat (268 ± 14 g) lungs were filled sequentially with perfluorocarbon and phosphate-buffered saline before being imaged in situ in a 7-T magnetic resonance (MR) scanner at a resolution of 0.2 × 0.2 × 0.27 mm. Airway length, diameter, gravitational, bifurcation, and rotational angles were measured for the first four airway generations from 3D geometric models built from the MR images. Minor interanimal variability [expressed by the relative standard deviation RSD (=SD/mean)] was found for length (0.18 ± 0.07), diameter (0.15 ± 0.15), and gravitational angle (0.12 ± 0.06). One rat model was extended to 16 airway generations. Organization of the airways using a diameter-defined Strahler ordering method resulted in lower interorder variability than conventional generation-based grouping for both diameter (RSD = 0.12 vs. 0.42) and length (0.16 vs. 0.67). Gravitational and rotational angles averaged 82.9 ± 37.9° and 53.6 ± 24.1°, respectively. Finally, the major daughter branch bifurcated at a smaller angle (19.3 ± 14.6°) than the minor branch (60.5 ± 19.4°). These data represent the most comprehensive set of rodent in situ measurements to date and can be used readily in computational studies of lung function and aerosol exposure.

In silico modelling of physiologic systems

In silico modelling, in which computer models are developed to model a pharmacologic or physiologic process, is a logical extension of controlled in vitro experimentation. It is the natural result of the explosive increase in computing power available to the research scientist at continually decreasing cost. In silico modelling combines the advantages of both in vivo and in vitro experimentation, without subjecting itself to the ethical considerations and lack of control associated with in vivo experiments. Unlike in vitro experiments, which exist in isolation, in silico models allow the researcher to include a virtually unlimited array of parameters, which render the results more applicable to the organism as a whole. In silico modelling is best known for its extensive use in pharmacokinetic experimentation, the best-known example of which is the development of the three-compartment model. In addition, complex in silico models have been applied to pathophysiological problems to provide information which cannot be obtained practically or ethically by traditional clinical research methods. These experiments have led to the development of significant insights in subject matters ranging from pure physiology to congenital heart surgery, obstetric anaesthesia airway management, mechanical ventilation and cardiopulmonary bypass/ventricular support devices. The utility of these models is based on both the validity of the model framework as well as the corresponding assumptions. In vivo experimentation has validated some, but not all of the in silico strategies employed. We present a review illustrating by example how in silico modelling has been applied to a number of cardio-respiratory problems in states of health and disease, the purpose of which is to give the reader a sense of the complexity and assumptions which underlie this diverse and underappreciated research strategy, as well as an introduction to a research strategy that will likely continue to grow in importance.

Accuracy vs. computational time: translating aortic simulations to the clinic

State of the art simulations of aortic haemodynamics feature full fluid-structure interaction (FSI) and coupled 0D boundary conditions. Such analyses require not only significant computational resource but also weeks to months of run time, which compromises the effectiveness of their translation to a clinical workflow. This article employs three computational fluid methodologies, of varying levels of complexity with coupled 0D boundary conditions, to simulate the haemodynamics within a patient-specific aorta. The most comprehensive model is a full FSI simulation. The simplest is a rigid walled incompressible fluid simulation while an alternative middle-ground approach employs a compressible fluid, tuned to elicit a response analogous to the compliance of the aortic wall. The results demonstrate that, in the context of certain clinical questions, the simpler analysis methods may capture the important characteristics of the flow field.

Virtual surgeries in patients with congenital heart disease: a multi-scale modelling test case

The objective of this work is to perform a virtual planning of surgical repairs in patients with congenital heart diseases--to test the predictive capability of a closed-loop multi-scale model. As a first step, we reproduced the pre-operative state of a specific patient with a univentricular circulation and a bidirectional cavopulmonary anastomosis (BCPA), starting from the patient's clinical data. Namely, by adopting a closed-loop multi-scale approach, the boundary conditions at the inlet and outlet sections of the three-dimensional model were automatically calculated by a lumped parameter network. Successively, we simulated three alternative surgical designs of the total cavopulmonary connection (TCPC). In particular, a T-junction of the venae cavae to the pulmonary arteries (T-TCPC), a design with an offset between the venae cavae (O-TCPC) and a Y-graft design (Y-TCPC) were compared. A multi-scale closed-loop model consisting of a lumped parameter network representing the whole circulation and a patient-specific three-dimensional finite volume model of the BCPA with detailed pulmonary anatomy was built. The three TCPC alternatives were investigated in terms of energetics and haemodynamics. Effects of exercise were also investigated. Results showed that the pre-operative caval flows should not be used as boundary conditions in post-operative simulations owing to changes in the flow waveforms post-operatively. The multi-scale approach is a possible solution to overcome this incongruence. Power losses of the Y-TCPC were lower than all other TCPC models both at rest and under exercise conditions and it distributed the inferior vena cava flow evenly to both lungs. Further work is needed to correlate results from these simulations with clinical outcomes.

Use of mathematical modeling to compare and predict hemodynamic effects between hybrid and surgical Norwood palliations for hypoplastic left heart syndrome

BACKGROUND

Combining bilateral pulmonary artery banding with arterial duct stenting, the hybrid approach achieves stage 1 palliation for hypoplastic left heart syndrome with different flow characteristics than those after the surgical Norwood procedures. Accordingly, we used computational modeling to assess some of these differences, including influence on systemic and cerebral oxygen deliveries.

METHODS AND RESULTS

A 3-dimensional computational model of hybrid palliation was developed by the finite volume method, along with models of the Norwood operation with a modified Blalock-Tausig or right ventricle-to-pulmonary artery shunt. Hybrid circulation was modeled with a 7-mm ductal stent and bilateral pulmonary artery banding to a 2-mm diameter. A 3.5-mm conduit was used in the Blalock-Tausig shunt model, whereas a 5-mm conduit was used in the right ventricle-to-pulmonary artery shunt model. Coupled to all the models was an identical hydraulic network that described the entire circulatory system based on pre-stage 2 hemodynamics. This clinically validated multiscale approach predicts flow dynamics, as well as global cardiac output, mixed venous oxygen saturation, and systemic and cerebral oxygen delivery. Compared with either of the Norwood models, the hybrid palliation had higher pulmonary-to-systemic flow ratio and lower cardiac output. Total systemic oxygen delivery was markedly reduced in the hybrid palliation (Blalock-Tausig shunt 591, right ventricle-to-pulmonary artery shunt 640, and hybrid 475 mL · min(-1) · m(-2)). Cerebral oxygen delivery was similarly lower in the hybrid palliation.

CONCLUSIONS

These computational results suggest that the hybrid approach may provide inferior systemic and cerebral oxygen deliveries compared with either of the 2 surgical Norwood procedures before stage 2 palliation.

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.

Norwood with right ventricle-to-pulmonary artery conduit is more effective than Norwood with Blalock-Taussig shunt for hypoplastic left heart syndrome: mathematic modeling of hemodynamics

OBJECTIVE

The introduction of right ventricle to pulmonary artery (RV-PA) conduit in the Norwood procedure for hypoplastic left heart syndrome resulted in a higher survival rate in many centers. A higher diastolic aortic pressure and a higher mean coronary perfusion pressure were suggested as the hemodynamic advantage of this source of pulmonary blood flow. The main objective of this study was the comparison of two models of Norwood physiology with different types of pulmonary blood flow sources and their hemodynamics.

METHOD

Based on anatomic details obtained from echocardiographic assessment and angiographic studies, two three-dimensional computer models of post-Norwood physiology were developed. The finite-element method was applied for computational hemodynamic simulations. Norwood physiology with RV-PA 5-mm conduit and Blalock-Taussig shunt (BTS) 3.5-mm shunt were compared. Right ventricle work, wall stress, flow velocity, shear rate stress, energy loss and turbulence eddy dissipation were analyzed in both models.

RESULTS

The total work of the right ventricle after Norwood procedure with the 5-mm RV-PA conduit was lower in comparison to the 3.5-mm BTS while establishing an identical systemic blood flow. The Qp/Qs ratio was higher in the BTS group.

CONCLUSIONS

Hemodynamic performance after Norwood with the RV-PA conduit is more effective than after Norwood with BTS. Computer simulations of complicated hemodynamics after the Norwood procedure could be helpful in establishing optimal post-Norwood physiology.

Review of zero-D and 1-D models of blood flow in the cardiovascular system

Background

Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models.

Method and Results

The purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/. They are freely available to the research and education communities.

Conclusion

Each published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

In vitro validation of finite element analysis of blood flow in deformable models

The purpose of this article is to validate numerical simulations of flow and pressure incorporating deformable walls using in vitro flow phantoms under physiological flow and pressure conditions. We constructed two deformable flow phantoms mimicking a normal and a restricted thoracic aorta, and used a Windkessel model at the outlet boundary. We acquired flow and pressure data in the phantom while it operated under physiological conditions. Next, in silico numerical simulations were performed, and velocities, flows, and pressures in the in silico simulations were compared to those measured in the in vitro phantoms. The experimental measurements and simulated results of pressure and flow waveform shapes and magnitudes compared favorably at all of the different measurement locations in the two deformable phantoms. The average difference between measured and simulated flow and pressure was approximately 3.5 cc/s (13% of mean) and 1.5 mmHg (1.8% of mean), respectively. Velocity patterns also showed good qualitative agreement between experiment and simulation especially in regions with less complex flow patterns. We demonstrated the capabilities of numerical simulations incorporating deformable walls to capture both the vessel wall motion and wave propagation by accurately predicting the changes in the flow and pressure waveforms at various locations down the length of the deformable flow phantoms.

Boundary conditions of patient-specific fluid dynamics modelling of cavopulmonary connections: possible adaptation of pulmonary resistances results in a critical issue for a virtual surgical planning

Cavopulmonary connections are surgical procedures used to treat a variety of complex congenital cardiac defects. Virtual pre-operative planning based on in silico patient-specific modelling might become a powerful tool in the surgical decision-making process. For this purpose, three-dimensional models can be easily developed from medical imaging data to investigate individual haemodynamics. However, the definition of patient-specific boundary conditions is still a crucial issue. The present study describes an approach to evaluate the vascular impedance of the right and left lungs on the basis of pre-operative clinical data and numerical simulations. Computational fluid dynamics techniques are applied to a patient with a bidirectional cavopulmonary anastomosis, who later underwent a total cavopulmonary connection (TCPC). Multi-scale models describing the surgical region and the lungs are adopted, while the flow rates measured in the venae cavae are used at the model inlets. Pre-operative and post-operative conditions are investigated; namely, TCPC haemodynamics, which are predicted using patient-specific pre-operative boundary conditions, indicates that the pre-operative balanced lung resistances are not compatible with the TCPC measured flows, suggesting that the pulmonary vascular impedances changed individually after the surgery. These modifications might be the consequence of adaptation to the altered pulmonary blood flows.

In Vitro Simulation and Validation of the Circulation with Congenital Heart Defects

Despite the recent advances in computational modeling, experimental simulation of the circulation with congenital heart defect using mock flow circuits remains an important tool for device testing, and for detailing the probable flow consequences resulting from surgical and interventional corrections. Validated mock circuits can be applied to qualify the results from novel computational models. New mathematical tools, coupled with advanced clinical imaging methods, allow for improved assessment of experimental circuit performance relative to human function, as well as the potential for patient-specific adaptation. In this review, we address the development of three in vitro mock circuits specific for studies of congenital heart defects. Performance of an in vitro right heart circulation circuit through a series of verification and validation exercises is described, including correlations with animal studies, and quantifying the effects of circuit inertiance on test results. We present our experience in the design of mock circuits suitable for investigations of the characteristics of the Fontan circulation. We use one such mock circuit to evaluate the accuracy of Doppler predictions in the presence of aortic coarctation.

Multiscale models of the hybrid palliation for hypoplastic left heart syndrome

A less-invasive procedure that combines interventional stent placement in the ductus arteriosus and surgical banding of the branch pulmonary arteries has been recently introduced in the treatment of the hypoplastic left heart syndrome (HLHS). The hemodynamic behaviour of this hybrid approach has not been examined before in a mathematical model. In this study, a mathematical model of the hybrid procedure for HLHS is described, applying a multiscale approach that couples 3D models of the area of the surgical operation and lumped parameter models of the remaining circulation. The effects of various degrees of pulmonary banding and different stent sizes inserted in the ductus arteriosus on pulmonary-systemic flow ratio, cardiac output and oxygen delivery were assessed. Computational results suggest that balanced systemic and pulmonary blood flow and optimal systemic oxygen delivery are sensitive to the degree of pulmonary arterial banding and not to the size of the ductal stent.

Outflow Conditions for Image-Based Hemodynamic Models of the Carotid Bifurcation: Implications for Indicators of Abnormal Flow

Computational fluid dynamics (CFD) models have become very effective tools for predicting the flow field within the carotid bifurcation, and for understanding the relationship between local hemodynamics, and the initiation and progression of vascular wall pathologies. As prescribing proper boundary conditions can affect the solutions of the equations governing blood flow, in this study, we investigated the influence to assumptions regarding the outflow boundary conditions in an image-based CFD model of human carotid bifurcation. Four simulations were conducted with identical geometry, inlet flow rate, and fluid parameters. In the first case, a physiological time-varying flow rate partition at branches along the cardiac cycle was obtained by coupling the 3D model of the carotid bifurcation at outlets with a lumped-parameter model of the downstream vascular network. Results from the coupled model were compared with those obtained by imposing three fixed flow rate divisions (50/50, 60/40, and 70/30) between the two branches of the isolated 3D model of the carotid bifurcation. Three hemodynamic wall parameters were considered as indicators of vascular wall dysfunction. Our findings underscore that the overall effect of the assumptions done in order to simulate blood flow within the carotid bifurcation is mainly in the hot-spot modulation of the hemodynamic descriptors of atherosusceptible areas, rather than in their distribution. In particular, the more physiological, time-varying flow rate division deriving from the coupled simulation has the effect of damping wall shear stress (WSS) oscillations (differences among the coupled and the three fixed flow partition models are up to 37.3% for the oscillating shear index). In conclusion, we recommend to adopt more realistic constraints, for example, by coupling models at different scales, as in this study, when the objective is the outcome prediction of alternate therapeutic interventions for individual patients, or to test hypotheses related to the role of local fluid dynamics and other biomechanical factors in vascular diseases.

Patient-specific modeling of blood flow and pressure in human coronary arteries

Coronary flow is different from the flow in other parts of the arterial system because it is influenced by the contraction and relaxation of the heart. To model coronary flow realistically, the compressive force of the heart acting on the coronary vessels needs to be included. In this study, we developed a method that predicts coronary flow and pressure of three-dimensional epicardial coronary arteries by considering models of the heart and arterial system and the interactions between the two models. For each coronary outlet, a lumped parameter coronary vascular bed model was assigned to represent the impedance of the downstream coronary vascular networks absent in the computational domain. The intramyocardial pressure was represented with either the left or right ventricular pressure depending on the location of the coronary arteries. The left and right ventricular pressure were solved from the lumped parameter heart models coupled to a closed loop system comprising a three-dimensional model of the aorta, three-element Windkessel models of the rest of the systemic circulation and the pulmonary circulation, and lumped parameter models for the left and right sides of the heart. The computed coronary flow and pressure and the aortic flow and pressure waveforms were realistic as compared to literature data.

Feasibility of in vivo pressure measurement using a pressure-tip catheter via transventricular puncture

Pressure-tip catheters (PTCs) are used to evaluate ventricular mechanics during surgical repair of congenital heart disease in children. Studies in infants require miniaturized sensors. We compared the safety and accuracy of a 2-Fr ultraminiature PTC with a 5-Fr PTC. In 10 piglets (weight 19-22 kg), a 5-Fr PTC was inserted through a 3-mm apical puncture with a #11 blade. A 20-gauge angiocatheter was inserted using a separate site. A 2-Fr PTC was threaded through the angiocatheter lumen. The angiocatheter was withdrawn, leaving the 2-Fr PTC within the left ventricle (LV). Left ventricular pressure (LVP) changes were measured during three inferior vena caval occlusions. Reliability coefficients demonstrated correlation between the 2-Fr PTC and 5-Fr PTC for LV end-diastolic pressure (0.90-0.95), peak LVP (0.92-0.99), and the maximal (0.87-0.93) and minimal (0.89-0.94) first derivatives of LVP. Bland-Altman analysis demonstrated agreement for all variables. Blood loss was trivial with pressure manipulation and catheter placement and removal. Pressure measurements using the 2-Fr PTC were accurate and comparable with those from the 5-Fr PTC. Transventricular placement of a 2-Fr PTC is feasible and should allow evaluation of ventricular mechanics during surgical repair of congenital heart disease.

On coupling a lumped parameter heart model and a three-dimensional finite element aorta model

Aortic flow and pressure result from the interactions between the heart and arterial system. In this work, we considered these interactions by utilizing a lumped parameter heart model as an inflow boundary condition for three-dimensional finite element simulations of aortic blood flow and vessel wall dynamics. The ventricular pressure-volume behavior of the lumped parameter heart model is approximated using a time varying elastance function scaled from a normalized elastance function. When the aortic valve is open, the coupled multidomain method is used to strongly couple the lumped parameter heart model and three-dimensional arterial models and compute ventricular volume, ventricular pressure, aortic flow, and aortic pressure. The shape of the velocity profiles of the inlet boundary and the outlet boundaries that experience retrograde flow are constrained to achieve a robust algorithm. When the aortic valve is closed, the inflow boundary condition is switched to a zero velocity Dirichlet condition. With this method, we obtain physiologically realistic aortic flow and pressure waveforms. We demonstrate this method in a patient-specific model of a normal human thoracic aorta under rest and exercise conditions and an aortic coarctation model under pre- and post-interventions.

Adaptation and development of software simulation methodologies for cardiovascular engineering: present and future challenges from an end-user perspective

This paper describes the use of diverse software tools in cardiovascular applications. These tools were primarily developed in the field of engineering and the applications presented push the boundaries of the software to address events related to venous and arterial valve closure, exploration of dynamic boundary conditions or the inclusion of multi-scale boundary conditions from protein to organ levels. The future of cardiovascular research and the challenges that modellers and clinicians face from validation to clinical uptake are discussed from an end-user perspective.

Patient-specific modeling of cardiovascular mechanics

Advances in numerical methods and three-dimensional imaging techniques have enabled the quantification of cardiovascular mechanics in subject-specific anatomic and physiologic models. Patient-specific models are being used to guide cell culture and animal experiments and test hypotheses related to the role of biomechanical factors in vascular diseases. Furthermore, biomechanical models based on noninvasive medical imaging could provide invaluable data on the in vivo service environment where cardiovascular devices are employed and on the effect of the devices on physiologic function. Finally, patient-specific modeling has enabled an entirely new application of cardiovascular mechanics, namely predicting outcomes of alternate therapeutic interventions for individual patients. We review methods to create anatomic and physiologic models, obtain properties, assign boundary conditions, and solve the equations governing blood flow and vessel wall dynamics. Applications of patient-specific models of cardiovascular mechanics are presented, followed by a discussion of the challenges and opportunities that lie ahead.

In vitro hemodynamic investigation of the embryonic aortic arch at late gestation

This study focuses on the dynamic flow through the fetal aortic arch driven by the concurrent action of right and left ventricles. We created a parametric pulsatile computational fluid dynamics (CFD) model of the fetal aortic junction with physiologic vessel geometries. To gain a better biophysical understanding, an in vitro experimental fetal flow loop for flow visualization was constructed for identical CFD conditions. CFD and in vitro experimental results were comparable. Swirling flow during the acceleration phase of the cardiac cycle and unidirectional flow following mid-deceleration phase were observed in pulmonary arteries (PA), head-neck vessels, and descending aorta. Right-to-left (oxygenated) blood flowed through the ductus arteriosus (DA) posterior relative to the antegrade left ventricular outflow tract (LVOT) stream and resembled jet flow. LVOT and right ventricular outflow tract flow mixing had not completed until approximately 3.5 descending aorta diameters downstream of the DA insertion into the aortic arch. Normal arch model flow patterns were then compared to flow patterns of four common congenital heart malformations that include aortic arch anomalies. Weak oscillatory reversing flow through the DA junction was observed only for the Tetralogy of Fallot configuration. PA and hypoplastic left heart syndrome configurations demonstrated complex, abnormal flow patterns in the PAs and head-neck vessels. Aortic coarctation resulted in large-scale recirculating flow in the aortic arch proximal to the DA. Intravascular flow patterns spatially correlated with abnormal vascular structures consistent with the paradigm that abnormal intravascular flow patterns associated with congenital heart disease influence vascular growth and function.